Principal Component Analysis 1.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "6c51286a",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.decomposition import PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "506cc0b3",
   "metadata": {},
   "outputs": [],
   "source": [
    "x1 = np.array([1, 2, 3, 4, 5, 6])\n",
    "x2 = np.array([7.5, 11, 16, 18, 20, 26])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "cd5fab35",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x1, x2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "3a3b5062",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. ,  7.5],\n",
       "       [ 2. , 11. ],\n",
       "       [ 3. , 16. ],\n",
       "       [ 4. , 18. ],\n",
       "       [ 5. , 20. ],\n",
       "       [ 6. , 26. ]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = np.c_[x1, x2]\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "3874438d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-9.26049952e+00,  9.65263430e-03],\n",
       "       [-5.62047228e+00,  2.38560598e-02],\n",
       "       [-5.36562171e-01, -3.68391297e-01],\n",
       "       [ 1.65958219e+00,  5.22629108e-02],\n",
       "       [ 3.85572654e+00,  4.72917119e-01],\n",
       "       [ 9.90222524e+00, -1.90297426e-01]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pca = PCA()\n",
    "X_transformed = pca.fit_transform(X)\n",
    "X_transformed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "c7cdf891",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.27096719,  0.96258858],\n",
       "       [ 0.96258858, -0.27096719]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pca.components_"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2dbfd390",
   "metadata": {},
   "source": [
    "the above 2 results show the unit vector of one vs the other(perpendicular) direction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "0c6e3be7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-9.26049952],\n",
       "       [-5.62047228],\n",
       "       [-0.53656217],\n",
       "       [ 1.65958219],\n",
       "       [ 3.85572654],\n",
       "       [ 9.90222524]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pca = PCA(n_components = 1)\n",
    "X_transformed = pca.fit_transform(X)\n",
    "X_transformed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "2f58b69f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.27096719, 0.96258858]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pca.components_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "f97f1592",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.99070848,  7.50261555],\n",
       "       [ 1.97703643, 11.00646421],\n",
       "       [ 3.35460926, 15.90017805],\n",
       "       [ 3.94969232, 18.01416153],\n",
       "       [ 4.54477538, 20.12814502],\n",
       "       [ 6.18317813, 25.94843564]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_approx = pca.inverse_transform(X_transformed)\n",
    "X_approx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "c24894a8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X_approx[:, 0], X_approx[:, 1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f8fa1e1",
   "metadata": {},
   "source": [
    "the points actually lie on a line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "4816e9ee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(X_approx[:, 0], X_approx[:, 1])\n",
    "plt.show()"
   ]
  }
 ],
 "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": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}