How to fit an exponential curve with known error bars

Hello,

i have X,Y data that i want to exponential curve the data with errorbars though im strugeling to find the command to do so, whenever i try the basic errorbar command it gives me a plot but not a curved plot, is there any 1 to do the command i need?

thx in advance!

[Scott MacKenzie]

It would help if you showed your code, data, and the plot that your code currently generates.

[Avichay Ashur]

X=[-504,-494,-490,-484,-478,-456,-436,-402,-364,-322]*10^-6 
Y=[4.85,4.37,4.07,3.65,3.24,2.16,1.52,0.8,0.4,0.18] %[V] +-0.002[V]
errX=[0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01]

Answer 1

[John D'Errico]

Some thoughts ...

Here is your data:

X=[-504,-494,-490,-484,-478,-456,-436,-402,-364,-322]*10^-6;
Y=[4.85,4.37,4.07,3.65,3.24,2.16,1.52,0.8,0.4,0.18]; %[V] +-0.002[V]
errX=[0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01];

First, those error bars are constant across X, but they are relatively HUGE, compared to the size of what is in X. As such, they would have almost no impact on the fit.

Next, you want to use the model

Y = a*exp(b*X)

If that model does fit, then suppose we log the model? We can see the impact of doing so by a semilogy plot. If that is a straight line, then your model is appropriate.

semilogy(X,Y,'o-')
grid on
xlabel 'X'
ylabel 'Y'

I see some tiny deviation from linearity at the left end, but a linear fit would do just fine in that domain. I'm actually surprised it was that close to a straight line.

We can do the fit on the logged model using a simple linear least squares (i.e., polyfit.) But since you are using the curve fitting toolbox, we can use that. First, I'll do the fit on the logged model. As a logged model, we would see:

log(Y) = log(a) + b*x

loggedmodel = fit(X',log(Y'),'poly1')

loggedmodel = 
     Linear model Poly1:
     loggedmodel(x) = p1*x + p2
     Coefficients (with 95% confidence bounds):
       p1 =  -1.835e+04  (-1.869e+04, -1.8e+04)
       p2 =      -7.602  (-7.756, -7.448)

aest = exp(loggedmodel.p2)

aest = 4.9945e-04

best = loggedmodel.p1

best = -1.8348e+04

Again, this totally ignores those error bars. But your error bars are IMMENSE, compared to the size of the numbers in X.

As well, the above model presumes that the noise is in log(Y), NOT in X. Is this important here? Possibly not, since your data is so nicely linear in this domain.

Can we estimate the model in a nonlinear form? Of course. But still, I'll be ignoring those error bars for now, both since they are so huge, and they are constant for all X.

ft = fittype('exp1')

ft = 
     General model Exp1:
     ft(a,b,x) = a*exp(b*x)

mdl = fit(X',Y',ft,'start',[aest,best])

mdl = 
     General model Exp1:
     mdl(x) = a*exp(b*x)
     Coefficients (with 95% confidence bounds):
       a =   0.0008243  (0.0002759, 0.001373)
       b =   -1.73e+04  (-1.866e+04, -1.594e+04)

plot(X,Y,'o')
hold on
plot(mdl)
grid on

As you can see, the linearized model estimates do lie within the confidence bounds we found for the nonlinear fit, however, they are not identical. A problem is, I don't know if your data really has errors in Y or in X. It looks like you have measured values on both axes. So I don't know how to best treat the estimation process.

[Scott MacKenzie]

@John D'Errico I'm just looking through your answer which I find very instructive. Thanks. However, I'm struggling with your use of the fit function. On my system, your code generates a syntax error: Check for incorrect argument data type or missing argument in call to function 'fit'. With a little digging, I discovered that that fit changed in R2020b:

I'm running R2021a. There doesn't seem backward compatibility to the forms of fit you are using, which are simpler and more intuitive than the current version of fit. Any suggestions?

[John D'Errico]

@Scott MacKenzie The code that I wrote in this answer WAS run using R2021a. If you look at the top right hand corner of my answer, it tells you: "Ran in: R2021a". I did not even use my own copy of MATLAB when I answered the question. I used the website to run my code, and it uses the recent release.

So whatever you did wrong is NOT relevant to the release. And the code I wrote would still have run in R2020b, since that Is what I have on my computer. (I just checked that claim on my computer.) Fit is indeed backwardly compatible in this respect. So you are incorrect in your claim about what happened. You dug wrongly. And that means if you had a problem, you need to show what you did. Otherwise, I would be left making wild guesses about what you may have done wrong.

[Scott MacKenzie]

@John D'Errico Gee, I am indeed puzzled, then. My code is just copied from this answer...

X=[-504,-494,-490,-484,-478,-456,-436,-402,-364,-322]*10^-6;
Y=[4.85,4.37,4.07,3.65,3.24,2.16,1.52,0.8,0.4,0.18]; %[V] +-0.002[V]
errX=[0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01];

loggedmodel = fit(X',log(Y'),'poly1')

My script is called test. Here's the error I see in the command window (in red):

Check for incorrect argument data type or missing argument in call to function 'fit'.

Error in test (line 10)

loggedmodel = fit(X',log(Y'),'poly1')

[Adam Danz]

@Scott MacKenzie

Check which fit function is being called using which fit -all

https://www.mathworks.com/matlabcentral/answers/589969-check-for-missing-argument-or-incorrect-argument-data-type-in-call-to-function-fit-using-two-1x1

[Scott MacKenzie]

@Adam Danz Thanks for adding your insight here.

>> which fit -all

C:\Program Files\MATLAB\R2021a\toolbox\stats\stats\@gmdistribution\fit.m % gmdistribution method

The problem is clearly stated in the link you posted. I need to install the curve fitting toolbox. -Thanks.

[John D'Errico]

I was wondering if you did not have the CFTB, or it was not installed. If you do not have it, then it will try to use fit from the stats toolbox, and then it gets confused, since it expects a specific input form.

[Scott MacKenzie]

Yes, that's it. I'm a bit surprised that the error message does not note this. It's not as though the missing toolbox is an open source add-on. It's a MathWorks product! But, in any event, I'm sorted out now. Thanks for your detailed answer here.

[Adam Danz]

The hard-problem is that Matlab has no way of knowing the intentions of the programmer so it chooses which function to run based on the Function Precedence Order (pardon the anthropomorphisms).

@gmdistribution/fit.m is designed to be run by fitgmdist() which calls the file explicitly using gm=gmdistribution.fit(X,k,varargin{:}) which is possible because @gmdistribution/fit.m is defined as a method in the class constructor gmdistribution.m. So fitgmdist() cannot accidentally choose another fit() function.

However, @gmdistribution/fit.m still resides on the matlab path and when no function named "fit" exists on the path and the user calls the fit() function, Matlab chooses the only one it knows, @gmdistribution/fit.m, unless the user has another function named fit.

When the Curve Fitting Toolbox is added to the Matlab path and the user directly calls fit(), Matlab chooses curvefit/fit.m since @gmdistribution/fit.m is buried in a method directory and is designed to be called explicitly.

I suppose the @gmdistribution/fit.m function could detect if input #2 is a vector of length size(x,1) in which case its error message could suggest that the curve fitting toolbox may be required since @gmdistribution/fit.m expects input #2 to be a positive integer. But too much defensive programming comes at the expense of run-time and maintenance costs so ultimately the user is responsible for knowing function precedences.

Fortunately, calling the wrong function usually produces an error message and after confirming that the inputs are all OK, I usually go directly to which <name> -all. It's a very frequent solution to many questions asked in the forum, primarily because users name custom functions things like sum or plot which is a whole other can of worms 😃.

[Scott MacKenzie]

Thanks, Adam. Excellent narative on this issue.

Answer 2

[Scott MacKenzie]

The plot looks curved to me.

The error bars in your chart appear as horizontal lines because they are really small. Here's a tweak to your data to increase the size of the error bars:

X = [-504,-494,-490,-484,-478,-456,-436,-402,-364,-322] * 10^-6 ;
Y = [4.85,4.37,4.07,3.65,3.24,2.16,1.52,0.8,0.4,0.18]; 
errX = [0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01,0.01];

errX = 15 * errX; % so the error bars are visible

errorbar(X, Y, errX);

[Avichay Ashur]

First of all thx u for answering my question i realy appreciate it. i think the problem is i didnt explain my self correctly, im trying to have a graph that looks like this:

this is from the curve fitting tool, smooth graph but i need to implement errorbars the graph i ploted first is not smooth and cant be used to for my needs. i hope i explained myself correctly this time

[Scott MacKenzie]

The general approach is to use hold on followed by errorbar. By default, errorbar includes lines connecting the points. But, I suspect you only want the error bars, because you've already got a smooth curve fitted to the data. So, you need to use errorbar with some additional code to remove the lines. Try adding this following the code that generated the graph in your comment:

hold on;
h = errorbar(X, Y, errX, errX); 
h.Color = [0 0 0]; % black error bars                         
h.LineStyle = 'none'; % remove lines connecting points

[Avichay Ashur]

again, thx u for spending ur time to help me:)

i understand the code that u wrote and what it does, but smooth curve that i showed above is inside the curve fitting app, so therefore i cant (or atleast i dont know how) to export it into a code so i can add the code that u listed above to make it work.

this is the curve fitting tool im using, but im not sure how i can export this smooth graph into a plot so i can just add the errorbars as u mentioned. thx u again for ur help!

[John D'Errico]

@Scott MacKenzie

Note that you are plotting the error bars in Y, not in X.

As well, this will cause significant problems in the fit, because the fitting tool does not ssume errors in X, but that all the errors are in y. In turn, that results in biased estimaes of the parametrs, IF you allow it to assume the error i in the wrong variable.

You very much want to be careful in this!

[Scott MacKenzie]

John: You're spot on correct here. Thanks. The original question said nothing about using a curve fitting tool. I was just thinking about getting errorbars in the graph. It had occurred to me that it was odd that the error variable was named errX, as that implicitly suggests putting error bars w.r.t. X (although that was not mentioned).

Avichay: Sorry, I'm not familiar with using the curve fitting tool.

[Avichay Ashur]

@Scott MacKenzie Hey scott, the code u gave me earlier actualy worked and helped me alot, i figured how to export the curve fitting code and using the error bars as u said, 1 last thing i could realy use ur help is how i keep the points on the graph, right now there is only the curve that i exported, and the errorbars, could u explain me what i need to add in order to make the data points larger?

[Scott MacKenzie]

I not familiar with the curve fitting tool. There's likely a property named MarkerSize which you can change to control the size of the markers (i.e., data points). See if you can find it in the tool. Good luck.