How to find coefficients in linear combination of functions

Hello

I have faced a problem: I have three function's made of a experimental data. Say f1(P), f2(P), f3(P) and they all depend on P. f1(P), f2(P) and f3(P) are in vector form, ie. : f1=[f1value1 f1value2 ... f1valuen] etc. and P=[p1 p2 ... pn]. Where f1valuen, f2valuen, f3valuen and pn are finite, discrete values.

Finally i have an equation like F(P) = ∝1*f1(P)+∝2*f1(P)+∝3*f3(P).

My question is: How to find ∝1, ∝2 and ∝3 ?

I know additionally that ∝1+∝2+∝3=1 and ∝1<∝2 and ∝3<∝2

Answer 1

[Matt J]

It's a linear least squares problem with linear constraints. You can use LSQLIN, if you have Optimization Toolbox.

   x = lsqlin(C,d,A,b,Aeq,beq,lb,ub)

The columns of the matrix C will come from your f_n(P) data and similary d will be a column vector composed of the F(P) samples.

[Daniel Wiczew]

I think, that i defined the question wrongly, those functions f1(P), f2(P), f3(P) are not exactly the same. For example f1(P) can be exponential, f2(P) can logarythmic and the thirds can be (a+x)/(b+x^2). Does it matter using this method from matlab ?

[Walter Roberson]

Is the question to find the alpha values for any given P ? If so then the solution given by Matt works.

If the question is to find the alpha values that will satisfy your F equation for all P, then that is not possible for non-linear functions.

[Daniel Wiczew]

@Walter Roberson I have four sets of data as shown below.

These sets are f1(P), f2(P), f3(P) and the F(P). The thing i would like to do, is to do a linear combination like: F(P) = ∝1*f1(P)+∝2*f2(P)+∝3*f3(P), where ∝1, ∝2 and ∝3 satisfy the equation for all P.

As you said "If the question is to find the alpha values that will satisfy your F equation for all P, then that is not possible for non-linear functions.", Have you any advice? I would be very grateful

[Matt J]

You have more points, P, than degrees of freedom, ∝_i. So, it may be impossible to do better than a least squares fit.

[Walter Roberson]

Given the particular values f1P, f2P, f3P, you can construct

residual = @(alpha) sum( (alpha(1)*f1P + alpha(2)*f2P + alpha(3)*f3P - fP).^2 )

and then do an optimization to try to minimize residual. This is a form of least squares fitting, and it should give the same as

alpha = [f1P(:), f2P(:), f3P(:)] \ fp(:);

[Matt J]

Although, to do least squares fitting subject to constraints on ∝_i, as in the original post, you will need lsqlin.