Package 'funcreg'

Functional Data Fit

Description

These are the main functions to fit a functional data object to a dataset. The model can be linear or nonlinear. The smoothing parameter and the number of basis is assumed to be known.

Usage

coefEst(y, t, lam, k, L=2,
        create_basis=create.bspline.basis, ...)

nlCoefEst(y, t, lam, k, L=2,
          create_basis=create.bspline.basis, maxit=100, tol=1e-8,
          typels=c("brent","grid"), addDat=FALSE, rangeval=NULL, ...)

makeLinFda(obj, npoints=200)

logFda(obj)

## S3 method for class 'myfda'
as.fd(x, fdnames=NULL, npoints=200, ...)

## S3 method for class 'myfda'
fitted(object, t=NULL, ...)

Arguments

y	A $T \times n$ matrix of data points, when $T$ is the number of periods and $n$ the number of individuals.
t	A numeric vector of time units. In coefEst and nlCoefEst, the number of elements is equal to the number of rows of y. For the fitted method, it can be a fine grid, and the range must be inside the range of the basis.
lam	The regularization parameter that penalizes for the absence of smoothness.
k	The number of basis.
L	Either a nonnegative integer defining an order of a derivative or a linear differential operator (see eval.penalty).
create_basis	The function used to create the basis object (see create.bspline.basis).
maxit	The maximum number of iteration for the Newton method
tol	The tolerance parameter for the stopping rule of the Newton method
...	Other argument that is passed to create_basis (see create.bspline.basis), or to other methods in the case of as.fd.myfda.
obj	Object of class "myfda"
npoints	Number of points to fit the linear FDA to the non-linear one
x	Object of class "myfda"
object	Object of class "myfda"
fdnames	Names of elements of the "fd" object (see fd)
typels	The nlCoefEst is based on Newton with line search. The argument refers to the method used for the line search
addDat	If TRUE, fake observations are added before and after assuming stationarity. for the estimation, the time span is expanded accordingly.
rangeval	This can be set if the basis is defined on a range other than range(t).

Details

In coefEst, the functional data is a linear combination of the basis, while in nlCoefEst it is the exponential of the linear combination of the basis. The latter is used when we want to force the functional data curve to be positive,

makeLinFda will transform the nonlinear fit to a linear FDA by fitting the nonlinear curve to a linear combination of the same basis. The curve is almost identical and it is transformed this way by funcreg

logFDA transforms the non-linear FDA into a linear FDA. It only redefine the link equation and take the log of the observed data.

Value

It returns an object of 'class' "myfda"

The object of class "myfda" is a list containing at least:

coefficients	$k\times n$ matrix of coefficients.
y	The dataset matrix y.
t	The input vector t.
basis	The basis object
link	The function that links y to the linear combination of the basis.
lambda	The value of lam.
convergence	Information on the convergence of the Newton method.

References

Ramsay, James O., & Silverman, Bernard W. (2005), Functional Data Analysis, Springer, New York.

Examples

data(GDPv56) 
t <- seq(0,1,len=nrow(GDPv56))

## Linear estimation
res <- coefEst(y=GDPv56, t=t, lam=0.003, k=15)
res

cbind(data=GDPv56[,1], fitted=fitted(res)[,1])

## Nonlinear Estimation
res <- nlCoefEst(y=GDPv56, t=t, lam=0.003, k=15)
res

resLin <- makeLinFda(res)
resLin

## Take the logarithm of the non-linear functional data
logres <- logFda(res)
plot(res,1:2)

## Transform the myfda object into an fd object
resfd <- as.fd(resLin)

## Nonlinear Estimation with zeros
data(simData)
t <- seq(0,1,length.out=5)
res <- nlCoefEst(y=simData, t=t, lam=0.003, k=7)
res
res <- nlCoefEst(y=simData, t=t, lam=0.003, k=7, addDat=TRUE)
res

---

Leave-one-out cross-validation

Description

It computes the cross-validation for a given lambda and k or find the optimal lambda for a given k (see below).

Usage

funcregCV(form, create_basis=create.bspline.basis, LD=2, lambda,
          k, CstInt=FALSE, regularized=TRUE, alpha=1e-5, data=NULL,
          obj=NULL, ...)

getFuncregLam(form, create_basis=create.bspline.basis, LD=2, lam0,
              k, regularized=TRUE, CstInt=FALSE, data=NULL, loglam=FALSE,
              method="BFGS", alpha=1e-5, optimArg=list(), ...)

Arguments

obj	An object of class "funcreg". If NULL, all other arguments must be set.
form	A formula for the functional regression. Arguments are objects of class "myfda" obtained either by coefEst or nlCoefEst, and must be in the local environment.
create_basis	The function used to create the basis object (see create.bspline.basis) for the functional parameters.
LD	Either a nonnegative integer defining an order of a derivative or a linear differential operator (see eval.penalty).
data	An optional list of myfda objects, which are the functional data included in the formula form
lambda	A vector of regularization parameters that penalizes for the absence of smoothness. The dimention is 2 times the number of regressors plus 1 if there is an intercept and if CstInt=FALSE. The first is for the intercept, the second and third for the first regressor and so on. If it is a scalar, it is assumed that the smoothing parameter is the same for all functional parameters.
lam0	Initial value of lambda for the minimization of the cross-validation
k	A vector of integers representing the number of basis for each functional parameter. The first is for the intercept (if there is an intercept and that Cstint=FALSE), the second for the first functional parameter and so on. If it is a scalar, it is assumed that the number of basis is the same for all functional parameters.
regularized	If TRUE, the system is regularized by add alpha to the main diagonal of the regression matrix
CstInt	If TRUE, the intercept is assumed constant. If FALSE, it is a one dimensional functional parameter.
loglam	If TRUE, lam0 is log10(lambda) and the function minimizes CV(10^lam). It prevents the lambda's from being negative.
method	The algorithm used by link{optim}
optimArg	A list of additional arguments to pass to link{optim}
alpha	The regularization parameter. The value is added to the diagonal of the regression matrix to make sure it is positive definite.
...	Additional arguments to be passed to create_basis (see create.bspline.basis).

Value

'funregCV' returns the value of the leave-one-out cross-validation. The attributes 'convergence', returns convergence code for each estimation.

'getFuncregLam' returns the fitted object of class 'funcreg' using the optimal lambda. The argument 'optimRes' gives the optim output from the minimization of the cross-validation.

References

Ramsay, James O., & Silverman, Bernard W. (2005), Functional Data Analysis, Springer, New York.

Examples

data(GDPv56) 

## We just create response and a covariate artificialy from the GDP
## series

y <- GDPv56[,1:30]
x <- GDPv56[,31:60]
t <- seq(0,1,len=nrow(y))

## First we create the "myfda" objects
yfd <- coefEst(y, t, .0004, 15, 2)
xfd <- coefEst(x, t, .0004, 15, 2)


## with the object
res <- funcreg(yfd~xfd, k=c(5,5), lambda=c(.001,.001,.001))
funcregCV(obj=res)

## without object
funcregCV(yfd~xfd, k=c(5,5), lambda=c(.001,.001,.001))

## With the default values
getFuncregLam(yfd~xfd, k=c(5,5), lam0=c(.001,.001,.001))

## With upper and lower bound
getFuncregLam(yfd~xfd, k=c(5,5), lam0=c(.001,.001,.001), method="L-BFGS-B",
             optimArg=list(lower=c(0,0,0),upper=c(200,200,200))) 

## With loglam
res <- getFuncregLam(yfd~xfd, k=c(5,5), lam0=c(-3,-3,-3), loglam=TRUE)
res

## Get info about the optim()
res$optimRes

---

Cross-Validation

Description

The functions compute the leave-one-out cross-validation of functional data. It applies also to already estimated models.

Usage

nlFdaCV(y, t, lamvec, kvec, L=2,
        create_basis=create.bspline.basis, maxit=100, tol=1e-8,
         obj=NULL, typels=c("brent","grid"), addDat=FALSE, ...)

fdaCV(y, t, lamvec, kvec, L=2,
      create_basis=create.bspline.basis, obj=NULL, ...)

Arguments

y	A $T \times n$ matrix of data points, when $T$ is the number of periods and $n$ the number of individuals.
t	A numeric vector of time units. The number of elements is equal to the number of rows of y. It must have a range inside rangeval.
lamvec	A vector of regularization parameters that penalizes for the absence of smoothness.
kvec	A vector of integers that indicates the number of basis.
L	Either a nonnegative integer defining an order of a derivative or a linear differential operator (see eval.penalty).
create_basis	The function used to create the basis object (see create.bspline.basis).
maxit	The maximum number of iteration for the Newton method
tol	The tolerance parameter for the stopping rule of the Newton method
obj	Object of class "myfda" if we want the cross-validation of an already fitted model
typels	The nlCoefEst is based on Newton with line search. The argument refers to the method used for the line search
addDat	If TRUE, fake observations are added before and after assuming stationarity. for the estimation, the time span is expanded accordingly.
...	Other argument that is passed to create_basis (see create.bspline.basis).

Details

It returns a matrix of cross-validations. The rows are for the lambda, and the columns for the number of basis.

Value

fdaCV returns a matrix of cross-validations. The rows are for the lambda, and the columns for the number of basis.

nlFdaCV returns a list with the following items:

cv	The cross-validation matrix (see details)
convergence	A matrix returning TRUE if there was no convergence problem while computing the cross-validation.
kvec	The vector of integers that indicates the number of basis.
lamvec	The vector of regularization parameters.
info	An array of convergence codes for all estimations in the process of computing the cross-validations

References

Ramsay, James O., & Silverman, Bernard W. (2005), Functional Data Analysis, Springer, New York.

Examples

data(GDPv56) 
t <- seq(0,1,len=nrow(GDPv56))

## Linear estimation
####################

# From an object
res <- coefEst(y=GDPv56, t=t, lam=0.003, k=15)
fdaCV(obj=res)

# CV with the estimation
cv <- fdaCV(y=GDPv56,t,10^seq(-5,1,len=4), k=5:7)
cv


## Nonlinear Estimation
#######################

res <- nlCoefEst(y=GDPv56, t=t, lam=0.003, k=10)
nlFdaCV(obj=res)

# CV with the estimation
cv <- nlFdaCV(y=GDPv56,t,10^seq(-5,1,len=4), k=5:7)

## Nonlinear Estimation with zeros
##################################

data(simData)
t <- seq(0,1,length.out=5)
cv <- nlFdaCV(y=simData, t=t, lam=c(1e-5, 1e-2), k=c(6:8), addDat=TRUE)
print(cv, conv=TRUE)

---

Linear Functional Regression

Description

Estimation of the functional parameters in a multiple functional regression. The smoothing parameters and the number of basis is assumed to be known.

Usage

funcreg(form, create_basis=create.bspline.basis, LD=2, lambda,
                    k, regularized=TRUE, CstInt=FALSE, CV=FALSE, data=NULL,
                    alpha=1e-5, optAlpha=FALSE,
                    controlAlpha=list(), ...)

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

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

## S3 method for class 'funcreg'
fitted(object, ...)

## S3 method for class 'funcreg'
residuals(object, ...)

Arguments

x	An object of class "funcreg"
object	An object of class "funcreg"
form	A formula for the functional regression. Arguments are objects of class "myfda" obtained either by coefEst or nlCoefEst, and must be in the local environment.
create_basis	The function used to create the basis object (see create.bspline.basis) for the functional parameters.
LD	Either a nonnegative integer defining an order of a derivative or a linear differential operator (see eval.penalty).
lambda	A vector of regularization parameters that penalizes for the absence of smoothness. The dimention is 2 times the number of regressors plus 1 if there is an intercept and if CstInt=FALSE. The first is for the intercept, the second and third for the first regressor and so on. If it is a scalar, it is assumed that the smoothing parameter is the same for all functional parameters.
k	A vector of integers representing the number of basis for each functional parameter. The first is for the intercept (if there is an intercept and that Cstint=FALSE), the second for the first functional parameter and so on. If it is a scalar, it is assumed that the number of basis is the same for all functional parameters.
regularized	If TRUE, the system is regularized by add alpha to the main diagonal of the regression matrix
CstInt	If TRUE, the intercept is assumed constant. If FALSE, it is a one dimensional functional parameter.
CV	if TRUE, it computes the cross-validation.
data	An optional list of myfda objects, which are the functional data included in the formula form
alpha	The regularization parameter. The value is added to the diagonal of the regression matrix to make sure it is positive definite.
optAlpha	If TRUE, the regularization parameter is the one that minimizes the cross-validation using the Brent method
controlAlpha	A list of tuning parameter for the choice of the optimal alpha. The default parameters are maxit=100 , tol=1e-5, and intAlpha=c(1e-11, 1e-4).
...	For funcreg, it is other argument that is passed to create_basis (see create.bspline.basis). For the methods, it is the other arguments to pass to other objects.

Value

It returns an object of 'class' "funcreg"

The object of class "funcreg" is a list containing at least:

X	The list of functional regressors.
Y	The functional response.
data	A list with element "t" for time units of the observed data and "Y" for the matrix of observed response varable.

References

Ramsay, James O., & Silverman, Bernard W. (2005), Functional Data Analysis, Springer, New York.

Examples

data(GDPv56) 

## We just create response and a covariate artificialy from the GDP
## series

y <- GDPv56[,1:30]
x <- GDPv56[,31:60]
t <- seq(0,1,len=nrow(y))

## First we create the "myfda" objects
yfd <- coefEst(y, t, .0004, 15, 2)
xfd <- coefEst(x, t, .0004, 15, 2)


## we just set lambda and k to arbitrary values
res <- funcreg(yfd~xfd, k=c(5,5), lambda=c(.001,.001,.001))
res

summary(res)

plot(fitted(res))

plot(residuals(res))

## Getting the optimal regularization parameter
res <- funcreg(yfd~xfd, k=c(5,5), lambda=c(.001,.001,.001),
               optAlpha=TRUE)
res

---

International Gross Domestic Product

Description

The dataset includes GDP's of 90 countries from 1960 to 1980. The frequency is yearly.

Usage

data(GDPv56)

Format

A matrix containing 90 columns, one per country. Dates are reported as rownames().

ARGENTINA

GDP of ARGENTINA

AUSTRALIA

GDP of AUSTRALIA

AUSTRIA

GDP of AUSTRIA

BURUNDI

GDP of BURUNDI

BELGIUM

GDP of BELGIUM

BENIN

GDP of BENIN

BURKINA FASO

GDP of BURKINA FASO

BOLIVIA

GDP of BOLIVIA

BRAZIL

GDP of BRAZIL

CANADA

GDP of CANADA

SWITZERLAND

GDP of SWITZERLAND

CHILE

GDP of CHILE

COTE D'IVOIRE

GDP of COTE D'IVOIRE

CAMEROON

GDP of CAMEROON

CONGO REPUBLIC OF

GDP of CONGO REPUBLIC OF

COLOMBIA

GDP of COLOMBIA

COMOROS

GDP of COMOROS

CAPE VERDE

GDP of CAPE VERDE

COSTA RICA

GDP of COSTA RICA

DENMARK

GDP of DENMARK

DOMINICAN REPUBLIC

GDP of DOMINICAN REPUBLIC

ALGERIA

GDP of ALGERIA

ECUADOR

GDP of ECUADOR

EGYPT

GDP of EGYPT

SPAIN

GDP of SPAIN

FINLAND

GDP of FINLAND

FRANCE

GDP of FRANCE

GABON

GDP of GABON

UNITED KINGDOM

GDP of UNITED KINGDOM

GHANA

GDP of GHANA

GUINEA

GDP of GUINEA

GAMBIA THE

GDP of GAMBIA THE

GUINEA-BISSAU

GDP of GUINEA-BISSAU

GREECE

GDP of GREECE

GUATEMALA

GDP of GUATEMALA

HONG KONG

GDP of HONG KONG

HONDURAS

GDP of HONDURAS

INDONESIA

GDP of INDONESIA

INDIA

GDP of INDIA

IRELAND

GDP of IRELAND

IRAN

GDP of IRAN

ICELAND

GDP of ICELAND

ISRAEL

GDP of ISRAEL

ITALY

GDP of ITALY

JAMAICA

GDP of JAMAICA

JORDAN

GDP of JORDAN

JAPAN

GDP of JAPAN

KENYA

GDP of KENYA

KOREA REPUBLIC OF

GDP of KOREA REPUBLIC OF

SRI LANKA

GDP of SRI LANKA

LESOTHO

GDP of LESOTHO

LUXEMBOURG

GDP of LUXEMBOURG

MOROCCO

GDP of MOROCCO

MADAGASCAR

GDP of MADAGASCAR

MEXICO

GDP of MEXICO

MALI

GDP of MALI

MOZAMBIQUE

GDP of MOZAMBIQUE

MAURITIUS

GDP of MAURITIUS

MALAWI

GDP of MALAWI

MALAYSIA

GDP of MALAYSIA

NIGERIA

GDP of NIGERIA

NICARAGUA

GDP of NICARAGUA

NETHERLANDS

GDP of NETHERLANDS

NORWAY

GDP of NORWAY

NEW ZEALAND

GDP of NEW ZEALAND

PAKISTAN

GDP of PAKISTAN

PANAMA

GDP of PANAMA

PERU

GDP of PERU

PHILIPPINES

GDP of PHILIPPINES

PORTUGAL

GDP of PORTUGAL

PARAGUAY

GDP of PARAGUAY

RWANDA

GDP of RWANDA

SENEGAL

GDP of SENEGAL

SINGAPORE

GDP of SINGAPORE

EL SALVADOR

GDP of EL SALVADOR

SWEDEN

GDP of SWEDEN

SYRIA

GDP of SYRIA

CHAD

GDP of CHAD

TOGO

GDP of TOGO

THAILAND

GDP of THAILAND

TRINIDAD AND TOBAGO

GDP of TRINIDAD AND TOBAGO

TURKEY

GDP of TURKEY

TAIWAN

GDP of TAIWAN

UGANDA

GDP of UGANDA

URUGUAY

GDP of URUGUAY

UNITED STATES

GDP of UNITED STATES

VENEZUELA

GDP of VENEZUELA

SOUTH AFRICA

GDP of SOUTH AFRICA

ZAMBIA

GDP of ZAMBIA

ZIMBABWE

GDP of ZIMBABWE

Source

TBA

---

Cross-Validation

Description

Several functions that computes the leave-one-out cross-validation of functional data. They include cases of linear and nonlinear fit.

Usage

nlGetLam(y, t, lamInt, k, L=2,
         create_basis=create.bspline.basis, maxit=100, tol=1e-8,
         maxitalgo=100, tolalgo=1e-7, type=c("Brent","GS"),
         typels=c("brent","grid"), addDat=FALSE, ...)

nlGetLandKopt(y, t, lamInt, kvec, L=2,
              create_basis=create.bspline.basis, maxit=100, tol=1e-7,
              maxitalgo=100, tolalgo=1e-6, type=c("Brent", "GS"),
              typels=c("brent","grid"), addDat=FALSE, ...)

nlGetLandK(y, t, lamvec, kvec, L=2,
           create_basis=create.bspline.basis, maxit=100, tol=1e-8,
           typels=c("brent","grid"), addDat=FALSE, ...)

getLam(y, t, lamInt, k, L=2,
       create_basis=create.bspline.basis, 
       maxitalgo=100, tolalgo=1e-7, type=c("Brent","GS"), ...)

getLandK(y, t, lamvec, kvec, L=2,
         create_basis=create.bspline.basis, ...)

getLandKopt(y, t, lamInt, kvec, L=2,
            create_basis=create.bspline.basis, 
            maxitalgo=100, tolalgo=1e-6, type=c("Brent", "GS"), ...)

Arguments

y	A $T \times n$ matrix of data points, when $T$ is the number of periods and $n$ the number of individuals.
t	A numeric vector of time units. The number of elements is equal to the number of rows of y. It must have a range inside rangeval.
lamvec	A vector of regularization parameters that penalizes for the absence of smoothness.
lamInt	A vector that contains the lower and upper bounds for Lambda that is passed to the Brent or Golden Section method.
kvec	A vector of integers that indicates the number of basis.
k	A scalar that indicates the number of basis.
L	Either a nonnegative integer defining an order of a derivative or a linear differential operator (see eval.penalty).
create_basis	The function used to create the basis object (see create.bspline.basis).
maxit	The maximum number of iteration for the Newton method
tol	The tolerance parameter for the stopping rule of the Newton method
maxitalgo	Maximum number of iteration for the brent and Golden-Section methods
tolalgo	Tolerance level for the stopping rule of the Brent or Golden-Section method.
type	Which algorithm should we use to get the lambda.
typels	The nlCoefEst is based on Newton with line search. The argument refers to the method used for the line search
addDat	If TRUE, fake observations are added before and after assuming stationarity. for the estimation, the time span is expanded accordingly.
...	Other argument that is passed to create_basis

(see create.bspline.basis).

Details

Function that starts with 'nl' are the nonlinear functional data (see coefEst).

getLam and nlGetLam, finds the Lambda that minimizes the cross-validation for a given k using either the Brent method or the Golden-Section method.

nlGetLandK and getLandK find the optimal Lambda and k using a grid search. The grid is provided by the user through lamvec and kvec.

getLandKopt and nlGetLandKopt find the optimal Lambda and k using a grid search for k only. Lambda is obtained using the Brent or Golden-Section method.

Value

getLam and nlGetLam returns a list with the following items:

lam	The optimal Lambda
info	A convergence code for the optimization method (0 for normal convergence).
iter	The number of iterations for the Brent or Golden-Section method.
cv	The minimum cross-validation

getLandK, codenlGetLandK, getLandKopt, and codenlGetLandKopt, returns an object of class "myfda" (see coefEst). In addition to the usual element, the list inludes information about the search.

References

Ramsay, James O., & Silverman, Bernard W. (2005), Functional Data Analysis, Springer, New York.

Examples

data(GDPv56) 
t <- seq(0,1,len=nrow(GDPv56))

## Linear estimation
####################

# Using a grid
nlam <- 5
lamvec <- 10^seq(-5,1,length.out=nlam)
kvec <- 5:7
res1 <- getLandK(GDPv56, t, lamvec, kvec)
res1

# Using Brent
res2 <- getLandKopt(GDPv56, t, lamvec, kvec, tolalgo=1e-5)
res2

## Nonlinear estimation
###########################

# Using a grid
res3 <- nlGetLandK(GDPv56, t, lamvec, kvec)
res3

# Using Brent
res4 <- nlGetLandKopt(GDPv56, t, lamvec, kvec, tolalgo=1e-5)
res3

---

Methods for Functional Data Fit

Description

These are the plot and print methods which are associated with the objects of class "myfda".

Usage

## S3 method for class 'myfda'
plot(x, which=NULL, addpoints=TRUE, npoints=100,
 add=FALSE, ...)

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

## S3 method for class 'myfdaCV'
print(x, digits=4, conv=FALSE, ...)

## S3 method for class 'myfda'
x[i=NULL]

Arguments

x	Object of class "myfda" (see coefEst).
which	Which curve shall we plot
addpoints	If TRUE and which is not null, observations are added to the graph
npoints	The number of points used to plot the curves
digits	The number of digits to be printed
conv	Shall the information about convergence be printed
add	If TRUE, the line associated with "which" is added to the existing graph
i	The selection index.
...	Other argument that is passed to the plot or print methods

Examples

data(GDPv56) 
t <- seq(0,1,len=nrow(GDPv56))

## Linear estimation
res <- coefEst(y=GDPv56, t=t, lam=0.003, k=15)
print(res)
plot(res)
plot(res,which=1:3)

##  Using the operator []
plot(res[1:4])

res2 <- coefEst(y=GDPv56, t=t, lam=0.3, k=15)
plot(res,which=1)
plot(res2,which=1, add=TRUE, col=2, lty=3)

---

Plot functions for functional coefficients

Description

It plot the estimated functional coefficients with confidence bands. The coefficients are estimated by funcreg.

Usage

plotCoef(obj, which, n = c(50, 50), fixeds=NULL, fixedt=NULL, plotWF=TRUE, ...)

plotConfInt(obj, which, type = c("beta_t", "beta_s", "beta_st"),
                       n = c(50, 50), level = 0.95, plotWF = TRUE, beta=NULL)

Arguments

obj	An object of class "funcreg"
which	Which functional coefficient we want to plot. 0 for the intercept, 1 for the first functional slope and so on.
n	The number of grid points for the plot. The first element is for the t dimension and the second for the s dimension.
fixeds	If NULL, a three dimensional plot is produced. Alternatively, we can set a value for s and plot beta(t,s) for a given s. It has no effect when which=0.
fixedt	If NULL, a three dimensional plot is produced. Alternatively, we can set a value for t and plot beta(t,s) for a given t. It has no effect when which=0.
plotWF	If FALSE, the wireframe is not shown. In that case, only the output of the function is returned. Only applicable for three dimensional plot.
type	The type of plot (see details).
level	The level for the confidence interval
beta	A one or two argument function (see details).
...	Other argument. to pass to either plot or wireframe.

Details

If type="beta_t", the default, the integral with respect to s is shown with pointwise confidence bands. It is simply the functional intercept with its pointwise confidence bands if which=0.

If type="beta_s" (not applicable for which=0), the integral with respect to t is shown with its pointwise confidence bands.

If type="beta_st" (not applicable for which=0), the whole three dimensional curve is shown with its pointwise confidence surfaces.

If beta is provided, the estimated curve is replaced by the one produced by the function beta, but the same confidence bands are produced. Useful for pointwise hypothesis testing. If type="beta_s" or type="beta_t", beta must be a function of t or s. If type="beta_st", it must be a function of both.

Value

If plotWF=FALSE and the expected plot is a surface, it returns an object of class "trellis" produced by wireframe.

References

Ramsay, James O., & Silverman, Bernard W. (2005), Functional Data Analysis, Springer, New York.

Examples

data(GDPv56) 

## We just create response and a covariate artificialy from the GDP
## series

y <- GDPv56[,1:30]
x <- GDPv56[,31:60]
t <- seq(0,1,len=nrow(y))

## First we create the "myfda" objects
yfd <- coefEst(y, t, .0004, 15, 2)
xfd <- coefEst(x, t, .0004, 15, 2)


## we just set lambda and k to arbitrary values
res <- funcreg(yfd~xfd, k=c(5,5), lambda=c(.001,.001,.001))

plotCoef(res, 1, type="beta_st")

plotConfInt(res, 1, type="beta_t")

## Assume we what to do pointwise hypothesis testing
plotConfInt(res, 1, type="beta_t", beta=function(t) .6-.7*t)

## Nice 3-D confidence interval
plotConfInt(res, 1, type="beta_st")

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Simulated panel with some zeros

Description

This dataset is used to show how the nonlinear fit can be used when some observations are zero.

Usage

data(simData)

Format

A matrix containing 293 columns, and 5 time units.

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Variance of functional coefficients

Description

It computes the functional variance of the estimated functional parameters obtained by funcreg

Usage

## S3 method for class 'funcreg'
vcov(object, which, type = c("beta", "beta_t", "beta_s", "beta_st"),
                         s = NULL, t = NULL, ...)

Arguments

object	An object of class "funcreg"
which	The variance of which functional parameters shall we compute? 0 for the intercept, 1 for the first functional slope and so on.
type	What type of variance to compute (see details).
s	If type is "beta_s", the variance is computed at each element of s. If NULL, it is computed at the same time units as for the observed observations.
t	If type is "beta_t", the variance is computed at each element of s. If NULL, it is computed at the same time units as for the observed observations.
...	Argument to pass to other objects

Details

The function computed the variance of the integral of the functional parameters (single for the intercept and double for the slopes) when type='beta'.

If type='beta_t', it is the pointwise variance of the functional intercept if which=0 and the pointwise variance of the integral of the slopes which respect to s. By default (t=NULL) the time units of the observed data is used. Alternatively, we can provide a vector of points.

If type='beta_s', (not available for which=0) it is the pointwise variance of the integral of the slopes which respect to y. By default (s=NULL), equally spaced grid points are generated using the range of s-basis and the number of time units equals to the one from the observed observations.

If type='beta_st' (not available for which=0), a two dimensional pointwise variance is computed.

For the last three cases, vcov returns either a one dimension functional data object or a two-dimensional one.

Value

It returns a list containing at least:

V	Either a numeric value when type='beta' or an 'fda' object when type='beta_t' or type='beta_s', or a matrix when type='beta_st'.

References

Ramsay, James O., & Silverman, Bernard W. (2005), Functional Data Analysis, Springer, New York.

Examples

data(GDPv56) 

## We just create response and a covariate artificialy from the GDP
## series

y <- GDPv56[,1:30]
x <- GDPv56[,31:60]
t <- seq(0,1,len=nrow(y))

## First we create the "myfda" objects
yfd <- coefEst(y, t, .0004, 15, 2)
xfd <- coefEst(x, t, .0004, 15, 2)


## we just set lambda and k to arbitrary values
res <- funcreg(yfd~xfd, k=c(5,5), lambda=c(.001,.001,.001))

vcov(res, which=0)

plot(vcov(res, 1, type="beta_t")$V)

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