PART III - Build a graph for the travel.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dream Team:  Xingce Bao, Sohyeong Kim, Guilio Masinelli, Silvio Zanoli\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PART III - Build a graph for the travel\n",
    "\n",
    "In this part, we are going to build an isochrone map by doing a breadth-first-search from Zurich HB station to find all the other stations that can be reach in the selected time interval. \n",
    "\n",
    "When building the graph, a probability threshold is set by the user. So, when computing the transfer, only the transfers which satisfy the probability constraint are considered. \n",
    "\n",
    "In the last section of this part, we create an adjacent matrix which stores the information of the direct connection between two stations. This matrix is then stored and used for routing algorithm in the next part."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "import getpass\n",
    "import pyspark\n",
    "from pyspark.sql import SparkSession\n",
    "import os\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from datetime import timedelta\n",
    "import pickle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we start by importing processed data created and stored from the previous sections. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import the lists of stations within 10km from Zürich HB\n",
    "station_data = pickle.load(open(\"./data/train_station_id.p\",\"rb\"))\n",
    "\n",
    "# Import the processed actual data containing mean and variance\n",
    "data_mean_variance_list = []\n",
    "data_mean_variance_list.append(pd.read_csv('./data/monday_processed.csv'))\n",
    "data_mean_variance_list.append(pd.read_csv('./data/tuesday_processed.csv'))\n",
    "data_mean_variance_list.append(pd.read_csv('./data/wednesday_processed.csv'))\n",
    "data_mean_variance_list.append(pd.read_csv('./data/thursday_processed.csv'))\n",
    "data_mean_variance_list.append(pd.read_csv('./data/friday_processed.csv'))\n",
    "data_mean_variance_list.append(pd.read_csv('./data/saturday_processed.csv'))\n",
    "data_mean_variance_list.append(pd.read_csv('./data/sunday_processed.csv'))\n",
    "\n",
    "# Import the schedule of the transportation for each day\n",
    "data_schedule_list = []\n",
    "data_schedule_list.append(pd.read_csv('./data/monday_schedule.csv'))\n",
    "data_schedule_list.append(pd.read_csv('./data/tuesday_schedule.csv'))\n",
    "data_schedule_list.append(pd.read_csv('./data/wednesday_schedule.csv'))\n",
    "data_schedule_list.append(pd.read_csv('./data/thursday_schedule.csv'))\n",
    "data_schedule_list.append(pd.read_csv('./data/friday_schedule.csv'))\n",
    "data_schedule_list.append(pd.read_csv('./data/saturday_schedule.csv'))\n",
    "data_schedule_list.append(pd.read_csv('./data/sunday_schedule.csv'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## CASE 1. Simple case"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start by implementing the simple case of the travel without considering the transfer. It will be like taking a one line of the transport and move until it reaches our expectation of the time and probability. \n",
    "\n",
    "To realize this case, we have implemented several functions as shown below. The descriptions of each function is commented along with the functions. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define a function to compute the distance with the longtitude and the latitude \n",
    "# This function is reused from the PART 0.\n",
    "from math import sin, cos, sqrt, atan2, radians\n",
    "zurich_longtitude = 8.540192\n",
    "zurich_latitude = 47.378177\n",
    "def compute_distance(parameter,longtitude2 = zurich_longtitude,latitude2 = zurich_latitude):\n",
    "    # approximate radius of earth in km\n",
    "    R = 6373.0\n",
    "    longtitude1,latitude1 = parameter\n",
    "    longtitude1 = float(longtitude1)\n",
    "    latitude1 = float(latitude1)\n",
    "    lat1 = radians(latitude1)\n",
    "    lon1 = radians(longtitude1)\n",
    "    lat2 = radians(latitude2)\n",
    "    lon2 = radians(longtitude2)\n",
    "\n",
    "    dlon = lon2 - lon1\n",
    "    dlat = lat2 - lat1\n",
    "\n",
    "    a = sin(dlat / 2)**2 + cos(lat1) * cos(lat2) * sin(dlon / 2)**2\n",
    "    c = 2 * atan2(sqrt(a), sqrt(1 - a))\n",
    "\n",
    "    distance = R * c\n",
    "    return distance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy.stats\n",
    "'''\n",
    "This function gives the probability that P(X<x) , where X is a gamma distribution random variable\n",
    "'''\n",
    "def compute_probability(parameter,x):\n",
    "    k = parameter[0]\n",
    "    theta = parameter[1]\n",
    "    if k*theta*theta < 0.01:\n",
    "        if k*theta < x:\n",
    "            return 1.0\n",
    "        else:\n",
    "            return 0.0\n",
    "    dist = scipy.stats.gamma(k, 0, theta)\n",
    "    return dist.cdf(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "'''\n",
    "Now we give the function to compute the probability that P(X<Y) where X and Y are both gamma distribution.This cannot be \n",
    "compute analytically so we use Monte Carlo simulation to simulate the probability.\n",
    "'''\n",
    "def compute_probability_sample(parameter_departure,parameter_arrival,N=1000):\n",
    "    k_arrival, theta_arrival = parameter_arrival\n",
    "    k_departure, theta_departure = parameter_departure\n",
    "    # If it never departures, k_departure will be -1, then we just give 0 for return which means there is no\n",
    "    # possibility that you catch any train from here\n",
    "    if k_departure < 0 or theta_departure < 0:\n",
    "        return 0.0\n",
    "    if np.isnan(k_departure) or np.isnan(theta_departure):\n",
    "        return 0.0\n",
    "    # Build two distribution\n",
    "    dist_arrival = scipy.stats.gamma(k_arrival, 0, theta_arrival+1e-20)\n",
    "    dist_departure = scipy.stats.gamma(k_departure, 0, theta_departure+1e-20)\n",
    "    # Draw samples\n",
    "    arrival_s = dist_arrival.rvs(size=N)\n",
    "    departure_s = dist_departure.rvs(size=N)\n",
    "    # Return the probability by simulation\n",
    "    return np.sum(arrival_s<departure_s)/N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "'''\n",
    "This function gives the train that you can catch providing the station number and the time distribution you arrive. \n",
    "'''\n",
    "def get_train_list(station_no,time_mean,time_variance,end_second,dayofweek,probability):\n",
    "    theta = time_variance/time_mean\n",
    "    k = time_mean/theta\n",
    "    # Filter only keeps the data with relevant day\n",
    "    data_mean_variance = data_mean_variance_list[dayofweek]\n",
    "    data_schedule = data_schedule_list[dayofweek]\n",
    "    # Filter only keeps the data with relevant station\n",
    "    station_frame = data_mean_variance[data_mean_variance.station_id == station_no]\n",
    "    # If no data , return -1 (-1 means no data in this program)\n",
    "    if station_frame.shape[0] == 0:\n",
    "        return -1\n",
    "    # Merge the schedule data\n",
    "    train_frame = pd.merge(station_frame, data_schedule, how='left', on=['train_number', 'station_id','line_id'])\n",
    "    # Filter the data which is not associate to our problem (whose departure time is already out of our time interval)\n",
    "    train_frame = train_frame[train_frame.departuretimeoffsetschedule<=end_second]\n",
    "    # Use the MC simulation function to compute the probability\n",
    "    train_frame_probability_temp = train_frame[['departure_k','departure_theta']].apply(compute_probability_sample, axis=1,args=((k,theta),))\n",
    "    # Keep the train which satisify the probability low bound\n",
    "    if train_frame_probability_temp.shape[0] == 0:\n",
    "        return -1\n",
    "    train_frame = train_frame[train_frame_probability_temp>probability]\n",
    "    return train_frame"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "'''\n",
    "Now we define another function to find the station it can reach using the trains it can be taken.Basically we \n",
    "find the train station after that station using the schedule.\n",
    "'''\n",
    "def get_reachable_station(time_mean,time_variance,end_second,station_no,dayofweek,probability):\n",
    "    # Find the train can be taken\n",
    "    train_list = get_train_list(station_no,time_mean,time_variance,end_second,dayofweek,probability)\n",
    "    # If no train(which train list == -1),return -1\n",
    "    if type(train_list) is int:\n",
    "        return -1\n",
    "    data_schedule = data_schedule_list[dayofweek]\n",
    "    train_list = train_list.reset_index()\n",
    "    train_dataFrame_list = []\n",
    "    # Search every train in the list\n",
    "    for i in range(train_list.shape[0]):\n",
    "        # Get the train number and the departure time\n",
    "        departure_time = train_list.at[i,\"departuretimeoffsetschedule\"]\n",
    "        train_number = train_list.at[i,\"train_number\"]\n",
    "        line_number = train_list.at[i,\"line_id\"]\n",
    "        # Filter the relevant data\n",
    "        train_reachable_place = data_schedule[data_schedule.train_number == train_number]\n",
    "        train_reachable_place = train_reachable_place[train_reachable_place.line_id == line_number]\n",
    "        # Keep the station after the departure\n",
    "        train_reachable_place = train_reachable_place[train_reachable_place.arrivaltimeoffsetschedule > departure_time]\n",
    "        train_reachable_place = train_reachable_place[train_reachable_place.arrivaltimeoffsetschedule <= end_second]\n",
    "        # If no data, use -1 to represent no data \n",
    "        if train_reachable_place.shape[0] == 0:\n",
    "            train_reachable_place = -1\n",
    "        train_dataFrame_list.append(train_reachable_place)\n",
    "    # Remove all -1\n",
    "    train_dataFrame_list = list(filter(lambda a: type(a) is not int, train_dataFrame_list))\n",
    "    # If no data, return -1 to represent no data \n",
    "    if train_dataFrame_list == []: \n",
    "        return -1\n",
    "    return pd.concat(train_dataFrame_list)\n",
    "        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "'''\n",
    "This is the function that compute all the station that you can get by one single departure station.\n",
    "Basically use the function above.\n",
    "'''\n",
    "def one_line_find(time_mean,time_variance,end_second,station_no,dayofweek,probability):\n",
    "    # Get station list\n",
    "    station_list = get_reachable_station(time_mean,time_variance,end_second,station_no,dayofweek,probability)\n",
    "    # Filter only keeps the data with relevant day\n",
    "    data_mean_variance = data_mean_variance_list[dayofweek]\n",
    "    # No train can be taken\n",
    "    if type(station_list) is int:\n",
    "        return -1\n",
    "    print (\"Search for the departure from station \",station_no)\n",
    "    # Join the data with the distribution data\n",
    "    result = pd.merge(station_list, data_mean_variance, how='left', on=['train_number', 'station_id','line_id'])\n",
    "    result = result.sort_values(by = [\"train_number\",'line_id',\"arrivaltimeoffsetschedule\"])\n",
    "    result[\"probability\"] = result[['arrival_k','arrival_theta']].apply(compute_probability, axis=1,args=(end_second,))\n",
    "    # Keep the data which satisfy the probability low bound\n",
    "    result = result[result.probability>probability].reset_index().drop(columns=\"index\")\n",
    "    # If the result is Null\n",
    "    if result.shape[0] == 0:\n",
    "        return -1\n",
    "    return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "'''\n",
    "Now we define the function to give the simple case --- No transfer!\n",
    "station_no is the departure station\n",
    "'''\n",
    "def simple_case(datetime,time_interval,station_no,probability):\n",
    "    # Change the data to pandas datatime\n",
    "    time = pd.to_datetime(datetime,dayfirst=True)\n",
    "    end_time = time + timedelta(seconds = time_interval)\n",
    "    start_second = time.timestamp()%(24*3600)\n",
    "    end_second = end_time.timestamp()%(24*3600)\n",
    "    dayofweek = time.dayofweek\n",
    "    # For the first station, it departs with a fix time\n",
    "    # So start_second is the mean of the departure time , and give a variance which is very small (0.01)\n",
    "    # In fact the variance is 0 , but 0 cause overflow here, and 0.01 second won't change anything\n",
    "    result = one_line_find(start_second,0.01,end_second,station_no,dayofweek,probability)\n",
    "    return result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example of simple case\n",
    "\n",
    "Here is the example of the list of all the transportation we can take from Zürich HB(8503000). \n",
    "\n",
    "1. \n",
    "   - Datetime : 28.05.2018 15:45:00 \n",
    "   - Allowed time : 10 minutes (= 600 seconds)\n",
    "   - Probability : 90%\n",
    "   \n",
    "2. \n",
    "    - Datetime : 06.06.2018 11:00:00 \n",
    "    - Allowed time : 7 minutes (= 420 seconds)\n",
    "    - Probability : 80%\n",
    "    \n",
    "3. \n",
    "    - Datetime : 06.06.2018 11:00:00 \n",
    "    - Allowed time : 7 minutes (= 420 seconds)\n",
    "    - Probability : 50%   \n",
    "    \n",
    "4. \n",
    "    - Datetime : 10.06.2018 11:00:00 \n",
    "    - Allowed time : 7 minutes (= 420 seconds)\n",
    "    - Probability : 50%   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Search for the departure from station  8503000\n"
     ]
    },
    {
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       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>train_number</th>\n",
       "      <th>station_id</th>\n",
       "      <th>line_id</th>\n",
       "      <th>arrivaltimeoffsetschedule</th>\n",
       "      <th>departuretimeoffsetschedule</th>\n",
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       "      <th>var(departuretimeoffset)</th>\n",
       "      <th>arrival_theta</th>\n",
       "      <th>departure_theta</th>\n",
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       "      <th>probability</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>85:11:18259:001</td>\n",
       "      <td>8503011</td>\n",
       "      <td>Zug:18259:S2</td>\n",
       "      <td>56940</td>\n",
       "      <td>56940</td>\n",
       "      <td>57011.0</td>\n",
       "      <td>57045.0</td>\n",
       "      <td>330.0</td>\n",
       "      <td>368.0</td>\n",
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       "      <td>0.006451</td>\n",
       "      <td>9.849255e+06</td>\n",
       "      <td>8.842750e+06</td>\n",
       "      <td>1.000000</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>85:11:18259:001</td>\n",
       "      <td>8503010</td>\n",
       "      <td>Zug:18259:S2</td>\n",
       "      <td>57120</td>\n",
       "      <td>57180</td>\n",
       "      <td>57120.0</td>\n",
       "      <td>57218.0</td>\n",
       "      <td>394.0</td>\n",
       "      <td>99.0</td>\n",
       "      <td>0.006898</td>\n",
       "      <td>0.001730</td>\n",
       "      <td>8.280950e+06</td>\n",
       "      <td>3.306969e+07</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>85:11:18259:002</td>\n",
       "      <td>8503011</td>\n",
       "      <td>Zug:18259:S2</td>\n",
       "      <td>56940</td>\n",
       "      <td>56940</td>\n",
       "      <td>57006.0</td>\n",
       "      <td>57046.0</td>\n",
       "      <td>245.0</td>\n",
       "      <td>324.0</td>\n",
       "      <td>0.004298</td>\n",
       "      <td>0.005680</td>\n",
       "      <td>1.326402e+07</td>\n",
       "      <td>1.004397e+07</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>85:11:18259:002</td>\n",
       "      <td>8503010</td>\n",
       "      <td>Zug:18259:S2</td>\n",
       "      <td>57120</td>\n",
       "      <td>57180</td>\n",
       "      <td>57125.0</td>\n",
       "      <td>57226.0</td>\n",
       "      <td>403.0</td>\n",
       "      <td>135.0</td>\n",
       "      <td>0.007055</td>\n",
       "      <td>0.002359</td>\n",
       "      <td>8.097433e+06</td>\n",
       "      <td>2.425789e+07</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>85:11:19259:001</td>\n",
       "      <td>8503003</td>\n",
       "      <td>Zug:19259:S12</td>\n",
       "      <td>56940</td>\n",
       "      <td>57000</td>\n",
       "      <td>56984.0</td>\n",
       "      <td>57055.0</td>\n",
       "      <td>2207.0</td>\n",
       "      <td>2035.0</td>\n",
       "      <td>0.038730</td>\n",
       "      <td>0.035667</td>\n",
       "      <td>1.471308e+06</td>\n",
       "      <td>1.599643e+06</td>\n",
       "      <td>1.000000</td>\n",
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       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>85:11:19659:001</td>\n",
       "      <td>8503003</td>\n",
       "      <td>Zug:19659:S16</td>\n",
       "      <td>56820</td>\n",
       "      <td>56880</td>\n",
       "      <td>56880.0</td>\n",
       "      <td>56963.0</td>\n",
       "      <td>3314.0</td>\n",
       "      <td>4931.0</td>\n",
       "      <td>0.058263</td>\n",
       "      <td>0.086565</td>\n",
       "      <td>9.762626e+05</td>\n",
       "      <td>6.580376e+05</td>\n",
       "      <td>1.000000</td>\n",
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       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>85:11:19659:001</td>\n",
       "      <td>8503004</td>\n",
       "      <td>Zug:19659:S16</td>\n",
       "      <td>57000</td>\n",
       "      <td>57060</td>\n",
       "      <td>57098.0</td>\n",
       "      <td>57156.0</td>\n",
       "      <td>5216.0</td>\n",
       "      <td>4662.0</td>\n",
       "      <td>0.091352</td>\n",
       "      <td>0.081566</td>\n",
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       "      <td>7.007311e+05</td>\n",
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       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>85:11:19959:001</td>\n",
       "      <td>8503006</td>\n",
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       "      <td>57300</td>\n",
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       "      <td>57350.0</td>\n",
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       "      <td>159.0</td>\n",
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       "      <td>2.068568e+07</td>\n",
       "      <td>0.983920</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      train_number  station_id        line_id  arrivaltimeoffsetschedule  \\\n",
       "0  85:11:18259:001     8503011   Zug:18259:S2                      56940   \n",
       "1  85:11:18259:001     8503010   Zug:18259:S2                      57120   \n",
       "2  85:11:18259:002     8503011   Zug:18259:S2                      56940   \n",
       "3  85:11:18259:002     8503010   Zug:18259:S2                      57120   \n",
       "4  85:11:19259:001     8503003  Zug:19259:S12                      56940   \n",
       "5  85:11:19659:001     8503003  Zug:19659:S16                      56820   \n",
       "6  85:11:19659:001     8503004  Zug:19659:S16                      57000   \n",
       "7  85:11:19959:001     8503006  Zug:19959:S19                      57240   \n",
       "\n",
       "   departuretimeoffsetschedule  avg(arrivaltimeoffset)  \\\n",
       "0                        56940                 57011.0   \n",
       "1                        57180                 57120.0   \n",
       "2                        56940                 57006.0   \n",
       "3                        57180                 57125.0   \n",
       "4                        57000                 56984.0   \n",
       "5                        56880                 56880.0   \n",
       "6                        57060                 57098.0   \n",
       "7                        57300                 57255.0   \n",
       "\n",
       "   avg(departuretimeoffset)  var(arrivaltimeoffset)  var(departuretimeoffset)  \\\n",
       "0                   57045.0                   330.0                     368.0   \n",
       "1                   57218.0                   394.0                      99.0   \n",
       "2                   57046.0                   245.0                     324.0   \n",
       "3                   57226.0                   403.0                     135.0   \n",
       "4                   57055.0                  2207.0                    2035.0   \n",
       "5                   56963.0                  3314.0                    4931.0   \n",
       "6                   57156.0                  5216.0                    4662.0   \n",
       "7                   57350.0                   441.0                     159.0   \n",
       "\n",
       "   arrival_theta  departure_theta     arrival_k   departure_k  probability  \n",
       "0       0.005788         0.006451  9.849255e+06  8.842750e+06     1.000000  \n",
       "1       0.006898         0.001730  8.280950e+06  3.306969e+07     1.000000  \n",
       "2       0.004298         0.005680  1.326402e+07  1.004397e+07     1.000000  \n",
       "3       0.007055         0.002359  8.097433e+06  2.425789e+07     1.000000  \n",
       "4       0.038730         0.035667  1.471308e+06  1.599643e+06     1.000000  \n",
       "5       0.058263         0.086565  9.762626e+05  6.580376e+05     1.000000  \n",
       "6       0.091352         0.081566  6.250348e+05  7.007311e+05     0.997398  \n",
       "7       0.007702         0.002772  7.433413e+06  2.068568e+07     0.983920  "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# First example\n",
    "simple_case('28.05.2018 15:45:00',600,8503000,0.9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Search for the departure from station  8503000\n"
     ]
    },
    {
     "data": {
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       "<div>\n",
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>train_number</th>\n",
       "      <th>station_id</th>\n",
       "      <th>line_id</th>\n",
       "      <th>arrivaltimeoffsetschedule</th>\n",
       "      <th>departuretimeoffsetschedule</th>\n",
       "      <th>avg(arrivaltimeoffset)</th>\n",
       "      <th>avg(departuretimeoffset)</th>\n",
       "      <th>var(arrivaltimeoffset)</th>\n",
       "      <th>var(departuretimeoffset)</th>\n",
       "      <th>arrival_theta</th>\n",
       "      <th>departure_theta</th>\n",
       "      <th>arrival_k</th>\n",
       "      <th>departure_k</th>\n",
       "      <th>probability</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>85:11:18341:001</td>\n",
       "      <td>8503003</td>\n",
       "      <td>Zug:18341:S3</td>\n",
       "      <td>39900</td>\n",
       "      <td>39960</td>\n",
       "      <td>39949.0</td>\n",
       "      <td>40027.0</td>\n",
       "      <td>3343.0</td>\n",
       "      <td>3690.0</td>\n",
       "      <td>0.083682</td>\n",
       "      <td>0.092188</td>\n",
       "      <td>4.773923e+05</td>\n",
       "      <td>4.341899e+05</td>\n",
       "      <td>0.890226</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>85:11:18640:001</td>\n",
       "      <td>8503020</td>\n",
       "      <td>Zug:18640:S6</td>\n",
       "      <td>39780</td>\n",
       "      <td>39780</td>\n",
       "      <td>39824.0</td>\n",
       "      <td>39876.0</td>\n",
       "      <td>800.0</td>\n",
       "      <td>886.0</td>\n",
       "      <td>0.020088</td>\n",
       "      <td>0.022219</td>\n",
       "      <td>1.982439e+06</td>\n",
       "      <td>1.794690e+06</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      train_number  station_id       line_id  arrivaltimeoffsetschedule  \\\n",
       "0  85:11:18341:001     8503003  Zug:18341:S3                      39900   \n",
       "1  85:11:18640:001     8503020  Zug:18640:S6                      39780   \n",
       "\n",
       "   departuretimeoffsetschedule  avg(arrivaltimeoffset)  \\\n",
       "0                        39960                 39949.0   \n",
       "1                        39780                 39824.0   \n",
       "\n",
       "   avg(departuretimeoffset)  var(arrivaltimeoffset)  var(departuretimeoffset)  \\\n",
       "0                   40027.0                  3343.0                    3690.0   \n",
       "1                   39876.0                   800.0                     886.0   \n",
       "\n",
       "   arrival_theta  departure_theta     arrival_k   departure_k  probability  \n",
       "0       0.083682         0.092188  4.773923e+05  4.341899e+05     0.890226  \n",
       "1       0.020088         0.022219  1.982439e+06  1.794690e+06     1.000000  "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# second example\n",
    "simple_case('06.06.2018 11:00:00',420,8503000,0.8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Search for the departure from station  8503000\n"
     ]
    },
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>train_number</th>\n",
       "      <th>station_id</th>\n",
       "      <th>line_id</th>\n",
       "      <th>arrivaltimeoffsetschedule</th>\n",
       "      <th>departuretimeoffsetschedule</th>\n",
       "      <th>avg(arrivaltimeoffset)</th>\n",
       "      <th>avg(departuretimeoffset)</th>\n",
       "      <th>var(arrivaltimeoffset)</th>\n",
       "      <th>var(departuretimeoffset)</th>\n",
       "      <th>arrival_theta</th>\n",
       "      <th>departure_theta</th>\n",
       "      <th>arrival_k</th>\n",
       "      <th>departure_k</th>\n",
       "      <th>probability</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>85:11:18341:001</td>\n",
       "      <td>8503003</td>\n",
       "      <td>Zug:18341:S3</td>\n",
       "      <td>39900</td>\n",
       "      <td>39960</td>\n",
       "      <td>39949.0</td>\n",
       "      <td>40027.0</td>\n",
       "      <td>3343.0</td>\n",
       "      <td>3690.0</td>\n",
       "      <td>0.083682</td>\n",
       "      <td>0.092188</td>\n",
       "      <td>4.773923e+05</td>\n",
       "      <td>4.341899e+05</td>\n",
       "      <td>0.890226</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>85:11:18639:001</td>\n",
       "      <td>8503003</td>\n",
       "      <td>Zug:18639:S6</td>\n",
       "      <td>39720</td>\n",
       "      <td>39780</td>\n",
       "      <td>39787.0</td>\n",
       "      <td>39857.0</td>\n",
       "      <td>26387.0</td>\n",
       "      <td>26628.0</td>\n",
       "      <td>0.663207</td>\n",
       "      <td>0.668088</td>\n",
       "      <td>5.999187e+04</td>\n",
       "      <td>5.965827e+04</td>\n",
       "      <td>0.924062</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>85:11:18639:001</td>\n",
       "      <td>8503004</td>\n",
       "      <td>Zug:18639:S6</td>\n",
       "      <td>39900</td>\n",
       "      <td>39960</td>\n",
       "      <td>39963.0</td>\n",
       "      <td>40015.0</td>\n",
       "      <td>435.0</td>\n",
       "      <td>453.0</td>\n",
       "      <td>0.010885</td>\n",
       "      <td>0.011321</td>\n",
       "      <td>3.671359e+06</td>\n",
       "      <td>3.534658e+06</td>\n",
       "      <td>0.996851</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>85:11:18640:001</td>\n",
       "      <td>8503020</td>\n",
       "      <td>Zug:18640:S6</td>\n",
       "      <td>39780</td>\n",
       "      <td>39780</td>\n",
       "      <td>39824.0</td>\n",
       "      <td>39876.0</td>\n",
       "      <td>800.0</td>\n",
       "      <td>886.0</td>\n",
       "      <td>0.020088</td>\n",
       "      <td>0.022219</td>\n",
       "      <td>1.982439e+06</td>\n",
       "      <td>1.794690e+06</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      train_number  station_id       line_id  arrivaltimeoffsetschedule  \\\n",
       "0  85:11:18341:001     8503003  Zug:18341:S3                      39900   \n",
       "1  85:11:18639:001     8503003  Zug:18639:S6                      39720   \n",
       "2  85:11:18639:001     8503004  Zug:18639:S6                      39900   \n",
       "3  85:11:18640:001     8503020  Zug:18640:S6                      39780   \n",
       "\n",
       "   departuretimeoffsetschedule  avg(arrivaltimeoffset)  \\\n",
       "0                        39960                 39949.0   \n",
       "1                        39780                 39787.0   \n",
       "2                        39960                 39963.0   \n",
       "3                        39780                 39824.0   \n",
       "\n",
       "   avg(departuretimeoffset)  var(arrivaltimeoffset)  var(departuretimeoffset)  \\\n",
       "0                   40027.0                  3343.0                    3690.0   \n",
       "1                   39857.0                 26387.0                   26628.0   \n",
       "2                   40015.0                   435.0                     453.0   \n",
       "3                   39876.0                   800.0                     886.0   \n",
       "\n",
       "   arrival_theta  departure_theta     arrival_k   departure_k  probability  \n",
       "0       0.083682         0.092188  4.773923e+05  4.341899e+05     0.890226  \n",
       "1       0.663207         0.668088  5.999187e+04  5.965827e+04     0.924062  \n",
       "2       0.010885         0.011321  3.671359e+06  3.534658e+06     0.996851  \n",
       "3       0.020088         0.022219  1.982439e+06  1.794690e+06     1.000000  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Third example\n",
    "simple_case('06.06.2018 11:00:00',420,8503000,0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Search for the departure from station  8503000\n"
     ]
    },
    {
     "data": {
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       "      <th>train_number</th>\n",
       "      <th>station_id</th>\n",
       "      <th>line_id</th>\n",
       "      <th>arrivaltimeoffsetschedule</th>\n",
       "      <th>departuretimeoffsetschedule</th>\n",
       "      <th>avg(arrivaltimeoffset)</th>\n",
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       "  <tbody>\n",
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       "      <th>0</th>\n",
       "      <td>85:11:18259:001</td>\n",
       "      <td>8503011</td>\n",
       "      <td>Zug:18259:S2</td>\n",
       "      <td>56940</td>\n",
       "      <td>56940</td>\n",
       "      <td>57016.0</td>\n",
       "      <td>57052.0</td>\n",
       "      <td>405.0</td>\n",
       "      <td>622.0</td>\n",
       "      <td>0.007103</td>\n",
       "      <td>0.010902</td>\n",
       "      <td>8.026727e+06</td>\n",
       "      <td>5.233008e+06</td>\n",
       "      <td>1.00000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>85:11:18259:001</td>\n",
       "      <td>8503010</td>\n",
       "      <td>Zug:18259:S2</td>\n",
       "      <td>57120</td>\n",
       "      <td>57180</td>\n",
       "      <td>57129.0</td>\n",
       "      <td>57219.0</td>\n",
       "      <td>613.0</td>\n",
       "      <td>193.0</td>\n",
       "      <td>0.010730</td>\n",
       "      <td>0.003373</td>\n",
       "      <td>5.324180e+06</td>\n",
       "      <td>1.696380e+07</td>\n",
       "      <td>1.00000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>85:11:18259:002</td>\n",
       "      <td>8503011</td>\n",
       "      <td>Zug:18259:S2</td>\n",
       "      <td>56940</td>\n",
       "      <td>56940</td>\n",
       "      <td>57008.0</td>\n",
       "      <td>57043.0</td>\n",
       "      <td>228.0</td>\n",
       "      <td>335.0</td>\n",
       "      <td>0.003999</td>\n",
       "      <td>0.005873</td>\n",
       "      <td>1.425400e+07</td>\n",
       "      <td>9.713146e+06</td>\n",
       "      <td>1.00000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>85:11:18259:002</td>\n",
       "      <td>8503010</td>\n",
       "      <td>Zug:18259:S2</td>\n",
       "      <td>57120</td>\n",
       "      <td>57180</td>\n",
       "      <td>57119.0</td>\n",
       "      <td>57216.0</td>\n",
       "      <td>338.0</td>\n",
       "      <td>19.0</td>\n",
       "      <td>0.005917</td>\n",
       "      <td>0.000332</td>\n",
       "      <td>9.652604e+06</td>\n",
       "      <td>1.722985e+08</td>\n",
       "      <td>1.00000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>85:11:18758:001</td>\n",
       "      <td>8503020</td>\n",
       "      <td>Zug:18758:S7</td>\n",
       "      <td>57060</td>\n",
       "      <td>57060</td>\n",
       "      <td>57089.0</td>\n",
       "      <td>57141.0</td>\n",
       "      <td>373.0</td>\n",
       "      <td>510.0</td>\n",
       "      <td>0.006534</td>\n",
       "      <td>0.008925</td>\n",
       "      <td>8.737678e+06</td>\n",
       "      <td>6.402145e+06</td>\n",
       "      <td>1.00000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>85:11:19259:001</td>\n",
       "      <td>8503003</td>\n",
       "      <td>Zug:19259:S12</td>\n",
       "      <td>56940</td>\n",
       "      <td>57000</td>\n",
       "      <td>56971.0</td>\n",
       "      <td>57046.0</td>\n",
       "      <td>207.0</td>\n",
       "      <td>337.0</td>\n",
       "      <td>0.003633</td>\n",
       "      <td>0.005908</td>\n",
       "      <td>1.567969e+07</td>\n",
       "      <td>9.656517e+06</td>\n",
       "      <td>1.00000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>85:11:19658:001</td>\n",
       "      <td>8503020</td>\n",
       "      <td>Zug:19658:S16</td>\n",
       "      <td>56880</td>\n",
       "      <td>56880</td>\n",
       "      <td>56914.0</td>\n",
       "      <td>56972.0</td>\n",
       "      <td>592.0</td>\n",
       "      <td>733.0</td>\n",
       "      <td>0.010402</td>\n",
       "      <td>0.012866</td>\n",
       "      <td>5.471627e+06</td>\n",
       "      <td>4.428116e+06</td>\n",
       "      <td>1.00000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>85:11:19658:001</td>\n",
       "      <td>8503006</td>\n",
       "      <td>Zug:19658:S16</td>\n",
       "      <td>57180</td>\n",
       "      <td>57240</td>\n",
       "      <td>57187.0</td>\n",
       "      <td>57277.0</td>\n",
       "      <td>698.0</td>\n",
       "      <td>76.0</td>\n",
       "      <td>0.012206</td>\n",
       "      <td>0.001327</td>\n",
       "      <td>4.685319e+06</td>\n",
       "      <td>4.316651e+07</td>\n",
       "      <td>0.99999</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>85:11:19659:001</td>\n",
       "      <td>8503003</td>\n",
       "      <td>Zug:19659:S16</td>\n",
       "      <td>56820</td>\n",
       "      <td>56880</td>\n",
       "      <td>56856.0</td>\n",
       "      <td>56925.0</td>\n",
       "      <td>167.0</td>\n",
       "      <td>171.0</td>\n",
       "      <td>0.002937</td>\n",
       "      <td>0.003004</td>\n",
       "      <td>1.935691e+07</td>\n",
       "      <td>1.895003e+07</td>\n",
       "      <td>1.00000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>85:11:19659:001</td>\n",
       "      <td>8503004</td>\n",
       "      <td>Zug:19659:S16</td>\n",
       "      <td>57000</td>\n",
       "      <td>57060</td>\n",
       "      <td>57059.0</td>\n",
       "      <td>57118.0</td>\n",
       "      <td>219.0</td>\n",
       "      <td>291.0</td>\n",
       "      <td>0.003838</td>\n",
       "      <td>0.005095</td>\n",
       "      <td>1.486634e+07</td>\n",
       "      <td>1.121122e+07</td>\n",
       "      <td>1.00000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>85:11:19659:001</td>\n",
       "      <td>8503100</td>\n",
       "      <td>Zug:19659:S16</td>\n",
       "      <td>57180</td>\n",
       "      <td>57180</td>\n",
       "      <td>57199.0</td>\n",
       "      <td>57244.0</td>\n",
       "      <td>345.0</td>\n",
       "      <td>537.0</td>\n",
       "      <td>0.006032</td>\n",
       "      <td>0.009381</td>\n",
       "      <td>9.483263e+06</td>\n",
       "      <td>6.102189e+06</td>\n",
       "      <td>1.00000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       train_number  station_id        line_id  arrivaltimeoffsetschedule  \\\n",
       "0   85:11:18259:001     8503011   Zug:18259:S2                      56940   \n",
       "1   85:11:18259:001     8503010   Zug:18259:S2                      57120   \n",
       "2   85:11:18259:002     8503011   Zug:18259:S2                      56940   \n",
       "3   85:11:18259:002     8503010   Zug:18259:S2                      57120   \n",
       "4   85:11:18758:001     8503020   Zug:18758:S7                      57060   \n",
       "5   85:11:19259:001     8503003  Zug:19259:S12                      56940   \n",
       "6   85:11:19658:001     8503020  Zug:19658:S16                      56880   \n",
       "7   85:11:19658:001     8503006  Zug:19658:S16                      57180   \n",
       "8   85:11:19659:001     8503003  Zug:19659:S16                      56820   \n",
       "9   85:11:19659:001     8503004  Zug:19659:S16                      57000   \n",
       "10  85:11:19659:001     8503100  Zug:19659:S16                      57180   \n",
       "\n",
       "    departuretimeoffsetschedule  avg(arrivaltimeoffset)  \\\n",
       "0                         56940                 57016.0   \n",
       "1                         57180                 57129.0   \n",
       "2                         56940                 57008.0   \n",
       "3                         57180                 57119.0   \n",
       "4                         57060                 57089.0   \n",
       "5                         57000                 56971.0   \n",
       "6                         56880                 56914.0   \n",
       "7                         57240                 57187.0   \n",
       "8                         56880                 56856.0   \n",
       "9                         57060                 57059.0   \n",
       "10                        57180                 57199.0   \n",
       "\n",
       "    avg(departuretimeoffset)  var(arrivaltimeoffset)  \\\n",
       "0                    57052.0                   405.0   \n",
       "1                    57219.0                   613.0   \n",
       "2                    57043.0                   228.0   \n",
       "3                    57216.0                   338.0   \n",
       "4                    57141.0                   373.0   \n",
       "5                    57046.0                   207.0   \n",
       "6                    56972.0                   592.0   \n",
       "7                    57277.0                   698.0   \n",
       "8                    56925.0                   167.0   \n",
       "9                    57118.0                   219.0   \n",
       "10                   57244.0                   345.0   \n",
       "\n",
       "    var(departuretimeoffset)  arrival_theta  departure_theta     arrival_k  \\\n",
       "0                      622.0       0.007103         0.010902  8.026727e+06   \n",
       "1                      193.0       0.010730         0.003373  5.324180e+06   \n",
       "2                      335.0       0.003999         0.005873  1.425400e+07   \n",
       "3                       19.0       0.005917         0.000332  9.652604e+06   \n",
       "4                      510.0       0.006534         0.008925  8.737678e+06   \n",
       "5                      337.0       0.003633         0.005908  1.567969e+07   \n",
       "6                      733.0       0.010402         0.012866  5.471627e+06   \n",
       "7                       76.0       0.012206         0.001327  4.685319e+06   \n",
       "8                      171.0       0.002937         0.003004  1.935691e+07   \n",
       "9                      291.0       0.003838         0.005095  1.486634e+07   \n",
       "10                     537.0       0.006032         0.009381  9.483263e+06   \n",
       "\n",
       "     departure_k  probability  \n",
       "0   5.233008e+06      1.00000  \n",
       "1   1.696380e+07      1.00000  \n",
       "2   9.713146e+06      1.00000  \n",
       "3   1.722985e+08      1.00000  \n",
       "4   6.402145e+06      1.00000  \n",
       "5   9.656517e+06      1.00000  \n",
       "6   4.428116e+06      1.00000  \n",
       "7   4.316651e+07      0.99999  \n",
       "8   1.895003e+07      1.00000  \n",
       "9   1.121122e+07      1.00000  \n",
       "10  6.102189e+06      1.00000  "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Fourth example\n",
    "simple_case('16.06.2018 15:45:00',600,8503000,0.9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see from the results of the examples, when we increase the probability, the more journeys that we can choose from (compare example 2 and example 3). \n",
    "\n",
    "The change of the data also affects on the result since some journey may not be operated during weekends. (compare example 1 and example 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## CASE 2. Complicated case"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we are implementing the complicated case of the travel by considering the transfer. In this case, we are not only considering the direct transfer(which is transfering to a different line from the same station) but also the transfer by walking to another stations. \n",
    "\n",
    "Here, we assumed that walking speed is 1.75km/h and we would only transfer to the other stations within 100m radius.\n",
    "\n",
    "To realize this case, we have implemented more functions as shown below. The `full_transfer` function searches for all the journeys with possible transfers that meets our time and probability criteria."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "'''\n",
    "This function gives a list of the station that you can find within radius.\n",
    "Also, it returns the time you need to get there\n",
    "'''\n",
    "def get_near_station(station_no,walk_speed = 1.75,radius = 0.1):\n",
    "    # Get this station data position\n",
    "    station = station_data[station_data.station_number == str(station_no)].reset_index()\n",
    "    station_longtitude = float(station.at[0,\"longtitude\"])\n",
    "    station_latitude = float(station.at[0,\"latitude\"])\n",
    "    # Apply the function to add the column distance_temp to store distance relative to this station\n",
    "    station_data['distance_temp'] = station_data[['longtitude','latitude']].apply(compute_distance, axis=1,args=(station_longtitude,station_latitude))\n",
    "    # Filter the station in the radius\n",
    "    station_in_radius = station_data[station_data.distance_temp < radius]\n",
    "    # Get the list of station number and the distance (divide by walk_speed we will have the time)\n",
    "    station_list = station_in_radius.station_number.values.tolist()\n",
    "    time_list = np.around(station_in_radius.distance_temp.values*1000/walk_speed).tolist()\n",
    "    # Change all the str to int for convenience\n",
    "    station_list = [int(i) for i in station_list]\n",
    "    # Delete the station itself or it will cause infinite loop in the function after\n",
    "    for i,now_station in enumerate(station_list):\n",
    "        if now_station == station_no:\n",
    "            del station_list[i]\n",
    "            del time_list[i]\n",
    "            break\n",
    "    return station_list,time_list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "'''\n",
    "This function gives you all the station that we can transfer with their arriving distribution\n",
    "It mainly contains a loop of dealing each station using the function before.\n",
    "'''\n",
    "def find_all_station(station_list,time_mean,time_variance):\n",
    "    all_station_list = station_list\n",
    "    all_time_mean = time_mean\n",
    "    all_time_variance = time_variance\n",
    "    for index,station_no in enumerate(station_list):\n",
    "        # Get the only this station data\n",
    "        near_station,time_list = get_near_station(station_no)\n",
    "        # For all station, walking do not change the variance that we arrive but the mean\n",
    "        time_mean_temp = np.array(time_list) + time_mean[index]\n",
    "        time_mean_temp = list(time_mean_temp)\n",
    "        # Create a variance list with the same length with the mean list\n",
    "        time_variance_temp = len(time_list)*[time_variance[index]]\n",
    "        # Concat all the lists\n",
    "        all_station_list = all_station_list + near_station\n",
    "        all_time_mean = all_time_mean + time_mean_temp \n",
    "        all_time_variance = all_time_variance + time_variance_temp\n",
    "    return all_station_list,all_time_mean,all_time_variance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "'''\n",
    "This function computes the case of transfering the transportations.\n",
    "'''\n",
    "def transfer_case(station_list,time_mean,time_variance,end_second,probability,dayofweek,\\\n",
    "                                            pass_station_no_list,pass_station_mean_list,pass_station_var_list):\n",
    "    result = []\n",
    "    result_fifo = []\n",
    "    depth = -1\n",
    "    while True:\n",
    "        # Find all the station in the radius\n",
    "        all_station_list,all_time_mean,all_time_variance = find_all_station(station_list,time_mean,time_variance)\n",
    "        # Do a loop for the station list to compute the simple case\n",
    "        for station_no,station_time_mean,station_time_variance in zip(all_station_list,all_time_mean,all_time_variance):\n",
    "            i = -1\n",
    "            # If already passed that station, then compute the probability for the first pass\n",
    "            if station_no in pass_station_no_list:\n",
    "                for i,no_temp in enumerate(pass_station_no_list):\n",
    "                    if no_temp == station_no:\n",
    "                        last_mean = pass_station_mean_list[i] \n",
    "                        last_var = pass_station_var_list[i]\n",
    "                        break\n",
    "                p = compute_probability_sample((station_time_mean**2/station_time_variance,station_time_variance\\\n",
    "                                                /station_time_mean),(last_mean**2/last_var,last_var/last_mean))\n",
    "                print ('The probability of first pass this station is ',p)\n",
    "                # If greater than 85 percent it is a duplicate pass, then remove it\n",
    "                if p > 0.85:\n",
    "                    continue\n",
    "\n",
    "            temp = one_line_find(station_time_mean,station_time_variance,end_second,station_no,dayofweek,probability)\n",
    "            # Update the passed station information\n",
    "            if i==-1:\n",
    "                pass_station_no_list.append(station_no) \n",
    "                pass_station_mean_list.append(station_time_mean) \n",
    "                pass_station_var_list.append(station_time_variance)\n",
    "            else:\n",
    "                pass_station_no_list[i] = station_no \n",
    "                pass_station_mean_list[i] = station_time_mean\n",
    "                pass_station_var_list[i] = station_time_variance\n",
    "            result.append((temp,depth+1))\n",
    "            result_fifo.append((temp,depth+1))\n",
    "            #print (\"depth of transfer: \",depth)\n",
    "        # Filter all -1\n",
    "        result = list(filter(lambda a: type(a[0]) is not int, result))\n",
    "        result_fifo = list(filter(lambda a: type(a[0]) is not int, result_fifo))\n",
    "        print (\"****** First IN First OUT ******* \",len(result_fifo))\n",
    "        if len(result_fifo) == 0:\n",
    "            break\n",
    "        # Get the next result in the breadth-first-search fifo\n",
    "        temp,depth = result_fifo[0]\n",
    "        del result_fifo[0]\n",
    "        if type(temp) is not int:\n",
    "            station_list = list(temp.station_id.values)\n",
    "            time_mean = list(temp[\"avg(arrivaltimeoffset)\"].values)\n",
    "            time_variance = list(temp[\"var(arrivaltimeoffset)\"].values)\n",
    "    # Filter all -1\n",
    "    result = list(filter(lambda a: type(a[0]) is not int, result))\n",
    "    if result == []:\n",
    "        return -1\n",
    "    return pd.concat([i[0] for i in result],axis=0)    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "'''\n",
    "This function gives you all the station that we can transfer with their arriving distribution\n",
    "It mainly contains a loop of dealing each station using the function before.\n",
    "'''\n",
    "def full_transfer(datetime,time_interval,station_no,probability):\n",
    "    # Change the data to pandas datatime\n",
    "    time = pd.to_datetime(datetime,dayfirst=True)\n",
    "    end_time = time + timedelta(seconds = time_interval)\n",
    "    dayofweek = time.dayofweek\n",
    "    # Get the time offset with integer\n",
    "    start_second = time.timestamp()%(24*3600)\n",
    "    end_second = end_time.timestamp()%(24*3600)\n",
    "    pass_station_no_list = []\n",
    "    pass_station_mean_list = []\n",
    "    pass_station_var_list = []\n",
    "    # Begin with only one station number at a fix time. Variance gives 0.01 in case of overflow\n",
    "    result = transfer_case([station_no],[start_second],[0.01],end_second,probability,dayofweek,\\\n",
    "                           pass_station_no_list,pass_station_mean_list,pass_station_var_list)\n",
    "    if type(result) is int:\n",
    "        return result\n",
    "    return result.drop_duplicates().reset_index().drop(columns=\"index\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example of complicated case\n",
    "\n",
    "Here is the example of the list of all the transportation we can take from Zürich HB(8503000). \n",
    "\n",
    "1. \n",
    "   - Datetime : 28.05.2018 15:45:00 \n",
    "   - Allowed time : 10 minutes (= 600 seconds)\n",
    "   - Probability : 90%\n",
    "   \n",
    "2. \n",
    "    - Datetime : 06.06.2018 11:00:00 \n",
    "    - Allowed time : 7 minutes (= 420 seconds)\n",
    "    - Probability : 80% \n",
    "\n",
    "3. \n",
    "    - Datetime : 15.06.2018 17:00:00 \n",
    "    - Allowed time : 15 minutes (= 900 seconds)\n",
    "    - Probability : 90%  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Search for the departure from station  8503000\n",
      "****** First IN First OUT *******  1\n",
      "Search for the departure from station  8503011\n",
      "The probability of first pass this station is  0.417\n",
      "Search for the departure from station  8503011\n",
      "The probability of first pass this station is  0.562\n",
      "The probability of first pass this station is  0.09\n",
      "Search for the departure from station  8503003\n",
      "Search for the departure from station  8573710\n",
      "Search for the departure from station  8591058\n",
      "The probability of first pass this station is  0.435\n",
      "Search for the departure from station  8573710\n",
      "The probability of first pass this station is  0.555\n",
      "Search for the departure from station  8591058\n",
      "The probability of first pass this station is  0.572\n",
      "Search for the departure from station  8503059\n",
      "The probability of first pass this station is  0.069\n",
      "Search for the departure from station  8503059\n",
      "Search for the departure from station  8576182\n",
      "****** First IN First OUT *******  5\n",
      "The probability of first pass this station is  0.436\n",
      "The probability of first pass this station is  0.552\n",
      "The probability of first pass this station is  0.405\n",
      "Search for the departure from station  8591058\n",
      "The probability of first pass this station is  0.472\n",
      "The probability of first pass this station is  0.563\n",
      "Search for the departure from station  8591058\n",
      "The probability of first pass this station is  0.564\n",
      "****** First IN First OUT *******  4\n",
      "The probability of first pass this station is  0.432\n",
      "The probability of first pass this station is  0.542\n",
      "The probability of first pass this station is  0.427\n",
      "Search for the departure from station  8591058\n",
      "The probability of first pass this station is  0.422\n",
      "The probability of first pass this station is  0.549\n",
      "Search for the departure from station  8591058\n",
      "The probability of first pass this station is  0.557\n",
      "****** First IN First OUT *******  3\n",
      "****** First IN First OUT *******  2\n",
      "The probability of first pass this station is  0.488\n",
      "****** First IN First OUT *******  1\n",
      "Search for the departure from station  8530813\n",
      "Search for the departure from station  8591364\n",
      "Search for the departure from station  8576193\n",
      "****** First IN First OUT *******  0\n"
     ]
    }
   ],
   "source": [
    "result1 = full_transfer('28.05.2018 15:45:00',600,8503000,0.9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Search for the departure from station  8503000\n",
      "****** First IN First OUT *******  1\n",
      "Search for the departure from station  8591060\n",
      "****** First IN First OUT *******  0\n"
     ]
    }
   ],
   "source": [
    "result2 = full_transfer('06.06.2018 11:00:00',420,8503000,0.8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Search for the departure from station  8503000\n",
      "****** First IN First OUT *******  1\n",
      "Search for the departure from station  8503003\n",
      "Search for the departure from station  8503020\n",
      "The probability of first pass this station is  0.021\n",
      "Search for the departure from station  8503003\n",
      "Search for the departure from station  8503004\n",
      "Search for the departure from station  8503100\n",
      "Search for the departure from station  8503141\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.0\n",
      "Search for the departure from station  8503020\n",
      "Search for the departure from station  8503006\n",
      "Search for the departure from station  8503007\n",
      "Search for the departure from station  8503010\n",
      "The probability of first pass this station is  0.998\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  0.497\n",
      "Search for the departure from station  8503006\n",
      "The probability of first pass this station is  0.442\n",
      "Search for the departure from station  8503006\n",
      "The probability of first pass this station is  0.998\n",
      "The probability of first pass this station is  0.998\n",
      "Search for the departure from station  8591065\n",
      "The probability of first pass this station is  0.017\n",
      "Search for the departure from station  8576182\n",
      "The probability of first pass this station is  0.0\n",
      "Search for the departure from station  8580449\n",
      "Search for the departure from station  8591062\n",
      "The probability of first pass this station is  0.996\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  0.515\n",
      "Search for the departure from station  8580449\n",
      "The probability of first pass this station is  0.499\n",
      "Search for the departure from station  8591062\n",
      "The probability of first pass this station is  0.425\n",
      "Search for the departure from station  8580449\n",
      "The probability of first pass this station is  0.44\n",
      "Search for the departure from station  8591062\n",
      "The probability of first pass this station is  0.995\n",
      "The probability of first pass this station is  0.997\n",
      "The probability of first pass this station is  0.997\n",
      "The probability of first pass this station is  1.0\n",
      "****** First IN First OUT *******  16\n",
      "The probability of first pass this station is  0.486\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  0.476\n",
      "Search for the departure from station  8591065\n",
      "****** First IN First OUT *******  15\n",
      "The probability of first pass this station is  0.494\n",
      "Search for the departure from station  8503147\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  0.511\n",
      "Search for the departure from station  8503004\n",
      "The probability of first pass this station is  0.509\n",
      "Search for the departure from station  8503100\n",
      "The probability of first pass this station is  0.506\n",
      "Search for the departure from station  8503141\n",
      "The probability of first pass this station is  0.513\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  0.468\n",
      "Search for the departure from station  8591065\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  0.473\n",
      "Search for the departure from station  8576182\n",
      "The probability of first pass this station is  0.503\n",
      "The probability of first pass this station is  0.491\n",
      "****** First IN First OUT *******  18\n",
      "The probability of first pass this station is  0.504\n",
      "Search for the departure from station  8503100\n",
      "The probability of first pass this station is  0.49\n",
      "Search for the departure from station  8503141\n",
      "The probability of first pass this station is  0.504\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  0.493\n",
      "The probability of first pass this station is  0.504\n",
      "The probability of first pass this station is  1.0\n",
      "****** First IN First OUT *******  19\n",
      "The probability of first pass this station is  0.509\n",
      "Search for the departure from station  8503141\n",
      "The probability of first pass this station is  0.49\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.474\n",
      "****** First IN First OUT *******  19\n",
      "The probability of first pass this station is  0.467\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.492\n",
      "****** First IN First OUT *******  18\n",
      "The probability of first pass this station is  0.616\n",
      "Search for the departure from station  8503006\n",
      "The probability of first pass this station is  0.495\n",
      "Search for the departure from station  8503007\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  0.6\n",
      "Search for the departure from station  8580449\n",
      "The probability of first pass this station is  0.635\n",
      "Search for the departure from station  8591062\n",
      "The probability of first pass this station is  1.0\n",
      "****** First IN First OUT *******  20\n",
      "The probability of first pass this station is  0.487\n",
      "Search for the departure from station  8503007\n",
      "The probability of first pass this station is  0.499\n",
      "****** First IN First OUT *******  19\n",
      "The probability of first pass this station is  0.535\n",
      "Search for the departure from station  8503007\n",
      "The probability of first pass this station is  0.496\n",
      "****** First IN First OUT *******  18\n",
      "The probability of first pass this station is  0.487\n",
      "Search for the departure from station  8503007\n",
      "The probability of first pass this station is  0.512\n",
      "The probability of first pass this station is  0.504\n",
      "The probability of first pass this station is  0.507\n",
      "****** First IN First OUT *******  17\n",
      "Search for the departure from station  8576202\n",
      "The probability of first pass this station is  0.399\n",
      "The probability of first pass this station is  0.462\n",
      "Search for the departure from station  8576202\n",
      "The probability of first pass this station is  0.43\n",
      "Search for the departure from station  8576203\n",
      "The probability of first pass this station is  0.334\n",
      "The probability of first pass this station is  0.335\n",
      "The probability of first pass this station is  0.345\n",
      "Search for the departure from station  8576198\n",
      "****** First IN First OUT *******  16\n",
      "Search for the departure from station  8591063\n",
      "Search for the departure from station  8591127\n",
      "The probability of first pass this station is  0.746\n",
      "Search for the departure from station  8591063\n",
      "****** First IN First OUT *******  15\n",
      "****** First IN First OUT *******  14\n",
      "The probability of first pass this station is  0.265\n",
      "Search for the departure from station  8591063\n",
      "The probability of first pass this station is  0.492\n",
      "Search for the departure from station  8591127\n",
      "The probability of first pass this station is  0.735\n",
      "Search for the departure from station  8591063\n",
      "****** First IN First OUT *******  13\n",
      "The probability of first pass this station is  0.48\n",
      "****** First IN First OUT *******  12\n",
      "The probability of first pass this station is  0.258\n",
      "Search for the departure from station  8591063\n",
      "The probability of first pass this station is  0.498\n",
      "Search for the departure from station  8591127\n",
      "The probability of first pass this station is  0.725\n",
      "Search for the departure from station  8591063\n",
      "****** First IN First OUT *******  11\n",
      "The probability of first pass this station is  0.498\n",
      "****** First IN First OUT *******  10\n",
      "The probability of first pass this station is  0.498\n",
      "Search for the departure from station  8503100\n",
      "The probability of first pass this station is  0.485\n",
      "Search for the departure from station  8503141\n",
      "The probability of first pass this station is  0.481\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  1.0\n",
      "The probability of first pass this station is  0.497\n",
      "The probability of first pass this station is  0.487\n",
      "The probability of first pass this station is  1.0\n",
      "****** First IN First OUT *******  11\n",
      "The probability of first pass this station is  0.524\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Search for the departure from station  8503141\n",
      "The probability of first pass this station is  0.464\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.499\n",
      "****** First IN First OUT *******  11\n",
      "The probability of first pass this station is  0.516\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.515\n",
      "****** First IN First OUT *******  10\n",
      "The probability of first pass this station is  0.548\n",
      "The probability of first pass this station is  0.552\n",
      "Search for the departure from station  8576202\n",
      "The probability of first pass this station is  0.548\n",
      "The probability of first pass this station is  0.638\n",
      "The probability of first pass this station is  0.655\n",
      "The probability of first pass this station is  0.679\n",
      "The probability of first pass this station is  0.429\n",
      "The probability of first pass this station is  0.429\n",
      "Search for the departure from station  8576202\n",
      "The probability of first pass this station is  0.41\n",
      "Search for the departure from station  8576203\n",
      "The probability of first pass this station is  0.33\n",
      "Search for the departure from station  8576200\n",
      "The probability of first pass this station is  0.328\n",
      "The probability of first pass this station is  0.297\n",
      "Search for the departure from station  8576198\n",
      "The probability of first pass this station is  0.515\n",
      "Search for the departure from station  8576197\n",
      "****** First IN First OUT *******  10\n",
      "The probability of first pass this station is  0.482\n",
      "Search for the departure from station  8503141\n",
      "The probability of first pass this station is  0.511\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.502\n",
      "****** First IN First OUT *******  10\n",
      "The probability of first pass this station is  0.504\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.516\n",
      "****** First IN First OUT *******  9\n",
      "The probability of first pass this station is  0.513\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.507\n",
      "****** First IN First OUT *******  8\n",
      "The probability of first pass this station is  0.501\n",
      "Search for the departure from station  8503007\n",
      "The probability of first pass this station is  0.502\n",
      "The probability of first pass this station is  0.484\n",
      "The probability of first pass this station is  0.5\n",
      "****** First IN First OUT *******  7\n",
      "The probability of first pass this station is  0.287\n",
      "Search for the departure from station  8591063\n",
      "The probability of first pass this station is  0.52\n",
      "Search for the departure from station  8591127\n",
      "The probability of first pass this station is  0.752\n",
      "Search for the departure from station  8591063\n",
      "****** First IN First OUT *******  6\n",
      "The probability of first pass this station is  0.513\n",
      "****** First IN First OUT *******  5\n",
      "The probability of first pass this station is  0.517\n",
      "Search for the departure from station  8503141\n",
      "The probability of first pass this station is  0.497\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.517\n",
      "****** First IN First OUT *******  5\n",
      "The probability of first pass this station is  0.51\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.483\n",
      "****** First IN First OUT *******  4\n",
      "The probability of first pass this station is  0.526\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.528\n",
      "****** First IN First OUT *******  3\n",
      "The probability of first pass this station is  0.703\n",
      "****** First IN First OUT *******  2\n",
      "The probability of first pass this station is  0.516\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.493\n",
      "****** First IN First OUT *******  1\n",
      "The probability of first pass this station is  0.502\n",
      "Search for the departure from station  8503101\n",
      "The probability of first pass this station is  0.506\n",
      "****** First IN First OUT *******  0\n"
     ]
    }
   ],
   "source": [
    "result3 = full_transfer('15.06.2018 17:00:00',900,8503000,0.7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As can be seen from the results above, we can see that more stations can be reached if we consider transfer. \n",
    "\n",
    "Next step is to visualize the stations where we can reach from Zürich HB considering the transfering cases. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For visualization we are using Google Map API and this can be done by importing the `gmaps` and by displaying on the map a marker for each of the station that can be reached withing the time interval selected and the probability choosen. We used the data we compute before, since the computation of the full transfer for a long time interval needs a long time. \n",
    "\n",
    "The result we show is:\n",
    "\n",
    "`result = full_transfer('08.01.2018 11:55:00',1200,8503000,0.9)`\n",
    "\n",
    "This travel is departing at 2018 January 8th,11.55, and we want to compute how far it can go within 20 minutes. We are going to use this result that we computed before."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "result = pd.read_csv('./data/1155MondayResult.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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       " (47.366611, 8.548466),\n",
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       " (47.420195, 8.619255),\n",
       " (47.366611, 8.548466),\n",
       " (47.366611, 8.548466),\n",
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       " (47.378177, 8.540192),\n",
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       " (47.378177, 8.540192),\n",
       " (47.378177, 8.540192),\n",
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       " (47.387575, 8.519445),\n",
       " (47.390795, 8.522398),\n",
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       " (47.390077, 8.487058),\n",
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       " (47.390077, 8.487058),\n",
       " (47.390077, 8.487058),\n",
       " (47.388763, 8.490144),\n",
       " (47.389433, 8.493575),\n",
       " (47.386564, 8.503372),\n",
       " (47.387772, 8.499211),\n",
       " (47.391129, 8.479171),\n",
       " (47.391802, 8.48435),\n",
       " (47.389433, 8.493575),\n",
       " (47.391802, 8.48435),\n",
       " (47.391129, 8.479171),\n",
       " (47.392885, 8.474887),\n",
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       " (47.36817, 8.539747),\n",
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       " (47.373066, 8.538457),\n",
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       " ...]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Get all the stations we are able to reach from the result\n",
    "stations = []\n",
    "for station in result['station_id']:\n",
    "    infos = {}\n",
    "    infos['name'] = station_data[station_data.station_number==str(station)]['name'].item()\n",
    "    infos['station_number'] = station_data[station_data.station_number==str(station)]['station_number'].item()\n",
    "    infos['position'] =(float(station_data[station_data.station_number==str(station)]['latitude'].item()), (float(station_data[station_data.station_number==str(station)]['longtitude'].item())))\n",
    "    infos['probability'] = round(result[result.station_id==(station)]['probability'].min(),2)\n",
    "    stations.append(infos)\n",
    "    \n",
    "# Get the location of the stations\n",
    "station_locations = [station['position'] for station in stations]\n",
    "station_locations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### WARNING:  This gmaps library has some inner bug in printing.If you get nothing when running it, reboot the computer and it will work.\n",
    "#### WARNING:   Here you need to change the api key to your google api key.You need a credit card to get it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "import gmaps\n",
    "key = 'AI...'\n",
    "if key == 'AI...':\n",
    "    raise Exception('You need to put your Gmap key here!')\n",
    "gmaps.configure(api_key=key)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the format of the info box\n",
    "info_box_template = \"\"\"\n",
    "<dl>\n",
    " <center><dt>Name</dt><dd>{name}</dd> \n",
    "<dt>Station numer</dt><dd>{station_number}</dd>\n",
    "<dt>Probability</dt><dd>{probability}</dd></center>\n",
    "</dl>\n",
    "\"\"\"\n",
    "stat_info = [info_box_template.format(**station) for station in stations]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Visualization 1.\n",
    "Here we are displaying each of the station that can be reached with a marker with an info box showing the name of the station, the station number and the probability of reaching that station (in the case the station can be reached by more than one means of transportation, the lowest probability is displayed)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "model_id": "1ac70305cdf840edae7bac5c3724f330",
       "version_major": 2,
       "version_minor": 0
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      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>Figure</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in the Jupyter Notebook or JupyterLab Notebook, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
      ],
      "text/plain": [
       "Figure(layout=FigureLayout(height='420px'))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "marker_layer = gmaps.marker_layer(station_locations, info_box_content=stat_info)\n",
    "fig = gmaps.figure()\n",
    "fig.add_layer(marker_layer)\n",
    "fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the captured image for above cell. \n",
    "<img src=\"./images/map1.png\">\n",
    "</img>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Visualization 2.\n",
    "\n",
    "We can also visualize an heatmap showing the density of the stations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "8592ce0db44d43c8a81e2f1116919d42",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>Figure</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in the Jupyter Notebook or JupyterLab Notebook, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
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       "Figure(layout=FigureLayout(height='420px'))"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "fig = gmaps.figure()\n",
    "\n",
    "heatmap = gmaps.heatmap_layer(station_locations, max_intensity=2)\n",
    "marker_layer = gmaps.marker_layer(station_locations, info_box_content=stat_info)\n",
    "fig.add_layer(marker_layer)\n",
    "heatmap.point_radius = 50\n",
    "fig.add_layer(heatmap)\n",
    "fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the captured image for above cell. \n",
    "<img src=\"./images/map2.png\">\n",
    "</img>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Visualization 3. Convex Hull\n",
    "This time, we tried to visualize the result by taking the furthest stations from the Zürich HB and draw the convex hull. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.spatial import ConvexHull\n",
    "hull = ConvexHull(station_locations)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Add the hull vertices\n",
    "loc = []\n",
    "for index in hull.vertices:\n",
    "    loc.append(station_locations[index])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "9082ab7b115e412892b9bb2c6c4eccde",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>Figure</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in the Jupyter Notebook or JupyterLab Notebook, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
      ],
      "text/plain": [
       "Figure(layout=FigureLayout(height='420px'))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = gmaps.figure(center=(47.378177, 8.540192), zoom_level=12)\n",
    "myMap = gmaps.Polygon(\n",
    "    loc,\n",
    "    stroke_color='blue',\n",
    "    fill_color='blue'\n",
    ")\n",
    "drawing = gmaps.drawing_layer(\n",
    "    features=[myMap],\n",
    "    show_controls=True\n",
    ")\n",
    "fig.add_layer(drawing)\n",
    "\n",
    "fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the captured image for above cell. \n",
    "<img src=\"./images/map3.png\">\n",
    "</img>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Visualization 4. \n",
    "Same as before, but including the markers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "stat_info = [info_box_template.format(**station) for station in stations if (station['position'] in loc)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "723698bbbdfb4e50af20a2ca28d68866",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>Figure</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in the Jupyter Notebook or JupyterLab Notebook, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
      ],
      "text/plain": [
       "Figure(layout=FigureLayout(height='420px'))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = gmaps.figure(center=(47.378177, 8.540192), zoom_level=12)\n",
    "myMap = gmaps.Polygon(\n",
    "    loc,\n",
    "    stroke_color='blue',\n",
    "    fill_color='blue'\n",
    ")\n",
    "drawing = gmaps.drawing_layer(\n",
    "    features=[myMap],\n",
    "    show_controls=True\n",
    ")\n",
    "fig.add_layer(drawing)\n",
    "\n",
    "marker_layer = gmaps.marker_layer(loc, info_box_content=list(set(stat_info)))\n",
    "fig.add_layer(marker_layer)\n",
    "\n",
    "fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the captured image for above cell. \n",
    "<img src=\"./images/map4.png\">\n",
    "</img>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Verification"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To make sure that our result is reasonable, not only we **compared the results manually**(if the result is small enough to be compared by human), we also **check the isochronic map from othersites**. [https://app.traveltimeplatform.com]\n",
    "\n",
    "We first start by using a small dataset and check if what we have computed is correct. Once we are confident that our algorithm is working correctly, we use the whole dataset and visualize them. \n",
    "\n",
    "In this part, we are plotting a heat map and the convex hull map to compare our result and the othersites' result at 15.06.2018 17:00."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Choose the current data result to compare. (15.06.2018 17:00, 900 seconds, 70%)\n",
    "result = result3\n",
    "\n",
    "# Get all the stations we are able to reach from the result\n",
    "stations = []\n",
    "for station in result['station_id']:\n",
    "    infos = {}\n",
    "    infos['name'] = station_data[station_data.station_number==str(station)]['name'].item()\n",
    "    infos['station_number'] = station_data[station_data.station_number==str(station)]['station_number'].item()\n",
    "    infos['position'] =(float(station_data[station_data.station_number==str(station)]['latitude'].item()), (float(station_data[station_data.station_number==str(station)]['longtitude'].item())))\n",
    "    infos['probability'] = round(result[result.station_id==(station)]['probability'].min(),2)\n",
    "    stations.append(infos)\n",
    "    \n",
    "# Get the location of the stations\n",
    "station_locations = [station['position'] for station in stations]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the format of the info box\n",
    "info_box_template = \"\"\"\n",
    "<dl>\n",
    " <center><dt>Name</dt><dd>{name}</dd> \n",
    "<dt>Station numer</dt><dd>{station_number}</dd>\n",
    "<dt>Probability</dt><dd>{probability}</dd></center>\n",
    "</dl>\n",
    "\"\"\"\n",
    "stat_info = [info_box_template.format(**station) for station in stations]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "dd921205fb2d4dc3a060b28dc7288e61",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>Figure</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in the Jupyter Notebook or JupyterLab Notebook, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
      ],
      "text/plain": [
       "Figure(layout=FigureLayout(height='420px'))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = gmaps.figure()\n",
    "\n",
    "heatmap = gmaps.heatmap_layer(station_locations, max_intensity=2)\n",
    "marker_layer = gmaps.marker_layer(station_locations, info_box_content=stat_info)\n",
    "fig.add_layer(marker_layer)\n",
    "heatmap.point_radius = 50\n",
    "fig.add_layer(heatmap)\n",
    "fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the captured image for above cell. \n",
    "\n",
    "<img src=\"./images/map_validation1.png\">\n",
    "</img>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "hull = ConvexHull(station_locations)\n",
    "\n",
    "# Add the hull vertices\n",
    "loc = []\n",
    "for index in hull.vertices:\n",
    "    loc.append(station_locations[index])\n",
    "    \n",
    "stat_info = [info_box_template.format(**station) for station in stations if (station['position'] in loc)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "b4c0dc9bc6e44c5eb01cfe4a80b0f3f6",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>Figure</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in the Jupyter Notebook or JupyterLab Notebook, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
      ],
      "text/plain": [
       "Figure(layout=FigureLayout(height='420px'))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = gmaps.figure(center=(47.378177, 8.540192), zoom_level=12)\n",
    "myMap = gmaps.Polygon(\n",
    "    loc,\n",
    "    stroke_color='blue',\n",
    "    fill_color='blue'\n",
    ")\n",
    "drawing = gmaps.drawing_layer(\n",
    "    features=[myMap],\n",
    "    show_controls=True\n",
    ")\n",
    "fig.add_layer(drawing)\n",
    "\n",
    "marker_layer = gmaps.marker_layer(loc, info_box_content=list(set(stat_info)))\n",
    "fig.add_layer(marker_layer)\n",
    "\n",
    "fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the captured image for above cell. \n",
    "\n",
    "<img src=\"./images/map_validation2.png\">\n",
    "</img>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we can see the results for the deterministic isochrone map obtained from the website below and the non-deterministic one computed by us (convex hull and heatmap). \n",
    "[https://app.traveltimeplatform.com/#/search/0_lng=8.53945&0_tt=15&0_time=d1529074809491&0_title=Z%C3%BCrich%20HB%20SZU%2C%20Zurich%2C%20Switzerland&0_lat=47.37737\"].\n",
    "<img src=\"./images/map_validation3.png\">\n",
    "</img>\n",
    "\n",
    "\n",
    "Both of them were computed for the 15/06/2018 at 17:00 and the confidence level used was 70%. We can see that the results are overall promising: Because of the confidence level we are missing some of the stations present on the west side of the centre of Zurich, that maybe because to travel in the city center implies taking buses that are statistical more prone to have delays and hence get excluded from our reachable stations. At the same time we can see that the furthest stations are equally reached, that's because they are probably connected by trains that statistically have less delays. The representation we obtained from the determistic isochrone map is a sparse representation of the reachable stations, that means that it only put in evidence a small area around the stations reached, such representation is more similar to the heat-map we computed instead of the convex hull. In this example we can observe that our algorithm computes a whole set of reachable stations along the east cost of the lake while the deterministic one only finds two. The two stops found by the determistic isochrone map are quite far from the south-east side of the lake and the absence of all the intermediate stations found by us may be because the algoritm that run in background of the web-site (to which we don't have access) is computing only the furthest reachable station without the intermediate stops."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Build an adjacent matrix of direct transfer"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before going into the next part which is finding the routing plan, we build a matrix which contains the information of the direct transfer between two stations. We also consider walking to close station as a direct transfer so that we can have full information of transfering between stations. In the case of the stations connected by the train, we put 10.0 to represent them. For stations which are connected by walking, we use 1.0 to represent them.\n",
    "To build this matrix, we use arrival and departure schedule(fixed) and we saved the matrix with pickle. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Get all the schedule data\n",
    "data_schedule_all = pd.concat(data_schedule_list,axis=0)\n",
    "\n",
    "# Make a list for stations\n",
    "stations = [int(i) for i in station_data.station_number.values]\n",
    "\n",
    "# Init a connection matrix\n",
    "inf = 1e9\n",
    "adj_map = np.zeros((len(stations),len(stations)))+inf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "''' \n",
    "This function gives whether there is direct connection between two stations. \n",
    "It considers walking to another station as direct transfer but with different weight.\n",
    "'''\n",
    "def find_connection(station1,station2):\n",
    "    # Get the station can be reach by walk\n",
    "    walk_station = get_near_station(station1)\n",
    "    # Use 1.0 to delegete can be reach by walk\n",
    "    if station2 in walk_station[0]:\n",
    "        return 1.0\n",
    "    # Find whether there is public transport connections in between\n",
    "    trains_1 = data_schedule_all[data_schedule_all.station_id == station1][[\"train_number\",\"line_id\"]]\n",
    "    trains_2 = data_schedule_all[data_schedule_all.station_id == station2][[\"train_number\",\"line_id\"]]\n",
    "    inter = pd.merge(trains_1,trains_2,how = \"inner\", on = [\"train_number\",\"line_id\"])\n",
    "    # If public transport connection exists, use 10 to delegate it\n",
    "    if inter.shape[0] != 0:\n",
    "        return 10.0\n",
    "    return inf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute the adjacent matrix \n",
    "# This block takes about 3 hours, we have already saved the results, you can directly use that\n",
    "k = 0\n",
    "for i in range(len(stations)):\n",
    "    for j in range(i+1,len(stations)):\n",
    "        k = k + 1\n",
    "        if k%100 == 0:\n",
    "            #print (\"processing, \" , k)\n",
    "            pass\n",
    "        connect = find_connection(stations[i],stations[j])\n",
    "        adj_map[i,j]=connect\n",
    "        adj_map[j,i]=connect"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pickle\n",
    "\n",
    "# Save the matrix to pickle\n",
    "pickle.dump(adj_map,open(\"connection.p\",\"wb\"))"
   ]
  }
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