Least square fit variable

This variable fits a parameter vector \(\mathbf{p}\) consisting of \(M\) parameters \(p_1, p_2,\dots, p_M\) to a model function \(f(\mathbf{x}, \mathbf{p})\) that depends on the model variables \(x_1, x_2,\dots, x_D\). The output of the variable is a least-square estimate of the fit parameters to data points \((\mathbf{x}_i, y_i), i=1,\dots,N\), where \(y_1,\dots,y_n\) are the mean values of the input. That is, the variable locally minimizes

\[\chi^2(\mathbf{p}) = \sum_{i=1}^N \frac{\left(f\left(\mathbf{x}_i, \mathbf{p}\right) - y_i\right)^2}{\eta_i^2},\]

where \(\eta_i^2\) is the variance of input \(i\) to the variable.

name (str)

The name of the variable under which it can be addressed by other variables or objectives. The name must be distinct from any surrogate name.

Default: "v"

output_dim (int)

Output dimension of the fit variable. The variables outputs the values of the fitted parameters in the first \(M\) entries and the minimum value of the mean squared error \({\rm MSE} = \chi^2/(N-M)\) found after the least-square fit in the last entry. Hence, the output_dim is \(M+1\).

Default: This value has no default and must be provided

output_names (cell{str})

Allows to assign names to each of the outputs of the variable. By specifying input names they can be accessed in variables as {output_names[0]}, {output_names[1]}, ..., {output_names[K-1]}, where K is the output dimension.

Default:

By default, the variables can be accessed as {name}0, {name}1, ...,{name}(K-1), where {name} is the name of the variable and K is the output dimension.

input (str)

Names of a multi-output surrogate or variable.

Default: This value has no default and must be provided

initial_parameters (cell{float})

Initial values of the fit parameters to perform the least-square fit.

Default: This value has no default and must be provided
Example

The initial value of \(M=2\) parameters.
{0.5,0.2}

prior_uncertainties (cell{float})

Prior uncertainties of the fit parameters. The posterior covariance of the parameters is determined as the likelihood times the prior.

Default: Each parameter has a uncertainty of 100.

model_variables (cell{str})

The names of the variables \(x_1, x_2,\dots, x_D\) of the model function. The number of variables \(D\) must correspond to the output dimensionality of the input surrogate or variable.

Default: By default, the variables are named ‘x0’, ‘x1’, … ‘x{D-1}’.

variable_values (cell{cell{float}})

The values of the model variables \(\mathbf{x}_1,\dots,\mathbf{x}_N\). This is a list of \(N\) vectors where each entry is a list of \(D\) values.

Default: This value has no default and must be provided
Example

A list with of six values for two variables.
{{1.0,0.0},...
 {1.0,0.2},...
 {1.0,0.4},...
 {2.0,0.0},...
 {2.0,0.2},...
 {2.0,0.4}}

expression (str)

The expression describing the model function \(f(\mathbf{x}, \mathbf{p})\). The names of the fit parameters correspond to the first \(M\) output names.

Default: This value has no default and must be provided
Example

Fit of the amplitude, frequency and phase of an oscillating input.
"p0*sin(p1*x0 + p2)"

ftol (float)

The least square problem is solved using scipy.optimize.least_squares. The parameter ftol is one of the convergence criteria. Smaller values correspond to more accurate fitting parameters, but longer fitting times that lead to longer sample computation times.

Default: 1e-05

diff_step (float)

Absolute step size for the finite difference approximation of the Jacobian.

Default: 1e-10

max_nfev (int)

Maximum number of function evaluations before the termination of the least-square fit.

Default: \(100\cdot M\).