python - 使用 matplotlib 和 np.linalg 绘制协方差矩阵的特征向量

Question

我正在尝试绘制从一堆点(3D 多面体)接收到的特征向量和协方差矩阵。这就是我所做的。

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from numpy import linalg as la
from matplotlib.patches import FancyArrowPatch
from mpl_toolkits.mplot3d import proj3d

class Arrow3D(FancyArrowPatch):
    def __init__(self, xs, ys, zs, *args, **kwargs):
        FancyArrowPatch.__init__(self, (0,0), (0,0), *args, **kwargs)
        self._verts3d = xs, ys, zs

    def draw(self, renderer):
        xs3d, ys3d, zs3d = self._verts3d
        xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, renderer.M)
        self.set_positions((xs[0],ys[0]),(xs[1],ys[1]))
        FancyArrowPatch.draw(self, renderer)
##################################################################################################
#here i start with drawing the actual polyhedron and the vector 
##################################################################################################
#generate num random points in 3d
num = 5
#coord = 10*np.random.rand(3,num)#num points in 3D #first axis is x, second = y, third = z
#xcod = np.array([1,2,3,2.7,2.4,1])
xcod = np.array([1,1,1,1,1,1])
ycod = np.array([1,1,4.5,5.,6,1])
zcod = np.array([1,-2,0,2,3,1])
#coord = np.concatenate(coord,coord[0])
#####plotting in 3d
fig = plt.figure()
ax = fig.add_subplot(111,projection = '3d')
#plotting all the points
ax.plot(xcod,ycod,zcod,'x-')
#adding labels for vertice
for i in range(num):
    ax.text(xcod[i],ycod[i],zcod[i],'%d(%.2f,%.2f,%.2f)'%(i,xcod[i],ycod[i],zcod[i]))
#supposed centroid
centroid = np.array([np.mean(xcod),np.mean(ycod),np.mean(zcod)])
ax.scatter(centroid[0],centroid[1],centroid[2],marker = 'o',color='r')
#labelling the axes
ax.set_xlabel("x axis")
ax.set_ylabel("y axis")
ax.set_zlabel("z axis")
#getting a stack of all vertices, while removing last repeat vertex
cod = np.vstack((np.delete(xcod,-1),np.delete(ycod,-1),np.delete(zcod,-1)))
#caculating covariance matrix
#ddof = 0 is using simple averages or normalising with N ; ddof = 1 means normalising with N-1
covmat = np.cov(cod,ddof=0)
#computing eigen values and eigen vectors
eigval,eigvec = la.eig(covmat)
#multiplying eigen value and eigen vec
#for counter in range(len(eigval)):
#    eigvec[counter]= eigval[counter]*eigvec[counter]
#####################################################################################
#plotting Eigen vectors
#####################################################################################
for vec in eigvec:#fetching one vector from list of eigvecs
    #drawing the vec, basically drawing a arrow form centroid to the end point of vec
    drawvec = Arrow3D([centroid[0],vec[0]],[centroid[1],vec[1]],[centroid[2],vec[2]],mutation_scale=20,lw=3,arrowstyle="-|>",color='r')
    #adding the arrow to the plot
    ax.add_artist(drawvec)
#plot show
plt.show()  

我这样做得到的情节并不令人满意。从两个不同角度观察特征向量。

我期待这样的事情。向量从质心弹出以给出最大方差的方向。但它似乎不起作用，也许 np.linalg 没有正确计算特征向量？
你能告诉我我错过了什么吗？

此外，我在获得特征向量后尝试绘制椭圆体。如果你也能就此向我提出建议，那就太好了:)

编辑:一点进步我认为 np.linalg 只是从原点给我特征向量的位置向量，所以我只是将它们转换为来自质心，

#getting tuples of x,y,z
verts = [zip(xcod,ycod,zcod)]
#plotting polyhedron surface
ax.add_collection3d(Poly3DCollection(verts,alpha=0.5))
#changing eigvec from origin to centroid
for counteri in range(len(eigvec)):
    eigvec[counteri][0]+=centroid[0]
    eigvec[counteri][1]+=centroid[1]
    eigvec[counteri][2]+=centroid[2]

在进入#plotting Eigen vectors 的绘图部分之前添加上面的代码

现在这给了我类似下面的东西

Answer 1

The eigenvectors in eigvec are column vectors .因此，要通过迭代检索特征向量，需要转置eigvec:

for vec in eigvec.T:  

结合您的观察，vec 需要移动 centroid:

vec += centroid

产量

为了完整性，

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from numpy import linalg as LA
from matplotlib.patches import FancyArrowPatch
from mpl_toolkits.mplot3d import proj3d


class Arrow3D(FancyArrowPatch):

    def __init__(self, xs, ys, zs, *args, **kwargs):
        FancyArrowPatch.__init__(self, (0, 0), (0, 0), *args, **kwargs)
        self._verts3d = xs, ys, zs

    def draw(self, renderer):
        xs3d, ys3d, zs3d = self._verts3d
        xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, renderer.M)
        self.set_positions((xs[0], ys[0]), (xs[1], ys[1]))
        FancyArrowPatch.draw(self, renderer)
##########################################################################
# here i start with drawing the actual polyhedron and the vector
##########################################################################
# generate num random points in 3d
num = 5
# coord = 10*np.random.rand(3,num)#num points in 3D #first axis is x, second = y, third = z
#xcod = np.array([1,2,3,2.7,2.4,1])
xcod = np.array([1, 1, 1, 1, 1, 1])
ycod = np.array([1, 1, 4.5, 5., 6, 1])
zcod = np.array([1, -2, 0, 2, 3, 1])
#coord = np.concatenate(coord,coord[0])
# plotting in 3d
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
# plotting all the points
ax.plot(xcod, ycod, zcod, 'x-')
# adding labels for vertice
for i in range(num):
    ax.text(xcod[i], ycod[i], zcod[i], '%d(%.2f,%.2f,%.2f)' %
            (i, xcod[i], ycod[i], zcod[i]))
# supposed centroid
centroid = np.array([np.mean(xcod), np.mean(ycod), np.mean(zcod)])
ax.scatter(centroid[0], centroid[1], centroid[2], marker='o', color='r')
# labelling the axes
ax.set_xlabel("x axis")
ax.set_ylabel("y axis")
ax.set_zlabel("z axis")
# getting a stack of all vertices, while removing last repeat vertex
cod = np.vstack(
    (np.delete(xcod, -1), np.delete(ycod, -1), np.delete(zcod, -1)))
# caculating covariance matrix
# ddof = 0 is using simple averages or normalising with N ; ddof = 1 means
# normalising with N-1
covmat = np.cov(cod, ddof=0)
# computing eigen values and eigen vectors
eigval, eigvec = LA.eig(covmat)
# multiplying eigen value and eigen vec
# for counter in range(len(eigval)):
#    eigvec[counter]= eigval[counter]*eigvec[counter]
##########################################################################
# plotting Eigen vectors
##########################################################################
for vec in eigvec.T:  # fetching one vector from list of eigvecs
    # drawing the vec, basically drawing a arrow form centroid to the end
    # point of vec
    vec += centroid
    drawvec = Arrow3D([centroid[0], vec[0]], [centroid[1], vec[1]], [centroid[2], vec[2]],
                      mutation_scale=20, lw=3, arrowstyle="-|>", color='r')
    # adding the arrow to the plot
    ax.add_artist(drawvec)
# plot show
plt.show()