Estimation of errors from solving A\B=X

I have a few vectors which by a linear combination fits a different vector quite nice. E.g. I solve the equation

[vec1 vec2 vec3 vec4]*X=vec_dat.

Now I would like to get a feeling of the confidence level on the coefficients returned by matlab.

One thing I have tried is

c=coef(1)
l=linspace(c-c/5,c+c/5,20);
for i=1:length(l)
  l(i);
  sim=l(i).*a+coef(2).*b+coef(3).*g+coef(4).*d;
  RMSa(i)=sum(sqrt((dat-sim).^2));
end

I observe parabolas with different 2nd derivatives for each of the coefficients i test. Obviously the 2nd derivative of such a parabola must be related to the confidence of the given coefficient tested?

Any one know how to extract it?

Answer 1

[Rahul]

Hi Kresten,

I am assuming that you are using the latest R2024b release of MATLAB. To assess the confidence level of the coefficients in a linear regression model, you can use the standard errors and confidence intervals of the coefficients. This could be a simpler way to observe the relation between confidence and coefficients, rather than relying on the second derivative of the RMS error with respect to the coefficients. Here's a general approach to achieve this:

1. Fit the Linear Model: Useing MATLAB's built-in functions to fit the linear model and obtain the coefficients and their statistics.
2. Extract Standard Errors and Confidence Intervals: MATLAB provides functions to calculate these directly from the fitted model.

You can use the‘fitlm’function to fit a linear model and get detailed statistics about the coefficients:

% Sample data 
vec1 = [1; 2; 3; 4; 5]; 
vec2 = [2; 3; 4; 5; 6]; 
vec3 = [3; 4; 5; 6; 7]; 
vec4 = [4; 5; 6; 7; 8]; 
vec_dat = [10; 14; 18; 22; 26]; % Linear combination of vec1, vec2, vec3, vec4 

X = [vec1, vec2, vec3, vec4]; 
y = vec_dat; 

% Fit linear model 
mdl = fitlm(X, y); 

Once you have the model, you can extract various statistics, as shown below:

% Display the model summary 
disp(mdl); 
 
% Extract coefficients 
coefficients = mdl.Coefficients.Estimate; 
 
% Extract standard errors 
standardErrors = mdl.Coefficients.SE; 
 
% Extract confidence intervals (default 95%) 
confidenceIntervals = mdl.coefCI;

Instead of manually testing the sensitivity of the coefficients by perturbing them and observing the RMS error, you can use the‘fitlm’function, whcih provides a more robust statistical framework for assessing the confidence in your coefficients. The standard errors and confidence intervals can give a clearer picture of the reliability of each coefficient estimate.

In case, you want to visualize how the RMS error changes with perturbations to a coefficient, you can use the following code:

% Choose a coefficient to perturb (e.g., the second one) 
coef_index = 2; 
c = coefficients(coef_index); 
l = linspace(c - c/5, c + c/5, 20); 

% Compute the RMS error for different values of the chosen coefficient 
RMSa = zeros(size(l)); 
for i = 1:length(l) 
    sim = l(i) * vec1 + coefficients(2) * vec2 + coefficients(3) * vec3 + coefficients(4) * vec4; 
    RMSa(i) = sqrt(mean((vec_dat - sim).^2)); 
end 

% Plot the parabola 
figure; 
plot(l, RMSa, '-o'); 
xlabel('Coefficient Value'); 
ylabel('RMS Error'); 
title('RMS Error vs. Coefficient Value'); 

The curvature of the RMS error plot gives an intuitive sense of the sensitivity of the fit to changes in the coefficient. A steeper parabola (greater second derivative) suggests that small changes in the coefficient lead to large changes in the error, indicating a more precise estimate (lower variance). However, the standard errors provided by‘fitlm’are a more direct and statistically sound measure of confidence in the coefficients.

To know more about the usage of ‘fitlm’ function, refer to the documentation link mentioned below:

Fit linear regression model using fitlm

Best!