MATLAB: Demo Weighted Nonlinear Regression (Statistics Toolbox)

nhanesnonlinearregressionStatistics and Machine Learning Toolboxweighted

In the Statistics toolbox Demo "Weighted Nonlinear Regression" the weights are normilized w = w / mean(w), but when applied to the data the square root of the normilized weights are used modelFunw = @(b,x) sqrt(w).*modelFun(b,x);
Why use the sqrt(w) and not just w? Should you always use sqrt(w)? The demo provided no explantion as to why sqrt(w) was used. Can someone provide an explantion?
I'm trying to apply this to the heavly weighted NHANES database.

Best Answer

Weighted least squares means we want to minimize
     sum over i of    w(i) * {y(i)-yfit(i)}^2
You can write this by multiplying both y and yfit by sqrt(w), inside the thing in {} that is squared.
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