How to do weighted least square rgression?

Hello

I want to fit an exponential curve to my data using MatLab. Could you please guide me how can I write a code to fit this through weighted least square reression? The weight matrix is as follows. I could not find any instruction or sample code to do so!

weight=[18.5204555170429; 24.8007492441765; 21.4204953742493; 12.0007299687922; 5.17482448096073;2.24987321147564]

x=[1.5;4.5;7.5;10.5;13.5;16.5]

y=[0.466132177250491;0.307694759285704;0.477550022494737;0.489169081512968;0.439027124884140;0.306938063199741]

Thanks

Answer 1

[Star Strider]

One option is to use fitlm —

weight=[18.5204555170429; 24.8007492441765; 21.4204953742493; 12.0007299687922; 5.17482448096073;2.24987321147564];
x=[1.5;4.5;7.5;10.5;13.5;16.5];
y=[0.466132177250491;0.307694759285704;0.477550022494737;0.489169081512968;0.439027124884140;0.306938063199741];

mdl = fitlm(x,y, 'Weights',weight)

mdl = 
Linear regression model:
    y ~ 1 + x1

Estimated Coefficients:
                   Estimate        SE        tStat      pValue 
                   _________    _________    ______    ________

    (Intercept)      0.39951      0.07398    5.4003    0.005691
    x1             0.0031898    0.0099772    0.3197     0.76519


Number of observations: 6, Error degrees of freedom: 4
Root Mean Squared Error: 0.353
R-squared: 0.0249,  Adjusted R-Squared: -0.219
F-statistic vs. constant model: 0.102, p-value = 0.765

figure
plot(mdl)
grid

I assume that you want to do a linear regression, since you did not mention a nonllinear model of any sort.

.

[Pooneh Shah Malekpoor]

Thanjs, however, I need a nonlinear exponential model , how about that?

[Star Strider]

It would have been nice to know that at the outset.

weight=[18.5204555170429; 24.8007492441765; 21.4204953742493; 12.0007299687922; 5.17482448096073;2.24987321147564];
x=[1.5;4.5;7.5;10.5;13.5;16.5];
y=[0.466132177250491;0.307694759285704;0.477550022494737;0.489169081512968;0.439027124884140;0.306938063199741];

fcn = @(b,x) b(1) .* exp(x-b(2)) - b(3) .* exp(-(x-b(4)));         % Nonlinear Exponential Model

mdl = fitnlm(x,y,fcn,[rand; 7; 12; rand], 'Weights',weight)

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.

mdl = 
Nonlinear regression model:
    y ~ F(b,x)

Estimated Coefficients:
          Estimate         SE          tStat        pValue  
          _________    __________    __________    _________

    b1    0.0020994      0.006279       0.33434      0.75492
    b2       11.349    1.3173e-05    8.6156e+05    1.089e-23
    b3      -1.0124       0.38895       -2.6029     0.059862
    b4      0.76701       0.39382        1.9476      0.12329


Number of observations: 6, Error degrees of freedom: 4
Root Mean Squared Error: 1.63
R-Squared: -19.8,  Adjusted R-Squared -25
F-statistic vs. zero model: 0.882, p-value = 0.482

xv = linspace(min(x), max(x))';
[yp,yci] = predict(mdl, xv);

figure
plot(x, y, 'p')
hold on
plot(xv, yp, '-r')
plot(xv, yci, '--r')
hold off
grid

I have no idea what sort of exponential model you want. This gives an approximation of one. Experiment with others, since you understand the process that created these data, and a mathematical model of that process would be the best option for ‘fcn’ here.

[Shiba shankar]

I have a doubt that if you don't know the weights, how you can get the weights for WLS

I have only two data sets.

[Torsten]

@Shiba shankar

The weights must be known and are usually derived from the "degree of certainty" of the data. The more certain a data point, the higher its weight.

[Shiba shankar]

how to know then,any steps to know

[Torsten]

The explanations given here might help:

https://quantifyinghealth.com/weighted-regression/