Problem/question with regression

Hi folks,

I have the following data:

Reflectance	R	G	B
7.51	    109	54	124
3.17	    57	30	63
1.24	    30	17	31

I am trying to get a relationship for reflectance based on the R, G and B values. Doing this using simultaneous equations yields the following coefficients:

0.452923977
-0.538011696
-0.103274854

which is problematic because it gives me negative values for fomr RGB values, which is incorrect. So I tried to regress the table above in Matlab's curve fitting app, but it doesn't let me as the matrix dimensions aren't compatible.

Is there a way to get around this problem and regress the data?

Thanks

Answer 1

[William Rose]

You need at least one more data point to do a standard regression, so that you have more equations than unknowns.

Even with more points, it is possible that the regression equation will predict negative reflectance for some combinatons of R, G, B. If you want a model that will never give values outside [0,1], then you need a nonlinear model. You could take the linear prediction and apply a hard or smooth limit funciton to it. Suppose your linear model is rinit=a*R+b*G+c*B. The code below shows final reflectance, rfinal, computed with hard and smooth limits.

rinit=-1:.05:2;
rfinalH=min(max(rinit,0),1);
rfinalS=exp(4*(rinit-.5))./(1+exp(4*(rinit-.5)));
plot(rinit,rfinalH,'rx-',rinit,rfinalS,'bo-');
xlabel('Linear Reflectance Prediction'); grid on;
ylabel('Final Reflectance'); legend('Hard','Soft');

Try it.