Weighted Least Squares

Problem

Fit y = a + b*x by weighted least squares, minimizing sum(w_i * (y_i - a - b*x_i)^2).

With W = sum(w_i), use the weighted means mx = sum(w_i*x_i)/W and my = sum(w_i*y_i)/W, then b = sum(w_i*(x_i - mx)*(y_i - my)) / sum(w_i*(x_i - mx)^2) and a = my - b*mx. Return [a, b]; if W is 0 or the input is empty return [0.0, 0.0], and if the weighted denominator is 0 return [my, 0.0].

Hint

Everything is the simple linear regression formula with each observation scaled by its weight.

Test cases

case 1

in: [[1,2,3,4,5],[3,5,7,9,11],[1,1,1,1,1]]

expected: [1,2]

case 2

in: [[1,2,3],[1,2,10],[1,1,1]]

expected: [-4.666666666666667,4.5]

case 3

in: [[1,2,3],[1,2,10],[1,1,9]]

expected: [-5.478260869565219,5.108695652173914]

case 4

in: [[2],[5],[3]]

expected: [5,0]

Starter codePython

def weighted_least_squares(X, y, weights):
    # Returns [intercept, slope]
    # Your code here
    pass

4 test cases · run your code against them in the editor