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I have been reading about linear regression a lot on internet. And people everywhere use a model: y = w*x+b and I have huge difficulties to understand why? As well I think they are overthinking with loss function. If you check the code bellow, my prediction model doesn't have constant, still it's working just fine.

import numpy as np

X = np.random.randint(100,1000,100).reshape(50,2) #features
y = np.mean(X,axis=1).reshape(50,1) #labels

w = np.random.random((2,1)) #weights

for i in range(100000):
    model = X.dot(w) #model
    loss = model-y #loss


    w = w- (X.T.dot(loss))*0.0000000001 #gradient

    if np.mean(np.abs(loss)) < 0.00001:
        break


prediction_input = np.array([[8880,9000]])

print(prediction_input.dot(w))

Linear models are convex function, so local minima is always it's global. I am sure there is a reason to include constant b, however I don't get why. Is there any good explanation please?

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This is a peculiar question. In the case of linear regression, the constant b represents the vertical offset or bias on the y-axis. Imagine fitting y=w*x + b on x=[0, 1, 2, 3] ,y=[500, 501, 502, 503]. Try fitting these points with only y=w*x...

The bias or y-intercept represented by b is the only way of reasonably fitting a relationship with a non-zero value of y when x is 0.

Let me know if I am misunderstanding the question! This is pretty fundamental to the form of linear regression. If you are not 100% sure that y=0 when x=0, there is no appropriate way to remove the constant term!

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  • $\begingroup$ Yes, you are absolutley correct, I have never thought it this way. $\endgroup$ – Stenga Mar 17 '18 at 17:31

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