正交向量未正交绘制（Python 中的 PCA）

Question

我正在尝试在 Python 中实现 PCA，但是当我用主轴注释我的图形时，我的向量似乎不是正交的。

import numpy as np
import matplotlib.pyplot as plt 
import scipy.linalg as LA

N = 1000
xTrue = np.linspace(0, 1000, N)
yTrue = 4 * xTrue
xData = xTrue + np.random.normal(0, 100, N)
yData = yTrue + np.random.normal(0, 100, N)
xData = np.reshape(xData, (N, 1))
yData = np.reshape(yData, (N, 1))
Data = np.hstack((xData, yData))

C = np.cov(Data, rowvar = False)
e_vals, e_vecs = LA.eig(C)
e_vals = np.real(e_vals)
e_vecs = -e_vecs

avg = (1/Data.shape[0] )*sum(Data, 0)

arrowprops = dict(arrowstyle = '->', linewidth = 3, shrinkA = 0, shrinkB = 0, color = 'r')
plt.scatter(Data[:,0], Data[:,1])
plt.title("Principal Component Analysis Example: Linear Data")
for i in [0,1]:
    ax = plt.gca()
    ax.annotate('', avg + np.sqrt(e_vals[i])*e_vecs[:,i], avg, arrowprops = arrowprops)

我可以通过显示 np.matmul(e_vecs, e_vecs.T) 大致是一个单位矩阵来验证我的特征向量是正交的。但是我得到的图像如下：

很明显，图像中的向量不是正交的，但它们为什么不正交是没有意义的，因为通过向量 avg 翻译它们不应该消除这个属性。任何人都知道这是怎么回事？这是缩放问题还是我遗漏了一些重要参数？

Answer 1

The vectors are mutually orthogonal, but when you plot them they do not appear orthogonal due to the scale of the plot (the x-axis range is much smaller than the y-axis range). This becomes clear if the aspect of the plot is set to 'eqaul' via matplotlib.axes.Axes.set_aspect:

for i in [0,1]:
    ax = plt.gca()
    ax.set_aspect('equal')
    ax.annotate('', avg + np.sqrt(e_vals[i])*e_vecs[:,i], avg, arrowprops = arrowprops)

You can of course reverse the scaling for the vectors to force them to be visually orthogonal on a set of axis for which the aspect is non-equal, but that ultimately would not be representative of the actual data. I would recommend either setting the aspect to 'equal' using set_aspect('equal') as above or simply noting that the vectors are orthogonal but don't plot orthogonally due to the aspect of the plot.

正交向量未正交绘制（Python 中的 PCA）

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pca