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Concept
Weighted least squares
Summary
Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (heteroscedasticity) is incorporated into the regression.
WLS is also a specialization of generalized least squares, when all the off-diagonal entries of the covariance matrix of the errors, are null.
Formulation
The fit of a model to a data point is measured by its residual, r_i , defined as the difference between a measured value of the dependent variable, y_i and the value predicted by the model, f(x_i, \boldsymbol\beta):
r_i(\boldsymbol\beta) = y_i - f(x_i, \boldsymbol\beta).
If the errors are uncorrelated and have equal variance, then the function
S(\boldsymbol\beta) = \sum_i r_i(\boldsymbol\beta)^2,
is minimised at \boldsymbol\hat\beta, such that \frac{\pa
Official source
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