Find and draw regression plane to a set of points

Question

I want to fit a plane to some data points and draw it. My current code is this:

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

points = [(1.1,2.1,8.1),
          (3.2,4.2,8.0),
          (5.3,1.3,8.2),
          (3.4,2.4,8.3),
          (1.5,4.5,8.0)]

xs, ys, zs = zip(*points)

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

ax.scatter(xs, ys, zs)

point  = np.array([0.0, 0.0, 8.1])
normal = np.array([0.0, 0.0, 1.0])
d = -point.dot(normal)
xx, yy = np.meshgrid([-5,10], [-5,10])
z = (-normal[0] * xx - normal[1] * yy - d) * 1. /normal[2]
ax.plot_surface(xx, yy, z, alpha=0.2, color=[0,1,0])

ax.set_xlim(-10,10)
ax.set_ylim(-10,10)
ax.set_zlim(  0,10)

plt.show()

which results in the following:

As you can see at the moment I create the plane manually. How can I calculate it? I guess it is possible with scipy.optimize.minimize somehow. The kind of error function is not that important to me at the moment. I think least squares (vertical point-plane-distance) would be fine. It would be cool if one of you could show me how to do it.

Answer 1

Oh, the idea just came to my mind. It's quite easy. :-)

import numpy as np
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
import scipy.optimize
import functools

def plane(x, y, params):
    a = params[0]
    b = params[1]
    c = params[2]
    z = a*x + b*y + c
    return z

def error(params, points):
    result = 0
    for (x,y,z) in points:
        plane_z = plane(x, y, params)
        diff = abs(plane_z - z)
        result += diff**2
    return result

def cross(a, b):
    return [a[1]*b[2] - a[2]*b[1],
            a[2]*b[0] - a[0]*b[2],
            a[0]*b[1] - a[1]*b[0]]

points = [(1.1,2.1,8.1),
          (3.2,4.2,8.0),
          (5.3,1.3,8.2),
          (3.4,2.4,8.3),
          (1.5,4.5,8.0)]

fun = functools.partial(error, points=points)
params0 = [0, 0, 0]
res = scipy.optimize.minimize(fun, params0)

a = res.x[0]
b = res.x[1]
c = res.x[2]

xs, ys, zs = zip(*points)

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

ax.scatter(xs, ys, zs)

point  = np.array([0.0, 0.0, c])
normal = np.array(cross([1,0,a], [0,1,b]))
d = -point.dot(normal)
xx, yy = np.meshgrid([-5,10], [-5,10])
z = (-normal[0] * xx - normal[1] * yy - d) * 1. /normal[2]
ax.plot_surface(xx, yy, z, alpha=0.2, color=[0,1,0])

ax.set_xlim(-10,10)
ax.set_ylim(-10,10)
ax.set_zlim(  0,10)

plt.show()

Sorry for asking unnecessarily.

Answer 2

Another way is with a straight forward least squares solution. The equation for a plane is: ax + by + c = z. So set up matrices like this with all your data:

    x_0   y_0   1  
A = x_1   y_1   1  
          ... 
    x_n   y_n   1  

And

    a  
x = b  
    c

And

    z_0   
B = z_1   
    ...   
    z_n

In other words: Ax = B. Now solve for x which are your coefficients. But since (I assume) you have more than 3 points, the system is over-determined so you need to use the left pseudo inverse. So the answer is:

a 
b = (A^T A)^-1 A^T B
c

And here is some simple Python code with an example:

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

N_POINTS = 10
TARGET_X_SLOPE = 2
TARGET_y_SLOPE = 3
TARGET_OFFSET  = 5
EXTENTS = 5
NOISE = 5

# create random data
xs = [np.random.uniform(2*EXTENTS)-EXTENTS for i in range(N_POINTS)]
ys = [np.random.uniform(2*EXTENTS)-EXTENTS for i in range(N_POINTS)]
zs = []
for i in range(N_POINTS):
    zs.append(xs[i]*TARGET_X_SLOPE + \
              ys[i]*TARGET_y_SLOPE + \
              TARGET_OFFSET + np.random.normal(scale=NOISE))

# plot raw data
plt.figure()
ax = plt.subplot(111, projection='3d')
ax.scatter(xs, ys, zs, color='b')

# do fit
tmp_A = []
tmp_b = []
for i in range(len(xs)):
    tmp_A.append([xs[i], ys[i], 1])
    tmp_b.append(zs[i])
b = np.matrix(tmp_b).T
A = np.matrix(tmp_A)
fit = (A.T * A).I * A.T * b
errors = b - A * fit
residual = np.linalg.norm(errors)

print "solution:"
print "%f x + %f y + %f = z" % (fit[0], fit[1], fit[2])
print "errors:"
print errors
print "residual:"
print residual

# plot plane
xlim = ax.get_xlim()
ylim = ax.get_ylim()
X,Y = np.meshgrid(np.arange(xlim[0], xlim[1]),
                  np.arange(ylim[0], ylim[1]))
Z = np.zeros(X.shape)
for r in range(X.shape[0]):
    for c in range(X.shape[1]):
        Z[r,c] = fit[0] * X[r,c] + fit[1] * Y[r,c] + fit[2]
ax.plot_wireframe(X,Y,Z, color='k')

ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
plt.show()