Package 'multisensi' reference manual

Title:	Multivariate Sensitivity Analysis
Description:	Functions to perform sensitivity analysis on a model with multivariate output.
Authors:	Caroline Bidot <[email protected]>, Matieyendou Lamboni <[email protected]>, Hervé Monod <[email protected]>
Maintainer:	Hervé Monod <[email protected]>
License:	CeCILL-2
Version:	2.1-1
Built:	2024-12-10 06:36:04 UTC
Source:	CRAN

Multivariate sensitivity Analysis

Description

Sensitivity Analysis (SA) for models with multivariate output

Details

This package generalises sensitivity analysis to simulation models with multivariate output. It makes it easy to run a series of independent sensitivity analyses on a set of output variables and to plot the results. Alternatively, it allows to apply sensitivity analyses to the variables resulting from the application of a multivariate method (such as PCA or splines or polynomial regression) to the output data (Lamboni et al., 2009).

The function multisensi integrates all the different possible methods implemented in the package. Besides, the user may consider the functions which have existed since the first version of the package:

i) gsi function for the Generalised Sensitivity Analysis (Lamboni et al., 2011, Xiao and Li, 2016) based on inertia decomposition. This method synthesizes the information that is spread between the time outputs or between the principal components and produces a unique sensitivity index for each factor.

ii) gsi function for the componentwise sensitivity analysis obtained by computing sensitivity indices on principal components (Campbell et al., 2006)

iii) dynsi function for the dynamic sensitivity analysis obtained by computing sensitivity indices on each output variable.

In the first version of multisensi, sensitivity indices were based on using a factorial design and a classical ANOVA decomposition. It is now possible to use other methods for the design and for the sensitivity analysis.

Simulation model management

The multisensi package works on simulation models coded either in R or using an external language (typically as an executable file). Models coded in R must be either functions or objects that have a predict method, such as lm objects. Models defined as functions will be called once with an expression of the form y <- f(X) where X is a vector containing a combination of levels of the input factors, and y is the output vector of length $q$ , where $q$ is the number of output variables. If the model is external to R, for instance a computational code, it must be analyzed with the decoupled approach: the methods require an input data frame (X) containing all the combinations of the input levels and the outputs data frame (Y) containing the response of the model corresponding to these combinations. The size of X is $n*p$ and the size of Y is $n*q$ where $p$ is the number of the input factor, $q$ is the number of the model outputs and $n$ is the number of all the combinations of the input levels. This approach can also be used on R models that do not fit the required specifications.

References

Lamboni, M., Makowski, D., Monod, H., 2009. Multivariate global sensitivity analysis for dynamic crop models. Field Crops Research, volume 113, pp. 312-320.

Lamboni, M., Makowski, D., Monod, H., 2011. Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models. Reliability Engineering & System Safety, volume 96, pp. 450-459.

Xiao, H., Li, L., 2016. Discussion of paper by M. Lamboni, H. Monod, D. Makowski Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models, Reliab. Eng. Syst. Saf. 96 (2011) 450-459. Reliability Engineering & System Safety, volume 147, pp. 194-195.

Saltelli, A., Chan, K., Scott, E.M. eds, 2000. Sensitivity Analysis Wiley, New York.

Runs a series of analyses of variance

Description

The analysis.anoasg function runs a series of analyses of variance on the columns of a data.frame, by using the aov function.

Usage

analysis.anoasg(Y, plan, nbcomp = 2, sigma.car = NULL,
                analysis.args = list(formula = 2,
                                     keep.outputs = FALSE))
analysis.anoasg(Y, plan, nbcomp = 2, sigma.car = NULL,
                analysis.args = list(formula = 2,
                                     keep.outputs = FALSE))

Arguments

`Y`	a data.frame of output variables or principal components.
`plan`	a data.frame containing the design.
`nbcomp`	the number of `Y` variables to analyse (the first `nbcomp` variables of `Y` will be analysed).
`sigma.car`	NULL or sum of squares of Y. If not NULL, compute the Generalised Sensitivity Indices (saved in the last column of the data.frame mSI/tSI/iSI outputs.
`analysis.args`	a list of arguments. The `formula` component is for ANOVA formula like `"A+B+c+A:B"` OR an integer giving the maximum interaction order (1 for main effects). If it contains `keep.outputs=TRUE`, the outputs associated with the analysis of each variable are returned (see section `Value`).

Value

A list containing:

`SI`	data.frame of sensitivity indices
`mSI`	data.frame of first-order sensitivity indices
`tSI`	data.frame of total sensitivity indices
`iSI`	data.frame of interaction sensitivity indices
`inertia`	vector of Inertia explained by the variables
`indic.fact`	0-1 matrix to indicate the factors associated with each factorial effect
`Hpredict`	prediction of outputs
`outputkept`	if `analysis.args$keep.outputs=TRUE`, list of the outputs returned by the sensitivity analysis performed on each variable
`call.info`	list with first element `analysis="anova"`

Examples

# Test case : the Winter Wheat Dynamic Models (WWDM)
#  input factors design
 data(biomasseX)
# output variables (precalculated to speed up the example)
 data(biomasseY)

 res <- analysis.anoasg(biomasseY, biomasseX, 
                        nbcomp = 2, sigma.car = NULL,
                        analysis.args = list(formula = 2,
                                             keep.outputs = FALSE))

# Test case : the Winter Wheat Dynamic Models (WWDM)
#  input factors design
 data(biomasseX)
# output variables (precalculated to speed up the example)
 data(biomasseY)

 res <- analysis.anoasg(biomasseY, biomasseX, 
                        nbcomp = 2, sigma.car = NULL,
                        analysis.args = list(formula = 2,
                                             keep.outputs = FALSE))

Runs a series of sensitivity analyses by a function from the sensitivity package

Description

The analysis.sensitivity function runs a series of sensitivity analyses on the columns of a data.frame, using a method implemented in the sensitivity package.

Usage

analysis.sensitivity(Y, plan, nbcomp = 2, sigma.car = NULL,
                     analysis.args = list(keep.outputs = FALSE))
analysis.sensitivity(Y, plan, nbcomp = 2, sigma.car = NULL,
                     analysis.args = list(keep.outputs = FALSE))

Arguments

`Y`	a data.frame of output variables or principal components.
`plan`	an object containing the design. It must be created by a function from the sensitivity package with argument `model=NULL`.
`nbcomp`	the number of `Y` variables to analyse (the first `nbcomp` variables of `Y` will be analysed).
`sigma.car`	NULL or sum of squares of Y. If not NULL, compute the Generalised Sensitivity Indices (saved in the last column of the data.frame mSI/tSI/iSI outputs.
`analysis.args`	a list of arguments. If it contains `keep.outputs=TRUE`, the outputs associated with the analysis of each variable are returned (see section `Value`).

Details

The argument plan must be an object created by a method implemented in the sensitivity package. Thus it belongs to a class such as morris or fast99. The name of the class is stored in the element call.info$fct of the output returned by analysis.sensitivity.

Value

A list containing:

`SI`	data.frame of sensitivity indices or other importance measures returned by the function from the sensitivity package used. Sometimes empty but kept for compatibility reasons.
`mSI`	data.frame of first-order sensitivity indices
`tSI`	data.frame of total sensitivity indices
`iSI`	data.frame of interaction sensitivity indices
`inertia`	empty (kept for compatibility reasons)
`indic.fact`	0-1 matrix to indicate the factors associated with each factorial effect
`Hpredict`	empty (kept for compatibility reasons)
`outputkept`	if `analysis.args$keep.outputs=TRUE`, list of the outputs returned by the sensitivity analysis performed on each variable
`call.info`	list with first element `analysis="sensitivity"` and second element `fct` storing the class name of the argument `plan`

Examples

# Test case : the Winter Wheat Dynamic Models (WWDM)
library(sensitivity) # to use fast99
# input factors design
data(biomasseX)
# input climate variable
data(Climat)

# example of the sensitivity:fast99 function
# design
newplan <- fast99(model = NULL, factors = names(biomasseX), n = 100, 
            q = "qunif", q.arg = list(list(min = 0.9, max = 2.8),
                         list(min = 0.9, max = 0.99),
                         list(min = 0.6, max = 0.8),
                         list(min = 3, max = 12),
                         list(min = 0.0035, max = 0.01),
                         list(min = 0.0011, max = 0.0025),
                         list(min = 700, max = 1100)))
# simulations
wwdm.Y <- simulmodel(model=biomasse, plan=newplan$X, climdata=Climat)
# analysis
res <- analysis.sensitivity(data.frame(wwdm.Y), plan=newplan, nbcomp=4)
# Test case : the Winter Wheat Dynamic Models (WWDM)
library(sensitivity) # to use fast99
# input factors design
data(biomasseX)
# input climate variable
data(Climat)

# example of the sensitivity:fast99 function
# design
newplan <- fast99(model = NULL, factors = names(biomasseX), n = 100, 
            q = "qunif", q.arg = list(list(min = 0.9, max = 2.8),
                         list(min = 0.9, max = 0.99),
                         list(min = 0.6, max = 0.8),
                         list(min = 3, max = 12),
                         list(min = 0.0035, max = 0.01),
                         list(min = 0.0011, max = 0.0025),
                         list(min = 700, max = 1100)))
# simulations
wwdm.Y <- simulmodel(model=biomasse, plan=newplan$X, climdata=Climat)
# analysis
res <- analysis.sensitivity(data.frame(wwdm.Y), plan=newplan, nbcomp=4)

A function to decompose multivariate data by principal components analysis (PCA)

Description

The basis.ACP function decomposes a multivariate data set according to principal components analysis.

Usage

basis.ACP(simuls, basis.args = list())
basis.ACP(simuls, basis.args = list())

Arguments

`simuls`	a data.frame of size `N x T`, typically a set of `N` simulation outputs of length `T`.
`basis.args`	an empty list of arguments for the PCA decomposition.

Details

This function uses prcomp.

Value

`H`	a data.frame of size `N x T`, containing the coefficients of the PCA decomposition. It is equal to the `x` output of function `prcomp`.
`L`	a matrix of size `T x T`. It contains the eigenvectors of the PCA decomposition.
`call.info`	list with the element `reduction="pca"`

Examples

data(biomasseY)
res <- basis.ACP(biomasseY)

data(biomasseY)
res <- basis.ACP(biomasseY)

A function to decompose multivariate data on a B-spline basis

Description

The basis.bsplines function decomposes a multivariate data set on a B-spline basis defined by its knots and mdegree parameters.

Usage

basis.bsplines(simuls, basis.args = list(knots = 5, mdegree = 3))
basis.bsplines(simuls, basis.args = list(knots = 5, mdegree = 3))

Arguments

`simuls`	a data.frame of size `N x T`, typically a set of `N` simulation outputs of length `T`.
`basis.args`	a list of arguments for the B-spline decomposition. The `knots` argument is the number of knots or the vector of knot positions. The `mdegree` argument is the polynomial degree. For the optional `x.coord` argument, see the Details section.

Details

The optional x.coord element of the list in basis.args can be used to specify the support of the B-spline decomposition, if different from 1:T. It must be a vector of length T.

Value

`H`	a data.frame of size `N x d`, where `d` is the dimension of the B-spline decomposition. It contains the coefficients of the decomposition for each row of the `simuls` data.frame.
`L`	a matrix of size `T x d`. It contains the vectors of the B-spline basis.
`call.info`	list with the element `reduction="b-splines"`

Examples

data(biomasseY)

res <- basis.bsplines(biomasseY,basis.args=list(knots=7,mdegree=3))

data(biomasseY)

res <- basis.bsplines(biomasseY,basis.args=list(knots=7,mdegree=3))

A function to decompose multivariate data on a user-defined basis

Description

The basis.mine function decomposes a multivariate data set on a user-defined basis.

Usage

basis.mine(simuls, basis.args = list(
        baseL=1*outer(sort(0:(ncol(simuls)-1)%%5),0:4,"==") ) )
basis.mine(simuls, basis.args = list(
        baseL=1*outer(sort(0:(ncol(simuls)-1)%%5),0:4,"==") ) )

Arguments

`simuls`	a data.frame of size `N x T`, typically a set of `N` simulation outputs of length `T`.
`basis.args`	a list of arguments for the polynomial decomposition. The `baseL` argument is a matrix of size `T x d` containing the coordinates of the `d` basis vectors.

Details

The default basis.args argument generates a projection on a moving-average basis. But if in the multisensi function this basis.args argument is not given for reduction=basis.mine, the execution will be stopped.

Value

`H`	a data.frame of size `N x d`, where `d` is the number of basis vectors. It contains the coefficients of the decomposition for each row of the `simuls` data.frame.
`L`	a matrix of size `T x d`. It contains the vectors of the user-defined basis.
`call.info`	list with the element `reduction="matrix"`

Examples

data(biomasseY)
M <- 1*outer(sort(0:(ncol(biomasseY)-1)%%5),0:4,"==")
norm.M <- sqrt(colSums(M^2)) 
for (i in 1:ncol(M)){
    M[,i]=M[,i]/norm.M[i]
}

res <- basis.mine(biomasseY, basis.args=list(baseL=M))

data(biomasseY)
M <- 1*outer(sort(0:(ncol(biomasseY)-1)%%5),0:4,"==")
norm.M <- sqrt(colSums(M^2)) 
for (i in 1:ncol(M)){
    M[,i]=M[,i]/norm.M[i]
}

res <- basis.mine(biomasseY, basis.args=list(baseL=M))

A function to decompose multivariate data on an orthogonal B-spline basis (O-spline)

Description

The basis.osplines function decomposes a multivariate data set on an orthogonalised B-spline (or O-spline) basis defined by its knots and mdegree parameters.

Usage

basis.osplines(simuls, basis.args = list(knots = 5, mdegree = 3))
basis.osplines(simuls, basis.args = list(knots = 5, mdegree = 3))

Arguments

`simuls`	a data.frame of size `N x T`, typically a set of `N` simulation outputs of length `T`.
`basis.args`	a list of arguments for the O-spline decomposition. The `knots` argument is the number of knots or the vector of knot positions. The `mdegree` argument is the polynomial degree. For the optional `x.coord` argument, see the Details section.

Details

The optional x.coord element of the list in basis.args can be used to specify the support of the O-spline decomposition, if different from 1:T. It must be a vector of length T.

Value

`H`	a data.frame of size `N x d`, where `d` is the dimension of the O-spline decomposition. It contains the coefficients of the decomposition for each row of the `simuls` data.frame.
`L`	a matrix of size `T x d`. It contains the vectors of the O-spline basis.
`call.info`	list with the element `reduction="o-splines"`

Examples

data(biomasseY)

res <- basis.osplines(biomasseY,basis.args=list(knots=7,mdegree=3))

data(biomasseY)

res <- basis.osplines(biomasseY,basis.args=list(knots=7,mdegree=3))

A function to decompose multivariate data on a polynomial basis

Description

The basis.poly function decomposes a multivariate data set on a polynomial basis.

Usage

basis.poly(simuls, basis.args = list(degree = 3))
basis.poly(simuls, basis.args = list(degree = 3))

Arguments

`simuls`	a data.frame of size `N x T`, typically a set of `N` simulation outputs of length `T`.
`basis.args`	a list of arguments for the polynomial decomposition. The `degree` argument is the maximum degree of the polynomial basis. For the optional `x.coord` argument, see the Details section.

Details

This function uses poly. The optional x.coord element of the list in basis.args can be used to specify the support of the polynomial decomposition, if different from 1:T. It must be a vector of length T.

Value

`H`	a data.frame of size `N x (d+1)`, where `d` is the degree of the polynomial decomposition. It contains the coefficients of the decomposition for each row of the `simuls` data.frame.
`L`	a matrix of size `T x (d+1)`. It contains the vectors of the polynomial basis.
`call.info`	list with the element `reduction="polynomial"`

Examples

data(biomasseY)

res <- basis.poly(biomasseY,basis.args=list(degree=3))

data(biomasseY)

res <- basis.poly(biomasseY,basis.args=list(degree=3))

The Winter Wheat Dynamic Model

Description

The Winter Wheat Dynamic Model, a toy model to illustrate the main multisensi methods

Usage

biomasse(input, climdata, annee = 3)
biomasse(input, climdata, annee = 3)

Arguments

`input`	vector of input values.
`annee`	year.
`climdata`	a meteorological data.frame specific to biomasse.

Details

The Winter Wheat Dry Matter model (WWDM) is a dynamic crop model running at a daily time step (Makowski et al, 2004). It has two state variables, the above-ground winter wheat dry matter $U(t)$ , in $g/m^2$ and the leaf area index LAI( $t$ ) with $t$ the day number from sowing ( $t=1$ ) to harvest ( $t=223$ ). In the multisensi package implementation, the biomasse function simulates the output for only one parameter set (the first row of input if it is a matrix or a data.frame).

Value

a vector of daily dry matter increase of the Winter Wheat biomass, over 223 days

References

Makowski, D., Jeuffroy, M.-H., Gu\'erif, M., 2004 Bayesian methods for updating crop model predictions, applications for predicting biomass and grain protein content. In: Bayesian Statistics and Quality Modelling in the Agro-Food Production Chain (van Boeakel et al. eds), pp. 57-68. Kluwer, Dordrecht

Monod, H., Naud, C., Makowski, D., 2006 Uncertainty and sensitivity analysis for crop models. In: Working with Dynamic Crop Models (Wallach D., Makowski D. and Jones J. eds), pp. 55-100. Elsevier, Amsterdam

A factorial input design for the main example

Description

Factorial design (resolution V) data for the 7 WWDM model input factors

Usage

data(biomasseX)data(biomasseX)

Format

A data frame with 2187 observations on the following 7 variables.

Eb: First WWDM input factor name
Eimax: Second WWDM input factor name
K: Thirth WWDM input factor name
Lmax: Fourth WWDM input factor name
A: Fifth WWDM input factor name
B: Sixth WWDM input factor name
TI: Seventh WWDM input factor name

Examples

data(biomasseX)
## maybe str(biomasseX) ; plot(biomasseX) ...
data(biomasseX)
## maybe str(biomasseX) ; plot(biomasseX) ...

Output of the biomasse model for the plan provided in the package

Description

Simplified output of the biomasse model (one column per decade), especially generated for examples in the package help files

Usage

data(biomasseY)data(biomasseY)

Format

A data frame with 2187 rows and 22 output variables (one per decade).

Examples

data(biomasseY)
dim(biomasseY)
data(biomasseY)
dim(biomasseY)

function to evaluate B-spline basis functions

Description

The bspline function evaluates ith B-spline basis function of order m at the values in x, given knot locations in k

Usage

bspline(x = seq(0, 1, len = 101), k = knots, i = 1, m = 2)
bspline(x = seq(0, 1, len = 101), k = knots, i = 1, m = 2)

Arguments

`x`	vector or scalar, coordinate where to calculate the B-spline functions
`k`	vector of knot locations
`i`	integer; from 0 to length(knots)+1-m
`m`	integer, degree of the B-Splines

Details

B-splines are defined by recursion : $b_{i,0}(x) = 1$ if $k_j \leq x < k_{j+1}$ ; 0 else.

$b_{i,m}(x) = \frac{x-k_i}{k_{i+m}-k_i} b_{i,m-1}(x) + \frac{k_{i+m+1}-x}{k_{i+m+1}-k_{i+1}} b_{i+1,m-1}(x)$

Value

values in x of the ith B-spline basis function of order m

Note

This is essentially an internal function for the multisensi package

References

Wood Simon, 2006. Generalized Additive Models: An Introduction with R Chapman and Hall/CRC.

Climate data

Description

Climate data for the WWDM model (needed by the biomasse function)

Usage

data(Climat)data(Climat)

Format

A data frame with 3126 observations on the following 4 variables.

ANNEE: a factor with levels 1 to 14, indicating 14 different years
RG: daily radiation variable
Tmin: daily maximum temperature
Tmax: daily minimum temperature

Source

Dynamic Sensitivity Indices: DSI

Description

dynsi implements the Dynamic Sensitivity Indices. This method allows to compute classical Sensitivity Indices on each output variable of a dynamic or multivariate model by using the ANOVA decomposition

Usage

dynsi(formula, model, factors, cumul = FALSE, simulonly=FALSE,
      nb.outp = NULL, Name.File=NULL, ...)
dynsi(formula, model, factors, cumul = FALSE, simulonly=FALSE,
      nb.outp = NULL, Name.File=NULL, ...)

Arguments

`formula`	ANOVA formula like `"A+B+c+A:B"` OR an integer equal to the maximum interaction order in the sensitivity model.
`model`	output data.frame OR the name of the R-function which calculates the model output. The only argument of this function must be a vector containing the input factors values.
`factors`	input data.frame (the design) if model is a data.frame OR a list of factors levels such as `factor.example <- list(A=c(0,1),B=c(0,1,4))`.
`cumul`	logical value. If TRUE the sensitivity analysis will be done on the cumalative outputs.
`simulonly`	logical value. If TRUE the program stops after calculating the design and the model outputs.
`nb.outp`	The first nb.outp number of model outputs to be considered. If NULL all the outputs are considered.
`Name.File`	optional name of a R script file containing the R-function that calculates the simulation model. e.g `"exc.ssc"`.
`...`	possible fixed parameters of the model function.

Details

If factors is a list of factors, the dynsi function generates a complete factorial design. If it is a data.frame, dynsi expects that each column is associated with an input factor.

Value

dynsi returns a list of class "dynsi" containing the following components:

`X`	a data.frame containing the experimental design (input samples)
`Y`	a data.frame containing the output (response)
`SI`	a data.frame containing the Sensitivity Indices (SI) on each output variable of the model and the Generalised SI (GSI)
`mSI`	a data.frame of first order SI on each output variable and first order GSI
`tSI`	a data.frame containing the total SI on each output variable and the total GSI
`iSI`	a data.frame of interaction SI on each output variable and interaction GSI
`Att`	0-1 matrix of association between input factors and factorial terms in the anovas
`call.info`	a list containing informations on the process (reduction=NULL, analysis, fct, call)
`inputdesign`	either the input data.frame or the sensitivity object used
`outputs`	a list of results on each output variable

...

Note

This function can now be replaced by a call to the multisensi function. It is kept for compatibility with Version 1 of the multisensi package.

References

M. Lamboni, D. Makowski and H. Monod, 2009. Multivariate global sensitivity analysis for dynamic crop models. Field Crops Research, 113, 312-320.

A. Saltelli, K. Chan and E. M. Scott eds, 2000. Sensitivity Analysis. Wiley, New York.

Examples

# Test case : the Winter Wheat Dynamic Models (WWDM)
# input factors design,
 data(biomasseX)
# input Climate variables
 data(Climat)
# output variables (precalculated to speed up the example)
 data(biomasseY)
#
 DYNSI <- dynsi(2, biomasseY, biomasseX)
 summary(DYNSI)
 print(DYNSI)
 plot(DYNSI, color=heat.colors)
 #graph.bar(DYNSI,col=1, beside=F)  # sensitivity bar plot
                                    # for the first output (col=1)
 #graph.bar(DYNSI,col=2, xmax=1)    #
  # Test case : the Winter Wheat Dynamic Models (WWDM)
# input factors design,
 data(biomasseX)
# input Climate variables
 data(Climat)
# output variables (precalculated to speed up the example)
 data(biomasseY)
#
 DYNSI <- dynsi(2, biomasseY, biomasseX)
 summary(DYNSI)
 print(DYNSI)
 plot(DYNSI, color=heat.colors)
 #graph.bar(DYNSI,col=1, beside=F)  # sensitivity bar plot
                                    # for the first output (col=1)
 #graph.bar(DYNSI,col=2, xmax=1)    #

Sensitivity index bar plot

Description

A function that plots sensitivity indices by a bar graph

Usage

graph.bar(x, col = 1, nb.plot = 15, xmax = NULL,
          beside = TRUE, xlab = NULL, ...)
graph.bar(x, col = 1, nb.plot = 15, xmax = NULL,
          beside = TRUE, xlab = NULL, ...)

Arguments

`x`	an object of class gsi or dynsi
`col`	the column number of GSI to represent in the bar graph
`nb.plot`	number of input factors to be considered
`xmax`	a user-defined maximal $x$ value ( $x\leq 1$ ) in all the bar graphs that show sensitivity indices; or NULL if the user wants to keep default values
`beside`	if TRUE, the main and total sensitivity indices are represented by two bars; if FALSE, they are represented by the same bar
`xlab`	a label for the x axis
`...`	graphical parameters

Principal Components graph for gsi objects

Description

A function that plots the Principal Components (PCs) and the sensitivity indices on each PC

Usage

graph.pc(x, nb.plot = 15, nb.comp = NULL, xmax = NULL,
         beside = TRUE, cor.plot=FALSE, xtick=TRUE, type="l", ...)
graph.pc(x, nb.plot = 15, nb.comp = NULL, xmax = NULL,
         beside = TRUE, cor.plot=FALSE, xtick=TRUE, type="l", ...)

Arguments

`x`	gsi object.
`nb.plot`	number of input factors to be considered.
`nb.comp`	number of PCs.
`xmax`	a user-defined maximal $x$ value ( $x\leq 1$ ) in all the bar graphs that show sensitivity indices; or NULL if the user wants to keep default values.
`beside`	if TRUE, the main and total sensitivity indices are represented by two bars; if FALSE, they are represented by the same bar.
`cor.plot`	if TRUE a correlation graph is made to represent the PCs ; if FALSE (default) a functionnal boxplot of the PCs is plotted.
`xtick`	if TRUE, put column names of outputs (Y) as ticks for the x axis.
`type`	what type of plot should be drawn for correlation graph ("l" for lines).
`...`	graphical parameters.

Group factor GSI, obsolete function

Description

An obsolete function that computed the GSI of a group factor as one factor

Usage

grpe.gsi(GSI, fact.interet)
grpe.gsi(GSI, fact.interet)

Arguments

`GSI`	a gsi or dynsi object
`fact.interet`	input factor to be grouped

Note

This is essentially an internal function for the multisensi package

Generalised Sensitivity Indices: GSI

Description

The gsi function implements the calculation of Generalised Sensitivity Indices. This method allows to compute a synthetic Sensitivity Index for the dynamic or multivariate models by using factorial designs and the MANOVA decomposition of inertia. It computes also the Sensitivity Indices on principal components

Usage

gsi(formula, model, factors, inertia = 0.95, normalized = TRUE,
    cumul = FALSE, simulonly = FALSE, Name.File = NULL, ...)gsi(formula, model, factors, inertia = 0.95, normalized = TRUE,
    cumul = FALSE, simulonly = FALSE, Name.File = NULL, ...)

Arguments

`formula`	ANOVA formula like `"A+B+C+A:B"` OR an integer equal to the maximum interaction order in the sensitivity model
`model`	output data.frame OR the name of the R-function which calculates the model output. The only argument of this function must be a vector containing the input factors values
`factors`	input data.frame (the design) if model is a data.frame OR a list of factors levels such as : `factor.example <- list(A=c(0,1),B=c(0,1,4))`
`inertia`	cumulated proportion of inertia (a scalar $< 1$ ) to be explained by the selected Principal components OR number of PCs to be used (e.g 3)
`normalized`	logical value. TRUE (default) computes a normalized Principal Component analysis.
`cumul`	logical value. If TRUE the PCA will be done on the cumulative outputs
`simulonly`	logical value. If TRUE the program stops after calculating the design and the model outputs
`Name.File`	optional name of a R script file containing the R-function that calculates the simulation model. e.g `"exc.ssc"`
`...`	possible fixed parameters of the model function

Details

If factors is a list of factors, the gsi function generates a complete factorial design. If it is a data.frame, gsi expects that each column is associated with an input factor.

Value

gsi returns a list of class "gsi", containing all the input arguments detailed before, plus the following components:

`X`	a data.frame containing the experimental design (input samples)
`Y`	a data.frame containing the output matrix (response)
`H`	a data.frame containing the principal components
`L`	a data.frame whose columns contain the basis eigenvectors (the variable loadings)
`lambda`	the variances of the principal components
`inertia`	vector of inertia percentages per PCs and global criterion
`cor`	a data.frame of correlation between PCs and outputs
`SI`	a data.frame containing the Sensitivity Indices (SI) on PCs and the Generalised SI (GSI)
`mSI`	a data.frame of first order SI on PCs and first order GSI
`tSI`	a data.frame containing the total SI on PCs and the total GSI
`iSI`	a data.frame of interaction SI on PCs and interaction GSI
`pred`	a data.frame containing the output predicted by the metamodel arising from the PCA and anova decompositions
`residuals`	a data.frame containing the residuals between actual and predicted outputs
`Rsquare`	vector of dynamic coefficient of determination
`Att`	0-1 matrix of association between input factors and factorial terms in the anovas
`scale`	logical value, see the arguments
`normalized`	logical value, see the arguments
`cumul`	logical value, see the arguments
`call.info`	a list containing informations on the process (reduction, analysis, fct, call)
`inputdesign`	either the input data.frame or the sensitivity object used
`outputs`	a list of results on each output variable

...

Note

This function can now be replaced by a call to the multisensi function. It is kept for compatibility with Version 1 of the multisensi package.

References

M. Lamboni, D. Makowski and H. Monod, 2009. Multivariate global sensitivity analysis for dynamic crop models. Field Crops Research, volume 113. pp. 312-320

M. Lamboni, D. Makowski and H. Monod, 2009. Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models. Submitted to Reliability Engineering and System Safety.

Examples

# Test case : the Winter Wheat Dynamic Models (WWDM)
#  input factors design
 data(biomasseX)
# input climate variable
 data(Climat)
# output variables (precalculated to speed up the example)
 data(biomasseY)
#
 GSI <- gsi(2, biomasseY, biomasseX, inertia=3, normalized=TRUE, cumul=FALSE,
            climdata=Climat)
 summary(GSI)
 print(GSI)
 plot(x=GSI, beside=FALSE)
 #plot(GSI, nb.plot=4)         # the 'nb.plot' most influent factors
                               # are represented in the plots
 #plot(GSI,nb.comp=2, xmax=1)  # nb.comp = number of principal components
 #plot(GSI,nb.comp=3, graph=1) # graph=1 for first figure; 2 for 2nd one
                               # and 3 for 3rd one; or 1:3 etc.
 #graph.bar(GSI,col=1, beside=F)  # sensitivity bar plot on the first PC
 #graph.bar(GSI,col=2, xmax=1)    #
  # Test case : the Winter Wheat Dynamic Models (WWDM)
#  input factors design
 data(biomasseX)
# input climate variable
 data(Climat)
# output variables (precalculated to speed up the example)
 data(biomasseY)
#
 GSI <- gsi(2, biomasseY, biomasseX, inertia=3, normalized=TRUE, cumul=FALSE,
            climdata=Climat)
 summary(GSI)
 print(GSI)
 plot(x=GSI, beside=FALSE)
 #plot(GSI, nb.plot=4)         # the 'nb.plot' most influent factors
                               # are represented in the plots
 #plot(GSI,nb.comp=2, xmax=1)  # nb.comp = number of principal components
 #plot(GSI,nb.comp=3, graph=1) # graph=1 for first figure; 2 for 2nd one
                               # and 3 for 3rd one; or 1:3 etc.
 #graph.bar(GSI,col=1, beside=F)  # sensitivity bar plot on the first PC
 #graph.bar(GSI,col=2, xmax=1)    #

A function with multiple options to perform multivariate sensitivity analysis

Description

The multisensi function can conduct the different steps of a multivariate sensitivity analysis (design, simulation, dimension reduction, analysis, plots). It includes different options for each of these steps.

Usage

multisensi(design = expand.grid, model, reduction = basis.ACP, 
           dimension = 0.95, center = TRUE, scale = TRUE, 
           analysis = analysis.anoasg, cumul = FALSE, 
           simulonly = FALSE, Name.File = NULL, 
           design.args = list(), basis.args = list(), 
           analysis.args = list(), ...)
multisensi(design = expand.grid, model, reduction = basis.ACP, 
           dimension = 0.95, center = TRUE, scale = TRUE, 
           analysis = analysis.anoasg, cumul = FALSE, 
           simulonly = FALSE, Name.File = NULL, 
           design.args = list(), basis.args = list(), 
           analysis.args = list(), ...)

Arguments

`design`	EITHER a function such as expand.grid to generate the design OR a data.frame of size `N x P` containing `N` combinations of levels of the `P` input factors OR a function from the sensitivity package such as fast99 OR an object generated by a function from the sensitivity package. The first and third cases require additional information to be given in the `design.args` argument.
`model`	EITHER a function to run the model simulations OR a data.frame of size `N x T` containing `N` realizations of `T` output variables.
`reduction`	EITHER a function to decompose the multivariate output on a basis of smaller dimension OR `NULL`. The first case requires additional information to be given in the `basis.args` argument. In the second case, sensitivity analyses are performed on the raw output variables.
`dimension`	EITHER the number of variables to analyse, specified by an integer or by the minimal proportion of inertia (a scalar < 1) to keep in the output decomposition OR a vector specifying a subset of columns in the output data.frame OR NULL if all variables must be analysed.
`center`	logical value. If TRUE (default value) the output variables are centred.
`scale`	logical value. If TRUE (default value) the output variables are normalized before applying the reduction function.
`analysis`	a function to run the sensitivity analysis. Additional information can be given in the `analysis.args` argument.
`cumul`	logical value. If TRUE the output variables are replaced by their cumulative sums.
`simulonly`	logical value. If TRUE the program stops after the model simulations.
`Name.File`	Name of file containing the R-function model.
`design.args`	a list of arguments for the function possibly given in the `design` argument.
`basis.args`	a list of arguments for the function given in the `reduction` argument. See the function help for more precision.
`analysis.args`	a list of arguments for the function possibly given in the `analysis` argument. See the function help for more precision.
`...`	optional parameters of the function possibly given in the `model` argument.

Value

an object of class dynsi if reduction=NULL, otherwise an object of class gsi. See the functions dynsi and gsi for more information.

Examples

## Test case : the Winter Wheat Dynamic Models (WWDM)
#  input factors design
 data(biomasseX)
# input climate variable
 data(Climat)
# output variables (precalculated to speed up the example)
 data(biomasseY)

# to do dynsi process
# argument reduction=NULL
resD <- multisensi(design=biomasseX, model=biomasseY, reduction=NULL,
                 dimension=NULL, analysis=analysis.anoasg,
                 analysis.args=list(formula=2,keep.outputs = FALSE))
summary(resD)


# to do gsi process
#------------
# with dimension reduction by PCA
# argument reduction=basis.ACP
resG1 <- multisensi(design=biomasseX, model=biomasseY, reduction=basis.ACP,
                   dimension=0.95, analysis=analysis.anoasg,
                   analysis.args=list(formula=2,keep.outputs = FALSE))
summary(resG1)

plot(x=resG1, beside=FALSE)

#------------
# with dimension reduction by o-splines basis
# arguments reduction=basis.osplines
# and basis.args=list(knots= ... , mdegree= ... )

resG2 <- multisensi(design=biomasseX, model=biomasseY, reduction=basis.osplines,
                   dimension=NULL, center=FALSE, scale=FALSE,
                   basis.args=list(knots=11, mdegree=3), analysis=analysis.anoasg, 
                   analysis.args=list(formula=2,keep.outputs = FALSE))
summary(resG2)

#------------
library(sensitivity) # to use fast99

# with dimension reduction by o-splines basis
# and sensitivity analysis with sensitivity:fast99
resG3 <- multisensi(design=fast99, model=biomasse,
                   analysis=analysis.sensitivity, 
                   design.args=list(factors = names(biomasseX), n = 100,
                   q = "qunif", q.arg = list(list(min = 0.9, max = 2.8),
                   list(min = 0.9, max = 0.99), list(min = 0.6, max = 0.8), 
                   list(min = 3, max = 12), list(min = 0.0035, max = 0.01),
                   list(min = 0.0011, max = 0.0025),
                   list(min = 700, max = 1100))), climdata=Climat,
                   reduction=basis.osplines,
                   basis.args=list(knots=7, mdegree=3),
                   center=FALSE,scale=FALSE,dimension=NULL)
summary(resG3)

## Test case : the Winter Wheat Dynamic Models (WWDM)
#  input factors design
 data(biomasseX)
# input climate variable
 data(Climat)
# output variables (precalculated to speed up the example)
 data(biomasseY)

# to do dynsi process
# argument reduction=NULL
resD <- multisensi(design=biomasseX, model=biomasseY, reduction=NULL,
                 dimension=NULL, analysis=analysis.anoasg,
                 analysis.args=list(formula=2,keep.outputs = FALSE))
summary(resD)


# to do gsi process
#------------
# with dimension reduction by PCA
# argument reduction=basis.ACP
resG1 <- multisensi(design=biomasseX, model=biomasseY, reduction=basis.ACP,
                   dimension=0.95, analysis=analysis.anoasg,
                   analysis.args=list(formula=2,keep.outputs = FALSE))
summary(resG1)

plot(x=resG1, beside=FALSE)

#------------
# with dimension reduction by o-splines basis
# arguments reduction=basis.osplines
# and basis.args=list(knots= ... , mdegree= ... )

resG2 <- multisensi(design=biomasseX, model=biomasseY, reduction=basis.osplines,
                   dimension=NULL, center=FALSE, scale=FALSE,
                   basis.args=list(knots=11, mdegree=3), analysis=analysis.anoasg, 
                   analysis.args=list(formula=2,keep.outputs = FALSE))
summary(resG2)

#------------
library(sensitivity) # to use fast99

# with dimension reduction by o-splines basis
# and sensitivity analysis with sensitivity:fast99
resG3 <- multisensi(design=fast99, model=biomasse,
                   analysis=analysis.sensitivity, 
                   design.args=list(factors = names(biomasseX), n = 100,
                   q = "qunif", q.arg = list(list(min = 0.9, max = 2.8),
                   list(min = 0.9, max = 0.99), list(min = 0.6, max = 0.8), 
                   list(min = 3, max = 12), list(min = 0.0035, max = 0.01),
                   list(min = 0.0011, max = 0.0025),
                   list(min = 700, max = 1100))), climdata=Climat,
                   reduction=basis.osplines,
                   basis.args=list(knots=7, mdegree=3),
                   center=FALSE,scale=FALSE,dimension=NULL)
summary(resG3)

A function to decompose the output data set and reduce its dimension

Description

The function multivar applies a multivariate method to decompose the output variables on a given basis.

Usage

multivar(simuls, dimension = NULL, reduction, centered = TRUE, 
         scale = TRUE, basis.args = list())
multivar(simuls, dimension = NULL, reduction, centered = TRUE, 
         scale = TRUE, basis.args = list())

Arguments

`simuls`	a data.frame of size `N x T`, typically a set of `N` simulation outputs of length `T`
`dimension`	the number of variables to analyse, specified by an integer (for example 3) or by the minimal proportion of inertia (for example 0.95) to keep in the output decomposition
`reduction`	a function to decompose the multivariate output on a basis of smaller dimension
`centered`	logical value. If TRUE the output variables are centred.
`scale`	logical value. If TRUE the output variables are normalized.
`basis.args`	a list of arguments for the function given in the `reduction` argument. See the function help for more precision.

Value

A list containing:

`H`	a data.frame of size `N x d`, where `d` is the number of basis vectors. It contains the coefficients of the decomposition for each row of the `simuls` data.frame.
`L`	a matrix of size `T x d`. It contains the vectors of the user-defined basis.
`sdev`	standard deviations of the columns of `H`
`nbcomp`	number of components kept from the decomposition
`SStot`	total sums of squares of the simulations (after application of `centered` and `scale`)
`centering`	either 0 or the column averages of `simuls`
`scaling`	either 1 or `sdY`, depending on the `scale` argument
`sdY`	standard deviations of the columns of `simuls`
`cor`	correlation matrix (L*sdev), of size `T x nbcomp`
`scale`	kept in case the option scale has been changed in the function
`importance`	cumulated percentage of SS_H (sdev^2) with respect to SStot
`call.info`	list with the element `reduction` storing the name of the argument `reduction`

Examples

data(biomasseY)

res <- multivar(biomasseY, dimension=0.95, reduction=basis.ACP)

data(biomasseY)

res <- multivar(biomasseY, dimension=0.95, reduction=basis.ACP)

Complete factorial design in lexical order

Description

Function that generates a complete factorial design in lexical order

Usage

planfact(nb.niv, make.factor = TRUE)
planfact(nb.niv, make.factor = TRUE)

Arguments

`nb.niv`	vector containing the number of each input levels
`make.factor`	logical value. If TRUE the columns of the output are of class factor

Value

plan

data frame of the complete factorial design

Note

This is essentially an internal function for the multisensi package

Complete factorial design

Description

Computation of a complete factorial design for model input factors.

Usage

planfact.as(input)
planfact.as(input)

Arguments

input

list of factor levels

Value

comp2

complete factorial design of model input

Note

This is essentially an internal function for the multisensi package. It is almost equivalent to the function expand.grid.

Plot method for dynamic sensitivity results

Description

Plot method for dynamic sensitivity results of class dynsi

Usage

## S3 method for class 'dynsi'
plot(x, normalized=FALSE, text.tuning = NULL, shade=FALSE,
                     color=NULL, xtick=TRUE, total.plot=FALSE, gsi.plot=TRUE, ...)
## S3 method for class 'dynsi'
plot(x, normalized=FALSE, text.tuning = NULL, shade=FALSE,
                     color=NULL, xtick=TRUE, total.plot=FALSE, gsi.plot=TRUE, ...)

Arguments

`x`	a dynsi object.
`normalized`	logical value, FALSE => SI plotted within var(Y).
`text.tuning`	NULL or a small integer to improve the position of input factor labels.
`shade`	if TRUE, put different shadings to enhance the different factorial effects in the plot (long).
`color`	a palette of colors to enhance the different factorial effects in the plot (for example `color=heat.colors`).
`xtick`	if TRUE, put column names of outputs (Y) as ticks for the x axis.
`total.plot`	logical value, TRUE => a new plot is produced with the total SI.
`gsi.plot`	logical value, TRUE => a new plot is produced for the Generalised Sensitivity Indice.
`...`	graphical parameters.

Details

For labels that would be partly positioned outside the plot frame, the argument "text.tuning" may allow to get a better positioning. If it is equal to $n$ , say, these labels are moved by $n$ positions inside the frame, where 1 position corresponds to 1 output variable on the x-axis.

Plot method for generalised sensitivity analysis

Description

Plot method for generalised sensitivity analysis of class gsi

Usage

## S3 method for class 'gsi'
plot(x, nb.plot = 10, nb.comp = 3, graph = 1:3, xmax=NULL,
                   beside=TRUE, cor.plot=FALSE, xtick=TRUE, type="l",...)
## S3 method for class 'gsi'
plot(x, nb.plot = 10, nb.comp = 3, graph = 1:3, xmax=NULL,
                   beside=TRUE, cor.plot=FALSE, xtick=TRUE, type="l",...)

Arguments

`x`	a gsi object.
`nb.plot`	number of input factors to be considered.
`nb.comp`	number of Principal Components to be plotted.
`graph`	figures number: 1 or 2 or 3. 1 is for plotting the PCs and their sensitivity indices, 2 is for plotting the Generalised Sensitivity Indice, 3 is for plotting the Rsquare.
`xmax`	a user-defined maximal $x$ value ( $x\leq 1$ ) in all the bar graphs that show sensitivity indices; or NULL if the user wants to keep default values.
`beside`	if TRUE, the main and total sensitivity indices are represented by two bars; if FALSE, they are represented by the same bar.
`cor.plot`	if TRUE a correlation graph is made to represent the PCs ; if FALSE (default) a functionnal boxplot of the PCs is plotted.
`xtick`	if TRUE, put column names of outputs (Y) as ticks for the x axis.
`type`	what type of plot should be drawn for correlation graph ("l" for lines).
`...`	graphical parameters.

A function to predict multivariate output

Description

The function predict.gsi generates predicted multivariate output for user-specified combinations of levels of the input factors.

Usage

## S3 method for class 'gsi'
predict(object, newdata, ...)
## S3 method for class 'gsi'
predict(object, newdata, ...)

Arguments

`object`	Object of class gsi.
`newdata`	An optional data frame in which to look for variables with which to predict. If omitted, the fitted values are used. need to be same factors and levels as for obtained the gsi object.
`...`	others parameters

Details

Only available if the gsi object was obtained with analysis.anoasg and analysis.args$keep.outputs=TRUE.

Value

a data.frame of predicted values for newdata

Examples

  data(biomasseX)
  data(biomasseY)
  x=multisensi(design=biomasseX,model=biomasseY,basis=basis.ACP,
               analysis=analysis.anoasg,
               analysis.args=list(formula=2, keep.outputs=TRUE))
  newdata=as.data.frame(apply(biomasseX,2,unique))
  predict(x,newdata)
data(biomasseX)
  data(biomasseY)
  x=multisensi(design=biomasseX,model=biomasseY,basis=basis.ACP,
               analysis=analysis.anoasg,
               analysis.args=list(formula=2, keep.outputs=TRUE))
  newdata=as.data.frame(apply(biomasseX,2,unique))
  predict(x,newdata)

print DYNSI

Description

A function to print DYNSI results

Usage

## S3 method for class 'dynsi'
print(x, ...)
## S3 method for class 'dynsi'
print(x, ...)

Arguments

`x`	a dynsi object
`...`	print parameters

print GSI

Description

function to print GSI results

Usage

## S3 method for class 'gsi'
print(x, ...)
## S3 method for class 'gsi'
print(x, ...)

Arguments

`x`	a gsi object
`...`	print parameters

quality of any approximation

Description

Function that computes the sensitivity quality after making some assumptions about the number of PCs and the number of interactions

Usage

quality(echsimul, echsimul.app)
quality(echsimul, echsimul.app)

Arguments

`echsimul`	model outputs
`echsimul.app`	Predicted model output

Value

A list with the following components:

moy.biais: mean of the residuals
residuals: biais
coef.det: R-square

Note

This is essentially an internal function for the multisensi package

normalized B-splines basis functions

Description

The sesBsplinesNORM evaluates B-Splines basis functions at some points.

Usage

sesBsplinesNORM(x = seq(0, 1, len = 101), knots = 5, m = 2)
sesBsplinesNORM(x = seq(0, 1, len = 101), knots = 5, m = 2)

Arguments

`x`	vector, coordinates where to calculate the B-spline functions
`knots`	number of knots or vector of knots locations
`m`	integer, degree of the B-Splines

Value

`x`	as input
`bsplines`	matrix, values in x of all B-spline basis functions of order m
`knots`	vector of knots locations
`projecteur`	inverse matrix of `bsplines`

Note

This is essentially an internal function for the multisensi package

orthogonalized B-splines basis functions

Description

The sesBsplinesORTHONORM evaluates O-Splines basis functions at some points.

Usage

sesBsplinesORTHONORM(x = seq(0, 1, len = 101), knots = 5, m = 2)
sesBsplinesORTHONORM(x = seq(0, 1, len = 101), knots = 5, m = 2)

Arguments

`x`	vector, coordinates where to calculate the B-spline functions
`knots`	number of knots or vector of knots locations
`m`	integer, degree of the B-Splines

Value

`x`	as input
`osplines`	matrix, values in x of all O-spline basis functions of order m
`knots`	vector of knots locations
`projecteur`	inverse matrix of `osplines`

Note

This is essentially an internal function for the multisensi package

Model simulation

Description

Function that simulates the model outputs

Usage

simulmodel(model, plan, nomFic = NULL, verbose = FALSE, ...)
simulmodel(model, plan, nomFic = NULL, verbose = FALSE, ...)

Arguments

`model`	name of R-function
`plan`	data frame of input design
`nomFic`	name of file that contains the model function
`verbose`	verbose
`...`	...

possible fixed parameters of the R-function

Details

The model function must be a R-function. Models defined as functions will be called once with an expression of the form y <- f(X) where X is a vector containing a combination of levels of the input factors, and y is the output vector of length $q$ , where $q$ is the number of output variables

Value

data frame of model outputs

Note

This is essentially an internal function for the multisensi package

dynsi summary

Description

Function to summarize the dynamic sensitivity results

Usage

## S3 method for class 'dynsi'
summary(object, ...)
## S3 method for class 'dynsi'
summary(object, ...)

Arguments

`object`	a dynsi object
`...`	summary parameters

summary of GSI results

Description

function to summarize the GSI results

Usage

## S3 method for class 'gsi'
summary(object, ...)## S3 method for class 'gsi'
summary(object, ...)

Arguments

`object`	a GSI object
`...`	summary parameters

Prediction based on PCA and anovas (NOT ONLY)

Description

A function that predicts the model output after PCA and aov analyses

Usage

yapprox(multivar.obj, nbcomp = 2, aov.obj)
yapprox(multivar.obj, nbcomp = 2, aov.obj)

Arguments

`multivar.obj`	output of the multivar function
`nbcomp`	number of columns
`aov.obj`	aov object

Value

model output predictions

Note

This is essentially an internal function for the multisensi package

Package 'multisensi'

Help Index

Multivariate sensitivity Analysis

Description

Details

References

Runs a series of analyses of variance

Description

Usage

Arguments

Value

See Also

Examples

Runs a series of sensitivity analyses by a function from the sensitivity package

Description

Usage

Arguments

Details

Value

Examples

A function to decompose multivariate data by principal components analysis (PCA)

Description

Usage

Arguments

Details

Value

See Also

Examples

A function to decompose multivariate data on a B-spline basis

Description

Usage

Arguments

Details

Value

See Also

Examples

A function to decompose multivariate data on a user-defined basis

Description

Usage

Arguments

Details

Value

Examples

A function to decompose multivariate data on an orthogonal B-spline basis (O-spline)

Description

Usage

Arguments

Details

Value

See Also

Examples

A function to decompose multivariate data on a polynomial basis

Description

Usage

Arguments

Details

Value

See Also

Examples

The Winter Wheat Dynamic Model

Description

Usage

Arguments

Details

Value

References

A factorial input design for the main example

Description

Usage

Format

See Also

Examples

Output of the biomasse model for the plan provided in the package

Description

Usage

Format

See Also

Examples

function to evaluate B-spline basis functions

Description