Custom curve fitting with constraint on parameters based on a value of estimated dependent variable (y)

Hello everyone,

I am working on developing deterioration model for road pavement condition data for my thesis. I have been using the Matlab curve fitting tool with a custom deterioration model. I have my x (age of pavement) and y (condition index) data. my model equation is y = o-exp(a-b*c^log(1/x)). I am estimating coefficient a, b, c and o based on some constraints. I was able to do all that but i am stuck with a constrain that I dont know how to formulate.

My current curve fitting code is giving parameters (a,b,c and o) that when i use them in the equation to estimate y, my y =2.7 is around x = 55. Since i have aready my x and y data, i am trying to develop a constraint that will force Matlab to estimate parameters a, b, c and o that will give a y value equal or around 30 at age x=2.7.

below is my code and attached the excel data. Thank you in advance and let me know if you have any question.

function [fitresult, gof] = bestCoefPDIcon(x, y)
S = uiimport('-file'); %this gives option to automatically select excel file 
% It will show an error message but wait until a pop window show
x = S.data(:,1);
y = S.data(:,2);

[xData, yData] = prepareCurveData( x, y );

% Set up fittype and options.
ft = fittype( 'o-exp(a-b*c^log(1/x))', 'independent', 'x', 'dependent', 'y' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Display = 'Off';

%[3.611 5.638 1.27 5] are actual parameters for PDI CON activity 16 (OC-CON)
% PDI CON min coef value [1.715 2.361 1.02 4.8]
% PDI CON max coef value [43.733 44.779 2.631 5]
aMin = 1.715;
bMin = 2.361;
cMin = 1.02;
oMin = 4.8;

aMax = 43.733;
bMax = 44.779;
cMax = 2.631;
oMax = 5;

opts.Lower = [aMin 0 1 0]; % set lower bond coef A = lowest value of coef A for BIT pavement
% set lower bond coef B & O = 0, and coef C =1

opts.StartPoint = [0 0 0 0];

opts.Upper = [aMax bMax cMax oMax]; % set upper bond coef A = highest value of coef B for BIT pavement
% set upper bond coef B & C = infinity, and coef O = max value of coef O for BIT pavement

%[3.611 5.638 1.27 5] are actual parameters for PDI CON activity 16 (OC-CON)

% Fit model to data.
[fitresult, gof] = fit( xData, yData, ft, opts )

% Plot fit with data.
figure( 'Name', 'PDI' );
h = plot( fitresult, xData, yData );
legend( h, 'Measured', 'Estimated', 'Location', 'NorthEast');
% Label axes
xlabel AGE(year)
ylabel PDI
grid on

Answer 1

[Cris LaPierre]

This is what fmincon does. Have you tried using that?

[Amara Kouyate]

Hi, thank you for your answer. No i never used or heard about that function. do you know or have an idea how i could incorporate that function to my code? thank you very much.

best,

Amara

[Cris LaPierre]

I might have ready the OP too quickly. fmincon allows you to place upper and lower contraints on the estimated parameters a, b, c and o. It does not allow you to impose a constraint on your coeffients to yield a specific prediction.

This might suggest you either need more data, or need to preprocess your data to minimize the impact of outliers.

[Amara Kouyate]

Thank you Cris for your helpI i hoping i could find some way to resolve this issue i am having. i did remove a lot of outlier in my data. Do you think i can use my model equation with fmincon, if i have a range of upper and lower constrain for parameter a, b, c and o. maybe such that i can obtain parameter that stay in that range and that do not over predict model parameters beyong what i already have???

best,

AMara

[Cris LaPierre]

Perhaps. I'd be concerned that you are no longer fitting a model to your data. There is a bigger issue you should explore about either how the data was collected, or the assumptions being made about what is "correct". Forcing an 'expected' fit is not a good solution.

[Amara Kouyate]

I think you bring up a good point that i did not think about before, i will see if i can obtaini a new set of data different from what i have now. because you said forcing the model is not a good option... maybe my data is not correct as you said. i will focus on trying to obtain a new set of data. Thank you again for your input.

Best,

Amara