{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Ab4iZfp4eXPk"
      },
      "source": [
        "# Bike Rides and the Poisson Model\n",
        "\n",
        "To help the urban planners, you are called to model the daily bike rides in NYC using [this dataset](https://gist.github.com/sachinsdate/c17931a3f000492c1c42cf78bf4ce9fe/archive/7a5131d3f02575668b3c7e8c146b6a285acd2cd7.zip).  The dataset contains date, day of the week, high and low temp, precipitation and bike ride counts as columns. \n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "!wget https://gist.github.com/sachinsdate/c17931a3f000492c1c42cf78bf4ce9fe/archive/7a5131d3f02575668b3c7e8c146b6a285acd2cd7.zip\n",
        "!unzip 7a5131d3f02575668b3c7e8c146b6a285acd2cd7.zip"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "gM-y0BWye7WN",
        "outputId": "ba5d103a-ec03-42ec-8bf2-1094d2ed99ee"
      },
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "--2023-02-26 21:10:52--  https://gist.github.com/sachinsdate/c17931a3f000492c1c42cf78bf4ce9fe/archive/7a5131d3f02575668b3c7e8c146b6a285acd2cd7.zip\n",
            "Resolving gist.github.com (gist.github.com)... 192.30.255.112\n",
            "Connecting to gist.github.com (gist.github.com)|192.30.255.112|:443... connected.\n",
            "HTTP request sent, awaiting response... 302 Found\n",
            "Location: https://codeload.github.com/gist/c17931a3f000492c1c42cf78bf4ce9fe/zip/7a5131d3f02575668b3c7e8c146b6a285acd2cd7 [following]\n",
            "--2023-02-26 21:10:53--  https://codeload.github.com/gist/c17931a3f000492c1c42cf78bf4ce9fe/zip/7a5131d3f02575668b3c7e8c146b6a285acd2cd7\n",
            "Resolving codeload.github.com (codeload.github.com)... 192.30.255.120\n",
            "Connecting to codeload.github.com (codeload.github.com)|192.30.255.120|:443... connected.\n",
            "HTTP request sent, awaiting response... 200 OK\n",
            "Length: unspecified [application/zip]\n",
            "Saving to: ‘7a5131d3f02575668b3c7e8c146b6a285acd2cd7.zip’\n",
            "\n",
            "7a5131d3f02575668b3     [ <=>                ]   2.56K  --.-KB/s    in 0s      \n",
            "\n",
            "2023-02-26 21:10:53 (27.7 MB/s) - ‘7a5131d3f02575668b3c7e8c146b6a285acd2cd7.zip’ saved [2623]\n",
            "\n",
            "Archive:  7a5131d3f02575668b3c7e8c146b6a285acd2cd7.zip\n",
            "7a5131d3f02575668b3c7e8c146b6a285acd2cd7\n",
            "   creating: c17931a3f000492c1c42cf78bf4ce9fe-7a5131d3f02575668b3c7e8c146b6a285acd2cd7/\n",
            "  inflating: c17931a3f000492c1c42cf78bf4ce9fe-7a5131d3f02575668b3c7e8c146b6a285acd2cd7/nyc_bb_bicyclist_counts.csv  \n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "id": "RxdI4hgDeXPr"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "import seaborn as sns\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "np.random.seed(0)\n",
        "sns.set_theme(style='whitegrid', palette='pastel')\n",
        "\n",
        "import warnings\n",
        "warnings.filterwarnings('ignore')"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "filename = \"c17931a3f000492c1c42cf78bf4ce9fe-7a5131d3f02575668b3c7e8c146b6a285acd2cd7/nyc_bb_bicyclist_counts.csv\"\n",
        "\n",
        "df = pd.read_csv(filename)\n",
        "df.head()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 206
        },
        "id": "ImGoQW97gLQN",
        "outputId": "9e481eb5-d4f1-4d14-c8b2-d128abdab273"
      },
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "       Date  HIGH_T  LOW_T  PRECIP  BB_COUNT\n",
              "0  1-Apr-17    46.0   37.0    0.00       606\n",
              "1  2-Apr-17    62.1   41.0    0.00      2021\n",
              "2  3-Apr-17    63.0   50.0    0.03      2470\n",
              "3  4-Apr-17    51.1   46.0    1.18       723\n",
              "4  5-Apr-17    63.0   46.0    0.00      2807"
            ],
            "text/html": [
              "\n",
              "  <div id=\"df-48098f5e-8a60-404c-87cc-cbf28fd48965\">\n",
              "    <div class=\"colab-df-container\">\n",
              "      <div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
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              "\n",
              "    .dataframe thead th {\n",
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              "</style>\n",
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              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>Date</th>\n",
              "      <th>HIGH_T</th>\n",
              "      <th>LOW_T</th>\n",
              "      <th>PRECIP</th>\n",
              "      <th>BB_COUNT</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>1-Apr-17</td>\n",
              "      <td>46.0</td>\n",
              "      <td>37.0</td>\n",
              "      <td>0.00</td>\n",
              "      <td>606</td>\n",
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              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>2-Apr-17</td>\n",
              "      <td>62.1</td>\n",
              "      <td>41.0</td>\n",
              "      <td>0.00</td>\n",
              "      <td>2021</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>3-Apr-17</td>\n",
              "      <td>63.0</td>\n",
              "      <td>50.0</td>\n",
              "      <td>0.03</td>\n",
              "      <td>2470</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>4-Apr-17</td>\n",
              "      <td>51.1</td>\n",
              "      <td>46.0</td>\n",
              "      <td>1.18</td>\n",
              "      <td>723</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>5-Apr-17</td>\n",
              "      <td>63.0</td>\n",
              "      <td>46.0</td>\n",
              "      <td>0.00</td>\n",
              "      <td>2807</td>\n",
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              "  </tbody>\n",
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              "              title=\"Convert this dataframe to an interactive table.\"\n",
              "              style=\"display:none;\">\n",
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              "    .colab-df-container {\n",
              "      display:flex;\n",
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              "      gap: 12px;\n",
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              "\n",
              "    .colab-df-convert {\n",
              "      background-color: #E8F0FE;\n",
              "      border: none;\n",
              "      border-radius: 50%;\n",
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              "      fill: #1967D2;\n",
              "      height: 32px;\n",
              "      padding: 0 0 0 0;\n",
              "      width: 32px;\n",
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              "\n",
              "    .colab-df-convert:hover {\n",
              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
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              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
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              "          document.querySelector('#df-48098f5e-8a60-404c-87cc-cbf28fd48965 button.colab-df-convert');\n",
              "        buttonEl.style.display =\n",
              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-48098f5e-8a60-404c-87cc-cbf28fd48965');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
              "          element.innerHTML = '';\n",
              "          dataTable['output_type'] = 'display_data';\n",
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              "          element.appendChild(docLink);\n",
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              "  </div>\n",
              "  "
            ]
          },
          "metadata": {},
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        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rdF7NfHKeXPo"
      },
      "source": [
        "## Maximum Likelihood I \n",
        " \n",
        "The obvious choice in distributions is the [Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution) which depends only on one parameter, λ, which is the average number of occurrences per interval. We want to estimate this parameter using Maximum Likelihood Estimation.\n",
        "\n",
        "Implement a Gradient Descent algorithm from scratch that will estimate the Poisson distribution according to the Maximum Likelihood criterion. Plot the estimated mean vs iterations to showcase convergence towards the true mean. \n",
        "\n",
        "References: \n",
        "\n",
        "1. [This blog post](https://towardsdatascience.com/the-poisson-process-everything-you-need-to-know-322aa0ab9e9a). \n",
        "\n",
        "2. [This blog post](https://towardsdatascience.com/understanding-maximum-likelihood-estimation-fa495a03017a) and note the negative  log likelihood function.  \n"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Negative log likelihood function for Poisson distribution\n",
        "\n",
        "$n \\lambda + \\sum_{i=1}^n ln(x_i!) - ln(\\lambda) \\sum_{i=1}^n x_i$\n",
        "\n",
        "Derivative of negative log likelihood in respect to lambda\n",
        "\n",
        "$n - {1 \\over \\lambda} \\sum_{i=1}^n x_i$"
      ],
      "metadata": {
        "id": "_TdBnxVrlRYH"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def gradient(l, x):\n",
        "  n = len(x)\n",
        "  lambda_gradient = n - (1 / l) * np.sum(x)\n",
        "  return lambda_gradient\n",
        "\n",
        "def gradient_descent(x, l, learning_rate, max_iter):\n",
        "  lam = l\n",
        "  lams = [] # keeping track of updated lambda\n",
        "  for _ in range(max_iter):\n",
        "    g = gradient(lam, x)\n",
        "\n",
        "    lams.append(lam)\n",
        "    lam = lam - learning_rate * g\n",
        "  return lam, lams"
      ],
      "metadata": {
        "id": "noR1cKD4hS2c"
      },
      "execution_count": 18,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "cyclists = np.array(df['BB_COUNT'])\n",
        "iterations = 100000\n",
        "alpha = 0.001\n",
        "lam = 1\n",
        "\n",
        "estimated, lams = gradient_descent(cyclists, lam, alpha, iterations)\n",
        "\n",
        "actual = (1 / len(cyclists)) * np.sum(cyclists)\n",
        "\n",
        "print(f\"Estimated mean={estimated}\")\n",
        "print(f\"Actual mean={actual}\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "WsUk4PlAmY7L",
        "outputId": "c4eea6f9-db9a-47cc-e856-128348dcc496"
      },
      "execution_count": 39,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Estimated mean=2679.715270113102\n",
            "Actual mean=2680.042056074766\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "sns.lineplot(x=range(len(lams)), y=lams)\n",
        "plt.xlabel('Iterations')\n",
        "plt.ylabel('Estimated mean')\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 285
        },
        "id": "Dz-7upxEoKno",
        "outputId": "41370016-06be-4c4c-fa64-7a5a294fe3f6"
      },
      "execution_count": 38,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XINkmCdVeXPr"
      },
      "source": [
        "## Maximum Likelihood II\n",
        "\n",
        "A colleague of yours suggest that the parameter $\\lambda$ must be itself dependent on the weather and other factors since people bike when its not raining. Assume that you model $\\lambda$ as \n",
        "\n",
        "$$\\lambda_i = \\exp(\\mathbf w^T \\mathbf x_i)$$\n",
        "\n",
        "where $\\mathbf x_i$ is one of the example features and $\\mathbf w$ is a set of parameters. \n",
        "\n",
        "Train the model with SGD with this assumption and compare the MSE of the predictions with the `Maximum Likelihood I` approach. \n",
        "\n",
        "You may want to use [this partial derivative of the log likelihood function](http://home.cc.umanitoba.ca/~godwinrt/7010/poissonregression.pdf)"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "${\\partial l \\over \\partial \\beta} = \\sum_{i=1}^n(y_i - exp(X'_i\\beta)X_i$"
      ],
      "metadata": {
        "id": "ni9r4QxeK4L3"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def s_gradient(x, y, w):\n",
        "  y_est = np.exp(np.dot(x, w))\n",
        "  res = (y - y_est) * x.T\n",
        "\n",
        "  r = []\n",
        "  for i in range(len(res)):\n",
        "    r.append(res[i][0])\n",
        "  return np.array(r)\n",
        "\n",
        "def s_gradient_descent(learning_rate, max_iter):\n",
        "  w = np.array([0.0 for _ in range(4)])\n",
        "  batch_size = 0.2\n",
        "\n",
        "  for _ in range(max_iter):\n",
        "    samples = df.sample(frac=batch_size)\n",
        "    n = len(samples)\n",
        "\n",
        "    x = np.hstack([\n",
        "        np.array([1 for _ in range(n)]).reshape(n, 1),\n",
        "        np.array(samples['HIGH_T']).reshape(n, 1),\n",
        "        np.array(samples['LOW_T']).reshape(n, 1),\n",
        "        np.array(samples['PRECIP']).reshape(n, 1)\n",
        "    ])\n",
        "    y = np.array(samples['BB_COUNT'])\n",
        "    g = s_gradient(x, y, w)\n",
        "    w = w - learning_rate * g\n",
        "\n",
        "  return w"
      ],
      "metadata": {
        "id": "9e8m9A9jsrO2"
      },
      "execution_count": 157,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "iterations = 10\n",
        "alpha = 0.0001\n",
        "lam = 1000\n",
        "\n",
        "w = s_gradient_descent(alpha, iterations)\n",
        "# print(w)\n",
        "\n",
        "n = len(df)\n",
        "x = np.hstack([\n",
        "    np.array([1 for _ in range(n)]).reshape(n, 1),\n",
        "    np.array(df['HIGH_T']).reshape(n, 1),\n",
        "    np.array(df['LOW_T']).reshape(n, 1),\n",
        "    np.array(df['PRECIP']).reshape(n, 1)\n",
        "])\n",
        "l = np.exp(w * x.T)\n",
        "print(l)\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 223
        },
        "id": "FMOiaAn7A86p",
        "outputId": "f07affe7-c717-424e-c5ed-75c9f388ca00"
      },
      "execution_count": 167,
      "outputs": [
        {
          "output_type": "error",
          "ename": "ValueError",
          "evalue": "ignored",
          "traceback": [
            "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
            "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
            "\u001b[0;32m<ipython-input-167-0e5add9133d1>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m     13\u001b[0m     \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdf\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'PRECIP'\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreshape\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     14\u001b[0m ])\n\u001b[0;32m---> 15\u001b[0;31m \u001b[0ml\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mw\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mT\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     16\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ml\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;31mValueError\u001b[0m: operands could not be broadcast together with shapes (4,) (4,214) "
          ]
        }
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "name": "python",
      "version": "3.10.9"
    },
    "orig_nbformat": 4,
    "vscode": {
      "interpreter": {
        "hash": "7d6993cb2f9ce9a59d5d7380609d9cb5192a9dedd2735a011418ad9e827eb538"
      }
    },
    "colab": {
      "provenance": []
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}