doe_model – FSTATS

type, public :: doe_model

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

$Y = \beta_{0} + \sum_{i=1}^{n} \beta_{i} X_{i} + \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \beta_{ij} X_{i} X_{j} + \sum_{i=1}^{n} \beta_{ii} X_{i}^{2} + \sum_{i=1}^{n-2} \sum_{j=i+1}^{n-1} \sum_{k=j+1}^{n} \beta_{ijk} X_{i} X_{j} X_{k} + \sum_{i=1}^{n} \sum_{j=1 \\ j \neq i}^{n} \beta_{iij} X_{i}^{2} X_{j} + \sum_{i=1}^{n} \beta_{iii} X_{i}^{3}$

The cross and squared terms are included for a two-way model, and the triple terms are added for a three-way model. Each unique combination of factors appears exactly once. For two factors, a two-way model is $Y = \beta_{0} + \beta_{1} X_{1} + \beta_{2} X_{2} + \beta_{12} X_{1} X_{2} + \beta_{11} X_{1}^{2} + \beta_{22} X_{2}^{2}$ .

Components

Type	Visibility	Attributes		Name		Initial
real(kind=real64),	public,	allocatable, dimension(:)	::	coefficients			The model coefficients, where N is the number of factors. The coefficients are ordered as follows, with each group included only if the model has a sufficient number of interaction levels. The constant term: $\beta_{0}$ (1 term). Main effects, $i = 1 \ldots N$ : $\beta_{1}, \beta_{2}, \ldots, \beta_{N}$ (N terms). Included for nway >= 1. Two-way cross terms, $i < j$ in lexicographic order: $\beta_{12}, \beta_{13}, \ldots, \beta_{1N}, \beta_{23}, \ldots, \beta_{N-1,N}$ (N(N-1)/2 terms). Included for nway >= 2. Squared terms: $\beta_{11}, \beta_{22}, \ldots, \beta_{NN}$ (N terms). Included for nway >= 2. Three-way cross terms, $i < j < k$ in lexicographic order: $\beta_{123}, \beta_{124}, \ldots, \beta_{N-2,N-1,N}$ (N(N-1)(N-2)/6 terms). Included for nway >= 3. Squared-by-linear terms $X_{i}^{2} X_{j}$ , $i \neq j$ , with i as the outer index and j as the inner index: $\beta_{112}, \beta_{113}, \ldots, \beta_{11N}, \beta_{221}, \beta_{223}, \ldots, \beta_{NN,N-1}$ (N(N-1) terms). Included for nway >= 3. Cubed terms: $\beta_{111}, \beta_{222}, \ldots, \beta_{NNN}$ (N terms). Included for nway >= 3. For example, with N = 2 and nway = 2, the order is $\beta_{0}, \beta_{1}, \beta_{2}, \beta_{12}, \beta_{11}, \beta_{22}$ .
real(kind=real64),	public,	allocatable, dimension(:,:)	::	covariance			The unscaled parameter covariance matrix $\left( X^{T} X \right)^{-1}$ , where $X$ is the design matrix used to fit the model. Only the parameters included in the model (map = true) are represented, in coefficient order. This is populated by doe_fit_model, and is used to compute prediction intervals.
integer(kind=int32),	public		::	dof	=	0	The residual degrees of freedom of the fit, $m - p$ . A value of zero indicates the fit information is not available (e.g. the model was not created by doe_fit_model).
logical,	public,	allocatable, dimension(:)	::	map			An array denoting if a model coefficient should be included as part of the model (true), or neglected (false).
real(kind=real64),	public		::	mse	=	0.0d0	The residual mean squared error of the fit, $SS_{res} / (m - p)$ . This is only meaningful if dof > 0.
integer(kind=int32),	public		::	nway			The number of interaction levels.
type(regression_statistics),	public,	allocatable, dimension(:)	::	stats			Statistical information for each model parameter.

doe_model Derived Type

Contents

Variables

type, public :: doe_model

Components