深度学习花书 笔记3 - 矩阵对角化、奇异值分解（SVD）、极大似然估计、误差的高斯分布与最小二乘估计的等价性、PCA原理与推导

1. 矩阵对角化

2. 奇异值分解

3. 极大似然估计、误差的高斯分布与最小二乘估计的等价性

4. PCA原理与推导

PCA实现代码：
首先，先创建数据

'''
Created on Apr 6, 2020
Author: ***
'''
import os, sys
import numpy as np
import matplotlib
import matplotlib.pyplot as plt

n = 400 #数据的个数
fw = open('data.txt', 'w')
classes = [0, 1]

x1 = np.random.uniform(1, 10, 200)
y1 = np.random.uniform(1, 10, 200)
for i in range(200):
    fw.write("%f\t%f\t%d\n" % (x1[i], y1[i], classes[0]))

x2 = np.random.uniform(9, 15, 200)
y2 = np.random.uniform(9, 15, 200)
for i in range(200):
    fw.write("%f\t%f\t%d\n" % (x2[i], y2[i], classes[1]))

fw.close()
figure = plt.figure()
ax = figure.add_subplot(111)
ax.scatter(x1, y1, marker='o',s=80, c='green')
ax.scatter(x2, y2, marker='^',s=80, c='red')
plt.show()

具体实现代码：

'''
Created on Jun 1, 2011

@author: Peter
'''
from numpy import *
import matplotlib
import matplotlib.pyplot as plt

def loadDataSet(fileName, delim='\t'):
    fr = open(fileName)
    stringArr = [line.strip().split(delim) for line in fr.readlines()]
    datArr = [list(map(float, line)) for line in stringArr]
    return mat(datArr)

def pca(data, max_n=99999):
    mean_values = mean(data, axis=0)  # 计算数据均值
    remove_mean = data - mean_values  # 数据减去均值
    cov_values = cov(remove_mean, rowvar=0)  # 计算协方差
    eigen_values, eigen_vectors = linalg.eig(mat(cov_values))  # 计算特征值和特征向量
    eigen_value_indexes = argsort(eigen_values)  # 从小到大排列
    eigen_value_indexes = eigen_value_indexes[:-(max_n + 1):-1]  # 保留最大的前n个特征值
    red_eigen_vectors = eigen_vectors[:, eigen_value_indexes]  # 获得对应的特征向量
    low_dimension_data = remove_mean * red_eigen_vectors  # 将数据转换到新的低维空间
    new_data = (low_dimension_data * red_eigen_vectors.T) + mean_values  # 对新的数据进行重构
    return low_dimension_data, new_data

if __name__ == '__main__':
    dataMat = loadDataSet('data.txt')
    lowDMat, reconMat = pca(dataMat.A, 1)
    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.scatter(dataMat[:, 0].tolist(), dataMat[:, 1].tolist(), marker='^', s=90)
    ax.scatter(reconMat[:, 0].tolist(), reconMat[:, 1].tolist(), marker='o', s=50, c='red')
    plt.show()

结果：