regression.ipynb

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    "colab": {
      "name": "Untitled11.ipynb",
      "provenance": [],
      "authorship_tag": "ABX9TyNaIUTQlHqkpiNSamZb4TKR",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/foxtrotmike/CS909/blob/master/regression.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VRIbruKhGuTE",
        "colab_type": "text"
      },
      "source": [
        "# CS909 Regression Exercise\n",
        "(Fayyaz Minhas)\n",
        "\n",
        "The objective of this exercise is to enable you to do regression with sklearn models. Please use the accompanying lecture slides for conceptural information. \n",
        "\n",
        "## Ordinary Least Squares Regression (OLS)\n",
        "\n",
        "Please try to understand the simple code for OLS given below and relate it to the concepts covere din the class.\n",
        "\n",
        "This is a simple implementation of OLS as covered in the class. Please try to understand what individual lines in the code are doing."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Cx2ZDXmv_QzO",
        "colab_type": "code",
        "outputId": "ff9d8b1d-f2f2-441d-861e-3703b607e8e0",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 591
        }
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "X0 = np.array([[1],[2]])\n",
        "y =np.array([3.5,4.75])\n",
        "X = np.hstack((X0,np.ones((X.shape[0],1)))) #append 1 to each example\n",
        "w = np.linalg.pinv(X)@y\n",
        "f = X@w\n",
        "e = f-y\n",
        "L = e@e\n",
        "plt.figure();plt.plot(X0,f,'ro');plt.plot(X0,y,'bx');plt.grid();plt.xlabel('x');plt.ylabel('output');plt.legend(['Prediction','True']);plt.title('Data Plot')\n",
        "plt.figure();plt.plot(y,f,'o');plt.grid();plt.xlabel('True value');plt.ylabel('Predicted value');plt.title('Regression Scatter Plot')\n"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.0, 'Regression Scatter Plot')"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 12
        },
        {
          "output_type": "display_data",
          "data": {
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CM7PEOQjMzBLnIDAzS5yDwMwscZkHgaSekp6V9Egbff5OUkiqzroeMzPbVTm2CK4AlrfW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\njjQKsyQU7n0AAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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MWAr0pHEzM5uAiiSG/4iI58iuL7hHVTOzCa7IqaS7JX1J0oGSNNoNSJokabWkZQ2WLZa0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\n6lTrdCHV0QAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4ZTVm3kKKkDO",
        "colab_type": "text"
      },
      "source": [
        "Please answer the following questions:\n",
        "0. Can you see a linear relationship between the input and output variables for the given data?\n",
        "1. What is the pseudo inverse? What is the pseudo inverse of the given Matrix?\n",
        "2. What does the model predict for an example with a feature value of 1.4?\n",
        "3. How do the model's predictions change when you add another example (say, 1.6) with a feature value (say, 4.6)?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6Sko572nLG6i",
        "colab_type": "text"
      },
      "source": [
        "### OLS using sklearn\n",
        "\n",
        "The code below uses sk-learn for OLS over the same data as above. Please try to understand the role of fit_intercept in the code below and answer the following questions:\n",
        "\n",
        "1. Are your weights from sklearn OLS the same as you got above?\n",
        "2. What is the role of fit_intercept? \n",
        "3. Generate a prediction for a new example using the trained model."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "7TICxq1kLCnr",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "glYFwJf1KSXL",
        "colab_type": "text"
      },
      "source": [
        ""
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "imIKESWe_esr",
        "colab_type": "code",
        "outputId": "dd466ad8-bd6c-45e1-c277-ad8b9005f6e3",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 625
        }
      },
      "source": [
        "from sklearn.linear_model import LinearRegression\n",
        "regr = LinearRegression(fit_intercept = False).fit(X, y)\n",
        "f = regr.predict(X)\n",
        "print('Weights:',regr.coef_)\n",
        "\n",
        "plt.figure();plt.plot(y,f,'o');plt.grid();plt.xlabel('True value');plt.ylabel('Predicted value');plt.title('Regression Scatter Plot')\n",
        "\n",
        "# No need to append 1 to feature vector using below\n",
        "from sklearn.linear_model import LinearRegression\n",
        "regr = LinearRegression(fit_intercept = True).fit(X0, y)\n",
        "f = regr.predict(X0)\n",
        "print('Weights:',regr.coef_)\n",
        "plt.figure();plt.plot(y,f,'o');plt.grid();plt.xlabel('True value');plt.ylabel('Predicted value');plt.title('Regression Scatter Plot')\n"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[1.25 2.25]\n",
            "[1.25]\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.0, 'Regression Scatter Plot')"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 16
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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MWAr0pHEzM5uAiiSG/4iI58iuL7hHVTOzCa7IqaS7JX1J0oGSNNoNSJokabWkZQ2WLZa0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\n6lTrdCHV0QAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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sP1bHPyZf14avAC+V9GtgMfDR6iIrpi7+84Ae4FuSbpa0tMLQCqtrQ1fqsu/z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\n6lTrdCHV0QAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qpZQXYivMDSB",
        "colab_type": "text"
      },
      "source": [
        "## Ridge Regression\n",
        "\n",
        "Ridge regression is OLS but with a regularizer. In the code below, we will be using both OLS and Ridge Regression for a simple task and try to understand the differences between the two. Here, we train both OLS and Ridge regression over a very small dataset but after adding noise to the example features to test the noise robustness of both models. Please go through the code and try to understand it."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "QI985L6FMCGn",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 433
        },
        "outputId": "fc54cc18-d211-4d72-978b-b348df934879"
      },
      "source": [
        "from sklearn import linear_model\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "X_train = np.c_[.5, 1].T #train features\n",
        "y_train = [.5, 1] #labels\n",
        "X_test = np.c_[0, 2].T #test features\n",
        "\n",
        "np.random.seed(0)\n",
        "regressors = dict(ols=linear_model.LinearRegression(),\n",
        "                   ridge=linear_model.Ridge(alpha=.1) )\n",
        "for name, regr in regressors.items(): # pick a regressor at a time\n",
        "    fig, ax = plt.subplots(figsize=(4, 3))\n",
        "\n",
        "    for _ in range(5): # do 5 runs\n",
        "        this_X = .1 * np.random.normal(size=(2, 1)) + X_train # adding random noise to training data\n",
        "        regr.fit(this_X, y_train) #training on noisy data\n",
        "        ax.scatter(this_X, y_train, s=10, c='gray', marker='o', zorder=10)#plot noisy data with target\n",
        "        ax.plot(X_test, regr.predict(X_test), color='gray') # testing and plotting on test data as a line\n",
        "        ax.scatter(X_test, regr.predict(X_test), s=10, c='blue', marker='o', zorder=10)#plot noisy data with target\n",
        "        \n",
        "\n",
        "    regr.fit(X_train, y_train) # training on non-noisy data       \n",
        "    ax.scatter(X_train, y_train, s=30, c='red', marker='+', zorder=10) #show training examples in red\n",
        "    ax.plot(X_test, regr.predict(X_test), marker = '.',linewidth=2, color='red') #plotting test predictions as a line with the non-noisy data predictor\n",
        "\n",
        "    ax.set_title(name)\n",
        "    #ax.set_xlim(-0.2, 2.5)\n",
        "    #ax.set_ylim((-0.2, 1.7))\n",
        "    ax.set_xlabel('X')\n",
        "    ax.set_ylabel('y')\n",
        "\n",
        "    fig.tight_layout()\n",
        "    plt.grid()\n",
        "plt.show()"
      ],
      "execution_count": 34,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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DT55gNBoxXb064zweHh4EBAQghGB8fBwpJVJKIiMj2bVrF8nJyerSR1lU6rdp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\n",
            "text/plain": [
              "<Figure size 288x216 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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BUVERSUlJuF6/zrcyMwkoLARAPvUUx/fsIaO6GtnY2L24dnt7OwDjx49n/fr1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\niqI4jCowiqI4zP8PR6LkU0FdNSoAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 288x216 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "iOrU0ug_Qpp-",
        "colab_type": "text"
      },
      "source": [
        "We do multiple runs in training the two methods with different noise added to the training examples and then plot the results for a simple test data. Answer the following questions:\n",
        "1. Do you see that each time the OLS is trained, it fits the training examples perfectly? \n",
        "2. Do you see that each time the OLS is trained with added noise, it fits a very different line as it fits the training examples perfectly? You can view the weight vectors of the fitted model after each training to understand this better as well.\n",
        "3. Do you see that the line you get when OLS is trained without added noise is significantly different from the lines you get with added noise?\n",
        "4. Do you see that each time the Ridge regression model is trained, it DOES NOT fit the training examples perfectly? You can view the weight vectors of the fitted model to understand this better as well.\n",
        "5. Do you see that each time the ridge regression is trained with added noise, it fits a somewhat similar line?\n",
        "6. Do you see that the line you get when ridge regression is trained without added noise is closer to the lines you get with added noise in comparison to the OLS model?\n",
        "7. Which model is more impacted by added noise?\n",
        "8. Try to understand the impact of regularization by changing the alpha hyper-parameter of the ridge regression model."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "B6IK8QVKTkzu",
        "colab_type": "text"
      },
      "source": [
        "## Support Vector Regression\n",
        "\n",
        "You can build a very simple SVR model in sklearn as follows."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "kqtrBAVNTkHV",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 312
        },
        "outputId": "c0dd14c8-e698-4737-e074-42823c87070a"
      },
      "source": [
        "from sklearn.svm import SVR\n",
        "regr = SVR(kernel = 'linear', C=1.0)\n",
        "X0 = np.array([[1],[2]])\n",
        "y =np.array([3.5,4.75])\n",
        "regr.fit(X0,y)\n",
        "f = regr.predict(X0)\n",
        "plt.figure();plt.plot(y,f,'o');plt.grid();plt.xlabel('True value');plt.ylabel('Predicted value');plt.title('Regression Scatter Plot')"
      ],
      "execution_count": 39,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.0, 'Regression Scatter Plot')"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 39
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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MC3hLRDwzUkURsU7SUHpK1pHAMWnVOcCKiLhQ0kzg2gabP0f+DH543wK+GRGf\nKtIYs63Bl5LMktTx/JikvdN1/PfWrP4R2WitAEjaf4RqLiN7DOT2EXFnWrYzL16mOmaE7dYC/anu\nt5CGC0/7/YCkaWndyyW9ahTNMhs1JwazvNPIRtO8gWwc/mEfBeYoe5j6nWSPe2zkcuB/kca/T84D\nlkj6Ofkzkvrt+iTdDiwC7gWIiNuAs4AfpdFwfwD0bUnDzIry6KpmZpbjMwYzM8txYjAzsxwnBjMz\ny3FiMDOzHCcGMzPLcWIwM7McJwYzM8v5/9NXMHcZB2iCAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lcLjySNLUw7n",
        "colab_type": "text"
      },
      "source": [
        "# Practical Exercise\n",
        "\n",
        "In the exercise below, we will try to predict house prices based on some features using the Boston dataset. We can try different models for this task.\n",
        "\n",
        "Can you find which model works best? Can you calculate some performance metrics (https://scikit-learn.org/stable/modules/model_evaluation.html) for the test set? (see class notes for more details)."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "J-vwzWd1Uxb0",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "outputId": "68889abf-f7bb-4f74-cc9a-20f173f00478"
      },
      "source": [
        "import matplotlib.pyplot as plt\n",
        "\n",
        "from sklearn import datasets\n",
        "# Loading some example data\n",
        "X, y = datasets.load_boston(return_X_y=True)\n",
        "Ntr = 400\n",
        "Xtr,ytr = X[:Ntr],y[:Ntr]\n",
        "Xtt,ytt = X[Ntr:],y[Ntr:]\n",
        "# Training classifiers\n",
        "reg1 = linear_model.LinearRegression().fit(Xtr, ytr)\n",
        "reg2 = linear_model.Ridge(alpha=.1).fit(Xtr, ytr)\n",
        "reg3 = SVR(kernel='linear', C= 1.0).fit(Xtr, ytr)\n",
        "reg4 = SVR(kernel='poly', degree = 1, C= 10.0).fit(Xtr, ytr)\n",
        "\n",
        "plt.figure(); plt.scatter(ytt,reg1.predict(Xtt)); plt.xlabel('True');plt.ylabel('Predicted');plt.grid();plt.title('OLS')\n",
        "plt.figure(); plt.scatter(ytt,reg2.predict(Xtt)); plt.xlabel('True');plt.ylabel('Predicted');plt.grid();plt.title('RR')\n",
        "plt.figure(); plt.scatter(ytt,reg3.predict(Xtt)); plt.xlabel('True');plt.ylabel('Predicted');plt.grid();plt.title('SVR-Linear')\n",
        "plt.figure(); plt.scatter(ytt,reg4.predict(Xtt)); plt.xlabel('True');plt.ylabel('Predicted');plt.grid();plt.title('SVR-Poly')\n",
        "plt.show()"
      ],
      "execution_count": 59,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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FJCIi2aQAICJSUgoAIiIlpQAgIlJSCgAiIiWlWUA9UIZN1kSkeBQAurSzYR+f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\nB15LWd5C7QuYAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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Wcs21pCHVokXlRHqDAkBKVJ9Zrxjo55mDh1hYPHoPX6VaRKSTFABSpPrMWhOu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\nABGRnFIAEBHJKQUAEZGcUgAQEcmp/w/kpy2lBRLU1QAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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VNWKmS9tfyijqjWBJGnf3c8p88Ac1YqZN21/KKAsBQOjNLFvdzl1cZJrlTMrI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\ngIhISSkAiIiU1P8H2mh0qYf/EYcAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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qaUNLFuTIzeomVyEohhkpJ7GaaNj0UfFF5cFTT9DKpiIxUeBvsqSqZnq7u7ht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\nAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "9Xl-S-8WHPuY",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    }
  ]
}