Accessing estimated parameters from nlinfit for weighted regression

Hello,

I am looking to perform a weighted, non-linear regression on some data. I am currently fitting the data using the nlinfit function. I have derived the analytical expression of the variance of the data as a function of the estimated parameters, and I would thus like to use the estimated beta parameters at each fitting step as inputs into my weighted function.

I.e., I essentially want argmin sum((ydata-F(beta,x)).^2./sigma(beta,x).^2)

However, according to the documentation, the weights will only accept the output of the fitted model as a vector. Is there a way to access the estimated parameters during the fitting procedure so that nlinfit can weight the data as desired?

Thank you for your time

(Edit: for now I have simply fitted the parameters using fminsearch for the above expression, but I was wondering if there is an option on nlinfit or otherwise that allows you to adjust the weights during the fitting process as described)

Answer 1

[Torsten]

Use "lsqnonlin" and define the functions f_i as

f_i = (ydata_i - F(beta,xdata_i))/sigma(beta,xdata_i)

"lsqnonlin" is the solver best suited for this kind of problem.

But you can also use nlinfit with the weights returned from a function.

From the documentation of nlinfit:

Weights:

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a vector of real positive weights or a function handle. You can use observation weights to down-weight the observations that you want to have less influence on the fitted model.

If W is a vector, then it must be the same size as Y.

If W is a function handle, then it must accept a vector of predicted response values as input, and return a vector of real positive weights as output.

[Maximilian Bailey]

That's perfect, thank you! Just quickly though, why is the lsqnonlin better suited for this problem than say nlinfit? For ease of use or computational reasons?

Also, regarding your comment on weighting with nlinfit, I am still fairly sure that what I would like to have done would not be possible, the term "vector of predicted response values as input" to me implies that the input vector into my weighting function would be F(beta,xdata_i) with the fitted beta, rather than the beta values themselves (by default). This appears to be reiterated in the weighted fitting section, where it states that "Specify a function handle for observation weights. The function accepts the model fitted values as input, and returns a vector of weights.", and is at least the default setting for the Hougen-Watson example provided (I checked). I was wondering if there was a way to easily access the beta values directly.

But in any case, the lsqnonlin function works as desired.

[Torsten]

As far as I understand the documentation of nlinfit, the weights function is called in every iteration step of the solver. The input to this function is F(beta_i,xdata) where beta_i are the fitting parameters obtained in iteration i. So if you can calculate the weights from this input without knowing the beta_i's themselves, this would be an option. The advantage over lsqnonlin or lsqcurvefit is that there is a choice between two solvers and that you have several statistical postprocessing facilities.

But lsqnonlin and lsqcurvefit are - as the name indicates - especially designed for curve fitting purposes and thus best suited for your application.

[Maximilian Bailey]

Great - thank you for the explanations. Unfortunately I need to know the beta_i's (because of the analytical form), but the alternative function you proposed works just fine for what I need.

[Torsten]

I think you only need your model function values y_i, evaluated with the xdata_i and the beta_i, to determine the weights in each step, and this is what nlinfit delivers.

But anyhow, nice to hear that the proposed method works out fine.