Nonlinear Regression.
min [EuclidianNorm (Y - modelfun (beta, X))] ^ 2 beta
X is a matrix of independents, Y is the observed output and modelfun is the nonlinear regression model function.
modelfun should be specified as a function handle, which
accepts two inputs: an array of coefficients and an array of
independents – in that order.
The first four input arguments must be provided with non-empty initial guess of the coefficients beta0.
Y and X must be the same size as the vector (or matrix) returned by fun.
options is a structure containing estimation algorithm options. It can be set using statset.
Follwing Matlab compatible options are recognized:
TolFun
Minimum fractional improvement in objective function in an iteration
(termination criterium). Default: 1e-6.
MaxIter
Maximum number of iterations allowed. Default: 400.
DerivStep
Step size factor. The default is eps^(1/3) for finite differences gradient
calculation.
Display
String indicating the degree of verbosity. Default: "off".
Currently only supported values are "off" (no messages) and "iter"
(some messages after each iteration).
Optional Name, Value pairs can be provided to set additional options. Currently the only applicable name-value pair is ’Weights’, w, where w is the array of real positive weight factors for the squared residuals.
Returned values:
Coefficients to best fit the nonlinear function modelfun (beta, X) to the observed values Y.
Value of solution residuals: modelfun (beta, X) - Y.
If observation weights are specified then R is the array of
weighted residuals: sqrt (weights) .* modelfun (beta, X) - Y.
A matrix where J(i,j) is the partial derivative of modelfun(i) with respect to beta(j).
If observation weights are specified, then J is the weighted
model function Jacobian: diag (sqrt (weights)) * J.
Estimated covariance matrix of the fitted coefficients.
Scalar valued estimate of the variance of error term. If the model Jacobian is full rank, then MSE = (R’ * R)/(N-p), where N is the number of observations and p is the number of estimated coefficients.
This function is a compatibility wrapper. It calls the more general nonlin_curvefit
and curvefit_stat functions internally.
See also: nonlin_residmin, nonlin_curvefit, residmin_stat, curvefit_stat.
The following code
modelfun = @(b, x) (b(1) + b(2) * exp (- b(3) * x));
%% actual value
beta_without_noise = [1; 3; 2];
x = [3.49622; 0.33751; 1.25675; 3.66981; 0.26237; 5.51095; ...
2.11407; 1.48774; 6.22436; 2.04519];
y_actual = modelfun (beta_without_noise, x);
noise = [0.176110; -0.066850; 0.231000; -0.047570; -0.108230; ...
0.122790; 0.062940; 0.151510; 0.116010; -0.097460];
y_noisy = y_actual + noise;
%% initial guess
beta0 = [2; 2; 2];
%% weights vector
weights = [5; 16; 1; 20; 12; 11; 17; 8; 11; 13];
[beta, R, J, covb, mse] = nlinfit (x, y_noisy, modelfun, beta0)
[beta_w, R_w, J_w, covb_w, mse_w] = nlinfit (x, y_noisy, modelfun, beta0, [], "weights", weights)
Produces the following output
beta = 1.0593 2.4516 1.6207 R = 1.1110e-01 -1.7384e-02 9.4882e-02 -1.1131e-01 5.1953e-03 6.3231e-02 -3.2298e-02 2.5370e-02 5.6636e-02 -1.9565e-01 J = 1.0000e+00 3.4603e-03 -2.9659e-02 1.0000e+00 5.7868e-01 -4.7882e-01 1.0000e+00 1.3044e-01 -4.0189e-01 1.0000e+00 2.6117e-03 -2.3498e-02 1.0000e+00 6.5362e-01 -4.2043e-01 1.0000e+00 1.3213e-04 -1.7852e-03 1.0000e+00 3.2506e-02 -1.6848e-01 1.0000e+00 8.9706e-02 -3.2719e-01 1.0000e+00 4.1577e-05 -6.3446e-04 1.0000e+00 3.6345e-02 -1.8224e-01 covb = 2.8309e-03 3.2125e-03 1.0622e-02 3.2125e-03 5.5470e-02 5.7435e-02 1.0622e-02 5.7435e-02 9.5325e-02 mse = 0.011605 beta_w = 1.0361 2.6290 1.8142 R_w = 3.0895e-01 -2.5185e-03 1.6898e-01 -3.8045e-01 -8.5216e-03 2.8737e-01 5.7045e-02 2.5925e-01 2.6505e-01 -5.3240e-01 J_w = 2.2361e+00 3.9338e-03 -3.6158e-02 4.0000e+00 2.1684e+00 -1.9240e+00 1.0000e+00 1.0228e-01 -3.3794e-01 4.4721e+00 5.7422e-03 -5.5399e-02 3.4641e+00 2.1521e+00 -1.4845e+00 3.3166e+00 1.5088e-04 -2.1859e-03 4.1231e+00 8.9033e-02 -4.9483e-01 2.8284e+00 1.9026e-01 -7.4416e-01 3.3166e+00 4.1355e-05 -6.7673e-04 3.6056e+00 8.8220e-02 -4.7434e-01 covb_w = 2.2105e-03 6.7625e-03 1.3396e-02 6.7625e-03 1.1812e-01 1.4646e-01 1.3396e-02 1.4646e-01 2.1109e-01 mse_w = 0.1108
Package: optim