Optimizing the coefficients of basis functions to fit a curve

I have 3 vectors F1, F2, and F3. I want to linearly combine these vectors such that I create a new function W(z)=a*F1+b*F2+c*F3, where a,b,c are just constant coefficients and the 3 vectors are essentially basis functions. These vectors are 1x70 doubles, and are shown plotted below (note that the horizontal axis 'Range' is the 'z'-variable in the equations below).

The problem I am having is finding a way to calculate the coefficients a,b,c so that W=a*F1+b*F2+c*F3 most closely matches a function given by:

Plotted together, W and r are shown below...

What sort of optimization process can I use to build a function W=a*F1+b*F2+c*F3 that fits 'W' to 'r' and returns the coefficients a,b,c used to do so? Please note that F1,F2, and F3 are all 1x70 doubles, and are not explicitly functions of z anymore. They have each individually been evaluated. And so I tried using the curvefitting app with no success because of this. I really just need a systematic way to optimize the coefficients a,b, and c. Is there a built in way to do this?

Answer 1

[Matt J]

You don't need any special curve fitting tool. Due to the linearity of the model, the coefficients can be found using mldivide, \

coefficients = [F1(:),F2(:),F3(:)]\r(:)

[Image Analyst]

Good answer Matt. I wish this trick was better known.

Shouldn't W be in the denominator instead of r?

Also note there is a similar version mrdivide, / that uses forward slash. Since I can never remember which to use I always have to look at the documentation and see which one best matches up with my data and coefficients. If you have a good mnemonic to avoid having to check the documentation, I'd like to hear it.

help mldivide

 \   Backslash or left matrix divide.
    A\B is the matrix division of A into B, which is roughly the
    same as INV(A)*B , except it is computed in a different way.
    If A is an N-by-N matrix and B is a column vector with N
    components, or a matrix with several such columns, then
    X = A\B is the solution to the equation A*X = B. A warning 
    message is printed if A is badly scaled or nearly singular.
    A\EYE(SIZE(A)) produces the inverse of A.
 
    If A is an M-by-N rectangular matrix with M~=N and B is a column
    vector with M components, or a matrix with several such columns,
    then X = A\B is the solution in the least squares sense to the
    under- or overdetermined system of equations A*X = B. The
    effective rank, K, of A is determined from the QR decomposition
    with pivoting. A solution X is computed which has at most K
    nonzero components per column. If K < N this will usually not
    be the same solution as PINV(A)*B.  A\EYE(SIZE(A)) produces a
    generalized inverse of A.
 
    X = MLDIVIDE(A,B) is called for the syntax 'A \ B' when A or B is an object.
 
    See MATLAB Operators and Special Characters for more details.
 
    See also LDIVIDE, RDIVIDE, MRDIVIDE, PAGEMLDIVIDE.

    Documentation for mldivide
       doc mldivide

    Other uses of mldivide

       codistributed/mldivide    laurpoly/mldivide
       comm/mldivide             StaticModel/mldivide
       distributed/mldivide      sym/mldivide
       duration/mldivide         symbolic/mldivide
       DynamicSystem/mldivide    tall/mldivide
       gpuArray/mldivide         timeseries/mldivide
       LagOp/mldivide            tsdata.datametadata/mldivide

help mrdivide

 / Right matrix divide.
    B/A is the matrix division of A into B, which is roughly the
    same as B*INV(A) , except it is computed in a different way.
    More precisely, B/A = (A'\B')'. See MLDIVIDE for details.
 
    C = MRDIVIDE(B,A) is called for the syntax 'B / A' when B or A is an object.
 
    See MATLAB Operators and Special Characters for more details.
 
    See also MLDIVIDE, RDIVIDE, LDIVIDE, PAGEMRDIVIDE.

    Documentation for mrdivide
       doc mrdivide

    Other uses of mrdivide

       codistributed/mrdivide    laurpoly/mrdivide
       distributed/mrdivide      se2/mrdivide
       dlarray/mrdivide          StaticModel/mrdivide
       duration/mrdivide         sym/mrdivide
       DynamicSystem/mrdivide    symbolic/mrdivide
       embedded.fi/mrdivide      tall/mrdivide
       gpuArray/mrdivide         timeseries/mrdivide
       LagOp/mrdivide            tsdata.datametadata/mrdivide

[Matt J]

Shouldn't W be in the denominator instead of r?

W=[F1(:),F2(:),F3(:)] is in the denominator. I jut didn't create a W variable explicitly.