The functions described in this section can be used to perform least-squares fits to a straight line model, Y = c_0 + c_1 X. For weighted data the best-fit is found by minimizing the weighted sum of squared residuals, chi^2,
chi^2 = sum_i w_i (y_i - (c0 + c1 x_i))^2
for the parameters c0, c1. For unweighted data the sum is computed with w_i = 1.
GSL::Fit::linear(x, y)[c0, c1, cov00, cov01, cov11, chisq, status], where c0, c1 are the estimated parameters, cov00, cov01, cov11 are the variance-covariance matrix elements, chisq is the sum of squares of the residuals, and status is the return code from the GSL function gsl_fit_linear().GSL::Fit::wlinear(x, w, y)linear.GSL::Fit::linear_est(x, c0, c1, c00, c01, c11)GSL::Fit::linear_est(x, [c0, c1, c00, c01, c11])[y, yerr].GSL::Fit::mul(x, y)c1 of the model Y = c1 X for the datasets (x, y), two vectors of equal length with stride 1. This returns an array of 4 elements, [c1, cov11, chisq, status].GSL::Fit::wmul(x, w, y)c1 of the model Y = c_1 X for the weighted datasets (x, y). The vector w specifies the weight of each datapoint. The weight is the reciprocal of the variance for each datapoint in y.GSL::Fit::mul_est(x, c1, c11)GSL::Fit::mul_est(x, [c1, c11])y and its standard deviation y_err for the model Y = c_1 X at the point x. The returned value is an array of [y, yerr].GSL::MultiFit::Workspace.new(n, p)GSL::MultiFit::linear(X, y, work)GSL::MultiFit::linear(X, y)This function computes the best-fit parameters c of the model y = X c for the observations y and the matrix of predictor variables X. The variance-covariance matrix of the model parameters cov is estimated from the scatter of the observations about the best-fit. The sum of squares of the residuals from the best-fit is also calculated. The returned value is an array of 4 elements, [c, cov, chisq, status], where c is a GSL::Vector object which contains the best-fit parameters, and cov is the variance-covariance matrix as a GSL::Matrix object.
The best-fit is found by singular value decomposition of the matrix X using the workspace provided in work (optional, if not given, it is allocated internally). The modified Golub-Reinsch SVD algorithm is used, with column scaling to improve the accuracy of the singular values. Any components which have zero singular value (to machine precision) are discarded from the fit.
GSL::MultiFit::wlinear(X, w, y, work)GSL::MultiFit::wlinear(X, w, y)c of the model y = X c for the observations y and the matrix of predictor variables X. The covariance matrix of the model parameters cov is estimated from the weighted data. The weighted sum of squares of the residuals from the best-fit is also calculated. The returned value is an array of 4 elements, [c: Vector, cov: Matrix, chisq: Float, status: Fixnum]. The best-fit is found by singular value decomposition of the matrix X using the workspace provided in work (optional). Any components which have zero singular value (to machine precision) are discarded from the fit.GSL::MultiFit::polyfit(x, y, order)Finds the coefficient of a polynomial of order order that fits the vector data (x, y) in a least-square sense.
Example:
#!/usr/bin/env ruby
require("gsl")
x = Vector[1, 2, 3, 4, 5]
y = Vector[5.5, 43.1, 128, 290.7, 498.4]
# The results are stored in a polynomial "coef"
coef, err, chisq, status = MultiFit.polyfit(x, y, 3)
x2 = Vector.linspace(1, 5, 20)
graph([x, y], [x2, coef.eval(x2)], "-C -g 3 -S 4")
#!/usr/bin/env ruby
require("gsl")
include GSL::Fit
n = 4
x = Vector.new(1970, 1980, 1990, 2000)
y = Vector.new(12, 11, 14, 13)
w = Vector.new(0.1, 0.2, 0.3, 0.4)
#for i in 0...n do
# printf("%e %e %e\n", x[i], y[i], 1.0/Math::sqrt(w[i]))
#end
c0, c1, cov00, cov01, cov11, chisq = wlinear(x, w, y)
printf("# best fit: Y = %g + %g X\n", c0, c1);
printf("# covariance matrix:\n");
printf("# [ %g, %g\n# %g, %g]\n",
cov00, cov01, cov01, cov11);
printf("# chisq = %g\n", chisq);
#!/usr/bin/env ruby
require("gsl")
# Create data
r = Rng.alloc("knuthran")
a = 2.0
b = -1.0
sigma = 0.01
N = 10
x = Vector.linspace(0, 5, N)
y = a*Sf::exp(b*x) + sigma*r.gaussian
# Fitting
a2, b2, = Fit.linear(x, Sf::log(y))
x2 = Vector.linspace(0, 5, 20)
A = Sf::exp(a2)
printf("Expect: a = %f, b = %f\n", a, b)
printf("Result: a = %f, b = %f\n", A, b2)
graph([x, y], [x2, A*Sf::exp(b2*x2)], "-C -g 3 -S 4")
#!/usr/bin/env ruby
require("gsl")
include GSL::MultiFit
Rng.env_setup()
r = GSL::Rng.new(Rng::DEFAULT)
n = 19
dim = 3
X = Matrix.new(n, dim)
y = Vector.new(n)
w = Vector.new(n)
a = 0.1
for i in 0...n
y0 = Math::exp(a)
sigma = 0.1*y0
val = r.gaussian(sigma)
X.set(i, 0, 1.0)
X.set(i, 1, a)
X.set(i, 2, a*a)
y[i] = y0 + val
w[i] = 1.0/(sigma*sigma)
#printf("%g %g %g\n", a, y[i], sigma)
a += 0.1
end
c, cov, chisq, status = MultiFit.wlinear(X, w, y)