如何在 python 中使用 PCA 绘制 3D 平面？

Question

X = np.array([[24,13,38],[8,3,17],[21,6,40],[1,14,-9],[9,3,21],[7,1,14],[8,7,11],[10,16,3],[1,3,2],
    [15,2,30],[4,6,1],[12,10,18],[1,9,-4],[7,3,19],[5,1,13],[1,12,-6],[21,9,34],[8,8,7],
  [1,18,-18],[15,8,25],[16,10,29],[7,0,17],[14,2,31],[3,7,0],[5,6,7]])
pca = PCA(n_components=1)

pca.fit(X)
a = pca.components_[0][0] # a
b = pca.components_[0][1] # b
c = pca.components_[0][2] # c

def average(values):
    if(values) ==0:
        return None
    return sum(values, 0.0) / len(values)

x_mean = average(x) # For an approximation
y_mean = average(y)
z_mean = average(z)
d = -(a * x_mean + b * y_mean + c * z_mean)

所以 -0.375978766054x + 0.10612154283y -0.920531469111z + 15.1366572005 = 0

实际上，我不确定它是否正确。

我想在这种情况下使用 matplotlib 库绘制平面。

我该如何编码？

Answer 1

每个主成分在特征空间中定义一个向量。 PCA 根据每个方向上数据的方差对这些向量进行排序。 所以第一个向量将代表数据的最大方差和最后一个向量的最小方差。 假设数据分布在一个平面上，第三个向量应该垂直于该平面。 这是代码：

import numpy as np
from sklearn.decomposition import PCA
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

X = np.array([[24,13,38],[8,3,17],[21,6,40],[1,14,-9],[9,3,21],[7,1,14],[8,7,11],[10,16,3],[1,3,2],
    [15,2,30],[4,6,1],[12,10,18],[1,9,-4],[7,3,19],[5,1,13],[1,12,-6],[21,9,34],[8,8,7],
  [1,18,-18],[15,8,25],[16,10,29],[7,0,17],[14,2,31],[3,7,0],[5,6,7]])
pca = PCA(n_components=3)

pca.fit(X)
eig_vec = pca.components_

print(pca.explained_variance_ratio_)
# [0.90946569 0.08816839 0.00236591]
# Percentage of variance explain by last vector is less 0.2%

# This is the normal vector of minimum variance
normal = eig_vec[2, :]  # (a, b, c)
centroid = np.mean(X, axis=0)

# Every point (x, y, z) on the plane should satisfy a*x+b*y+c*z = d
# Taking centroid as a point on the plane
d = -centroid.dot(normal)

# Draw plane
xx, yy = np.meshgrid(np.arange(np.min(X[:, 0]), np.max(X[:, 0])), np.arange(np.min(X[:, 1]), np.max(X[:, 1])))

z = (-normal[0] * xx - normal[1] * yy - d) * 1. / normal[2]

# plot the surface
plt3d = plt.figure().gca(projection='3d')
plt3d.plot_surface(xx, yy, z)
plt3d.scatter(*(X.T))
plt.show()

Answer 2

第一个主成分没有定义平面，它定义了一个三维向量。 以下是在 3D 中对其进行可视化的方法：代码从您的代码开始，然后是绘图步骤：

import numpy as np
from sklearn.decomposition import PCA

X = np.array([[24, 13, 38], [8, 3, 17], [21, 6, 40], [1, 14, -9], [9, 3, 21], [7, 1, 14],
              [8, 7, 11], [10, 16, 3], [1, 3, 2], [15, 2, 30], [4, 6, 1], [12, 10, 18], [1, 9, -4],
              [7, 3, 19], [5, 1, 13], [1, 12, -6], [21, 9, 34], [8, 8, 7], [1, 18, -18],
              [15, 8, 25], [16, 10, 29], [7, 0, 17], [14, 2, 31], [3, 7, 0], [5, 6, 7]])

pca = PCA(n_components=1)
pca.fit(X)

## New code below
p = pca.components_
centroid = np.mean(X, 0)
segments = np.arange(-40, 40)[:, np.newaxis] * p

import matplotlib
matplotlib.use('TkAgg') # might not be necessary for you
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
plt.ion()

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
scatterplot = ax.scatter(*(X.T))
lineplot = ax.plot(*(centroid + segments).T, color="red")
plt.xlabel('x')
plt.ylabel('y')
plt.savefig('result.png', dpi=150)

（注意上面的代码是用yapf自动格式化的，我强烈推荐。）结果图：