我正在使用部分鸢尾花数据集来更好地理解PCA。
这是我的代码：
from sklearn.datasets import load_irisimport numpy as npimport matplotlib.pyplot as pltfrom sklearn import decompositiondataset = load_iris()X = dataset.data[:20,]pca = decomposition.PCA(n_components=4)pca.fit(X)X = pca.transform(X)print(X)print()print(pca.explained_variance_ratio_)print(pca.explained_variance_)print(pca.noise_variance_)print()print(pca.components_)print()pca = decomposition.PCA(n_components=3)pca.fit(X)X = pca.transform(X)print(X)print()print(pca.explained_variance_ratio_)print(pca.explained_variance_)print(pca.noise_variance_)print()print(pca.components_)print()pca = decomposition.PCA(n_components=2)pca.fit(X)X = pca.transform(X)print(X)print()print(pca.explained_variance_ratio_)print(pca.explained_variance_)print(pca.noise_variance_)print()print(pca.components_)print()pca = decomposition.PCA(n_components=1)pca.fit(X)X = pca.transform(X)print(X)print()print(pca.explained_variance_ratio_)print(pca.explained_variance_)print(pca.noise_variance_)print()print(pca.components_)print()输出：
| F1 | F2 | F3 | F4 | Label ||5.1 |3.5 |1.4 |0.2 | 0 ||4.9 |3.0 |1.4 |0.2 | 0 ||4.7 |3.2 |1.3 |0.2 | 0 ||4.6 |3.1 |1.5 |0.2 | 0 ||5.0 |3.6 |1.4 |0.2 | 0 ||5.4 |3.9 |1.7 |0.4 | 0 ||4.6 |3.4 |1.4 |0.3 | 0 ||5.0 |3.4 |1.5 |0.2 | 0 ||4.4 |2.9 |1.4 |0.2 | 0 ||4.9 |3.1 |1.5 |0.1 | 0 ||5.4 |3.7 |1.5 |0.2 | 0 ||4.8 |3.4 |1.6 |0.2 | 0 ||4.8 |3.0 |1.4 |0.1 | 0 ||4.3 |3.0 |1.1 |0.1 | 0 ||5.8 |4.0 |1.2 |0.2 | 0 ||5.7 |4.4 |1.5 |0.4 | 0 ||5.4 |3.9 |1.3 |0.4 | 0 ||5.1 |3.5 |1.4 |0.3 | 0 ||5.7 |3.8 |1.7 |0.3 | 0 ||5.1 |3.8 |1.5 |0.3 | 0 |[[ -5.35882132e-02 2.13091549e-02 5.63776995e-02 2.38909674e-02] [ 4.31102885e-01 2.27802156e-01 7.74776903e-02 -8.56077547e-02] [ 4.46437821e-01 -6.48981661e-02 7.80252213e-02 -2.16463511e-02] [ 5.70213598e-01 1.37832371e-02 -1.17201913e-01 -2.27730577e-03] [ -4.99837824e-02 -1.06433448e-01 1.11801355e-02 6.42148516e-02] [ -5.88493547e-01 1.19234918e-02 -2.42112963e-01 -4.46036896e-02] [ 3.62588639e-01 -2.42562846e-01 -9.89230051e-02 -3.13366123e-02] [ 7.83136388e-02 6.27754417e-02 -4.79067754e-02 2.65736478e-02] [ 8.58395527e-01 -1.49295381e-02 -5.29428852e-02 -4.69710396e-02] [ 3.65880852e-01 2.20160693e-01 -4.51271386e-03 5.21066893e-02] [ -4.13586321e-01 1.11767646e-01 2.13883619e-02 5.54246013e-02] [ 2.13819922e-01 -2.35008745e-02 -1.97388814e-01 6.95802124e-02] [ 5.14034854e-01 1.87196747e-01 7.30881295e-02 2.14166399e-02] [ 8.97493973e-01 -2.33177183e-01 1.99567657e-01 3.71580447e-02] [ -8.81108056e-01 4.91145021e-02 3.63511477e-01 3.42164603e-02] [ -1.12874867e+00 -2.07254026e-01 -5.20579454e-02 1.83622028e-02] [ -5.55989247e-01 -1.36936973e-01 1.21657674e-01 -1.11349149e-01] [ -6.47040031e-02 1.68848098e-04 3.14975704e-02 -6.99733273e-02] [ -7.24614545e-01 2.84297834e-01 -1.13495890e-01 -1.73834789e-02] [ -2.77465322e-01 -1.60606696e-01 -1.07228711e-01 2.82043907e-02]][ 0.87954353 0.06300167 0.05039505 0.00705974][ 0.31612993 0.02264438 0.01811324 0.00253745]0.0[[-0.71816179 -0.68211748 -0.08126075 -0.1111579 ] [ 0.61745716 -0.65996887 0.37215116 -0.21140307] [ 0.2926969 -0.15927874 -0.90942659 -0.24880129] [-0.131601 0.27163784 0.16686365 -0.93864295]][[ -5.35882132e-02 2.13091549e-02 -5.63776995e-02] [ 4.31102885e-01 2.27802156e-01 -7.74776903e-02] [ 4.46437821e-01 -6.48981661e-02 -7.80252213e-02] [ 5.70213598e-01 1.37832371e-02 1.17201913e-01] [ -4.99837824e-02 -1.06433448e-01 -1.11801355e-02] [ -5.88493547e-01 1.19234918e-02 2.42112963e-01] [ 3.62588639e-01 -2.42562846e-01 9.89230051e-02] [ 7.83136388e-02 6.27754417e-02 4.79067754e-02] [ 8.58395527e-01 -1.49295381e-02 5.29428852e-02] [ 3.65880852e-01 2.20160693e-01 4.51271386e-03] [ -4.13586321e-01 1.11767646e-01 -2.13883619e-02] [ 2.13819922e-01 -2.35008745e-02 1.97388814e-01] [ 5.14034854e-01 1.87196747e-01 -7.30881295e-02] [ 8.97493973e-01 -2.33177183e-01 -1.99567657e-01] [ -8.81108056e-01 4.91145021e-02 -3.63511477e-01] [ -1.12874867e+00 -2.07254026e-01 5.20579454e-02] [ -5.55989247e-01 -1.36936973e-01 -1.21657674e-01] [ -6.47040031e-02 1.68848098e-04 -3.14975704e-02] [ -7.24614545e-01 2.84297834e-01 1.13495890e-01] [ -2.77465322e-01 -1.60606696e-01 1.07228711e-01]][ 0.87954353 0.06300167 0.05039505][ 0.31612993 0.02264438 0.01811324]0.00253744874373[[ 1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00] [ -0.00000000e+00 1.00000000e+00 -3.33066907e-15 0.00000000e+00] [ 0.00000000e+00 -3.10862447e-15 -1.00000000e+00 -3.60822483e-16]][[ -5.35882132e-02 2.13091549e-02] [ 4.31102885e-01 2.27802156e-01] [ 4.46437821e-01 -6.48981661e-02] [ 5.70213598e-01 1.37832371e-02] [ -4.99837824e-02 -1.06433448e-01] [ -5.88493547e-01 1.19234918e-02] [ 3.62588639e-01 -2.42562846e-01] [ 7.83136388e-02 6.27754417e-02] [ 8.58395527e-01 -1.49295381e-02] [ 3.65880852e-01 2.20160693e-01] [ -4.13586321e-01 1.11767646e-01] [ 2.13819922e-01 -2.35008745e-02] [ 5.14034854e-01 1.87196747e-01] [ 8.97493973e-01 -2.33177183e-01] [ -8.81108056e-01 4.91145021e-02] [ -1.12874867e+00 -2.07254026e-01] [ -5.55989247e-01 -1.36936973e-01] [ -6.47040031e-02 1.68848098e-04] [ -7.24614545e-01 2.84297834e-01] [ -2.77465322e-01 -1.60606696e-01]][ 0.88579703 0.06344961][ 0.31612993 0.02264438]0.0181132415475[[ 1.00000000e+00 0.00000000e+00 0.00000000e+00] [ -0.00000000e+00 1.00000000e+00 -5.55111512e-16]][[-0.05358821] [ 0.43110288] [ 0.44643782] [ 0.5702136 ] [-0.04998378] [-0.58849355] [ 0.36258864] [ 0.07831364] [ 0.85839553] [ 0.36588085] [-0.41358632] [ 0.21381992] [ 0.51403485] [ 0.89749397] [-0.88110806] [-1.12874867] [-0.55598925] [-0.064704 ] [-0.72461455] [-0.27746532]][ 0.93315793][ 0.31612993]0.0226443764968[[ 1. 0.]]在我的数据集中，F1具有最高的方差。这在PCA的输出中是如何体现的？
这里的“解释方差”具体是什么意思？这是否意味着原始特征对新计算值的方差有多大影响？
为什么在使用4个成分的第一个例子中噪声方差为0？
components_具体是什么？它们是n维特征向量吗？
回答：
F1具有最高的方差。这在PCA的输出中是如何体现的？
PCA是一种特征转换技术，它旋转您的原始数据维度并转换到新的正交特征空间。在新的特征空间中，主成分（您数据的标准化协方差矩阵的正交特征向量）构成了空间的维度。这些成分是您原始特征维度的线性组合。考虑以下代码，主导主成分PC1（捕捉数据中最高方差）可以表示为特征的线性组合，如PC1=-0.718162*F1+0.292697*F3-0.131601*F4。
import pandas as pdpd.DataFrame(pca.components_, columns=['PC1', 'PC2', 'PC3', 'PC4'], index=['F1', 'F2', 'F3', 'F4'])# PC1 PC2 PC3 PC4#F1 -0.718162 -0.682117 -0.081261 -0.111158#F2 0.617457 -0.659969 0.372151 -0.211403#F3 0.292697 -0.159279 -0.909427 -0.248801#F4 -0.131601 0.271638 0.166864 -0.938643这里的“解释方差”具体是什么意思？这是否意味着原始特征对新计算值的方差有多大影响？
每个选定成分解释的方差量，通过简单地获取PCA加载列的方差（即pca.transform返回的列的方差，即转换后的特征的方差，而不是原始的），见以下代码：
X = pca.transform(X)print(np.var(X, axis=0))#[ 0.31612993 0.02264438 0.01811324 0.00253745]print(pca.explained_variance_)#[ 0.31612993 0.02264438 0.01811324 0.00253745]为什么在使用4个成分的第一个例子中噪声方差为0？
因为在第一种情况下我们没有进行任何降维，我们只是将特征空间转换到另一个空间并使用了所有4个成分，没有排除任何成分（因此没有信息丢失）。
components_具体是什么？它们是n维特征向量吗？
成分可以被认为是缩放数据的协方差矩阵的正交特征向量，尽管如文档所说，它是使用奇异值分解以更数值稳定的方式计算的，在这种情况下，它们是从右奇异向量计算得出的。
