fstats_experimental_design

Uses

Interfaces

public interface doe_evaluate_model

private function doe_evaluate_model_1(nway, beta, x, map) result(rst)

Evaluates the model of the following form.

$Y = \beta_{0} + \sum_{i=1}^{n} \beta_{i} X_{i} + \sum_{i=1}^{n} \sum_{j=1 \\ i \neq j}^{n} \beta_{ij} X_{i} X_{j} + \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1 \\ i \neq j \neq k}^{n} \beta_{ijk} X_{i} X_{j} X_{k} + ...$

Arguments

Type	Intent	Optional	Attributes		Name
integer(kind=int32),	intent(in)			::	nway	The number of interaction levels. Currently, this algorithm supports a maximum of three-way interaction.
real(kind=real64),	intent(in),		dimension(:)	::	beta	The model coefficients.
real(kind=real64),	intent(in),		dimension(:,:)	::	x	The M-by-N matrix containing the M values of each of the N factors at which to evaluate the model.
logical,	intent(in),	optional,	target, dimension(:)	::	map	An optional array of the same size as beta that can be used to eliminate a parameter from the model (false), or keep a parameter in the model (true). If not supplied, all parameters will be assumed to be part of the model as if the array were filled with all true values.

Return Value real(kind=real64), allocatable, dimension(:)

The resulting M-element array.

private function doe_evaluate_model_2(mdl, x) result(rst)

Evaluates the model of the following form.

Arguments

Type	Intent	Optional	Attributes		Name
class(doe_model),	intent(in)			::	mdl	The model to evaluate.
real(kind=real64),	intent(in),		dimension(:,:)	::	x	The M-by-N matrix containing the M values of each of the N factors at which to evaluate the model.

Return Value real(kind=real64), allocatable, dimension(:)

The resulting M-element array.

Derived Types

type, public :: doe_anova_table

ANOVA table for overall model fit assessment.

Components

Type	Visibility		Name
integer(kind=int32),	public	::	df_model	Model degrees of freedom.
integer(kind=int32),	public	::	df_residual	Residual degrees of freedom.
integer(kind=int32),	public	::	df_total	Total degrees of freedom.
real(kind=real64),	public	::	f_statistic	F-statistic.
real(kind=real64),	public	::	ms_model	Model mean square.
real(kind=real64),	public	::	ms_residual	Residual mean square.
real(kind=real64),	public	::	p_value	P-value for the F-test.
real(kind=real64),	public	::	r_squared	R-squared value.
real(kind=real64),	public	::	ss_model	Model sum of squares.
real(kind=real64),	public	::	ss_residual	Residual sum of squares.
real(kind=real64),	public	::	ss_total	Total sum of squares.

type, public :: doe_comparison_result

Results from comparing two models.

Components

Type	Visibility		Name
character(len=256),	public	::	conclusion	Interpretation of comparison results.
integer(kind=int32),	public	::	df_diff	Difference in degrees of freedom.
integer(kind=int32),	public	::	df_full	Degrees of freedom for full model.
integer(kind=int32),	public	::	df_reduced	Degrees of freedom for reduced model.
real(kind=real64),	public	::	f_statistic	F-statistic for model comparison.
real(kind=real64),	public	::	p_value	P-value for the F-test.
real(kind=real64),	public	::	rss_full	Residual sum of squares for full model.
real(kind=real64),	public	::	rss_reduced	Residual sum of squares for reduced model.
logical,	public	::	significant_difference	True if models differ significantly (p < 0.05).

type, public :: doe_diagnostics

Model diagnostics and goodness-of-fit metrics.

Components

Type	Visibility		Name
real(kind=real64),	public	::	f_p_value	P-value for the overall F-statistic.
real(kind=real64),	public	::	f_statistic	Overall F-statistic for the model.
real(kind=real64),	public	::	mean_response	Mean of the response variable.
integer(kind=int32),	public	::	n_observations	Number of observations.
integer(kind=int32),	public	::	n_parameters	Number of model parameters (including intercept).
real(kind=real64),	public	::	r_squared	The coefficient of determination (RÂ²), range [0, 1].
real(kind=real64),	public	::	r_squared_adjusted	The adjusted RÂ² accounting for model complexity.
real(kind=real64),	public	::	residual_std_error	Residual standard error (standard deviation of residuals).
real(kind=real64),	public	::	rmse	Root mean square error.

type, public :: doe_efficiency_metrics

Design efficiency metrics for evaluating design quality.

Components

Type	Visibility		Name
real(kind=real64),	public	::	a_efficiency	Higher is better.
real(kind=real64),	public	::	d_efficiency	Range [0,1]. Higher is better (max=1 for orthogonal designs).
real(kind=real64),	public	::	g_efficiency	G-efficiency: 1 - (max_prediction_variance / avg_prediction_variance)
logical,	public	::	is_orthogonal	True if design is perfectly orthogonal.
integer(kind=int32),	public	::	n_factors	Number of factors.
integer(kind=int32),	public	::	n_parameters	Number of model parameters.
integer(kind=int32),	public	::	n_runs	Number of design runs.
real(kind=real64),	public	::	orthogonality	Orthogonality measure: $1.0$ if perfectly orthogonal, $<1.0$ otherwise.

type, public :: doe_model

A model used to represent a design of experiments result. The model is of the following form.

Components

Type	Visibility	Attributes		Name
real(kind=real64),	public,	allocatable, dimension(:)	::	coefficients	The model coefficients.
logical,	public,	allocatable, dimension(:)	::	map	An array denoting if a model coefficient should be included as part of the model (true), or neglected (false).
integer(kind=int32),	public		::	nway	The number of interaction levels.
type(regression_statistics),	public,	allocatable, dimension(:)	::	stats	Statistical information for each model parameter.

type, public :: doe_optimization_result

Results from RSM-based optimization.

Components

Type	Visibility	Attributes		Name
logical,	public		::	converged	Whether optimization converged.
real(kind=real64),	public		::	convergence_tolerance	Tolerance used for convergence.
integer(kind=int32),	public		::	iteration_count	Number of iterations to converge.
character(len=256),	public		::	method	Optimization method used.
real(kind=real64),	public,	allocatable, dimension(:)	::	optimal_coded_factors	Optimal factor settings (in coded scale).
real(kind=real64),	public,	allocatable, dimension(:)	::	optimal_natural_factors	Optimal factor settings (in natural scale).
real(kind=real64),	public		::	optimal_response	Predicted response at optimal point.

type, public :: doe_prediction

Prediction with uncertainty quantification.

Components

Type	Visibility	Attributes		Name
real(kind=real64),	public		::	confidence_level	Confidence level (e.g., 0.95 for 95% CI).
real(kind=real64),	public,	allocatable, dimension(:)	::	confidence_lower	Lower confidence interval bounds.
real(kind=real64),	public,	allocatable, dimension(:)	::	confidence_upper	Upper confidence interval bounds.
real(kind=real64),	public,	allocatable, dimension(:)	::	predicted_values	Predicted response values.
real(kind=real64),	public,	allocatable, dimension(:)	::	prediction_lower	Lower prediction interval bounds.
real(kind=real64),	public,	allocatable, dimension(:)	::	prediction_upper	Upper prediction interval bounds.

type, public :: doe_residuals

Residual analysis data.

Components

Type	Visibility	Attributes		Name
real(kind=real64),	public,	allocatable, dimension(:)	::	observed_values	Observed response values.
real(kind=real64),	public,	allocatable, dimension(:)	::	predicted_values	Predicted response values.
real(kind=real64),	public		::	residual_mean	Mean of residuals (should be ~0).
real(kind=real64),	public		::	residual_std	Standard deviation of residuals.
real(kind=real64),	public,	allocatable, dimension(:)	::	residuals	Raw residuals: y - y_predicted.
real(kind=real64),	public,	allocatable, dimension(:)	::	standardized_residuals	Standardized residuals for outlier detection.

type, public :: doe_rsm_model

Response Surface Model (quadratic model for RSM).

Components

Type	Visibility	Attributes		Name
type(doe_model),	public		::	base_model	Base fitted model.
real(kind=real64),	public,	allocatable, dimension(:)	::	interaction_coeff	Interaction coefficients.
real(kind=real64),	public		::	intercept	Model intercept.
real(kind=real64),	public,	allocatable, dimension(:)	::	linear_coeff	Linear coefficients for each factor.
integer(kind=int32),	public		::	n_factors	Number of factors.
real(kind=real64),	public,	allocatable, dimension(:)	::	quadratic_coeff	Quadratic coefficients (main effects squared).
real(kind=real64),	public		::	response_at_center	Predicted response at design center (coded 0,0,...,0).

Functions

public function doe_compare_models(mdl1, mdl2, x, y) result(comp)

Arguments

Type	Intent	Attributes		Name
class(doe_model),	intent(in)		::	mdl1
class(doe_model),	intent(in)		::	mdl2
real(kind=real64),	intent(in),	dimension(:,:)	::	x
real(kind=real64),	intent(in),	dimension(:)	::	y

Return Value type(doe_comparison_result)

public pure function doe_design_efficiency(x) result(metrics)

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in),		dimension(:,:)	::	x

Return Value type(doe_efficiency_metrics)

public function doe_fit_model(nway, x, y, map, alpha) result(rst)

Fits a Taylor series model to the provided data.

Arguments

Type	Intent	Optional	Attributes		Name
integer(kind=int32),	intent(in)			::	nway	The number of interaction levels.
real(kind=real64),	intent(in),		dimension(:,:)	::	x	The M-by-N matrix containing the M values of each of the N factors used to produce the results.
real(kind=real64),	intent(in),		dimension(:)	::	y	An M-element array containing the results from the M experiments.
logical,	intent(in),	optional,	target, dimension(:)	::	map	An optional array of the same size as beta that can be used to eliminate a parameter from the model (false), or keep a parameter in the model (true). If not supplied, all parameters will be assumed to be part of the model as if the array were filled with all true values.
real(kind=real64),	intent(in),	optional		::	alpha	The significance level at which to evaluate the confidence intervals. The default value is 0.05 such that a 95% confidence interval is calculated.

Return Value type(doe_model)

The resulting model.

public function doe_model_anova(mdl, x, y) result(anova)

Arguments

Type	Intent	Attributes		Name
class(doe_model),	intent(in)		::	mdl
real(kind=real64),	intent(in),	dimension(:,:)	::	x
real(kind=real64),	intent(in),	dimension(:)	::	y

Return Value type(doe_anova_table)

public function doe_model_diagnostics(mdl, x, y) result(diag)

Computes model diagnostics and goodness-of-fit metrics.

Arguments

Type	Intent	Attributes		Name
class(doe_model),	intent(in)		::	mdl	The fitted DOE model.
real(kind=real64),	intent(in),	dimension(:,:)	::	x	The M-by-N matrix of factor values used in model fitting.
real(kind=real64),	intent(in),	dimension(:)	::	y	The M-element array of observed responses.

Return Value type(doe_diagnostics)

The resulting diagnostics.

public pure function doe_optimize_rsm(mdl, x_low, x_high, method, tol) result(opt)

Arguments

Type	Intent	Optional	Attributes		Name
class(doe_model),	intent(in)			::	mdl
real(kind=real64),	intent(in),		dimension(:)	::	x_low
real(kind=real64),	intent(in),		dimension(:)	::	x_high
character(len=*),	intent(in),	optional		::	method
real(kind=real64),	intent(in),	optional		::	tol

Return Value type(doe_optimization_result)

public function doe_predict(mdl, x, alpha) result(pred)

Computes predictions with confidence and prediction intervals.

Arguments

Type	Intent	Optional	Attributes		Name
class(doe_model),	intent(in)			::	mdl	The fitted DOE model.
real(kind=real64),	intent(in),		dimension(:,:)	::	x	The M-by-N matrix at which to evaluate the model.
real(kind=real64),	intent(in),	optional		::	alpha	Significance level (default 0.05 for 95% CI).

Return Value type(doe_prediction)

The predictions with intervals.

public function doe_predict_enhanced(mdl, x, alpha, residual_mse) result(pred)

Arguments

Type	Intent	Optional	Attributes		Name
class(doe_model),	intent(in)			::	mdl
real(kind=real64),	intent(in),		dimension(:,:)	::	x
real(kind=real64),	intent(in),	optional		::	alpha
real(kind=real64),	intent(in),	optional		::	residual_mse

Return Value type(doe_prediction)

public function doe_residuals_analysis(mdl, x, y) result(resid)

Computes residual analysis data for model diagnostics.

Arguments

Type	Intent	Attributes		Name
class(doe_model),	intent(in)		::	mdl	The fitted DOE model.
real(kind=real64),	intent(in),	dimension(:,:)	::	x	The M-by-N matrix of factor values.
real(kind=real64),	intent(in),	dimension(:)	::	y	The M-element array of observed responses.

Return Value type(doe_residuals)

The residual analysis results.

Subroutines

public subroutine central_composite_design(nfactors, alpha_type, tbl)

Generates a central composite design in coded variables.

Arguments

Type	Intent	Optional		Name
integer(kind=int32),	intent(in)		::	nfactors	Number of factors.
character(len=*),	intent(in),	optional	::	alpha_type	Type: "orthogonal" (default), "rotatable", "uniform".
real(kind=real64),	intent(out)		::	tbl(:,:)	Design table (coded variables in [-1, +1] range).

public pure subroutine central_composite_design_size(nfactors, alpha_type, m, n)

Computes the size of a central composite design.

Arguments

Type	Intent	Optional		Name
integer(kind=int32),	intent(in)		::	nfactors	Number of factors.
character(len=*),	intent(in),	optional	::	alpha_type	Type of CCD: "orthogonal" (default), "rotatable", or "uniform".
integer(kind=int32),	intent(out)		::	m	Number of runs (rows).
integer(kind=int32),	intent(out)		::	n	Number of factors (columns).

public pure subroutine decode_variables(x_coded, x_low, x_high, x_natural)

Converts coded variable values (-1, +1) to natural scale.

Arguments

Type	Intent	Attributes		Name
real(kind=real64),	intent(in),	dimension(:,:)	::	x_coded	M-by-N matrix of coded values in range [-1, +1].
real(kind=real64),	intent(in),	dimension(:)	::	x_low	N-element array of low values for each factor.
real(kind=real64),	intent(in),	dimension(:)	::	x_high	N-element array of high values for each factor.
real(kind=real64),	intent(out),	dimension(:,:)	::	x_natural	M-by-N matrix of natural (physical) variable values.

public pure subroutine encode_variables(x_natural, x_low, x_high, x_coded)

Converts natural variable values to coded (-1, +1) scale.

Arguments

Type	Intent	Attributes		Name
real(kind=real64),	intent(in),	dimension(:,:)	::	x_natural	M-by-N matrix of natural (physical) variable values.
real(kind=real64),	intent(in),	dimension(:)	::	x_low	N-element array of low values for each factor.
real(kind=real64),	intent(in),	dimension(:)	::	x_high	N-element array of high values for each factor.
real(kind=real64),	intent(out),	dimension(:,:)	::	x_coded	M-by-N matrix of coded values in range [-1, +1].

public subroutine fractional_factorial(nfactors, fraction, tbl)

Generates a 2-level fractional factorial design.

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	nfactors	Number of factors.
integer(kind=int32),	intent(in)	::	fraction	Fraction level (1 for 1/2, 2 for 1/4, etc.).
integer(kind=int32),	intent(out)	::	tbl(:,:)	Design table (runs Ã— factors), coded as 1 and 2.

public subroutine fractional_factorial_size(nfactors, fraction, m, n)

Computes the size of a fractional factorial design.

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	nfactors	Number of factors (k).
integer(kind=int32),	intent(in)	::	fraction	Fraction level (p): 1 for 1/2, 2 for 1/4, 3 for 1/8, etc.
integer(kind=int32),	intent(out)	::	m	Number of runs (rows).
integer(kind=int32),	intent(out)	::	n	Number of factors (columns), same as nfactors.

public subroutine full_factorial(vars, tbl)

Computes a table with values scaled from 1 to N describing a full-factorial design.

Arguments

Type	Intent	Optional	Attributes		Name
integer(kind=int32),	intent(in)			::	vars(:)	An M-element array containing the M factors to study. Each of the M entries to the array is expected to contain the number of options for that particular factor to explore. This value must be greater than or equal to 1.
integer(kind=int32),	intent(out)			::	tbl(:,:)	A table where the design will be written. Use get_full_factorial_matrix_size to determine the appropriate table size.

public subroutine get_full_factorial_matrix_size(vars, m, n)

Computes the appropriate size for a full-factorial design table.

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	vars(:)	An M-element array containing the M factors to study. Each of the M entries to the array is expected to contain the number of options for that particular factor to explore. This value must be greater than or equal to 1.
integer(kind=int32),	intent(out)	::	m	The number of rows for the table.
integer(kind=int32),	intent(out)	::	n	The number of columns for the table.

public subroutine latin_hypercube_design(nfactors, nsamples, seed, tbl)

Generates a Latin hypercube design for factor space exploration.

Arguments

Type	Intent	Optional		Name
integer(kind=int32),	intent(in)		::	nfactors	Number of factors.
integer(kind=int32),	intent(in)		::	nsamples	Number of samples (runs).
integer(kind=int32),	intent(inout),	optional	::	seed	Random seed for reproducibility.
real(kind=real64),	intent(out)		::	tbl(:,:)	Latin hypercube design (nsamples Ã— nfactors).

fstats_experimental_design Module

Contents

Interfaces

Derived Types

Functions

Subroutines

Uses

Interfaces

public interface doe_evaluate_model

private function doe_evaluate_model_1(nway, beta, x, map) result(rst)

Arguments

Return Value real(kind=real64), allocatable, dimension(:)

private function doe_evaluate_model_2(mdl, x) result(rst)

Arguments

Return Value real(kind=real64), allocatable, dimension(:)

Derived Types

type, public :: doe_anova_table

Components

type, public :: doe_comparison_result

Components

type, public :: doe_diagnostics

Components

type, public :: doe_efficiency_metrics

Components

type, public :: doe_model

Components

type, public :: doe_optimization_result

Components

type, public :: doe_prediction

Components

type, public :: doe_residuals

Components

type, public :: doe_rsm_model

Components

Functions

public function doe_compare_models(mdl1, mdl2, x, y) result(comp)

Arguments

Return Value type(doe_comparison_result)

public pure function doe_design_efficiency(x) result(metrics)

Arguments

Return Value type(doe_efficiency_metrics)

public function doe_fit_model(nway, x, y, map, alpha) result(rst)

Arguments

Return Value type(doe_model)

public function doe_model_anova(mdl, x, y) result(anova)

Arguments

Return Value type(doe_anova_table)

public function doe_model_diagnostics(mdl, x, y) result(diag)

Arguments

Return Value type(doe_diagnostics)

public pure function doe_optimize_rsm(mdl, x_low, x_high, method, tol) result(opt)

Arguments

Return Value type(doe_optimization_result)

public function doe_predict(mdl, x, alpha) result(pred)

Arguments

Return Value type(doe_prediction)

public function doe_predict_enhanced(mdl, x, alpha, residual_mse) result(pred)

Arguments

Return Value type(doe_prediction)

public function doe_residuals_analysis(mdl, x, y) result(resid)

Arguments

Return Value type(doe_residuals)

Subroutines

public subroutine central_composite_design(nfactors, alpha_type, tbl)

Arguments

public pure subroutine central_composite_design_size(nfactors, alpha_type, m, n)

Arguments

public pure subroutine decode_variables(x_coded, x_low, x_high, x_natural)

Arguments

public pure subroutine encode_variables(x_natural, x_low, x_high, x_coded)

Arguments

public subroutine fractional_factorial(nfactors, fraction, tbl)

Arguments

public subroutine fractional_factorial_size(nfactors, fraction, m, n)

Arguments

public subroutine full_factorial(vars, tbl)

Arguments

public subroutine get_full_factorial_matrix_size(vars, m, n)

Arguments

public subroutine latin_hypercube_design(nfactors, nsamples, seed, tbl)