When I use nlinfit, the sintax nlinfit(t0,p0 ,@(b,t) ...),but if Ireplace nlinfit with fmincon, the sintax is not valid, why?

When I use nlinfit, the sintax nlinfit(t0,p0 ,@(b,t) ...) works, but if Ireplace nlinfit with fmincon, the sintax is not valid, why?

Answer 1

[Matt J]

Because it is a different function. The syntax of fmincon is documented here,

https://www.mathworks.com/help/optim/ug/fmincon.html

[Marcos]

Thank you for the answer. I have to figure out how to solve this. I will try to use global variables and see if it works

[John D'Errico]

@Marcos

It is not just a question of directly replacing nlinfit with fmincon. nlinfit assumes you will be minimizing a sum of squares of the residuals. So nlinfit assumes a function form that approximates f(x), then it implicitly forms the sum of squares of residuals. (Well, actually, it does not truly need to do that, but in terms of the mathematics, you can think of it like that.)

HOWEVER, fmincon is designed to mimimize a scalar function of some parameters. It has no clue that you want it to form a sum of squares of the residuals. It is you who needs to do that.

Answer 2

[John D'Errico]

@Marcos there is no need to use global variables. For example...

x = rand(20,1);
y = 3 + 4*exp(x*1.5) + randn(size(x))/3;
plot(x,y,'.')

The model is

y = a + b*exp(c*x);

help nlinfit

 nlinfit - Nonlinear regression
    This MATLAB function returns a vector of estimated coefficients for the
    nonlinear regression of the responses in Y on the predictors in X using
    the model specified by modelfun.

    Syntax
      beta = nlinfit(X,Y,modelfun,beta0)
      beta = nlinfit(X,Y,modelfun,beta0,options)
      beta = nlinfit(___,Name=Value)
      [beta,R,J,CovB,MSE,ErrorModelInfo] = nlinfit(___)

    Input Arguments
      X - Predictor variables
        matrix
      Y - Response values
        vector
      modelfun - Nonlinear regression model function
        function handle
      beta0 - Initial coefficient values
        vector
      options - Estimation algorithm options
        structure created using statset

    Name-Value Arguments
      ErrorModel - Form of error term
        'constant' (default) | 'proportional' | 'combined'
      ErrorParameters - Initial estimates for error model parameters
        1 or [1,1] (default) | scalar value | two-element vector
      ReduceJacobian - Flag to reduce Jacobian rank
        true or 1 (default) | false or 0
      Weights - Observation weights
        vector | function handle

    Output Arguments
      beta - Estimated regression coefficients
        vector
      R - Residuals
        vector
      J - Jacobian
        matrix
      CovB - Estimated variance-covariance matrix
        matrix
      MSE - Mean squared error
        scalar value
      ErrorModelInfo - Information about error model fit
        structure

    Examples
      openExample('stats/NonlinearRegressionModelUsingDefaultOptionsExample')
      openExample('stats/NonlinearRegressionUsingRobustOptionsExample')
      openExample('stats/NonlinearRegressionUsingObservationWeightsExample')
      openExample('stats/NonlinearRegressionUsingWeightsFunctionHandleExample')
      openExample('stats/NonlinearRegressionUsingNonconstantErrorModelExample')

    See also fitnlm, nlparci, nlpredci, Nonlinear Regression Fitter

    Introduced in Statistics and Machine Learning Toolbox before R2006a
    Documentation for nlinfit
       doc nlinfit

bstart = rand(1,3)*2;
fun = @(b,X) b(1)+b(2)*exp(b(3)*X);
paramsnl = nlinfit(x,y,fun,bstart)

paramsnl = 1×3

3.5314 3.5085 1.6182

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The solution should be something close to the vector [3,4,1.5]. As you can see, it did well. Now we can try fmincon.

ssqobj = @(Bfit) sum((y - fun(Bfit,x)).^2);

Note that the variables X and Y are buried inside the functions fun and ssqobj. There is no need for any global variables.

paramsfmincon = fmincon(ssqobj,bstart)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

paramsfmincon = 1×3

3.5314 3.5085 1.6182

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I could have employed constraints on the variables of course, since fmincon was used. But what you need to see was nothing special was done, EXCEPT for the creation of a wrapper like ssqobj to convert the function in fun into a scalar objective that fmincon can use.