Regression_lineaire_ec_normale.ipynb

{
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  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.8.3"
    },
    "colab": {
      "name": "Regression_lineaire_ec_normale.ipynb",
      "provenance": [],
      "include_colab_link": true
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/roxin/psm-machine-learning/blob/master/Regression_lineaire_ec_normale.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CxrbsNLO_THE",
        "colab_type": "text"
      },
      "source": [
        "# Régression linéaire\n",
        "## Avec l'équation normale"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "sypok7nV_THF",
        "colab_type": "code",
        "colab": {},
        "outputId": "f70b14db-91fd-4727-e2e3-19c65a1c019f"
      },
      "source": [
        "# importation bibliothéques\n",
        "import numpy as np\n",
        "from matplotlib import pyplot\n",
        "# definition dataset\n",
        "data = np.array([\n",
        "[0.05, 0.12],\n",
        "[0.18, 0.22],\n",
        "[0.31, 0.35],\n",
        "[0.42, 0.38],\n",
        "[0.54, 0.49],\n",
        "[0.67, 0.84],\n",
        "[0.83, 1.05]])\n",
        "print(data)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[0.05 0.12]\n",
            " [0.18 0.22]\n",
            " [0.31 0.35]\n",
            " [0.42 0.38]\n",
            " [0.54 0.49]\n",
            " [0.67 0.84]\n",
            " [0.83 1.05]]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "1y0BxQS3_THK",
        "colab_type": "code",
        "colab": {},
        "outputId": "516617f0-139c-4340-ee58-8ffce3cfe6d6"
      },
      "source": [
        "# trancher (split) les données en variables d'entrées et de sorties\n",
        "X, y = data[:,0], data[:,1]\n",
        "print('X = ', X)\n",
        "print(X.shape)\n",
        "print('X : vactor ligne avec ', X.shape[0], ' colonnes')\n",
        "print('y =', y)\n",
        "X = X.reshape((len(X), 1))\n",
        "print('X : lignes = ', X.shape[0], ' ; colonnes = ', X.shape[1])\n",
        "print(X.shape)\n",
        "# print(X)\n",
        "# scatter plot\n",
        "#pyplot.scatter(X, y)\n",
        "#pyplot.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "X =  [0.05 0.18 0.31 0.42 0.54 0.67 0.83]\n",
            "(7,)\n",
            "X : vactor ligne avec  7  colonnes\n",
            "y = [0.12 0.22 0.35 0.38 0.49 0.84 1.05]\n",
            "X : lignes =  7  ; colonnes =  1\n",
            "(7, 1)\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TJXyL2yI_THN",
        "colab_type": "text"
      },
      "source": [
        "### Equation normale"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TkSUqsFG_THO",
        "colab_type": "text"
      },
      "source": [
        "\\begin{equation*}\n",
        "Xb = y\n",
        "\\end{equation*}"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vCgLpXIb_THO",
        "colab_type": "text"
      },
      "source": [
        "\\begin{equation*}\n",
        "X^T.X.b = X^T.y\n",
        "\\end{equation*}"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GGjALwPN_THP",
        "colab_type": "text"
      },
      "source": [
        "\\begin{equation}\n",
        "b = (X^T.X)^{-1}.X^T.y\n",
        "\\end{equation}"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-J6yuVxu_THP",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "# Solution avec l'équation normale\n",
        "# linalg - linear algebra\n",
        "b = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(y)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "PWGA2cpq_THS",
        "colab_type": "code",
        "colab": {},
        "outputId": "3f6da823-de90-408a-84a6-dc9c15bcfa40"
      },
      "source": [
        "y_chapeau = X.dot(b)\n",
        "print(y_chapeau)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[0.05793989 0.2085836  0.35922731 0.48669507 0.62575081 0.77639452\n",
            " 0.96180216]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dvlVLfdr_THV",
        "colab_type": "code",
        "colab": {},
        "outputId": "bcec5c0b-caf7-4f22-9739-89fe39fed94f"
      },
      "source": [
        "pyplot.scatter(X, y)\n",
        "pyplot.plot(X, y_chapeau, color='red')\n",
        "pyplot.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "vW6jdZtS_THZ",
        "colab_type": "code",
        "colab": {},
        "outputId": "3f1bfa98-c815-4f3d-f9df-7873096d5cf6"
      },
      "source": [
        "diff = y_chapeau - y\n",
        "print(diff)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[-0.06206011 -0.0114164   0.00922731  0.10669507  0.13575081 -0.06360548\n",
            " -0.08819784]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "rcZjlLoo_THb",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "# Erreur quadratique moyenne"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4JyojsLu_THe",
        "colab_type": "text"
      },
      "source": [
        "###  Mean Square Error  - MSE"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "fkI7R_IB_THe",
        "colab_type": "text"
      },
      "source": [
        "\\begin{equation*}\n",
        "MSE = \\frac{1}{2m}\\sum_{i=1}^m(yh^i - y^i)^2\n",
        "\\end{equation*}"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "GEsWxI2F_THf",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "diff_carre = np.sum(diff ** 2)\n",
        "MSE = diff_carre / len(y)\n",
        "print(\"MSE =\", MSE)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Lca2JADn_THh",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": null,
      "outputs": []
    }
  ]
}