Nlinfit: Warning: The Jacobian at the solution is ill-conditioned, and some model parameters may not be estimated well (they are not identifiable). Use caution in making predictions.

I wrote a Matlab program for fitting some experimental data using nlinfit. The fit looks very nice but, Matlab returns the following warning : Warning: The Jacobian at the solution is ill-conditioned, and some model parameters may not be estimated well (they are not identifiable). Use caution in making predictions.

I cannot understand why and correct it. Could someone help me ? Thanks !

Answer 1

[John D'Errico]

Perhaps an example is simplest. In fact, I can give you a linear example that would probably cause the same error from nlinfit.

Consider the simple linear model:

z = b*x + c*y

Looks simple. Who could possibly complain?

x = randn(10,1);
y = x;

z = x + 2*y;

So I have a two parameter model, with 10 data points. I know in advance that I would EXPECT the coefficients to be [1,2].

But will nlinfit have a problem? Why would it have a problem? Look at the data that went into the problem. Given that x and y are identical, I could as easily have estimated the coefficients as [3,0], [0,3], [1.5 1.5], or [-5,8]. ANY of those sets of parameters will be equally valid. The result is the Jacobian matrix ends up as a singular one. I.e., it is ill-conditioned. If nlinfit DOES converge, I would hope it fails with the same error that you got, and for good reason. You cannot trust the coefficients.

So, the problem here is mainly with my data, in particular, how it relates to the model i have chosen. The model would be fine had my data been of sufficient quality to fit the model.

In your case, I do not have the model to look at, or the data that went into fitting the model. Lets try nlinfit out on my data with my model.

modelfun = @(beta,X) X*beta(:);

[BETA,R,~,COVB,MSE] = nlinfit([x,y],z,modelfun,[1,1])

Warning: The Jacobian at the solution is ill-conditioned, and some model parameters may not be estimated well (they are not identifiable).  Use
caution in making predictions. 
> In nlinfit (line 376) 
BETA =
          1.5          1.5
R =
   4.4409e-16
   1.7764e-15
  -8.8818e-16
            0
   2.2204e-16
  -8.8818e-16
  -2.2204e-16
            0
   1.7764e-15
   1.7764e-15
COVB =
   9.8161e-33   9.8161e-33
   9.8161e-33   9.8161e-33
MSE =
     1.26e-30

So we got the same error that you did. The residuals are ZERO, or at least as close as MATLAB can come in floating point arithmetic. The MSE is zero. COVB is singular.

Essentially, the fit is great, as was yours. But it is not a good model for this data. Or the data is not good for this model. (Take your pick.)

Did we get the same solution as that we started with? NO. It is as good a solution. But it is different. There are infinitely many equally good solutions to the problem I posed. Should I trust the one I got? Of course not. And that is what nlinfit told you.

Your problem is likely similar. You may be trying to fit a model that is too complex for your data, one that is not supported by that data. This is often called overfitting. You may have a variety of numerical issues with the data that MIGHT result in the same error. You may just have poor data, as was the case that I created, where the data was simply insufficient to fit two parameters. For example, in the case I created, were I to recognize that x and y were identical, then I might have chosen to reduce the model to

z = a*(x+y)

So now one parameter to fit. This fit will also be perfect, with essentially zero residuals, but it will not generate any warning messages. The solution will be unique, with a==3. Everybody will be happy.

So, as I said, since I cannot see your data or your model, I cannot know "why" you got that warning message, in the sense that I do not know exactly what sin you have committed in creating this fit.

[Anna Levy]

Hello

thanks a lot for these answers and this is clear for me now. However, I have a real problem because my model is given by a physical model. I cannot change it. My data should be equal to a sum of two gaussians and one exponential. I cannot fix the relative height or position or width of the gaussians..

So, I have 8 parameters for my fit : mdl = @(b,x)(((b(1)*exp(-(x-b(2)).^2/(2*b(3)^2))) + (b(4)*exp(-(x-b(5)).^2/(2*b(6)^2)))) + b(7) * exp(-b(8) * x));

What can I do in this case ? Thanks !

[Anna Levy]

Here is the file containing the experimental data. More easy to read :)

[Anna Levy]

And, actually, the confidence intervals are not equal to zero see attached file.

[Alex Sha]

Hi, Levy, the above result you obtained from Matlab is actually a local solution, the global solution is as follow:

Root of Mean Square Error (RMSE): 42.3206655472139
Sum of Squared Residual: 359998.785204187
Correlation Coef. (R): 0.998695719683172
R-Square: 0.997393140513488
Adjusted R-Square: 0.997366808599483

Parameter	Best Estimate
----------	-------------
b1	1264.24758950612
b2	33.1768834472874
b3	-1.28693903949025
b4	2517.61090242376
b5	36.2909656085989
b6	-1.77281786426151
b7	135705.564914938
b8	0.189920882137506

[Andrea Arguelles]

Alex, can you expand what you meant by a global vs. a local solution? Was Anna doing the fit incorrectly? I am having a similar issue and am curious how you were able to obtain the above result.

[Alex Sha]

Hi, refer to: What Is Global Optimization? - MATLAB & Simulink (mathworks.com)

[Bryce Gadogbe]

Hi Alex, what function did you use?

[Alex Sha]

Hi, the results I provided above had been obtained by 1stOpt, an package other than Matlab, with greate ability on global optimization.

[Chinmayee L M]

I am facing the same problem as Anna while using nlinfit. I am trying to fit a single exponential decay function to a noisy time series. I run into the error being discuss, while using nlinfit function. But, when I use fit function to fit the same data with the same exponential decay function, I do not see that error. Why is that? Are implementaions of the fit different? Which is better?