PCA降维练习

【题目】

1.现有我国大陆30个省、直辖市、自治区的经济发展状况数据集如表所示，包括8项经济指标：国民生产总值（A1）；居民消费水平（A2）；固定资产投资（A3）；职工平均工资（A4）；货物周转量（A5）；居民消费指数（A6）；商品零售价格指数（A7）；工业总产值（A8），试用基本PCA方法将这8项经济指标融合成3项综合指标。

【要求】

1.写出PCA完成降维的主要步骤；

2.详细写出题目降维的计算过程；

3.请大家在博客中直接完成或在作业本上完成后拍照上传。

import pandas as pd
import openpyxl
import numpy as np
data = pd.read_excel("D:\User\ROG\jupyter_root_directory\data/我国大陆经济发展状况数据.xlsx",header=None,engine='openpyxl')
data = data [2:]
data = data[[1,2,3,4,5,6,7,8]]
data

#减去平均值
sample,feature=data.shape
data = data - np.mean(data)
data

S = np.mat(data)
S = S.astype('float16') 
# 计算协方差矩阵
X = np.cov(data1.T)
X

array([[ 2.17512816e+06,  3.39017180e+05,  5.64795310e+05,
         3.66799624e+05,  4.18740435e+05, -8.14159678e+02,
        -7.37804742e+02,  7.53426315e+05],
       [ 3.39017180e+05,  7.42673545e+05,  1.47954656e+05,
         8.10174225e+05, -5.98597476e+04, -4.10269647e+02,
        -9.69473039e+02,  1.82942939e+05],
       [ 5.64795310e+05,  1.47954656e+05,  1.62302951e+05,
         2.10470018e+05,  7.98055667e+04, -2.28857444e+02,
        -2.74499481e+02,  1.86527121e+05],
       [ 3.66799624e+05,  8.10174225e+05,  2.10470018e+05,
         1.71571948e+06, -2.14593340e+05, -3.56041630e+02,
        -1.33849453e+03,  7.91434689e+04],
       [ 4.18740435e+05, -5.98597476e+04,  7.98055667e+04,
        -2.14593340e+05,  2.11547288e+05, -2.35784325e+02,
         1.89958719e+01,  1.77085901e+05],
       [-8.14159678e+02, -4.10269647e+02, -2.28857444e+02,
        -3.56041630e+02, -2.35784325e+02,  4.10102506e+00,
         2.93249666e+00, -1.48336671e+02],
       [-7.37804742e+02, -9.69473039e+02, -2.74499481e+02,
        -1.33849453e+03,  1.89958719e+01,  2.93249666e+00,
         3.60350766e+00, -2.13093770e+02],
       [ 7.53426315e+05,  1.82942939e+05,  1.86527121e+05,
         7.91434689e+04,  1.77085901e+05, -1.48336671e+02,
        -2.13093770e+02,  3.41794042e+05]])

#求特征值和特征向量
eig_val, eig_vec = np.linalg.eig(X)
eig_pairs = [(np.abs(eig_val[i]), eig_vec[:, i]) for i in range(feature)]
eig_val

array([3.00989343e+06, 1.90990488e+06, 3.00900997e+05, 8.22843300e+04,
4.08689937e+04, 5.31590865e+03, 4.22552725e+00, 4.17682352e-01])

index = np.argsort(-eig_val)
# 将特征值按从大到小进行排序
np.argsort(eig_val)

#进行降维
k = 3
selectVec = np.matrix(eig_vec.T[index[:k]])
finalData = data1 * selectVec.T # (30, 8) * (8, 3) = (30, 3)
finalData.shape
finalData

matrix([[  991.02791011, -2598.05143659,  -442.07545458],
        [ -145.34532529, -1734.65887357,   457.16158882],
        [  557.61661571,  1483.60034716,   -93.14172337],
        [-1197.70506441,   311.50193131,    20.75043138],
        [-1691.41478306,   542.6983315 ,   364.01896739],
        [  925.58426428,   922.66694957,   961.42758107],
        [-1105.78714784,   282.27855119,   652.96927436],
        [ -230.8764751 ,   915.2909667 ,  1154.93539315],
        [ 3479.73362389, -3833.86882421,  1432.59322724],
        [ 3279.28517562,  1443.82207397,  -340.60839295],
        [ 2020.42780485,  -236.33723265,  -368.15435258],
        [ -459.95372537,   899.6555233 ,   -51.4321421 ],
        [  421.88446529,  -480.08094189,   143.32609056],
        [-1424.33703514,   671.74965252,    96.75028233],
        [ 2712.10215068,  2166.91458621,  -234.05003709],
        [  370.55219993,  1915.41058406,  -103.89839129],
        [   75.40542303,   988.63141429,   171.25143746],
        [ -179.38146112,   819.30680274,   -18.32663648],
        [ 4553.43459221,  -648.74796439,  -936.50316474],
        [ -631.0710909 ,   148.57756477,  -260.22529535],
        [-1543.48297854,  -931.20547131,   148.74331019],
        [  963.19113519,  1714.01295905,  -193.78601902],
        [-1924.84798562,   212.88195738,  -181.92664169],
        [ -942.34945313,   -52.02992477,  -208.50404981],
        [-1165.55041791, -2423.78738221, -1276.80765054],
        [-1469.95701957,   456.57292109,    92.47839383],
        [-1461.05431309,  -532.72325834,  -483.85639992],
        [-1662.97477831, -1243.25114115,  -294.80964414],
        [-1974.30737332,  -688.33978176,   -78.22032709],
        [-1139.6165708 ,  -491.32802526,  -129.98141111]])