使用matplotlib和np.linalg绘制协方差矩阵的特征向量

问题描述

我试图绘制从一堆点(3D中的多面体)接收的特征向量和协方差矩阵.这是我的工作.

I am trying to draw eigenvector and of covariance matrix received from a bunch of points (polyhedron in 3D). Here is what i do.

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()  

我这样做的结果并不令人满意.特征向量从两个不同角度的视图.

The plot I get by doing this is less than satisfactory. The view of the eigenvectors from two different angles.

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.但是似乎不起作用，也许特征向量没有正确地由np.linalg计算出来?
你能建议我我想念什么吗?

I was expecting something like this. Vectors popping form the centroid to give direction of largest variance. But it seems it is not working, perhaps the eigenvectors are not correctly calculated by np.linalg?
Can you suggest me what am i missing?

此外，在我拥有特征向量之后，我正在尝试绘制椭圆体.如果您也可以建议我，那将是很棒的:)

Also, I am trying to draw the ellipsoid after I have the eigenvectors. If you can suggest me on that too, it would be great :)

有点进步我认为np.linalg只是给了我本征向量的位置向量，所以我只是将它们转换为质心，

bit of progress I think the np.linalg is just giving me position vector of eigenvectors from origin, so i just transformed them to be from centroid,

#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以上代码

adding above code before going into the plotting part of #plotting Eigen vectors

现在，这给了我类似下面的内容

This now gives me something like below

推荐答案

eigvec中的特征向量是列向量 .因此，要通过迭代检索特征向量，您需要转置eigvec:

for vec in eigvec.T:  

根据您的观察，将vec需要移动centroid:

Coupling that with your observation that vec needs to be shifted by centroid:

vec += centroid

收益

出于完整性考虑，

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()

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