Package 'ReMFPCA'

Title: Regularized Multivariate Functional Principal Component Analysis
Description: Methods and tools for implementing regularized multivariate functional principal component analysis ('ReMFPCA') for multivariate functional data whose variables might be observed over different dimensional domains. 'ReMFPCA' is an object-oriented interface leveraging the extensibility and scalability of R6. It employs a parameter vector to control the smoothness of each functional variable. By incorporating smoothness constraints as penalty terms within a regularized optimization framework, 'ReMFPCA' generates smooth multivariate functional principal components, offering a concise and interpretable representation of the data. For detailed information on the methods and techniques used in 'ReMFPCA', please refer to Haghbin et al. (2023) <doi:10.48550/arXiv.2306.13980>.
Authors: Hossein Haghbin [aut, cre] , Yue Zhao [aut] , Mehdi Maadooliat [aut]
Maintainer: Hossein Haghbin <[email protected]>
License: GPL (>= 2)
Version: 1.1.0
Built: 2025-01-16 09:34:52 UTC
Source: https://github.com/haghbinh/remfpca

Help Index

Subtract two 'mfd' objects

Description

Subtract two 'mfd' objects

Usage

## S3 method for class 'mfd'
obj1 - obj2 = NULL

Arguments

obj1

An 'mfd' object
obj2

An 'mfd' object or a scalar

Value

The difference between the two 'mfd' objects

See Also

basismfd, mfd

Subtraction of two 'mvmfd' objects

Description

Subtraction of two 'mvmfd' objects

Usage

## S3 method for class 'mvmfd'
obj1 - obj2 = NULL

Arguments

obj1

An 'mvmfd' object
obj2

An optional 'mvmfd' object

Value

An 'mvmfd' object

See Also

mvmfd,mvbasismfd

Extract subsets of an 'mfd' object

Description

Extract subsets of an 'mfd' object

Usage

## S3 method for class 'mfd'
mfd_obj[i = "index"]

Arguments

mfd_obj

An 'mfd' object
i

An index or indices specifying the subsets to extract

Value

An 'mfd' object containing the specified subsets

See Also

basismfd, mfd

Extract subsets of an 'mvmfd' object

Description

Extract subsets of an 'mvmfd' object

Usage

## S3 method for class 'mvmfd'
mvmfd_obj[i = "index", j = "index"]

Arguments

mvmfd_obj

An 'mvmfd' object
i

An index or indices specifying the subsets to extract for the first dimension
j

An index or indices specifying the subsets to extract for the second dimension

Value

An 'mvmfd' object containing the specified subsets

See Also

mvmfd,mvbasismfd

Scalar multiplication of an 'mfd' object

Description

Scalar multiplication of an 'mfd' object. One object must be an 'mfd', and the other one a scalar

Usage

## S3 method for class 'mfd'
obj1 * obj2

Arguments

obj1

An 'mfd' object or an scalar
obj2

An 'mfd' object or an scalar

Value

An 'mfd' object

See Also

basismfd, mfd

Multiplication of an 'mvmfd' object with a scalar

Description

Multiplication of an 'mvmfd' object with a scalar

Usage

## S3 method for class 'mvmfd'
obj1 * obj2

Arguments

obj1

An 'mvmfd' object or a scalar
obj2

An 'mvmfd' object or a scalar

Value

An 'mvmfd' object

See Also

mvmfd,mvbasismfd

Add two 'mfd' objects

Description

Add two 'mfd' objects

Usage

## S3 method for class 'mfd'
obj1 + obj2 = NULL

Arguments

obj1

An 'mfd' object
obj2

An 'mfd' object or a scalar

Value

The sum of the two 'mfd' objects

See Also

basismfd, mfd

Addition of two 'mvmfd' objects

Description

Addition of two 'mvmfd' objects

Usage

## S3 method for class 'mvmfd'
obj1 + obj2 = NULL

Arguments

obj1

An 'mvmfd' object
obj2

An optional 'mvmfd' object

Value

An 'mvmfd' object

See Also

mvmfd,mvbasismfd

Define a Set of Multidimensional Functional Basis

Description

The 'basismfd' class represents a set of multidimensional basis functions. This class utilizes basis objects from the 'fda' package, such as B-splines and Fourier bases.

Constructor for 'basismfd' objects (same as Basismfd(...) )

Usage

Basismfd(...)

Basismfd(...)

Arguments

...

A list of 'basisfd' objects

Active bindings

basis

A list of basis objects from the 'fda' package.

dimSupp

The dimension of the support domain of the 'basismfd' object.

supp

The matrix representing the ranges of the dimensions.

gram

The Gram matrix.

nbasis

A numeric vector containing the number of bases.

Methods

Public methods

Method new()

The constructor function for objects of the class 'basismfd' (same as Basismfd(...) )

Usage
basismfd$new(...)
Arguments
...

A list of 'basisfd' objects

Method eval()

Evaluate the 'basismfd' object at given argument values

Usage
basismfd$eval(evalarg)
Arguments
evalarg

A list of numeric vectors of argument values at which the 'basismfd' is to be evaluated

Returns

A list of evaluated values

Method print()

Print method for 'basismfd' objects

Usage
basismfd$print(...)
Arguments
...

Additional arguments to be passed to 'print' Getter and setter for 'basis' field Getter and setter for 'dimSupp' field Getter and setter for 'nbasis' field Getter and setter for 'supp' field Getter and setter for 'gram' field

Method clone()

The objects of this class are cloneable with this method.

Usage
basismfd$clone(deep = FALSE)
Arguments
deep

Whether to make a deep clone.

Examples

require(fda)
bs1 <- create.fourier.basis(c(0, 2 * pi), 5)
bs2 <- create.bspline.basis(c(0, 1), 7)
bs3 <- create.exponential.basis(c(0, 2), 3)
# 1-D Basis ######## (similar to the fd features)
mdbs1 <- Basismfd(bs1)
mdbs1$basis
mdbs1$dimSupp
mdbs1$nbasis
mdbs1$supp
mdbs1$gram
mdbs1$eval(1:7 / 10)
image(as.matrix(mdbs1$gram))

####### 2-D Basis ######## (fd cannot handle this)
mdbs2 <- Basismfd(bs1, bs2)
mdbs2$basis
mdbs2$dimSupp
mdbs2$nbasis
mdbs2$supp
dim(mdbs2$gram)
arg_mdbs <- list(1:10, 1:9 / 10)
mdbs2$eval(arg_mdbs)
image(as.matrix(mdbs2$gram))

Bivariate plot for 'mvmfd' objects

Description

Bivariate plot for 'mvmfd' objects

Usage

bimfdplot(mvmfd_obj, type = "l", lty = 1, xlab = "", ylab = "", main = "", ...)

Arguments

mvmfd_obj

An 'mvmfd' object
type

Type of plot ('l' for lines, 'p' for points, etc.)
lty

Line type
xlab

Label for the x-axis
ylab

Label for the y-axis
main

Main title
...

Additional arguments for the matplot function

See Also

mvmfd, mvbasismfd

Electrical Power Dataset

Description

This dataset contains measurements of voltage and electrical power consumption recorded from a household between December 2006 and November 2010.

Format

A bivariate functional data object of class 'mvmfd' with the following fields:

voltage

Voltage measurements in volts.

power

Calculated power consumption in watts.

Source

The dataset was collected from https://weather.com.

Examples

## Not run: 
# Load the Electrical Power Dataset
data("electrical_power_data")
head(electrical_power_data)

## End(Not run)

Compute the inner product between two objects of class 'mfd'

Description

Compute the inner product between two objects of class 'mfd'

Usage

inprod_mfd(mfd_obj1, mfd_obj2)

Arguments

mfd_obj1

An 'mfd' object
mfd_obj2

An 'mfd' object

Value

The inner products matrix between the two 'mfd' objects

See Also

basismfd, mfd

Compute the inner product between two objects of class 'mvmfd'

Description

Compute the inner product between two objects of class 'mvmfd'

Usage

inprod_mvmfd(mvmfd_obj1, mvmfd_obj2)

Arguments

mvmfd_obj1

An 'mvmfd' object
mvmfd_obj2

An 'mvmfd' object

Value

The inner products matrix between the two 'mvmfd' objects

See Also

mvmfd,mvbasismfd

Check if an object is of class 'basismfd'

Description

Check if an object is of class 'basismfd'

Usage

is.basismfd(fdobj)

Arguments

fdobj

The object to check.

Value

TRUE if the object is of class 'basismfd', FALSE otherwise.

See Also

is.mvbasismfd, is.mfd, is.mvmfd

Check if an object is of class 'mfd'

Description

Check if an object is of class 'mfd'

Usage

is.mfd(fdobj)

Arguments

fdobj

The object to check.

Value

TRUE if the object is of class 'mfd', FALSE otherwise.

See Also

is.mvbasismfd, is.basismfd, is.mvmfd

Check if an object is of class 'mvbasismfd'

Description

Check if an object is of class 'mvbasismfd'

Usage

is.mvbasismfd(fdobj)

Arguments

fdobj

The object to check.

Value

TRUE if the object is of class 'mvbasismfd', FALSE otherwise.

See Also

is.basismfd, is.mfd, is.mvmfd

Check if an object is of class 'mvmfd'

Description

Check if an object is of class 'mvmfd'

Usage

is.mvmfd(fdobj)

Arguments

fdobj

The object to check.

Value

TRUE if the object is of class 'mvmfd', FALSE otherwise.

See Also

is.mvbasismfd, is.mfd, is.basismfd

Length of an object of classes 'mfd'or 'mvmfd'.

Description

Length of an object of an object of classes 'mfd' or 'mvmfd'.

Usage

length(x, ...)

Arguments

x

An object of classes 'mfd' or 'mvmfd'.
...

all 'length' function arguments.

mean of an object of classes 'mfd'or 'mvmfd'.

Description

mean of an object of classes 'mfd'or 'mvmfd'.

Usage

mean(x, ...)

Arguments

x

An object of classes 'mfd' or 'mvmfd'.
...

all 'mean' function arguments.

Value

An object of class 'mfd'

Define a Set of Multidimensional Functional Data objects

Description

The 'mfd' class represents a set of multidimensional functional data with 'basismfd' object. Functional data objects are constructed by specifying a set of basis functions and a set of coefficients defining a linear combination of these basis functions.

Constructor for 'mfd' objects (same as Mfd(...) )

Usage

Mfd(argval = NULL, X, mdbs, method = "data")

Arguments

argval

A list of numeric vectors of argument values at which the 'mfd' object is to be evaluated
X

A numeric matrix corresponds to basis expansion coefficients if 'method="coefs"' and discrete observations if 'method="data"'.
mdbs

a basismfd object
method

determine the 'X' matrix type as "coefs" and "data".

Active bindings

basis

an object of the class 'basismfd'.

coefs

a matrix of the coefficients.

nobs

number of the observation

Methods

Public methods

Method new()

Constructor for 'mfd' objects (same as Mfd(...) )

Usage
mfd$new(argval = NULL, X, mdbs, method = "data")
Arguments
argval

A list of numeric vectors of argument values at which the 'mfd' object is to be evaluated

X

A numeric matrix corresponds to basis expansion coefficients if 'method="coefs"' and discrete observations if 'method="data"'.

mdbs

a basismfd object

method

determine the 'X' matrix type as "coefs" and "data".

Method eval()

Evaluation an 'mfd' object in some arguments.

Usage
mfd$eval(evalarg)
Arguments
evalarg

a list of numeric vector of argument values at which the mfd is to be evaluated.

Returns

A matrix of evaluated values

Method print()

Print method for 'mfd' objects

Usage
mfd$print(...)
Arguments
...

Additional arguments to be passed to 'print'

Method clone()

The objects of this class are cloneable with this method.

Usage
mfd$clone(deep = FALSE)
Arguments
deep

Whether to make a deep clone.

See Also

basismfd

Examples

require(fda)
bs1 <- create.fourier.basis(c(0,2*pi),5)
bs2 <- create.bspline.basis(c(0,1),7)
bs3 <- create.exponential.basis(c(0,2),3)

#1-D mfd :_____________________________
argval <- seq(0,2*pi,length.out=100)
nobs <- 10;
X <- outer(sin(argval),seq(0.5,1.5,length.out=nobs))
mdbs1 <- Basismfd(bs1)
mfd1 <- Mfd(X=X, mdbs = mdbs1)
inprod_mfd(mfd1,mfd1)
norm_mfd(mfd1)
mfd0 <- 2.5*mfd1
mfd1-mfd0
mfd1[1:3]

mfd1$eval(argval)
mfd1c <- Mfd(X=mfd1$coefs, mdbs = mdbs1, method = "coefs")
all.equal(c(mfd1$basis,mfd1$coefs,mfd1$nobs),c(mfd1c$basis,mfd1c$coefs,mfd1c$nobs))
length(mfd1)
mean(mfd1)
plot(mfd1)

Motion Sense Dataset: Measurements of user acceleration and pitch attitude collected by smartphones from 24 individuals performing four distinct activities: jogging, walking, sitting, and standing.

Description

This dataset includes time-series data generated by accelerometer and gyroscope sensors. A total of 24 participants in a range of gender, age, weight, and height performed 4 activities in the same environment and conditions: downstairs, upstairs, walking, jogging, sitting, and standing.

Format

A bivariate functional data object of class 'mvmfd' with the following fields:

user_acceleration

Time-series data of user acceleration.

pitch_attitude

Time-series data of pitch attitude.

Source

The MotionSense dataset collected by https://github.com/mmalekzadeh.

Examples

## Not run: 
# Load the Motion Sense Dataset
data("motion_sense_data")

## End(Not run)

Define a Set of Multivariate Multidimensional Functional Basis

Description

The 'mvbasismfd' a set of multivariate multidimensional basis functions. This class utilizes basis objects 'basismfd'.

Constructor for 'mvbasismfd' objects (same as 'Mvbasismfd')

Usage

Mvbasismfd(basis)

Mvbasismfd(basis)

## S3 method for class 'mvbasismfd'
mvbasismfd_obj[i = "index"]

Arguments

basis

A list of basisfd objects
mvbasismfd_obj

An 'mvmfd' object
i

An index or indices specifying the subsets to extract for the first dimension

Value

An 'mvbasismfd' object containing the specified subsets

Active bindings

nvar

number of variables

basis

A list of 'mvbasisfd' objects

dimSupp

A sequence of positive integers specifying support domain of the 'mvbasismfd' object.

nbasis

A list of integers specifying the number of basis functions

supp

A list of matrices specifying the support of basis functions

gram

The Gram matrix.

Methods

Public methods

Method new()

Constructor for 'mvbasismfd' objects (same as Mvbasismfd(...) )

Usage
mvbasismfd$new(basis)
Arguments
basis

A list of 'basismfd' objects

Method eval()

Evaluate the 'mvbasismfd' object at given argument values

Usage
mvbasismfd$eval(evalarg)
Arguments
evalarg

A list of numeric vectors of argument values at which the 'mvbasismfd' is to be evaluated

Returns

A list of evaluated values

Method clone()

The objects of this class are cloneable with this method.

Usage
mvbasismfd$clone(deep = FALSE)
Arguments
deep

Whether to make a deep clone.

See Also

mvmfd, basismfd

Define a Set of Multivariate Multidimensional Functional Data objects

Description

The 'mvmfd' class represents functional data ...

Constructor for 'mvmfd' objects (same as 'Mvmfd')

Usage

Mvmfd(...)

Arguments

...

A 'mfd' objects which have separated by comma

Active bindings

basis

A 'mvbasismfd' object

coefs

a matrix of the coefficients.

nobs

number of observation

nvar

number of variables

Methods

Public methods

Method new()

Constructor for ‘mvmfd' objects (same as ’Mvmfd')

Usage
mvmfd$new(...)
Arguments
...

A 'mfd' objects which have separated by comma

Method eval()

Eval method for 'mvmfd' objects

Usage
mvmfd$eval(evalarg)
Arguments
evalarg

A list of numeric vectors of argument values at which the 'mvmfd' is to be evaluated.

Returns

A list of evaluated values

Method print()

Print method for 'mvmfd' objects

Usage
mvmfd$print(...)
Arguments
...

Additional arguments to be passed to 'print'

Method clone()

The objects of this class are cloneable with this method.

Usage
mvmfd$clone(deep = FALSE)
Arguments
deep

Whether to make a deep clone.

See Also

mvbasismfd, mfd

Examples

require(fda)
bs1 <- create.fourier.basis(c(0, 2 * pi), 5)
bs2 <- create.bspline.basis(c(0, 1), 7)
bs3 <- create.exponential.basis(c(0, 2), 3)
nobs <- 10
argval1 <- seq(0, 2 * pi, length.out = 12)
X1 <- outer(sin(argval1), seq(0.5, 1.5, length.out = nobs))
mdbs1 <- Basismfd(bs1)
mfd1 <- Mfd(argval1, X1, mdbs1)
mdbs2 <- Basismfd(bs1)
argval2 <- argval1
X2 <- outer(cos(argval2), seq(0.2, 1.5, length.out = nobs))
mfd2 <- Mfd(argval2, X2, mdbs1)
mvmfd1 <- Mvmfd(mfd1, mfd2)
mvmfd1[1]
mvmfd1[1, 1]
mvmfd1[1:5, 2]
mvmfd1[, 1]
mvmfd1[1:5, ]
evalarg <- list(argval1, argval2)
mvmfd1$eval(evalarg)
mvmfd1 + mvmfd1
mean(mvmfd1)
inprod_mvmfd(mvmfd1, mvmfd1)
norm_mvmfd(mvmfd1)
plot(mvmfd1)
bimfdplot(mvmfd1)

Compute the norm of an object of class 'mfd'

Description

Compute the norm of an object of class 'mfd'

Usage

norm_mfd(mfd_obj)

Arguments

mfd_obj

An object of class 'mfd'

Value

The norm vector of the an object of class 'mfd'

See Also

basismfd, mfd

Compute the norm of an object of class 'mvmfd'

Description

Compute the norm of an object of class 'mvmfd'

Usage

norm_mvmfd(mvmfd_obj)

Arguments

mvmfd_obj

An 'mvmfd' object

Value

The norm vector of the an object of class 'mvmfd'

See Also

mvmfd,mvbasismfd

Penalty Function

Description

Calculate the penalty matrix for 'mvmfd' objects.

Usage

pen_fun(data, devorder = 2, type)

Arguments

data

an object of class 'mvmfd'.
devorder

The order of the derivative.
type

The type of penalty. The types "coefpen" and "basispen" is supported.

Value

The penalty matrix.

plots an object of classes 'mfd', 'mvmfd' or 'remfpca'

Description

plot an object of classes 'mfd', 'mvmfd' or 'remfpca'

Usage

plot(x, ...)

Arguments

x

An object of classes 'mfd', 'mvmfd' or 'remfpca'
...

all 'plot' function arguments.

A Class for 'ReMFPCA' objects

Description

The 'remfpca' class represents regularized functional principal components components.

The ‘remfpca' class represents regularized functional principal components (’ReMFPCs') components.

Usage

Remfpca(
  mvmfd_obj,
  method = "power",
  ncomp,
  smooth_tuning = NULL,
  sparse_tuning = NULL,
  centerfns = TRUE,
  alpha_orth = FALSE,
  smoothing_type = "basispen",
  sparse_type = "soft",
  K_fold = 30,
  sparse_CV = TRUE,
  smooth_GCV = TRUE
)

Arguments

mvmfd_obj

An 'mvmfd' object representing the multivariate functional data.
method

A character string specifying the approach to be used for MFPCA computation. Options are "power" (the default), which uses the power algorithm, or "eigen", which uses the eigen decomposition approach.
ncomp

The number of functional principal components to retain.
smooth_tuning

A list or vector specifying the smoothing regularization parameter(s) for each variable. If NULL, non-smoothing MFPCA is estimated.
sparse_tuning

A list or vector specifying the sparsity regularization parameter(s) for each variable. If NULL, non-sparse MFPCA is estimated.
centerfns

Logical indicating whether to center the functional data before analysis. Default is TRUE.
alpha_orth

Logical indicating whether to perform orthogonalization of the regularization parameters. If 'method' is "power", setting 'alpha_orth = FALSE' (default) uses the sequential power approach, while setting 'alpha_orth = TRUE' uses the joint power approach.
smoothing_type

The type of smoothing penalty to be applied on the estimated functional PCs. The types "basispen" and "coefpen" is supported. Default is "basispen".
sparse_type

The type of sparse penalty to be applied on the estimated functional PCs. The types "soft-threshold", "hard-threshold" and "SCAD" is supported. Default is "soft-threshold".
K_fold

An integer specifying the number of folds in the sparse cross-validation process. Default is 30.
sparse_CV

@param sparse_CV Logical indicating whether cross-validation should be applied to select the optimal sparse tuning parameter in sequential power approach. If 'sparse_CV = TRUE', a series of tuning parameters should be provided as a vector with positive number with max equals to number of subjects. If 'sparse_CV = FALSE', specific tuning parameters are given directly to each principal components. Tuning parameters should be provided as a vector with length equal to 'ncomp'. If the dimensions of input tuning parameters are incorrect, it will be converted to a list internally, and a warning will be issued.
smooth_GCV

@param smooth_GCV Logical indicating whether generalized cross-validation should be applied to select the optimal smooth tuning parameter. If 'smooth_GCV = TRUE', a series of tuning parameters should be provided as a list with length equal to the number of variables. If a list with incorrect dimensions is provided, it will be converted to a correct list internally, and a warning will be issued. If 'smooth_GCV = FALSE', specific tuning parameters are given directly. If 'method' is "power" and 'alpha_orth = FALSE' (sequential power), tuning parameters should be provided as a list with length equal to the number of variables, where each element is a vector of length 'ncomp'. If 'method' is "power" and 'alpha_orth = TRUE' (joint power), tuning parameters should be provided as a vector with length equal to the number of variables. If the dimensions of input tuning parameters are incorrect, it will be converted to a list internally, and a warning will be issued.

Active bindings

pc_mfd

An object of class 'mvmfd' where the first indices (fields) represents harmonics and second indices represents variables

lsv

= Left singular values vectors

values

= The set of eigenvalues

smooth_tuning

= The list of smoothing penalties parameters

sparse_tuning

= The list of sparse penalties parameters

GCVs

= Generalized cross validations scores of smoothing penalties parameters. If both smoothing and sparse tuning penalties are used in the ReMFPCA method, this represents the conditional generalized cross-validation scores, which means it is computed based on the optimal sparse tuning parameter selected via cross validation.

CVs

= Cross validations scores of sparse penalties parameters

mean_mfd

A multivariate functional data object giving the mean function

Methods

Public methods

Method new()

Initialize the 'remfpca' class.

Usage
remfpca$new(
  mvmfd_obj,
  method = "power",
  ncomp,
  smooth_tuning = NULL,
  sparse_tuning = NULL,
  centerfns = TRUE,
  alpha_orth = FALSE,
  smoothing_type = "coefpen",
  sparse_type = "soft",
  K_fold = 30,
  sparse_CV,
  smooth_GCV
)
Arguments
mvmfd_obj

An 'mvmfd' object representing the multivariate functional data.

method

A character string specifying the approach to be used for MFPCA computation. Options are "power" (the default) or "eigen".

ncomp

The number of functional principal components to retain.

smooth_tuning

A list or vector specifying the smoothing regularization parameter(s).

sparse_tuning

A list or vector specifying the sparsity regularization parameter(s).

centerfns

Logical. Whether to center the functional data before analysis.

alpha_orth

Logical. Whether to perform orthogonalization of the regularization parameters.

smoothing_type

The type of smoothing penalty to be applied.

sparse_type

The type of sparse penalty to be applied.

K_fold

An integer specifying the number of folds in cross-validation.

sparse_CV

Logical. Whether cross-validation should be applied for sparse tuning.

smooth_GCV

Logical. Whether generalized cross-validation should be applied for smoothing tuning.

Method clone()

The objects of this class are cloneable with this method.

Usage
remfpca$clone(deep = FALSE)
Arguments
deep

Whether to make a deep clone.

See Also

mvmfd

Examples

require(fda)
# Brownian Bridge simulation on [0,1]
M <- 110 # number of components
N <- 20 # number of instances
n <- 100 # number of grides
t0 <- seq(0, 1, len = n)
j <- 1:M
alpha1 <- list(a1 = 2^seq(0, 1, length.out = 3), a2 = 2^seq(0, 1, length.out = 3))
sparse_tuning = as.integer(seq(1, N-1, length.out = 10))
psi_1 <- function(t, m) sin(m * pi * t) # eigenfunction of BB
psi_2 <- function(t, m) sin((2 * m - 1) * pi / 2 * t) # eigenfunction of BM
PC_1 <- outer(t0, j, FUN = psi_1) # n by M matrix
PC_2 <- outer(t0, j, FUN = psi_2) # n by M matrix
Z <- matrix(rnorm(N * M), nr = M)
lambda <- matrix(2 / (pi * (2 * j - 1)), nr = M, nc = N)
X_1t <- PC_1 %*% (lambda * Z)
X_2t <- PC_2 %*% (lambda * Z)
noise <- rnorm(n * N, 0, 0.1)
X_1 <- X_1t + noise
X_2 <- X_2t + noise
bs <- create.bspline.basis(c(0, 1), 51)
mdbs <- Basismfd(bs)
mfd1 <- Mfd(X = X_1, mdbs = mdbs)
mfd2 <- Mfd(X = X_2, mdbs = mdbs)
mvmfd_obj <- Mvmfd(mfd1, mfd2)
k <- 2
# Non Regularized MFPCA based on sequential power algorithm
Re0 <- Remfpca(mvmfd_obj, ncomp = k, smooth_GCV = FALSE, sparse_CV = FALSE)
fpc0 <- Re0$pc_mfd
scores0 <- inprod_mvmfd(mvmfd_obj, fpc0)
dim(scores0)
# Smooth MFPCA based on sequential power algorithm
Re1 <- Remfpca(mvmfd_obj, ncomp = k, smooth_tuning = alpha1)
# Smooth and sparse MFPCA based on sequential power algorithm
Re2 <- Remfpca(mvmfd_obj, ncomp = k, smooth_tuning = alpha1, sparse_tuning = sparse_tuning)
# Smooth MFPCA based on joint power algorithm
Re3 <- Remfpca(mvmfd_obj, ncomp = k, smooth_tuning = alpha1, alpha_orth = TRUE)
# Smooth MFPCA based on eigen decomposition algorithm
Re4 <- Remfpca(mvmfd_obj, ncomp = k, smooth_tuning = alpha1, method = "eigen")

Standard deviation of an object of class 'mfd'.

Description

Standard deviation an object of class 'mfd'.

Usage

sd(x, ...)

Arguments

x

An object of class 'mfd'
...

all 'sd' function arguments.

Value

An object of class 'mfd'