What is wrong with my simple linear regression?

Question

I fail to find the theta using this code.

I added plotting code to help visualize the issue.

Please, help me find the bug in this short block of code

Thanks

import numpy as np import matplotlib.pyplot as plt N = 20 def arr(n): return np.arange(n) + 1 def linear(features, y): x = np.vstack(features).T xT = np.transpose(x) xTx = xT.dot(x) return np.linalg.inv(xTx).dot(xT).dot(y) def plot(x, y, dots_y): plt.plot(x, y) plt.plot(x, dots_y, marker='o', linestyle=' ', color='r') plt.show() y = arr(N) ** 2 + 3 theta = linear((np.ones(N), arr(N), arr(N) ** 2), y) plot(arr(N), arr(N) ** theta[1] + theta[0], y)

score 1 · Accepted Answer · 2017-11-25 07:55:08Z

The error is in the plotting line, which should be

plot(arr(N), arr(N)**2 * theta[2] + arr(N) * theta[1] + theta[0], y)

according to the quadratic polynomial model.

Also; I suppose you did the computation of least-square solution this way for expository reasons, but in practice, linear least squares fit would be obtained with np.linalg.lstsq as follows, with much shorter and more efficient code:

N = 20 x = np.arange(1, N+1) y = x**2 + 3 basis = np.vstack((x**0, x**1, x**2)).T # basis for the space of quadratic polynomials theta = np.linalg.lstsq(basis, y)[0] # least squares approximation to y in this basis plt.plot(x, y, 'ro') # original points plt.plot(x, basis.dot(theta)) # best fit plt.show()

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