002/002-pca.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Пример понижения размерности с помощью PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "train = np.loadtxt('./data/digit/train.csv', delimiter=',', skiprows=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# сохраняем разметку в отдельную переменную\n",
    "train_label = train[:, 0]\n",
    "# приводим размерность к удобному для обаботки виду\n",
    "train_img = np.reshape(train[:, 1:], (len(train[:, 1:]), 28, 28))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from matplotlib import pyplot as plt\n",
    "fig = plt.figure(figsize=(20, 10))\n",
    "for i, img in enumerate(train_img[0:5], 1):\n",
    "    subplot = fig.add_subplot(1, 7, i)\n",
    "    plt.imshow(img, cmap='gray');\n",
    "    subplot.set_title('%s' % train_label[i - 1]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [],
   "source": [
    "# выбираем семпл данных для обработки\n",
    "choices = np.random.choice(train_img.shape[0], 10000)\n",
    "\n",
    "y = train_label[choices]\n",
    "X = train_img[choices].reshape(-1, 28 * 28).astype(np.float32)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "# центрируем данные\n",
    "X_mean = X.mean(axis=0)\n",
    "X -= X_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "# матрица ковариации признаков\n",
    "cov = np.dot(X.T, X) / X.shape[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "U, S, _ = np.linalg.svd(cov)\n",
    "# U - собсвенные вектора матрицы ковариации\n",
    "# S - собственные значения"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Собственные числа"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# накопленная сумма собственных значений\n",
    "S_cumsum = np.cumsum(S) / np.sum(S)\n",
    "plt.plot(S_cumsum, 'o')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(5):\n",
    "    print('[%03d] %.3f' % (i, S_cumsum[i]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Понижаем размерность"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "S_thr = 0.75  # задаем порог для накопленной суммы собственных значений\n",
    "\n",
    "# определяем необходимое число компонент для заданного порога\n",
    "n_comp = np.argmax(np.where(S_cumsum > S_thr, 1, 0))\n",
    "\n",
    "print('n_comp=%d S=%.3f' % (n_comp, S_cumsum[n_comp]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "# получаем сжатое представление объектов\n",
    "Xrot_reduced = np.dot(X, U[:, :n_comp])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Восстанавливаем изображение после понижения размерности"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "Xrot_restored = np.dot(Xrot_reduced, U[:,:n_comp].T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "fig = plt.figure(figsize=(20, 10))\n",
    "for i, img in enumerate(X[0:6], 1):\n",
    "    subplot = fig.add_subplot(1, 7, i)\n",
    "    img_ = img + X_mean\n",
    "    plt.title('%s' % y[i-1])\n",
    "    plt.imshow(img_.reshape((28,28)), cmap='gray');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "fig = plt.figure(figsize=(20, 10))\n",
    "for i, img in enumerate(Xrot_restored[0:6], 1):\n",
    "    subplot = fig.add_subplot(1, 7, i)\n",
    "    img_ = img + X_mean\n",
    "    plt.title('%s' % y[i-1])\n",
    "    plt.imshow(img_.reshape((28,28)) + X_mean.reshape((28,28)), cmap='gray');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Визуализация собственных векторов"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = plt.figure(figsize=(20, 10))\n",
    "for i, img in enumerate(U.T[:5, :], 1):\n",
    "    subplot = fig.add_subplot(1, 5, i)\n",
    "    plt.imshow(img.reshape((28,28)), cmap='gray');"
   ]
  }
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