Initial values in nlinfit or fitnlm

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wesleynotwise 2017-6-26

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评论： Priya Goel 2020-8-22

I am trying to run a non-linear multiple variable model in Matlab. The model has about 20 coefficients. I have been using 1s as my initial values in developing the model, and my model has an acceptable R2 value and good residual plots. However, I am not sure if the generated coefficients are sensitive to the initial values that were assigned by me.

This begs the question of whether one can check if the generated coefficient values are highly sensitive to the assigned initial values? Or one should do it manually, ie test the model with different set of initial values and compare the RMSE of the model?

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Star Strider 2017-6-26

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A model with 20 parameters is likely going to be a challenge. If you have any doubts — and if you have the Global Optimization Toolbox — use the patternsearch (link) function to find the best parameter set. Another option is the genetic algorithm, the Global Optimization Toolbox ga (link) function.

I would also use the coefCI (link) function to determine if any of the confidence intervals for the coefficients (parameters) include zero, i.e. have opposite signs. If they do, they are not required for the model, since they are not statistically different from zero. This can help you ‘trim’ your model.

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wesleynotwise 2017-6-26

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Hello Star Strider, nice to hear from you again!!! I've seen your reply hours ago, but I got distracted and couldn't reply to you quickly.

I have the Global Optimization Toolbox and I tried to run the code as below:

beta1 = ones (22,1)                 % initial value for 22 coefficients
x = patternsearch(modelfun1, beta1) % the patternsearch code
x = ga(modelfun1, 22)               % the ga code
mdl = fitnlm(tbl,modelfun1,beta1)   % fit into nonlinear

However, both ran into the same problem "Not enough input arguments", which is caused by "Failure in initial user-supplied fitness function evaluation. PATTERNSEARCH (or GA) cannot continue.", Just to let you know I have no problem when I run my model.

Can I check with you, if the above two codes work, does it mean I should use their coefficient values as my initial values in my regression model?

mdl = fitnlm(tbl,modelfun1,beta2)  
% beta2 is the results from GA or PATTERNSEARCH

Also, correct me if I'm wrong, I realise that the PATTERNSEARCH function requires the initial values, does that mean it may also giving me the same problem, i.e. the coefficients depend on what you assigned.

And, thank you for the suggestion for the coefCI function. I have it in my codes, as I wanted to use it to round the coefficient values to either 1 or 2 decimal points. I've never thought that it can also be used to trim my model! What a brilliant suggestion!!!!!

Star Strider 2017-6-26

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My pleasure.

Since you’re fitting your function to data, you have to introduce a cost function and minimise it with patternsearch or ga.

Example —

x = ...;                                    % Independent Variable
y = ...;                                    % Dependent Variable
RNCF = @(b) norm(y - modelfun1(b,x));       % Residual Norm Cost Function

where ‘b’ is your parameter vector.

Both functions will search as exhaustively as you let them to find the optimal parameter estimates.

You do not require the fitnlm function to estimate your parameters later, since the Global Optimization Toolbox functions that you decide to use will fit them about as well as can be expected. They are most likely to find the global optimum without your having to guess the initial values, so your fitnlm call will simply provide you with a model to use to present to coefCI.

You can round the coefficient estimates using the round function, to the number of decimal places you want. (I don’t remember when this option was introduced, so if your documentation for round doesn’t include it, I can post a one-line anonymous function that does the same thing.)

‘What a brilliant suggestion!!!!!’

Thank you!

wesleynotwise 2017-6-27

编辑：wesleynotwise 2017-6-27

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Ah... no wonder. But, at the moment, I fit the model from a table but not a matrix. See the codes below. And the table that I built has more input than what I actually need for my model, as it is still pretty much in the development state. I assume I need to build a matrix for your code in order to incorporate in the existing one?

tbl = table(CN, CR, CON, FON, ANT, ART, AS, NAK, RSK, A_AN,...
              A_AR, ACa, SMPa, SAN, SAR, ANonly, CS, CNS,...
              EMO); % The table has more input than i actually need
modelfun1 = @(b,x)(((SMPa < 10).*b(1).*(x(:,13).^b(2))+...
                   (SMPa >= 10).* (x(:,13).^b(3))).*...
% the equation is very long, I only showed part of it   
X = [CN, CR, CON ...]; % I assume this is needed ?
y = EMO;               % And this?
beta1 = ones (22,1)                 % initial value for 22 coefficients
RNCF = @(b) norm(y - modelfun1(b,x))% Residual Norm Cost Function
xp = patternsearch(RNCF, beta1)     % the patternsearch code
xg = ga(RNCF, 22)                   % the ga code
mdlp = fitnlm(tbl,modelfun1,xp)     % fit into nonlinear
mdlg = fitnlm(tbl,modelfun1,xg)     % fit into nonlinear

Excuse the messiness in my codes. Need a good housekeeping.

Star Strider 2017-6-27

My pleasure!

You can always sidetrack as you please. I will answer within the areas of my knowledge.

They have their own roles. The patternsearch and ga functions search the entire (or a very large part of the) parameter space for the best parameter estimates. The fitnlm function searches in the region near the initial estimates you’ve given it. The advantage of fitnlm is that it then allows you to calculate the statistics on the fit.
The parameters estimated by ga are more likely to be the most accurate, because it searches more widely. In a parameter space with a global minimum that is relatively straightforward to find, all parameter estimation routines will work optimally, and find essentially the same parameter estimates. The problem arises when there are several local minima that fitnlm, using a gradient-descent approach, could become ‘trapped’ in. Since ga does not use a gradient-descent approach, it is more likely to find the global minimum without getting trapped in local minima. When you then give those parameter estimates to fitnlm, it will converge quickly on the optimal parameter estimates, and give you the statistics on the fit.

Star Strider 2020-8-21

This should be a new Question.

It does not directly relate to the current thread. I am not going to respond to it further here.

Priya Goel 2020-8-22

Yes. I am also of the opinion that it is deviation from the original topic.

Anyways, Thank you for your valuable time and inputs. These are very helpful for beginners (like me). You clarify doubts which otherwise remain unanswered for months.

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