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A few posts ago, we saw how to use the function numpy.linalg.lstsq(...) to solve an over-determined system. This time, we'll use it to estimate the parameters of a regression line.

A linear regression line is of the form w₁x+w₂=y and it is the line that minimizes the sum of the squares of the distance from each data point to the line. So, given n pairs of data (x_i, y_i), the parameters that we are looking for are w₁ and w₂ which minimize the error



and we can compute the parameter vector w = (w₁ , w₂)^T as the least-squares solution of the following over-determined system



Let's use numpy to compute the regression line:

from numpy import arange,array,ones,random,linalg
from pylab import plot,show

xi = arange(0,9)
A = array([ xi, ones(9)])
# linearly generated sequence
y = [19, 20, 20.5, 21.5, 22, 23, 23, 25.5, 24]
w = linalg.lstsq(A.T,y)[0] # obtaining the parameters

# plotting the line
line = w[0]*xi+w[1] # regression line
plot(xi,line,'r-',xi,y,'o')
show()

We can see the result in the plot below.



You can find more about data fitting using numpy in the following posts:

• Polynomial curve fitting
• Curve fitting using fmin