notes/09-pca.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "c5c708d9",
   "metadata": {},
   "source": [
    "Today, we are going to get a little deeper into what is going on with PCA.  In fact, over the course of this notebook, we are going to implement it. First, let's do some feature selection warmups\n",
    "\n",
    "## Feature selection\n",
    "\n",
    "We will start off by loading a simple data set and then performing some simple approaches to selecting features"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "cf2956f8",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.datasets import load_linnerud\n",
    "linnerud_data = load_linnerud()\n",
    "        \n",
    "D = linnerud_data['data']"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6dc7ecc0",
   "metadata": {},
   "source": [
    "First let's check out some info about the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "7bf98cf5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['Chins', 'Situps', 'Jumps']"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linnerud_data['feature_names']"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e76c1e8",
   "metadata": {},
   "source": [
    "Super!  This data set has something to do with exercising (and is generally used for leaning about linear regression, but will be fine for our purposes).  Next, let's grab the data and look at some info about it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "32a008d8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(20, 3)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = linnerud_data['data']\n",
    "D.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "f919e882",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  5., 162.,  60.],\n",
       "       [  2., 110.,  60.],\n",
       "       [ 12., 101., 101.],\n",
       "       [ 12., 105.,  37.],\n",
       "       [ 13., 155.,  58.],\n",
       "       [  4., 101.,  42.],\n",
       "       [  8., 101.,  38.],\n",
       "       [  6., 125.,  40.],\n",
       "       [ 15., 200.,  40.],\n",
       "       [ 17., 251., 250.],\n",
       "       [ 17., 120.,  38.],\n",
       "       [ 13., 210., 115.],\n",
       "       [ 14., 215., 105.],\n",
       "       [  1.,  50.,  50.],\n",
       "       [  6.,  70.,  31.],\n",
       "       [ 12., 210., 120.],\n",
       "       [  4.,  60.,  25.],\n",
       "       [ 11., 230.,  80.],\n",
       "       [ 15., 225.,  73.],\n",
       "       [  2., 110.,  43.]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e483f34a",
   "metadata": {},
   "source": [
    "We saw that one way we can quantify the useful featuers of our data is by variance.  So, let's look at the estimated covariance matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "3cb5a98f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  27.94473684,  230.10789474,  134.38421053],\n",
       "       [ 230.10789474, 3914.57631579, 2146.98421053],\n",
       "       [ 134.38421053, 2146.98421053, 2629.37894737]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "np.cov(D.T, ddof=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "86f2277b",
   "metadata": {},
   "source": [
    "Great!  so let's keep only the meaningful attributes.  I wasn't sure how to tell what a good cutoff would be so I asked Prof Barktreuse.  She exercises a ton (and infact go a little nuts if she doesn't get at least 30 minutes of exercise per day).  She was suspect of my idea, but said to try 100 as the variance theshold."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "d945aced",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[162.,  60.],\n",
       "       [110.,  60.],\n",
       "       [101., 101.],\n",
       "       [105.,  37.],\n",
       "       [155.,  58.],\n",
       "       [101.,  42.],\n",
       "       [101.,  38.],\n",
       "       [125.,  40.],\n",
       "       [200.,  40.],\n",
       "       [251., 250.],\n",
       "       [120.,  38.],\n",
       "       [210., 115.],\n",
       "       [215., 105.],\n",
       "       [ 50.,  50.],\n",
       "       [ 70.,  31.],\n",
       "       [210., 120.],\n",
       "       [ 60.,  25.],\n",
       "       [230.,  80.],\n",
       "       [225.,  73.],\n",
       "       [110.,  43.]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.feature_selection import VarianceThreshold\n",
    "\n",
    "threshold = 100\n",
    "feature_selector = VarianceThreshold(threshold=threshold)\n",
    "feature_selector.fit_transform(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6663dbb",
   "metadata": {},
   "source": [
    "So it looks like we got the features in the situps and jumps columns.\n",
    "\n",
    "I asked Prof Barktreuse why she was suspect, and she said that I have not taken scale into account. ... Of course... Prof Barktrues saved the day again. Let's min-max scale our data and try again with a threashold of .6."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "58f46d4f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.25      , 0.55721393, 0.15555556],\n",
       "       [0.0625    , 0.29850746, 0.15555556],\n",
       "       [0.6875    , 0.25373134, 0.33777778],\n",
       "       [0.6875    , 0.27363184, 0.05333333],\n",
       "       [0.75      , 0.52238806, 0.14666667],\n",
       "       [0.1875    , 0.25373134, 0.07555556],\n",
       "       [0.4375    , 0.25373134, 0.05777778],\n",
       "       [0.3125    , 0.37313433, 0.06666667],\n",
       "       [0.875     , 0.74626866, 0.06666667],\n",
       "       [1.        , 1.        , 1.        ],\n",
       "       [1.        , 0.34825871, 0.05777778],\n",
       "       [0.75      , 0.7960199 , 0.4       ],\n",
       "       [0.8125    , 0.82089552, 0.35555556],\n",
       "       [0.        , 0.        , 0.11111111],\n",
       "       [0.3125    , 0.09950249, 0.02666667],\n",
       "       [0.6875    , 0.7960199 , 0.42222222],\n",
       "       [0.1875    , 0.04975124, 0.        ],\n",
       "       [0.625     , 0.89552239, 0.24444444],\n",
       "       [0.875     , 0.87064677, 0.21333333],\n",
       "       [0.0625    , 0.29850746, 0.08      ]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.preprocessing import MinMaxScaler\n",
    "D_normalized = MinMaxScaler().fit_transform(D)\n",
    "D_normalized"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "dd2bbfb3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.10915913, 0.07155096, 0.03732895],\n",
       "       [0.07155096, 0.09689306, 0.04747339],\n",
       "       [0.03732895, 0.04747339, 0.05193835]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "np.cov(D_normalized.T, ddof=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "e892d564",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.25      , 0.55721393],\n",
       "       [0.0625    , 0.29850746],\n",
       "       [0.6875    , 0.25373134],\n",
       "       [0.6875    , 0.27363184],\n",
       "       [0.75      , 0.52238806],\n",
       "       [0.1875    , 0.25373134],\n",
       "       [0.4375    , 0.25373134],\n",
       "       [0.3125    , 0.37313433],\n",
       "       [0.875     , 0.74626866],\n",
       "       [1.        , 1.        ],\n",
       "       [1.        , 0.34825871],\n",
       "       [0.75      , 0.7960199 ],\n",
       "       [0.8125    , 0.82089552],\n",
       "       [0.        , 0.        ],\n",
       "       [0.3125    , 0.09950249],\n",
       "       [0.6875    , 0.7960199 ],\n",
       "       [0.1875    , 0.04975124],\n",
       "       [0.625     , 0.89552239],\n",
       "       [0.875     , 0.87064677],\n",
       "       [0.0625    , 0.29850746]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "threshold = .06\n",
    "feature_selector = VarianceThreshold(threshold=threshold)\n",
    "feature_selector.fit_transform(D_normalized)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e0bda82d",
   "metadata": {},
   "source": [
    "And this time, it looks like we got the chins and situps features.  So, this demonstrates when performing features selection using variance, we have to be careful as the scale of the fields of the data can matter.\n",
    "\n",
    "## PCA\n",
    "Recall from last time, the PCA algorithm:\n",
    "\n",
    "![pca-algo.png](09-imgs/pca-algo.png)\n",
    "\n",
    "We will walk though some background and then implement the algo.  We will start by loading in the iris data set"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "5a768d39",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.datasets import load_iris\n",
    "D_iris = load_iris()['data']"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f29c877",
   "metadata": {},
   "source": [
    "And lets get some info about the data and take a look at a projection of the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "cadbd212",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(150, 4)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D_iris.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "52444db2",
   "metadata": {},
   "source": [
    "Let's just grab 2 dimensions of the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "a11dbd3e",
   "metadata": {},
   "outputs": [],
   "source": [
    "D = D_iris[:, 1:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "01fbb775",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Scatter plot of data matrix D_iris')"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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nvxRBqR7pqCozS0OqZ2Lsu6p6pAgK/NJRVXLQIZVDMfZdz+iVIijVIx2VOQed9WZwWSuHYu27ntErsemKXzoq68zSGA88Keu+i8yUAr90VNYcdIwHnpR130VmSqke6aisD3yJ8cCTsu67yEwp8Mu0rh/b+GzJ4fjEJNePbewY/GI8bFwPPBEJp1SPdHThVXdyx/3b9mq74/5tXHjVnS2Xj/Ww8VNf9PxM7SGlmVUpZRWZSoFfOpoa9Kdrj/Ww8Qd/27qypl17SGlmVUpZRabKNdVjZg8CO4E9wNPuPpJnf3kpa4ogRkolq1gPG89aahlSmlnmUlaRmYhxxf8adz+xn4N+Gd/ux0qpZBXyIJYQWR+sElKaqXJOqSqleqZR1rf7sVIqpx97RKb2kAexhMj6YJWQ0kyVc0pV5R34HfiRma0xs6WtFjCzpWY2ZmZjW7duzXk42ZX17X6slMq1F53GS1548F5tL3nhwVx70Wktl7988cJ9XhROP/aInqegsj5YJeSBJ7qdglRV3uWcp7v7FjN7IXCzma1399uaF3D3FcAKgJGRkYIfhLevst49cT+DZ1ocrf16fH/50bXjbH7sib3aNj/2BKNrx1sGwNG149y9cftebXdv3N52+VBZH6wS+sAT3U5BqijXK35335L++wjwbeDkPPvLQ1nf7h+4f+tT1649VNZUV6zUWNYHq5Q1ZSdShNwCv5kdbGaHNv4PvAH4eV795aWsb/ef2P1MpvZQMapnQmR9sEpZU3YiRcgz1XMU8G1LLsH2B77u7j/Msb/clPHtfqwUVNZ+qjauspbyisxEblf87v6Au788/Xqpu38qr77qKFYKKms/C17QOpC2aw/1muNnZ2ov64NYRIqgcs4+FSsFlbWfVQ88lqk91C3rW1eAtWsv64NYRIqgm7T1sVgpqCz9hJaZZk2pxMjZh/ah9JCUnQK/9FRImWkjpdK4um6kVIC2ATNrzj6kj6GDZvFYiw+L280ODu1HJDaleqSnQspMQ1IqWXP2IX1knR0c2o9IbAr80lMhZaYhKZWsM3FD+sg6Ozi0H5HYFPilp7LePC10nXYzcdtV3MS6SZtu7Cb9QIFfeiokPRIjpRLrJm1lnekt0kwf7kpPhaRHYqRUQp6fG2sdkdgU+LtQ1vK8Mo4rZIZsrHXKWP4K5TyPUm1K9UyjrLM3yzqurDNqQ9eJMUM4xjEu63mUalPgn0ZZy/PKOq6sM2pD14kxQzjGMS7reZRqU6pnGmWdvVmlcYWsEzJDuEqzg0VmQlf80wgpz4vx9r1K4wpZJ+v9+EP2PUZppso/pQgK/NMIKc+L8fa9SuMKWWewzUzgdu0xZgeHUPmnFEGpnmmElOfFePtepXEtXjTM9WMbueP+bc+2vWL+4R3XmWwzE7hde+jsYMi3NFPln1IEBf4uZC3Pi/UwkqqMa9nour2CPsAd929j2ei6tg9pj/UglhgloGV80I9Um1I9OSjr2/eyjuu61ZsytUP2fSnrvosUQVf8OSjr2/eyjiukQifrvpR130WKoMCfk7K+fQ8ZV94loANmLYP8QLsSndTYhm08vP0JHHh4+xOMbdg27WcJZTwnIrEp1SMdxSgBPfVFz8/UDsnnAitXbdzr7pwrV21k2ei6no1LpKoU+KWjGCWgD/62dWVNu3YI+1xARBJK9UhHIWWQMWbIhj7bV0R0xS/TyPqQlFgzZLPO3BWR5yjwS0dZH5ISa4Zs1pm7IvKc3FM9ZjYAjAHj7n5e3v3lIaSqJcY91mOMK+tDUmLNkM06c1dEnhMjx38xcC9wWIS+eq6RumhcxTZSF0DbwBSyTlnHlXXG69BBs3js8X1fFDo9P7fRfxlnIYtUUcf3xWZ2mJkd26L9Zd1s3MzmAm8Erg4bXvFCUhdlvY97jDRMyPNzQ2gmrki4toHfzP4AWA/cYGa/MLNXNv34mi63fyXwEaDt+28zW2pmY2Y2tnVr+wdvFCXW/eWzijWuxYuGWX7+QoaHBjFgeGiQ5ecvbHt1HvL83BBZxyUiz+mU6rkMOMndHzKzk4G/M7PL3P1GYNraCTM7D3jE3deY2RntlnP3FcAKgJGRkdLV4sV6HmyZx5VlhmzMFIxm4oqE6ZTq2d/dHwJw97uA1wB/amYfALoJ0KcDbzazB4FvAGea2coZjje6WPeXzyrkObUh48o6Q1YpGJHy6xT4dzTn99MXgTOA3wdeOt2G3f1Sd5/r7guAC4Afu/vbZjbc+EJSCjHSECHPqQ0ZV9YZskrBiJRfp1TPR5mS0nH3nWZ2NnBprqMqmZCUQt43Qwv9HCHruELvnKlAL1Jena74vwq8xcyefXEws6OArwBvytKJu9/arzX8sWSd8RrrWa3t7pA53Z0zRaS8OgX+k4B/Dqw1szPN7GLgLuBO4JQYg6uTrKWWsXLpS06Zl6ldRMqvbarH3R8D3psG/P8NbAFOdffNsQZXJ1lTN7EeLNJ49OF1qzexx50BM5acMq/tIxFFpPzaBn4zGwI+TXJ1fzZwLvADM7vY3X8cZ3j1cfjgLCZa1LofPth+xmusXPrlixcq0ItUSKcPd+8GvgD8sbs/DfzIzE4EvmBmG9x9SYwB1oXuNikisXQK/L83Na3j7vcA/9rMLsp1VDU00eL+Np3aRURCdcrxt83lu/tV+Qynvup+07EYdzMVkYRuXl4SdZ7xGuO5viLyHAX+kqjzjNcYdzMVkef0/TN3q5QiiFGlU8bjFeNupiLynL4O/DEeeFIlZT1edf98QyS2vk71KEWQTVmPV50/3xApQl9f8StFkE1Zj1esWcgikujrwK8UQTZlPl66o6dIPH2d6lGKIBsdLxGBPr/iV4ogGx0vEQEw7/BAjdhGRkZ8bGys6GH0REjZZBlLLUWk3MxsjbuPZFmnr6/4yyqkbLKspZYiUj19neMvq5CyybKWWopI9Sjw5yCkbLKspZYiUj0K/DkIeR5urGfoiogo8OcgpGxSpZYiEos+3M1BSNmkSi1FJBaVc4qI9LGQck6lekREakaBX0SkZnLL8ZvZ84DbgAPTfr7l7h/Pq7860kxfEQmR54e7TwJnuvsuM5sF3G5mP3D3VTn2WRua6SsioXJL9XhiV/rtrPSrPJ8k9znN9BWRULnm+M1swMzuAR4Bbnb31S2WWWpmY2Y2tnXr1jyHUyma6SsioXIN/O6+x91PBOYCJ5vZCS2WWeHuI+4+Mnv27DyHUyma6SsioaJU9bj7BHArcHaM/upAM31FJFRugd/MZpvZUPr/QeB1wPq8+qubxYuGWX7+QoaHBjFgeGiQ5ecv1Ae7IjKtPKt6jga+amYDJC8w33T37+bYX+3oObUiEiK3wO/uPwMW5bV9EREJo5m7IiI1o8AvIlIzCvwiIjWjwC8iUjMK/CIiNaPALyJSMwr8IiI1o8AvIlIzCvwiIjWjwC8iUjMK/CIiNaPALyJSMwr8IiI1o8AvIlIzCvwiIjWjwC8iUjMK/CIiNaPALyJSMwr8IiI1o8AvIlIzCvwiIjWjwC8iUjMK/CIiNaPALyJSM7kFfjObZ2a3mNm9ZvYLM7s4r75ERKR7++e47aeBD7n73WZ2KLDGzG5291/m2KeIiEwjtyt+d3/I3e9O/78TuBcYzqs/ERHpTpQcv5ktABYBq1v8bKmZjZnZ2NatW2MMR0Sk1vJM9QBgZocANwAfdPcdU3/u7iuAFQAjIyOe93hEymZ07ThX3HQfWyYmmTM0yCVnHcfiRf355rhK+1JluQZ+M5tFEvSvdfcb8+xLpB+Nrh3n0hvXMbl7DwDjE5NceuM6gL4LmFXal6rLs6rHgC8D97r7Z/PqR6SfXXHTfc8GyobJ3Xu44qb7ChpRuCrtS9XlmeM/HXg7cKaZ3ZN+nZtjfyJ9Z8vEZKb2MqvSvlRdbqked78dsLy2L1IFc4YGGW8RGOcMDRYwmpmp0r5UnWbuihTokrOOY3DWwF5tg7MGuOSs4woaUbgq7UvV5V7VI1I3WSpbFi8aZmzDNq5bvYk97gyY8ZaThvvyw9DQfVElUHy64hfpoUZly/jEJM5zlS2ja8fbLn/DmnH2eFLJvMedG9aMt12+zEL2Jevxkt5Q4BfpoayVLVWqhAnZlyrtfz9R4BfpoayVLVWqhAnZlyrtfz9Rjl/6VtbccEguOes6WStbQithypgXD9kXVQIVQ1f80pdCculZc8kh62StbFnwgtYBrl176LhieM3xszO1gyqBiqLAL30pRi49ZJ3Fi4ZZfv5ChocGMWB4aJDl5y9sezW+6oHHMrWHjiuGW9a3vsliu3bIfrykN5Tqkb4UI5cemn9evKj7csxGBUy37TMZV1YXXnUnd9y/7dnvTz/2CK696LSejyvL8QpVxtRYkePSFb/0pU4581ZmDbSeRN6uPaSPWIYOmpWpPcTUoA9wx/3buPCqOwsdV4iypsaKHJcCv/SlrLnhp/a0voJu1x7SRyzt3gx0eJOQ2dSgP117p/57Oa4QZU2NFTkupXqkLzXeDuf5NjlGHyG2T+7O1B5LWcdV1pLRIselwC+lEJLrvH5s47OlgOMTk1w/trHwoAz5521DSyCXja7b63YKS06Zx+WLFxY+rryVtWS2yOOlVI8ULiTXmTUHffqxR2RqDx1X1nWOOvSATO0QVgK6bHQdK1dt3Ot2CitXbWTZ6LqWy4ccr7KmxkLGFSP/XuTxUuCXwoXkOrPmoK+96LR9gtZ0VSoxSkAf3dU6DdKuHcJKQK9bvSlTe8jxKmtpZsi4YuTfizxeSvXkpKzlY2UUK9fZKWhl6b+XJaAh5Zyx1sl6vCCsNDPG30rWccX6nYxRytqKrvhzUNbysbIqa9nkQQcMZGqH7PsyYK3LSdu1h4rVT1Zl/Vsp6+9kryjw56Cs5WNlFZLrPKBN/X279hCPP7UnUztk35clp8zL1A7tH2vXac9D+omhrH8rZf28oleU6slBWcvHyiqkbHJ3m/r7du0h2m2pUw9Z96VRVZOl2iZkXCH9xBDytxIrNQTlK+XtFQX+HJS1rK3MsuY6Yxxjs9aTj6bLjmTdl8sXL8wUgAfMWubmp0vbZO0nhqznsZEaarxLaKSGgMI/F+gnSvXkoOpvE8sgxjEe3L/1n0e79ljKmrYJkfU8ljU11G90xZ+Dqr9NLIMYx3hy9zOZ2mMpa9omRNbzqDRqbyjw56TKbxProswpuzKmbUJl+Vsp8znpJ0r1SF+q+sxKaU3npDcU+KUvVX1mpbSmc9IbuaV6zOxvgfOAR9z9hLz6kXqq+szKPFRlNnmVzklR8rzivwY4O8ftS41VfWZlr5V1hqwUI7fA7+63Ae2f2iAyA8r1ZqMySGlWeFWPmS0FlgLMnz+/4NFIv1DJbDYqg5RmhQd+d18BrAAYGRkp+CFt0k+U6+2eyiClmap6RGpAqTFpVvgVv4jkT6kxaZZnOed1wBnAkWa2Gfi4u385r/5EpDOlxqQht8Dv7kvy2raIiIRTjl9EpGYU+EVEakaBX0SkZhT4RURqxrzVs+UKYmZbgQ2Bqx8JPNrD4fSTOu871Hv/te/11dj/Y9x9dpYVSxX4Z8LMxtx9pOhxFKHO+w713n/tez33HWa2/0r1iIjUjAK/iEjNVCnwryh6AAWq875Dvfdf+15fwftfmRy/iIh0p0pX/CIi0gUFfhGRmumrwG9m88zsFjO718x+YWYXt1jGzOx/mNmvzOxnZvaKIsbaa13u+xlmtt3M7km//qyIsfaamT3PzO4ys39K9/2TLZap5HmHrve/kue+wcwGzGytmX23xc8qe+5h2n0POu/9dj/+p4EPufvdZnYosMbMbnb3XzYtcw7wkvTrFOCL6b/9rpt9B/ipu59XwPjy9CRwprvvMrNZwO1m9gN3X9W0TFXPO3S3/1DNc99wMXAvcFiLn1X53EPnfYeA895XV/zu/pC7353+fyfJwZh6g/HfB77miVXAkJkdHXmoPdflvldSei53pd/OSr+mViVU8rxD1/tfWWY2F3gjcHWbRSp77rvY9yB9FfibmdkCYBGwesqPhoFNTd9vpmIBssO+A5yWpgR+YGYvjTuy/KRvd+8BHgFudvdanfcu9h8qeu6BK4GPAM+0+XmVz/2VdN53CDjvfRn4zewQ4Abgg+6+Y+qPW6xSmaujafb9bpL7drwc+GtgNPLwcuPue9z9RGAucLKZnTBlkUqf9y72v5Ln3szOAx5x9zWdFmvR1vfnvst9DzrvfRf40xznDcC17n5ji0U2A/Oavp8LbIkxtrxNt+/uvqOREnD37wOzzOzIyMPMlbtPALcCZ0/5UWXPe7N2+1/hc3868GYzexD4BnCmma2cskxVz/20+x563vsq8JuZAV8G7nX3z7ZZ7DvAO9JP+k8Ftrv7Q9EGmZNu9t3M/lm6HGZ2Msn5/W28UebDzGab2VD6/0HgdcD6KYtV8rxDd/tf1XPv7pe6+1x3XwBcAPzY3d82ZbFKnvtu9j30vPdbVc/pwNuBdWm+E+AyYD6Au38J+D5wLvAr4HHgXfGHmYtu9v3fAe8zs6eBSeACr8bU7KOBr5rZAMkv9jfd/btm9l6o/HmH7va/que+pRqd+3304rzrlg0iIjXTV6keERGZOQV+EZGaUeAXEakZBX4RkZpR4BcRqRkFfpEmltwF9ddmdkT6/fPT748xsx+a2USruySK9BMFfpEm7r6J5O6Of5k2/SWwwt03AFeQzKUQ6WsK/CL7+u/AqWb2QeBVwF8BuPs/ADsLHJdIT/TbzF2R3Ln7bjO7BPgh8AZ3f6roMYn0kq74RVo7B3gImHoXTJG+p8AvMoWZnQi8HjgV+E9VeaiHSIMCv0iT9E6HXyR53sFGkg90P1PsqER6S4FfZG8XARvd/eb0+y8Ax5vZq83sp8D1wGvNbLOZnVXYKEVmQHfnFBGpGV3xi4jUjAK/iEjNKPCLiNSMAr+ISM0o8IuI1IwCv4hIzSjwi4jUzP8HbJbcbBNTjO0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.scatter(D[:,0], D[:,1])\n",
    "plt.xlabel('X1')\n",
    "plt.ylabel('X2')\n",
    "plt.title('Scatter plot of data matrix D_iris')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "21a82f1f",
   "metadata": {},
   "source": [
    "### Mean centering\n",
    "\n",
    "Recall that the first thing we needed to do to our data was mean center it.  We have done that a few times, but, for completeness, let's mean center our data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "b6b7f91c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3.05733333, 3.758     ])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multi_d_mean = np.mean(D, axis=0)\n",
    "multi_d_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "e37e6c0f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.44266667, -2.358     ],\n",
       "       [-0.05733333, -2.358     ],\n",
       "       [ 0.14266667, -2.458     ],\n",
       "       [ 0.04266667, -2.258     ],\n",
       "       [ 0.54266667, -2.358     ],\n",
       "       [ 0.84266667, -2.058     ],\n",
       "       [ 0.34266667, -2.358     ],\n",
       "       [ 0.34266667, -2.258     ],\n",
       "       [-0.15733333, -2.358     ],\n",
       "       [ 0.04266667, -2.258     ],\n",
       "       [ 0.64266667, -2.258     ],\n",
       "       [ 0.34266667, -2.158     ],\n",
       "       [-0.05733333, -2.358     ],\n",
       "       [-0.05733333, -2.658     ],\n",
       "       [ 0.94266667, -2.558     ],\n",
       "       [ 1.34266667, -2.258     ],\n",
       "       [ 0.84266667, -2.458     ],\n",
       "       [ 0.44266667, -2.358     ],\n",
       "       [ 0.74266667, -2.058     ],\n",
       "       [ 0.74266667, -2.258     ],\n",
       "       [ 0.34266667, -2.058     ],\n",
       "       [ 0.64266667, -2.258     ],\n",
       "       [ 0.54266667, -2.758     ],\n",
       "       [ 0.24266667, -2.058     ],\n",
       "       [ 0.34266667, -1.858     ],\n",
       "       [-0.05733333, -2.158     ],\n",
       "       [ 0.34266667, -2.158     ],\n",
       "       [ 0.44266667, -2.258     ],\n",
       "       [ 0.34266667, -2.358     ],\n",
       "       [ 0.14266667, -2.158     ],\n",
       "       [ 0.04266667, -2.158     ],\n",
       "       [ 0.34266667, -2.258     ],\n",
       "       [ 1.04266667, -2.258     ],\n",
       "       [ 1.14266667, -2.358     ],\n",
       "       [ 0.04266667, -2.258     ],\n",
       "       [ 0.14266667, -2.558     ],\n",
       "       [ 0.44266667, -2.458     ],\n",
       "       [ 0.54266667, -2.358     ],\n",
       "       [-0.05733333, -2.458     ],\n",
       "       [ 0.34266667, -2.258     ],\n",
       "       [ 0.44266667, -2.458     ],\n",
       "       [-0.75733333, -2.458     ],\n",
       "       [ 0.14266667, -2.458     ],\n",
       "       [ 0.44266667, -2.158     ],\n",
       "       [ 0.74266667, -1.858     ],\n",
       "       [-0.05733333, -2.358     ],\n",
       "       [ 0.74266667, -2.158     ],\n",
       "       [ 0.14266667, -2.358     ],\n",
       "       [ 0.64266667, -2.258     ],\n",
       "       [ 0.24266667, -2.358     ],\n",
       "       [ 0.14266667,  0.942     ],\n",
       "       [ 0.14266667,  0.742     ],\n",
       "       [ 0.04266667,  1.142     ],\n",
       "       [-0.75733333,  0.242     ],\n",
       "       [-0.25733333,  0.842     ],\n",
       "       [-0.25733333,  0.742     ],\n",
       "       [ 0.24266667,  0.942     ],\n",
       "       [-0.65733333, -0.458     ],\n",
       "       [-0.15733333,  0.842     ],\n",
       "       [-0.35733333,  0.142     ],\n",
       "       [-1.05733333, -0.258     ],\n",
       "       [-0.05733333,  0.442     ],\n",
       "       [-0.85733333,  0.242     ],\n",
       "       [-0.15733333,  0.942     ],\n",
       "       [-0.15733333, -0.158     ],\n",
       "       [ 0.04266667,  0.642     ],\n",
       "       [-0.05733333,  0.742     ],\n",
       "       [-0.35733333,  0.342     ],\n",
       "       [-0.85733333,  0.742     ],\n",
       "       [-0.55733333,  0.142     ],\n",
       "       [ 0.14266667,  1.042     ],\n",
       "       [-0.25733333,  0.242     ],\n",
       "       [-0.55733333,  1.142     ],\n",
       "       [-0.25733333,  0.942     ],\n",
       "       [-0.15733333,  0.542     ],\n",
       "       [-0.05733333,  0.642     ],\n",
       "       [-0.25733333,  1.042     ],\n",
       "       [-0.05733333,  1.242     ],\n",
       "       [-0.15733333,  0.742     ],\n",
       "       [-0.45733333, -0.258     ],\n",
       "       [-0.65733333,  0.042     ],\n",
       "       [-0.65733333, -0.058     ],\n",
       "       [-0.35733333,  0.142     ],\n",
       "       [-0.35733333,  1.342     ],\n",
       "       [-0.05733333,  0.742     ],\n",
       "       [ 0.34266667,  0.742     ],\n",
       "       [ 0.04266667,  0.942     ],\n",
       "       [-0.75733333,  0.642     ],\n",
       "       [-0.05733333,  0.342     ],\n",
       "       [-0.55733333,  0.242     ],\n",
       "       [-0.45733333,  0.642     ],\n",
       "       [-0.05733333,  0.842     ],\n",
       "       [-0.45733333,  0.242     ],\n",
       "       [-0.75733333, -0.458     ],\n",
       "       [-0.35733333,  0.442     ],\n",
       "       [-0.05733333,  0.442     ],\n",
       "       [-0.15733333,  0.442     ],\n",
       "       [-0.15733333,  0.542     ],\n",
       "       [-0.55733333, -0.758     ],\n",
       "       [-0.25733333,  0.342     ],\n",
       "       [ 0.24266667,  2.242     ],\n",
       "       [-0.35733333,  1.342     ],\n",
       "       [-0.05733333,  2.142     ],\n",
       "       [-0.15733333,  1.842     ],\n",
       "       [-0.05733333,  2.042     ],\n",
       "       [-0.05733333,  2.842     ],\n",
       "       [-0.55733333,  0.742     ],\n",
       "       [-0.15733333,  2.542     ],\n",
       "       [-0.55733333,  2.042     ],\n",
       "       [ 0.54266667,  2.342     ],\n",
       "       [ 0.14266667,  1.342     ],\n",
       "       [-0.35733333,  1.542     ],\n",
       "       [-0.05733333,  1.742     ],\n",
       "       [-0.55733333,  1.242     ],\n",
       "       [-0.25733333,  1.342     ],\n",
       "       [ 0.14266667,  1.542     ],\n",
       "       [-0.05733333,  1.742     ],\n",
       "       [ 0.74266667,  2.942     ],\n",
       "       [-0.45733333,  3.142     ],\n",
       "       [-0.85733333,  1.242     ],\n",
       "       [ 0.14266667,  1.942     ],\n",
       "       [-0.25733333,  1.142     ],\n",
       "       [-0.25733333,  2.942     ],\n",
       "       [-0.35733333,  1.142     ],\n",
       "       [ 0.24266667,  1.942     ],\n",
       "       [ 0.14266667,  2.242     ],\n",
       "       [-0.25733333,  1.042     ],\n",
       "       [-0.05733333,  1.142     ],\n",
       "       [-0.25733333,  1.842     ],\n",
       "       [-0.05733333,  2.042     ],\n",
       "       [-0.25733333,  2.342     ],\n",
       "       [ 0.74266667,  2.642     ],\n",
       "       [-0.25733333,  1.842     ],\n",
       "       [-0.25733333,  1.342     ],\n",
       "       [-0.45733333,  1.842     ],\n",
       "       [-0.05733333,  2.342     ],\n",
       "       [ 0.34266667,  1.842     ],\n",
       "       [ 0.04266667,  1.742     ],\n",
       "       [-0.05733333,  1.042     ],\n",
       "       [ 0.04266667,  1.642     ],\n",
       "       [ 0.04266667,  1.842     ],\n",
       "       [ 0.04266667,  1.342     ],\n",
       "       [-0.35733333,  1.342     ],\n",
       "       [ 0.14266667,  2.142     ],\n",
       "       [ 0.24266667,  1.942     ],\n",
       "       [-0.05733333,  1.442     ],\n",
       "       [-0.55733333,  1.242     ],\n",
       "       [-0.05733333,  1.442     ],\n",
       "       [ 0.34266667,  1.642     ],\n",
       "       [-0.05733333,  1.342     ]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "centered_data = D - multi_d_mean\n",
    "centered_data"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6c1ff587",
   "metadata": {},
   "source": [
    "Next, let's plot the original data and the mean centered data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "c45b70c9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Scatter plot of centered and original data')"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111)\n",
    "ax.scatter(D[:,0], D[:,1], s=10, c='b', marker='s', label='original data')\n",
    "ax.scatter(centered_data[:,0], centered_data[:,1], s=10, c='r', marker='x', label='centered data')\n",
    "plt.xlabel('X1')\n",
    "plt.ylabel('X2')\n",
    "plt.legend(loc='upper left')\n",
    "plt.title('Scatter plot of centered and original data')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "465b03ba",
   "metadata": {},
   "source": [
    "**Question**: *What do you think the mean of the centered data is?*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "9bd9fd09",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-7.60872846e-16, -2.55203266e-15])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(centered_data, axis = 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bba59786",
   "metadata": {},
   "source": [
    "Yep, that is numerically zero!  \n",
    "\n",
    "### Linearly Transorming Data\n",
    "\n",
    "Next, let's recall linear transforamtions as a rotaton and a scaling.  Let's pick an arbitrary linear transormation and move some points around."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "d7610a0f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2,  1],\n",
       "       [ 1, -1]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[2,1],[1,-1]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "2c1ffe9e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3.5, 1.4])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D[0,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "fbfe7115",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([8.4, 2.1])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dot(D[0,:])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "39114548",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3. , 1.4])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D[1,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "2d0a38c5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([7.4, 1.6])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dot(D[1,:])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "63eec32e",
   "metadata": {},
   "source": [
    "Great!  So let's transform all the data in one go!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "11f94f60",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 8.4,  2.1],\n",
       "       [ 7.4,  1.6],\n",
       "       [ 7.7,  1.9],\n",
       "       [ 7.7,  1.6],\n",
       "       [ 8.6,  2.2],\n",
       "       [ 9.5,  2.2],\n",
       "       [ 8.2,  2. ],\n",
       "       [ 8.3,  1.9],\n",
       "       [ 7.2,  1.5],\n",
       "       [ 7.7,  1.6],\n",
       "       [ 8.9,  2.2],\n",
       "       [ 8.4,  1.8],\n",
       "       [ 7.4,  1.6],\n",
       "       [ 7.1,  1.9],\n",
       "       [ 9.2,  2.8],\n",
       "       [10.3,  2.9],\n",
       "       [ 9.1,  2.6],\n",
       "       [ 8.4,  2.1],\n",
       "       [ 9.3,  2.1],\n",
       "       [ 9.1,  2.3],\n",
       "       [ 8.5,  1.7],\n",
       "       [ 8.9,  2.2],\n",
       "       [ 8.2,  2.6],\n",
       "       [ 8.3,  1.6],\n",
       "       [ 8.7,  1.5],\n",
       "       [ 7.6,  1.4],\n",
       "       [ 8.4,  1.8],\n",
       "       [ 8.5,  2. ],\n",
       "       [ 8.2,  2. ],\n",
       "       [ 8. ,  1.6],\n",
       "       [ 7.8,  1.5],\n",
       "       [ 8.3,  1.9],\n",
       "       [ 9.7,  2.6],\n",
       "       [ 9.8,  2.8],\n",
       "       [ 7.7,  1.6],\n",
       "       [ 7.6,  2. ],\n",
       "       [ 8.3,  2.2],\n",
       "       [ 8.6,  2.2],\n",
       "       [ 7.3,  1.7],\n",
       "       [ 8.3,  1.9],\n",
       "       [ 8.3,  2.2],\n",
       "       [ 5.9,  1. ],\n",
       "       [ 7.7,  1.9],\n",
       "       [ 8.6,  1.9],\n",
       "       [ 9.5,  1.9],\n",
       "       [ 7.4,  1.6],\n",
       "       [ 9.2,  2.2],\n",
       "       [ 7.8,  1.8],\n",
       "       [ 8.9,  2.2],\n",
       "       [ 8. ,  1.9],\n",
       "       [11.1, -1.5],\n",
       "       [10.9, -1.3],\n",
       "       [11.1, -1.8],\n",
       "       [ 8.6, -1.7],\n",
       "       [10.2, -1.8],\n",
       "       [10.1, -1.7],\n",
       "       [11.3, -1.4],\n",
       "       [ 8.1, -0.9],\n",
       "       [10.4, -1.7],\n",
       "       [ 9.3, -1.2],\n",
       "       [ 7.5, -1.5],\n",
       "       [10.2, -1.2],\n",
       "       [ 8.4, -1.8],\n",
       "       [10.5, -1.8],\n",
       "       [ 9.4, -0.7],\n",
       "       [10.6, -1.3],\n",
       "       [10.5, -1.5],\n",
       "       [ 9.5, -1.4],\n",
       "       [ 8.9, -2.3],\n",
       "       [ 8.9, -1.4],\n",
       "       [11.2, -1.6],\n",
       "       [ 9.6, -1.2],\n",
       "       [ 9.9, -2.4],\n",
       "       [10.3, -1.9],\n",
       "       [10.1, -1.4],\n",
       "       [10.4, -1.4],\n",
       "       [10.4, -2. ],\n",
       "       [11. , -2. ],\n",
       "       [10.3, -1.6],\n",
       "       [ 8.7, -0.9],\n",
       "       [ 8.6, -1.4],\n",
       "       [ 8.5, -1.3],\n",
       "       [ 9.3, -1.2],\n",
       "       [10.5, -2.4],\n",
       "       [10.5, -1.5],\n",
       "       [11.3, -1.1],\n",
       "       [10.9, -1.6],\n",
       "       [ 9. , -2.1],\n",
       "       [10.1, -1.1],\n",
       "       [ 9. , -1.5],\n",
       "       [ 9.6, -1.8],\n",
       "       [10.6, -1.6],\n",
       "       [ 9.2, -1.4],\n",
       "       [ 7.9, -1. ],\n",
       "       [ 9.6, -1.5],\n",
       "       [10.2, -1.2],\n",
       "       [10. , -1.3],\n",
       "       [10.1, -1.4],\n",
       "       [ 8. , -0.5],\n",
       "       [ 9.7, -1.3],\n",
       "       [12.6, -2.7],\n",
       "       [10.5, -2.4],\n",
       "       [11.9, -2.9],\n",
       "       [11.4, -2.7],\n",
       "       [11.8, -2.8],\n",
       "       [12.6, -3.6],\n",
       "       [ 9.5, -2. ],\n",
       "       [12.1, -3.4],\n",
       "       [10.8, -3.3],\n",
       "       [13.3, -2.5],\n",
       "       [11.5, -1.9],\n",
       "       [10.7, -2.6],\n",
       "       [11.5, -2.5],\n",
       "       [10. , -2.5],\n",
       "       [10.7, -2.3],\n",
       "       [11.7, -2.1],\n",
       "       [11.5, -2.5],\n",
       "       [14.3, -2.9],\n",
       "       [12.1, -4.3],\n",
       "       [ 9.4, -2.8],\n",
       "       [12.1, -2.5],\n",
       "       [10.5, -2.1],\n",
       "       [12.3, -3.9],\n",
       "       [10.3, -2.2],\n",
       "       [12.3, -2.4],\n",
       "       [12.4, -2.8],\n",
       "       [10.4, -2. ],\n",
       "       [10.9, -1.9],\n",
       "       [11.2, -2.8],\n",
       "       [11.8, -2.8],\n",
       "       [11.7, -3.3],\n",
       "       [14. , -2.6],\n",
       "       [11.2, -2.8],\n",
       "       [10.7, -2.3],\n",
       "       [10.8, -3. ],\n",
       "       [12.1, -3.1],\n",
       "       [12.4, -2.2],\n",
       "       [11.7, -2.4],\n",
       "       [10.8, -1.8],\n",
       "       [11.6, -2.3],\n",
       "       [11.8, -2.5],\n",
       "       [11.3, -2. ],\n",
       "       [10.5, -2.4],\n",
       "       [12.3, -2.7],\n",
       "       [12.3, -2.4],\n",
       "       [11.2, -2.2],\n",
       "       [10. , -2.5],\n",
       "       [11.2, -2.2],\n",
       "       [12.2, -2. ],\n",
       "       [11.1, -2.1]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linearly_transformed_data = A.dot(D.T)\n",
    "linearly_transformed_data.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "18b571d3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3.5, 1.4],\n",
       "       [3. , 1.4],\n",
       "       [3.2, 1.3],\n",
       "       [3.1, 1.5],\n",
       "       [3.6, 1.4],\n",
       "       [3.9, 1.7],\n",
       "       [3.4, 1.4],\n",
       "       [3.4, 1.5],\n",
       "       [2.9, 1.4],\n",
       "       [3.1, 1.5],\n",
       "       [3.7, 1.5],\n",
       "       [3.4, 1.6],\n",
       "       [3. , 1.4],\n",
       "       [3. , 1.1],\n",
       "       [4. , 1.2],\n",
       "       [4.4, 1.5],\n",
       "       [3.9, 1.3],\n",
       "       [3.5, 1.4],\n",
       "       [3.8, 1.7],\n",
       "       [3.8, 1.5],\n",
       "       [3.4, 1.7],\n",
       "       [3.7, 1.5],\n",
       "       [3.6, 1. ],\n",
       "       [3.3, 1.7],\n",
       "       [3.4, 1.9],\n",
       "       [3. , 1.6],\n",
       "       [3.4, 1.6],\n",
       "       [3.5, 1.5],\n",
       "       [3.4, 1.4],\n",
       "       [3.2, 1.6],\n",
       "       [3.1, 1.6],\n",
       "       [3.4, 1.5],\n",
       "       [4.1, 1.5],\n",
       "       [4.2, 1.4],\n",
       "       [3.1, 1.5],\n",
       "       [3.2, 1.2],\n",
       "       [3.5, 1.3],\n",
       "       [3.6, 1.4],\n",
       "       [3. , 1.3],\n",
       "       [3.4, 1.5],\n",
       "       [3.5, 1.3],\n",
       "       [2.3, 1.3],\n",
       "       [3.2, 1.3],\n",
       "       [3.5, 1.6],\n",
       "       [3.8, 1.9],\n",
       "       [3. , 1.4],\n",
       "       [3.8, 1.6],\n",
       "       [3.2, 1.4],\n",
       "       [3.7, 1.5],\n",
       "       [3.3, 1.4],\n",
       "       [3.2, 4.7],\n",
       "       [3.2, 4.5],\n",
       "       [3.1, 4.9],\n",
       "       [2.3, 4. ],\n",
       "       [2.8, 4.6],\n",
       "       [2.8, 4.5],\n",
       "       [3.3, 4.7],\n",
       "       [2.4, 3.3],\n",
       "       [2.9, 4.6],\n",
       "       [2.7, 3.9],\n",
       "       [2. , 3.5],\n",
       "       [3. , 4.2],\n",
       "       [2.2, 4. ],\n",
       "       [2.9, 4.7],\n",
       "       [2.9, 3.6],\n",
       "       [3.1, 4.4],\n",
       "       [3. , 4.5],\n",
       "       [2.7, 4.1],\n",
       "       [2.2, 4.5],\n",
       "       [2.5, 3.9],\n",
       "       [3.2, 4.8],\n",
       "       [2.8, 4. ],\n",
       "       [2.5, 4.9],\n",
       "       [2.8, 4.7],\n",
       "       [2.9, 4.3],\n",
       "       [3. , 4.4],\n",
       "       [2.8, 4.8],\n",
       "       [3. , 5. ],\n",
       "       [2.9, 4.5],\n",
       "       [2.6, 3.5],\n",
       "       [2.4, 3.8],\n",
       "       [2.4, 3.7],\n",
       "       [2.7, 3.9],\n",
       "       [2.7, 5.1],\n",
       "       [3. , 4.5],\n",
       "       [3.4, 4.5],\n",
       "       [3.1, 4.7],\n",
       "       [2.3, 4.4],\n",
       "       [3. , 4.1],\n",
       "       [2.5, 4. ],\n",
       "       [2.6, 4.4],\n",
       "       [3. , 4.6],\n",
       "       [2.6, 4. ],\n",
       "       [2.3, 3.3],\n",
       "       [2.7, 4.2],\n",
       "       [3. , 4.2],\n",
       "       [2.9, 4.2],\n",
       "       [2.9, 4.3],\n",
       "       [2.5, 3. ],\n",
       "       [2.8, 4.1],\n",
       "       [3.3, 6. ],\n",
       "       [2.7, 5.1],\n",
       "       [3. , 5.9],\n",
       "       [2.9, 5.6],\n",
       "       [3. , 5.8],\n",
       "       [3. , 6.6],\n",
       "       [2.5, 4.5],\n",
       "       [2.9, 6.3],\n",
       "       [2.5, 5.8],\n",
       "       [3.6, 6.1],\n",
       "       [3.2, 5.1],\n",
       "       [2.7, 5.3],\n",
       "       [3. , 5.5],\n",
       "       [2.5, 5. ],\n",
       "       [2.8, 5.1],\n",
       "       [3.2, 5.3],\n",
       "       [3. , 5.5],\n",
       "       [3.8, 6.7],\n",
       "       [2.6, 6.9],\n",
       "       [2.2, 5. ],\n",
       "       [3.2, 5.7],\n",
       "       [2.8, 4.9],\n",
       "       [2.8, 6.7],\n",
       "       [2.7, 4.9],\n",
       "       [3.3, 5.7],\n",
       "       [3.2, 6. ],\n",
       "       [2.8, 4.8],\n",
       "       [3. , 4.9],\n",
       "       [2.8, 5.6],\n",
       "       [3. , 5.8],\n",
       "       [2.8, 6.1],\n",
       "       [3.8, 6.4],\n",
       "       [2.8, 5.6],\n",
       "       [2.8, 5.1],\n",
       "       [2.6, 5.6],\n",
       "       [3. , 6.1],\n",
       "       [3.4, 5.6],\n",
       "       [3.1, 5.5],\n",
       "       [3. , 4.8],\n",
       "       [3.1, 5.4],\n",
       "       [3.1, 5.6],\n",
       "       [3.1, 5.1],\n",
       "       [2.7, 5.1],\n",
       "       [3.2, 5.9],\n",
       "       [3.3, 5.7],\n",
       "       [3. , 5.2],\n",
       "       [2.5, 5. ],\n",
       "       [3. , 5.2],\n",
       "       [3.4, 5.4],\n",
       "       [3. , 5.1]])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7acb5e6e",
   "metadata": {},
   "source": [
    "But let's look at how the data changed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "13d5f588",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Scatter plot of original data and linearly transformed data')"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111)\n",
    "ax.scatter(linearly_transformed_data[0,:], linearly_transformed_data[1,:], s=10, c='r', marker='x', label='linearly transformed data')\n",
    "ax.scatter(D[:,0], D[:,1], s=10, c='b', marker='s', label='original data')\n",
    "plt.xlabel('X1')\n",
    "plt.ylabel('X2')\n",
    "plt.legend(loc='lower right')\n",
    "plt.title('Scatter plot of original data and linearly transformed data')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "707f4804",
   "metadata": {},
   "source": [
    "Great, now, let's compute the cov matrix.  The easiest way to do that, is just do get the cov of the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "e9ca263b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.18997942, -0.32965638],\n",
       "       [-0.32965638,  3.11627785]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sigma = np.cov(D.T, ddof=1)\n",
    "Sigma"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e439425a",
   "metadata": {},
   "source": [
    "Or, we an compute it the way we saw in the notes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "3ebb7606",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.18997942, -0.32965638],\n",
       "       [-0.32965638,  3.11627785]])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = D.shape[0]\n",
    "1/(n-1)*np.dot(centered_data.T, centered_data)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4acb6c1d",
   "metadata": {},
   "source": [
    "Notice that they are the same!  ... which does make sense.  Steps 1-3 were to compute the covariance matrix, which with numpy we can compute directly with the `cov` function."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1abe5cc3",
   "metadata": {},
   "source": [
    "### Eigenvalues\n",
    "\n",
    "Next up is one of the major steps, we will get the eigenvalues and engenvectors of the covariance matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "d27139ee",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy.linalg as LA\n",
    "evalues, evectors = LA.eig(Sigma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "1aa15dc1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.15330233, 3.15295494])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evalues"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "046526be",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.99386764,  0.11057628],\n",
       "       [-0.11057628, -0.99386764]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evectors"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b768fb9d",
   "metadata": {},
   "source": [
    "Note that in class, and the algorithm, the eigenvalues (and corresponding vectors) were assumed to be sorted in decending order by eigenvalue.  Numpy does not do that, so let's sort them! "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "73664e13",
   "metadata": {},
   "outputs": [],
   "source": [
    "idx = evalues.argsort()[::-1]\n",
    "evalues = evalues[idx]\n",
    "evectors = evectors[:, idx]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3f008f7",
   "metadata": {},
   "source": [
    "and now they are sorted in decending order by eigenvalue"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "01ac3e46",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3.15295494 0.15330233]\n",
      "[[ 0.11057628 -0.99386764]\n",
      " [-0.99386764 -0.11057628]]\n"
     ]
    }
   ],
   "source": [
    "print(evalues)\n",
    "print(evectors)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce8fe217",
   "metadata": {},
   "source": [
    "Recall that the sum of the eigenvalues is the total variance of the dataset.  That is "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "b3311bc3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.3062572706935125"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "total_var = sum(np.diag(Sigma))\n",
    "total_var"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "47461992",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.3062572706935125"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(evalues)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23837f6a",
   "metadata": {},
   "source": [
    "And, we can find the proportion of the total variance attributred to each eigenvector.  For example, in just the direction of the largest eigenvalues, we capture over 95% of the total variance:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "c5a06019",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9536326686835204"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evalues[0]/total_var"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f624a6f5",
   "metadata": {},
   "source": [
    "Let's project our data on to the largest eigenvector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "9dc6f94a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2.39248833,  2.33720019,  2.45870221,  2.24887105,  2.40354596,\n",
       "        2.13855855,  2.3814307 ,  2.28204394,  2.32614256,  2.24887105,\n",
       "        2.31521682,  2.18265717,  2.33720019,  2.63536048,  2.64655   ,\n",
       "        2.39262021,  2.5361056 ,  2.39248833,  2.12750092,  2.32627445,\n",
       "        2.08327041,  2.31521682,  2.80109301,  2.07221278,  1.88449688,\n",
       "        2.13842666,  2.18265717,  2.29310156,  2.3814307 ,  2.16054192,\n",
       "        2.14948429,  2.28204394,  2.35944733,  2.46989172,  2.24887105,\n",
       "        2.55808897,  2.49187509,  2.40354596,  2.43658695,  2.28204394,\n",
       "        2.49187509,  2.35918356,  2.45870221,  2.1937148 ,  1.92872739,\n",
       "        2.33720019,  2.22688768,  2.35931545,  2.31521682,  2.37037307,\n",
       "       -0.92044777, -0.72167424, -1.13027892, -0.32425907, -0.86529152,\n",
       "       -0.76590475, -0.90939014,  0.38250591, -0.85423389, -0.18064179,\n",
       "        0.13950187, -0.4456292 , -0.3353167 , -0.95362065,  0.13963375,\n",
       "       -0.6333451 , -0.7437895 , -0.37941532, -0.83225052, -0.20275705,\n",
       "       -1.01983453, -0.26897093, -1.19662469, -0.96467828, -0.5560736 ,\n",
       "       -0.64440273, -1.06406504, -1.24072332, -0.75484712,  0.20584763,\n",
       "       -0.11442791, -0.01504115, -0.18064179, -1.37328296, -0.7437895 ,\n",
       "       -0.69955898, -0.9315054 , -0.72180613, -0.34624244, -0.30214381,\n",
       "       -0.68863324, -0.84317626, -0.29108619,  0.37144828, -0.47880209,\n",
       "       -0.4456292 , -0.45668683, -0.5560736 ,  0.69172383, -0.3683577 ,\n",
       "       -2.20141807, -1.37328296, -2.13520419, -1.84810153, -2.03581743,\n",
       "       -2.83091154, -0.79907763, -2.54380888, -2.09110557, -2.26763195,\n",
       "       -1.31799482, -1.57205649, -1.73765714, -1.29601145, -1.36222534,\n",
       "       -1.51676835, -1.73765714, -2.84183728, -3.17330234, -1.32918434,\n",
       "       -1.91431541, -1.16345181, -2.95241356, -1.17450944, -1.90325778,\n",
       "       -2.2124757 , -1.06406504, -1.14133655, -1.85915916, -2.03581743,\n",
       "       -2.35609298, -2.54367699, -1.85915916, -1.36222534, -1.88127441,\n",
       "       -2.33397772, -1.79281339, -1.72659951, -1.04194979, -1.62721274,\n",
       "       -1.82598627, -1.32905245, -1.37328296, -2.11308894, -1.90325778,\n",
       "       -1.43949684, -1.29601145, -1.43949684, -1.59403986, -1.34011008])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coords_along_eig0 = evectors[:,0].T.dot(centered_data.T)\n",
    "coords_along_eig0"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f84df4c",
   "metadata": {},
   "source": [
    "Observe that now, we now have a 1 dimensional representation of our data that captures about 95% of the total variance of the original data set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "67f6b582",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(150,)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coords_along_eig0.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb61a6aa",
   "metadata": {},
   "source": [
    "It can be helpful to visuzlize the data projected on to the eigenvector.  Below, we plot the centered data in blue and the projected data in red"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "0448c68e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Scatter plot of centered and PCA-transformed data')"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "coords_along_eig0 = evectors[:,0].T.dot(centered_data.T)\n",
    "\n",
    "projected_along_eig0 = np.zeros((n,2))\n",
    "for i in range(len(coords_along_eig0)):\n",
    "    projected_along_eig0[i,:] = coords_along_eig0[i]*evectors[:,0]\n",
    "\n",
    "pjdev0 = projected_along_eig0\n",
    "# plot of centered data and its projection onto the first principial component (evec corresponding to largest eval)\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111)\n",
    "ax.scatter(centered_data[:,0], centered_data[:,1], s=10, c='b', marker='s', label='centered data')\n",
    "ax.scatter(pjdev0[:,0], pjdev0[:,1], s=10, c='r', marker='x', label='1-dimensional pca-transformed data')\n",
    "plt.xlabel('X1')\n",
    "plt.ylabel('X2')\n",
    "plt.xlim(-3,3)\n",
    "plt.ylim(-3,3)\n",
    "plt.legend(loc='upper left')\n",
    "plt.title('Scatter plot of centered and PCA-transformed data')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5feab7b0",
   "metadata": {},
   "source": [
    "And we can also look at what would happen if we projected on to the other eigenvector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "e55adf7c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Scatter plot of centered and PCA-transformed data')"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "coords_along_eig1 = evectors[:,1].T.dot(centered_data.T)\n",
    "\n",
    "projected_along_eig1 = np.zeros((n,2))\n",
    "for i in range(len(coords_along_eig1)):\n",
    "    projected_along_eig1[i,:] = coords_along_eig1[i]*evectors[:,1]\n",
    "\n",
    "pjdev1 = projected_along_eig1\n",
    "# plot of centered data and its projection onto the first principial component (evec corresponding to largest eval)\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111)\n",
    "ax.scatter(centered_data[:,0], centered_data[:,1], s=10, c='b', marker='s', label='centered data')\n",
    "ax.scatter(pjdev0[:,0], pjdev0[:,1], s=10, c='r', marker='x', label='1-dimensional pca-transformed data')\n",
    "ax.scatter(pjdev1[:,0], pjdev1[:,1], s=10, c='c', marker='x', label='1-dimensional transformed data onto evec 1')\n",
    "plt.xlabel('X1')\n",
    "plt.ylabel('X2')\n",
    "plt.xlim(-3,3)\n",
    "plt.ylim(-3,3)\n",
    "plt.legend(loc='upper left')\n",
    "plt.title('Scatter plot of centered and PCA-transformed data')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93570b3a",
   "metadata": {},
   "source": [
    "Note that we don't just have to project on to one of the vectors, we can project on to multiple axes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "a82b7eb9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 150)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coords_in_evec_axes = evectors.T.dot(centered_data.T)\n",
    "coords_in_evec_axes.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "1dc92787",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'scatter plot of transformed data with eigenvectors as axes')"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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(coords_in_evec_axes[0,:], coords_in_evec_axes[1,:])\n",
    "plt.xlabel('PC0 (evec corresponding to largest eval of Sigma) ')\n",
    "plt.ylabel('PC1 (evec corresponding to second largest eval of Sigma) ')\n",
    "plt.xlim(-3,3)\n",
    "plt.ylim(-3,3)\n",
    "plt.title('scatter plot of transformed data with eigenvectors as axes')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3aa9831a",
   "metadata": {},
   "source": [
    "### PCA in the wild\n",
    "\n",
    "PCA is one of the most popular dimensionality reduction algorithms.  So, it is natrual to assume that it would be in all the standard libraries.  So, let's look at how to use PCA in scikit-learn!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "21d2fade",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.decomposition import PCA\n",
    "pca = PCA(n_components=2)\n",
    "pca_transformed_D = pca.fit_transform(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3463ad55",
   "metadata": {},
   "source": [
    "After we run PCA, we can get some info about what the decomposition did:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "b85d1caf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.95363267, 0.04636733])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pca.explained_variance_ratio_"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e11aceb",
   "metadata": {},
   "source": [
    "**Question**: *Any idea about what those mean?*\n",
    "\n",
    "Yep, those are the propotion of the variance in the direction of each principal component."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "4b316967",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9536326686835204\n",
      "0.046367331316479625\n"
     ]
    }
   ],
   "source": [
    "print(evalues[0] / sum(evalues))\n",
    "print(evalues[1] / sum(evalues))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f7f70d4",
   "metadata": {},
   "source": [
    "We can also get the eigenvectors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "1ce64b18",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.11057628,  0.99386764],\n",
       "       [ 0.99386764,  0.11057628]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pca.components_ "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9a04ad7",
   "metadata": {},
   "source": [
    "**WARNING**: note that for `pca.components_` the eigenvectors are listed row wise in decreasing order of eigenvalue magnitudes.  So we have:\n",
    "- 0th eigenvector is [-0.11057628,  0.99386764] \n",
    "- 1st eigenvector is [ 0.99386764,  0.11057628]\n",
    "\n",
    "If we compare with our eigenvalues, we get:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "bada339f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.11057628, -0.99386764],\n",
       "       [-0.99386764, -0.11057628]])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evectors"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3648e37c",
   "metadata": {},
   "source": [
    "Which are listed column wise.  Recall that they are also in decreasing order of eignvalue magnitudes, however, we processed the results from the eigenvalue decomposition so that it was the case.  So we have:\n",
    "\n",
    "- 0th eigenvector is [0.11057628, -0.99386764]\n",
    "- 1st eigenvector is [-0.99386764, -0.11057628]\n",
    "\n",
    "Notice the difference? ... the vectors are facing opposite directions.  This is okay though.  Notice that we select the 0th eigenvector because it maximizes variance.  And therefore, the opposite would also maximize variance.  Then we get the next eigenvector, which maximizes variance subject to being ortogonal to the first direction.  As before the opposite direction has the same property!  Note that if that all sounded like a bunch of gobbly-gook, that is okay.  But, if you are interested in exploring these properties futher, we do that in the Grad Data Mining course... which will be offered in the Fall!\n",
    "\n",
    "We can, however, explore this idea a little further with pictures."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "b663f35a",
   "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": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111)\n",
    "ax.scatter(coords_in_evec_axes[0,:], coords_in_evec_axes[1,:], s=10, c='b', marker='s', label='transformed data using pca')\n",
    "ax.scatter(pca_transformed_D[:,0], pca_transformed_D[:,1], s=10, c='r', marker='x', label='transformed data using sklearn PCA')\n",
    "plt.xlabel('PC1')\n",
    "plt.ylabel('PC2')\n",
    "plt.xlim(-3,3)\n",
    "plt.ylim(-3,3)\n",
    "plt.legend(loc='lower right')\n",
    "plt.title('Scatter plot of centered and sklearn PCA-transformed data using 2 principal components')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "98cbde15",
   "metadata": {},
   "source": [
    "So, we can see that the blue points and the red points are reflectison of eachother.  That is becuase the directions we projected on to are the opposite of the directions that sklearn projected on to.  To use the same directions, just multiply each coordinate by -1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "19045c13",
   "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": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111)\n",
    "ax.scatter(-1*coords_in_evec_axes[0,:], -1*coords_in_evec_axes[1,:], s=10, c='b', marker='s', label='transformed data using pca')\n",
    "ax.scatter(pca_transformed_D[:,0], pca_transformed_D[:,1], s=10, c='r', marker='x', label='transformed data using sklearn PCA')\n",
    "plt.xlabel('PC1')\n",
    "plt.ylabel('PC2')\n",
    "plt.xlim(-3,3)\n",
    "plt.ylim(-3,3)\n",
    "plt.legend(loc='lower right')\n",
    "plt.title('Scatter plot of centered and sklearn PCA-transformed data using 2 principal components')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a9ec62b",
   "metadata": {},
   "source": [
    "One other thing that we discussed a few times is that once we have our transformed points, we have a representation in which the variance in each directdion is independed.  We can confim that now:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "c5a3f549",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3.15295494e+00, 4.76874319e-17],\n",
       "       [4.76874319e-17, 1.53302326e-01]])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.cov(coords_in_evec_axes)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a965ec29",
   "metadata": {},
   "source": [
    "(and we can observe that the off diagonal values are numerically 0!)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68af5476",
   "metadata": {},
   "source": [
    "### Using a real data set\n",
    "\n",
    "Now, let's look at using PCA to visualize a dataset with more than 3 dimensions.  Recall that we started off looking at specific features of the iris data set in 2D. That data set has 4 features, which made it hard to plot.  But we could instead use PCA to visualize the dataset in 2D!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "85ca64d6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.92461872, 0.05306648])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D_iris = load_iris()['data']\n",
    "pca = PCA(n_components=2)\n",
    "pca_transformed_D_iris = pca.fit_transform(D_iris)\n",
    "pca.explained_variance_ratio_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "e11f9519",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(150, 4)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D_iris.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "407b7e96",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(150, 2)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pca_transformed_D_iris.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "a09b5628",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'PCA-transformed iris data using 2 principal components')"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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(pca_transformed_D_iris[:, 0], pca_transformed_D_iris[:, 1])\n",
    "plt.xlabel('PC0')\n",
    "plt.ylabel('PC1')\n",
    "plt.title('PCA-transformed iris data using 2 principal components')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9959add6",
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   "source": [
    "Cool!!  Aside from visualizing, we can also run all the cool algos that we have studied so far (and will study for the rest of the semester) on the lower dimensional dataset!!  We still captured most of the variance of the data but with the reduced dimension, we have some algorithmic speedups!"
   ]
  }
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