The multi-dimension fitting library NDLINEAR is not included in GSL, but is provided as an extension library. This is available at the Patric Alken's page.
Contents:
The NDLINEAR extension provides support for general linear least squares
fitting to data which is a function of more than one variable (multi-linear or
multi-dimensional least squares fitting). This model has the form where
x is a vector of independent variables, a_i are the fit coefficients,
and F_i are the basis functions of the fit. This GSL extension computes the
design matrix X_{ij = F_j(x_i) in the special case that the basis functions
separate: Here the superscript value j indicates the basis function
corresponding to the independent variable x_j. The subscripts (i_1, i_2, i_3,
c) refer to which basis function to use from the complete set. These
subscripts are related to the index i in a complex way, which is the main
problem this extension addresses. The model then becomes where n is the
dimension of the fit and N_i is the number of basis functions for the variable
x_i. Computationally, it is easier to supply the individual basis functions
u^{(j) than the total basis functions F_i(x). However the design matrix X is
easiest to construct given F_i(x). Therefore the routines below allow the user
to specify the individual basis functions u^{(j) and then automatically
construct the design matrix X.
GSL::MultiFit::Ndlinear.alloc(n_dim, N, u, params)GSL::MultiFit::Ndlinear::Workspace.alloc(n_dim, N, u, params)Creates a workspace for solving multi-parameter, multi-dimensional linear
least squares problems. n_dim specifies the dimension of the fit
(the number of independent variables in the model). The array N of
length n_dim specifies the number of terms in each sum, so that
N[i]
specifies the number of terms in the sum of the i-th independent variable.
The array of Proc objects u of length n_dim specifies
the basis functions for each independent fit variable, so that u[i]
is a procedure to calculate the basis function for the i-th
independent variable.
Each of the procedures u takes three block parameters: a point
x at which to evaluate the basis function, an array y of length
N[i] which is filled on output with the basis function values at
x for all i, and a params argument which contains parameters needed
by the basis function. These parameters are supplied in the params
argument to this method.
Ex)
N_DIM = 3
N_SUM_R = 10
N_SUM_THETA = 11
N_SUM_PHI = 9
basis_r = Proc.new { |r, y, params|
params.eval(r, y)
}
basis_theta = Proc.new { |theta, y, params|
for i in 0...N_SUM_THETA do
y[i] = GSL::Sf::legendre_Pl(i, Math::cos(theta));
end
}
basis_phi = Proc.new { |phi, y, params|
for i in 0...N_SUM_PHI do
if i%2 == 0
y[i] = Math::cos(i*0.5*phi)
else
y[i] = Math::sin((i+1.0)*0.5*phi)
end
end
}
N = [N_SUM_R, N_SUM_THETA, N_SUM_PHI]
u = [basis_r, basis_theta, basis_phi]
bspline = GSL::BSpline.alloc(4, N_SUM_R - 2)
ndlinear = GSL::MultiFit::Ndlinear.alloc(N_DIM, N, u, bspline)GSL::MultiFit::Ndlinear.design(vars, X, w)GSL::MultiFit::Ndlinear.design(vars, w)GSL::MultiFit::Ndlinear::Workspace#design(vars, X)GSL::MultiFit::Ndlinear::Workspace#design(vars)GSL::MultiFit::Ndlinear.est(x, c, cov, w)GSL::MultiFit::Ndlinear::Workspace#est(x, c, cov)GSL::MultiFit::linear,
this method can be used to evaluate the model at the data point x.
The coefficient vector c and covariance matrix cov are
outputs from GSL::MultiFit::linear. The model output value and
its error [y, yerr] are returned as an array.GSL::MultiFit::Ndlinear.calc(x, c, w)GSL::MultiFit::Ndlinear::Workspace#calc(x, c)GSL::MultiFit::Ndlinear.est, but does not compute the model error. It computes the model value at the data point x using the coefficient vector c and returns the model value.This example program generates data from the 3D isotropic harmonic oscillator wavefunction (real part) and then fits a model to the data using B-splines in the r coordinate, Legendre polynomials in theta, and sines/cosines in phi. The exact form of the solution is (neglecting the normalization constant for simplicity) The example program models psi by default.
#!/usr/bin/env ruby
require("rbgsl")
N_DIM = 3
N_SUM_R = 10
N_SUM_THETA = 10
N_SUM_PHI = 9
R_MAX = 3.0
def psi_real_exact(k, l, m, r, theta, phi)
rr = GSL::pow(r, l)*Math::exp(-r*r)*GSL::Sf::laguerre_n(k, l + 0.5, 2 * r * r)
tt = GSL::Sf::legendre_sphPlm(l, m, Math::cos(theta))
pp = Math::cos(m*phi)
rr*tt*pp
end
basis_r = Proc.new { |r, y, params|
params.eval(r, y)
}
basis_theta = Proc.new { |theta, y, params|
for i in 0...N_SUM_THETA do
y[i] = GSL::Sf::legendre_Pl(i, Math::cos(theta));
end
}
basis_phi = Proc.new { |phi, y, params|
for i in 0...N_SUM_PHI do
if i%2 == 0
y[i] = Math::cos(i*0.5*phi)
else
y[i] = Math::sin((i+1.0)*0.5*phi)
end
end
}
GSL::Rng::env_setup()
k = 5
l = 4
m = 2
NDATA = 3000
N = [N_SUM_R, N_SUM_THETA, N_SUM_PHI]
u = [basis_r, basis_theta, basis_phi]
rng = GSL::Rng.alloc()
bspline = GSL::BSpline.alloc(4, N_SUM_R - 2)
bspline.knots_uniform(0.0, R_MAX)
ndlinear = GSL::MultiFit::Ndlinear.alloc(N_DIM, N, u, bspline)
multifit = GSL::MultiFit.alloc(NDATA, ndlinear.n_coeffs)
vars = GSL::Matrix.alloc(NDATA, N_DIM)
data = GSL::Vector.alloc(NDATA)
for i in 0...NDATA do
r = rng.uniform()*R_MAX
theta = rng.uniform()*Math::PI
phi = rng.uniform()*2*Math::PI
psi = psi_real_exact(k, l, m, r, theta, phi)
dpsi = rng.gaussian(0.05*psi)
vars[i][0] = r
vars[i][1] = theta
vars[i][2] = phi
data[i] = psi + dpsi
end
X = GSL::MultiFit::Ndlinear::design(vars, ndlinear)
coeffs, cov, chisq, = GSL::MultiFit::linear(X, data, multifit)
rsq = 1.0 - chisq/data.tss
STDERR.printf("chisq = %e, Rsq = %f\n", chisq, rsq)
eps_rms = 0.0
volume = 0.0
dr = 0.05;
dtheta = 5.0 * Math::PI / 180.0
dphi = 5.0 * Math::PI / 180.0
x = GSL::Vector.alloc(N_DIM)
r = 0.01
while r < R_MAX do
theta = 0.0
while theta < Math::PI do
phi = 0.0
while phi < 2*Math::PI do
dV = r*r*Math::sin(theta)*r*dtheta*dphi
x[0] = r
x[1] = theta
x[2] = phi
psi_model, err = GSL::MultiFit::Ndlinear.calc(x, coeffs, ndlinear)
psi = psi_real_exact(k, l, m, r, theta, phi)
err = psi_model - psi
eps_rms += err * err * dV;
volume += dV;
if phi == 0.0
printf("%e %e %e %e\n", r, theta, psi, psi_model)
end
phi += dphi
end
theta += dtheta
end
printf("\n");
r += dr
end
eps_rms /= volume
eps_rms = Math::sqrt(eps_rms)
STDERR.printf("rms error over all parameter space = %e\n", eps_rms)