ipynb/clarman_cruz-nonlinearicp.ipynb

{
    "cells": [
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "This jupyter lab showcases the **nonlinear** ICP package. Lab [Linear Invariant Causal Prediction Using Employment Data From The Work Bank](https://notes.quantecon.org/submission/5e851bfecc00b7001acde469) shows the **linear** Invariant Causal Prediction (ICP) package. Both labs are self contained so you can read one lab only or both in any order."
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "# Nonlinear Invariant Causal Prediction Using Unemployment Data and Inflation Adjusted Prices from the United States Bureau of Labor Statistics\n",
                "\n",
                "By Clarman Cruz\n",
                "\n",
                "April, 2020\n",
                "\n",
                "Correlation does not imply causation.  Here is a funny one. \"The classic example of correlation not equaling causation can be found with ice cream and -- murder. That is, the rates of violent crime and murder have been known to jump when ice cream sales do. But, presumably, buying ice cream doesn't turn you into a killer (unless they're out of your favorite kind?).\"\n",
                "\n",
                "[Wiki](https://en.wikipedia.org/wiki/Correlation_does_not_imply_causation) defines it as \"the inability to legitimately deduce a cause-and-effect relationship between two variables solely on the basis of an observed association or correlation between them.[1]The idea that \"correlation implies causation\" is an example of a questionable-cause logical fallacy, in which two events occurring together are taken to have established a cause-and-effect relationship. This fallacy is also known by the Latin phrase cum hoc ergo propter hoc (\"with this, therefore because of this\").\"  \n",
                "\n",
                "A number of researchers have pointed out the limitations of the classical machine learning models for casual interference.  Some examples from the literature follow.\n",
                "\n",
                "*  [Beyond prediction: Using big data for policy problems](https://science.sciencemag.org/content/355/6324/483) (2017) says that \"Off-the-shelf applications of SML techniques are designed to answer the following type of question: If an analyst is told that on a particular day, prices were unusually high, what is the best prediction of occupancy on that day? The correct answer is that occupancy is likely to be high. By contrast, the question of the effect of changing the pricing policy is a causal question, and common experience indicates that if the firm implemented a new policy to systematically raise prices by 5% everywhere, it would be unlikely to sell more hotel rooms.\" \n",
                "\n",
                "* [From Dependency to Causality: A Machine Learning Approach](https://arxiv.org/abs/1412.6285) (2014) explains that \"the unidirectional link between these two notions (correlation and causation) has been used by many formal approaches to causality to justify the adoption of statistical methods for detecting or inferring causal links from observational data. The most inuential one is the Causal Bayesian Network approach, detailed in Koller and Friedman, 2009 which relies on notions of independence and conditional independence to detect causal patterns in the data.\"\n",
                " \n",
                "*  [Causal inference by using invariant prediction: identification and confidence intervals](https://arxiv.org/abs/1501.01332) (2016) asks \"what is the difference between a prediction that is made with a causal model and that with a non-causal model? Suppose that we intervene on the predictor variables or change the whole environment. The predictions from a causal model will in general work as well under interventions as for observational data. In contrast, predictions from a non-causal model can potentially be very wrong if we actively intervene on variables.\"\n",
                "\n",
                "*  [Invariant Causal Prediction for Nonlinear Models](https://arxiv.org/abs/1706.08576) (2018) points out that \"classical prediction models, however, do not answer whether an active intervention on some of the covariates leads to a change in the [observed system]. This can only be answered by exploiting causal knowledge of the system.\"\n"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## The Nonlinear Invariant Causal Prediction Package\n",
                "\n",
                "Some researchers have defined causal inference in mathematical form.  The task is not easy.  The authors of [Causal inference by using invariant prediction: identification and confidence intervals](https://arxiv.org/abs/1501.01332) not only define a rigorous mathematical model for causal inference but also create a predictive model on top of it.  It is a supervised machine learning model.  They describe their beautiful mathematics in their paper.  To very briefly summarize their approach, the authors \"use the invariance of the causal relationships under different settings e ∈ **E** (Environment set) for statistical estimation, which opens a new road for causal discovery and inference.  For simplicity, we shall mostly focus on a linear model with a target or response variable and various predictor variables, where expression (1) is unchanged and expression (2) then reads **Y**e= μ + **X**e **V** + error, with μ a constant intercept term.\" **V** is a vector relating to the predictors computed by the machine learning model.  \n",
                "\n",
                "The authors of the paper implemented their machine learning model in a R package named [InvariantCausalPrediction](https://cran.r-project.org/web/packages/InvariantCausalPrediction/index.html) (ICP). It is a great package allowing us to do both.  One is to find causality between the target variable and the predictors of the dataset. And two, one is able to predict using some of the dataset variables.  One easily defines two (or more) environments within the dataset. One environment is before and the second environment is after the experiment or intervention or treatment.  The idea is that predictors are associated with environments.  Casual interference logic is guided by the different environments.  This is awesome because in many applications as in economics one is not able to precisely define a control group and treatment group to conduct an experiment.  Often, one must use historical data to study casual interference without modifying the observed economical system.  \n",
                "\n",
                "\n",
                "The Invariant Causal Prediction package R package is computationally expensive.  It works better in small datasets that are very clean. The ICP algorithm has limitations. The authors later published paper [Invariant Causal Prediction for Nonlinear Models](https://arxiv.org/abs/1706.08576) (2018) extending the model to handle better non linear datasets. Many real world dataset are not linear.  The second paper also considers environments that are not discrete but continuous.  The nonlinear version handles more predictors that ICP. \n",
                "\n",
                "The nonlinearICP paper includes some fascinating mathematics involving probability, graphs, trees, and random forest. The task is complex.  Briefly, let the predictors be set **P** containing {p1, p2, ... pp}.  The nonlinear ICP model lists all subsets of set **P**.  Then, it computes the average causal effects for each subset within the confidence level. Then, a few of the subsets are accepted as good candidates of causality for target variable **Y**.  Let set element *pi* be a casual predictor found in each accepted subset.  Then, *pi* it is an invariant causal predictor of the target variable **Y**.  Other high level concepts of nonlinear IPC model are:\n",
                "\n",
                "* Use of nonlinear and nonparametric empirical evaluation of conditional independence tests\n",
                "* \"if we know that the parental set of a variable **Y** is either S = {1, 3} or S = {2, 3}, we know that {3} has to be a parent of Y. Yet we also want to explore the information that one variable out of the set {1, 2} also has to be causal for the target variable Y , even if we do not know which one out of the two.\"\n",
                "* Use of nonparametric or nonlinear confidence bands for the strength of the causal effects\n",
                "\n",
                "Try not to include many predictors since there are 2^p subsets unless you have computer and memory resources for it.  One can use the *maxSizeSets* argument to the nonlinearIPC function to reduce the number of subsets.  I am writing this lab in a four years old Lenovo Win10 ThinkPad.\n",
                "\n",
                "The purpose of this lab is to showcase, in [Julia](https://julialang.org) 1.4.0, the [nonlinearICP](https://cran.r-project.org/web/packages/nonlinearICP/index.html) R package accompanying the the nonlinear ICP paper. The main steps to use the InvariantCausalPrediction package are the same to use the nonlinear ICP package:  \n",
                "\n",
                "1. Define a matrix **X** containing rows as observations and columns as predictors.  No missing values are allowed in the package\n",
                "\n",
                "2. Define the two of environments or experimental settings **E**1, and **E**2\n",
                "\n",
                "3. Assign each row to an environment **E**i\n",
                "\n",
                "4. Define the answer to the prediction, **Y**.  That is, the target values used to train the supervised model\n",
                "\n",
                "5. Call function nonlinearICP(X,Y,E)  along any other input arguments\n",
                "\n",
                "6. Review the model results"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 1,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\n",
                        "R version 3.6.1 (2019-07-05) -- \"Action of the Toes\"\n",
                        "Copyright (C) 2019 The R Foundation for Statistical Computing\n",
                        "Platform: x86_64-w64-mingw32/x64 (64-bit)\n",
                        "\n",
                        "R is free software and comes with ABSOLUTELY NO WARRANTY.\n",
                        "You are welcome to redistribute it under certain conditions.\n",
                        "Type 'license()' or 'licence()' for distribution details.\n",
                        "\n",
                        "R is a collaborative project with many contributors.\n",
                        "Type 'contributors()' for more information and\n",
                        "'citation()' on how to cite R or R packages in publications.\n",
                        "\n",
                        "Type 'demo()' for some demos, 'help()' for on-line help, or\n",
                        "'help.start()' for an HTML browser interface to help.\n",
                        "Type 'q()' to quit R.\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "using RCall, Queryverse, DataFrames, CSV, Suppressor, Statistics\n",
                "@rlibrary nonlinearICP"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Unemployment Dataset\n",
                "\n",
                "This jupyter lab uses real unemployment data from [United States Bureau of Labor Statistics](https://beta.bls.gov/labs/).  The unemployment rate is by quarter.  "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 2,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/html": [
                            "<table class=\"data-frame\"><thead><tr><th></th><th>Series_ID</th><th>Year</th><th>Period</th><th>Label</th><th>Value</th></tr><tr><th></th><th>String</th><th>Int64</th><th>String</th><th>String</th><th>Float64</th></tr></thead><tbody><p>600 rows × 5 columns</p><tr><th>1</th><td>LNS14000000</td><td>1970</td><td>M01</td><td>1970 Jan</td><td>3.9</td></tr><tr><th>2</th><td>LNS14000000</td><td>1970</td><td>M02</td><td>1970 Feb</td><td>4.2</td></tr><tr><th>3</th><td>LNS14000000</td><td>1970</td><td>M03</td><td>1970 Mar</td><td>4.4</td></tr><tr><th>4</th><td>LNS14000000</td><td>1970</td><td>M04</td><td>1970 Apr</td><td>4.6</td></tr><tr><th>5</th><td>LNS14000000</td><td>1970</td><td>M05</td><td>1970 May</td><td>4.8</td></tr><tr><th>6</th><td>LNS14000000</td><td>1970</td><td>M06</td><td>1970 Jun</td><td>4.9</td></tr><tr><th>7</th><td>LNS14000000</td><td>1970</td><td>M07</td><td>1970 Jul</td><td>5.0</td></tr><tr><th>8</th><td>LNS14000000</td><td>1970</td><td>M08</td><td>1970 Aug</td><td>5.1</td></tr><tr><th>9</th><td>LNS14000000</td><td>1970</td><td>M09</td><td>1970 Sep</td><td>5.4</td></tr><tr><th>10</th><td>LNS14000000</td><td>1970</td><td>M10</td><td>1970 Oct</td><td>5.5</td></tr><tr><th>11</th><td>LNS14000000</td><td>1970</td><td>M11</td><td>1970 Nov</td><td>5.9</td></tr><tr><th>12</th><td>LNS14000000</td><td>1970</td><td>M12</td><td>1970 Dec</td><td>6.1</td></tr><tr><th>13</th><td>LNS14000000</td><td>1971</td><td>M01</td><td>1971 Jan</td><td>5.9</td></tr><tr><th>14</th><td>LNS14000000</td><td>1971</td><td>M02</td><td>1971 Feb</td><td>5.9</td></tr><tr><th>15</th><td>LNS14000000</td><td>1971</td><td>M03</td><td>1971 Mar</td><td>6.0</td></tr><tr><th>16</th><td>LNS14000000</td><td>1971</td><td>M04</td><td>1971 Apr</td><td>5.9</td></tr><tr><th>17</th><td>LNS14000000</td><td>1971</td><td>M05</td><td>1971 May</td><td>5.9</td></tr><tr><th>18</th><td>LNS14000000</td><td>1971</td><td>M06</td><td>1971 Jun</td><td>5.9</td></tr><tr><th>19</th><td>LNS14000000</td><td>1971</td><td>M07</td><td>1971 Jul</td><td>6.0</td></tr><tr><th>20</th><td>LNS14000000</td><td>1971</td><td>M08</td><td>1971 Aug</td><td>6.1</td></tr><tr><th>21</th><td>LNS14000000</td><td>1971</td><td>M09</td><td>1971 Sep</td><td>6.0</td></tr><tr><th>22</th><td>LNS14000000</td><td>1971</td><td>M10</td><td>1971 Oct</td><td>5.8</td></tr><tr><th>23</th><td>LNS14000000</td><td>1971</td><td>M11</td><td>1971 Nov</td><td>6.0</td></tr><tr><th>24</th><td>LNS14000000</td><td>1971</td><td>M12</td><td>1971 Dec</td><td>6.0</td></tr><tr><th>25</th><td>LNS14000000</td><td>1972</td><td>M01</td><td>1972 Jan</td><td>5.8</td></tr><tr><th>26</th><td>LNS14000000</td><td>1972</td><td>M02</td><td>1972 Feb</td><td>5.7</td></tr><tr><th>27</th><td>LNS14000000</td><td>1972</td><td>M03</td><td>1972 Mar</td><td>5.8</td></tr><tr><th>28</th><td>LNS14000000</td><td>1972</td><td>M04</td><td>1972 Apr</td><td>5.7</td></tr><tr><th>29</th><td>LNS14000000</td><td>1972</td><td>M05</td><td>1972 May</td><td>5.7</td></tr><tr><th>30</th><td>LNS14000000</td><td>1972</td><td>M06</td><td>1972 Jun</td><td>5.7</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
                        ],
                        "text/latex": [
                            "\\begin{tabular}{r|ccccc}\n",
                            "\t& Series\\_ID & Year & Period & Label & Value\\\\\n",
                            "\t\\hline\n",
                            "\t& String & Int64 & String & String & Float64\\\\\n",
                            "\t\\hline\n",
                            "\t1 & LNS14000000 & 1970 & M01 & 1970 Jan & 3.9 \\\\\n",
                            "\t2 & LNS14000000 & 1970 & M02 & 1970 Feb & 4.2 \\\\\n",
                            "\t3 & LNS14000000 & 1970 & M03 & 1970 Mar & 4.4 \\\\\n",
                            "\t4 & LNS14000000 & 1970 & M04 & 1970 Apr & 4.6 \\\\\n",
                            "\t5 & LNS14000000 & 1970 & M05 & 1970 May & 4.8 \\\\\n",
                            "\t6 & LNS14000000 & 1970 & M06 & 1970 Jun & 4.9 \\\\\n",
                            "\t7 & LNS14000000 & 1970 & M07 & 1970 Jul & 5.0 \\\\\n",
                            "\t8 & LNS14000000 & 1970 & M08 & 1970 Aug & 5.1 \\\\\n",
                            "\t9 & LNS14000000 & 1970 & M09 & 1970 Sep & 5.4 \\\\\n",
                            "\t10 & LNS14000000 & 1970 & M10 & 1970 Oct & 5.5 \\\\\n",
                            "\t11 & LNS14000000 & 1970 & M11 & 1970 Nov & 5.9 \\\\\n",
                            "\t12 & LNS14000000 & 1970 & M12 & 1970 Dec & 6.1 \\\\\n",
                            "\t13 & LNS14000000 & 1971 & M01 & 1971 Jan & 5.9 \\\\\n",
                            "\t14 & LNS14000000 & 1971 & M02 & 1971 Feb & 5.9 \\\\\n",
                            "\t15 & LNS14000000 & 1971 & M03 & 1971 Mar & 6.0 \\\\\n",
                            "\t16 & LNS14000000 & 1971 & M04 & 1971 Apr & 5.9 \\\\\n",
                            "\t17 & LNS14000000 & 1971 & M05 & 1971 May & 5.9 \\\\\n",
                            "\t18 & LNS14000000 & 1971 & M06 & 1971 Jun & 5.9 \\\\\n",
                            "\t19 & LNS14000000 & 1971 & M07 & 1971 Jul & 6.0 \\\\\n",
                            "\t20 & LNS14000000 & 1971 & M08 & 1971 Aug & 6.1 \\\\\n",
                            "\t21 & LNS14000000 & 1971 & M09 & 1971 Sep & 6.0 \\\\\n",
                            "\t22 & LNS14000000 & 1971 & M10 & 1971 Oct & 5.8 \\\\\n",
                            "\t23 & LNS14000000 & 1971 & M11 & 1971 Nov & 6.0 \\\\\n",
                            "\t24 & LNS14000000 & 1971 & M12 & 1971 Dec & 6.0 \\\\\n",
                            "\t25 & LNS14000000 & 1972 & M01 & 1972 Jan & 5.8 \\\\\n",
                            "\t26 & LNS14000000 & 1972 & M02 & 1972 Feb & 5.7 \\\\\n",
                            "\t27 & LNS14000000 & 1972 & M03 & 1972 Mar & 5.8 \\\\\n",
                            "\t28 & LNS14000000 & 1972 & M04 & 1972 Apr & 5.7 \\\\\n",
                            "\t29 & LNS14000000 & 1972 & M05 & 1972 May & 5.7 \\\\\n",
                            "\t30 & LNS14000000 & 1972 & M06 & 1972 Jun & 5.7 \\\\\n",
                            "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
                            "\\end{tabular}\n"
                        ],
                        "text/plain": [
                            "600×5 DataFrame\n",
                            "│ Row │ Series_ID   │ Year  │ Period │ Label    │ Value   │\n",
                            "│     │ \u001b[90mString\u001b[39m      │ \u001b[90mInt64\u001b[39m │ \u001b[90mString\u001b[39m │ \u001b[90mString\u001b[39m   │ \u001b[90mFloat64\u001b[39m │\n",
                            "├─────┼─────────────┼───────┼────────┼──────────┼─────────┤\n",
                            "│ 1   │ LNS14000000 │ 1970  │ M01    │ 1970 Jan │ 3.9     │\n",
                            "│ 2   │ LNS14000000 │ 1970  │ M02    │ 1970 Feb │ 4.2     │\n",
                            "│ 3   │ LNS14000000 │ 1970  │ M03    │ 1970 Mar │ 4.4     │\n",
                            "│ 4   │ LNS14000000 │ 1970  │ M04    │ 1970 Apr │ 4.6     │\n",
                            "│ 5   │ LNS14000000 │ 1970  │ M05    │ 1970 May │ 4.8     │\n",
                            "│ 6   │ LNS14000000 │ 1970  │ M06    │ 1970 Jun │ 4.9     │\n",
                            "│ 7   │ LNS14000000 │ 1970  │ M07    │ 1970 Jul │ 5.0     │\n",
                            "│ 8   │ LNS14000000 │ 1970  │ M08    │ 1970 Aug │ 5.1     │\n",
                            "│ 9   │ LNS14000000 │ 1970  │ M09    │ 1970 Sep │ 5.4     │\n",
                            "│ 10  │ LNS14000000 │ 1970  │ M10    │ 1970 Oct │ 5.5     │\n",
                            "⋮\n",
                            "│ 590 │ LNS14000000 │ 2019  │ M02    │ 2019 Feb │ 3.8     │\n",
                            "│ 591 │ LNS14000000 │ 2019  │ M03    │ 2019 Mar │ 3.8     │\n",
                            "│ 592 │ LNS14000000 │ 2019  │ M04    │ 2019 Apr │ 3.6     │\n",
                            "│ 593 │ LNS14000000 │ 2019  │ M05    │ 2019 May │ 3.6     │\n",
                            "│ 594 │ LNS14000000 │ 2019  │ M06    │ 2019 Jun │ 3.7     │\n",
                            "│ 595 │ LNS14000000 │ 2019  │ M07    │ 2019 Jul │ 3.7     │\n",
                            "│ 596 │ LNS14000000 │ 2019  │ M08    │ 2019 Aug │ 3.7     │\n",
                            "│ 597 │ LNS14000000 │ 2019  │ M09    │ 2019 Sep │ 3.5     │\n",
                            "│ 598 │ LNS14000000 │ 2019  │ M10    │ 2019 Oct │ 3.6     │\n",
                            "│ 599 │ LNS14000000 │ 2019  │ M11    │ 2019 Nov │ 3.5     │\n",
                            "│ 600 │ LNS14000000 │ 2019  │ M12    │ 2019 Dec │ 3.5     │"
                        ]
                    },
                    "execution_count": 2,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "fileName = \"C:\\\\Users\\\\drcxcruz\\\\Documents\\\\Julia\\\\nonlinearICP\\\\SesonalityUmemploymentRateSince1970.csv\"\n",
                "dfUmemploymentFile = CSV.File(fileName; normalizenames=true) |> DataFrame"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "We like to give the machine learning model different kinds of time series.  Therefore, we must normalize each time series. The result of normalization is to scale the original values to a range between 0 and 1. The normalise logic is also found in lab [A Data Science Example In Julia Using Education Statistics From The Work Bank](https://notes.quantecon.org/submission/5e49753efe86cc001a911eef)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 3,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "3-element Array{Float64,1}:\n",
                            " 0.0\n",
                            " 0.5\n",
                            " 1.0"
                        ]
                    },
                    "execution_count": 3,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "function normalizer(v)\n",
                "  fMax = partialsort(v,1,rev=true) \n",
                "  fMin = partialsort(v,1,rev=false) \n",
                "  if ismissing(fMax - fMin)    \n",
                "      n = Vector{Union{Missing, Float64}}(missing, length(v))\n",
                "  elseif (fMax - fMin) == 0\n",
                "      n = v\n",
                "  else \n",
                "      n = (v .- fMin) ./ (fMax - fMin)\n",
                "  end\n",
                "  return n\n",
                "end\n",
                "\n",
                "v = [1,2,3]\n",
                "normalizer(v)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Let us compute the unemployment rate by year and then normalize it.  "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 4,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/html": [
                            "<table class=\"data-frame\"><thead><tr><th></th><th>Year</th><th>Rate</th><th>RateNormUnemployment</th></tr><tr><th></th><th>Int64</th><th>Float64</th><th>Float64</th></tr></thead><tbody><p>50 rows × 3 columns</p><tr><th>1</th><td>1970</td><td>4.98333</td><td>0.217931</td></tr><tr><th>2</th><td>1971</td><td>5.95</td><td>0.377931</td></tr><tr><th>3</th><td>1972</td><td>5.6</td><td>0.32</td></tr><tr><th>4</th><td>1973</td><td>4.85833</td><td>0.197241</td></tr><tr><th>5</th><td>1974</td><td>5.64167</td><td>0.326897</td></tr><tr><th>6</th><td>1975</td><td>8.475</td><td>0.795862</td></tr><tr><th>7</th><td>1976</td><td>7.7</td><td>0.667586</td></tr><tr><th>8</th><td>1977</td><td>7.05</td><td>0.56</td></tr><tr><th>9</th><td>1978</td><td>6.06667</td><td>0.397241</td></tr><tr><th>10</th><td>1979</td><td>5.85</td><td>0.361379</td></tr><tr><th>11</th><td>1980</td><td>7.175</td><td>0.58069</td></tr><tr><th>12</th><td>1981</td><td>7.61667</td><td>0.653793</td></tr><tr><th>13</th><td>1982</td><td>9.70833</td><td>1.0</td></tr><tr><th>14</th><td>1983</td><td>9.6</td><td>0.982069</td></tr><tr><th>15</th><td>1984</td><td>7.50833</td><td>0.635862</td></tr><tr><th>16</th><td>1985</td><td>7.19167</td><td>0.583448</td></tr><tr><th>17</th><td>1986</td><td>7.0</td><td>0.551724</td></tr><tr><th>18</th><td>1987</td><td>6.175</td><td>0.415172</td></tr><tr><th>19</th><td>1988</td><td>5.49167</td><td>0.302069</td></tr><tr><th>20</th><td>1989</td><td>5.25833</td><td>0.263448</td></tr><tr><th>21</th><td>1990</td><td>5.61667</td><td>0.322759</td></tr><tr><th>22</th><td>1991</td><td>6.85</td><td>0.526897</td></tr><tr><th>23</th><td>1992</td><td>7.49167</td><td>0.633103</td></tr><tr><th>24</th><td>1993</td><td>6.90833</td><td>0.536552</td></tr><tr><th>25</th><td>1994</td><td>6.1</td><td>0.402759</td></tr><tr><th>26</th><td>1995</td><td>5.59167</td><td>0.318621</td></tr><tr><th>27</th><td>1996</td><td>5.40833</td><td>0.288276</td></tr><tr><th>28</th><td>1997</td><td>4.94167</td><td>0.211034</td></tr><tr><th>29</th><td>1998</td><td>4.5</td><td>0.137931</td></tr><tr><th>30</th><td>1999</td><td>4.21667</td><td>0.0910345</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
                        ],
                        "text/latex": [
                            "\\begin{tabular}{r|ccc}\n",
                            "\t& Year & Rate & RateNormUnemployment\\\\\n",
                            "\t\\hline\n",
                            "\t& Int64 & Float64 & Float64\\\\\n",
                            "\t\\hline\n",
                            "\t1 & 1970 & 4.98333 & 0.217931 \\\\\n",
                            "\t2 & 1971 & 5.95 & 0.377931 \\\\\n",
                            "\t3 & 1972 & 5.6 & 0.32 \\\\\n",
                            "\t4 & 1973 & 4.85833 & 0.197241 \\\\\n",
                            "\t5 & 1974 & 5.64167 & 0.326897 \\\\\n",
                            "\t6 & 1975 & 8.475 & 0.795862 \\\\\n",
                            "\t7 & 1976 & 7.7 & 0.667586 \\\\\n",
                            "\t8 & 1977 & 7.05 & 0.56 \\\\\n",
                            "\t9 & 1978 & 6.06667 & 0.397241 \\\\\n",
                            "\t10 & 1979 & 5.85 & 0.361379 \\\\\n",
                            "\t11 & 1980 & 7.175 & 0.58069 \\\\\n",
                            "\t12 & 1981 & 7.61667 & 0.653793 \\\\\n",
                            "\t13 & 1982 & 9.70833 & 1.0 \\\\\n",
                            "\t14 & 1983 & 9.6 & 0.982069 \\\\\n",
                            "\t15 & 1984 & 7.50833 & 0.635862 \\\\\n",
                            "\t16 & 1985 & 7.19167 & 0.583448 \\\\\n",
                            "\t17 & 1986 & 7.0 & 0.551724 \\\\\n",
                            "\t18 & 1987 & 6.175 & 0.415172 \\\\\n",
                            "\t19 & 1988 & 5.49167 & 0.302069 \\\\\n",
                            "\t20 & 1989 & 5.25833 & 0.263448 \\\\\n",
                            "\t21 & 1990 & 5.61667 & 0.322759 \\\\\n",
                            "\t22 & 1991 & 6.85 & 0.526897 \\\\\n",
                            "\t23 & 1992 & 7.49167 & 0.633103 \\\\\n",
                            "\t24 & 1993 & 6.90833 & 0.536552 \\\\\n",
                            "\t25 & 1994 & 6.1 & 0.402759 \\\\\n",
                            "\t26 & 1995 & 5.59167 & 0.318621 \\\\\n",
                            "\t27 & 1996 & 5.40833 & 0.288276 \\\\\n",
                            "\t28 & 1997 & 4.94167 & 0.211034 \\\\\n",
                            "\t29 & 1998 & 4.5 & 0.137931 \\\\\n",
                            "\t30 & 1999 & 4.21667 & 0.0910345 \\\\\n",
                            "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
                            "\\end{tabular}\n"
                        ],
                        "text/plain": [
                            "50×3 DataFrame\n",
                            "│ Row │ Year  │ Rate    │ RateNormUnemployment │\n",
                            "│     │ \u001b[90mInt64\u001b[39m │ \u001b[90mFloat64\u001b[39m │ \u001b[90mFloat64\u001b[39m              │\n",
                            "├─────┼───────┼─────────┼──────────────────────┤\n",
                            "│ 1   │ 1970  │ 4.98333 │ 0.217931             │\n",
                            "│ 2   │ 1971  │ 5.95    │ 0.377931             │\n",
                            "│ 3   │ 1972  │ 5.6     │ 0.32                 │\n",
                            "│ 4   │ 1973  │ 4.85833 │ 0.197241             │\n",
                            "│ 5   │ 1974  │ 5.64167 │ 0.326897             │\n",
                            "│ 6   │ 1975  │ 8.475   │ 0.795862             │\n",
                            "│ 7   │ 1976  │ 7.7     │ 0.667586             │\n",
                            "│ 8   │ 1977  │ 7.05    │ 0.56                 │\n",
                            "│ 9   │ 1978  │ 6.06667 │ 0.397241             │\n",
                            "│ 10  │ 1979  │ 5.85    │ 0.361379             │\n",
                            "⋮\n",
                            "│ 40  │ 2009  │ 9.28333 │ 0.929655             │\n",
                            "│ 41  │ 2010  │ 9.60833 │ 0.983448             │\n",
                            "│ 42  │ 2011  │ 8.93333 │ 0.871724             │\n",
                            "│ 43  │ 2012  │ 8.075   │ 0.729655             │\n",
                            "│ 44  │ 2013  │ 7.35833 │ 0.611034             │\n",
                            "│ 45  │ 2014  │ 6.15833 │ 0.412414             │\n",
                            "│ 46  │ 2015  │ 5.275   │ 0.266207             │\n",
                            "│ 47  │ 2016  │ 4.875   │ 0.2                  │\n",
                            "│ 48  │ 2017  │ 4.34167 │ 0.111724             │\n",
                            "│ 49  │ 2018  │ 3.89167 │ 0.0372414            │\n",
                            "│ 50  │ 2019  │ 3.66667 │ 0.0                  │"
                        ]
                    },
                    "execution_count": 4,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "dfUmemployment = dfUmemploymentFile |>\n",
                "            @groupby(_.Year) |>\n",
                "            @map({\n",
                "                  Year=key(_),\n",
                "                  Rate=mean(_.Value)\n",
                "                }) |>\n",
                "            @orderby(_.Year) |>\n",
                "            DataFrame\n",
                "\n",
                "dfUmemployment[!,:RateNormUnemployment] .= normalizer(dfUmemployment[!,:Rate])\n",
                "\n",
                "dfUmemployment"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Inflation Adjusted Prices\n",
                "\n",
                "This jupyter lab uses real [Inflation Rates Pricing](https://www.in2013dollars.com/inflation-cpi-categories).  The prices are computed with the United States Bureau of Labor Statistics. We back filled the college tuition data from 1977 to 1970 to make all time series from 1970 to 2019.  We like to check if low housing resulted in increased employment rates in the United States.  The [S&P 500 Index](https://www.officialdata.org/us-economy/s-p-500-price-inflation-adjusted) is included in the analysis to see the effect of the stock market."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 5,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/html": [
                            "<table class=\"data-frame\"><thead><tr><th></th><th>YearMed</th><th>USDvalueMed</th><th>RateMed</th><th>ValueNormMed</th></tr><tr><th></th><th>Int64</th><th>Float64</th><th>String</th><th>Float64</th></tr></thead><tbody><p>50 rows × 4 columns</p><tr><th>1</th><td>1970</td><td>1000.0</td><td>6.40%</td><td>0.0</td></tr><tr><th>2</th><td>1971</td><td>1064.05</td><td>6.40%</td><td>0.00468291</td></tr><tr><th>3</th><td>1972</td><td>1099.14</td><td>3.30%</td><td>0.00724845</td></tr><tr><th>4</th><td>1973</td><td>1141.35</td><td>3.84%</td><td>0.0103346</td></tr><tr><th>5</th><td>1974</td><td>1247.61</td><td>9.31%</td><td>0.0181036</td></tr><tr><th>6</th><td>1975</td><td>1398.53</td><td>12.10%</td><td>0.0291378</td></tr><tr><th>7</th><td>1976</td><td>1531.29</td><td>9.49%</td><td>0.0388444</td></tr><tr><th>8</th><td>1977</td><td>1677.79</td><td>9.57%</td><td>0.0495555</td></tr><tr><th>9</th><td>1978</td><td>1818.4</td><td>8.38%</td><td>0.0598359</td></tr><tr><th>10</th><td>1979</td><td>1986.99</td><td>9.27%</td><td>0.0721621</td></tr><tr><th>11</th><td>1980</td><td>2204.42</td><td>10.94%</td><td>0.0880591</td></tr><tr><th>12</th><td>1981</td><td>2441.96</td><td>10.78%</td><td>0.105426</td></tr><tr><th>13</th><td>1982</td><td>2727.36</td><td>11.69%</td><td>0.126293</td></tr><tr><th>14</th><td>1983</td><td>2964.17</td><td>8.68%</td><td>0.143607</td></tr><tr><th>15</th><td>1984</td><td>3150.18</td><td>6.28%</td><td>0.157207</td></tr><tr><th>16</th><td>1985</td><td>3345.77</td><td>6.21%</td><td>0.171507</td></tr><tr><th>17</th><td>1986</td><td>3597.55</td><td>7.53%</td><td>0.189915</td></tr><tr><th>18</th><td>1987</td><td>3833.37</td><td>6.56%</td><td>0.207157</td></tr><tr><th>19</th><td>1988</td><td>4082.21</td><td>6.49%</td><td>0.225351</td></tr><tr><th>20</th><td>1989</td><td>4395.09</td><td>7.66%</td><td>0.248226</td></tr><tr><th>21</th><td>1990</td><td>4794.36</td><td>9.08%</td><td>0.277418</td></tr><tr><th>22</th><td>1991</td><td>5212.76</td><td>8.73%</td><td>0.308009</td></tr><tr><th>23</th><td>1992</td><td>5596.81</td><td>7.37%</td><td>0.336088</td></tr><tr><th>24</th><td>1993</td><td>5930.8</td><td>5.97%</td><td>0.360507</td></tr><tr><th>25</th><td>1994</td><td>6214.23</td><td>4.78%</td><td>0.38123</td></tr><tr><th>26</th><td>1995</td><td>6491.78</td><td>4.47%</td><td>0.401522</td></tr><tr><th>27</th><td>1996</td><td>6721.96</td><td>3.55%</td><td>0.418351</td></tr><tr><th>28</th><td>1997</td><td>6908.22</td><td>2.77%</td><td>0.43197</td></tr><tr><th>29</th><td>1998</td><td>7130.06</td><td>3.21%</td><td>0.448189</td></tr><tr><th>30</th><td>1999</td><td>7378.4</td><td>3.48%</td><td>0.466346</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
                        ],
                        "text/latex": [
                            "\\begin{tabular}{r|cccc}\n",
                            "\t& YearMed & USDvalueMed & RateMed & ValueNormMed\\\\\n",
                            "\t\\hline\n",
                            "\t& Int64 & Float64 & String & Float64\\\\\n",
                            "\t\\hline\n",
                            "\t1 & 1970 & 1000.0 & 6.40\\% & 0.0 \\\\\n",
                            "\t2 & 1971 & 1064.05 & 6.40\\% & 0.00468291 \\\\\n",
                            "\t3 & 1972 & 1099.14 & 3.30\\% & 0.00724845 \\\\\n",
                            "\t4 & 1973 & 1141.35 & 3.84\\% & 0.0103346 \\\\\n",
                            "\t5 & 1974 & 1247.61 & 9.31\\% & 0.0181036 \\\\\n",
                            "\t6 & 1975 & 1398.53 & 12.10\\% & 0.0291378 \\\\\n",
                            "\t7 & 1976 & 1531.29 & 9.49\\% & 0.0388444 \\\\\n",
                            "\t8 & 1977 & 1677.79 & 9.57\\% & 0.0495555 \\\\\n",
                            "\t9 & 1978 & 1818.4 & 8.38\\% & 0.0598359 \\\\\n",
                            "\t10 & 1979 & 1986.99 & 9.27\\% & 0.0721621 \\\\\n",
                            "\t11 & 1980 & 2204.42 & 10.94\\% & 0.0880591 \\\\\n",
                            "\t12 & 1981 & 2441.96 & 10.78\\% & 0.105426 \\\\\n",
                            "\t13 & 1982 & 2727.36 & 11.69\\% & 0.126293 \\\\\n",
                            "\t14 & 1983 & 2964.17 & 8.68\\% & 0.143607 \\\\\n",
                            "\t15 & 1984 & 3150.18 & 6.28\\% & 0.157207 \\\\\n",
                            "\t16 & 1985 & 3345.77 & 6.21\\% & 0.171507 \\\\\n",
                            "\t17 & 1986 & 3597.55 & 7.53\\% & 0.189915 \\\\\n",
                            "\t18 & 1987 & 3833.37 & 6.56\\% & 0.207157 \\\\\n",
                            "\t19 & 1988 & 4082.21 & 6.49\\% & 0.225351 \\\\\n",
                            "\t20 & 1989 & 4395.09 & 7.66\\% & 0.248226 \\\\\n",
                            "\t21 & 1990 & 4794.36 & 9.08\\% & 0.277418 \\\\\n",
                            "\t22 & 1991 & 5212.76 & 8.73\\% & 0.308009 \\\\\n",
                            "\t23 & 1992 & 5596.81 & 7.37\\% & 0.336088 \\\\\n",
                            "\t24 & 1993 & 5930.8 & 5.97\\% & 0.360507 \\\\\n",
                            "\t25 & 1994 & 6214.23 & 4.78\\% & 0.38123 \\\\\n",
                            "\t26 & 1995 & 6491.78 & 4.47\\% & 0.401522 \\\\\n",
                            "\t27 & 1996 & 6721.96 & 3.55\\% & 0.418351 \\\\\n",
                            "\t28 & 1997 & 6908.22 & 2.77\\% & 0.43197 \\\\\n",
                            "\t29 & 1998 & 7130.06 & 3.21\\% & 0.448189 \\\\\n",
                            "\t30 & 1999 & 7378.4 & 3.48\\% & 0.466346 \\\\\n",
                            "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
                            "\\end{tabular}\n"
                        ],
                        "text/plain": [
                            "50×4 DataFrame\n",
                            "│ Row │ YearMed │ USDvalueMed │ RateMed │ ValueNormMed │\n",
                            "│     │ \u001b[90mInt64\u001b[39m   │ \u001b[90mFloat64\u001b[39m     │ \u001b[90mString\u001b[39m  │ \u001b[90mFloat64\u001b[39m      │\n",
                            "├─────┼─────────┼─────────────┼─────────┼──────────────┤\n",
                            "│ 1   │ 1970    │ 1000.0      │ 6.40%   │ 0.0          │\n",
                            "│ 2   │ 1971    │ 1064.05     │ 6.40%   │ 0.00468291   │\n",
                            "│ 3   │ 1972    │ 1099.14     │ 3.30%   │ 0.00724845   │\n",
                            "│ 4   │ 1973    │ 1141.35     │ 3.84%   │ 0.0103346    │\n",
                            "│ 5   │ 1974    │ 1247.61     │ 9.31%   │ 0.0181036    │\n",
                            "│ 6   │ 1975    │ 1398.53     │ 12.10%  │ 0.0291378    │\n",
                            "│ 7   │ 1976    │ 1531.29     │ 9.49%   │ 0.0388444    │\n",
                            "│ 8   │ 1977    │ 1677.79     │ 9.57%   │ 0.0495555    │\n",
                            "│ 9   │ 1978    │ 1818.4      │ 8.38%   │ 0.0598359    │\n",
                            "│ 10  │ 1979    │ 1986.99     │ 9.27%   │ 0.0721621    │\n",
                            "⋮\n",
                            "│ 40  │ 2009    │ 11060.9     │ 3.17%   │ 0.735582     │\n",
                            "│ 41  │ 2010    │ 11438.2     │ 3.41%   │ 0.763174     │\n",
                            "│ 42  │ 2011    │ 11786.2     │ 3.04%   │ 0.788618     │\n",
                            "│ 43  │ 2012    │ 12218.6     │ 3.67%   │ 0.820229     │\n",
                            "│ 44  │ 2013    │ 12519.2     │ 2.46%   │ 0.84221      │\n",
                            "│ 45  │ 2014    │ 12818.8     │ 2.39%   │ 0.864114     │\n",
                            "│ 46  │ 2015    │ 13156.2     │ 2.63%   │ 0.888778     │\n",
                            "│ 47  │ 2016    │ 13654.5     │ 3.79%   │ 0.925215     │\n",
                            "│ 48  │ 2017    │ 13997.3     │ 2.51%   │ 0.950273     │\n",
                            "│ 49  │ 2018    │ 14273.6     │ 1.97%   │ 0.970476     │\n",
                            "│ 50  │ 2019    │ 14677.4     │ 2.83%   │ 1.0          │"
                        ]
                    },
                    "execution_count": 5,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "fileName = \"C:\\\\Users\\\\drcxcruz\\\\Documents\\\\Julia\\\\nonlinearICP\\\\PriceInflationforMedicalCareServicesSince1970.txt\"\n",
                "dfMedicalCareServices = CSV.File(fileName; normalizenames=true, delim='\\t', decimal='.') |> DataFrame\n",
                "dfMedicalCareServices[!,:USD_Value] .= replace.(dfMedicalCareServices[!,:USD_Value], \"\\$\" => \"\")\n",
                "dfMedicalCareServices[!,:USD_Value] .= parse.( Float64, replace.(dfMedicalCareServices[!,:USD_Value], \",\" => \"\"))\n",
                "rename!(dfMedicalCareServices, [:YearMed,:USDvalueMed,:RateMed])\n",
                "dfMedicalCareServices[!,:ValueNormMed] .= normalizer(dfMedicalCareServices[!,:USDvalueMed])\n",
                "\n",
                "dfMedicalCareServices"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 6,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/html": [
                            "<table class=\"data-frame\"><thead><tr><th></th><th>YearCollege</th><th>USDvalueCollege</th><th>RateCollege</th><th>ValueNormCollege</th></tr><tr><th></th><th>Int64</th><th>Float64</th><th>String</th><th>Float64</th></tr></thead><tbody><p>50 rows × 4 columns</p><tr><th>1</th><td>1970</td><td>20000.0</td><td>-</td><td>0.0</td></tr><tr><th>2</th><td>1971</td><td>20000.0</td><td>-</td><td>0.0</td></tr><tr><th>3</th><td>1972</td><td>20000.0</td><td>-</td><td>0.0</td></tr><tr><th>4</th><td>1973</td><td>20000.0</td><td>-</td><td>0.0</td></tr><tr><th>5</th><td>1974</td><td>20000.0</td><td>-</td><td>0.0</td></tr><tr><th>6</th><td>1975</td><td>20000.0</td><td>-</td><td>0.0</td></tr><tr><th>7</th><td>1976</td><td>20000.0</td><td>-</td><td>0.0</td></tr><tr><th>8</th><td>1977</td><td>20000.0</td><td>-</td><td>0.0</td></tr><tr><th>9</th><td>1978</td><td>20693.5</td><td>3.47%</td><td>0.00248644</td></tr><tr><th>10</th><td>1979</td><td>22338.2</td><td>7.95%</td><td>0.00838343</td></tr><tr><th>11</th><td>1980</td><td>24466.6</td><td>9.53%</td><td>0.0160143</td></tr><tr><th>12</th><td>1981</td><td>27503.7</td><td>12.41%</td><td>0.0269032</td></tr><tr><th>13</th><td>1982</td><td>31199.7</td><td>13.44%</td><td>0.0401547</td></tr><tr><th>14</th><td>1983</td><td>34446.7</td><td>10.41%</td><td>0.0517962</td></tr><tr><th>15</th><td>1984</td><td>37975.3</td><td>10.24%</td><td>0.0644476</td></tr><tr><th>16</th><td>1985</td><td>41424.2</td><td>9.08%</td><td>0.0768132</td></tr><tr><th>17</th><td>1986</td><td>44777.5</td><td>8.09%</td><td>0.0888357</td></tr><tr><th>18</th><td>1987</td><td>48146.7</td><td>7.52%</td><td>0.100915</td></tr><tr><th>19</th><td>1988</td><td>51813.5</td><td>7.62%</td><td>0.114062</td></tr><tr><th>20</th><td>1989</td><td>55924.0</td><td>7.93%</td><td>0.1288</td></tr><tr><th>21</th><td>1990</td><td>60438.4</td><td>8.07%</td><td>0.144986</td></tr><tr><th>22</th><td>1991</td><td>66592.3</td><td>10.18%</td><td>0.167049</td></tr><tr><th>23</th><td>1992</td><td>73753.2</td><td>10.75%</td><td>0.192724</td></tr><tr><th>24</th><td>1993</td><td>80653.6</td><td>9.36%</td><td>0.217464</td></tr><tr><th>25</th><td>1994</td><td>86284.0</td><td>6.98%</td><td>0.237651</td></tr><tr><th>26</th><td>1995</td><td>91473.4</td><td>6.01%</td><td>0.256257</td></tr><tr><th>27</th><td>1996</td><td>96649.4</td><td>5.66%</td><td>0.274814</td></tr><tr><th>28</th><td>1997</td><td>101589.0</td><td>5.11%</td><td>0.292524</td></tr><tr><th>29</th><td>1998</td><td>105880.0</td><td>4.22%</td><td>0.30791</td></tr><tr><th>30</th><td>1999</td><td>110084.0</td><td>3.97%</td><td>0.322981</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
                        ],
                        "text/latex": [
                            "\\begin{tabular}{r|cccc}\n",
                            "\t& YearCollege & USDvalueCollege & RateCollege & ValueNormCollege\\\\\n",
                            "\t\\hline\n",
                            "\t& Int64 & Float64 & String & Float64\\\\\n",
                            "\t\\hline\n",
                            "\t1 & 1970 & 20000.0 & - & 0.0 \\\\\n",
                            "\t2 & 1971 & 20000.0 & - & 0.0 \\\\\n",
                            "\t3 & 1972 & 20000.0 & - & 0.0 \\\\\n",
                            "\t4 & 1973 & 20000.0 & - & 0.0 \\\\\n",
                            "\t5 & 1974 & 20000.0 & - & 0.0 \\\\\n",
                            "\t6 & 1975 & 20000.0 & - & 0.0 \\\\\n",
                            "\t7 & 1976 & 20000.0 & - & 0.0 \\\\\n",
                            "\t8 & 1977 & 20000.0 & - & 0.0 \\\\\n",
                            "\t9 & 1978 & 20693.5 & 3.47\\% & 0.00248644 \\\\\n",
                            "\t10 & 1979 & 22338.2 & 7.95\\% & 0.00838343 \\\\\n",
                            "\t11 & 1980 & 24466.6 & 9.53\\% & 0.0160143 \\\\\n",
                            "\t12 & 1981 & 27503.7 & 12.41\\% & 0.0269032 \\\\\n",
                            "\t13 & 1982 & 31199.7 & 13.44\\% & 0.0401547 \\\\\n",
                            "\t14 & 1983 & 34446.7 & 10.41\\% & 0.0517962 \\\\\n",
                            "\t15 & 1984 & 37975.3 & 10.24\\% & 0.0644476 \\\\\n",
                            "\t16 & 1985 & 41424.2 & 9.08\\% & 0.0768132 \\\\\n",
                            "\t17 & 1986 & 44777.5 & 8.09\\% & 0.0888357 \\\\\n",
                            "\t18 & 1987 & 48146.7 & 7.52\\% & 0.100915 \\\\\n",
                            "\t19 & 1988 & 51813.5 & 7.62\\% & 0.114062 \\\\\n",
                            "\t20 & 1989 & 55924.0 & 7.93\\% & 0.1288 \\\\\n",
                            "\t21 & 1990 & 60438.4 & 8.07\\% & 0.144986 \\\\\n",
                            "\t22 & 1991 & 66592.3 & 10.18\\% & 0.167049 \\\\\n",
                            "\t23 & 1992 & 73753.2 & 10.75\\% & 0.192724 \\\\\n",
                            "\t24 & 1993 & 80653.6 & 9.36\\% & 0.217464 \\\\\n",
                            "\t25 & 1994 & 86284.0 & 6.98\\% & 0.237651 \\\\\n",
                            "\t26 & 1995 & 91473.4 & 6.01\\% & 0.256257 \\\\\n",
                            "\t27 & 1996 & 96649.4 & 5.66\\% & 0.274814 \\\\\n",
                            "\t28 & 1997 & 101589.0 & 5.11\\% & 0.292524 \\\\\n",
                            "\t29 & 1998 & 105880.0 & 4.22\\% & 0.30791 \\\\\n",
                            "\t30 & 1999 & 110084.0 & 3.97\\% & 0.322981 \\\\\n",
                            "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
                            "\\end{tabular}\n"
                        ],
                        "text/plain": [
                            "50×4 DataFrame\n",
                            "│ Row │ YearCollege │ USDvalueCollege │ RateCollege │ ValueNormCollege │\n",
                            "│     │ \u001b[90mInt64\u001b[39m       │ \u001b[90mFloat64\u001b[39m         │ \u001b[90mString\u001b[39m      │ \u001b[90mFloat64\u001b[39m          │\n",
                            "├─────┼─────────────┼─────────────────┼─────────────┼──────────────────┤\n",
                            "│ 1   │ 1970        │ 20000.0         │ -           │ 0.0              │\n",
                            "│ 2   │ 1971        │ 20000.0         │ -           │ 0.0              │\n",
                            "│ 3   │ 1972        │ 20000.0         │ -           │ 0.0              │\n",
                            "│ 4   │ 1973        │ 20000.0         │ -           │ 0.0              │\n",
                            "│ 5   │ 1974        │ 20000.0         │ -           │ 0.0              │\n",
                            "│ 6   │ 1975        │ 20000.0         │ -           │ 0.0              │\n",
                            "│ 7   │ 1976        │ 20000.0         │ -           │ 0.0              │\n",
                            "│ 8   │ 1977        │ 20000.0         │ -           │ 0.0              │\n",
                            "│ 9   │ 1978        │ 20693.5         │ 3.47%       │ 0.00248644       │\n",
                            "│ 10  │ 1979        │ 22338.2         │ 7.95%       │ 0.00838343       │\n",
                            "⋮\n",
                            "│ 40  │ 2009        │ 209559.0        │ 6.01%       │ 0.679635         │\n",
                            "│ 41  │ 2010        │ 220445.0        │ 5.19%       │ 0.718664         │\n",
                            "│ 42  │ 2011        │ 231520.0        │ 5.02%       │ 0.758372         │\n",
                            "│ 43  │ 2012        │ 242749.0        │ 4.85%       │ 0.798631         │\n",
                            "│ 44  │ 2013        │ 253002.0        │ 4.22%       │ 0.835391         │\n",
                            "│ 45  │ 2014        │ 262347.0        │ 3.69%       │ 0.868898         │\n",
                            "│ 46  │ 2015        │ 271451.0        │ 3.47%       │ 0.901537         │\n",
                            "│ 47  │ 2016        │ 278757.0        │ 2.69%       │ 0.927731         │\n",
                            "│ 48  │ 2017        │ 284328.0        │ 2.00%       │ 0.947706         │\n",
                            "│ 49  │ 2018        │ 290615.0        │ 2.21%       │ 0.97025          │\n",
                            "│ 50  │ 2019        │ 298913.0        │ 2.86%       │ 1.0              │"
                        ]
                    },
                    "execution_count": 6,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "fileName = \"C:\\\\Users\\\\drcxcruz\\\\Documents\\\\Julia\\\\nonlinearICP\\\\PriceInflationforCollegeTuitionAndFeesSince1977.txt\"\n",
                "dfCollegeTuition = CSV.File(fileName; normalizenames=true, delim='\\t', decimal='.') |> DataFrame\n",
                "dfCollegeTuition[!,:USD_Value] .= replace.(dfCollegeTuition[!,:USD_Value], \"\\$\" => \"\")\n",
                "dfCollegeTuition[!,:USD_Value] .= parse.( Float64, replace.(dfCollegeTuition[!,:USD_Value], \",\" => \"\"))\n",
                "rename!(dfCollegeTuition, [:YearCollege,:USDvalueCollege,:RateCollege])\n",
                "dfCollegeTuition[!,:ValueNormCollege] .= normalizer(dfCollegeTuition[!,:USDvalueCollege])\n",
                "\n",
                "dfCollegeTuition"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 7,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/html": [
                            "<table class=\"data-frame\"><thead><tr><th></th><th>YearHousing</th><th>USDvalueHousing</th><th>RateHousing</th><th>ValueNormHousing</th></tr><tr><th></th><th>Int64</th><th>Float64</th><th>String</th><th>Float64</th></tr></thead><tbody><p>50 rows × 4 columns</p><tr><th>1</th><td>1970</td><td>100000.0</td><td>6.93%</td><td>0.0</td></tr><tr><th>2</th><td>1971</td><td>104445.0</td><td>4.45%</td><td>0.00704174</td></tr><tr><th>3</th><td>1972</td><td>108501.0</td><td>3.88%</td><td>0.0134664</td></tr><tr><th>4</th><td>1973</td><td>113222.0</td><td>4.35%</td><td>0.0209437</td></tr><tr><th>5</th><td>1974</td><td>125940.0</td><td>11.23%</td><td>0.041089</td></tr><tr><th>6</th><td>1975</td><td>139230.0</td><td>10.55%</td><td>0.0621416</td></tr><tr><th>7</th><td>1976</td><td>147777.0</td><td>6.14%</td><td>0.0756806</td></tr><tr><th>8</th><td>1977</td><td>157860.0</td><td>6.82%</td><td>0.0916516</td></tr><tr><th>9</th><td>1978</td><td>171677.0</td><td>8.75%</td><td>0.113539</td></tr><tr><th>10</th><td>1979</td><td>192621.0</td><td>12.20%</td><td>0.146715</td></tr><tr><th>11</th><td>1980</td><td>222846.0</td><td>15.69%</td><td>0.194592</td></tr><tr><th>12</th><td>1981</td><td>248488.0</td><td>11.51%</td><td>0.235209</td></tr><tr><th>13</th><td>1982</td><td>266407.0</td><td>7.21%</td><td>0.263594</td></tr><tr><th>14</th><td>1983</td><td>273511.0</td><td>2.67%</td><td>0.274846</td></tr><tr><th>15</th><td>1984</td><td>284899.0</td><td>4.16%</td><td>0.292886</td></tr><tr><th>16</th><td>1985</td><td>296173.0</td><td>3.96%</td><td>0.310744</td></tr><tr><th>17</th><td>1986</td><td>304927.0</td><td>2.96%</td><td>0.32461</td></tr><tr><th>18</th><td>1987</td><td>314207.0</td><td>3.04%</td><td>0.33931</td></tr><tr><th>19</th><td>1988</td><td>325848.0</td><td>3.70%</td><td>0.35775</td></tr><tr><th>20</th><td>1989</td><td>338222.0</td><td>3.80%</td><td>0.37735</td></tr><tr><th>21</th><td>1990</td><td>353483.0</td><td>4.51%</td><td>0.401525</td></tr><tr><th>22</th><td>1991</td><td>367369.0</td><td>3.93%</td><td>0.423521</td></tr><tr><th>23</th><td>1992</td><td>378071.0</td><td>2.91%</td><td>0.440472</td></tr><tr><th>24</th><td>1993</td><td>388336.0</td><td>2.72%</td><td>0.456733</td></tr><tr><th>25</th><td>1994</td><td>398098.0</td><td>2.51%</td><td>0.472196</td></tr><tr><th>26</th><td>1995</td><td>408181.0</td><td>2.53%</td><td>0.488167</td></tr><tr><th>27</th><td>1996</td><td>420142.0</td><td>2.93%</td><td>0.507115</td></tr><tr><th>28</th><td>1997</td><td>431141.0</td><td>2.62%</td><td>0.524537</td></tr><tr><th>29</th><td>1998</td><td>440994.0</td><td>2.29%</td><td>0.540145</td></tr><tr><th>30</th><td>1999</td><td>450687.0</td><td>2.20%</td><td>0.555499</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
                        ],
                        "text/latex": [
                            "\\begin{tabular}{r|cccc}\n",
                            "\t& YearHousing & USDvalueHousing & RateHousing & ValueNormHousing\\\\\n",
                            "\t\\hline\n",
                            "\t& Int64 & Float64 & String & Float64\\\\\n",
                            "\t\\hline\n",
                            "\t1 & 1970 & 100000.0 & 6.93\\% & 0.0 \\\\\n",
                            "\t2 & 1971 & 104445.0 & 4.45\\% & 0.00704174 \\\\\n",
                            "\t3 & 1972 & 108501.0 & 3.88\\% & 0.0134664 \\\\\n",
                            "\t4 & 1973 & 113222.0 & 4.35\\% & 0.0209437 \\\\\n",
                            "\t5 & 1974 & 125940.0 & 11.23\\% & 0.041089 \\\\\n",
                            "\t6 & 1975 & 139230.0 & 10.55\\% & 0.0621416 \\\\\n",
                            "\t7 & 1976 & 147777.0 & 6.14\\% & 0.0756806 \\\\\n",
                            "\t8 & 1977 & 157860.0 & 6.82\\% & 0.0916516 \\\\\n",
                            "\t9 & 1978 & 171677.0 & 8.75\\% & 0.113539 \\\\\n",
                            "\t10 & 1979 & 192621.0 & 12.20\\% & 0.146715 \\\\\n",
                            "\t11 & 1980 & 222846.0 & 15.69\\% & 0.194592 \\\\\n",
                            "\t12 & 1981 & 248488.0 & 11.51\\% & 0.235209 \\\\\n",
                            "\t13 & 1982 & 266407.0 & 7.21\\% & 0.263594 \\\\\n",
                            "\t14 & 1983 & 273511.0 & 2.67\\% & 0.274846 \\\\\n",
                            "\t15 & 1984 & 284899.0 & 4.16\\% & 0.292886 \\\\\n",
                            "\t16 & 1985 & 296173.0 & 3.96\\% & 0.310744 \\\\\n",
                            "\t17 & 1986 & 304927.0 & 2.96\\% & 0.32461 \\\\\n",
                            "\t18 & 1987 & 314207.0 & 3.04\\% & 0.33931 \\\\\n",
                            "\t19 & 1988 & 325848.0 & 3.70\\% & 0.35775 \\\\\n",
                            "\t20 & 1989 & 338222.0 & 3.80\\% & 0.37735 \\\\\n",
                            "\t21 & 1990 & 353483.0 & 4.51\\% & 0.401525 \\\\\n",
                            "\t22 & 1991 & 367369.0 & 3.93\\% & 0.423521 \\\\\n",
                            "\t23 & 1992 & 378071.0 & 2.91\\% & 0.440472 \\\\\n",
                            "\t24 & 1993 & 388336.0 & 2.72\\% & 0.456733 \\\\\n",
                            "\t25 & 1994 & 398098.0 & 2.51\\% & 0.472196 \\\\\n",
                            "\t26 & 1995 & 408181.0 & 2.53\\% & 0.488167 \\\\\n",
                            "\t27 & 1996 & 420142.0 & 2.93\\% & 0.507115 \\\\\n",
                            "\t28 & 1997 & 431141.0 & 2.62\\% & 0.524537 \\\\\n",
                            "\t29 & 1998 & 440994.0 & 2.29\\% & 0.540145 \\\\\n",
                            "\t30 & 1999 & 450687.0 & 2.20\\% & 0.555499 \\\\\n",
                            "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
                            "\\end{tabular}\n"
                        ],
                        "text/plain": [
                            "50×4 DataFrame\n",
                            "│ Row │ YearHousing │ USDvalueHousing │ RateHousing │ ValueNormHousing │\n",
                            "│     │ \u001b[90mInt64\u001b[39m       │ \u001b[90mFloat64\u001b[39m         │ \u001b[90mString\u001b[39m      │ \u001b[90mFloat64\u001b[39m          │\n",
                            "├─────┼─────────────┼─────────────────┼─────────────┼──────────────────┤\n",
                            "│ 1   │ 1970        │ 100000.0        │ 6.93%       │ 0.0              │\n",
                            "│ 2   │ 1971        │ 104445.0        │ 4.45%       │ 0.00704174       │\n",
                            "│ 3   │ 1972        │ 108501.0        │ 3.88%       │ 0.0134664        │\n",
                            "│ 4   │ 1973        │ 113222.0        │ 4.35%       │ 0.0209437        │\n",
                            "│ 5   │ 1974        │ 125940.0        │ 11.23%      │ 0.041089         │\n",
                            "│ 6   │ 1975        │ 139230.0        │ 10.55%      │ 0.0621416        │\n",
                            "│ 7   │ 1976        │ 147777.0        │ 6.14%       │ 0.0756806        │\n",
                            "│ 8   │ 1977        │ 157860.0        │ 6.82%       │ 0.0916516        │\n",
                            "│ 9   │ 1978        │ 171677.0        │ 8.75%       │ 0.113539         │\n",
                            "│ 10  │ 1979        │ 192621.0        │ 12.20%      │ 0.146715         │\n",
                            "⋮\n",
                            "│ 40  │ 2009        │ 596859.0        │ 0.37%       │ 0.78704          │\n",
                            "│ 41  │ 2010        │ 594648.0        │ -0.37%      │ 0.783536         │\n",
                            "│ 42  │ 2011        │ 602486.0        │ 1.32%       │ 0.795953         │\n",
                            "│ 43  │ 2012        │ 612412.0        │ 1.65%       │ 0.811676         │\n",
                            "│ 44  │ 2013        │ 625160.0        │ 2.08%       │ 0.831869         │\n",
                            "│ 45  │ 2014        │ 641287.0        │ 2.58%       │ 0.857415         │\n",
                            "│ 46  │ 2015        │ 654612.0        │ 2.08%       │ 0.878521         │\n",
                            "│ 47  │ 2016        │ 670874.0        │ 2.48%       │ 0.904281         │\n",
                            "│ 48  │ 2017        │ 690833.0        │ 2.98%       │ 0.935897         │\n",
                            "│ 49  │ 2018        │ 710764.0        │ 2.89%       │ 0.967468         │\n",
                            "│ 50  │ 2019        │ 731301.0        │ 2.89%       │ 1.0              │"
                        ]
                    },
                    "execution_count": 7,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "fileName = \"C:\\\\Users\\\\drcxcruz\\\\Documents\\\\Julia\\\\nonlinearICP\\\\PriceInflationOfHousingSince1970.txt\"\n",
                "dfHousing = CSV.File(fileName; normalizenames=true, delim='\\t', decimal='.') |> DataFrame\n",
                "dfHousing[!,:USD_Value] .= replace.(dfHousing[!,:USD_Value], \"\\$\" => \"\")\n",
                "dfHousing[!,:USD_Value] .= parse.( Float64, replace.(dfHousing[!,:USD_Value], \",\" => \"\"))\n",
                "rename!(dfHousing, [:YearHousing,:USDvalueHousing,:RateHousing])\n",
                "dfHousing[!,:ValueNormHousing] .= normalizer(dfHousing[!,:USDvalueHousing])\n",
                "\n",
                "dfHousing"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 8,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/html": [
                            "<table class=\"data-frame\"><thead><tr><th></th><th>YearCars</th><th>USDvalueCars</th><th>RateCars</th><th>ValueNormCars</th></tr><tr><th></th><th>Int64</th><th>Float64</th><th>String</th><th>Float64</th></tr></thead><tbody><p>50 rows × 4 columns</p><tr><th>1</th><td>1970</td><td>15000.0</td><td>3.11%</td><td>0.0</td></tr><tr><th>2</th><td>1971</td><td>15607.8</td><td>4.05%</td><td>0.0234032</td></tr><tr><th>3</th><td>1972</td><td>15475.9</td><td>-0.85%</td><td>0.0183234</td></tr><tr><th>4</th><td>1973</td><td>15480.6</td><td>0.03%</td><td>0.0185048</td></tr><tr><th>5</th><td>1974</td><td>16378.2</td><td>5.80%</td><td>0.0530655</td></tr><tr><th>6</th><td>1975</td><td>17784.7</td><td>8.59%</td><td>0.10722</td></tr><tr><th>7</th><td>1976</td><td>18910.8</td><td>6.33%</td><td>0.150579</td></tr><tr><th>8</th><td>1977</td><td>19914.4</td><td>5.31%</td><td>0.189222</td></tr><tr><th>9</th><td>1978</td><td>21429.2</td><td>7.61%</td><td>0.247548</td></tr><tr><th>10</th><td>1979</td><td>23130.2</td><td>7.94%</td><td>0.313041</td></tr><tr><th>11</th><td>1980</td><td>24979.6</td><td>8.00%</td><td>0.384249</td></tr><tr><th>12</th><td>1981</td><td>26503.8</td><td>6.10%</td><td>0.442938</td></tr><tr><th>13</th><td>1982</td><td>27526.3</td><td>3.86%</td><td>0.482307</td></tr><tr><th>14</th><td>1983</td><td>28233.1</td><td>2.57%</td><td>0.50952</td></tr><tr><th>15</th><td>1984</td><td>29050.6</td><td>2.90%</td><td>0.540996</td></tr><tr><th>16</th><td>1985</td><td>29978.8</td><td>3.20%</td><td>0.576736</td></tr><tr><th>17</th><td>1986</td><td>31272.2</td><td>4.31%</td><td>0.626536</td></tr><tr><th>18</th><td>1987</td><td>32398.3</td><td>3.60%</td><td>0.669895</td></tr><tr><th>19</th><td>1988</td><td>33039.1</td><td>1.98%</td><td>0.694569</td></tr><tr><th>20</th><td>1989</td><td>33710.5</td><td>2.03%</td><td>0.720421</td></tr><tr><th>21</th><td>1990</td><td>34200.6</td><td>1.45%</td><td>0.739289</td></tr><tr><th>22</th><td>1991</td><td>35430.3</td><td>3.60%</td><td>0.786639</td></tr><tr><th>23</th><td>1992</td><td>36290.2</td><td>2.43%</td><td>0.819749</td></tr><tr><th>24</th><td>1993</td><td>37173.7</td><td>2.43%</td><td>0.853765</td></tr><tr><th>25</th><td>1994</td><td>38443.5</td><td>3.42%</td><td>0.902658</td></tr><tr><th>26</th><td>1995</td><td>39301.1</td><td>2.23%</td><td>0.935676</td></tr><tr><th>27</th><td>1996</td><td>39979.6</td><td>1.73%</td><td>0.961801</td></tr><tr><th>28</th><td>1997</td><td>40052.6</td><td>0.18%</td><td>0.964613</td></tr><tr><th>29</th><td>1998</td><td>39777.0</td><td>-0.69%</td><td>0.954</td></tr><tr><th>30</th><td>1999</td><td>39451.9</td><td>-0.82%</td><td>0.941482</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
                        ],
                        "text/latex": [
                            "\\begin{tabular}{r|cccc}\n",
                            "\t& YearCars & USDvalueCars & RateCars & ValueNormCars\\\\\n",
                            "\t\\hline\n",
                            "\t& Int64 & Float64 & String & Float64\\\\\n",
                            "\t\\hline\n",
                            "\t1 & 1970 & 15000.0 & 3.11\\% & 0.0 \\\\\n",
                            "\t2 & 1971 & 15607.8 & 4.05\\% & 0.0234032 \\\\\n",
                            "\t3 & 1972 & 15475.9 & -0.85\\% & 0.0183234 \\\\\n",
                            "\t4 & 1973 & 15480.6 & 0.03\\% & 0.0185048 \\\\\n",
                            "\t5 & 1974 & 16378.2 & 5.80\\% & 0.0530655 \\\\\n",
                            "\t6 & 1975 & 17784.7 & 8.59\\% & 0.10722 \\\\\n",
                            "\t7 & 1976 & 18910.8 & 6.33\\% & 0.150579 \\\\\n",
                            "\t8 & 1977 & 19914.4 & 5.31\\% & 0.189222 \\\\\n",
                            "\t9 & 1978 & 21429.2 & 7.61\\% & 0.247548 \\\\\n",
                            "\t10 & 1979 & 23130.2 & 7.94\\% & 0.313041 \\\\\n",
                            "\t11 & 1980 & 24979.6 & 8.00\\% & 0.384249 \\\\\n",
                            "\t12 & 1981 & 26503.8 & 6.10\\% & 0.442938 \\\\\n",
                            "\t13 & 1982 & 27526.3 & 3.86\\% & 0.482307 \\\\\n",
                            "\t14 & 1983 & 28233.1 & 2.57\\% & 0.50952 \\\\\n",
                            "\t15 & 1984 & 29050.6 & 2.90\\% & 0.540996 \\\\\n",
                            "\t16 & 1985 & 29978.8 & 3.20\\% & 0.576736 \\\\\n",
                            "\t17 & 1986 & 31272.2 & 4.31\\% & 0.626536 \\\\\n",
                            "\t18 & 1987 & 32398.3 & 3.60\\% & 0.669895 \\\\\n",
                            "\t19 & 1988 & 33039.1 & 1.98\\% & 0.694569 \\\\\n",
                            "\t20 & 1989 & 33710.5 & 2.03\\% & 0.720421 \\\\\n",
                            "\t21 & 1990 & 34200.6 & 1.45\\% & 0.739289 \\\\\n",
                            "\t22 & 1991 & 35430.3 & 3.60\\% & 0.786639 \\\\\n",
                            "\t23 & 1992 & 36290.2 & 2.43\\% & 0.819749 \\\\\n",
                            "\t24 & 1993 & 37173.7 & 2.43\\% & 0.853765 \\\\\n",
                            "\t25 & 1994 & 38443.5 & 3.42\\% & 0.902658 \\\\\n",
                            "\t26 & 1995 & 39301.1 & 2.23\\% & 0.935676 \\\\\n",
                            "\t27 & 1996 & 39979.6 & 1.73\\% & 0.961801 \\\\\n",
                            "\t28 & 1997 & 40052.6 & 0.18\\% & 0.964613 \\\\\n",
                            "\t29 & 1998 & 39777.0 & -0.69\\% & 0.954 \\\\\n",
                            "\t30 & 1999 & 39451.9 & -0.82\\% & 0.941482 \\\\\n",
                            "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
                            "\\end{tabular}\n"
                        ],
                        "text/plain": [
                            "50×4 DataFrame\n",
                            "│ Row │ YearCars │ USDvalueCars │ RateCars │ ValueNormCars │\n",
                            "│     │ \u001b[90mInt64\u001b[39m    │ \u001b[90mFloat64\u001b[39m      │ \u001b[90mString\u001b[39m   │ \u001b[90mFloat64\u001b[39m       │\n",
                            "├─────┼──────────┼──────────────┼──────────┼───────────────┤\n",
                            "│ 1   │ 1970     │ 15000.0      │ 3.11%    │ 0.0           │\n",
                            "│ 2   │ 1971     │ 15607.8      │ 4.05%    │ 0.0234032     │\n",
                            "│ 3   │ 1972     │ 15475.9      │ -0.85%   │ 0.0183234     │\n",
                            "│ 4   │ 1973     │ 15480.6      │ 0.03%    │ 0.0185048     │\n",
                            "│ 5   │ 1974     │ 16378.2      │ 5.80%    │ 0.0530655     │\n",
                            "│ 6   │ 1975     │ 17784.7      │ 8.59%    │ 0.10722       │\n",
                            "│ 7   │ 1976     │ 18910.8      │ 6.33%    │ 0.150579      │\n",
                            "│ 8   │ 1977     │ 19914.4      │ 5.31%    │ 0.189222      │\n",
                            "│ 9   │ 1978     │ 21429.2      │ 7.61%    │ 0.247548      │\n",
                            "│ 10  │ 1979     │ 23130.2      │ 7.94%    │ 0.313041      │\n",
                            "⋮\n",
                            "│ 40  │ 2009     │ 38642.2      │ 0.95%    │ 0.910308      │\n",
                            "│ 41  │ 2010     │ 39041.0      │ 1.03%    │ 0.925661      │\n",
                            "│ 42  │ 2011     │ 40209.6      │ 2.99%    │ 0.970656      │\n",
                            "│ 43  │ 2012     │ 40761.0      │ 1.37%    │ 0.99189       │\n",
                            "│ 44  │ 2013     │ 40971.7      │ 0.52%    │ 1.0           │\n",
                            "│ 45  │ 2014     │ 40842.8      │ -0.31%   │ 0.995037      │\n",
                            "│ 46  │ 2015     │ 40824.0      │ -0.05%   │ 0.994313      │\n",
                            "│ 47  │ 2016     │ 40623.9      │ -0.49%   │ 0.986611      │\n",
                            "│ 48  │ 2017     │ 40330.4      │ -0.72%   │ 0.975311      │\n",
                            "│ 49  │ 2018     │ 40136.0      │ -0.48%   │ 0.967825      │\n",
                            "│ 50  │ 2019     │ 40370.3      │ 0.58%    │ 0.976846      │"
                        ]
                    },
                    "execution_count": 8,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "fileName = \"C:\\\\Users\\\\drcxcruz\\\\Documents\\\\Julia\\\\nonlinearICP\\\\PriceInflationOfCarsSince1970.txt\"\n",
                "dfCars = CSV.File(fileName; normalizenames=true, delim='\\t', decimal='.') |> DataFrame\n",
                "dfCars[!,:USD_Value] .= replace.(dfCars[!,:USD_Value], \"\\$\" => \"\")\n",
                "dfCars[!,:USD_Value] .= replace.(dfCars[!,:USD_Value], \"\\$\" => \"\")\n",
                "dfCars[!,:USD_Value] .= parse.( Float64, replace.(dfCars[!,:USD_Value], \",\" => \"\"))\n",
                "rename!(dfCars, [:YearCars,:USDvalueCars,:RateCars])\n",
                "dfCars[!,:ValueNormCars] .= normalizer(dfCars[!,:USDvalueCars])\n",
                "\n",
                "dfCars"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 9,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/html": [
                            "<table class=\"data-frame\"><thead><tr><th></th><th>YearBananas</th><th>USDvalueBananas</th><th>RateBananas</th><th>ValueNormBananas</th></tr><tr><th></th><th>Int64</th><th>Float64</th><th>String</th><th>Float64</th></tr></thead><tbody><p>50 rows × 4 columns</p><tr><th>1</th><td>1970</td><td>3.5</td><td>0.14%</td><td>0.0130548</td></tr><tr><th>2</th><td>1971</td><td>3.3</td><td>-5.82%</td><td>0.0</td></tr><tr><th>3</th><td>1972</td><td>3.51</td><td>6.43%</td><td>0.0137076</td></tr><tr><th>4</th><td>1973</td><td>3.66</td><td>4.43%</td><td>0.0234987</td></tr><tr><th>5</th><td>1974</td><td>4.11</td><td>12.31%</td><td>0.0528721</td></tr><tr><th>6</th><td>1975</td><td>5.15</td><td>25.26%</td><td>0.120757</td></tr><tr><th>7</th><td>1976</td><td>5.22</td><td>1.33%</td><td>0.125326</td></tr><tr><th>8</th><td>1977</td><td>5.68</td><td>8.70%</td><td>0.155352</td></tr><tr><th>9</th><td>1978</td><td>6.35</td><td>11.81%</td><td>0.199086</td></tr><tr><th>10</th><td>1979</td><td>7.17</td><td>12.93%</td><td>0.252611</td></tr><tr><th>11</th><td>1980</td><td>8.22</td><td>14.64%</td><td>0.321149</td></tr><tr><th>12</th><td>1981</td><td>8.77</td><td>6.69%</td><td>0.35705</td></tr><tr><th>13</th><td>1982</td><td>8.63</td><td>-1.61%</td><td>0.347911</td></tr><tr><th>14</th><td>1983</td><td>9.52</td><td>10.31%</td><td>0.406005</td></tr><tr><th>15</th><td>1984</td><td>8.79</td><td>-7.61%</td><td>0.358355</td></tr><tr><th>16</th><td>1985</td><td>8.97</td><td>2.07%</td><td>0.370104</td></tr><tr><th>17</th><td>1986</td><td>9.43</td><td>5.07%</td><td>0.400131</td></tr><tr><th>18</th><td>1987</td><td>9.35</td><td>-0.80%</td><td>0.394909</td></tr><tr><th>19</th><td>1988</td><td>10.71</td><td>14.46%</td><td>0.483681</td></tr><tr><th>20</th><td>1989</td><td>11.79</td><td>10.11%</td><td>0.554178</td></tr><tr><th>21</th><td>1990</td><td>12.4</td><td>5.24%</td><td>0.593995</td></tr><tr><th>22</th><td>1991</td><td>13.02</td><td>4.95%</td><td>0.634465</td></tr><tr><th>23</th><td>1992</td><td>12.56</td><td>-3.51%</td><td>0.604439</td></tr><tr><th>24</th><td>1993</td><td>12.17</td><td>-3.15%</td><td>0.578982</td></tr><tr><th>25</th><td>1994</td><td>12.89</td><td>5.94%</td><td>0.625979</td></tr><tr><th>26</th><td>1995</td><td>13.81</td><td>7.10%</td><td>0.686031</td></tr><tr><th>27</th><td>1996</td><td>14.28</td><td>3.41%</td><td>0.71671</td></tr><tr><th>28</th><td>1997</td><td>14.33</td><td>0.35%</td><td>0.719974</td></tr><tr><th>29</th><td>1998</td><td>14.44</td><td>0.80%</td><td>0.727154</td></tr><tr><th>30</th><td>1999</td><td>14.31</td><td>-0.89%</td><td>0.718668</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
                        ],
                        "text/latex": [
                            "\\begin{tabular}{r|cccc}\n",
                            "\t& YearBananas & USDvalueBananas & RateBananas & ValueNormBananas\\\\\n",
                            "\t\\hline\n",
                            "\t& Int64 & Float64 & String & Float64\\\\\n",
                            "\t\\hline\n",
                            "\t1 & 1970 & 3.5 & 0.14\\% & 0.0130548 \\\\\n",
                            "\t2 & 1971 & 3.3 & -5.82\\% & 0.0 \\\\\n",
                            "\t3 & 1972 & 3.51 & 6.43\\% & 0.0137076 \\\\\n",
                            "\t4 & 1973 & 3.66 & 4.43\\% & 0.0234987 \\\\\n",
                            "\t5 & 1974 & 4.11 & 12.31\\% & 0.0528721 \\\\\n",
                            "\t6 & 1975 & 5.15 & 25.26\\% & 0.120757 \\\\\n",
                            "\t7 & 1976 & 5.22 & 1.33\\% & 0.125326 \\\\\n",
                            "\t8 & 1977 & 5.68 & 8.70\\% & 0.155352 \\\\\n",
                            "\t9 & 1978 & 6.35 & 11.81\\% & 0.199086 \\\\\n",
                            "\t10 & 1979 & 7.17 & 12.93\\% & 0.252611 \\\\\n",
                            "\t11 & 1980 & 8.22 & 14.64\\% & 0.321149 \\\\\n",
                            "\t12 & 1981 & 8.77 & 6.69\\% & 0.35705 \\\\\n",
                            "\t13 & 1982 & 8.63 & -1.61\\% & 0.347911 \\\\\n",
                            "\t14 & 1983 & 9.52 & 10.31\\% & 0.406005 \\\\\n",
                            "\t15 & 1984 & 8.79 & -7.61\\% & 0.358355 \\\\\n",
                            "\t16 & 1985 & 8.97 & 2.07\\% & 0.370104 \\\\\n",
                            "\t17 & 1986 & 9.43 & 5.07\\% & 0.400131 \\\\\n",
                            "\t18 & 1987 & 9.35 & -0.80\\% & 0.394909 \\\\\n",
                            "\t19 & 1988 & 10.71 & 14.46\\% & 0.483681 \\\\\n",
                            "\t20 & 1989 & 11.79 & 10.11\\% & 0.554178 \\\\\n",
                            "\t21 & 1990 & 12.4 & 5.24\\% & 0.593995 \\\\\n",
                            "\t22 & 1991 & 13.02 & 4.95\\% & 0.634465 \\\\\n",
                            "\t23 & 1992 & 12.56 & -3.51\\% & 0.604439 \\\\\n",
                            "\t24 & 1993 & 12.17 & -3.15\\% & 0.578982 \\\\\n",
                            "\t25 & 1994 & 12.89 & 5.94\\% & 0.625979 \\\\\n",
                            "\t26 & 1995 & 13.81 & 7.10\\% & 0.686031 \\\\\n",
                            "\t27 & 1996 & 14.28 & 3.41\\% & 0.71671 \\\\\n",
                            "\t28 & 1997 & 14.33 & 0.35\\% & 0.719974 \\\\\n",
                            "\t29 & 1998 & 14.44 & 0.80\\% & 0.727154 \\\\\n",
                            "\t30 & 1999 & 14.31 & -0.89\\% & 0.718668 \\\\\n",
                            "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
                            "\\end{tabular}\n"
                        ],
                        "text/plain": [
                            "50×4 DataFrame\n",
                            "│ Row │ YearBananas │ USDvalueBananas │ RateBananas │ ValueNormBananas │\n",
                            "│     │ \u001b[90mInt64\u001b[39m       │ \u001b[90mFloat64\u001b[39m         │ \u001b[90mString\u001b[39m      │ \u001b[90mFloat64\u001b[39m          │\n",
                            "├─────┼─────────────┼─────────────────┼─────────────┼──────────────────┤\n",
                            "│ 1   │ 1970        │ 3.5             │ 0.14%       │ 0.0130548        │\n",
                            "│ 2   │ 1971        │ 3.3             │ -5.82%      │ 0.0              │\n",
                            "│ 3   │ 1972        │ 3.51            │ 6.43%       │ 0.0137076        │\n",
                            "│ 4   │ 1973        │ 3.66            │ 4.43%       │ 0.0234987        │\n",
                            "│ 5   │ 1974        │ 4.11            │ 12.31%      │ 0.0528721        │\n",
                            "│ 6   │ 1975        │ 5.15            │ 25.26%      │ 0.120757         │\n",
                            "│ 7   │ 1976        │ 5.22            │ 1.33%       │ 0.125326         │\n",
                            "│ 8   │ 1977        │ 5.68            │ 8.70%       │ 0.155352         │\n",
                            "│ 9   │ 1978        │ 6.35            │ 11.81%      │ 0.199086         │\n",
                            "│ 10  │ 1979        │ 7.17            │ 12.93%      │ 0.252611         │\n",
                            "⋮\n",
                            "│ 40  │ 2009        │ 18.61           │ -0.07%      │ 0.999347         │\n",
                            "│ 41  │ 2010        │ 17.55           │ -5.70%      │ 0.930157         │\n",
                            "│ 42  │ 2011        │ 18.48           │ 5.33%       │ 0.990862         │\n",
                            "│ 43  │ 2012        │ 18.41           │ -0.39%      │ 0.986292         │\n",
                            "│ 44  │ 2013        │ 18.26           │ -0.82%      │ 0.976501         │\n",
                            "│ 45  │ 2014        │ 18.14           │ -0.65%      │ 0.968668         │\n",
                            "│ 46  │ 2015        │ 18.03           │ -0.62%      │ 0.961488         │\n",
                            "│ 47  │ 2016        │ 17.85           │ -0.96%      │ 0.949739         │\n",
                            "│ 48  │ 2017        │ 17.52           │ -1.85%      │ 0.928198         │\n",
                            "│ 49  │ 2018        │ 17.66           │ 0.79%       │ 0.937337         │\n",
                            "│ 50  │ 2019        │ 17.6            │ -0.37%      │ 0.93342          │"
                        ]
                    },
                    "execution_count": 9,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "fileName = \"C:\\\\Users\\\\drcxcruz\\\\Documents\\\\Julia\\\\nonlinearICP\\\\PriceInflationForBananasSince1970.txt\"\n",
                "dfBananas = CSV.File(fileName; normalizenames=true, delim='\\t', decimal='.') |> DataFrame\n",
                "dfBananas[!,:USD_Value] .= replace.(dfBananas[!,:USD_Value], \"\\$\" => \"\")\n",
                "dfBananas[!,:USD_Value] .= replace.(dfBananas[!,:USD_Value], \"\\$\" => \"\")\n",
                "dfBananas[!,:USD_Value] .= parse.( Float64, replace.(dfBananas[!,:USD_Value], \",\" => \"\"))\n",
                "rename!(dfBananas, [:YearBananas,:USDvalueBananas,:RateBananas])\n",
                "dfBananas[!,:ValueNormBananas] .= normalizer(dfBananas[!,:USDvalueBananas])\n",
                "\n",
                "dfBananas"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "The [S&P 500 Index](https://www.officialdata.org/us-economy/s-p-500-price-inflation-adjusted) prices are inflation adjusted also.  "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 10,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/html": [
                            "<table class=\"data-frame\"><thead><tr><th></th><th>YearSPX</th><th>USDvalueSPX</th><th>ChangeSPX</th><th>RateSPX</th><th>ValueNormSPX</th></tr><tr><th></th><th>Int64</th><th>Float64</th><th>Float64</th><th>Float64</th><th>Float64</th></tr></thead><tbody><p>50 rows × 5 columns</p><tr><th>1</th><td>1970</td><td>613.53</td><td>-122.24</td><td>-16.61</td><td>0.114396</td></tr><tr><th>2</th><td>1971</td><td>603.22</td><td>-10.31</td><td>-1.68</td><td>0.110385</td></tr><tr><th>3</th><td>1972</td><td>645.43</td><td>42.21</td><td>7.0</td><td>0.126804</td></tr><tr><th>4</th><td>1973</td><td>713.73</td><td>68.3</td><td>10.58</td><td>0.15337</td></tr><tr><th>5</th><td>1974</td><td>529.63</td><td>-184.1</td><td>-25.79</td><td>0.0817613</td></tr><tr><th>6</th><td>1975</td><td>357.64</td><td>-171.99</td><td>-32.47</td><td>0.0148625</td></tr><tr><th>7</th><td>1976</td><td>447.37</td><td>89.72</td><td>25.09</td><td>0.0497647</td></tr><tr><th>8</th><td>1977</td><td>455.65</td><td>8.29</td><td>1.85</td><td>0.0529853</td></tr><tr><th>9</th><td>1978</td><td>370.82</td><td>-84.84</td><td>-18.62</td><td>0.0199891</td></tr><tr><th>10</th><td>1979</td><td>374.9</td><td>4.08</td><td>1.1</td><td>0.0215761</td></tr><tr><th>11</th><td>1980</td><td>366.05</td><td>-8.84</td><td>-2.36</td><td>0.0181337</td></tr><tr><th>12</th><td>1981</td><td>392.58</td><td>26.52</td><td>7.25</td><td>0.0284531</td></tr><tr><th>13</th><td>1982</td><td>319.43</td><td>-73.14</td><td>-18.63</td><td>0.0</td></tr><tr><th>14</th><td>1983</td><td>378.9</td><td>59.46</td><td>18.62</td><td>0.023132</td></tr><tr><th>15</th><td>1984</td><td>419.35</td><td>40.45</td><td>10.68</td><td>0.0388658</td></tr><tr><th>16</th><td>1985</td><td>417.69</td><td>-1.65</td><td>-0.39</td><td>0.0382201</td></tr><tr><th>17</th><td>1986</td><td>487.82</td><td>70.13</td><td>16.79</td><td>0.0654985</td></tr><tr><th>18</th><td>1987</td><td>610.82</td><td>123.0</td><td>25.21</td><td>0.113342</td></tr><tr><th>19</th><td>1988</td><td>555.99</td><td>-54.83</td><td>-8.98</td><td>0.0920145</td></tr><tr><th>20</th><td>1989</td><td>605.2</td><td>49.21</td><td>8.85</td><td>0.111156</td></tr><tr><th>21</th><td>1990</td><td>685.27</td><td>80.07</td><td>13.23</td><td>0.1423</td></tr><tr><th>22</th><td>1991</td><td>620.99</td><td>-64.28</td><td>-9.38</td><td>0.117297</td></tr><tr><th>23</th><td>1992</td><td>773.71</td><td>152.71</td><td>24.59</td><td>0.176701</td></tr><tr><th>24</th><td>1993</td><td>783.78</td><td>10.07</td><td>1.3</td><td>0.180618</td></tr><tr><th>25</th><td>1994</td><td>830.8</td><td>47.03</td><td>6.0</td><td>0.198907</td></tr><tr><th>26</th><td>1995</td><td>794.91</td><td>-35.89</td><td>-4.32</td><td>0.184947</td></tr><tr><th>27</th><td>1996</td><td>1021.91</td><td>226.99</td><td>28.56</td><td>0.273243</td></tr><tr><th>28</th><td>1997</td><td>1236.73</td><td>214.83</td><td>21.02</td><td>0.356801</td></tr><tr><th>29</th><td>1998</td><td>1530.88</td><td>294.14</td><td>23.78</td><td>0.471216</td></tr><tr><th>30</th><td>1999</td><td>1951.81</td><td>420.93</td><td>27.5</td><td>0.634945</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table>"
                        ],
                        "text/latex": [
                            "\\begin{tabular}{r|ccccc}\n",
                            "\t& YearSPX & USDvalueSPX & ChangeSPX & RateSPX & ValueNormSPX\\\\\n",
                            "\t\\hline\n",
                            "\t& Int64 & Float64 & Float64 & Float64 & Float64\\\\\n",
                            "\t\\hline\n",
                            "\t1 & 1970 & 613.53 & -122.24 & -16.61 & 0.114396 \\\\\n",
                            "\t2 & 1971 & 603.22 & -10.31 & -1.68 & 0.110385 \\\\\n",
                            "\t3 & 1972 & 645.43 & 42.21 & 7.0 & 0.126804 \\\\\n",
                            "\t4 & 1973 & 713.73 & 68.3 & 10.58 & 0.15337 \\\\\n",
                            "\t5 & 1974 & 529.63 & -184.1 & -25.79 & 0.0817613 \\\\\n",
                            "\t6 & 1975 & 357.64 & -171.99 & -32.47 & 0.0148625 \\\\\n",
                            "\t7 & 1976 & 447.37 & 89.72 & 25.09 & 0.0497647 \\\\\n",
                            "\t8 & 1977 & 455.65 & 8.29 & 1.85 & 0.0529853 \\\\\n",
                            "\t9 & 1978 & 370.82 & -84.84 & -18.62 & 0.0199891 \\\\\n",
                            "\t10 & 1979 & 374.9 & 4.08 & 1.1 & 0.0215761 \\\\\n",
                            "\t11 & 1980 & 366.05 & -8.84 & -2.36 & 0.0181337 \\\\\n",
                            "\t12 & 1981 & 392.58 & 26.52 & 7.25 & 0.0284531 \\\\\n",
                            "\t13 & 1982 & 319.43 & -73.14 & -18.63 & 0.0 \\\\\n",
                            "\t14 & 1983 & 378.9 & 59.46 & 18.62 & 0.023132 \\\\\n",
                            "\t15 & 1984 & 419.35 & 40.45 & 10.68 & 0.0388658 \\\\\n",
                            "\t16 & 1985 & 417.69 & -1.65 & -0.39 & 0.0382201 \\\\\n",
                            "\t17 & 1986 & 487.82 & 70.13 & 16.79 & 0.0654985 \\\\\n",
                            "\t18 & 1987 & 610.82 & 123.0 & 25.21 & 0.113342 \\\\\n",
                            "\t19 & 1988 & 555.99 & -54.83 & -8.98 & 0.0920145 \\\\\n",
                            "\t20 & 1989 & 605.2 & 49.21 & 8.85 & 0.111156 \\\\\n",
                            "\t21 & 1990 & 685.27 & 80.07 & 13.23 & 0.1423 \\\\\n",
                            "\t22 & 1991 & 620.99 & -64.28 & -9.38 & 0.117297 \\\\\n",
                            "\t23 & 1992 & 773.71 & 152.71 & 24.59 & 0.176701 \\\\\n",
                            "\t24 & 1993 & 783.78 & 10.07 & 1.3 & 0.180618 \\\\\n",
                            "\t25 & 1994 & 830.8 & 47.03 & 6.0 & 0.198907 \\\\\n",
                            "\t26 & 1995 & 794.91 & -35.89 & -4.32 & 0.184947 \\\\\n",
                            "\t27 & 1996 & 1021.91 & 226.99 & 28.56 & 0.273243 \\\\\n",
                            "\t28 & 1997 & 1236.73 & 214.83 & 21.02 & 0.356801 \\\\\n",
                            "\t29 & 1998 & 1530.88 & 294.14 & 23.78 & 0.471216 \\\\\n",
                            "\t30 & 1999 & 1951.81 & 420.93 & 27.5 & 0.634945 \\\\\n",
                            "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
                            "\\end{tabular}\n"
                        ],
                        "text/plain": [
                            "50×5 DataFrame\n",
                            "│ Row │ YearSPX │ USDvalueSPX │ ChangeSPX │ RateSPX │ ValueNormSPX │\n",
                            "│     │ \u001b[90mInt64\u001b[39m   │ \u001b[90mFloat64\u001b[39m     │ \u001b[90mFloat64\u001b[39m   │ \u001b[90mFloat64\u001b[39m │ \u001b[90mFloat64\u001b[39m      │\n",
                            "├─────┼─────────┼─────────────┼───────────┼─────────┼──────────────┤\n",
                            "│ 1   │ 1970    │ 613.53      │ -122.24   │ -16.61  │ 0.114396     │\n",
                            "│ 2   │ 1971    │ 603.22      │ -10.31    │ -1.68   │ 0.110385     │\n",
                            "│ 3   │ 1972    │ 645.43      │ 42.21     │ 7.0     │ 0.126804     │\n",
                            "│ 4   │ 1973    │ 713.73      │ 68.3      │ 10.58   │ 0.15337      │\n",
                            "│ 5   │ 1974    │ 529.63      │ -184.1    │ -25.79  │ 0.0817613    │\n",
                            "│ 6   │ 1975    │ 357.64      │ -171.99   │ -32.47  │ 0.0148625    │\n",
                            "│ 7   │ 1976    │ 447.37      │ 89.72     │ 25.09   │ 0.0497647    │\n",
                            "│ 8   │ 1977    │ 455.65      │ 8.29      │ 1.85    │ 0.0529853    │\n",
                            "│ 9   │ 1978    │ 370.82      │ -84.84    │ -18.62  │ 0.0199891    │\n",
                            "│ 10  │ 1979    │ 374.9       │ 4.08      │ 1.1     │ 0.0215761    │\n",
                            "⋮\n",
                            "│ 40  │ 2009    │ 1052.74     │ -624.65   │ -37.24  │ 0.285235     │\n",
                            "│ 41  │ 2010    │ 1331.57     │ 278.82    │ 26.49   │ 0.393691     │\n",
                            "│ 42  │ 2011    │ 1495.64     │ 164.07    │ 12.32   │ 0.457509     │\n",
                            "│ 43  │ 2012    │ 1473.48     │ -22.16    │ -1.48   │ 0.448889     │\n",
                            "│ 44  │ 2013    │ 1650.88     │ 177.4     │ 12.04   │ 0.517893     │\n",
                            "│ 45  │ 2014    │ 2000.63     │ 349.75    │ 21.19   │ 0.653934     │\n",
                            "│ 46  │ 2015    │ 2228.58     │ 227.95    │ 11.39   │ 0.7426       │\n",
                            "│ 47  │ 2016    │ 2079.61     │ -148.96   │ -6.68   │ 0.684655     │\n",
                            "│ 48  │ 2017    │ 2405.9      │ 326.29    │ 15.69   │ 0.811572     │\n",
                            "│ 49  │ 2018    │ 2890.33     │ 484.42    │ 20.13   │ 1.0          │\n",
                            "│ 50  │ 2019    │ 2660.08     │ -230.25   │ -7.97   │ 0.91044      │"
                        ]
                    },
                    "execution_count": 10,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "fileName = \"C:\\\\Users\\\\drcxcruz\\\\Documents\\\\Julia\\\\nonlinearICP\\\\SPXSince1970.csv\"\n",
                "dfSPX = CSV.File(fileName; normalizenames=true) |> DataFrame\n",
                "rename!(dfSPX, [:YearSPX,:USDvalueSPX,:ChangeSPX,:RateSPX])\n",
                "dfSPX[!,:USDvalueSPX] .= parse.( Float64, replace.(dfSPX[!,:USDvalueSPX], \",\" => \"\"))\n",
                "dfSPX[!,:ValueNormSPX] .= normalizer(dfSPX[!,:USDvalueSPX])\n",
                "\n",
                "dfSPX"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Combining the Datasets\n",
                "\n",
                "For simplicity, let us make one single table with all the datasets.  Notice that each dataset is ordered by year."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 11,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "application/vnd.dataresource+json": {
                            "data": [
                                {
                                    "ChangeSPX": -122.24,
                                    "Rate": 4.983333333333333,
                                    "RateBananas": "0.14%",
                                    "RateNormUnemployment": 0.2179310344827587,
                                    "RateSPX": -16.61,
                                    "USDvalueBananas": 3.5,
                                    "USDvalueCars": 15000,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 100000,
                                    "USDvalueMed": 1000,
                                    "USDvalueSPX": 613.53,
                                    "ValueNormBananas": 0.013054830287206278,
                                    "ValueNormCars": 0,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0,
                                    "ValueNormMed": 0,
                                    "ValueNormSPX": 0.11439573690147417,
                                    "Year": 1970,
                                    "YearBananas": 1970,
                                    "YearCars": 1970,
                                    "YearCollege": 1970,
                                    "YearHousing": 1970,
                                    "YearMed": 1970,
                                    "YearSPX": 1970
                                },
                                {
                                    "ChangeSPX": -10.31,
                                    "Rate": 5.95,
                                    "RateBananas": "-5.82%",
                                    "RateNormUnemployment": 0.3779310344827587,
                                    "RateSPX": -1.68,
                                    "USDvalueBananas": 3.3,
                                    "USDvalueCars": 15607.82,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 104445.46,
                                    "USDvalueMed": 1064.05,
                                    "USDvalueSPX": 603.22,
                                    "ValueNormBananas": 0,
                                    "ValueNormCars": 0.0234031927866017,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.007041740273222625,
                                    "ValueNormMed": 0.004682907570152219,
                                    "ValueNormSPX": 0.11038546812400327,
                                    "Year": 1971,
                                    "YearBananas": 1971,
                                    "YearCars": 1971,
                                    "YearCollege": 1971,
                                    "YearHousing": 1971,
                                    "YearMed": 1971,
                                    "YearSPX": 1971
                                },
                                {
                                    "ChangeSPX": 42.21,
                                    "Rate": 5.6000000000000005,
                                    "RateBananas": "6.43%",
                                    "RateNormUnemployment": 0.3200000000000001,
                                    "RateSPX": 7,
                                    "USDvalueBananas": 3.51,
                                    "USDvalueCars": 15475.89,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 108501.37,
                                    "USDvalueMed": 1099.14,
                                    "USDvalueSPX": 645.43,
                                    "ValueNormBananas": 0.013707571801566577,
                                    "ValueNormCars": 0.018323427026448413,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.013466421811593516,
                                    "ValueNormMed": 0.0072484536534721585,
                                    "ValueNormSPX": 0.1268038430121747,
                                    "Year": 1972,
                                    "YearBananas": 1972,
                                    "YearCars": 1972,
                                    "YearCollege": 1972,
                                    "YearHousing": 1972,
                                    "YearMed": 1972,
                                    "YearSPX": 1972
                                },
                                {
                                    "ChangeSPX": 68.3,
                                    "Rate": 4.8583333333333325,
                                    "RateBananas": "4.43%",
                                    "RateNormUnemployment": 0.19724137931034474,
                                    "RateSPX": 10.58,
                                    "USDvalueBananas": 3.66,
                                    "USDvalueCars": 15480.6,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 113221.81,
                                    "USDvalueMed": 1141.35,
                                    "USDvalueSPX": 713.73,
                                    "ValueNormBananas": 0.0234986945169713,
                                    "ValueNormCars": 0.018504778475931676,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.020943738547168907,
                                    "ValueNormMed": 0.01033456651117902,
                                    "ValueNormSPX": 0.15337041502975612,
                                    "Year": 1973,
                                    "YearBananas": 1973,
                                    "YearCars": 1973,
                                    "YearCollege": 1973,
                                    "YearHousing": 1973,
                                    "YearMed": 1973,
                                    "YearSPX": 1973
                                },
                                {
                                    "ChangeSPX": -184.1,
                                    "Rate": 5.641666666666667,
                                    "RateBananas": "12.31%",
                                    "RateNormUnemployment": 0.32689655172413795,
                                    "RateSPX": -25.79,
                                    "USDvalueBananas": 4.11,
                                    "USDvalueCars": 16378.2,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 125939.51,
                                    "USDvalueMed": 1247.61,
                                    "USDvalueSPX": 529.63,
                                    "ValueNormBananas": 0.05287206266318541,
                                    "ValueNormCars": 0.05306551330738458,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.041088951927283276,
                                    "ValueNormMed": 0.018103586939038116,
                                    "ValueNormSPX": 0.081761250923801,
                                    "Year": 1974,
                                    "YearBananas": 1974,
                                    "YearCars": 1974,
                                    "YearCollege": 1974,
                                    "YearHousing": 1974,
                                    "YearMed": 1974,
                                    "YearSPX": 1974
                                },
                                {
                                    "ChangeSPX": -171.99,
                                    "Rate": 8.475,
                                    "RateBananas": "25.26%",
                                    "RateNormUnemployment": 0.7958620689655174,
                                    "RateSPX": -32.47,
                                    "USDvalueBananas": 5.15,
                                    "USDvalueCars": 17784.67,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 139230.06,
                                    "USDvalueMed": 1398.53,
                                    "USDvalueSPX": 357.64,
                                    "ValueNormBananas": 0.120757180156658,
                                    "ValueNormCars": 0.10721952034659298,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.06214157666989233,
                                    "ValueNormMed": 0.029137847836577125,
                                    "ValueNormSPX": 0.014862499513788936,
                                    "Year": 1975,
                                    "YearBananas": 1975,
                                    "YearCars": 1975,
                                    "YearCollege": 1975,
                                    "YearHousing": 1975,
                                    "YearMed": 1975,
                                    "YearSPX": 1975
                                },
                                {
                                    "ChangeSPX": 89.72,
                                    "Rate": 7.699999999999999,
                                    "RateBananas": "1.33%",
                                    "RateNormUnemployment": 0.6675862068965518,
                                    "RateSPX": 25.09,
                                    "USDvalueBananas": 5.22,
                                    "USDvalueCars": 18910.79,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 147777.27,
                                    "USDvalueMed": 1531.29,
                                    "USDvalueSPX": 447.37,
                                    "ValueNormBananas": 0.12532637075718014,
                                    "ValueNormCars": 0.15057907327484144,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.07568061039884075,
                                    "ValueNormMed": 0.03884437100618538,
                                    "ValueNormSPX": 0.049764673849624644,
                                    "Year": 1976,
                                    "YearBananas": 1976,
                                    "YearCars": 1976,
                                    "YearCollege": 1976,
                                    "YearHousing": 1976,
                                    "YearMed": 1976,
                                    "YearSPX": 1976
                                },
                                {
                                    "ChangeSPX": 8.29,
                                    "Rate": 7.050000000000001,
                                    "RateBananas": "8.70%",
                                    "RateNormUnemployment": 0.5600000000000002,
                                    "RateSPX": 1.85,
                                    "USDvalueBananas": 5.68,
                                    "USDvalueCars": 19914.4,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 157859.76,
                                    "USDvalueMed": 1677.79,
                                    "USDvalueSPX": 455.65,
                                    "ValueNormBananas": 0.15535248041775457,
                                    "ValueNormCars": 0.189221563341903,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.09165157310851858,
                                    "ValueNormMed": 0.049555471069062836,
                                    "ValueNormSPX": 0.05298533587459643,
                                    "Year": 1977,
                                    "YearBananas": 1977,
                                    "YearCars": 1977,
                                    "YearCollege": 1977,
                                    "YearHousing": 1977,
                                    "YearMed": 1977,
                                    "YearSPX": 1977
                                },
                                {
                                    "ChangeSPX": -84.84,
                                    "Rate": 6.066666666666666,
                                    "RateBananas": "11.81%",
                                    "RateNormUnemployment": 0.39724137931034487,
                                    "RateSPX": -18.62,
                                    "USDvalueBananas": 6.35,
                                    "USDvalueCars": 21429.24,
                                    "USDvalueCollege": 20693.5,
                                    "USDvalueHousing": 171677.36,
                                    "USDvalueMed": 1818.4,
                                    "USDvalueSPX": 370.82,
                                    "ValueNormBananas": 0.19908616187989556,
                                    "ValueNormCars": 0.24754819385892404,
                                    "ValueNormCollege": 0.0024864357490302544,
                                    "ValueNormHousing": 0.1135390606574518,
                                    "ValueNormMed": 0.05983593373009491,
                                    "ValueNormSPX": 0.019989108872379317,
                                    "Year": 1978,
                                    "YearBananas": 1978,
                                    "YearCars": 1978,
                                    "YearCollege": 1978,
                                    "YearHousing": 1978,
                                    "YearMed": 1978,
                                    "YearSPX": 1978
                                },
                                {
                                    "ChangeSPX": 4.08,
                                    "Rate": 5.8500000000000005,
                                    "RateBananas": "12.93%",
                                    "RateNormUnemployment": 0.36137931034482773,
                                    "RateSPX": 1.1,
                                    "USDvalueBananas": 7.17,
                                    "USDvalueCars": 23130.2,
                                    "USDvalueCollege": 22338.25,
                                    "USDvalueHousing": 192621.45,
                                    "USDvalueMed": 1986.99,
                                    "USDvalueSPX": 374.9,
                                    "ValueNormBananas": 0.25261096605744127,
                                    "ValueNormCars": 0.3130410943924669,
                                    "ValueNormCollege": 0.008383429546027386,
                                    "ValueNormHousing": 0.14671511938680695,
                                    "ValueNormMed": 0.07216210683317005,
                                    "ValueNormSPX": 0.021576101754249472,
                                    "Year": 1979,
                                    "YearBananas": 1979,
                                    "YearCars": 1979,
                                    "YearCollege": 1979,
                                    "YearHousing": 1979,
                                    "YearMed": 1979,
                                    "YearSPX": 1979
                                },
                                {
                                    "ChangeSPX": -8.84,
                                    "Rate": 7.175000000000001,
                                    "RateBananas": "14.64%",
                                    "RateNormUnemployment": 0.580689655172414,
                                    "RateSPX": -2.36,
                                    "USDvalueBananas": 8.22,
                                    "USDvalueCars": 24979.58,
                                    "USDvalueCollege": 24466.59,
                                    "USDvalueHousing": 222846.01,
                                    "USDvalueMed": 2204.42,
                                    "USDvalueSPX": 366.05,
                                    "ValueNormBananas": 0.3211488250652742,
                                    "ValueNormCars": 0.3842486832768167,
                                    "ValueNormCollege": 0.01601425962835046,
                                    "ValueNormHousing": 0.194591717397459,
                                    "ValueNormMed": 0.08805913404594441,
                                    "ValueNormSPX": 0.01813372748842818,
                                    "Year": 1980,
                                    "YearBananas": 1980,
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                            "<table><thead><tr><th>Year</th><th>Rate</th><th>RateNormUnemployment</th><th>YearMed</th><th>USDvalueMed</th><th>ValueNormMed</th><th>YearCollege</th><th>USDvalueCollege</th><th>ValueNormCollege</th><th>YearHousing</th><th>USDvalueHousing</th><th>ValueNormHousing</th><th>YearCars</th><th>USDvalueCars</th><th>ValueNormCars</th><th>YearBananas</th><th>USDvalueBananas</th><th>RateBananas</th><th>ValueNormBananas</th><th>YearSPX</th><th>USDvalueSPX</th><th>ChangeSPX</th><th>RateSPX</th><th>ValueNormSPX</th></tr></thead><tbody><tr><td>1970</td><td>4.98333</td><td>0.217931</td><td>1970</td><td>1000.0</td><td>0.0</td><td>1970</td><td>20000.0</td><td>0.0</td><td>1970</td><td>100000.0</td><td>0.0</td><td>1970</td><td>15000.0</td><td>0.0</td><td>1970</td><td>3.5</td><td>&quot;0.14&#37;&quot;</td><td>0.0130548</td><td>1970</td><td>613.53</td><td>-122.24</td><td>-16.61</td><td>0.114396</td></tr><tr><td>1971</td><td>5.95</td><td>0.377931</td><td>1971</td><td>1064.05</td><td>0.00468291</td><td>1971</td><td>20000.0</td><td>0.0</td><td>1971</td><td>104445.0</td><td>0.00704174</td><td>1971</td><td>15607.8</td><td>0.0234032</td><td>1971</td><td>3.3</td><td>&quot;-5.82&#37;&quot;</td><td>0.0</td><td>1971</td><td>603.22</td><td>-10.31</td><td>-1.68</td><td>0.110385</td></tr><tr><td>1972</td><td>5.6</td><td>0.32</td><td>1972</td><td>1099.14</td><td>0.00724845</td><td>1972</td><td>20000.0</td><td>0.0</td><td>1972</td><td>108501.0</td><td>0.0134664</td><td>1972</td><td>15475.9</td><td>0.0183234</td><td>1972</td><td>3.51</td><td>&quot;6.43&#37;&quot;</td><td>0.0137076</td><td>1972</td><td>645.43</td><td>42.21</td><td>7.0</td><td>0.126804</td></tr><tr><td>1973</td><td>4.85833</td><td>0.197241</td><td>1973</td><td>1141.35</td><td>0.0103346</td><td>1973</td><td>20000.0</td><td>0.0</td><td>1973</td><td>113222.0</td><td>0.0209437</td><td>1973</td><td>15480.6</td><td>0.0185048</td><td>1973</td><td>3.66</td><td>&quot;4.43&#37;&quot;</td><td>0.0234987</td><td>1973</td><td>713.73</td><td>68.3</td><td>10.58</td><td>0.15337</td></tr><tr><td>1974</td><td>5.64167</td><td>0.326897</td><td>1974</td><td>1247.61</td><td>0.0181036</td><td>1974</td><td>20000.0</td><td>0.0</td><td>1974</td><td>125940.0</td><td>0.041089</td><td>1974</td><td>16378.2</td><td>0.0530655</td><td>1974</td><td>4.11</td><td>&quot;12.31&#37;&quot;</td><td>0.0528721</td><td>1974</td><td>529.63</td><td>-184.1</td><td>-25.79</td><td>0.0817613</td></tr><tr><td>1975</td><td>8.475</td><td>0.795862</td><td>1975</td><td>1398.53</td><td>0.0291378</td><td>1975</td><td>20000.0</td><td>0.0</td><td>1975</td><td>139230.0</td><td>0.0621416</td><td>1975</td><td>17784.7</td><td>0.10722</td><td>1975</td><td>5.15</td><td>&quot;25.26&#37;&quot;</td><td>0.120757</td><td>1975</td><td>357.64</td><td>-171.99</td><td>-32.47</td><td>0.0148625</td></tr><tr><td>1976</td><td>7.7</td><td>0.667586</td><td>1976</td><td>1531.29</td><td>0.0388444</td><td>1976</td><td>20000.0</td><td>0.0</td><td>1976</td><td>147777.0</td><td>0.0756806</td><td>1976</td><td>18910.8</td><td>0.150579</td><td>1976</td><td>5.22</td><td>&quot;1.33&#37;&quot;</td><td>0.125326</td><td>1976</td><td>447.37</td><td>89.72</td><td>25.09</td><td>0.0497647</td></tr><tr><td>1977</td><td>7.05</td><td>0.56</td><td>1977</td><td>1677.79</td><td>0.0495555</td><td>1977</td><td>20000.0</td><td>0.0</td><td>1977</td><td>157860.0</td><td>0.0916516</td><td>1977</td><td>19914.4</td><td>0.189222</td><td>1977</td><td>5.68</td><td>&quot;8.70&#37;&quot;</td><td>0.155352</td><td>1977</td><td>455.65</td><td>8.29</td><td>1.85</td><td>0.0529853</td></tr><tr><td>1978</td><td>6.06667</td><td>0.397241</td><td>1978</td><td>1818.4</td><td>0.0598359</td><td>1978</td><td>20693.5</td><td>0.00248644</td><td>1978</td><td>171677.0</td><td>0.113539</td><td>1978</td><td>21429.2</td><td>0.247548</td><td>1978</td><td>6.35</td><td>&quot;11.81&#37;&quot;</td><td>0.199086</td><td>1978</td><td>370.82</td><td>-84.84</td><td>-18.62</td><td>0.0199891</td></tr><tr><td>1979</td><td>5.85</td><td>0.361379</td><td>1979</td><td>1986.99</td><td>0.0721621</td><td>1979</td><td>22338.2</td><td>0.00838343</td><td>1979</td><td>192621.0</td><td>0.146715</td><td>1979</td><td>23130.2</td><td>0.313041</td><td>1979</td><td>7.17</td><td>&quot;12.93&#37;&quot;</td><td>0.252611</td><td>1979</td><td>374.9</td><td>4.08</td><td>1.1</td><td>0.0215761</td></tr><tr><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table><p>... with more rows.</p>"
                        ],
                        "text/plain": [
                            "?x24 query result\n",
                            "Year │ Rate    │ RateNormUnemployment │ YearMed │ USDvalueMed │ ValueNormMed\n",
                            "─────┼─────────┼──────────────────────┼─────────┼─────────────┼─────────────\n",
                            "1970 │ 4.98333 │ 0.217931             │ 1970    │ 1000.0      │ 0.0         \n",
                            "1971 │ 5.95    │ 0.377931             │ 1971    │ 1064.05     │ 0.00468291  \n",
                            "1972 │ 5.6     │ 0.32                 │ 1972    │ 1099.14     │ 0.00724845  \n",
                            "1973 │ 4.85833 │ 0.197241             │ 1973    │ 1141.35     │ 0.0103346   \n",
                            "1974 │ 5.64167 │ 0.326897             │ 1974    │ 1247.61     │ 0.0181036   \n",
                            "1975 │ 8.475   │ 0.795862             │ 1975    │ 1398.53     │ 0.0291378   \n",
                            "1976 │ 7.7     │ 0.667586             │ 1976    │ 1531.29     │ 0.0388444   \n",
                            "1977 │ 7.05    │ 0.56                 │ 1977    │ 1677.79     │ 0.0495555   \n",
                            "1978 │ 6.06667 │ 0.397241             │ 1978    │ 1818.4      │ 0.0598359   \n",
                            "1979 │ 5.85    │ 0.361379             │ 1979    │ 1986.99     │ 0.0721621   \n",
                            "... with more rows, and 18 more columns: YearCollege, USDvalueCollege, ValueNormCollege, YearHousing, USDvalueHousing, ValueNormHousing, YearCars, USDvalueCars, ValueNormCars, YearBananas, USDvalueBananas, RateBananas, ValueNormBananas, YearSPX, USDvalueSPX, ChangeSPX, RateSPX, ValueNormSPX"
                        ]
                    },
                    "execution_count": 11,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "dfCombine = copy(dfUmemployment)\n",
                "\n",
                "for c in names(dfMedicalCareServices)\n",
                "    dfCombine[!,c] = dfMedicalCareServices[!,c]\n",
                "end\n",
                "for c in names(dfCollegeTuition)\n",
                "    dfCombine[!,c] = dfCollegeTuition[!,c]\n",
                "end\n",
                "for c in names(dfHousing)\n",
                "    dfCombine[!,c] = dfHousing[!,c]\n",
                "end\n",
                "for c in names(dfCars)\n",
                "    dfCombine[!,c] = dfCars[!,c]\n",
                "end\n",
                "for c in names(dfBananas)\n",
                "    dfCombine[!,c] = dfBananas[!,c]\n",
                "end\n",
                "for c in names(dfSPX)\n",
                "    dfCombine[!,c] = dfSPX[!,c]\n",
                "end\n",
                "\n",
                "select!(dfCombine, Not([:RateMed,:RateCollege,:RateHousing,:RateCars]))\n",
                "\n",
                "dfCombine |> @filter(_.Year >= 1970)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Defining the Invariant Causal Prediction Model\n",
                "\n",
                "Now, we are ready to run the Nonlinear Invariant Causal Prediction model.  We need to create **X** which is the observations, **Y** which is the learning target, and **E** are the environment assignments.  Here, we set a new environment in 2008 when the financial crisis occured in the United States. Year 2008 is the intervention or experiment point. Observations after 2008 are post intervention or treatment.   "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 12,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "application/vnd.dataresource+json": {
                            "data": [
                                {
                                    "ChangeSPX": -122.24,
                                    "Environment": 1,
                                    "Rate": 4.983333333333333,
                                    "RateBananas": "0.14%",
                                    "RateNormUnemployment": 0.2179310344827587,
                                    "RateSPX": -16.61,
                                    "USDvalueBananas": 3.5,
                                    "USDvalueCars": 15000,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 100000,
                                    "USDvalueMed": 1000,
                                    "USDvalueSPX": 613.53,
                                    "ValueNormBananas": 0.013054830287206278,
                                    "ValueNormCars": 0,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0,
                                    "ValueNormMed": 0,
                                    "ValueNormSPX": 0.11439573690147417,
                                    "Year": 1970,
                                    "YearBananas": 1970,
                                    "YearCars": 1970,
                                    "YearCollege": 1970,
                                    "YearHousing": 1970,
                                    "YearMed": 1970,
                                    "YearSPX": 1970
                                },
                                {
                                    "ChangeSPX": -10.31,
                                    "Environment": 1,
                                    "Rate": 5.95,
                                    "RateBananas": "-5.82%",
                                    "RateNormUnemployment": 0.3779310344827587,
                                    "RateSPX": -1.68,
                                    "USDvalueBananas": 3.3,
                                    "USDvalueCars": 15607.82,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 104445.46,
                                    "USDvalueMed": 1064.05,
                                    "USDvalueSPX": 603.22,
                                    "ValueNormBananas": 0,
                                    "ValueNormCars": 0.0234031927866017,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.007041740273222625,
                                    "ValueNormMed": 0.004682907570152219,
                                    "ValueNormSPX": 0.11038546812400327,
                                    "Year": 1971,
                                    "YearBananas": 1971,
                                    "YearCars": 1971,
                                    "YearCollege": 1971,
                                    "YearHousing": 1971,
                                    "YearMed": 1971,
                                    "YearSPX": 1971
                                },
                                {
                                    "ChangeSPX": 42.21,
                                    "Environment": 1,
                                    "Rate": 5.6000000000000005,
                                    "RateBananas": "6.43%",
                                    "RateNormUnemployment": 0.3200000000000001,
                                    "RateSPX": 7,
                                    "USDvalueBananas": 3.51,
                                    "USDvalueCars": 15475.89,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 108501.37,
                                    "USDvalueMed": 1099.14,
                                    "USDvalueSPX": 645.43,
                                    "ValueNormBananas": 0.013707571801566577,
                                    "ValueNormCars": 0.018323427026448413,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.013466421811593516,
                                    "ValueNormMed": 0.0072484536534721585,
                                    "ValueNormSPX": 0.1268038430121747,
                                    "Year": 1972,
                                    "YearBananas": 1972,
                                    "YearCars": 1972,
                                    "YearCollege": 1972,
                                    "YearHousing": 1972,
                                    "YearMed": 1972,
                                    "YearSPX": 1972
                                },
                                {
                                    "ChangeSPX": 68.3,
                                    "Environment": 1,
                                    "Rate": 4.8583333333333325,
                                    "RateBananas": "4.43%",
                                    "RateNormUnemployment": 0.19724137931034474,
                                    "RateSPX": 10.58,
                                    "USDvalueBananas": 3.66,
                                    "USDvalueCars": 15480.6,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 113221.81,
                                    "USDvalueMed": 1141.35,
                                    "USDvalueSPX": 713.73,
                                    "ValueNormBananas": 0.0234986945169713,
                                    "ValueNormCars": 0.018504778475931676,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.020943738547168907,
                                    "ValueNormMed": 0.01033456651117902,
                                    "ValueNormSPX": 0.15337041502975612,
                                    "Year": 1973,
                                    "YearBananas": 1973,
                                    "YearCars": 1973,
                                    "YearCollege": 1973,
                                    "YearHousing": 1973,
                                    "YearMed": 1973,
                                    "YearSPX": 1973
                                },
                                {
                                    "ChangeSPX": -184.1,
                                    "Environment": 1,
                                    "Rate": 5.641666666666667,
                                    "RateBananas": "12.31%",
                                    "RateNormUnemployment": 0.32689655172413795,
                                    "RateSPX": -25.79,
                                    "USDvalueBananas": 4.11,
                                    "USDvalueCars": 16378.2,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 125939.51,
                                    "USDvalueMed": 1247.61,
                                    "USDvalueSPX": 529.63,
                                    "ValueNormBananas": 0.05287206266318541,
                                    "ValueNormCars": 0.05306551330738458,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.041088951927283276,
                                    "ValueNormMed": 0.018103586939038116,
                                    "ValueNormSPX": 0.081761250923801,
                                    "Year": 1974,
                                    "YearBananas": 1974,
                                    "YearCars": 1974,
                                    "YearCollege": 1974,
                                    "YearHousing": 1974,
                                    "YearMed": 1974,
                                    "YearSPX": 1974
                                },
                                {
                                    "ChangeSPX": -171.99,
                                    "Environment": 1,
                                    "Rate": 8.475,
                                    "RateBananas": "25.26%",
                                    "RateNormUnemployment": 0.7958620689655174,
                                    "RateSPX": -32.47,
                                    "USDvalueBananas": 5.15,
                                    "USDvalueCars": 17784.67,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 139230.06,
                                    "USDvalueMed": 1398.53,
                                    "USDvalueSPX": 357.64,
                                    "ValueNormBananas": 0.120757180156658,
                                    "ValueNormCars": 0.10721952034659298,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.06214157666989233,
                                    "ValueNormMed": 0.029137847836577125,
                                    "ValueNormSPX": 0.014862499513788936,
                                    "Year": 1975,
                                    "YearBananas": 1975,
                                    "YearCars": 1975,
                                    "YearCollege": 1975,
                                    "YearHousing": 1975,
                                    "YearMed": 1975,
                                    "YearSPX": 1975
                                },
                                {
                                    "ChangeSPX": 89.72,
                                    "Environment": 1,
                                    "Rate": 7.699999999999999,
                                    "RateBananas": "1.33%",
                                    "RateNormUnemployment": 0.6675862068965518,
                                    "RateSPX": 25.09,
                                    "USDvalueBananas": 5.22,
                                    "USDvalueCars": 18910.79,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 147777.27,
                                    "USDvalueMed": 1531.29,
                                    "USDvalueSPX": 447.37,
                                    "ValueNormBananas": 0.12532637075718014,
                                    "ValueNormCars": 0.15057907327484144,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.07568061039884075,
                                    "ValueNormMed": 0.03884437100618538,
                                    "ValueNormSPX": 0.049764673849624644,
                                    "Year": 1976,
                                    "YearBananas": 1976,
                                    "YearCars": 1976,
                                    "YearCollege": 1976,
                                    "YearHousing": 1976,
                                    "YearMed": 1976,
                                    "YearSPX": 1976
                                },
                                {
                                    "ChangeSPX": 8.29,
                                    "Environment": 1,
                                    "Rate": 7.050000000000001,
                                    "RateBananas": "8.70%",
                                    "RateNormUnemployment": 0.5600000000000002,
                                    "RateSPX": 1.85,
                                    "USDvalueBananas": 5.68,
                                    "USDvalueCars": 19914.4,
                                    "USDvalueCollege": 20000,
                                    "USDvalueHousing": 157859.76,
                                    "USDvalueMed": 1677.79,
                                    "USDvalueSPX": 455.65,
                                    "ValueNormBananas": 0.15535248041775457,
                                    "ValueNormCars": 0.189221563341903,
                                    "ValueNormCollege": 0,
                                    "ValueNormHousing": 0.09165157310851858,
                                    "ValueNormMed": 0.049555471069062836,
                                    "ValueNormSPX": 0.05298533587459643,
                                    "Year": 1977,
                                    "YearBananas": 1977,
                                    "YearCars": 1977,
                                    "YearCollege": 1977,
                                    "YearHousing": 1977,
                                    "YearMed": 1977,
                                    "YearSPX": 1977
                                },
                                {
                                    "ChangeSPX": -84.84,
                                    "Environment": 1,
                                    "Rate": 6.066666666666666,
                                    "RateBananas": "11.81%",
                                    "RateNormUnemployment": 0.39724137931034487,
                                    "RateSPX": -18.62,
                                    "USDvalueBananas": 6.35,
                                    "USDvalueCars": 21429.24,
                                    "USDvalueCollege": 20693.5,
                                    "USDvalueHousing": 171677.36,
                                    "USDvalueMed": 1818.4,
                                    "USDvalueSPX": 370.82,
                                    "ValueNormBananas": 0.19908616187989556,
                                    "ValueNormCars": 0.24754819385892404,
                                    "ValueNormCollege": 0.0024864357490302544,
                                    "ValueNormHousing": 0.1135390606574518,
                                    "ValueNormMed": 0.05983593373009491,
                                    "ValueNormSPX": 0.019989108872379317,
                                    "Year": 1978,
                                    "YearBananas": 1978,
                                    "YearCars": 1978,
                                    "YearCollege": 1978,
                                    "YearHousing": 1978,
                                    "YearMed": 1978,
                                    "YearSPX": 1978
                                },
                                {
                                    "ChangeSPX": 4.08,
                                    "Environment": 1,
                                    "Rate": 5.8500000000000005,
                                    "RateBananas": "12.93%",
                                    "RateNormUnemployment": 0.36137931034482773,
                                    "RateSPX": 1.1,
                                    "USDvalueBananas": 7.17,
                                    "USDvalueCars": 23130.2,
                                    "USDvalueCollege": 22338.25,
                                    "USDvalueHousing": 192621.45,
                                    "USDvalueMed": 1986.99,
                                    "USDvalueSPX": 374.9,
                                    "ValueNormBananas": 0.25261096605744127,
                                    "ValueNormCars": 0.3130410943924669,
                                    "ValueNormCollege": 0.008383429546027386,
                                    "ValueNormHousing": 0.14671511938680695,
                                    "ValueNormMed": 0.07216210683317005,
                                    "ValueNormSPX": 0.021576101754249472,
                                    "Year": 1979,
                                    "YearBananas": 1979,
                                    "YearCars": 1979,
                                    "YearCollege": 1979,
                                    "YearHousing": 1979,
                                    "YearMed": 1979,
                                    "YearSPX": 1979
                                },
                                {
                                    "ChangeSPX": -8.84,
                                    "Environment": 1,
                                    "Rate": 7.175000000000001,
                                    "RateBananas": "14.64%",
                                    "RateNormUnemployment": 0.580689655172414,
                                    "RateSPX": -2.36,
                                    "USDvalueBananas": 8.22,
                                    "USDvalueCars": 24979.58,
                                    "USDvalueCollege": 24466.59,
                                    "USDvalueHousing": 222846.01,
                                    "USDvalueMed": 2204.42,
                                    "USDvalueSPX": 366.05,
                                    "ValueNormBananas": 0.3211488250652742,
                                    "ValueNormCars": 0.3842486832768167,
                                    "ValueNormCollege": 0.01601425962835046,
                                    "ValueNormHousing": 0.194591717397459,
                                    "ValueNormMed": 0.08805913404594441,
                                    "ValueNormSPX": 0.01813372748842818,
                                    "Year": 1980,
                                    "YearBananas": 1980,
                                    "YearCars": 1980,
                                    "YearCollege": 1980,
                                    "YearHousing": 1980,
                                    "YearMed": 1980,
                                    "YearSPX": 1980
                                },
                                {
                                    "ChangeSPX": 26.52,
                                    "Environment": 1,
                                    "Rate": 7.616666666666667,
                                    "RateBananas": "6.69%",
                                    "RateNormUnemployment": 0.653793103448276,
                                    "RateSPX": 7.25,
                                    "USDvalueBananas": 8.77,
                                    "USDvalueCars": 26503.85,
                                    "USDvalueCollege": 27503.65,
                                    "USDvalueHousing": 248487.63,
                                    "USDvalueMed": 2441.96,
                                    "USDvalueSPX": 392.58,
                                    "ValueNormBananas": 0.35704960835509136,
                                    "ValueNormCars": 0.4429384017277287,
                                    "ValueNormCollege": 0.02690316309763644,
                                    "ValueNormHousing": 0.23520880274400818,
                                    "ValueNormMed": 0.10542646994311784,
                                    "ValueNormSPX": 0.02845307090902018,
                                    "Year": 1981,
                                    "YearBananas": 1981,
                                    "YearCars": 1981,
                                    "YearCollege": 1981,
                                    "YearHousing": 1981,
                                    "YearMed": 1981,
                                    "YearSPX": 1981
                                },
                                {
                                    "ChangeSPX": -73.14,
                                    "Environment": 1,
                                    "Rate": 9.708333333333332,
                                    "RateBananas": "-1.61%",
                                    "RateNormUnemployment": 1,
                                    "RateSPX": -18.63,
                                    "USDvalueBananas": 8.63,
                                    "USDvalueCars": 27526.31,
                                    "USDvalueCollege": 31199.68,
                                    "USDvalueHousing": 266406.97,
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                            "<table><thead><tr><th>Environment</th><th>Year</th><th>Rate</th><th>RateNormUnemployment</th><th>YearMed</th><th>USDvalueMed</th><th>ValueNormMed</th><th>YearCollege</th><th>USDvalueCollege</th><th>ValueNormCollege</th><th>YearHousing</th><th>USDvalueHousing</th><th>ValueNormHousing</th><th>YearCars</th><th>USDvalueCars</th><th>ValueNormCars</th><th>YearBananas</th><th>USDvalueBananas</th><th>RateBananas</th><th>ValueNormBananas</th><th>YearSPX</th><th>USDvalueSPX</th><th>ChangeSPX</th><th>RateSPX</th><th>ValueNormSPX</th></tr></thead><tbody><tr><td>CategoricalValue&#123;Int64,UInt32&#125; 1</td><td>1970</td><td>4.98333</td><td>0.217931</td><td>1970</td><td>1000.0</td><td>0.0</td><td>1970</td><td>20000.0</td><td>0.0</td><td>1970</td><td>100000.0</td><td>0.0</td><td>1970</td><td>15000.0</td><td>0.0</td><td>1970</td><td>3.5</td><td>&quot;0.14&#37;&quot;</td><td>0.0130548</td><td>1970</td><td>613.53</td><td>-122.24</td><td>-16.61</td><td>0.114396</td></tr><tr><td>CategoricalValue&#123;Int64,UInt32&#125; 1</td><td>1971</td><td>5.95</td><td>0.377931</td><td>1971</td><td>1064.05</td><td>0.00468291</td><td>1971</td><td>20000.0</td><td>0.0</td><td>1971</td><td>104445.0</td><td>0.00704174</td><td>1971</td><td>15607.8</td><td>0.0234032</td><td>1971</td><td>3.3</td><td>&quot;-5.82&#37;&quot;</td><td>0.0</td><td>1971</td><td>603.22</td><td>-10.31</td><td>-1.68</td><td>0.110385</td></tr><tr><td>CategoricalValue&#123;Int64,UInt32&#125; 1</td><td>1972</td><td>5.6</td><td>0.32</td><td>1972</td><td>1099.14</td><td>0.00724845</td><td>1972</td><td>20000.0</td><td>0.0</td><td>1972</td><td>108501.0</td><td>0.0134664</td><td>1972</td><td>15475.9</td><td>0.0183234</td><td>1972</td><td>3.51</td><td>&quot;6.43&#37;&quot;</td><td>0.0137076</td><td>1972</td><td>645.43</td><td>42.21</td><td>7.0</td><td>0.126804</td></tr><tr><td>CategoricalValue&#123;Int64,UInt32&#125; 1</td><td>1973</td><td>4.85833</td><td>0.197241</td><td>1973</td><td>1141.35</td><td>0.0103346</td><td>1973</td><td>20000.0</td><td>0.0</td><td>1973</td><td>113222.0</td><td>0.0209437</td><td>1973</td><td>15480.6</td><td>0.0185048</td><td>1973</td><td>3.66</td><td>&quot;4.43&#37;&quot;</td><td>0.0234987</td><td>1973</td><td>713.73</td><td>68.3</td><td>10.58</td><td>0.15337</td></tr><tr><td>CategoricalValue&#123;Int64,UInt32&#125; 1</td><td>1974</td><td>5.64167</td><td>0.326897</td><td>1974</td><td>1247.61</td><td>0.0181036</td><td>1974</td><td>20000.0</td><td>0.0</td><td>1974</td><td>125940.0</td><td>0.041089</td><td>1974</td><td>16378.2</td><td>0.0530655</td><td>1974</td><td>4.11</td><td>&quot;12.31&#37;&quot;</td><td>0.0528721</td><td>1974</td><td>529.63</td><td>-184.1</td><td>-25.79</td><td>0.0817613</td></tr><tr><td>CategoricalValue&#123;Int64,UInt32&#125; 1</td><td>1975</td><td>8.475</td><td>0.795862</td><td>1975</td><td>1398.53</td><td>0.0291378</td><td>1975</td><td>20000.0</td><td>0.0</td><td>1975</td><td>139230.0</td><td>0.0621416</td><td>1975</td><td>17784.7</td><td>0.10722</td><td>1975</td><td>5.15</td><td>&quot;25.26&#37;&quot;</td><td>0.120757</td><td>1975</td><td>357.64</td><td>-171.99</td><td>-32.47</td><td>0.0148625</td></tr><tr><td>CategoricalValue&#123;Int64,UInt32&#125; 1</td><td>1976</td><td>7.7</td><td>0.667586</td><td>1976</td><td>1531.29</td><td>0.0388444</td><td>1976</td><td>20000.0</td><td>0.0</td><td>1976</td><td>147777.0</td><td>0.0756806</td><td>1976</td><td>18910.8</td><td>0.150579</td><td>1976</td><td>5.22</td><td>&quot;1.33&#37;&quot;</td><td>0.125326</td><td>1976</td><td>447.37</td><td>89.72</td><td>25.09</td><td>0.0497647</td></tr><tr><td>CategoricalValue&#123;Int64,UInt32&#125; 1</td><td>1977</td><td>7.05</td><td>0.56</td><td>1977</td><td>1677.79</td><td>0.0495555</td><td>1977</td><td>20000.0</td><td>0.0</td><td>1977</td><td>157860.0</td><td>0.0916516</td><td>1977</td><td>19914.4</td><td>0.189222</td><td>1977</td><td>5.68</td><td>&quot;8.70&#37;&quot;</td><td>0.155352</td><td>1977</td><td>455.65</td><td>8.29</td><td>1.85</td><td>0.0529853</td></tr><tr><td>CategoricalValue&#123;Int64,UInt32&#125; 1</td><td>1978</td><td>6.06667</td><td>0.397241</td><td>1978</td><td>1818.4</td><td>0.0598359</td><td>1978</td><td>20693.5</td><td>0.00248644</td><td>1978</td><td>171677.0</td><td>0.113539</td><td>1978</td><td>21429.2</td><td>0.247548</td><td>1978</td><td>6.35</td><td>&quot;11.81&#37;&quot;</td><td>0.199086</td><td>1978</td><td>370.82</td><td>-84.84</td><td>-18.62</td><td>0.0199891</td></tr><tr><td>CategoricalValue&#123;Int64,UInt32&#125; 1</td><td>1979</td><td>5.85</td><td>0.361379</td><td>1979</td><td>1986.99</td><td>0.0721621</td><td>1979</td><td>22338.2</td><td>0.00838343</td><td>1979</td><td>192621.0</td><td>0.146715</td><td>1979</td><td>23130.2</td><td>0.313041</td><td>1979</td><td>7.17</td><td>&quot;12.93&#37;&quot;</td><td>0.252611</td><td>1979</td><td>374.9</td><td>4.08</td><td>1.1</td><td>0.0215761</td></tr><tr><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table><p>... with more rows.</p>"
                        ],
                        "text/plain": [
                            "?x25 query result\n",
                            "Environment                      │ Year │ Rate    │ RateNormUnemployment\n",
                            "─────────────────────────────────┼──────┼─────────┼─────────────────────\n",
                            "CategoricalValue{Int64,UInt32} 1 │ 1970 │ 4.98333 │ 0.217931            \n",
                            "CategoricalValue{Int64,UInt32} 1 │ 1971 │ 5.95    │ 0.377931            \n",
                            "CategoricalValue{Int64,UInt32} 1 │ 1972 │ 5.6     │ 0.32                \n",
                            "CategoricalValue{Int64,UInt32} 1 │ 1973 │ 4.85833 │ 0.197241            \n",
                            "CategoricalValue{Int64,UInt32} 1 │ 1974 │ 5.64167 │ 0.326897            \n",
                            "CategoricalValue{Int64,UInt32} 1 │ 1975 │ 8.475   │ 0.795862            \n",
                            "CategoricalValue{Int64,UInt32} 1 │ 1976 │ 7.7     │ 0.667586            \n",
                            "CategoricalValue{Int64,UInt32} 1 │ 1977 │ 7.05    │ 0.56                \n",
                            "CategoricalValue{Int64,UInt32} 1 │ 1978 │ 6.06667 │ 0.397241            \n",
                            "CategoricalValue{Int64,UInt32} 1 │ 1979 │ 5.85    │ 0.361379            \n",
                            "... with more rows, and 21 more columns: YearMed, USDvalueMed, ValueNormMed, YearCollege, USDvalueCollege, ValueNormCollege, YearHousing, USDvalueHousing, ValueNormHousing, YearCars, USDvalueCars, ValueNormCars, YearBananas, USDvalueBananas, RateBananas, ValueNormBananas, YearSPX, USDvalueSPX, ChangeSPX, RateSPX, ValueNormSPX"
                        ]
                    },
                    "execution_count": 12,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "insertcols!(dfCombine, 1, Environment=[y >= 2008 ? 2 : 1 for y in dfCombine.Year])\n",
                "categorical!(dfCombine, [:Environment])\n",
                "\n",
                "dfCombine |> @filter(_.Year >= 1970)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Executing the Nonlinear Invariant Causal Prediction \n",
                "\n",
                "We now have defined the environment, now we define **X** and **Y**.  The column *RateNormUnemployment* is our main concern and it is our **Y**.  The matrix **X** are all the columns except the **Y** and the **E**.   The input argument *alpha* defines the confidence intervals for the variables or predictors. Use the *alpha*=0.1 to obtain 90% confidence intervals."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 13,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Accepted Sets\n",
                        "RObject{VecSxp}\n",
                        "[[1]]\n",
                        "[1] 2\n",
                        "\n",
                        "[[2]]\n",
                        "[1] 1 2\n",
                        "\n",
                        "[[3]]\n",
                        "[1] 1 3\n",
                        "\n",
                        "[[4]]\n",
                        "[1] 1 4\n",
                        "\n",
                        "[[5]]\n",
                        "[1] 4 5\n",
                        "\n",
                        "\n",
                        "Retrieved Causal Variables\n",
                        "RObject{IntSxp}\n",
                        "integer(0)\n",
                        "\n",
                        "pvalues.accepted\n",
                        "RObject{RealSxp}\n",
                        "[1] 0.5287159 0.3342816 0.9893191 0.2140740 1.0446330\n",
                        "\n",
                        "Defining sets \n",
                        "RObject{VecSxp}\n",
                        "[[1]]\n",
                        "[1] 1 2 4\n",
                        "\n",
                        "[[2]]\n",
                        "[1] 1 2 5\n",
                        "\n",
                        "[[3]]\n",
                        "[1] 2 3 4\n",
                        "\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "E = dfCombine.Environment\n",
                "Y = dfCombine.RateNormUnemployment\n",
                "X = select(dfCombine, [:ValueNormCollege, :ValueNormHousing, :ValueNormCars, :ValueNormBananas, :ValueNormSPX])\n",
                "\n",
                "icpUSA = @suppress nonlinearICP(X, Y, E, alpha = 0.10)  #maxSizeSets = 10, speedUp = true \n",
                "\n",
                "println( \"Accepted Sets\\n\", icpUSA[\"acceptedSets\"]) \n",
                "println( \"Retrieved Causal Variables\\n\", icpUSA[\"retrievedCausalVars\"]) \n",
                "println( \"pvalues.accepted\\n\", icpUSA[\"pvalues.accepted\"]) \n",
                "println( \"Defining sets \\n\", icpUSA[\"definingSets\"])\n"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Discussing the Results of the NonLinear Invariant Causal Prediction \n",
                "\n",
                "Here, we like to see what variables caused the higher unemployment numbers between 2008 and 2010. We know that the housing market resulted in a financial recession in the United States.  It is reasonable to expect that variable *ValueNormHousing* caused higher unemployment numbers between 2008 and 2010.  The unemployment rate is the target **Y** while the inflation adjusted prices are matrix **X**.  \n",
                "\n",
                "In the lab, none of the variables are strongly representing any casualty of the target **Y**, within the 90% confidence.  There is not a variable found in each of the accepted sets generated by the nonlinearICP package.  But not all is lost.  There are defining sets to look at.  A defining set implies that at least one of its members is in fact a casual predictor to target **Y**.  We just do not know which member it is.  It is helpful to study all the defining sets.  One could obtain the interaction of all defining sets. The interaction of the defining sets are variables that more likely represent casualty of the target **Y**.\n",
                "\n",
                "We used our favorite graphing Julia package [VegaLite](https://www.queryverse.org/VegaLite.jl/stable) to graph the intersection of the defining sets.  The graphs show that *ValueNormHousing* is very likely to represent casualty of the target **Y**. "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 14,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Defining Sets Interception: Set([2.0])\n"
                    ]
                }
            ],
            "source": [
                "inter = Set([i for i in 1:ncol(X)])\n",
                "for s in icpUSA[\"definingSets\"]\n",
                "    inter = intersect(Set(s), inter)\n",
                "end\n",
                "println(\"Defining Sets Interception: \", inter)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 15,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "application/vnd.vegalite.v4+json": {
                            "data": {
                                "values": [
                                    {
                                        "x": "ValueNormCollege",
                                        "y": 0
                                    },
                                    {
                                        "x": "ValueNormHousing",
                                        "y": 1
                                    },
                                    {
                                        "x": "ValueNormCars",
                                        "y": 0
                                    },
                                    {
                                        "x": "ValueNormBananas",
                                        "y": 0
                                    },
                                    {
                                        "x": "ValueNormSPX",
                                        "y": 0
                                    }
                                ]
                            },
                            "encoding": {
                                "x": {
                                    "axis": {
                                        "title": "Predictor"
                                    },
                                    "field": "x",
                                    "title": null,
                                    "type": "nominal"
                                },
                                "y": {
                                    "axis": {
                                        "title": "In All Definining Sets"
                                    },
                                    "field": "y",
                                    "title": null,
                                    "type": "quantitative"
                                }
                            },
                            "mark": "bar"
                        },
                        "image/png": 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",
                        "image/svg+xml": [
                            "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n",
                            "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
                            "<svg class=\"marks\" width=\"145\" height=\"323\" viewBox=\"0 0 145 323\" version=\"1.1\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><rect width=\"145\" height=\"323\" style=\"fill: white;\"></rect><g transform=\"translate(40,10)\"><g class=\"mark-group role-frame root\"><g transform=\"translate(0,0)\"><path class=\"background\" d=\"M0.5,0.5h100v200h-100Z\" style=\"fill: none; stroke: #ddd;\"></path><g><g class=\"mark-group role-axis\"><g transform=\"translate(0.5,0.5)\"><path class=\"background\" d=\"M0,0h0v0h0Z\" style=\"pointer-events: none; fill: none;\"></path><g><g class=\"mark-rule role-axis-grid\" style=\"pointer-events: none;\"><line transform=\"translate(0,200)\" x2=\"100\" y2=\"0\" style=\"fill: none; stroke: #ddd; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,160)\" x2=\"100\" y2=\"0\" style=\"fill: none; stroke: #ddd; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,120)\" x2=\"100\" y2=\"0\" style=\"fill: none; stroke: #ddd; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,80)\" x2=\"100\" y2=\"0\" style=\"fill: none; stroke: #ddd; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,40)\" x2=\"100\" y2=\"0\" style=\"fill: none; stroke: #ddd; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,0)\" x2=\"100\" y2=\"0\" style=\"fill: none; stroke: #ddd; stroke-width: 1; opacity: 1;\"></line></g></g><path class=\"foreground\" d=\"\" style=\"pointer-events: none; display: none; fill: none;\"></path></g></g><g class=\"mark-group role-axis\"><g transform=\"translate(0.5,200.5)\"><path class=\"background\" d=\"M0,0h0v0h0Z\" style=\"pointer-events: none; fill: none;\"></path><g><g class=\"mark-rule role-axis-tick\" style=\"pointer-events: none;\"><line transform=\"translate(10,0)\" x2=\"0\" y2=\"5\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(30,0)\" x2=\"0\" y2=\"5\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(50,0)\" x2=\"0\" y2=\"5\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(70,0)\" x2=\"0\" y2=\"5\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(90,0)\" x2=\"0\" y2=\"5\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line></g><g class=\"mark-text role-axis-label\" style=\"pointer-events: none;\"><text text-anchor=\"end\" transform=\"translate(9.5,7) rotate(270) translate(0,3)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">ValueNormBananas</text><text text-anchor=\"end\" transform=\"translate(29.5,7) rotate(270) translate(0,3)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">ValueNormCars</text><text text-anchor=\"end\" transform=\"translate(49.5,7) rotate(270) translate(0,3)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">ValueNormCollege</text><text text-anchor=\"end\" transform=\"translate(69.5,7) rotate(270) translate(0,3)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">ValueNormHousing</text><text text-anchor=\"end\" transform=\"translate(89.5,7) rotate(270) translate(0,3)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">ValueNormSPX</text></g><g class=\"mark-rule role-axis-domain\" style=\"pointer-events: none;\"><line transform=\"translate(0,0)\" x2=\"100\" y2=\"0\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line></g><g class=\"mark-text role-axis-title\" style=\"pointer-events: none;\"><text text-anchor=\"middle\" transform=\"translate(50,106)\" style=\"font-family: sans-serif; font-size: 11px; font-weight: bold; fill: #000; opacity: 1;\">Predictor</text></g></g><path class=\"foreground\" d=\"\" style=\"pointer-events: none; display: none; fill: none;\"></path></g></g><g class=\"mark-group role-axis\"><g transform=\"translate(0.5,0.5)\"><path class=\"background\" d=\"M0,0h0v0h0Z\" style=\"pointer-events: none; fill: none;\"></path><g><g class=\"mark-rule role-axis-tick\" style=\"pointer-events: none;\"><line transform=\"translate(0,200)\" x2=\"-5\" y2=\"0\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,160)\" x2=\"-5\" y2=\"0\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,120)\" x2=\"-5\" y2=\"0\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,80)\" x2=\"-5\" y2=\"0\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,40)\" x2=\"-5\" y2=\"0\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,0)\" x2=\"-5\" y2=\"0\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line></g><g class=\"mark-text role-axis-label\" style=\"pointer-events: none;\"><text text-anchor=\"end\" transform=\"translate(-7,203)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">0.0</text><text text-anchor=\"end\" transform=\"translate(-7,163)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">0.2</text><text text-anchor=\"end\" transform=\"translate(-7,123)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">0.4</text><text text-anchor=\"end\" transform=\"translate(-7,83)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">0.6</text><text text-anchor=\"end\" transform=\"translate(-7,42.99999999999999)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">0.8</text><text text-anchor=\"end\" transform=\"translate(-7,3)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">1.0</text></g><g class=\"mark-rule role-axis-domain\" style=\"pointer-events: none;\"><line transform=\"translate(0,200)\" x2=\"0\" y2=\"-200\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line></g><g class=\"mark-text role-axis-title\" style=\"pointer-events: none;\"><text text-anchor=\"middle\" transform=\"translate(-24,100) rotate(-90) translate(0,-2)\" style=\"font-family: sans-serif; font-size: 11px; font-weight: bold; fill: #000; opacity: 1;\">In All Definining Sets</text></g></g><path class=\"foreground\" d=\"\" style=\"pointer-events: none; display: none; fill: none;\"></path></g></g><g class=\"mark-rect role-mark marks\"><path d=\"M41,200h18v0h-18Z\" style=\"fill: #4c78a8;\"></path><path d=\"M61,0h18v200h-18Z\" style=\"fill: #4c78a8;\"></path><path d=\"M21,200h18v0h-18Z\" style=\"fill: #4c78a8;\"></path><path d=\"M1,200h18v0h-18Z\" style=\"fill: #4c78a8;\"></path><path d=\"M81,200h18v0h-18Z\" style=\"fill: #4c78a8;\"></path></g></g><path class=\"foreground\" d=\"\" style=\"display: none; fill: none;\"></path></g></g></g></svg>\n"
                        ],
                        "text/plain": [
                            "@vlplot(\n",
                            "    mark=\"bar\",\n",
                            "    encoding={\n",
                            "        y={\n",
                            "            axis={\n",
                            "                title=\"In All Definining Sets\"\n",
                            "            },\n",
                            "            field=\"y\",\n",
                            "            title=nothing\n",
                            "        },\n",
                            "        x={\n",
                            "            axis={\n",
                            "                title=\"Predictor\"\n",
                            "            },\n",
                            "            field=\"x\",\n",
                            "            title=nothing\n",
                            "        }\n",
                            "    },\n",
                            "    data={\n",
                            "        values=...\n",
                            "    }\n",
                            ")"
                        ]
                    },
                    "execution_count": 15,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "xAxis = [String(n) for n in names(X)]\n",
                "yAxis = [n in inter ? 1 : 0 for n in 1:ncol(X)]\n",
                "@vlplot(:bar, \n",
                "    y={yAxis, axis={title=\"In All Definining Sets\"}},\n",
                "    x={xAxis, axis={title=\"Predictor\"}}\n",
                ")"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Let us graph the defining sets in a bubble plot."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 16,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "application/vnd.vegalite.v4+json": {
                            "data": {
                                "values": [
                                    {
                                        "predictor": "ValueNormCollege",
                                        "value": 2,
                                        "variable": "1"
                                    },
                                    {
                                        "predictor": "ValueNormHousing",
                                        "value": 3,
                                        "variable": "2"
                                    },
                                    {
                                        "predictor": "ValueNormCars",
                                        "value": 1,
                                        "variable": "3"
                                    },
                                    {
                                        "predictor": "ValueNormBananas",
                                        "value": 2,
                                        "variable": "4"
                                    },
                                    {
                                        "predictor": "ValueNormSPX",
                                        "value": 1,
                                        "variable": "5"
                                    }
                                ]
                            },
                            "encoding": {
                                "color": {
                                    "field": "predictor",
                                    "legend": null,
                                    "type": "nominal"
                                },
                                "size": {
                                    "field": "value",
                                    "type": "quantitative"
                                },
                                "x": {
                                    "field": "predictor",
                                    "type": "ordinal"
                                },
                                "y": {
                                    "axis": {
                                        "title": "Count in Defining Sets Interceptions"
                                    },
                                    "field": "value",
                                    "type": "ordinal"
                                }
                            },
                            "height": 200,
                            "mark": "circle",
                            "width": 300
                        },
                        "image/png": 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font-size: 10px; fill: #000; opacity: 1;\">1.5</text></g></g><path class=\"foreground\" d=\"\" style=\"pointer-events: none; display: none; fill: none;\"></path></g><g transform=\"translate(0,50)\"><path class=\"background\" d=\"M0,0h38v18.25671751881339h-38Z\" style=\"pointer-events: none; fill: none; opacity: 1;\"></path><g><g class=\"mark-symbol role-legend-symbol\" style=\"pointer-events: none;\"><path transform=\"translate(10.5,9)\" d=\"M7.756717518813397,0A7.756717518813397,7.756717518813397,0,1,1,-7.756717518813397,0A7.756717518813397,7.756717518813397,0,1,1,7.756717518813397,0\" style=\"fill: black; fill-opacity: 0.7; stroke: none; stroke-width: 1.5; opacity: 0.7;\"></path></g><g class=\"mark-text role-legend-label\" style=\"pointer-events: none;\"><text text-anchor=\"start\" transform=\"translate(25,12)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">2.0</text></g></g><path class=\"foreground\" d=\"\" style=\"pointer-events: none; display: none; fill: none;\"></path></g><g transform=\"translate(0,72)\"><path class=\"background\" d=\"M0,0h38v19.672273827165128h-38Z\" style=\"pointer-events: none; fill: none; opacity: 1;\"></path><g><g class=\"mark-symbol role-legend-symbol\" style=\"pointer-events: none;\"><path transform=\"translate(10.5,9.5)\" d=\"M8.67227382716513,0A8.67227382716513,8.67227382716513,0,1,1,-8.67227382716513,0A8.67227382716513,8.67227382716513,0,1,1,8.67227382716513,0\" style=\"fill: black; fill-opacity: 0.7; stroke: none; stroke-width: 1.5; opacity: 0.7;\"></path></g><g class=\"mark-text role-legend-label\" style=\"pointer-events: none;\"><text text-anchor=\"start\" transform=\"translate(25,12.5)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">2.5</text></g></g><path class=\"foreground\" d=\"\" style=\"pointer-events: none; display: none; fill: none;\"></path></g><g transform=\"translate(0,95)\"><path class=\"background\" d=\"M0,0h38v21.5h-38Z\" style=\"pointer-events: none; fill: none; opacity: 1;\"></path><g><g class=\"mark-symbol role-legend-symbol\" style=\"pointer-events: none;\"><path transform=\"translate(10.5,10.5)\" d=\"M9.5,0A9.5,9.5,0,1,1,-9.5,0A9.5,9.5,0,1,1,9.5,0\" style=\"fill: black; fill-opacity: 0.7; stroke: none; stroke-width: 1.5; opacity: 0.7;\"></path></g><g class=\"mark-text role-legend-label\" style=\"pointer-events: none;\"><text text-anchor=\"start\" transform=\"translate(25,13.5)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">3.0</text></g></g><path class=\"foreground\" d=\"\" style=\"pointer-events: none; display: none; fill: none;\"></path></g></g></g><path class=\"foreground\" d=\"\" style=\"pointer-events: none; display: none; fill: none;\"></path></g></g><g class=\"mark-text role-legend-title\" style=\"pointer-events: none;\"><text text-anchor=\"start\" transform=\"translate(0,9)\" style=\"font-family: sans-serif; font-size: 11px; font-weight: bold; fill: #000; opacity: 1;\">value</text></g></g><path class=\"foreground\" d=\"\" style=\"pointer-events: none; display: none; fill: none;\"></path></g></g></g><path class=\"foreground\" d=\"\" style=\"display: none; fill: none;\"></path></g></g></g></svg>\n"
                        ],
                        "text/plain": [
                            "@vlplot(\n",
                            "    width=300,\n",
                            "    height=200,\n",
                            "    mark=\"circle\",\n",
                            "    encoding={\n",
                            "        color={\n",
                            "            legend=nothing,\n",
                            "            field=\"predictor\",\n",
                            "            type=\"nominal\"\n",
                            "        },\n",
                            "        y={\n",
                            "            axis={\n",
                            "                title=\"Count in Defining Sets Interceptions\"\n",
                            "            },\n",
                            "            field=\"value\",\n",
                            "            type=\"ordinal\"\n",
                            "        },\n",
                            "        x={\n",
                            "            field=\"predictor\",\n",
                            "            type=\"ordinal\"\n",
                            "        },\n",
                            "        size={\n",
                            "            field=\"value\"\n",
                            "        }\n",
                            "    },\n",
                            "    data={\n",
                            "        values=...\n",
                            "    }\n",
                            ")"
                        ]
                    },
                    "execution_count": 16,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "hash = Dict{Int64,Int64}()\n",
                "for s in icpUSA[\"definingSets\"]\n",
                "  for e in collect(s)\n",
                "        if haskey(hash, Int(e))\n",
                "            hash[Int(e)] += 1\n",
                "        else\n",
                "          hash[Int(e)] = 1\n",
                "        end \n",
                "    end\n",
                "end\n",
                "\n",
                "dfGraph = DataFrames.stack(DataFrame(hash), 1:ncol(X)) |> @orderby(_.variable) |> DataFrame\n",
                "dfGraph.predictor = [String(n) for n in names(X)]\n",
                "\n",
                "dfGraph|> \n",
                "@vlplot(\n",
                "    :circle,\n",
                "    width=300, \n",
                "    height=200,\n",
                "    color={ \"predictor:n\", legend=nothing},\n",
                "    y={\"value:o\", axis={title=\"Count in Defining Sets Interceptions\"}},\n",
                "    x=\"predictor:o\",\n",
                "    size=:value\n",
                ")"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Bananas:  'A Causing B' Does Not Mean 'B Causing A'\n",
                "\n",
                "Be very careful when deciding what are the appropriate target **Y** and the variables in **X** before running the ICP model.  Next, we switch target **Y** from *RateNormUnemployment* to *ValueNormHousing*.  And, we forget to add *ValueNormSPX*.   The result is that the banana pricing caused housing problems in the United States.  Variable 4 is *ValueNormBananas* and it is found in all accepted sets. Thus, the Retrieved Causal Variables set generated by the nonlinear ICP package includes 4. And according to the model, this is so with \"90% confidence\". "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 17,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Accepted Sets\n",
                        "RObject{VecSxp}\n",
                        "[[1]]\n",
                        "[1] 4\n",
                        "\n",
                        "[[2]]\n",
                        "[1] 1 4\n",
                        "\n",
                        "\n",
                        "Retrieved Causal Variables\n",
                        "RObject{IntSxp}\n",
                        "[1] 4\n",
                        "\n"
                    ]
                }
            ],
            "source": [
                "E = dfCombine.Environment\n",
                "Y = dfCombine.ValueNormHousing\n",
                "X = select(dfCombine, [:ValueNormCollege, :RateNormUnemployment, :ValueNormCars, :ValueNormBananas])\n",
                "\n",
                "icpUSA = @suppress nonlinearICP(X, Y, E, alpha = 0.10) \n",
                "\n",
                "println( \"Accepted Sets\\n\", icpUSA[\"acceptedSets\"]) \n",
                "println( \"Retrieved Causal Variables\\n\", icpUSA[\"retrievedCausalVars\"]) "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 18,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "application/vnd.vegalite.v4+json": {
                            "data": {
                                "values": [
                                    {
                                        "x": "ValueNormCollege",
                                        "y": 0
                                    },
                                    {
                                        "x": "RateNormUnemployment",
                                        "y": 0
                                    },
                                    {
                                        "x": "ValueNormCars",
                                        "y": 0
                                    },
                                    {
                                        "x": "ValueNormBananas",
                                        "y": 1
                                    }
                                ]
                            },
                            "encoding": {
                                "x": {
                                    "axis": {
                                        "title": "Predictor"
                                    },
                                    "field": "x",
                                    "title": null,
                                    "type": "nominal"
                                },
                                "y": {
                                    "axis": {
                                        "title": "In All Accepted Sets"
                                    },
                                    "field": "y",
                                    "title": null,
                                    "type": "quantitative"
                                }
                            },
                            "mark": "bar"
                        },
                        "image/png": 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",
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                            "<svg class=\"marks\" width=\"125\" height=\"349\" viewBox=\"0 0 125 349\" version=\"1.1\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><rect width=\"125\" height=\"349\" style=\"fill: white;\"></rect><g transform=\"translate(40,10)\"><g class=\"mark-group role-frame root\"><g transform=\"translate(0,0)\"><path class=\"background\" d=\"M0.5,0.5h80v200h-80Z\" style=\"fill: none; stroke: #ddd;\"></path><g><g class=\"mark-group role-axis\"><g transform=\"translate(0.5,0.5)\"><path class=\"background\" d=\"M0,0h0v0h0Z\" style=\"pointer-events: none; fill: none;\"></path><g><g class=\"mark-rule role-axis-grid\" style=\"pointer-events: none;\"><line transform=\"translate(0,200)\" x2=\"80\" y2=\"0\" style=\"fill: none; stroke: #ddd; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,160)\" x2=\"80\" y2=\"0\" style=\"fill: none; stroke: #ddd; stroke-width: 1; opacity: 1;\"></line><line transform=\"translate(0,120)\" x2=\"80\" y2=\"0\" style=\"fill: none; 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font-size: 10px; fill: #000; opacity: 1;\">0.4</text><text text-anchor=\"end\" transform=\"translate(-7,83)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">0.6</text><text text-anchor=\"end\" transform=\"translate(-7,42.99999999999999)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">0.8</text><text text-anchor=\"end\" transform=\"translate(-7,3)\" style=\"font-family: sans-serif; font-size: 10px; fill: #000; opacity: 1;\">1.0</text></g><g class=\"mark-rule role-axis-domain\" style=\"pointer-events: none;\"><line transform=\"translate(0,200)\" x2=\"0\" y2=\"-200\" style=\"fill: none; stroke: #888; stroke-width: 1; opacity: 1;\"></line></g><g class=\"mark-text role-axis-title\" style=\"pointer-events: none;\"><text text-anchor=\"middle\" transform=\"translate(-24,100) rotate(-90) translate(0,-2)\" style=\"font-family: sans-serif; font-size: 11px; font-weight: bold; fill: #000; opacity: 1;\">In All Accepted Sets</text></g></g><path class=\"foreground\" d=\"\" style=\"pointer-events: none; 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                        ],
                        "text/plain": [
                            "@vlplot(\n",
                            "    mark=\"bar\",\n",
                            "    encoding={\n",
                            "        y={\n",
                            "            axis={\n",
                            "                title=\"In All Accepted Sets\"\n",
                            "            },\n",
                            "            field=\"y\",\n",
                            "            title=nothing\n",
                            "        },\n",
                            "        x={\n",
                            "            axis={\n",
                            "                title=\"Predictor\"\n",
                            "            },\n",
                            "            field=\"x\",\n",
                            "            title=nothing\n",
                            "        }\n",
                            "    },\n",
                            "    data={\n",
                            "        values=...\n",
                            "    }\n",
                            ")"
                        ]
                    },
                    "execution_count": 18,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "xAxis = [String(n) for n in names(X)]\n",
                "yAxis = [n in icpUSA[\"retrievedCausalVars\"] ? 1 : 0 for n in 1:ncol(X)]\n",
                "@vlplot(:bar, \n",
                "    y={yAxis, axis={title=\"In All Accepted Sets\"}},\n",
                "    x={xAxis, axis={title=\"Predictor\"}}\n",
                ")"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Conclusion\n",
                "\n",
                "Correlation does not imply causation.  The purpose of this jupyter lab is to encourage conversation about how and why classical machine learning models do not handle casual interference.  The authors of [Invariant Causal Prediction for Nonlinear Models](https://arxiv.org/abs/1706.08576) (2018) not only define a sophisticated mathematical model for causal inference but also create a predictive model on top of it.  The paper's authors implemented their machine learning model in a R package named  [nonlinearICP](https://cran.r-project.org/web/packages/nonlinearICP/index.html).  The Nonlinear Invariant Causal Prediction is an interesting package.  One is able to find causality between the target variable and the predictors of the dataset. The lab shows the Nonlinear Invariant Causal Prediction package in [Julia](https://julialang.org) 1.4.0. along [VegaLite](https://www.queryverse.org/VegaLite.jl/stable) plots of the model results.  The real dataset is from [United States Bureau of Labor Statistics](https://beta.bls.gov/labs/).  The [S&P 500 Index](https://www.officialdata.org/us-economy/s-p-500-price-inflation-adjusted) prices are inflation adjusted also.  \n",
                "\n",
                "It is a challenge to create such causal inference modeling.   It requires new ways of thinking and advanced mathematics. The computation is expensive in time and/or memory.  One must be very careful defining the target and the predictors of the model. That is some of the reasons casual interference is missing from many popular machine learning libraries such as Python’s Scikit-learn. "
            ]
        },
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                "\n",
                "# About the Author\n",
                "\n",
                "Clarman Cruz completed his BS/MS in Computer Science from State University of New York at Stony Brook/Albany. He has more than 15 years technology experience in the banking industry.  He plans to start in 2020 his Master of Data Science from [Central Connecticut State University](https://www2.ccsu.edu/program/DataScience_MS]).\n"
            ]
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