PCA.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 0. Dependências"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/arnaldo/miniconda3/envs/ml/lib/python3.5/site-packages/sklearn/utils/fixes.py:313: FutureWarning: numpy not_equal will not check object identity in the future. The comparison did not return the same result as suggested by the identity (`is`)) and will change.\n",
      "  _nan_object_mask = _nan_object_array != _nan_object_array\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.datasets import load_iris\n",
    "from sklearn.preprocessing import StandardScaler\n",
    "from sklearn.decomposition import PCA\n",
    "\n",
    "%matplotlib inline\n",
    "pd.options.display.max_rows = 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1. Introdução "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**O objetivo principal do PCA é analisar os dados para identificar padrões visando reduzir a dimensionalidade dos dados com uma perda mínima de informação**. Uma possível aplicação seria o reconhecimento de padrões, onde queremos reduzir os custos computacionais e o erro da estimação de parâmetros pela redução da dimensionalidade do nosso espaço de atributos extraindo um subespaço que descreve os nosso dados \"de forma satisfatória\". **A redução de dimensionalidade se torna importante quando temos um número de atributos significativamente maior que o número de amostras de treinamento**.\n",
    "\n",
    "Nós aplicamos o PCA para projetar todos os nossos dados (sem rótulos de classe) em um subespaço diferente, procurando encontrar os eixos com a máxima variância onde os dados são mais distribuídos. A questão principal é: **\"Qual o subespaço que representa *bem* os nossos dados?\"**.\n",
    "\n",
    "**Primeiramente, calculamos os autovetores (componentes principais) dos nossos dados e organizamos em uma matriz de projeção. Cada autovetor (*eigenvector*) é associada a um autovalor (*eigenvalue*) que pode ser interpretado como o \"tamanho\" ou \"magnitude\" do autovetor correspondente**. Em geral, consideramos apenas o autovalores que tem uma magnitude significativamente maior que os outros e desconsideramos os autopares (autovetores-autovalores) que consideramos *menos informativos*. \n",
    "\n",
    "Se observamos que todos os autovalores tem uma magnitude similar, isso pode ser um bom indicador que nossos dados já estão em um *bom* subespaço. Por outro lado, **se alguns autovalores tem a magnitude muito maior que a dos outros, devemos escolher seus autovetores já que eles contém mais informação sobre a distribuição dos nossos dados**. Da mesma forma, autovalores próximos a zero são menos informativos e devemos desconsiderá-los na construção do nosso subespaço.\n",
    "\n",
    "Em geral, a aplicação do PCA envolve os seguintes passos:\n",
    "    \n",
    "1. Padronização dos dados\n",
    "2. Obtenção dos autovetores e autovalores através da:\n",
    "    - Matriz de Covariância; ou\n",
    "    - Matriz de Correlação; ou\n",
    "    - *Singular Vector Decomposition*\n",
    "3. Construção da matriz de projeção a partir dos autovetores selecionados\n",
    "4. Transformação dos dados originais X via a matriz de projeção para obtenção do subespaço Y\n",
    "\n",
    "## 1.1 PCA vs LDA\n",
    "\n",
    "Ambos PCA e LDA (*Linear Discrimant Analysis*) são métodos de transformação linear. Por uma lado, o PCA fornece as direções (autovetores ou componentes principais) que maximiza a variância dos dados, enquanto o LDA visa as direções que maximizam a separação (ou discriminação) entre classes diferentes. Maximizar a variância, no caso do PCA, também significa reduzir a perda de informação, que é representada pela soma das distâncias de projeção dos dados nos eixos dos componentes principais.\n",
    "\n",
    "Enquanto o PCA projeta os dados em um subespaço diferente, o LDA tenta determinar um subespaço adequado para distinguir os padrões pertencentes as classes diferentes.\n",
    "\n",
    "<img src=\"images/PCAvsLDA.png\" width=600>\n",
    "\n",
    "## 1.2 Autovetores e autovalores\n",
    "\n",
    "Os autovetores e autovalores de uma matriz de covariância (ou correlação) representam a base do PCA: os autovetores (componentes principais) determinam a direção do novo espaço de atributos, e os autovalores determinam sua magnitude. Em outras palavras, os autovalores explicam a variância dos dados ao longo dos novos eixos de atributos.\n",
    "\n",
    "### 1.2.1 Matriz de Covariância\n",
    "\n",
    "A abordagem clássica do PCA calcula a matriz de covariância, onde cada elemento representa a covariância entre dois atributos. A covariância entre dois atributos é calculada da seguinte forma:\n",
    "\n",
    "$$\\sigma_{jk} = \\frac{1}{n-1}\\sum_{i=1}^N(x_{ij}-\\overline{x}_j)(x_{ik}-\\overline{x}_k)$$\n",
    "\n",
    "Que podemos simplificar na forma vetorial através da fórmula:\n",
    "\n",
    "$$S=\\frac{1}{n-1}((x-\\overline{x})^T(x-\\overline{x}))$$\n",
    "\n",
    "onde $\\overline{x}$ é um vetor d-dimensional onde cada valor representa a média de cada atributo, e $n$ representa o número de atributos por amostra. Vale ressaltar ainda que x é um vetor onde cada amostra está organizada em linhas e cada coluna representa um atributo. Caso se tenha um vetor onde as amostras estão organizadas em colunas e cada linha representa um atributo, a transposta passa para o segundo elemento da multiplicação.\n",
    "\n",
    "Na prática, o resultado da matriz de covariância representa basicamente a seguinte estrutura:\n",
    "\n",
    "$$\\begin{bmatrix}var(1) & cov(1,2) & cov(1,3) & cov(1,4) \n",
    "\\\\ cov(1,2) & var(2) & cov(2,3) & cov(2,4)\n",
    "\\\\ cov(1,3) & cov(2,3) & var(3) & cov(3,4)\n",
    "\\\\ cov(1,4) & cov(2,4) & cov(3,4) & var(4)\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "Onde a diagonal principal representa a variância em cada dimensão e os demais elementos são a covariância entre cada par de dimensões.\n",
    "\n",
    "Para se calcular os autovalores e autovetores, só precisamos chamar a função *np.linalg.eig*, onde cada autovetor estará representado por uma coluna.\n",
    "\n",
    "> Uma propriedade interessante da matriz de covariância é que **a soma da diagonal principal da matriz (variância para cada dimensão) é igual a soma dos autovalores**.\n",
    " \n",
    "### 1.2.2 Matriz de Correlação\n",
    "\n",
    "Outra maneira de calcular os autovalores e autovetores é utilizando a matriz de correlação. Apesar das matrizes serem diferentes, elas vão resultar nos mesmos autovalores e autovetores (mostrado mais a frente) já que a matriz de correlação é dada pela normalização da matriz de covariância.\n",
    "\n",
    "$$corr(x,y) = \\frac{cov(x,y)}{\\sigma_x \\sigma_y}$$\n",
    "\n",
    "\n",
    "### 1.2.3 Singular Vector Decomposition\n",
    "\n",
    "Apesar da autodecomposição (cálculo dos autovetores e autovalores) efetuada pelas matriz de covariância ou correlação ser mais intuitiva, a maior parte das implementações do PCA executam a *Singular Vector Decomposition* (SVD) para melhorar o desempenho computacional. Para calcular a SVD, podemos utilizar a biblioteca numpy, através do método *np.linalg.svd*.\n",
    "\n",
    "Note que a autodecomposição resulta nos mesmos autovalores e autovetores utilizando qualquer uma das matrizes abaixo:\n",
    "\n",
    "- Matriz de covariânca após a padronização dos dados\n",
    "- Matriz de correlação\n",
    "- Matriz de correlação após a padronização dos dados\n",
    "\n",
    "Mas qual a relação entre a SVD e o PCA? Dado que a matriz de covariância $C = \\frac{X^TX}{n-1}$ é uma matriz simétrica, ela pode ser diagonalizada da seguinte forma:\n",
    "\n",
    "$$C = VLV^T$$\n",
    "\n",
    "onde $V$ é a matriz de autovetores (cada coluna é um autovetor) e $L$ é a matriz diagonal com os autovalores $\\lambda_i$ na ordem decrescente na diagonal. Se executarmos o SVD em X, nós obtemos a seguinte decomposição:\n",
    "\n",
    "$$X = USV^T$$\n",
    "\n",
    "onde $U$ é a matriz unitária e $S$ é a matriz diagonal de *singular values* $s_i$. A partir disso, pode-se calcular que:\n",
    "\n",
    "$$C = VSU^TUSV^T = V\\frac{S^2}{n-1}V^T$$\n",
    "\n",
    "Isso significa que os *right singular vectors* V são as *principal directions* e que os *singular values* estão relacionados aos autovalores da matriz de covariância via $\\lambda_i = \\frac{s_i^2}{n-1}$. Os componentes principais são dados por $XV = USV^TV = US$\n",
    "\n",
    "Resumindo:\n",
    "\n",
    "1. Se $X = USV^T$, então as colunas de V são as direções/eixos principais;\n",
    "2. As colunas de $US$ são os componentes principais;\n",
    "3. *Singular values* estão relacionados aos autovalores da matriz de covariância via $\\lambda_i = \\frac{s_i^2}{n-1}$;\n",
    "4. Scores padronizados (*standardized*) são dados pelas colunas de $\\sqrt{n-1}U$ e *loadings* são dados pelas colunas de $\\frac{VS}{\\sqrt{n-1}}$. Veja [este link](https://stats.stackexchange.com/questions/125684) e [este](https://stats.stackexchange.com/questions/143905) para entender as diferenças entre *loadings* e *principal directions*;\n",
    "5. As fórmulas acima só são válidas se $X$ está centralizado, ou seja, somente quando a matriz de covariância é igual a $\\frac{X^TX}{n-1}$;\n",
    "6. As proposições acima estão corretas somente quando $X$ for representado por uma matriz onde as linhas são amostras e as colunas são atributos. Caso contrário, $U$ e $V$ tem interpretações contrárias. Isto é, $U, V = V, U$;\n",
    "7. Para reduzir a dimensionalidade com o PCA baseado no SVD, selecione as *k*-ésimas primeiras colunas de U, e $k\\times k$ parte superior de S. O produto $U_kS_k$ é a matriz $n \\times k$ necessária para conter os primeiros $k$ PCs.\n",
    "8. Para reconstruir os dados originais a partir dos primeiros $k$ PCs, multiplicá-los pelo eixos principais correspondentes $V_k^T$ produz a matriz $X_k = U_kS_kV_k^T$ que tem o tamanho original $n \\times p$. Essa forma gera a matriz reconstruída com o menor erro de reconstrução possível dos dados originais. [Veja esse link](https://stats.stackexchange.com/questions/130721);\n",
    "9. Estritamente falando, $U$ é de tamanho $n \\times n$ e $V$ é de tamanho $p \\times p$. Entretanto, se $n > p$ então as últimas $n-p$ colunas de $U$ são arbitrárias (e as linhas correspondentes de $S$ são constantes e iguais a zero).\n",
    "\n",
    "### 1.2.4 Verificação dos autovetores e autovalores\n",
    "\n",
    "Para verificar se os autovetores e autovalores calculados na autodecomposição estão corretos, devemos verificar se eles satisfazem a equação para cada autovetor e autovalor correspondente:\n",
    "\n",
    "$$\\Sigma \\overrightarrow{v} = \\lambda \\overrightarrow{v}$$\n",
    "\n",
    "onde:\n",
    "$$\\Sigma = Matriz\\,de\\,Covariância$$\n",
    "$$\\overrightarrow{v} = autovetor$$\n",
    "$$\\lambda = autovalor$$\n",
    "\n",
    "### 1.2.5 Escolha dos autovetores e autovalores\n",
    "\n",
    "Como dito, o objetivo típico do PCA é reduzir a dimensionalidade dos dados pela projeção em um subespaço menor, onde os autovetores formam os eixos. Entretando, os autovetores definem somente as direções dos novos eixos, já que todos eles tem tamanho 1. Logo, para decidir qual(is) autovetor(es) podemos descartar sem perder muita informação na construção do nosso subespaço, precisamos verificar os autovalores correspondentes. **Os autovetores com os maiores valores são os que contém mais informação sobre a distribuição dos nossos dados**. Esse são os autovetores que queremos.\n",
    "\n",
    "Para fazer isso, devemos ordenar os autovalores em ordem decrescente para escolher o top k autovetores.\n",
    "\n",
    "### 1.2.6 Cálculo da Informação\n",
    "\n",
    "Após ordenar os autovalores, o próximo passo é **definir quantos componentes principais serão escolhidos para o nosso novo subespaço**. Para fazer isso, podemos utilizar o método da *variância explicada*, que calcula quanto de informação (variância) é atribuida a cada componente principal.\n",
    "\n",
    "## 1. 3 Matriz de Projeção\n",
    "\n",
    "Na prática, a matriz de projeção nada mais é que os top k autovetores concatenados. Portanto, se queremos reduzir o nosso espaço 4-dimensional para um espaço 2-dimensional, devemos escolher os 2 autovetores com os 2 maiores autovalores para construir nossa matriz W (d$\\times$k).\n",
    "\n",
    "## 1.4 Projeção no novo subespaço\n",
    "\n",
    "O último passo do PCA é utilizar a nossa matriz de projeção dimensional W (4x2, onde cada coluna representa um autovetor) para transformar nossas amostras em um novo subespaço. Para isso, basta aplicar a seguinte equação:\n",
    "\n",
    "$$S = (X-\\mu_X) \\times W$$\n",
    "\n",
    "Onde cada linha em S contém os pesos para cada atributo (coluna da matriz) no novo subespaço.\n",
    "\n",
    "A título de curiosidade, repare que se W representasse todos os autovetores - e não somente os escolhidos -, poderíamos recompor cada instância em X pela seguinte fórmula:\n",
    "\n",
    "$$X = (S \\times W^{-1}) + \\mu_X$$\n",
    "\n",
    "Novamente, cada linha em S representa os pesos para cada atributo, só que dessa vez seria possível representar X pela soma de cada autovetor multiplicado por um peso.\n",
    "\n",
    "## 1.5 Recomendações\n",
    "\n",
    "- Sempre normalize os atributos antes de aplicar o PCA (StandarScaler);\n",
    "- Lembre-se de armazenar a média para efetuar a ida e volta;\n",
    "- Não aplique o PCA após outros algoritmos de seleção de atributos ([fonte](https://www.quora.com/Should-I-apply-PCA-before-or-after-feature-selection));\n",
    "- O número de componentes principais que você quer manter deve ser escolhido através da análise entre o número de componentes e a precisão do sistema. Nem sempre mais componentes principais ocasionam em melhor precisão!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. Dados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>sepal length (cm)</th>\n",
       "      <th>sepal width (cm)</th>\n",
       "      <th>petal length (cm)</th>\n",
       "      <th>petal width (cm)</th>\n",
       "      <th>class</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>5.1</td>\n",
       "      <td>3.5</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>4.9</td>\n",
       "      <td>3.0</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>4.7</td>\n",
       "      <td>3.2</td>\n",
       "      <td>1.3</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4.6</td>\n",
       "      <td>3.1</td>\n",
       "      <td>1.5</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5.0</td>\n",
       "      <td>3.6</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>145</th>\n",
       "      <td>6.7</td>\n",
       "      <td>3.0</td>\n",
       "      <td>5.2</td>\n",
       "      <td>2.3</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>146</th>\n",
       "      <td>6.3</td>\n",
       "      <td>2.5</td>\n",
       "      <td>5.0</td>\n",
       "      <td>1.9</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>147</th>\n",
       "      <td>6.5</td>\n",
       "      <td>3.0</td>\n",
       "      <td>5.2</td>\n",
       "      <td>2.0</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>148</th>\n",
       "      <td>6.2</td>\n",
       "      <td>3.4</td>\n",
       "      <td>5.4</td>\n",
       "      <td>2.3</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>149</th>\n",
       "      <td>5.9</td>\n",
       "      <td>3.0</td>\n",
       "      <td>5.1</td>\n",
       "      <td>1.8</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>150 rows × 5 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "     sepal length (cm)  sepal width (cm)  petal length (cm)  petal width (cm)  \\\n",
       "0                  5.1               3.5                1.4               0.2   \n",
       "1                  4.9               3.0                1.4               0.2   \n",
       "2                  4.7               3.2                1.3               0.2   \n",
       "3                  4.6               3.1                1.5               0.2   \n",
       "4                  5.0               3.6                1.4               0.2   \n",
       "..                 ...               ...                ...               ...   \n",
       "145                6.7               3.0                5.2               2.3   \n",
       "146                6.3               2.5                5.0               1.9   \n",
       "147                6.5               3.0                5.2               2.0   \n",
       "148                6.2               3.4                5.4               2.3   \n",
       "149                5.9               3.0                5.1               1.8   \n",
       "\n",
       "     class  \n",
       "0        0  \n",
       "1        0  \n",
       "2        0  \n",
       "3        0  \n",
       "4        0  \n",
       "..     ...  \n",
       "145      2  \n",
       "146      2  \n",
       "147      2  \n",
       "148      2  \n",
       "149      2  \n",
       "\n",
       "[150 rows x 5 columns]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "iris = load_iris()\n",
    "\n",
    "df = pd.DataFrame(data=iris.data, columns=iris.feature_names)\n",
    "df['class'] = iris.target\n",
    "df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>sepal length (cm)</th>\n",
       "      <th>sepal width (cm)</th>\n",
       "      <th>petal length (cm)</th>\n",
       "      <th>petal width (cm)</th>\n",
       "      <th>class</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>150.000000</td>\n",
       "      <td>150.000000</td>\n",
       "      <td>150.000000</td>\n",
       "      <td>150.000000</td>\n",
       "      <td>150.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>5.843333</td>\n",
       "      <td>3.057333</td>\n",
       "      <td>3.758000</td>\n",
       "      <td>1.199333</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>0.828066</td>\n",
       "      <td>0.435866</td>\n",
       "      <td>1.765298</td>\n",
       "      <td>0.762238</td>\n",
       "      <td>0.819232</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>4.300000</td>\n",
       "      <td>2.000000</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.100000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>5.100000</td>\n",
       "      <td>2.800000</td>\n",
       "      <td>1.600000</td>\n",
       "      <td>0.300000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>5.800000</td>\n",
       "      <td>3.000000</td>\n",
       "      <td>4.350000</td>\n",
       "      <td>1.300000</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>6.400000</td>\n",
       "      <td>3.300000</td>\n",
       "      <td>5.100000</td>\n",
       "      <td>1.800000</td>\n",
       "      <td>2.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>7.900000</td>\n",
       "      <td>4.400000</td>\n",
       "      <td>6.900000</td>\n",
       "      <td>2.500000</td>\n",
       "      <td>2.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       sepal length (cm)  sepal width (cm)  petal length (cm)  \\\n",
       "count         150.000000        150.000000         150.000000   \n",
       "mean            5.843333          3.057333           3.758000   \n",
       "std             0.828066          0.435866           1.765298   \n",
       "min             4.300000          2.000000           1.000000   \n",
       "25%             5.100000          2.800000           1.600000   \n",
       "50%             5.800000          3.000000           4.350000   \n",
       "75%             6.400000          3.300000           5.100000   \n",
       "max             7.900000          4.400000           6.900000   \n",
       "\n",
       "       petal width (cm)       class  \n",
       "count        150.000000  150.000000  \n",
       "mean           1.199333    1.000000  \n",
       "std            0.762238    0.819232  \n",
       "min            0.100000    0.000000  \n",
       "25%            0.300000    0.000000  \n",
       "50%            1.300000    1.000000  \n",
       "75%            1.800000    2.000000  \n",
       "max            2.500000    2.000000  "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(150, 4) (150,)\n"
     ]
    }
   ],
   "source": [
    "x = df.drop(labels='class', axis=1).values\n",
    "y = df['class'].values\n",
    "\n",
    "print(x.shape, y.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3. Implementação "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "class MyPCACov():\n",
    "    def __init__(self, n_components=None):\n",
    "        self.n_components = n_components\n",
    "        self.eigen_values = None\n",
    "        self.eigen_vectors = None\n",
    "    \n",
    "    def fit(self, x):\n",
    "        self.n_components = x.shape[1] if self.n_components is None else self.n_components\n",
    "        self.mean_ = np.mean(x, axis=0)\n",
    "        \n",
    "        cov_matrix = np.cov(x - self.mean_, rowvar=False)\n",
    "        \n",
    "        self.eigen_values, self.eigen_vectors = np.linalg.eig(cov_matrix)\n",
    "        self.eigen_vectors = self.eigen_vectors.T\n",
    "        \n",
    "        self.sorted_components_ = np.argsort(self.eigen_values)[::-1]\n",
    "        \n",
    "        self.projection_matrix_ = self.eigen_vectors[self.sorted_components_[:self.n_components]]\n",
    "\n",
    "        self.explained_variance_ = self.eigen_values[self.sorted_components_]\n",
    "        self.explained_variance_ratio_ = self.explained_variance_ / self.eigen_values.sum()\n",
    "        \n",
    "    def transform(self, x):\n",
    "        return np.dot(x - self.mean_, self.projection_matrix_.T)\n",
    "    \n",
    "    def inverse_transform(self, x):\n",
    "        return np.dot(x, self.projection_matrix_) + self.mean_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "class MyPCASVD():\n",
    "    def __init__(self, n_components=None):\n",
    "        self.n_components = n_components\n",
    "        self.eigen_values = None\n",
    "        self.eigen_vectors = None\n",
    "    \n",
    "    def fit(self, x):\n",
    "        self.n_components = x.shape[1] if self.n_components is None else self.n_components\n",
    "        self.mean_ = np.mean(x, axis=0)\n",
    "        \n",
    "        U, s, Vt = np.linalg.svd(x - self.mean_, full_matrices=False) # a matriz s já retorna ordenada\n",
    "#         S = np.diag(s)\n",
    "        \n",
    "        self.eigen_vectors = Vt\n",
    "        self.eigen_values = s\n",
    "        \n",
    "        self.projection_matrix = self.eigen_vectors[:self.n_components]\n",
    "\n",
    "        self.explained_variance_ = (self.eigen_values ** 2) / (x.shape[0] - 1)\n",
    "        self.explained_variance_ratio_ = self.explained_variance_ / self.explained_variance_.sum()\n",
    "        \n",
    "    def transform(self, x):\n",
    "        return np.dot(x - self.mean_, self.projection_matrix.T)\n",
    "    \n",
    "    def inverse_transform(self, x):\n",
    "        return np.dot(x, self.projection_matrix) + self.mean_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 4. Teste "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "std = StandardScaler()\n",
    "x_std = StandardScaler().fit_transform(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### PCA implementado pela matriz de covariância"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Autovetores: \n",
      " [[ 0.52106591 -0.26934744  0.5804131   0.56485654]\n",
      " [-0.37741762 -0.92329566 -0.02449161 -0.06694199]\n",
      " [-0.71956635  0.24438178  0.14212637  0.63427274]\n",
      " [ 0.26128628 -0.12350962 -0.80144925  0.52359713]]\n",
      "Autovalores: \n",
      " [ 2.93808505  0.9201649   0.14774182  0.02085386]\n",
      "Variância explicada: \n",
      " [ 2.93808505  0.9201649   0.14774182  0.02085386]\n",
      "Variância explicada (ratio): \n",
      " [ 0.72962445  0.22850762  0.03668922  0.00517871]\n",
      "Componentes ordenados: \n",
      " [0 1 2 3]\n"
     ]
    }
   ],
   "source": [
    "pca_cov = MyPCACov(n_components=2)\n",
    "pca_cov.fit(x_std)\n",
    "\n",
    "print('Autovetores: \\n', pca_cov.eigen_vectors)\n",
    "print('Autovalores: \\n', pca_cov.eigen_values)\n",
    "print('Variância explicada: \\n', pca_cov.explained_variance_)\n",
    "print('Variância explicada (ratio): \\n', pca_cov.explained_variance_ratio_)\n",
    "print('Componentes ordenados: \\n', pca_cov.sorted_components_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x1a19adf4a8>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a0db80be0>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_std_proj = pca_cov.transform(x_std)\n",
    "\n",
    "plt.figure()\n",
    "plt.scatter(x_std_proj[:, 0], x_std_proj[:, 1], c=y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.90068117  1.01900435 -1.34022653 -1.3154443 ]\n",
      " [-1.14301691 -0.13197948 -1.34022653 -1.3154443 ]\n",
      " [-1.38535265  0.32841405 -1.39706395 -1.3154443 ]\n",
      " [-1.50652052  0.09821729 -1.2833891  -1.3154443 ]\n",
      " [-1.02184904  1.24920112 -1.34022653 -1.3154443 ]]\n",
      "[[-0.99888895  1.05319838 -1.30270654 -1.24709825]\n",
      " [-1.33874781 -0.06192302 -1.22432772 -1.22057235]\n",
      " [-1.36096129  0.32111685 -1.38060338 -1.35833824]\n",
      " [-1.42359795  0.0677615  -1.34922386 -1.33881298]\n",
      " [-1.00113823  1.24091818 -1.37125365 -1.30661752]]\n"
     ]
    }
   ],
   "source": [
    "x_std_back = pca_cov.inverse_transform(x_std_proj)\n",
    "\n",
    "print(x_std[:5])\n",
    "print(x_std_back[:5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### PCA implementado pela SVD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Autovetores: \n",
      " [[ 0.52106591 -0.26934744  0.5804131   0.56485654]\n",
      " [-0.37741762 -0.92329566 -0.02449161 -0.06694199]\n",
      " [ 0.71956635 -0.24438178 -0.14212637 -0.63427274]\n",
      " [ 0.26128628 -0.12350962 -0.80144925  0.52359713]]\n",
      "Autovalores: \n",
      " [ 20.92306556  11.7091661    4.69185798   1.76273239]\n",
      "Variância explicada: \n",
      " [ 2.93808505  0.9201649   0.14774182  0.02085386]\n",
      "Variância explicada (ratio): \n",
      " [ 0.72962445  0.22850762  0.03668922  0.00517871]\n"
     ]
    }
   ],
   "source": [
    "pca_svd = MyPCASVD(n_components=2)\n",
    "pca_svd.fit(x_std)\n",
    "\n",
    "print('Autovetores: \\n', pca_svd.eigen_vectors)\n",
    "print('Autovalores: \\n', pca_svd.eigen_values)\n",
    "print('Variância explicada: \\n', pca_svd.explained_variance_)\n",
    "print('Variância explicada (ratio): \\n', pca_svd.explained_variance_ratio_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x1a19bec9b0>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a0db80048>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_std_proj = pca_svd.transform(x_std)\n",
    "\n",
    "plt.figure()\n",
    "plt.scatter(x_std_proj[:, 0], x_std_proj[:, 1], c=y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.90068117  1.01900435 -1.34022653 -1.3154443 ]\n",
      " [-1.14301691 -0.13197948 -1.34022653 -1.3154443 ]\n",
      " [-1.38535265  0.32841405 -1.39706395 -1.3154443 ]\n",
      " [-1.50652052  0.09821729 -1.2833891  -1.3154443 ]\n",
      " [-1.02184904  1.24920112 -1.34022653 -1.3154443 ]]\n",
      "[[-0.99888895  1.05319838 -1.30270654 -1.24709825]\n",
      " [-1.33874781 -0.06192302 -1.22432772 -1.22057235]\n",
      " [-1.36096129  0.32111685 -1.38060338 -1.35833824]\n",
      " [-1.42359795  0.0677615  -1.34922386 -1.33881298]\n",
      " [-1.00113823  1.24091818 -1.37125365 -1.30661752]]\n"
     ]
    }
   ],
   "source": [
    "x_std_back = pca_svd.inverse_transform(x_std_proj)\n",
    "\n",
    "print(x_std[:5])\n",
    "print(x_std_back[:5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Comparação com o Scikit-learn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Autovetores: \n",
      " [[ 0.52106591 -0.26934744  0.5804131   0.56485654]\n",
      " [ 0.37741762  0.92329566  0.02449161  0.06694199]]\n",
      "Autovalores: \n",
      " [ 20.92306556  11.7091661 ]\n",
      "Variância explicada: \n",
      " [ 2.93808505  0.9201649 ]\n",
      "Variância explicada (ratio): \n",
      " [ 0.72962445  0.22850762]\n"
     ]
    }
   ],
   "source": [
    "pca_sk = PCA(n_components=2)\n",
    "pca_sk.fit(x_std)\n",
    "\n",
    "print('Autovetores: \\n', pca_sk.components_)\n",
    "print('Autovalores: \\n', pca_sk.singular_values_)\n",
    "print('Variância explicada: \\n', pca_sk.explained_variance_)\n",
    "print('Variância explicada (ratio): \\n', pca_sk.explained_variance_ratio_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x1a19cbefd0>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1a0db700f0>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_std_proj_sk = pca_sk.transform(x_std)\n",
    "\n",
    "plt.figure()\n",
    "plt.scatter(x_std_proj_sk[:, 0], x_std_proj_sk[:, 1], c=y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.90068117  1.01900435 -1.34022653 -1.3154443 ]\n",
      " [-1.14301691 -0.13197948 -1.34022653 -1.3154443 ]\n",
      " [-1.38535265  0.32841405 -1.39706395 -1.3154443 ]\n",
      " [-1.50652052  0.09821729 -1.2833891  -1.3154443 ]\n",
      " [-1.02184904  1.24920112 -1.34022653 -1.3154443 ]]\n",
      "[[-0.99888895  1.05319838 -1.30270654 -1.24709825]\n",
      " [-1.33874781 -0.06192302 -1.22432772 -1.22057235]\n",
      " [-1.36096129  0.32111685 -1.38060338 -1.35833824]\n",
      " [-1.42359795  0.0677615  -1.34922386 -1.33881298]\n",
      " [-1.00113823  1.24091818 -1.37125365 -1.30661752]]\n"
     ]
    }
   ],
   "source": [
    "x_std_back_sk = pca_sk.inverse_transform(x_std_proj_sk)\n",
    "\n",
    "print(x_std[:5])\n",
    "print(x_std_back_sk[:5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Observação sobre a implementação do Scikit-learn\n",
    "\n",
    "Por algum motivo (que não sei qual), a [implementação do scikit-learn](https://github.com/scikit-learn/scikit-learn/blob/master/sklearn/decomposition/pca.py) inverte os sinais de alguns valores na matriz de autovetores. Na implementação, as matrizes $U$ e $V$ são passada para um método ```svd_flip``` (implementada [nesse arquivo](https://github.com/scikit-learn/scikit-learn/blob/master/sklearn/utils/extmath.py)):\n",
    "\n",
    "```py\n",
    "U, V = svd_flip(U[:, ::-1], V[::-1])\n",
    "```\n",
    "\n",
    "Repare que isso muda apenas os dados projetados. No gráfico, isso inverte os eixos correspondentes apenas. No entanto, os **autovalores**, a ```explained_variance```, ```explained_variance_ratio``` e os dados reprojetados ao espaço original são exatamente iguais."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Referências"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- [Antigo notebook do PCA com explicações passo-a-passo](https://github.com/arnaldog12/Machine_Learning/blob/62b628bd3c37ec2fa52e349f38da24751ef67313/PCA.ipynb)\n",
    "- [Principal Component Analysis in Python](https://plot.ly/ipython-notebooks/principal-component-analysis/)\n",
    "- [Implementing a Principal Component Analysis (PCA)](https://sebastianraschka.com/Articles/2014_pca_step_by_step.html)\n",
    "- [Relationship between SVD and PCA. How to use SVD to perform PCA?](https://stats.stackexchange.com/questions/134282/relationship-between-svd-and-pca-how-to-use-svd-to-perform-pca)\n",
    "- [How to reverse PCA and reconstruct original variables from several principal components?](https://stats.stackexchange.com/questions/229092/how-to-reverse-pca-and-reconstruct-original-variables-from-several-principal-com)\n",
    "- [Everything you did and didn't know about PCA](http://alexhwilliams.info/itsneuronalblog/2016/03/27/pca/)\n",
    "- [Unpacking (** PCA )](https://towardsdatascience.com/unpacking-pca-b5ea8bec6aa5)"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "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": "3.5.3"
  },
  "nav_menu": {},
  "toc": {
   "navigate_menu": true,
   "number_sections": true,
   "sideBar": true,
   "threshold": 6,
   "toc_cell": false,
   "toc_section_display": "block",
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}