Machine Learning/multi_linear_regession.ipynb

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    "colab": {
      "name": "multi_linear_regession.ipynb",
      "provenance": []
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    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
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    "language_info": {
      "name": "python"
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4kl2JokbO7ME"
      },
      "source": [
        "###Linear Regression using only numpy"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "7tLaRecaHmXX"
      },
      "source": [
        "import numpy as np"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NBLApYBhHhVF"
      },
      "source": [
        "###Creation of data and labels \n",
        "Linerar relation : y = mx + q"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "-MlFblUhHffQ",
        "outputId": "9a44caa3-345b-4448-9725-bdad032889bf"
      },
      "source": [
        "X = np.random.rand(10000,5)\n",
        "m = np.random.randint(low = 1, high = 20,size = (5,1))  #parametri random tra low e high\n",
        "q = np.random.rand(1)\n",
        "y = (X @ m) + q \n",
        "\n",
        "noise = np.random.randn(y.shape[0], y.shape[1])\n",
        "y = y + noise\n",
        "\n",
        "X.shape, m.shape, q.shape, y.shape"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "((10000, 5), (5, 1), (1,), (10000, 1))"
            ]
          },
          "metadata": {},
          "execution_count": 2
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1J3cjMGAIpl0"
      },
      "source": [
        "q might be interpreted as a parameter within the m vector.\n",
        "\n",
        "y = m1x1 + m2x2 + m3x3 + m4x4 + m5x5 + m6x6\n",
        "\n",
        "Provided that x6 is always equal to one. We then add another column of \"1\" to the data and put q at the end of the vector m."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "28QK47wfIpYn",
        "outputId": "76f9dfd3-56b7-4f26-a1b0-a492b1b1f329"
      },
      "source": [
        "X = np.concatenate([X , np.ones((X.shape[0],1))], axis = 1)\n",
        "m = np.concatenate([m,q.reshape(1,-1)],axis = 0)\n",
        "\n",
        "X.shape, m.shape"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "((10000, 6), (6, 1))"
            ]
          },
          "metadata": {},
          "execution_count": 3
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hDwCHiorKXiE"
      },
      "source": [
        "def partial_derivative(X_batch, y_batch, m_stat):\n",
        "\n",
        "  y_pred = X_batch @ m_stat\n",
        "  n = len(X_batch)\n",
        "\n",
        "  df_dm =  (-2/n) * (X_batch.T @ (y_batch - y_pred))\n",
        "  df_dm = df_dm.reshape(len(df_dm),-1)\n",
        "\n",
        "  return df_dm\n",
        "\n",
        "\n",
        "def mean_squared_error(X,y,m_stat):\n",
        "  y_pred = X @ m_stat\n",
        "  mse = np.sum(((y_pred - y)**2),axis = 0) / len(X)\n",
        "  \n",
        "  return mse\n",
        "\n",
        "  \n",
        "\n",
        "def training(X, y, batch_size, lr, epochs):\n",
        "  \n",
        "  for epoch in range(epochs):\n",
        "\n",
        "    #random initial statistics\n",
        "    if epoch == 0:\n",
        "      m_stat = np.random.rand(X.shape[1],1)\n",
        "\n",
        "    #shuffle X and y using same permutation\n",
        "    indices = np.arange(X.shape[0])\n",
        "    np.random.shuffle(indices)\n",
        "\n",
        "    X = X[indices]\n",
        "    y = y[indices]\n",
        "\n",
        "    #store comulative derivative\n",
        "    cumulative_derivative = np.zeros((X.shape[1],1))\n",
        "\n",
        "    for batch in range(len(X)//batch_size):\n",
        "      start = batch*batch_size\n",
        "      stop = (batch*batch_size) + batch_size\n",
        "\n",
        "      X_batch = X[start:stop]\n",
        "      y_batch = y[start:stop]\n",
        "      \n",
        "      #derivative\n",
        "      cumulative_derivative = cumulative_derivative + partial_derivative(X_batch, y_batch, m_stat)\n",
        "\n",
        "      #updating rule\n",
        "      m_stat = m_stat - (lr*cumulative_derivative)\n",
        "    \n",
        "    print(f\"epoch: {epoch} ----> MSE: {mean_squared_error(X,y,m_stat)}\")\n",
        "      \n",
        "  return m_stat\n",
        "\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_bHg0oEALT43"
      },
      "source": [
        "We use the training function to find the \"hidden\" parameters"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "DopOFF6DYqCG"
      },
      "source": [
        "batch_size = 1024\n",
        "lr = 0.01\n",
        "epochs = 500\n",
        "\n",
        "m_stat = training(X,y, batch_size,lr,epochs)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qG3zx4s-Livg"
      },
      "source": [
        "Lets print the real and estimated parameters to see if they are equal (similar)\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WKKqIOWlbAXL",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "3429426f-77f0-41b6-d02b-21d9fa803470"
      },
      "source": [
        "print(m_stat,\"\\n\")\n",
        "print(m)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[[ 7.02023943]\n",
            " [12.92578848]\n",
            " [ 7.97939421]\n",
            " [ 0.97164595]\n",
            " [ 2.06477057]\n",
            " [ 0.59971073]] \n",
            "\n",
            "[[ 7.        ]\n",
            " [13.        ]\n",
            " [ 8.        ]\n",
            " [ 1.        ]\n",
            " [ 2.        ]\n",
            " [ 0.59567936]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IJfcnqhCLqL0"
      },
      "source": [
        "Lets create a test set, and try to predict the label using the hyperplane we just found"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "CjMyo0LaJ9kf"
      },
      "source": [
        "X_test = np.random.rand(500,5)\n",
        "X_test = np.concatenate([X_test,np.ones(shape = (500,1))] , axis = 1)\n",
        "y_test = X_test @ m"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "oD6jBhnNOkSX"
      },
      "source": [
        "I make my expeditions with the \"found\" parameters"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dHgjGgfuJ-XS"
      },
      "source": [
        "y_preds = X_test @ m_stat"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Wz97_XDVOppb"
      },
      "source": [
        "I calculate the MSE and notice that it is low"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "nr4XFQiXON11",
        "outputId": "83edcb81-e9cf-4b44-970a-f87db3d87dc8"
      },
      "source": [
        "mean_squared_error(X_test, y_test, m_stat)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([0.00044625])"
            ]
          },
          "metadata": {},
          "execution_count": 67
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eowCSRWlOun_"
      },
      "source": [
        "If I compare some of the real outputs with those of the prediction I notice that they are very similar"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "JZUCumvZOdNf",
        "outputId": "e2365b1f-881e-4d60-beb3-bae03521f270"
      },
      "source": [
        "y_test[:5]"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[55.68984025],\n",
              "       [27.89823867],\n",
              "       [54.46623722],\n",
              "       [29.61535891],\n",
              "       [56.02561976]])"
            ]
          },
          "metadata": {},
          "execution_count": 68
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Va35VHFUOes9",
        "outputId": "5a04ae66-1dd7-4348-c94d-30374ccf7dbe"
      },
      "source": [
        "y_preds[:5]"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[55.69776397],\n",
              "       [27.85997063],\n",
              "       [54.46251329],\n",
              "       [29.58735955],\n",
              "       [56.04068571]])"
            ]
          },
          "metadata": {},
          "execution_count": 69
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import matplotlib.pyplot as plt\n",
        "import seaborn as sns"
      ],
      "metadata": {
        "id": "Um_33mdoXfwN"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "cvNNLH0YOiQB"
      },
      "source": [
        "X = np.random.rand(100)\n",
        "y = 10*X + 5 \n",
        "y = y + np.random.randn(100)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "sns.regplot(X,y,line_kws={\"color\": \"red\"},)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 337
        },
        "id": "-g_MxoVAXcz2",
        "outputId": "4198e656-0633-45bc-af66-c6c02b8e4a14"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/seaborn/_decorators.py:43: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.\n",
            "  FutureWarning\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[]"
            ]
          },
          "metadata": {},
          "execution_count": 13
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}