Kernel_Regression_from_Scratch.ipynb

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  "metadata": {
    "colab": {
      "provenance": [],
      "authorship_tag": "ABX9TyM62Ec3W9DoC9+UhowhIIpc",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/SrivinaySridhar/MLT_workshop/blob/main/Kernel_Regression_from_Scratch.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "id": "3RQLKkM58BiF"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import random"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Data\n",
        "\n",
        "$\\mathbf{X}$ is a feature matrix corresponding to $n$ training examples, each represented with $d$ features and has shape $d \\times n$.  \n",
        "\\begin{equation}\n",
        "  \\mathbf{X}_{d \\times n} = \\begin{bmatrix}\n",
        "       | & | &\\dots & |\\\\\n",
        "       \\mathbf{x}_{1}&  \n",
        "       \\mathbf{x}_{2} &\n",
        "      \\dots &\n",
        "       \\mathbf{x}_{n}  \\\\\n",
        "        | & | &\\dots & |\\\\\n",
        "    \\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "* here each $\\mathbf{x}_i \\in \\mathbb{R}^d$\n",
        "\n",
        "*  $\\mathbf{y}$ is a label vector of shape $n \\times 1$.  \n",
        "\\begin{equation}\n",
        "    \\mathbf{y} = \\begin{bmatrix}\n",
        "        y_{1}  \\\\\n",
        "        y_{2} \\\\\n",
        "        \\vdots  \\\\\n",
        "        y_{n} \\\\\n",
        "      \\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "*  The $i$-th entry in this vector, $\\mathbf{y}[i]$ gives label for $i$-th example, which is denoted by $y_{i} \\in \\mathbb{R}$."
      ],
      "metadata": {
        "id": "FAlr8M-Z8Zor"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Training data\n",
        "\n",
        "Generated from a deterministic distribution with Noise ~ Normal(0, 1)\n",
        "\n",
        "$$y_{train} = 2X^3 + 3X^2 + 4 + Noise$$"
      ],
      "metadata": {
        "id": "5xKGE4b6hTq5"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "rng = np.random.default_rng(seed = 101)\n",
        "\n",
        "X_train = (np.arange(-2, 2, 0.01).reshape(1, -1))\n",
        "\n",
        "y_train = (2 * (X_train)**3 + 3 * (X_train)**2 + 4 + rng.normal(0, 1, X_train.shape[1]).reshape(1, -1)).T"
      ],
      "metadata": {
        "id": "GbDDcyLKiGsq"
      },
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "X_train.shape, y_train.shape"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "W5D5VxgK_A6o",
        "outputId": "be19a980-35ee-4ff3-eb37-42c103697149"
      },
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "((1, 400), (400, 1))"
            ]
          },
          "metadata": {},
          "execution_count": 3
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "plt.scatter(X_train, y_train);"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 430
        },
        "id": "OUqra1H_A_vb",
        "outputId": "ffc12edf-50ac-4206-d023-9ed8c1498ba2"
      },
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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Ul6YXJES+B8CcDyIiooDZ7ALKaxtwrqkNvTI7114RAwg5Qyg/viUPU4bH79Radww+iIiIAlBaVYeSTdUuQysGnQazx/ZF/5wM5GRoYNBpUW9pg+DlGCoAv581Ct8d2TssbY4WDD6IiIhkKq2qw7zVFV2CCpPFiuVbjjn+n5WeAgGdQYanAGTlfaMwZXhiBR4Acz6IiIhkEWexeOvNcNbY0rl2S4bG9be+Ua/FW3NGJ2TgATD4ICIikkXuLBYAuGS97Ph3hiYJP7j+akzMNyjdtJjB4IOIiEiGYAuBNVtteH3rcYz55WaUVtUp1KrYwuCDiIhIBqUKgTW2dGDe6oqEDEAYfBAREckwNi8bRr0WSlTkEACUbKqGzS4lgyR+MPggIiKSIUmtwtJp+QCgSABSZ25DeW2DAkeKHQw+iIiIZCouMOLNOaNh0CszBJMoC8qJWOeDiIgoAMUFRkzMN6C8tgEmcyuWbDzsMqtFjkRZUE7E4IOIiChASWoVCgf2AACkpSZ5LDzmiwqAQa9NmAXlRBx2ISIiUsDEfAOeLLoGGalJsvZbOi0/YRaUE7Hng4iIKEie1nnxJys9Bb+eUQB9Wio2Vp7psjBdPGPwQUREFARv67z481Bhf7z04RGXgMWo12LptHwUF8T3CrccdiEiIgqQnHVeRCp09nq8sfVYl54Sk7ktIQqPMfggIiIKkNx1XpxXt/UUsIi3xXvhMQYfREREAZJbn8Og12Jh0WDHareeCIj/wmPM+SAiIgqQ1PocC24fhPGDcjA2LxsfHDwraZ94LjzG4IOIiChAF5utfrcx6rVYOPEaxywWqQFLPBce47ALERFRAGx2AS99eMTvdsVDO6ugijkc/hamU6EzYInnwmMMPoiIiAIgNdl01acnMPvtvbj5lW0orarzuTCd+P94LzzG4IOIiCgAcnMynKfReluYzqDX4s05o+O+zgdzPoiIiAJw4nyzrO0FdPZslGyqxsR8g8vCdOea2hKqwqmsno9ly5bhhhtuQGZmJnr16oUZM2bg6NGjLtu0tbVh/vz56NGjB7p164Z77rkH9fX1ijaaiIgokmx2AWvLT8nez30arbgw3fSRV6FwYI+ECDwAmcHHzp07MX/+fOzduxebN29GR0cH7rzzTjQ3fxv9LVy4EJs2bcL69euxc+dOnD17FjNnzlS84URERJFSXtsAk8X/TBdv4nkarRSyhl1KS0td/v/OO++gV69e2L9/P2655RaYzWb85S9/wZo1azBhwgQAwKpVq3Dddddh7969uPHGG5VrORERUYQEGzzE8zRaKYJKODWbzQCA7OzO6UD79+9HR0cHioqKHNtce+216Nu3L8rKyjwew2q1wmKxuPwRERFFs0CDh0SYRitFwMGH3W7Hk08+ifHjx6OgoAAAYDKZkJqaiqysLJdtc3NzYTKZPB5n2bJl0Ov1jr8+ffoE2iQiIqKw8Ferw5NEmUYrRcDBx/z581FVVYV169YF1YDFixfDbDY7/r7++uugjkdERBRqUmp1ZKWnuNyeKNNopQhoqu2CBQvwwQcfYNeuXbj66qsdtxsMBrS3t6OxsdGl96O+vh4Gg8HjsTQaDTQaTSDNICIiihixVkfJpmqXYmMGvRZLp+Un7DRaKWQFH4Ig4PHHH8eGDRuwY8cO5OXludw/ZswYpKSkYOvWrbjnnnsAAEePHsWpU6dQWFioXKuJiIgiwGYXXAKKifkGn0FG4cAeEW5xdJIVfMyfPx9r1qzBxo0bkZmZ6cjj0Ov1SEtLg16vx9y5c7Fo0SJkZ2dDp9Ph8ccfR2FhIWe6EBFRzLLZBazYdhyr9tSisbXDcbvxSi8Hh1LkUQmCIEjeWOW5u2jVqlV46KGHAHQWGXvqqaewdu1aWK1WTJo0CX/4wx+8Dru4s1gs0Ov1MJvN0Ol0UptGRESkGOcejhPnW7oEHSLxqshcDnnXb1nBRzgw+CAiokgqrarrksfhiwqdeR67n5mQ0Dkdcq7fXFiOiIjoitKqOsxbXSE58AC6lkwn/xh8EBERoXOopWRTNQIdDkj0kulyMPggIiJC53otcno83CV6yXQ5GHwQEREhuJ4LlkyXh8EHERERgJxugRe8ZMl0eRh8EBFRwiutqsNT/6wMaF99WjIm5ksrJ0GdGHwQEVFCE2e4mCzWgPY3t17Gim3HFW5VfGPwQURECSvYGS6i5Vu+QmlVnSJtSgQMPoiIKGEFO8PFWcmmatjsUVW3M2ox+CAiooSlZG0OFhqTjsEHERElLKVrc7DQmDQMPoiIKGGNzcuGUa+FUpNkT5xvUehI8Y3BBxERJRybXUBZzQV8cPAsZt3QFwAUCUBeZ+KpJMmRbgAREVE4eVq1NistGZftwCXr5aCPX7KpGhPzDSw65gN7PoiIKGF4W7W2sfWyS+CRqU0K6Phc4VYaBh9ERJQQ5NT0aGqzBfVYTDz1jcEHERElBCVrevjDFW59Y84HERElhHD0RqgAGLjCrV/s+SAiooQQ6t4IMb2UK9z6x+CDiIgSgtI1PbLSUlz+b9Br8eac0SguMCr0CPGLwy5ERJQQktQqLJ2Wj3mrK6ACgl5MbuV9o6FWq3CuqQ29MjuHWtjjIQ2DDyIiShjFBUa8OWd0lzofcoh5HTcO7MFgI0AMPoiIKKEUFxgxMd+A8toGnGtqw4nzLVhbfgomi1PRsfQUNLZ0dOkhYV6HMhh8EBFRXLHZBUdg4W04JEmtwti8bEcxsP+9dwSgAs5fsjr22Vxt6tJDYtBrsXRaPvM6gsTgg4iI4oan0ulGDwGD5xLrKXh4fH98d3hvJKlVXXpImNehHJUgCMHm3CjKYrFAr9fDbDZDp9NFujlERBQjxNLp7hc1MVQQZ6J4206UlZ6Cl2cOY++GTHKu35xqS0REMc9X6XThyt+z/z6E/3z1DV5833eJ9caWDsxbXcHVaUOIwQcREcU8KaXTG1s78MBfy10SS70R0Lk6rc0eVYMDcYPBBxERxbxQlE7n6rShw+CDiIhiXqhKp3N12tBg8EFERDFP6dLpopwMjcJHJIDBBxERxQGxdDoARQOQp9Z/wcTTEGDwQUREcUEsnW7QKzcEU29p48yXEGDwQUREcaO4wIglU/P9bqcCkJWWjG4a37U2xbkunPmiLAYfREQUN2x2Ac+9d8jvdgKAxtbLuGS9LGlbznxRFoMPIiKKGyu2HUNjS4ff7fz1eHjCmS/KYfBBRERxwWYXsGrPCUnbSunxcBeq6byJiAvLERFRXCivbUBjq/9eD7lU6FzNdmxetuLHTlTs+SAiophmswsoq7mAjyXOSMlIlX7pE6ftLp2Wz9VsFSQ7+Ni1axemTZuG3r17Q6VS4b333nO5/6GHHoJKpXL5Ky4uVqq9RESU4MRgY2PlGbyx5RjGv7wNs9/ei7+VnZR4BOlBhEGvdayGS8qRPezS3NyMESNG4Ec/+hFmzpzpcZvi4mKsWrXK8X+NhhXiiIgoeKVVdSjZVO13ETlfmtttfreZO74/ivINGJuXzR6PEJAdfEyePBmTJ0/2uY1Go4HBYAi4UURERO5Kq+owb3UFQlltw6jXYum0fPZ0hFhIcj527NiBXr16YciQIZg3bx4uXLjgdVur1QqLxeLyR0RE5MxmF1CyqTrgwCMjNUnSdq99bwQDjzBQPPgoLi7G3/72N2zduhWvvPIKdu7cicmTJ8Nm89zNtWzZMuj1esdfnz59lG4SERHFuPLahoCGWtKvBB1ShloA4HyzVfZjkHyKT7WdNWuW49/Dhg3D8OHDMXDgQOzYsQN33HFHl+0XL16MRYsWOf5vsVgYgBARkYtAC3y1SAw6RKzlER4hn2o7YMAA5OTk4Pjx4x7v12g00Ol0Ln9ERETOQh0UqNCZ78FaHuER8uDj9OnTuHDhAoxGjqEREVFgxuZlw6jXypgkKx1reYSf7ODj0qVLqKysRGVlJQCgtrYWlZWVOHXqFC5duoSnn34ae/fuxYkTJ7B161ZMnz4dgwYNwqRJk5RuOxERJYgktQpLp3WuVqt0eMBaHuEnO/j4/PPPMWrUKIwaNQoAsGjRIowaNQovvPACkpKScPDgQdx111245pprMHfuXIwZMwb/+c9/WOuDiIiCUlxgxJtzRsOgdx2CyUoLLn3x+9f3YeARZipBEEI5ZVo2i8UCvV4Ps9nM/A8iIurCZhdQXtuAc01tyMnQ4Kn1X8BkCbzoWFZ6CvY/P5FDLkGSc/3m2i5ERBRTktQqFA7sgekjr4JarQoq8ACAxpYOlNc2KNQ6koKr2hIRUdRz7u3olal1lD0PNvAQBTqVlwLD4IOIiKKap/VcjHot7hphxPr9pxV5DNb3CC8GH0REFLW8redSZ27DH3fVSjqGSgV4y25UoXO2C+t7hBdzPoiIKCoFu56L6srfo9/J8zg9l/U9Ioc9H0REFJUCXc9FlKFJxmv3DkdxgRGj+nbvMnRj4Aq2EcPgg4iIoo7NLuBvZSeCOsYl62XHv4sLjJiYb/CYtErhx+CDiIiiSmlVHZ599xAaWzqCPlbJpmpMzDcgSa1yTNGlyGPOBxERRY3Sqjo8trpCkcAD6ExMZQ2P6MPgg4iIooKYYKo01vCIPgw+iIgoKuz974WgEky9YQ2P6MOcDyIiirjSqjo8++9Dih6TNTyiF4MPIiIKG09l0jdXmzwWEgsGa3hENwYfREQUFp7KpBt0GrRdtisaeACs4RHtGHwQEVHIeSuTbrJYZR0nIzUJze02r/ffcW1P/M93BrKGR5Rj8EFERCEVbJl0AEhPTcKPbxmABRMGY3O1qUsPSnZGCn45vQBThvcOvsEUcgw+iIgopIItkw4Abz9wPcYPzgHAaqXxgMEHERGFVDB1NsQZKze6VSZltdLYxjofREQUUoHW2eCMlfjF4IOIiEJqbF42jHqtx2XtfTHotXhzzmjOWIlDDD6IiCikktQqLJ2WL2ufJVOvw86nb4c+LRUbK8+grOYCbHalJ+RSpDDng4iIQq64wIg354zGi+8f9jm9VszxMOrTcOtvtrskqhpZuyNusOeDiIjCYmK+Af9770hMLsj1eL84LHPXCCPmr6noMkPGZG7DvNUVKK2qC3FLKdTY80FERLJ4KpHuLyHUU3VTtQpwHkkx6LVYMvU6vPThEY81QcTbXnz/MCbmG5iEGsMYfBARkWSegghPwyE2u4C9NRdQ9t/zqPmmGR9XmbocSww85o7vj6J8A8bmZUuqCWKyWLFi23E8UTRYmZOisGPwQUREkngtkX5lOEScmVJaVYdn3z2ExpYOv8dUAfioyoTnpnZOp91S3TVI8WT5lq8wxNCN+R8xisEHERH55atEuoDOIKJkUzXsduAnayokH1cAUGduQ3ltA8yt7fjLnhOS9y3ZVM3hlxjFhFMiIvLL33CIGET8/L1DAR3/w0Nn8dyGKln7iEELxR72fBARkV9SS6RflDDU4snqvacC2i+Y0u0UOez5ICIivwItkR5q0dou8o3BBxER+eWvRLoKncvah4sKnbNsxuZlh+0xSTkMPoiIyC/nEunuAYj4/19OL4BBpwl5W7jgXOxj8EFERJKIJdINetehDnEBuCnDe+PFu4b6Pc5t1+QE1Q4uOBf7VIIgRNVKPRaLBXq9HmazGTqdLtLNISIiN/4qnHqr89E9PQXLZg6DPi0Vs9/eK/nxDDoNZo/ti/45GZIrqlL4ybl+c7YLERHJkqRWoXBgD6/3FxcYMTHf4KhwCnRuf+OAHkhSq2CzCzDqtX4rmT58U3/cOdTAYCMOseeDiIjC7oPKs1iw7oDPbYx6LXY/M4GBR4xgzwcREUUN92Gai83teGGT/4JideY27K25gPGDg8sRoejD4IOIiBTlHGycON+CteWnYLIEVgxs/poKvHzPMCaXxhkGH0REpBhPq94Go7G1w2XROooPsqfa7tq1C9OmTUPv3r2hUqnw3nvvudwvCAJeeOEFGI1GpKWloaioCMeOHVOqvUREFKXEVW+VCjyclWyqhs0eVSmKFATZwUdzczNGjBiBlStXerz/1Vdfxe9+9zu89dZb2LdvHzIyMjBp0iS0tbH+PhFRvLLZBbz4vudVb4PlvPItxQfZwy6TJ0/G5MmTPd4nCAJef/11PP/885g+fToA4G9/+xtyc3Px3nvvYdasWcG1loiIotKKbccCzuuQiovIxQ9FK5zW1tbCZDKhqKjIcZter8e4ceNQVlbmcR+r1QqLxeLyR0REsaO0qg7Lt4R+eJ2LyMUPRRNOTSYTACA3N9fl9tzcXMd97pYtW4aSkhIlm0FERAoTZ7CYzK1oaG5HdjcNDDotRvbJwnMb/E+bDYYKnSXVuYhc/Ij4bJfFixdj0aJFjv9bLBb06dMngi0iIoo//kqi++JrBosKCEmeh/PxAS4iF28UDT4MBgMAoL6+Hkbjt1Oi6uvrMXLkSI/7aDQaaDShXwWRiChReQoejHotlk7L9zt9VZzB4i3AUDLwyEpPgTY5ySV3xCCxnRRbFA0+8vLyYDAYsHXrVkewYbFYsG/fPsybN0/JhyIiIgm8BQ8mc5vH+hnOPSQ5GZqQzWBxJvZnvDxzGCbmGwLuoaHYITv4uHTpEo4fP+74f21tLSorK5GdnY2+ffviySefxC9/+UsMHjwYeXl5WLJkCXr37o0ZM2Yo2W4iIvLDZhdQsslz8CCg86JfsqkaE/MNSFKrFC8QJpV774avResoPsgOPj7//HPcfvvtjv+L+RoPPvgg3nnnHfzsZz9Dc3MzHn30UTQ2NuLmm29GaWkptFpmKRMRhVN5bYPPQMK5foa5td3n8IrSsjNSsOS7Q2HQsXcjEckOPm677Tb4WghXpVLhF7/4BX7xi18E1TAiIgqO1LoYJnMrXv3kaNgCDwBoaO6AQadlL0eCUrTOBxERRQ+pdTEamtvDPtQCsGhYImPwQUQUp8bmZcOo18LbgIYKnbNeTl9sVewx5QyesGhY4mLwQUQUp5LUKiydlg+ga1Ag/v+uEUas+vSEYo+pT0vBE3cMgkHnP+hh0bDExeCDiCiOFRcY8eac0TDoXXsZDHotVt43Cu9/UafI43TTJAEAGls78MbW42i7bHPMqHHGomEEREGFUyIiCq3iAqPH+hn+ZsPIcclqc/m/uaUDAKBPT0HjlX8DLBpGnRh8EBElgCS1yhFwnGvqnF4bylVoxV4PbbIaf/+fcTh/ycqiYeTA4IOIKAF4KiCWnZES0scUAJgsVqhVKkwfeVVIH4tiC3M+iIjinFhi3X2IpaG5w8seytpc7XlVc0pcDD6IiOKYrxLrwZLac/LXPSdQWqVMYivFBwYfRERxTMmkUlFWWgr+/j/jsHdxEYx6/7U6xDVkbPZw1lClaMbgg4gojoWiimhjawfUKhVSk9WOOiK+OK8hQwQw+CAiimuhqiIqBjXFBUbMHd9f1j5EnO1CRBQHbHahSx0PcXqtUa+FydymaN5HToYGZTUXcK6pDb2z0iTtw3LqJGLwQUQU5bwFFiJP02iNTsW8lk7Lx7zVFVABigQgWekpeGr9Fy51QtQqwFtKhwqdxcVYTp1EDD6IiKKYv8BCnEbrft03mdswb3UF3pwz2lFi/cX3D8NksQbdps6Kpa7TdH0FHgDLqZMr5nwQEUUpb/U56sxteGx1BZZvPooX3/c8jVa8rWRTNdov26FPS8U9o68OeZvd4wuDXusIgIhEKkEQomruk8VigV6vh9lshk6ni3RziIgiwmYXcPMr2xSZJpudkYqG5nYFWiXNkqnXISdTw3LqCUbO9ZvDLkREUUjJ+hzhDDwA4ExjK+Z+Z0BYH5NiC4MPiln+kvCIYon7+zmUi74FQk6y6sbKs/j5VOZ4kHcMPijslAga/CXhEcWSSCz6JpdBr8WSqdfh+Y1VfteEudDcjvLaBhQO7BGm1lGsYfBBYaVE0CA1u58oFnh7P4dr0TcpsjNSsPPp25GarMb+kxfxlz0n/O7DgmLkC2e7UNh4y9wXgwYpC0/5WiTLObufa0hQLAjlom8ZqUmKHauhuQP7T14EABTlGyTtw4Ji5At7Pigs/AUN4sJTE/MNPodg/CXhOa8hMTYv2zG8k5OhAVTA+UvWgId6mGNCSpOaVNo9PQUXW7z3hKgA5Oo0mHVDX7zz6Qk0tnagud2mYEu/7cnwVzGVBcVICgYfFBZyggZf48RSu3I3V5uw6J+VXh8zkKEe5piQ0qS+n2eM7I31+8/gkvVyl/vE8Hf6yN54Y+uxkPSiAN/2ZCSpVV4rprKgGEnFYRcKC6lfsv62k9qV+9c9J3wGO3KGepQYLoomNruAspoL2Fh5BmU1FzhEFUFS38+rPj3pMfAAOkudr7xvFP7x+emQBB4qdAbazj0ZYsVUg961/SwoRlKx54PCQuqXrL/tpCyS5WuNCZHUoR6lhouiBXtwoosS72dNshpf1V+6UvJcWb56MooLjJiYb+BQJAWEPR9xJtp+1YrtMZlbkZ2RCm9fS55+XXkidvmK+7gfA/AfeIich3q8kTNcFO3irQcnHijxfjZZrFi547jibQP892QkqVUoHNgD00dehcKBPRh4kGTs+Ygj4fhVKyfp0lN7PJE7Tix2+bof26DXYkqBQdI0QGe+hnqUGi6KtHjrwYknSryfO2zK/chYcPtADM7NZE8GhRSDjygn9WIf6toXNruAFduOY9WeWjS2ftu96y248dYeTwx+AiRPz4G3Lt/y2gbZwUdOhsbrfcEMF0l57cI1g0aphF8KDSXfz8EaP6gn3wMUcgw+opjUnoxQ/6otrarDs+8e8jim7Cm4kVK7ICM1CQ+N74+bBubgxgHeu2v9PQfuX5JSxtDdPbX+C7x4l+fgJ9BphVJeu3DmX0jtmdlz/Bv+2o0QcQgD+DYoNZlbkalNRlOb52RTJXGKLIUTcz6ilJzx+VDmJZRW1eGx1RVek9k8FfaSUrugud2Gldtr8NP1X2BztcnrY8vNUfA1hu5NvSWw43kbLpLS7kDOLZh8Hqk9OCu21+DmV7ZFTf5HtOUwhUNpVR1ufmUbZr+9Fwv/+UXYAg+AU2QpfNjzEYXk9mQEk5fgq9tfbIc/7l32cvIfvA0LBdOb420M3Vf73Y/n/rysvG80Xvqw65i8p16oF9/33e4X3z8MQCXr3ILtJZHTIxQtZerDPTPH3xCYt/uVHDqTM1wZCBUAfXoKtMlJLgvX+Rv6JFIag48oJHd8PtC8BH9f7nKX9D7X1AabXcD5JqvkfbxdbOU+B+4XgIn5Bpcx9HMWK3710RFJxzO3tnt8XpZMvQ7dMzQ+Lz7/+eq8z9VIBXTOTvD3nDifm7cLUp25DY+trsBbEoIEX4WhPD1+pBNQw71+j7/Pgrf77xphxPtf1AUVIDkPsbz04ZGQBh4A8PLMYZwiSxHH4CMKye3JCCQvQcqXu/WyXVa7//PVN1i68bBLQqoU4sX2nT21yMnUyFpO/FxTm6RfyBsrz0g63uZqE1btOeHxeZm/5gDenDMa00de5bhd6oyeQIjBnL/8mWffPSQpSJDTIxTJBNRwz8zxF9w98p08/Pk/tR7v/+Ou2i7HkxMUhvL9o1IBglOj3Xs3mFRKkcTgIwrJ7clIUquwZGo+frKmoss2nsZypX65v3bvCMltVgH4V4W0C7w3L334bc+E1OXET5xvwetbvvL7C1nqc7p+v+cqkZ4ueqHuIu+VqZXU+9TY0oEV247jiaLBfo8pzqpYvvkrrNjuvzZEJKYQBzozJ5DhDynB3dv/6RpgSOEvKAz1+2fl7FEee+qIogETTqOQ2JMhtSBXaVUdXvrQc26GpyJBUr/cIcBnO9z3UZK/5cRVAAw6DdaWn/IaLAjozK+w2QW/z6nIV3Kf80UvlKuROr++Ui/+qz6tlZyMmaRWYfygHEnbHqu/FHSip9yk0UBymJyTNJ9YV4nZb++VlDgrd2hRDjEo9MRXbpA/3xt9FbLSvAfn3dNT8Nac0ZgyvDcLgFHUYs9HFJKzcJO/X09Lpl7XpetX6pf7+War3zwB967dUPD2HMwe2xfLtxzzua/JYnX0CkjNefDnXFNbyC5a7q/vifPNkvZrbOnwOkTiqUdAagLqiu3HsWL7cb95DN56HQJJGpXb8xdMfkioe3ZWfVqLBRMGdbnwr9h2TPLQojOjXotXvtfZIynmiZy/ZEVjawdU6Jyq62vqOlG0YPARpbyNz+vTUvDw+P6YmG/w++tbhc6hjEkFRpcvIzlf7oUDe3hsR1ZaCoqu6yV7qCWQC383tzoH4ti11JyU5Vu+whBDN9mzYLw5cb4l4H39cR6Xt9kFrC0/JXnfPce/kXXxlxOMiXkMC4sGY8GEzuEdMdg4cb4Fa8tPuVxMxWTMP+3qmivhLyiQk8MUbH6I1M9CoBpbOvDOnlo8UNgf+09edDxf/oJmb2bd0NdxHszZoFimEoRQ/26Vx2KxQK/Xw2w2Q6fTRbo5EWezC/j91mP48+7/4pLV5rjdqNdi1g19JH2JrX3kxi5j4ze/ss3vl/vuZyY4ZnPsrbmAsv+eB5x+XX1w8CyeWFcp+Vy6aZKRnKQKaAGsDE0SZl3fB0X5Bozp1x37T17EnuPfYMX2Gkn7G93OR/zV+ML7h2XXUTDqtXjt3hG4/8/7ZJ+HJ0umXudItHUely+ruYDZb+8N6Ji+Lv7iJfjNOaMBQHYwlpGaBKCzVkug3N9j7sTeDMBzr5cYuEh9jtw/AyLxsxCqoReRlMUOpXhj1kiXhGeiaCLn+q14z8eLL76IkpISl9uGDBmCL7/8UumHSgivlh7xmlEv9deTe9ey3GEd94vTvytOY+m0fNm/GputlyEAWFg0GP1zMnC+yeqSZOp7Xxv+sucEkpNUWPTPStkXC+cERbGSZFnNhYAKONWZ2/B/y07I3s+T7IwUPFDYH6nJXdOvghkS8DYTA3DtEdj9zATHtEupwVwwQYdzG3zNpvG13onzkI3U58hkbvV4u69kbSUpVRvtfJMVNrvAYRWKeSEZdhk6dCi2bNny7YMkc3QnEB8dPOv1AiJHr0ytxzoYK+8bjec3VqGhud2xrfOwjr+x9JX3jZJVylzcZm35Kez62QR8VtuArLQUWVNzg3k+3C9UW7xUVpWi9HB9wPs6a2juwK2/2e4xByKUQwLuF3+5xeGU4usxPa13IvZ6baw8g16ZWuR0874uj7OXPjyCtNQkj2sQeUvWjkYvfXgEf95d67G4na9igazpQdEmJFFBcnIyDAZDKA6dMGx2Ac9vrArqGGLX9sXm9i5dy1npndny7kMgja0dWL7lGNaWn0LbZbvPsfSXPjyCJVPzMX+NvCROk8WK0S9txiVr6MtGO3O+mJdW1YV9wS5v3HMgnIeFsjNScbG5PWTTMZ0v/qHOf/DkP1+ddzy2v4qiF5vbcetvtrvlQCVLSnq+2NzeJc/ko4N1Ie/xCAX394uvvB6g67BaKKvEEkkVkuDj2LFj6N27N7RaLQoLC7Fs2TL07dvX47ZWqxVW67cVHy0WSyiaFHPKaxv8Tjf1RwAwpl93zF/TtffCX96F1Cqc3TNS8eac0Xjx/cN+93EW7sAjKy0FdkFwTPOUUjY+XJyHQex2dCnjHko5GRqU1VzAuaY25GRoYNBpUW+RvihfsP5VcRr/qjgNwH9FUU/MrdLeR+7Jp59U1WHB2gNBtj4y3N8vnj7fpisJwp5ES/l8SmyKJ5x+/PHHuHTpEoYMGYK6ujqUlJTgzJkzqKqqQmZmZpftPeWIAEiohFNP3aJykzkjRUyA23P8vGIJmKEkJ1FXSdkZKUEHk0rytsZHVnpKQAnBSrUJAB69Jc9joqxSFhZdg+VbvgrR0cMrOyPVZdhUKn8Jv0SBkJNwGvLZLo2NjejXrx9++9vfYu7cuV3u99Tz0adPn4QJPrx1mUbiAhkIcabGsfpLkipmhlqyWoXLEV75VAUgV6fB/35/JM5fsjrKxS/8R2VQx83OSMHPp+TjxU3yZuh4Sir2NT1bQOdwhtReBV80yWrZZfqVmhnijdw8o0hIS1GjtUPe8xYIb7OAiAIR0dku7rKysnDNNdfg+HHPFyaNRgONRlrSWLzxldC5fMuxiP4KlUKlguTZKuESDYEHALx411CXKqJlNReCPnZDcwcaW9olBx4qdPYiuC98lqvToO2y3eN7S+zST0tJwoOF/fE7LxU6pZIbeAChDTwARH3gASAsgQcQmfL5REAYgo9Lly6hpqYGDzzwQKgfKqq5D62M6dfdb3EkILCiXOESqj6zQH4tR4us9BQsmzmsy1j62LxsGHQaWXkxnpxskFbgLCM1Cf/7/REoLjDiZ8XXubz37ILgc4hMXHlX76OEt7tofp/Gqqy0FJhbO7zW4umuwFBeJJKMiYAQrO3y05/+FDt37sSJEyfw6aef4u6770ZSUhJmz56t9EPFDE/rTty4bIvf9VUaWzpwz+irYNC5fkF0T0+GPi0+py+rVcCPvzMg0s0I2EUvPVVJahVmj/WcdC1Hv+x0Sdv96YHrHQGQWNdEXOPj/CVpAZDUQGds/+4MPELg4fF5Hm8Xf5j8cnqB5LWXPB3DeX0oonBT/Ap2+vRpzJ49GxcuXEDPnj1x8803Y+/evejZs6fSDxUTvA2tSP3F8q+KMzDoNI7CXGJC6uZqk9ds9lhmF4DfbT+OrPQUmFs8/+qLds9tOITWDjsMOtfpo/1zMgI+ppgg+EBhf/x5d63P2ipGvRY3+hjHl/prV2qgMzavB8pPXJS0LfknvtaDe3WD3sPQq3Pvmlqt8losUPDwb/H/gOtK10ThpnjPx7p163D27FlYrVacPn0a69atw8CBA5V+mJgQzMqVzuotVry+5Rg0yWpHhc7iAiPemjPaUa8j2iy4fSDemDUSC4uu6dJzI9Zm8Ebl5d+RNmNkb0nbNTR3YOE/uq6sGkwXt4DOi0VqstpRv8H9uVFd+fN3UZG6avIDhf39/rLOSk9hwmII3DXCiPlrKjzm5Tj3romVYA161/eWQa/FW3NG4y0v93GaLUUa13YJoTe2fKXYjBVvU+Oc110pr22Iml+gzln07vku/nIORAuLrsG6z06FreaFN+Jz/7Pia2XPWHFei2RivsHnmjq+/Gh8f7wwbajj/4GsFutM6toppVV1fnvY/nDfKLz04ZGAzive6LTJsARQsl/UIyMVL00fipc+POL1fe/pu4AVTikaRNVsl0RVWlWn6FRZb2thJKlVGD84B+MH58BmFzDml5sjPkOmmybJUdArSa1y5ByINlZKWwm3w2bDzqdvx/6TF/GnXTXYfvSbUDUZ6alJaL2yZon7BVQAMKXAgAaJuRLu+zoXtxLX1JFrYr5rxWDn0uMmcysamtuR3U0DfVqq17U/3C9CK+8b3aWgWa5Og9lj+8J62Y6ymguYcG2u31lXv/igGi98d6jHSrfi/2N5GE2Opd/Nx2ubvwooYM7OSEHZ4juw/+RFv/lg7t8F7p8xZ77uI4oUBh8hIC7zLZWcQkG+psYlqVV4eeawiOeCXLLacP+f93n9JS51+GHF9hr8u+IMCq7ShTTwAIAf3zIQQwzduvQmiDUnxFLsgczqEC8W//v/jqJHRipuu7Yntn8p/Xx6ZKR6TAxMUqtgbm3Hq58c9dsD4q2nZMnU69A9Q+NY6n1t+SmXoLm7hOneJosVx85d8rgQXPeMFPxyegHUalXI35dZHoqmhVvv7ulYMvU6/GSN9OqpYpj467uHITVZLXn6K6fJUixTPOeDOkujS/3lY9RrsXfxHVgy9TpJ258473sGwsR8Q9TMhBHLOIs5DyJ/OQfO6sxt2Fx9zuc23nqQM1KTUDw0F9kZqX7zFhZMGITiAiN2PzMBax+5ET8a3x9A15oTwfxy/8OOGrz04RFZgQcAFFylQ3ltg6M0vEgcOnF/r4nP+0cHz6Ks5gJ+sekwHvOwXZ25DT9ZcwDm1nZoktV4fctXXS7c3mbvuBMrhi6Zmo/sjFTH7Q3NHY5aMAuLBks74QD9ekYB9jzb+fp5yzcKFefZI90z5NUtcs/BkBqcc5osxbLouErFGH9jqHJ+kYhJhCaL5yW/3a377BQWTBjkdcy2vLZBkcqUSnAfchDbnKRWBTz84Ild6Ky0KvYgZXfTuMw0ES/S3notXp45zKVt4no40WLnV+ex86vzLj0aYu+atzoxADB/zQFJwdIz/z6ItJTkoIdEnn33kMehlbor64zMvRLQhcpLHx6B+koytmjBhEEun9VtX9bj7f/IXxnZV9VV99kjUj//Pyzsh8kFxi7fH2Jw7i2HRsz54DRZimUMPmSSkugn9RdJRmoSjpqasPe/F/DOpycl7eMp78NZtHXFestVKS4w4tFb8vDHXfIvBJ7kZGowfeRVHu8TZwRISdAsrarDcxsORdU6LCLnBcH0aal+e9ekBhPm1suKBKz+hmdW75P2Hg+UpwXTnPMdbHYBi/5ZKfu4C24fiP9zxzXYf/IiNleb8F7lWZdhUkOAn//JBUaPn2Pn4JzTZCleMfiQwVc5dOcvPX+/XETN7baAklJ9BRjR2hV7rqnNpccoJ0ODjZV1/neUyN95OydoOleZ3X/yIjZWnnEs2e5phdBo4dyT9LPiayPdHNmsl0P7zLr3tAHoMssqkETQ8YN6IvXKNPfCgT3w86n5Pns+lei58BYwuwc6RLGKwYdE/rq53YcXlBxWcOfrQqtUGW+gMxeiw2ZHs9UW9LFOnG/Bza9sC8m0WW8Jme6cfwWXVtXh1t9s75JcGq2Bh0jsSQpk5k08mDGyN96rPOv1fvH5eeZfX2DLkXMu67hkySgXD3gPEvzNHlGq58JTwMxpshQvmHAqkb8kUvFLb/nmr1BWcwET8w1Yed9oZGqVje/8lUROUqvw4l1Dvd4vxQ8L+2HtIzdi//MTcWDJnS4JhJ6oVd6LganQGcS8viWw6YdSvDS9QNYXsrdEzQivSSdLdkZqwKW1Y9nV3aVVXf1XxZkuC8jJWVAu2OENb8W/umek4Efj+zumRPvjXhqfgQfFCwYfEknNpVix/Thmv70XY176f/jZvw/KWvrcHxU6EyvLaxuwsfIMymouePwCE6ufpqcmBfQ44lh0klqF1GQ1fn13gaN6pnt7VAAe+U6e4//u94utC9V1/ce35GHKcOld0L56sCLNqNdKnhFyqqHVUek0URj12rDVq1CiCqj77KnOhOgO/GXPiS7Vb4kSDYddJJKbS9Go8IyT7ukp+P71V3epfJidkYK7R16FonyDS5fsxHwDdNpqtLRLHzLx1s0sZfx5VN/uHu+fdUMfRYqtuddCyb5SP2LKcGklz0VypkEHw9fsCHdzx/d3vH4AsLb8lN9hs3WfncLuZyZg5X2jJM9qiXVLpubjxgE9JOVTBf9Y1/kMPKRWDRVrsazac8JvrhhRImHwIZHUJNJQmDbcgMkFvT0mQ4q/pP6y54TL7I3y2gZZxZb8dTP7G3/2dv8HB72Pz8uxZOp1MOjTgh77DmY2UDdNkiP/xVMVT3Hxv/NNVkdtC196ZKTiV3cXdLnw3NC/OzYdNPncV5xB1D1DE9b3o1iYrLquCSu2Hw/jIwPdM1JDnk8FdL6eL314BJMKjB7fY3JK28vNFSNKFAw+JPKVRBYq4q/7SQVG3PzKNr+P6fxLynrZLuuxpGTRe0q08/cLUKnZNwZ9miJd7oG0Rzyb1+4dAQB+ZyBILR//vIdf16VVdX4DD1E4p1VPLjDgh4X9Ha/vsXNfBXwsOb1CzsTzVXqatjtv08MB6TPeRFJzxXxNnyeKRww+ZPA2/KA05274JLUKZTUXJD2e8y8p8ULpz4LbB2H8oJyAehKk/AKU0mPkr4CTkgWVAmmPe3DhbwaC1ADHoE9z+b/csvznm6y42CKtLH+wSqtMmD6yt6NoW6BDaSoAK2aPQvcMjceaGb6Iz6vNLuD9L0KfK+Ee3AXSi8FS6USeMfiAvFUfnYcX9hz/Biu21wT0mN8bfRV2H7/gMjTiretWzheT+EsKAiTVGlg48ZqAunu9/QIUq1m+deUXoJRph498Jw9/uvIrNtQFlXx124uPIF4cvb0f/E21DLTOg5x8FLUKkoZ2lFSyqRoTrs2VHCB10yTjkvXb3CcxP6l7hgZj87IdNTN+t/UYfrf1mNdg0P35ClfejnsQGUgvBkulE3mW8MFHIEuTixefsXnZ+HfFGdl5ID0yUvHK9zp7JvbWXEDZf88D6DzmjQO6XtQC+WI632wNWZVEKTNGnn33kOMXYDAJq6EqqKT3sGBaVnoKls0cFvTjBVrnQU6QGe5pweKF9f+WnZB04V9YdI2jtLlzD4d7fhIAn4GHyPn52lItbVgqUN6Cw0B6MVgqncizhA4+5I7fuhMvMnJX6xTrUrgHPiu2H/cY+ASS7Nors3NaYiiqJEr55dnY0oEV247jiStTRwNNWFU6Cc/baw5IX0RNikAqVIbi129WWjJuHtwTu4+dd6lzYdRr8fPJ1+KFTdWShz0A4GSD74UNRf1z0v3O9HhsdQWy0lN8vqfVKmDF7G8/h6VVdY4VhkPBV3AYSC8GS6UTeZawwYeUhbme23AIrR12l0XK3E3MNyBLwrLjIrEuhZzAR26Gv3MhslBc1KX+Alz1aa3LInhSKkOGMunOX4+N0jMP5D73Ss2omjGyN67unu7oSUtSq7wOLSYnq70GY570y5ZW5KtXplbSZ8zf58YudM5yAeTlxGRnpODnU/Kx9P3DLkM//vgKDgPtxWCpdKKuEjb4kPLrvaG5Awv/UQnA+1BMeW2DpMBDBeD3s0bhuyN7S05cm3BtLvafvOi4YPx+1ij8n38c8NvlvmRqvqwcBbmk/gJsbOmIqiz+SMw8kPPcKzWj6vZre3VZZM9bO8QL44vvH/ZZW0S8sD5Q2B9/3l0r6QKsVG6GGOzKOd6v7+4cPsvQJEkKrtyTvD0JpheDpdKJXCVs8CE3u9zbUIzUWhoCgB6ZGgDSL4I3LtvapbCWlLH+7n7KoQdrbF42stJSJJWrjqYs/liYeSA1GPBF7vCNeGFcse04lm/pOoXW+cKamqz2GSAJAKYUGGTXmfFFPB+pr8uPxvd3fEb9zVDzl9/lLphejFD37BHFkoQNPuR+QXuaSldaVYeXPjgs+Rjil6fUhDn3sXipy7yH+uKZpFbh4fH9JU23jKYs/liZeVBcYESmNgX3/3mfrP2CSV5MUqvwRNFgDDF083th9XYBFqcoi0ml2RnyFnLzdz5SXxdxRVuRc6+DydyKhuZ2ZHfT+BxO9YW9GETBS9jgI5DxdedueXNru6yxcqDzyzPUCXPi44TaggmDserTE16HnKIxiz+WZh6cl7lqrVLJi1IvrM7bba424a97TnTplfMXLKvQOevIfOU95G8YI5jXT+leB/ZiEAUnYReWE8dvAxlX/8+xc3j234dk7WvUazGmX3dZRaQC4W/VW6UkqVV4eeYwj6uqerpw2OwCymou+FwQL9TE19y5jaJom3kgN4BUYiE0kdSVVJPUKozNy8bHVf578rw93y/PHOZx9VdP5xNLrx8R+ZawPR+A/Jkqoj/s+K+s7VXo/FLcf/JiyIsjzbqhb9i+fKWOfwdSSyXSbY40KT1z2RkpWPLdoQEPHyhBahJod7eFAeVWjRXFyutHRL4ldPAhdaZKMJwvslLX/AhG/xxpUyGV4q+bPthaKpFoczSQMrNCnNERSVLzi/wtDChnGCMWXj8i8i2hg49QJ2YumXodHhqf5/hSlNqVnp2RiovN7QENCUUiWdLbhSOaV/SMhTH7WPiVL2cdG+ZcEJEooYOPUF2oxcQ358ADAC42+08i7FyyPB/z18ir9RBNyZIirugZvGj/lR9LSbxEFD0SMuFUTH40mVuRoUlS/PgCuia+2ewCnnuvyu++S6bmY8pwo8ckvKz0zqmLsZJsFwt1NWKB1ATQSGASKBEFIuF6PjwlPyrNuciRaMW2Y5LyS8QCYd5+8W6uNkV1N7yzWKmrQcGJheEhIoouCRV8+FpUTEl3XJuLspoLjqBhTL/uWCWxtodzL4Cnce1o74Z3xi75xBFL70siiryECT6kLAMfLLFo0lPrv3ApLZ2dIa0UOSCtFyBWku24omdiiZX3JRFFXsLkfCi1yJUvAjoXU3Nf00JqWfSs9JS46wUQu+SlFJEiIqLEkDA9H1LXUxFlapPQ1GaTtY9KBQhBdK08fFNeXPYCsEueiIicJUTwYbML2CCzwJfcwAMILvDISk/BggmDAj9AlGOXPBERiRJi2KW8tkHy0EekvDxzGHsCiIgoISRE8BFoHYnu6cEtCe5J9pWptCKjXou3mPtAREQJJCGGXQKpI5GdkYLpI6+SPEXWH3Fa6c6nb8f+kxeZ+0BERAkrIYIPKSuEumto7nAEHnLKnHviPK00NVnN3AciIkpoCTHs4qsEtBRi4DGlIBc/uW0gsjNSfR7HvSOD00qJiIi+FbLgY+XKlejfvz+0Wi3GjRuH8vLyUD2UJGK9CX2AeRwqAAe+NuOpO4fg13cXOG5z30YFYMXsUVj7yI14Y9ZIrH3kRux+ZgIDDyIioitCEnz84x//wKJFi7B06VJUVFRgxIgRmDRpEs6dOxeKh5NsYr4B2uTAFpITV2B9Z08trJfteLLoGuTqPBfOmjK8d9QuBEZERBRpKkEIpjqFZ+PGjcMNN9yAFStWAADsdjv69OmDxx9/HM8++6zPfS0WC/R6PcxmM3Q6naLtKqu5gNlv71XseAadBrPH9kX/nAwmjxIRUUKTc/1WvOejvb0d+/fvR1FR0bcPolajqKgIZWVlXba3Wq2wWCwuf6Gi9NLt9RYrXt9yDJorSaQMPIiIiPxTPPg4f/48bDYbcnNzXW7Pzc2FydS1xPmyZcug1+sdf3369FG6SQ5KL90udhmVbKqGzR7qtXKJiIjiQ8RnuyxevBhms9nx9/XXX4fsscQpt0r2T4i5IOW1DQoelYiIKH4pHnzk5OQgKSkJ9fX1LrfX19fDYDB02V6j0UCn07n8hYqvKbfiTJVHvtO/y1RZKZQe0iEiIopXigcfqampGDNmDLZu3eq4zW63Y+vWrSgsLFT64SSz2QWU1VzwO1NlwrUGBDKCovSQDhERUbwKSYXTRYsW4cEHH8T111+PsWPH4vXXX0dzczMefvjhUDycX6VVdSjZVI0687e9EwadBguLBneZqbJR5uq3Ytn0sXnZCreaiIgoPoUk+PjBD36Ab775Bi+88AJMJhNGjhyJ0tLSLkmo4VBaVYd5qyu6lEcXZ6q8OWe0S7lzOT0YzmXTOdOFiIhImpDU+QiGknU+bHYBN7+yzaXHw5nYa7H7mQmO4EHcR8o6MEa9Fkun5bN6KRERJTw51++4XliuvLbBa+ABuM5UEXs/xKTUeasrvC4oN3d8fxTlG1hUjIiIKAARn2obSlJnoLhvJ64DY9C7DsEY9Vq8NWc0lkwbyqJiREREAYrrng+p+RuetisuMGJivgHltQ0419TG8ulEREQKievgQywq5i1/w99MlSS1yiUZlYiIiIIX18Mu/oqKAZypQkREFG5xHXwA3vM3xKJinKlCREQUXnE97CJi/gYREVH0iPueD1GSWoWxednolanFuabO6bVciZaIiCj8EqLnA/BcYp1FwoiIiMIvIXo+xBLr7gXHTOY2zFtdgdKqugi1jIiIKPHEffBhswso2VTtcaqteFvJpmoOwRAREYVJ3AcfckqsExERUejFffARaIl1IiIiCo24Dz6CKbFOREREyov74EMsse6toocKnbNevJVYJyIiImXFffDBEutERETRJe6DD4Al1omIiKJJwhQZY4l1IiKi6JAwwQfQOQRTOLBHpJtBRESU0BJi2IWIiIiiB4MPIiIiCisGH0RERBRWDD6IiIgorBh8EBERUVgx+CAiIqKwYvBBREREYcXgg4iIiMKKwQcRERGFVdRVOBUEAQBgsVgi3BIiIiKSSrxui9dxX6Iu+GhqagIA9OnTJ8ItISIiIrmampqg1+t9bqMSpIQoYWS323H27FlkZmZCpVJ20TeLxYI+ffrg66+/hk6nU/TY0SDezw+I/3OM9/MD4v8c4/38gPg/x3g/PyA05ygIApqamtC7d2+o1b6zOqKu50OtVuPqq68O6WPodLq4fUMB8X9+QPyfY7yfHxD/5xjv5wfE/znG+/kByp+jvx4PERNOiYiIKKwYfBAREVFYJVTwodFosHTpUmg0mkg3JSTi/fyA+D/HeD8/IP7PMd7PD4j/c4z38wMif45Rl3BKRERE8S2hej6IiIgo8hh8EBERUVgx+CAiIqKwYvBBREREYRXXwceJEycwd+5c5OXlIS0tDQMHDsTSpUvR3t7uc7+2tjbMnz8fPXr0QLdu3XDPPfegvr4+TK2W51e/+hVuuukmpKenIysrS9I+Dz30EFQqlctfcXFxaBsahEDOURAEvPDCCzAajUhLS0NRURGOHTsW2oYGqKGhAffffz90Oh2ysrIwd+5cXLp0yec+t912W5fX8LHHHgtTi/1buXIl+vfvD61Wi3HjxqG8vNzn9uvXr8e1114LrVaLYcOG4aOPPgpTSwMj5/zeeeedLq+VVqsNY2vl2bVrF6ZNm4bevXtDpVLhvffe87vPjh07MHr0aGg0GgwaNAjvvPNOyNsZDLnnuGPHji6voUqlgslkCk+DZVq2bBluuOEGZGZmolevXpgxYwaOHj3qd79wfg7jOvj48ssvYbfb8cc//hGHDx/G8uXL8dZbb+G5557zud/ChQuxadMmrF+/Hjt37sTZs2cxc+bMMLVanvb2dtx7772YN2+erP2Ki4tRV1fn+Fu7dm2IWhi8QM7x1Vdfxe9+9zu89dZb2LdvHzIyMjBp0iS0tbWFsKWBuf/++3H48GFs3rwZH3zwAXbt2oVHH33U736PPPKIy2v46quvhqG1/v3jH//AokWLsHTpUlRUVGDEiBGYNGkSzp0753H7Tz/9FLNnz8bcuXNx4MABzJgxAzNmzEBVVVWYWy6N3PMDOqtIOr9WJ0+eDGOL5WlubsaIESOwcuVKSdvX1tZi6tSpuP3221FZWYknn3wS//M//4NPPvkkxC0NnNxzFB09etTldezVq1eIWhicnTt3Yv78+di7dy82b96Mjo4O3HnnnWhubva6T9g/h0KCefXVV4W8vDyv9zc2NgopKSnC+vXrHbcdOXJEACCUlZWFo4kBWbVqlaDX6yVt++CDDwrTp08PaXtCQeo52u12wWAwCL/5zW8ctzU2NgoajUZYu3ZtCFsoX3V1tQBA+Oyzzxy3ffzxx4JKpRLOnDnjdb9bb71VeOKJJ8LQQvnGjh0rzJ8/3/F/m80m9O7dW1i2bJnH7b///e8LU6dOdblt3Lhxwo9//OOQtjNQcs9Pzmcz2gAQNmzY4HObn/3sZ8LQoUNdbvvBD34gTJo0KYQtU46Uc9y+fbsAQLh48WJY2qS0c+fOCQCEnTt3et0m3J/DuO758MRsNiM7O9vr/fv370dHRweKiooct1177bXo27cvysrKwtHEsNixYwd69eqFIUOGYN68ebhw4UKkm6SY2tpamEwml9dQr9dj3LhxUfcalpWVISsrC9dff73jtqKiIqjVauzbt8/nvn//+9+Rk5ODgoICLF68GC0tLaFurl/t7e3Yv3+/y3OvVqtRVFTk9bkvKytz2R4AJk2aFHWvFRDY+QHApUuX0K9fP/Tp0wfTp0/H4cOHw9HcsIil1y9YI0eOhNFoxMSJE7Fnz55IN0cys9kMAD6vfeF+HaNuYblQOn78OH7/+9/jtdde87qNyWRCampql9yC3NzcqB3fk6u4uBgzZ85EXl4eampq8Nxzz2Hy5MkoKytDUlJSpJsXNPF1ys3Ndbk9Gl9Dk8nUpes2OTkZ2dnZPtt63333oV+/fujduzcOHjyIZ555BkePHsW7774b6ib7dP78edhsNo/P/ZdffulxH5PJFBOvFRDY+Q0ZMgR//etfMXz4cJjNZrz22mu46aabcPjw4ZAvohkO3l4/i8WC1tZWpKWlRahlyjEajXjrrbdw/fXXw2q14s9//jNuu+027Nu3D6NHj45083yy2+148sknMX78eBQUFHjdLtyfw5js+Xj22Wc9Jv84/7l/EZw5cwbFxcW499578cgjj0So5dIEcn5yzJo1C3fddReGDRuGGTNm4IMPPsBnn32GHTt2KHcSfoT6HCMt1Of36KOPYtKkSRg2bBjuv/9+/O1vf8OGDRtQU1Oj4FmQEgoLC/HDH/4QI0eOxK233op3330XPXv2xB//+MdIN40kGjJkCH784x9jzJgxuOmmm/DXv/4VN910E5YvXx7ppvk1f/58VFVVYd26dZFuiouY7Pl46qmn8NBDD/ncZsCAAY5/nz17Frfffjtuuukm/OlPf/K5n8FgQHt7OxobG116P+rr62EwGIJptmRyzy9YAwYMQE5ODo4fP4477rhDseP6EspzFF+n+vp6GI1Gx+319fUYOXJkQMeUS+r5GQyGLomKly9fRkNDg6z327hx4wB09u4NHDhQdnuVkpOTg6SkpC6zw3x9fgwGg6ztIymQ83OXkpKCUaNG4fjx46FoYth5e/10Ol1c9Hp4M3bsWOzevTvSzfBpwYIFjiR2f71s4f4cxmTw0bNnT/Ts2VPStmfOnMHtt9+OMWPGYNWqVVCrfXf2jBkzBikpKdi6dSvuueceAJ0ZzqdOnUJhYWHQbZdCzvkp4fTp07hw4YLLhTrUQnmOeXl5MBgM2Lp1qyPYsFgs2Ldvn+xZQYGSen6FhYVobGzE/v37MWbMGADAtm3bYLfbHQGFFJWVlQAQ1tfQk9TUVIwZMwZbt27FjBkzAHR2+27duhULFizwuE9hYSG2bt2KJ5980nHb5s2bw/Z5kyOQ83Nns9lw6NAhTJkyJYQtDZ/CwsIuUzKj9fVTUmVlZcQ/b94IgoDHH38cGzZswI4dO5CXl+d3n7B/DkOSxholTp8+LQwaNEi44447hNOnTwt1dXWOP+dthgwZIuzbt89x22OPPSb07dtX2LZtm/D5558LhYWFQmFhYSROwa+TJ08KBw4cEEpKSoRu3boJBw4cEA4cOCA0NTU5thkyZIjw7rvvCoIgCE1NTcJPf/pToaysTKitrRW2bNkijB49Whg8eLDQ1tYWqdPwSe45CoIgvPzyy0JWVpawceNG4eDBg8L06dOFvLw8obW1NRKn4FNxcbEwatQoYd++fcLu3buFwYMHC7Nnz3bc7/4ePX78uPCLX/xC+Pzzz4Xa2lph48aNwoABA4RbbrklUqfgYt26dYJGoxHeeecdobq6Wnj00UeFrKwswWQyCYIgCA888IDw7LPPOrbfs2ePkJycLLz22mvCkSNHhKVLlwopKSnCoUOHInUKPsk9v5KSEuGTTz4RampqhP379wuzZs0StFqtcPjw4Uidgk9NTU2OzxgA4be//a1w4MAB4eTJk4IgCMKzzz4rPPDAA47t//vf/wrp6enC008/LRw5ckRYuXKlkJSUJJSWlkbqFPySe47Lly8X3nvvPeHYsWPCoUOHhCeeeEJQq9XCli1bInUKPs2bN0/Q6/XCjh07XK57LS0tjm0i/TmM6+Bj1apVAgCPf6La2loBgLB9+3bHba2trcJPfvIToXv37kJ6erpw9913uwQs0eTBBx/0eH7O5wNAWLVqlSAIgtDS0iLceeedQs+ePYWUlBShX79+wiOPPOL44oxGcs9REDqn2y5ZskTIzc0VNBqNcMcddwhHjx4Nf+MluHDhgjB79myhW7dugk6nEx5++GGXwMr9PXrq1CnhlltuEbKzswWNRiMMGjRIePrppwWz2RyhM+jq97//vdC3b18hNTVVGDt2rLB3717Hfbfeeqvw4IMPumz/z3/+U7jmmmuE1NRUYejQocKHH34Y5hbLI+f8nnzySce2ubm5wpQpU4SKiooItFoacVqp+594Tg8++KBw6623dtln5MiRQmpqqjBgwACXz2I0knuOr7zyijBw4EBBq9UK2dnZwm233SZs27YtMo2XwNt1z/l1ifTnUHWloURERERhEZOzXYiIiCh2MfggIiKisGLwQURERGHF4IOIiIjCisEHERERhRWDDyIiIgorBh9EREQUVgw+iIiIKKwYfBAREVFYMfggIiKisGLwQURERGHF4IOIiIjC6v8DxB/lL2H02PsAAAAASUVORK5CYII=\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Test data\n",
        "\n",
        "Generated from a deterministic distribution with Noise ~ Normal(0, 2)\n",
        "\n",
        "$$y_{test} = 2X^3 + 3X^2 + 4 + Noise$$"
      ],
      "metadata": {
        "id": "ffSQPM_4BQya"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "rng = np.random.default_rng(seed = 102)\n",
        "\n",
        "X_test = np.arange(-2, 2, 0.02).reshape(1, -1)\n",
        "\n",
        "y_test = (2 * (X_test)**3 + 3 * (X_test)**2 + 4 + rng.normal(0, 2, X_test.shape[1]).reshape(1, -1)).T"
      ],
      "metadata": {
        "id": "kACpsUAh8Xpx"
      },
      "execution_count": 5,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "X_test.shape, y_test.shape"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "7qbkXfUKwmF3",
        "outputId": "deb91cb4-4a94-4851-9753-c20caee218e4"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "((1, 200), (200, 1))"
            ]
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Model\n",
        "\n",
        "Linear regression model for the dataset ${\\mathbb{x}, y}$ is given as\n",
        "$$h_w(\\mathbb{x}) = w_1x^{1}+w_2x^{2}+...+w_dx^{d} =  \\mathbf{x}^Tw\n",
        "$$\n",
        "\n",
        "where $x^{i}$ is the $i^{th}$ feature of the data point $\\mathbf{x}$ and $w = [w_1, w_2, ...w_d]^T$ is the weight vector.\n",
        "\n",
        "\n",
        "Notice that above model always pass through the origin but for a given dataset, best fit model need not pass through the origin. To tackle this issue, we add an intercept $w_0$ in the model and set the corresponding featrue $x^{0}$ to $1$. That is\n",
        "\n",
        "$$h_w(\\mathbb{x}) =w_0x^{0}+ w_1x^{1}+w_2x^{2}+...+w_dx^{n} =  \\mathbf{x}^Tw\n",
        "$$\n",
        "\n",
        "We call $x^{0}$ the dummy feature and set its value to 1 for each examples. Now $w$ is of shape $(d+1, 1)$ and $\\mathbf{X}$ is of shape $(d+1, n)$ where the first row of $\\mathbf{X}$ has entries as 1.\n"
      ],
      "metadata": {
        "id": "EmK77bhCkKI-"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Add dummy feature"
      ],
      "metadata": {
        "id": "A156drNBk37x"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "dummy_feature = np.ones(X_train.shape[1])\n",
        "X = np.row_stack((dummy_feature, X_train))"
      ],
      "metadata": {
        "id": "h1SiEEmyCgCx"
      },
      "execution_count": 7,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "X.shape"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "jFzjXIybC6Ij",
        "outputId": "5abc9e85-5732-437c-face-1e0e36566501"
      },
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(2, 400)"
            ]
          },
          "metadata": {},
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Test data"
      ],
      "metadata": {
        "id": "ivJZNL_aC8_Q"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "dummy_feature_t = np.ones(X_test.shape[1])\n",
        "X_t = np.row_stack((dummy_feature_t, X_test))"
      ],
      "metadata": {
        "id": "vIZA0Ls6DA-3"
      },
      "execution_count": 9,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "X_t.shape"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "BhLm2D8iDXpn",
        "outputId": "003022f1-8edc-4e3c-e985-3a5d42581168"
      },
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(2, 200)"
            ]
          },
          "metadata": {},
          "execution_count": 10
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Loss function\n",
        "\n",
        "The total loss is the sum of square of errors between actual and predicted labels for each training point.  \n",
        "\n",
        "The error for $i$-th training point is calculated as follows:\n",
        "\\begin{eqnarray}\n",
        "e^{(i)} &=& (\\mathrm{{actual\\ label} - {predicted\\ label}})^2 \\\\\n",
        "        &=& \\left ({y_{i}} -\n",
        "        {h_{\\mathbf{w}}(\\mathbf{x}_{i})} \\right)^2 \\\\\n",
        "        &=& \\left ({y_{i}} -\n",
        "        {\\mathbf{w}^T \\mathbf{x}_{i}} \\right)^2\n",
        "\\end{eqnarray}\n",
        "\n",
        "\n",
        "\n",
        "The total loss $L(\\mathbf{w})$ is sum of errors at each training point:\n",
        "\\begin{equation}\n",
        "L(\\mathbf{w}) = \\sum_{i=1}^{n} e^{(i)}\n",
        "\\end{equation}\n",
        "\n",
        "We divide this by $\\frac{1}{2}$ for mathematical convenience in later use:\n",
        "\n",
        "\n",
        "\\begin{eqnarray}\n",
        "  L(\\mathbf{w}) &=& \\frac{1}{2} \\sum_{i=1}^{n} e^{(i)} \\\\ &=&  \\frac{1}{2} \\left({{\\mathbf{X}^T \\mathbf{w}}} - \\mathbf{y} \\right)^T \\left({{\\mathbf{X}^T \\mathbf{w}}} - \\mathbf{y} \\right))\n",
        "\\end{eqnarray}\n"
      ],
      "metadata": {
        "id": "Wp1v00V9lBMd"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Optimization\n",
        "\n"
      ],
      "metadata": {
        "id": "MQtN8F2j9r42"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 1. Normal equation\n",
        "\n",
        "Let's set $\\dfrac{\\partial L(\\mathbf{w})}{\\partial \\mathbf{w}}$ to 0 and solve for $\\mathbf{w}$:\n",
        "\\begin{eqnarray}\n",
        "\\frac{\\partial L(\\mathbf{w})}{\\partial \\mathbf{w}} &=& \\mathbf{X} \\mathbf{X}^T \\mathbf{w} - \\mathbf{X} \\mathbf{y} = 0 \\\\\n",
        "\\mathbf{X}^T \\mathbf{X} \\mathbf{w} &=& \\mathbf{X}^T \\mathbf{Y} \\\\\n",
        "\\mathbf{w} &=& \\left( \\mathbf{X} \\mathbf{X}^T \\right)^{-1} \\mathbf{X} \\mathbf{y}\n",
        "\\end{eqnarray}\n",
        "\n"
      ],
      "metadata": {
        "id": "GvxZvoJO-OIn"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Find weight vector and prediction for training and test dataset"
      ],
      "metadata": {
        "id": "gwcJYj2Ml71z"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "### weight\n",
        "def weights(X, y):\n",
        "    return np.linalg.pinv(X@(X.T))@X@y"
      ],
      "metadata": {
        "id": "UiO64nmAeSkz"
      },
      "execution_count": 11,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "w = weights(X, y_train)\n",
        "w"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "IUswb7ATD1l4",
        "outputId": "a51ed8e9-419d-4836-c55b-9bfbc47d66c4"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[7.93982825],\n",
              "       [4.79910241]])"
            ]
          },
          "metadata": {},
          "execution_count": 12
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Evaluation\n",
        "\n",
        "##### RMSE $$ \\sqrt{\\dfrac{1}{n}\\sum\\limits_{i=1}^{n} (y_i- \\hat{y}_i)^2}\n",
        "$$\n",
        "\n"
      ],
      "metadata": {
        "id": "Sk1KvxpCn1Xr"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "### Loss function\n",
        "def loss(y, y_hat):\n",
        "    return np.sqrt(np.mean((y - y_hat)**2))"
      ],
      "metadata": {
        "id": "zVMo6BF_ediw"
      },
      "execution_count": 13,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Kernel Regression\n",
        "\n"
      ],
      "metadata": {
        "id": "Nco8j8s1VOFa"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "* The optimal weight vector can be written as linear combination of data points.\n",
        "\n",
        "---\n",
        "\n",
        "Let $\\phi$ be the trainsformation mapping that trasforms the $d$-dimensional data points to $D$-dimension.\n",
        "\n",
        "$$\\phi: \\mathbb{R}^d \\rightarrow \\mathbb{R}^D$$\n",
        "\n",
        "Then\n",
        "\n",
        "$$\\mathbf{w} = \\phi(\\mathbf{X})\\alpha$$\n",
        "\n",
        "where,\n",
        "\n",
        "* $\\mathbf{X}$ = Feature matrix of shape $(d, n)$\n",
        "* $\\alpha = [\\alpha_1, \\alpha_2, ..., \\alpha_n]^T \\in \\mathbb{R}^n$\n",
        "\n"
      ],
      "metadata": {
        "id": "-lEj5azOBtAd"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Here\n",
        "\n",
        "$$\\alpha = K^{-1}y$$\n",
        "\n",
        "where,\n",
        "\\begin{equation}\n",
        " K_{n \\times n} = \\begin{bmatrix}\n",
        "       k(\\mathbf{x}_1,\\mathbf{x}_1) & k(\\mathbf{x}_1,\\mathbf{x}_2) &\\dots & k(\\mathbf{x}_1,\\mathbf{x}_n)\\\\\n",
        "        k(\\mathbf{x}_2,\\mathbf{x}_1) & k(\\mathbf{x}_2,\\mathbf{x}_2) &\\dots & k(\\mathbf{x}_2,\\mathbf{x}_n)\\\\\n",
        "        \\vdots & \\vdots & \\dots & \\vdots\\\\\n",
        "       k(\\mathbf{x}_n,\\mathbf{x}_1) & k(\\mathbf{x}_n,\\mathbf{x}_2) &\\dots & k(\\mathbf{x}_n,\\mathbf{x}_n)\\\\\n",
        "    \\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "Here, $k$ is the kernel function\n",
        "\n",
        "$$k: \\mathbb{R}^d \\times \\mathbb{R}^d \\rightarrow \\mathbb{R}\\\\ k(x_1, x_2) = \\phi(x_1)^T\\phi(x_2)\n",
        "$$"
      ],
      "metadata": {
        "id": "kp20Xt_GDT3B"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Prediction"
      ],
      "metadata": {
        "id": "__g386ueGBoH"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "For a test point $\\mathbf{x}_t$, prediction is given by\n",
        "\n",
        "$$\\sum\\limits_{i=1}^{n}k(\\mathbf{x}_i, \\mathbf{x}_t)\\alpha_i$$\n",
        "\n"
      ],
      "metadata": {
        "id": "WEkJCpdVGFQ3"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Polynomial kernel of degree $p$:\n",
        "\n",
        "\n",
        "$k(\\mathbf{x}_1, \\mathbf{x}_2) = (\\mathbf{x}_1^T\\mathbf{x}_2+1)^p$"
      ],
      "metadata": {
        "id": "AGPoE7q3Hx-d"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\\begin{equation}\n",
        "  \\mathbf{X} = \\begin{bmatrix}\n",
        "       | & | &\\dots & |\\\\\n",
        "       \\mathbf{x}_{1}&  \n",
        "       \\mathbf{x}_{2} &\n",
        "      \\dots &\n",
        "       \\mathbf{x}_{n}  \\\\\n",
        "        | & | &\\dots & |\\\\\n",
        "    \\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "\\begin{equation}\n",
        "  \\mathbf{X^TX} =\\begin{bmatrix}\n",
        "       | & | &\\dots & |\\\\\n",
        "       \\mathbf{x}_{1}&  \n",
        "       \\mathbf{x}_{2} &\n",
        "      \\dots &\n",
        "       \\mathbf{x}_{n}  \\\\\n",
        "        | & | &\\dots & |\\\\\n",
        "    \\end{bmatrix}^T\\begin{bmatrix}\n",
        "       | & | &\\dots & |\\\\\n",
        "       \\mathbf{x}_{1}&  \n",
        "       \\mathbf{x}_{2} &\n",
        "      \\dots &\n",
        "       \\mathbf{x}_{n}  \\\\\n",
        "        | & | &\\dots & |\\\\\n",
        "    \\end{bmatrix}\n",
        "    =  \\begin{bmatrix}\n",
        "       \\mathbf{x}_1^T\\mathbf{x}_{1} &  \\mathbf{x}_1^T\\mathbf{x}_{2}  &\\dots &  \\mathbf{x}_1^T\\mathbf{x}_{m} \\\\\n",
        "      \\mathbf{x}_2^T\\mathbf{x}_{1} &  \\mathbf{x}_2^T\\mathbf{x}_{2}  &\\dots &  \\mathbf{x}_2^T\\mathbf{x}_{m} \\\\\n",
        "        \\vdots & \\vdots &\\dots & \\vdots\\\\\n",
        "         \\mathbf{x}_n^T\\mathbf{x}_{1} &  \\mathbf{x}_n^T\\mathbf{x}_{2}  &\\dots &  \\mathbf{x}_n^T\\mathbf{x}_{m} \\\\\n",
        "    \\end{bmatrix}\n",
        "\\end{equation}\n"
      ],
      "metadata": {
        "id": "0VlFDad_Dloi"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Polynomial Kernel\n",
        "def poly_kernel(X, p):\n",
        "    return (X.T@X+1)**p"
      ],
      "metadata": {
        "id": "EhpPUAOuLNkp"
      },
      "execution_count": 14,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# coef vector alpha\n",
        "def coef(X_train, y,  p):\n",
        "    K = poly_kernel(X_train, p)\n",
        "    return (np.linalg.pinv(K))@y"
      ],
      "metadata": {
        "id": "fUTLgwU0TBj0"
      },
      "execution_count": 15,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### For prediction\n",
        "\n",
        "Let the training data matrix with $n$ examples is given by\n",
        " \\begin{equation}\n",
        "  \\mathbf{X}_{train} = \\begin{bmatrix}\n",
        "       | & | &\\dots & |\\\\\n",
        "       \\mathbf{x}_{1}&  \n",
        "       \\mathbf{x}_{2} &\n",
        "      \\dots &\n",
        "       \\mathbf{x}_{n}  \\\\\n",
        "        | & | &\\dots & |\\\\\n",
        "    \\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "and the test data matrix with $m$ examples is given by\n",
        " \\begin{equation}\n",
        "  \\mathbf{X}_{test} = \\begin{bmatrix}\n",
        "       | & | &\\dots & |\\\\\n",
        "       \\mathbf{x}_{t1}&  \n",
        "       \\mathbf{x}_{t2} &\n",
        "      \\dots &\n",
        "       \\mathbf{x}_{tm}  \\\\\n",
        "        | & | &\\dots & |\\\\\n",
        "    \\end{bmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "----\n",
        "\n",
        "Then\n",
        "\n",
        "\\begin{equation}\n",
        "  \\mathbf{X}_{train}^T \\mathbf{X}_{test}  = \\begin{bmatrix}\n",
        "       \\mathbf{x}_1^T\\mathbf{x}_{t1} &  \\mathbf{x}_1^T\\mathbf{x}_{t2}  &\\dots &  \\mathbf{x}_1^T\\mathbf{x}_{tm} \\\\\n",
        "      \\mathbf{x}_2^T\\mathbf{x}_{t1} &  \\mathbf{x}_2^T\\mathbf{x}_{t2}  &\\dots &  \\mathbf{x}_2^T\\mathbf{x}_{tm} \\\\\n",
        "        \\vdots & \\vdots &\\dots & \\vdots\\\\\n",
        "         \\mathbf{x}_n^T\\mathbf{x}_{t1} &  \\mathbf{x}_n^T\\mathbf{x}_{t2}  &\\dots &  \\mathbf{x}_n^T\\mathbf{x}_{tm} \\\\\n",
        "    \\end{bmatrix}\n",
        "\\end{equation}\n"
      ],
      "metadata": {
        "id": "TkjwkpKpOXxH"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Prediction\n",
        "def prediction(X_train, y_train, X_test, p):\n",
        "    alpha = coef(X_train, y_train, p)\n",
        "    K = (X_train.T @ X_test + 1)**p\n",
        "    return K.T @ alpha"
      ],
      "metadata": {
        "id": "72M-Xoiwe8Ew"
      },
      "execution_count": 16,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Training"
      ],
      "metadata": {
        "id": "m1Ljxa6qTmGA"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "y_hat = prediction(X_train, y_train, X_train, 3)"
      ],
      "metadata": {
        "id": "Ru9U9p80Tawj"
      },
      "execution_count": 17,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "loss(y_train, y_hat)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "dnjHvfqPTeX3",
        "outputId": "5b4178e7-9804-4f9f-e8fc-b8ebd272cfce"
      },
      "execution_count": 18,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.0061212525621974"
            ]
          },
          "metadata": {},
          "execution_count": 18
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Test"
      ],
      "metadata": {
        "id": "r4g2p56NVCoS"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "y_hat_t = prediction(X_train, y_train, X_test, 3)"
      ],
      "metadata": {
        "id": "H6taeBlrVD3P"
      },
      "execution_count": 19,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "loss(y_test, y_hat_t)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "BuL2gGmSVINT",
        "outputId": "775f49f3-6328-46a1-a70b-2717aa45d449"
      },
      "execution_count": 20,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "2.1103921105921506"
            ]
          },
          "metadata": {},
          "execution_count": 20
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### degree vs loss"
      ],
      "metadata": {
        "id": "5OnaPAgPU29n"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "### Plot degree vs loss\n",
        "degrees = [2, 3, 4, 5, 6]\n",
        "loss_train = []\n",
        "loss_test = []\n",
        "\n",
        "for d in degrees:\n",
        "    # Training\n",
        "    y_hat = prediction(X_train, y_train, X_train, d)\n",
        "    loss_train_iter = loss(y_train, y_hat)\n",
        "    loss_train.append(loss_train_iter)\n",
        "\n",
        "    # Testing\n",
        "    y_hat_t = prediction(X_train, y_train, X_test, d)\n",
        "    loss_test_iter = loss(y_test, y_hat_t)\n",
        "    loss_test.append(loss_test_iter)"
      ],
      "metadata": {
        "id": "70cn-98Se9lH"
      },
      "execution_count": 21,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "plt.plot(degrees, loss_train, label = \"Training loss\")\n",
        "plt.plot(degrees, loss_test, label = \"Test loss\")\n",
        "plt.legend();"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 430
        },
        "id": "VP46eOrsX_TS",
        "outputId": "14969d08-7b79-440e-8d44-c72f2726da64"
      },
      "execution_count": 22,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    }
  ]
}