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I've generated two clouds of 3d points from multivariate_normal 

data = np.random.multivariate_normal([2,2,2],[[1,0,0],[0,5,0],[0,0,10]], 
size=500)
data = np.vstack((data, np.random.multivariate_normal([-2,-2,-2], [[1,0,0],[0,5,0],[0,0,10]], size=500)))
data = data - data.mean(axis=0)

And try to do PCA like this

covmat = np.cov(data.T)
v, W = np.linalg.eig(covmat)

And draw:

def get_vec(eig_v, eig_vec):
    t = np.linspace(0, eig_v)
    return np.array([np.array(v * eig_vec) for v in t])

def ang(v1, v2):
    return np.rad2deg(np.arccos(np.dot(v1,v2)/np.linalg.norm(v1)/np.linalg.norm(v2)))

l1 = get_vec(v[0], W[:,0])
l2 = get_vec(v[1], W[:,1])
l3 = get_vec(v[2], W[:,2])
x = data[:,0]
y = data[:,1]
z = data[:,2]
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot(l1[:,0],l1[:,1],l1[:,2], c='r')
ax.plot(l2[:,0],l2[:,1],l2[:,2], c='b')
ax.plot(l3[:,0],l3[:,1],l3[:,2], c='y')
ax.scatter(x,y,z,c='g')
plt.show()

This is what I get: enter image description here

It's clearly visible that axes are not orthogonal. I've checked it and they seem to be orthogonal with regard to numbers:

print(ang(W[:,0], W[:,1]))
print(ang(W[:,0], W[:,2]))
print(ang(W[:,1], W[:,2]))

90.00000000000003 89.99999999999999 90.0 Could it be that such a tiny error makes that much visual difference?

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  • $\begingroup$ The axes are in 3D space; they are orthogonal as you've found. Maybe you're just seeing them at an angle in your plot? $\endgroup$ – Sean Owen Jan 11 '19 at 20:07
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    $\begingroup$ @SeanOwen Well, that may be the case but I was not able to find the point of view where they are orthogonal. Actually there should be at least 3 such points(looking from the top of each vector) but even from there vectors are not orthogonal. I suspect that the problem here is projection matrix which distorts orthogonality. $\endgroup$ – s0nicYouth Jan 14 '19 at 11:38
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The PCA projections do not look not orthogonal because your figure axes are not equal.

Set all axis equal with something like this:

ax.axis('equal')

or 

ax.xlim(-5, 5)
ax.ylim(-5, 5)
ax.zlim(-5, 5)
ax.gca().set_aspect('equal')
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  • $\begingroup$ Yeah, you are right, thanks. Was it caused by the way projection matrix distorts the plot? $\endgroup$ – s0nicYouth Jan 11 '19 at 19:14

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