05-linear-regression-simple.md

2.5 Linear regression

Slides

Notes

Model for solving regression tasks, in which the objective is to adjust a line for the data and make predictions on new values. The input of this model is the feature matrix X and a y vector of predictions is obtained, trying to be as close as possible to the actual y values. The linear regression formula is the sum of the bias term ( $w_0$ ), which refers to the predictions if there is no information, and each of the feature values times their corresponding weights as ( $x_{i1} \cdot w_1 + x_{i2} \cdot w_2 + ... + x_{in} \cdot w_n$ ).

So the simple linear regression formula looks like:

$g(x_i) = w_0 + x_{i1} \cdot w_1 + x_{i2} \cdot w_2 + ... + x_{in} \cdot w_n$.

And that can be further simplified as:

$g(x_i) = w_0 + \displaystyle\sum_{j=1}^{n} w_j \cdot x_{ij}$

Here is a simple implementation of Linear Regression in python:

w0 = 7.1
def linear_regression(xi):
    
    n = len(xi)
    
    pred = w0
    w = [0.01, 0.04, 0.002]
    for j in range(n):
        pred = pred + w[j] * xi[j]
    return pred

If we look at the $\displaystyle\sum_{j=1}^{n} w_j \cdot x_{ij}$ part in the above equation, we know that this is nothing else but a vector-vector multiplication. Hence, we can rewrite the equation as $g(x_i) = w_0 + x_i^T \cdot w$

We need to assure that the result is shown on the untransformed scale by using the inverse function exp().

The entire code of this project is available in this jupyter notebook.

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