pca.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "from typing import List, Dict\n",
    "from collections import Counter\n",
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "from Vector_operations_on_data import vector_mean\n",
    "import tqdm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x27310fed358>"
      ]
     },
     "execution_count": 10,
     "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": [
    "# Dimensionality reduction (PCA)\n",
    "pca_data = [\n",
    "[20.9666776351559,-13.1138080189357],\n",
    "[22.7719907680008,-19.8890894944696],\n",
    "[25.6687103160153,-11.9956004517219],\n",
    "[18.0019794950564,-18.1989191165133],\n",
    "[21.3967402102156,-10.8893126308196],\n",
    "[0.443696899177716,-19.7221132386308],\n",
    "[29.9198322142127,-14.0958668502427],\n",
    "[19.0805843080126,-13.7888747608312],\n",
    "[16.4685063521314,-11.2612927034291],\n",
    "[21.4597664701884,-12.4740034586705],\n",
    "[3.87655283720532,-17.575162461771],\n",
    "[34.5713920556787,-10.705185165378],\n",
    "[13.3732115747722,-16.7270274494424],\n",
    "[20.7281704141919,-8.81165591556553],\n",
    "[24.839851437942,-12.1240962157419],\n",
    "[20.3019544741252,-12.8725060780898],\n",
    "[21.9021426929599,-17.3225432396452],\n",
    "[23.2285885715486,-12.2676568419045],\n",
    "[28.5749111681851,-13.2616470619453],\n",
    "[29.2957424128701,-14.6299928678996],\n",
    "[15.2495527798625,-18.4649714274207],\n",
    "[26.5567257400476,-9.19794350561966],\n",
    "[30.1934232346361,-12.6272709845971],\n",
    "[36.8267446011057,-7.25409849336718],\n",
    "[32.157416823084,-10.4729534347553],\n",
    "[5.85964365291694,-22.6573731626132],\n",
    "[25.7426190674693,-14.8055803854566],\n",
    "[16.237602636139,-16.5920595763719],\n",
    "[14.7408608850568,-20.0537715298403],\n",
    "[6.85907008242544,-18.3965586884781],\n",
    "[26.5918329233128,-8.92664811750842],\n",
    "[-11.2216019958228,-27.0519081982856],\n",
    "[8.93593745011035,-20.8261235122575],\n",
    "[24.4481258671796,-18.0324012215159],\n",
    "[2.82048515404903,-22.4208457598703],\n",
    "[30.8803004755948,-11.455358009593],\n",
    "[15.4586738236098,-11.1242825084309],\n",
    "[28.5332537090494,-14.7898744423126],\n",
    "[40.4830293441052,-2.41946428697183],\n",
    "[15.7563759125684,-13.5771266003795],\n",
    "[19.3635588851727,-20.6224770470434],\n",
    "[13.4212840786467,-19.0238227375766],\n",
    "[7.77570680426702,-16.6385739839089],\n",
    "[21.4865983854408,-15.290799330002],\n",
    "[12.6392705930724,-23.6433305964301],\n",
    "[12.4746151388128,-17.9720169566614],\n",
    "[23.4572410437998,-14.602080545086],\n",
    "[13.6878189833565,-18.9687408182414],\n",
    "[15.4077465943441,-14.5352487124086],\n",
    "[20.3356581548895,-10.0883159703702],\n",
    "[20.7093833689359,-12.6939091236766],\n",
    "[11.1032293684441,-14.1383848928755],\n",
    "[17.5048321498308,-9.2338593361801],\n",
    "[16.3303688220188,-15.1054735529158],\n",
    "[26.6929062710726,-13.306030567991],\n",
    "[34.4985678099711,-9.86199941278607],\n",
    "[39.1374291499406,-10.5621430853401],\n",
    "[21.9088956482146,-9.95198845621849],\n",
    "[22.2367457578087,-17.2200123442707],\n",
    "[10.0032784145577,-19.3557700653426],\n",
    "[14.045833906665,-15.871937521131],\n",
    "[15.5640911917607,-18.3396956121887],\n",
    "[24.4771926581586,-14.8715313479137],\n",
    "[26.533415556629,-14.693883922494],\n",
    "[12.8722580202544,-21.2750596021509],\n",
    "[24.4768291376862,-15.9592080959207],\n",
    "[18.2230748567433,-14.6541444069985],\n",
    "[4.1902148367447,-20.6144032528762],\n",
    "[12.4332594022086,-16.6079789231489],\n",
    "[20.5483758651873,-18.8512560786321],\n",
    "[17.8180560451358,-12.5451990696752],\n",
    "[11.0071081078049,-20.3938092335862],\n",
    "[8.30560561422449,-22.9503944138682],\n",
    "[33.9857852657284,-4.8371294974382],\n",
    "[17.4376502239652,-14.5095976075022],\n",
    "[29.0379635148943,-14.8461553663227],\n",
    "[29.1344666599319,-7.70862921632672],\n",
    "[32.9730697624544,-15.5839178785654],\n",
    "[13.4211493998212,-20.150199857584],\n",
    "[11.380538260355,-12.8619410359766],\n",
    "[28.672631499186,-8.51866271785711],\n",
    "[16.4296061111902,-23.3326051279759],\n",
    "[25.7168371582585,-13.8899296143829],\n",
    "[13.3185154732595,-17.8959160024249],\n",
    "[3.60832478605376,-25.4023343597712],\n",
    "[39.5445949652652,-11.466377647931],\n",
    "[25.1693484426101,-12.2752652925707],\n",
    "[25.2884257196471,-7.06710309184533],\n",
    "[6.77665715793125,-22.3947299635571],\n",
    "[20.1844223778907,-16.0427471125407],\n",
    "[25.5506805272535,-9.33856532270204],\n",
    "[25.1495682602477,-7.17350567090738],\n",
    "[15.6978431006492,-17.5979197162642],\n",
    "[37.42780451491,-10.843637288504],\n",
    "[22.974620174842,-10.6171162611686],\n",
    "[34.6327117468934,-9.26182440487384],\n",
    "[34.7042513789061,-6.9630753351114],\n",
    "[15.6563953929008,-17.2196961218915],\n",
    "[25.2049825789225,-14.1592086208169]\n",
    "]\n",
    "\n",
    "plt.scatter(*zip(*pca_data))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "from Vector_operations_on_data import subtract\n",
    "def de_mean(data: List[Vector]) -> List[Vector]:\n",
    "    \"\"\"Recenters the data to have 0 mean in every dimension\"\"\"\n",
    "    mean = vector_mean(data)\n",
    "    return [subtract(vector, mean) for vector in data]\n",
    "pca_data = de_mean(pca_data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x273110376a0>"
      ]
     },
     "execution_count": 12,
     "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(*zip(*pca_data))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "from Vector_operations_on_data import magnitude, dot\n",
    "\n",
    "def direction(w: Vector) -> Vector:\n",
    "    \"\"\"Computes a unit vector\"\"\"\n",
    "    mag = magnitude(w)\n",
    "    return [w_i / mag for w_i in w]\n",
    "\n",
    "def directional_variance(data: List[Vector], w: Vector) -> float:\n",
    "    \"\"\"Returns the variance of each vector of data matrix\n",
    "    in the direction of vector w\"\"\"\n",
    "    w_dir = direction(w)\n",
    "    return [dot(v,w_dir) ** 2 for v in data]\n",
    "\n",
    "def directional_variance_gradient(data: List[Vector], w: Vector) -> Vector:\n",
    "    \"\"\"The gradient of directional variance w.r.t w\"\"\"\n",
    "    w_dir = direction(w)\n",
    "    return [sum(2 * dot(v, w_dir) * v[i] for v in data)\n",
    "           for i in range(len(w))]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "from gradient_descent import gradient_step;\n",
    "\n",
    "# Use gradient ascent to find the direction that maximizes data varaince\n",
    "# gradient ascent will used + step size \n",
    "def first_principal_component(data: List[Vector],\n",
    "                              n: int = 100,\n",
    "                              step_size: float = 0.1) -> Vector:\n",
    "    # Start with a random direction\n",
    "    guess = [1.0 for _ in data[0]]\n",
    "    \n",
    "    with tqdm.trange(n) as t:\n",
    "        for _ in t:\n",
    "            dv = directional_variance(data, guess)\n",
    "            gradient = directional_variance_gradient(data, guess)\n",
    "            guess = gradient_step(guess, gradient, step_size)\n",
    "            t.set_description(f\"dv: {dv}\")\n",
    "           \n",
    "    return direction(guess)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "dv: [1.5919452889207217, 0.11162039957636091, 36.4025547649421, 11.728489502502454, 6.3054693783325995, 409.1411976464874, 83.82938013669161, 0.5462659150557589, 4.768848492476459, 3.850536813148179, 263.53718173994434, 217.6040428594683, 50.931247036952826, 7.232500531801361, 27.233703089683054, 0.5477907044993009, 0.26388630980894356, 13.507324864098399, 67.78295966206943, 70.13649830855653, 36.833917659713805, 62.81150607293555, 99.42289309223996, 329.6646373521476, 159.02482358322735, 267.28038443536576, 25.2543636846306, 19.704656821177355, 51.08779881606417, 190.25852253860643, 64.99131775830777, 1143.1476522469352, 163.9832972890635, 6.726703383013999, 363.4921548740514, 122.20171442360089, 9.388707436498311, 57.899598294900905, 546.9038273357406, 13.902881092109721, 9.580837565714676, 63.552565331962946, 150.63412511151375, 0.824646397014724, 109.49196159980754, 71.29373158385135, 8.953570007916698, 59.362114009082894, 19.516082109120678, 3.3772730813077505, 1.403973058739719, 67.92877745685063, 0.20239562311218734, 14.317676332982543, 41.95900480411703, 225.2135768620539, 361.9118645821048, 11.176992185804046, 0.74313940334468, 126.70764051058602, 38.28828971693292, 32.83932429207744, 14.678081985618979, 33.62421029055763, 87.26298274022683, 11.656980623140662, 3.4694385381847574, 292.69255265012595, 63.35003007149911, 1.7486970401733233, 2.0418886129863987, 115.06422272919981, 201.68392222982325, 270.87246604789516, 6.415047270595669, 64.86426613033342, 118.30873335950083, 130.1016363197281, 70.61991094064652, 56.201952117317504, 102.82144203558677, 46.83452079418191, 28.647042585897434, 58.29133222871509, 379.461370598992, 363.04456755942664, 29.866807217491147, 57.3051436681881, 237.19189185523382, 0.3396422361927264, 48.194045160835145, 54.774686936102455, 28.333003084391915, 300.5690004331994, 16.58824446404271, 235.95801882390666, 265.93572436883727, 27.209626364342242, 22.813346265058268]: 100%|█| 100/100 [00:00<00:00, 424.84it/s]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x273111990f0>"
      ]
     },
     "execution_count": 20,
     "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": [
    "# directional vector of the first principal component\n",
    "pca1 = first_principal_component(pca_data)\n",
    "\n",
    "ax = plt.axes()\n",
    "ax.arrow(-1, -1.0, 8*pca1[0], 8*pca1[1], \n",
    "         head_width=0.5, head_length=0.7, \n",
    "         fc='lightblue', ec='black')\n",
    "ax.scatter(*zip(*pca_data))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x273111deb00>"
      ]
     },
     "execution_count": 21,
     "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": [
    "# First prinical component is the projection of v \n",
    "from Vector_operations_on_data import scalar_multiply\n",
    "def project(v: Vector, w: Vector) -> Vector:\n",
    "    \"\"\"Return the projection of v onto the direction w\"\"\"\n",
    "    projection_length = dot(v,w)\n",
    "    return scalar_multiply(projection_length, w)\n",
    "\n",
    "# For finding other principal components, just remove the projection of \n",
    "# previous components from the data matrix\n",
    "def remove_projection_from_vector(v: Vector, w: Vector) -> Vector:\n",
    "    \"\"\"projects v onto w and subtracts the projection from v\"\"\"\n",
    "    return subtract(v,project(v,w))\n",
    "\n",
    "def remove_projection(data: List[Vector], w: Vector) -> List[Vector]:\n",
    "    return [remove_projection_from_vector(v, w) for v in data]\n",
    "\n",
    "pr_data = remove_projection(pca_data, pca1)\n",
    "plt.scatter(*zip(*pr_data))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Iteratively find multiple principal components from a high-dimensional dataset\n",
    "def pca(data: List[Vector], num_components: int) -> List[Vector]:\n",
    "    components: List[Vector] = []\n",
    "    for _ in range(num_components):\n",
    "        component = first_principal_component(data)\n",
    "        components.append(component)\n",
    "        data = remove_projection(data, component)\n",
    "    return components\n",
    "\n",
    "def transform_vector(v: Vector, components: List[Vector]) -> Vector:\n",
    "    return [dot(v,w) for w in components]\n",
    "\n",
    "def transform(data: List[Vector], components: List[Vector]) -> List[Vector]:\n",
    "    return [transform_vector(v, components) for v in data]"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}