Least-square methods

 
least-square-solutions

Numpy and Least-square solutions

Assume we have the following line, and also assume we are not that clever and can’t guess its slope and intercept…

In [103]:
import matplotlib.pyplot as plt
import numpy as np

%matplotlib inline

x=np.array([1., 2. , 3. , 4. ,5.,6])
y=np.array([6.,8.,10.,12.,14.,16.])


plt.plot(x,y)
plt.show()

The obvious solution is:
$$
coeffs= X^{-1}y
$$
where $X$ is:
$$
\left( \begin{array}{ccc}
1 & 1 \\
2 & 1 \\
3 & 1 \\
4 & 1 \\
5 & 1 \\
6 & 1 \\
\end{array} \right)
$$

In [104]:
X=np.mat(  np.vstack((x,np.ones(len(x)))).T)
y=np.mat(y.reshape(6,1))
X.I * y
Out[104]:
matrix([[ 2.],
        [ 4.]])

Now suppose we have:

In [105]:
x=np.array([1., 2. , 3. , 4. ,5.,6])
y=np.array([5.,7.,10.,11.,13.,14.])

plt.plot(x,y)
plt.show()

We can find approximate solutions.

Using covariation

$$
slope=\frac{ \sum_i{(x_i – \bar{x})(y_i – \bar{y}) } }{\sum_i{{(x_i – \bar{x})}^2}}
$$
In [106]:
x_bar=np.mean(x)
y_bar=np.mean(y)

covariation = np.sum( (x-x_bar) * (y - y_bar) )
x_variation=np.sum((x-x_bar)**2)

slope=covariation/x_variation
intercept = y_bar - (slope * x_bar)
print("Slope = {:.3f} and intercept = {:.3f}".format(slope,intercept) )
Slope = 1.829 and intercept = 3.600

Using normal equations

In [107]:
X=np.mat(X) ; y=np.mat(y.reshape(6,1))
slope_and_intercept = (X.T * X).I * X.T * y 
slope_and_intercept
Out[107]:
matrix([[ 1.82857143],
        [ 3.6       ]])

Finally, let Numpy do the job …

In [108]:
np.linalg.lstsq(X,y)[0]
Out[108]:
matrix([[ 1.82857143],
        [ 3.6       ]])

 

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