3D-plot of the error function in a linear regression

Question

I would like to visually plot a 3D graph of the error function calculated for a given slope and y-intercept for a linear regression. This graph will be used to illustrate a gradient descent application. Let’s suppose we want to model a set of points with a line. To do this we’ll use the standard y=mx+b line equation where m is the line’s slope and b is the line’s y-intercept. To find the best line for our data, we need to find the best set of slope m and y-intercept b values. A standard approach to solving this type of problem is to define an error function (also called a cost function) that measures how “good” a given line is. This function will take in a (m,b) pair and return an error value based on how well the line fits the data. To compute this error for a given line, we’ll iterate through each (x,y) point in the data set and sum the square distances between each point’s y value and the candidate line’s y value (computed at mx+b). It’s conventional to square this distance to ensu…

Accepted Answer

Simply replace `fun` with `computeErrorForLineGivenPoints`:

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

def error(m, b, points):
    totalError = 0
    for i in range(0, len(points)):
        totalError += (points[i].y - (m * points[i].x + b)) ** 2
    return totalError / float(len(points))

x = y = np.arange(-3.0, 3.0, 0.05)
Point = collections.namedtuple('Point', ['x', 'y'])

m, b = 3, 2
noise = np.random.random(x.size)
points = [Point(xp, m*xp+b+err) for xp,err in zip(x, noise)]

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

ms = np.linspace(2.0, 4.0, 10)
bs = np.linspace(1.5, 2.5, 10)

M, B = np.meshgrid(ms, bs)
zs = np.array([error(mp, bp, points) 
               for mp, bp in zip(np.ravel(M), np.ravel(B))])
Z = zs.reshape(M.shape)

ax.plot_surface(M, B, Z, rstride=1, cstride=1, color='b', alpha=0.5)

ax.set_xlabel('m')
ax.set_ylabel('b')
ax.set_zlabel('error')

plt.show()
```

yields

Tip: I renamed `computeErrorForLineGivenPoints` as `error`. Generally, there is no need to name a function `compute...` since almost all functions compute something. You also do not need to specify "GivenPoints" since the function signature shows that `points` is an argument. If you have other error functions or variables in your program, `line_error` or `total_error` might be a better name for this function.

3D-plot of the error function in a linear regression

Solution

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