Data points with sigma = 0 and -1 are ignored. Points with
sigma = 0 should be treated as constraints. Currently, only linear
fit with linear known constraints are handled.
Given a set of N data points (measurements), (x(i), y(i)) i = 0,...,N-1, along with a set of standard deviations, sigma(i), for the data points, and M specified functions, f(j)(x) j = 0,...,M-1, we form a linear combination of the functions:
z(i) = a(0)f(0)(x(i)) + a(1)f(1)(x(i)) + ... + a(M-1)f(M-1)(x(i)),
where a(j) j = 0,...,M-1 are a set of parameters to be determined.
Besides the M specified functions, we are also given a set of P linear equations (constraints) for the parameters a(i) to satisfy:
e(0,j)a(0) + e(1,j)a(1) + ... + e(M-1,j)a(M-1) + c(j) = 0, j=0,P-1,
where e(i,j) and c are known constants. The linear least-squares fit with
constraints tries to minimize chi-square
chi-square = [(y(0)-z(0))/sigma(0)]^2 + [(y(1)-z(1))/sigma(1)]^2 + ...
+ [(y(N-1)-z(N-1))/sigma(N-1)]^2.
by adjusting parameters a(i) within the limit imposed by the constraint
equations.
If we define
A(i,j) = function(j)(x(i))/sigma(i)
b(i) = y(i)/sigma(i)
The chi-square we are trying to minimize becomes:
chi-square = |A*a -b|^2
The constraint equations are
e(0,j)a(0) + e(1,j)a(1) + ... + e(M-1,j)a(M-1) + c(j) = 0; j = 0, P-1
The matrix equation to be solved then becomes
|(trans(A)*A) e | | a | = |trans(A)*b|
| trans(e) 0 | |lambda| | -c |
where lambda = (lambda(0), lambda(1), ..., lambda(P-1)) are coefficients.
The above matrix equation can be expressed as D*d = g. This class solves
the equation using the LU factorization method. The covariance
matrix is the inverse of (trans(A)*A). If no constraints are imposed,
this class solves the normal equation of linear fit problem using the LU
factorization method.
The following data types can be used to instantiate the
LinearFitConstraintLU template class:
If the class is instantaited with Float, all computations are carried out in single precision. If more precise results are desired, Double should be used.
If there are a large Number of unknowns or a large number of data points machine memory limit may not allow a complete in core fitting (calling fit(...) function) to be performed. In this case one can incrementally build the normal equations.
fit() performs least-squares fit on the supplied x, y, and sigma. For complex data, user should supply sigma for both real and imaginary parts. Returns fitted parameters in a vector.
This function is for fitting onedimensional (1D) function to data points. Here 1D means the fitted function f(x) is a function of one variable, x.
fit() performs least-squares fit on the supplied x, y, and sigma. For complex data, user should supply sigma for both real and imaginary parts. Returns fitted parameters in a vector.
This function is for fitting multidimensional (ND) function to data points. Here ND means the fitted function f(x0,x1,...,xn) is a function of n variables, x0,x1,...,xn. As input, a row of x, x(i,l) l = 0, L-1, where L denotes the number of independent variables, contains the variables corresponding to a single mesurement y(i).