Dear Sirs,
I want to invert following set of equations:
Y(i)=a(i)X1+b(i)X2+c(i)X3+d(i)X1X2+e(i)X1X3+f(i)X2X3+g(i)X1^2+h(i)X2^2+i(i)X3^2 i=[1,4]
It consists of four outputs (Y1-Y4), 3 inputs (X1-X3), and (9*4) coeff.
I intend to generate X1,X2,X3=function(Y1,Y2,Y3,Y4).i.e. Equations of X1,X2,X3. We may assume coefficients of your choice to simplify the solution. Please help me with possible guidance.
Even if you have a simpler problem, when i belongs to the interval [1,3] (Y=XAX+BX where A and B are square matrices and X and Y are vectors), the analytical solution will be known for some specific relations between matrices A and B. For more details, see Gantmaher "The Theory of Matrices". In your case, I would not think of analytical solution, try to recompose the problem.
Is this just a simple problem of multi-variable regression of order 2? Do you have a set of data and want a single set of coefficients? If so, you can use this free tool. It works for y(x)=c1+c2*x+... or y(x)=p(x)/q(x), or y(x1,x2,x3,x...)=c1+c2*x+... There's a bunch of other free stuff at the second link.
http://dudleybenton.altervista.org/software/CurveFit517.zip
http://dudleybenton.altervista.org/software/index.html
Respected Sir,
Thanks for your valuable reply. I donot think that the software will help me much as I am already using MATLAB for multivariable regression type of problems(sum square error, stepwise regression,Lasso, Elastic Net).
Actually I have developed 36 equations using multivariable regression each consisting of one output (say Y1,Y2,Y3...Y36) using around 9 input variables (X1,X2,X3...X9). Each equation comprises of same number of coefficients (say 126+) but different magnitudes. Now I want to invert these equations i.e. I want to get all 9 variables in terms of every output parameters (Y(1-36)).
One solution I considered is using sets of 9 equations (so 4 sets). From them I will get 4 equations for each variable (say X1=func(Y1,Y2..Y9), X1=func(Y10,Y11...18).....). Then I can add all 4 equations to get single X1 in terms of Y1,Y2...Y36. But This method seems unrealistic to me as it may be difficult to say whether to add/multiply/divide/sub these 4 equations.
Could you please guide me further in this problem
Inverting the functions would simply require modifying a few do loops, if you were using C instead of MATLAB, which I can't recommend strongly enough. If you're using the Windows O/S, I'll send you a little program to do this and even a link to a free compiler.
I don't believe there is general case for the setup of your example problem. MATLAB's fminsearch might be useful in approximating solutions, given an initial solution close to the actual. Given input vectors of the form:
psi = [y1;y2;y3;y4]
and coefficients
A = [a1,b1,c1,d1,e1,f1,g1,h1,i1; ...]
Let:
% x = [x1,x2,x3]
X = @(x) [x(1); x(2); x(3); x(1)*x(2); x(1)*x(3); x(2)*x(3); x(1)^2; x(2)^2; x(3)^2]
Y = @(X) A * X
ferr= @(Y) sum( (psi - Y).^2 )
xbest = fminsearch( @(x) ferr( Y( X( x ) ) ), xi )
Using randomly large signed elements for A and x, fminsearch finds near-exact answers given a suitably close initial. The coefficients of your actual data may warrant useful simplifications, esp. if the initial condition can be approximated via the squares of your domain.
Thank You Sir. Really interesting. I will try to implement your guidance. Regards
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