ex_0404.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "l8JaOdnCXH7F"
      },
      "source": [
        "## Coding Exercise #0304"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kGgktgfiXH7G"
      },
      "source": [
        "### 1. Linear regression prediction and confidence interval:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "id": "qC7mt98yXH7H"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "import matplotlib.pyplot as plt\n",
        "import scipy.stats as st\n",
        "from sklearn.linear_model import LinearRegression\n",
        "from sklearn import metrics\n",
        "%matplotlib inline"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OFOX4kYfXH7H"
      },
      "source": [
        "#### 1.1. Data: \n",
        "<br>\n",
        "study = an array that contains the hours of study. This is the explanatory variable. <br>\n",
        "score = an array that contains the test scores. This is the response variable."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "id": "DG3WcRGFXH7I"
      },
      "outputs": [],
      "source": [
        "study = np.array([ 3, 4.5, 6, 1.2, 2, 6.9, 6.7, 5.5])\n",
        "score = np.array([ 88, 85, 90, 80, 81, 92, 95, 90])\n",
        "n = study.size"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KJTWn6BEXH7I"
      },
      "source": [
        "#### 1.2. Training:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "id": "_7Kdg9BdXH7I"
      },
      "outputs": [],
      "source": [
        "# Instantiate a linear regression object.\n",
        "lm = LinearRegression()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "scrolled": true,
        "id": "xVUFzUNeXH7I",
        "outputId": "683ecadd-1577-47ea-9ba8-1714efb90949",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "LinearRegression()"
            ]
          },
          "metadata": {},
          "execution_count": 7
        }
      ],
      "source": [
        "# Train.\n",
        "lm.fit(study.reshape(-1,1), score.reshape(-1,1))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "id": "lYvMv_rzXH7I",
        "outputId": "e0af8f92-d182-463b-fbfb-c1d1b7660352",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "77.68466977583873\n",
            "2.2213028433879947\n"
          ]
        }
      ],
      "source": [
        "# Get the parameters.\n",
        "b0 = lm.intercept_[0]\n",
        "b1 = lm.coef_[0][0]\n",
        "print(b0)\n",
        "print(b1)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {
        "id": "fAs9MYVtXH7I",
        "outputId": "3b0c2e98-d1e3-4630-9e06-184e79459fe3",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.93"
            ]
          },
          "metadata": {},
          "execution_count": 9
        }
      ],
      "source": [
        "# Calculate the in-sample RMSE.\n",
        "predScore = lm.predict(study.reshape(-1,1))\n",
        "mse = metrics.mean_squared_error(score, predScore)\n",
        "rmse=np.sqrt(mse)\n",
        "np.round(rmse,2)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Z5qV6-UuXH7I"
      },
      "source": [
        "####  1.3. Confidence interval and visualization:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {
        "id": "LrLmQztYXH7J"
      },
      "outputs": [],
      "source": [
        "# We define here the function that calculates standard error.\n",
        "# Refer to the formula given in the lecture note.\n",
        "def StdError(x_star, x_vec, mse, n):\n",
        "    x_mean = np.mean(x_vec)\n",
        "    return (np.sqrt(mse*(1/n+(x_star-x_mean)**2/np.sum((x_vec-x_mean)**2))))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {
        "id": "NslbLYgKXH7J"
      },
      "outputs": [],
      "source": [
        "# y_hat : the predicted y.\n",
        "# y_low : lower bound of the confidence interval (95%).\n",
        "# y_up : upper bound of the confidence interval (95%).\n",
        "x_star = np.linspace(1,9,10)\n",
        "y_hat = b0 + b1*x_star\n",
        "y_low = y_hat - st.t.ppf(0.975,n-2)*StdError(x_star,study,mse,n)\n",
        "y_up = y_hat + st.t.ppf(0.975,n-2)*StdError(x_star,study,mse,n)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 12,
      "metadata": {
        "id": "QmcwFeecXH7J",
        "outputId": "1b503b6c-ad2d-474d-b306-18e6f9a1ce72",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        }
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "# Now, disaply.\n",
        "plt.scatter(study, score, c='red')\n",
        "plt.plot(x_star,y_low,c = 'blue',linestyle='--',linewidth=1)\n",
        "plt.plot(x_star,y_hat,c = 'green',linewidth = 1.5, alpha=0.5)\n",
        "plt.plot(x_star,y_up,c = 'blue',linestyle='--',linewidth=1)\n",
        "plt.xlabel('Study')\n",
        "plt.ylabel('Score')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "vmVbB0HLXH7J"
      },
      "outputs": [],
      "source": []
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3 (ipykernel)",
      "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": "3.9.13"
    },
    "colab": {
      "provenance": []
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}