Type:	Package
Title:	Robust Analysis of High Dimensional Data
Version:	1.4.3
Description:	A collection of methods for the robust analysis of univariate and multivariate functional data, possibly in high-dimensional cases, and hence with attention to computational efficiency and simplicity of use. See the R Journal publication of Ieva et al. (2019) <doi:10.32614/RJ-2019-032> for an in-depth presentation of the 'roahd' package. See Aleman-Gomez et al. (2021) <doi:10.48550/arXiv.2103.08874> for details about the concept of depthgram.
Depends:	R (≥ 2.10)
License:	GPL-3
Encoding:	UTF-8
LazyData:	yes
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown, withr, vdiffr
Imports:	scales, robustbase, magrittr, dplyr, ggplot2, plotly
VignetteBuilder:	knitr
RoxygenNote:	7.1.2
URL:	https://astamm.github.io/roahd/, https://github.com/astamm/roahd
BugReports:	https://github.com/astamm/roahd/issues
Language:	en-US
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2021-11-03 16:59:47 UTC; stamm-a
Author:	Nicholas Tarabelloni [aut], Ana Arribas-Gil [aut], Francesca Ieva [aut], Anna Maria Paganoni [aut], Juan Romo [aut], Francesco Palma [ctb], Aymeric Stamm [ctb, cre], Antonio Elias-Fernandez [ctb]
Maintainer:	Aymeric Stamm <aymeric.stamm@math.cnrs.fr>
Repository:	CRAN
Date/Publication:	2021-11-04 00:10:02 UTC

roahd: RObust Analysis for High dimensional Data.

Description

A package meant to collect and provide methods for the analysis of univariate and multivariate functional datasets through the use of robust methods, with special focus on computation of depths and outlier detection. Special care was devoted to the efficient implementation of robust methods, so that they can be employed also in high-dimensional datasets,

Author(s)

Maintainer: Aymeric Stamm aymeric.stamm@math.cnrs.fr (ORCID) [contributor]

Authors:

• Nicholas Tarabelloni nicholas.tarabelloni@polimi.it

• Ana Arribas-Gil aarribas@est-econ.uc3m.es

• Francesca Ieva francesca.ieva@polimi.it

• Anna Maria Paganoni anna.paganoni@polimi.it

• Juan Romo romo@est-econ.uc3m.es

Other contributors:

• Francesco Palma frapalma7892@gmail.com [contributor]

• Antonio Elias-Fernandez [contributor]

See Also

Useful links:

• https://astamm.github.io/roahd/

• https://github.com/astamm/roahd

• Report bugs at https://github.com/astamm/roahd/issues

Pipe operator

Description

See magrittr::%>% for details.

Usage

lhs %>% rhs

Arguments

Value

The result of calling rhs(lhs).

Bootstrap Confidence Interval on Spearman's Correlation Coefficient between Univariate Functional Datasets

Description

This function computes the bootstrap confidence interval of coverage probability 1 - \alpha for the Spearman correlation coefficient between two univariate functional samples.

Usage

BCIntervalSpearman(
  fD1,
  fD2,
  ordering = "MEI",
  bootstrap_iterations = 1000,
  alpha = 0.05,
  verbose = FALSE
)

Arguments

Details

The function takes two samples of compatible functional data (i.e., they must be defined over the same grid and have same number of observations) and computes a bootstrap confidence interval for their Spearman correlation coefficient.

Value

The function returns a list of two elements, lower and upper, representing the lower and upper end of the bootstrap confidence interval.

See Also

cor_spearman, cor_spearman_accuracy, fData, mfData, BCIntervalSpearmanMultivariate

Examples

set.seed(1)

N <- 200
P <- 100

grid <- seq(0, 1, length.out = P)

# Creating an exponential covariance function to simulate Gaussian data
Cov <- exp_cov_function(grid, alpha = 0.3, beta = 0.4)

# Simulating (independent) Gaussian functional data with given center and covariance function
Data_1 <- generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = Cov )
Data_2 <- generate_gauss_fdata(
  N = N,
  centerline = sin(2 * pi * grid),
  Cov = Cov
)

# Using the simulated data as (independent) components of a bivariate functional dataset
mfD <- mfData(grid, list(Data_1, Data_2))


BCIntervalSpearman(mfD$fDList[[1]], mfD$fDList[[2]], ordering = "MEI")
BCIntervalSpearman(mfD$fDList[[1]], mfD$fDList[[2]], ordering = "MHI")


# BC intervals contain zero since the functional samples are uncorrelated.

Bootstrap Confidence Interval on Spearman's Correlation Coefficient of a Multivariate Functional Dataset

Description

This function computes the bootstrap confidence intervals of coverage probability 1 - \alpha for the Spearman correlation coefficients within a multivariate functional dataset.

Usage

BCIntervalSpearmanMultivariate(
  mfD,
  ordering = "MEI",
  bootstrap_iterations = 1000,
  alpha = 0.05,
  verbose = FALSE
)

Arguments

Details

The function takes a multivariate functional dataset and computes a matrix of bootstrap confidence intervals for its Spearman correlation coefficients.

Value

The function returns a list of two elements, lower and upper, representing the matrices of lower and upper ends of the bootstrap confidence intervals for each pair of components. The elements on the main diagonal are set to 1.

See Also

cor_spearman, cor_spearman_accuracy, fData, mfData, BCIntervalSpearman

Examples

set.seed(1)

N <- 200
P <- 100
grid <- seq(0, 1, length.out = P)

# Creating an exponential covariance function to simulate Gaussian data
Cov <- exp_cov_function(grid, alpha = 0.3, beta = 0.4)

# Simulating (independent) Gaussian functional data with given center and covariance function
Data_1 <- generate_gauss_fdata(
  N = N,
  centerline = sin(2 * pi * grid),
  Cov = Cov
)
Data_2 <- generate_gauss_fdata(
  N = N,
  centerline = sin(4 * pi * grid),
  Cov = Cov
)
Data_3 <- generate_gauss_fdata(
  N = N,
  centerline = sin(6 * pi * grid),
  Cov = Cov
)

# Using the simulated data as (independent) components of a multivariate functional dataset
mfD <- mfData(grid, list(Data_1, Data_2, Data_3))


BCIntervalSpearmanMultivariate(mfD, ordering = "MEI")


# BC intervals contain zero since the functional samples are uncorrelated.

Band Depth for univariate functional data

Description

This function computes the Band Depth (BD) of elements of a functional dataset.

Usage

BD(Data)

## S3 method for class 'fData'
BD(Data)

## Default S3 method:
BD(Data)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t), this function computes the sample BD of each element with respect to the other elements of the dataset, i.e.:

BD( X( t ) ) = {N \choose 2 }^{-1} \sum_{1 \leq i_1 < i_2 \leq N} I( G(X) \subset B( X_{i_1}, X_{i_2} ) ),

where G(X) is the graphic of X(t), B(X_{i_1},X_{i_2}) is the envelope of X_{i_1}(t) and X_{i_2}(t), and X \in \left\{X_1, \ldots, X_N\right\}.

See the References section for more details.

Value

The function returns a vector containing the values of BD for the given dataset.

References

Lopez-Pintado, S. and Romo, J. (2009). On the Concept of Depth for Functional Data, Journal of the American Statistical Association, 104, 718-734.

Lopez-Pintado, S. and Romo. J. (2007). Depth-based inference for functional data, Computational Statistics & Data Analysis 51, 4957-4968.

See Also

MBD, BD_relative, MBD_relative, fData

Examples

grid = seq( 0, 1, length.out = 1e2 )


D = matrix( c( 1 + sin( 2 * pi * grid ),
               0 + sin( 4 * pi * grid ),
               1 - sin( pi * ( grid - 0.2 ) ),
               0.1 + cos( 2 * pi * grid ),
               0.5 + sin( 3 * pi + grid ),
               -2 + sin( pi * grid ) ),
            nrow = 6, ncol = length( grid ), byrow = TRUE )

fD = fData( grid, D )

BD( fD )

BD( D )

Relative Band Depth of functions in a univariate functional dataset

Description

This function computes Band Depth (BD) of elements of a univariate functional dataset with respect to another univariate functional dataset.

Usage

BD_relative(Data_target, Data_reference)

## S3 method for class 'fData'
BD_relative(Data_target, Data_reference)

## Default S3 method:
BD_relative(Data_target, Data_reference)

Arguments

Details

Given a univariate functional dataset of elements X_1(t), X_2(t), \ldots, X_N(t), and another univariate functional dataset of elements Y_1(t), Y_2(t) \ldots, Y_M(t), this function computes the BD of elements of the former with respect to elements of the latter, i.e.:

BD( X_i( t ) ) = {M \choose 2 }^{-1} \sum_{1 \leq i_1 < i_2 \leq M} I( G(X_i) \subset B( Y_{i_1}, Y_{i_2} ) ), \quad \forall i = 1, \ldots, N,

where G(X_i) is the graphic of X_i(t) and B(Y_{i_1},Y_{i_2}) is the envelope of Y_{i_1}(t) and Y_{i_2}(t).

Value

The function returns a vector containing the BD of elements in Data_target with respect to elements in Data_reference.

See Also

BD, MBD, MBD_relative, fData

Examples

grid = seq( 0, 1, length.out = 1e2 )

Data_ref = matrix( c( 0  + sin( 2 * pi * grid ),
                      1  + sin( 2 * pi * grid ),
                      -1 + sin( 2 * pi * grid ) ),
                   nrow = 3, ncol = length( grid ), byrow = TRUE )

Data_test_1 = matrix( c( 0.6 + sin( 2 * pi * grid ) ),
                      nrow = 1, ncol = length( grid ), byrow = TRUE )

Data_test_2 = matrix( c( 0.6 + sin( 2 * pi * grid ) ),
                      nrow = length( grid ), ncol = 1, byrow = TRUE )

Data_test_3 = 0.6 + sin( 2 * pi * grid )

Data_test_4 = array( 0.6 + sin( 2 * pi * grid ), dim = length( grid ) )

Data_test_5 = array( 0.6 + sin( 2 * pi * grid ), dim = c( 1, length( grid ) ) )

Data_test_6 = array( 0.6 + sin( 2 * pi * grid ), dim = c( length( grid ), 1 ) )

Data_test_7 = matrix( c( 0.5  + sin( 2 * pi * grid ),
                         -0.5 + sin( 2 * pi * grid ),
                         1.1 + sin( 2 * pi * grid ) ),
                      nrow = 3, ncol = length( grid ), byrow = TRUE )

fD_ref = fData( grid, Data_ref )
fD_test_1 = fData( grid, Data_test_1 )
fD_test_2 = fData( grid, Data_test_2 )
fD_test_3 = fData( grid, Data_test_3 )
fD_test_4 = fData( grid, Data_test_4 )
fD_test_5 = fData( grid, Data_test_5 )
fD_test_6 = fData( grid, Data_test_6 )
fD_test_7 = fData( grid, Data_test_7 )

BD_relative( fD_test_1, fD_ref )
BD_relative( Data_test_1, Data_ref )

BD_relative( fD_test_2, fD_ref )
BD_relative( Data_test_2, Data_ref )

BD_relative( fD_test_3, fD_ref )
BD_relative( Data_test_3, Data_ref )

BD_relative( fD_test_4, fD_ref )
BD_relative( Data_test_4, Data_ref )

BD_relative( fD_test_5, fD_ref )
BD_relative( Data_test_5, Data_ref )

BD_relative( fD_test_6, fD_ref )
BD_relative( Data_test_6, Data_ref )

BD_relative( fD_test_7, fD_ref )
BD_relative( Data_test_7, Data_ref )

Bootstrap Hypothesis Test on Spearman Correlation Coefficients for Multivariate Functional Data

Description

This function performs a bootstrap test that checks whether the Spearman correlation structures (e.g. matrices) of two populations of compatible multivariate functional data are equal or not.

Usage

BTestSpearman(
  mfD1,
  mfD2,
  bootstrap_iterations = 1000,
  ordering = "MEI",
  normtype = "f",
  verbose = FALSE
)

Arguments

Details

Given a first multivariate functional population, X_1^(i), \ldots, X_n^(i) with i=1, \ldots, L, defined on the grid I = t_1, \ldots, t_P, and a second multivariate functional population, Y_1^(i), \ldots, Y_m^(i) with i=1, \ldots, L, defined on the same grid I, the function performs a bootstrap test to check the hypothesis:

H_0: R_X = R_Y

H_1: R_X \neq R_Y,

where R_X and R_Y denote the L x L matrices of Spearman correlation coefficients of the two populations.

• The two functional samples must have the same number of components and must be defined over the same discrete interval t_1, \ldots, t_P.

• The test is performed through a bootstrap argument, so a number of bootstrap iterations must be specified as well. A high value for this parameter may result in slow performances of the test (you may consider setting verbose to TRUE to get hints on the process).

Value

The function returns the estimates of the test's p-value and statistics.

See Also

BCIntervalSpearman, BCIntervalSpearmanMultivariate, mfData

Examples

set.seed(1)

N  <- 200
P <- 100
L <- 2

grid <- seq(0, 1, length.out = P)

# Creating an exponential covariance function to simulate Gaussian data
Cov <- exp_cov_function(grid, alpha = 0.3, beta = 0.4)

# Simulating two populations of bivariate functional data
#
# The first population has very high correlation between first and second component
centerline_1 <- matrix(
  data = rep(sin(2 * pi * grid)),
  nrow = L,
  ncol = P,
  byrow = TRUE
)
values1 <- generate_gauss_mfdata(
  N = N,
  L = L,
  correlations = 0.9,
  centerline = centerline_1,
  listCov = list(Cov, Cov)
)
mfD1 <- mfData(grid, values1)

# Pointwise estimate
cor_spearman(mfD1)

# The second population has zero correlation between first and second component
centerline_2 <- matrix(
  data = rep(cos(2 * pi * grid)),
  nrow = L,
  ncol = P,
  byrow = TRUE
)
values2 <- generate_gauss_mfdata(
  N = N,
  L = L,
  correlations = 0,
  centerline = centerline_2,
  listCov = list(Cov, Cov)
)
mfD2 <- mfData(grid, values2)

# Pointwise estimate
cor_spearman(mfD2)

# Applying the test

BTestSpearman(mfD1, mfD2)

Epigraph Index of univariate functional dataset

Description

This function computes the Epigraphic Index (EI) of elements of a univariate functional dataset.

Usage

EI(Data)

## S3 method for class 'fData'
EI(Data)

## Default S3 method:
EI(Data)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t), defined over a compact interval I=[a,b], this function computes the EI, i.e.:

EI( X(t) ) = \frac{1}{N} \sum_{i=1}^N I( G( X_i(t) ) \subset epi( X(t) ) ) = \frac{1}{N} \sum_{i=1}^N I( X_i(t) \geq X(t), \ \ \forall t \in I),

where G(X_i(t)) indicates the graph of X_i(t), epi( X(t)) indicates the epigraph of X(t).

Value

The function returns a vector containing the values of EI for each element of the functional dataset provided in Data.

References

Lopez-Pintado, S. and Romo, J. (2012). A half-region depth for functional data, Computational Statistics and Data Analysis, 55, 1679-1695.

Arribas-Gil, A., and Romo, J. (2014). Shape outlier detection and visualization for functional data: the outliergram, Biostatistics, 15(4), 603-619.

See Also

MEI, HI, MHI, fData

Examples

N = 20
P = 1e2

grid = seq( 0, 1, length.out = P )

C = exp_cov_function( grid, alpha = 0.2, beta = 0.3 )

Data = generate_gauss_fdata( N,
                             centerline = sin( 2 * pi * grid ),
                             C )
fD = fData( grid, Data )

EI( fD )

EI( Data )

Hypograph Index of univariate functional dataset

Description

This function computes the Hypograph Index (HI) of elements of a univariate functional dataset.

Usage

HI(Data)

## S3 method for class 'fData'
HI(Data)

## Default S3 method:
HI(Data)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t), defined over a compact interval I=[a,b], this function computes the HI, i.e.:

HI( X(t) ) = \frac{1}{N} \sum_{i=1}^N I( G( X_i(t) ) \subset hyp( X(t) ) ) = \frac{1}{N} \sum_{i=1}^N I( X_i(t) \leq X(t), \ \ \forall t \in I),

where G(X_i(t)) indicates the graph of X_i(t), hyp( X(t)) indicates the hypograph of X_i(t).

Value

The function returns a vector containing the values of HI for each element of the functional dataset provided in Data.

References

Lopez-Pintado, S. and Romo, J. (2012). A half-region depth for functional data, Computational Statistics and Data Analysis, 55, 1679-1695.

Arribas-Gil, A., and Romo, J. (2014). Shape outlier detection and visualization for functional data: the outliergram, Biostatistics, 15(4), 603-619.

See Also

MHI, EI, MEI, fData

Examples

N = 20
P = 1e2

grid = seq( 0, 1, length.out = P )

C = exp_cov_function( grid, alpha = 0.2, beta = 0.3 )

Data = generate_gauss_fdata( N,
                             centerline = sin( 2 * pi * grid ),
                             C )
fD = fData( grid, Data )

HI( fD )

HI( Data )

Half-Region Depth for univariate functional data

Description

This function computes the Half-Region Depth (HRD) of elements of a univariate functional dataset.

Usage

HRD(Data)

## S3 method for class 'fData'
HRD(Data)

## Default S3 method:
HRD(Data)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t), defined over a compact interval I=[a,b], this function computes the HRD of its elements, i.e.:

HRD(X(t)) = \min( EI( X(t) ), HI(X(t)) ),

where EI(X(t)) indicates the Epigraph Index (EI) of X(t) with respect to the dataset, and HI(X(t)) indicates the Hypograph Index of X(t) with respect to the dataset.

Value

The function returns a vector containing the values of HRD for each element of the functional dataset provided in Data.

References

Lopez-Pintado, S. and Romo, J. (2012). A half-region depth for functional data, Computational Statistics and Data Analysis, 55, 1679-1695.

Arribas-Gil, A., and Romo, J. (2014). Shape outlier detection and visualization for functional data: the outliergram, Biostatistics, 15(4), 603-619.

See Also

MHRD, EI, HI, fData

Examples

N = 20
P = 1e2

grid = seq( 0, 1, length.out = P )

C = exp_cov_function( grid, alpha = 0.2, beta = 0.3 )

Data = generate_gauss_fdata( N,
                             centerline = sin( 2 * pi * grid ),
                             C )

fD = fData( grid, Data )

HRD( fD )

HRD( Data )

Modified Band Depth for univariate functional data

Description

This function computes the Modified Band Depth (MBD) of elements of a functional dataset.

Usage

MBD(Data, manage_ties = FALSE)

## S3 method for class 'fData'
MBD(Data, manage_ties = FALSE)

## Default S3 method:
MBD(Data, manage_ties = FALSE)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t), defined over a compact interval I= [a,b], this function computes the sample MBD of each element with respect to the other elements of the dataset, i.e.:

MBD( X( t ) ) = {N \choose 2 }^{-1} \sum_{1 \leq i_1 < i_2 \leq N} \tilde{\lambda}\big( {t : \min( X_{i_1}(t), X_{i_2}(t) ) \leq X(t) \leq \max( X_{i_1}(t), X_{i_2}(t) ) } \big),

where \tilde{\lambda}(\cdot) is the normalized Lebesgue measure over I=[a,b], that is \tilde{\lambda(A)} = \lambda( A ) / ( b - a ).

See the References section for more details.

Value

The function returns a vector containing the values of MBD for the given dataset.

References

Lopez-Pintado, S. and Romo, J. (2009). On the Concept of Depth for Functional Data, Journal of the American Statistical Association, 104, 718-734.

Lopez-Pintado, S. and Romo. J. (2007). Depth-based inference for functional data, Computational Statistics & Data Analysis 51, 4957-4968.

See Also

BD, MBD_relative, BD_relative, fData

Examples

grid = seq( 0, 1, length.out = 1e2 )


D = matrix( c( 1 + sin( 2 * pi * grid ),
               0 + sin( 4 * pi * grid ),
               1 - sin( pi * ( grid - 0.2 ) ),
               0.1 + cos( 2 * pi * grid ),
               0.5 + sin( 3 * pi + grid ),
               -2 + sin( pi * grid ) ),
            nrow = 6, ncol = length( grid ), byrow = TRUE )

fD = fData( grid, D )

MBD( fD )

MBD( D )

Relative Modified Band Depth of functions in a univariate functional dataset

Description

This function computes Modified Band Depth (BD) of elements of a univariate functional dataset with respect to another univariate functional dataset.

Usage

MBD_relative(Data_target, Data_reference)

## S3 method for class 'fData'
MBD_relative(Data_target, Data_reference)

## Default S3 method:
MBD_relative(Data_target, Data_reference)

Arguments

Details

Given a univariate functional dataset of elements X_1(t), X_2(t), \ldots, X_N(t), and another univariate functional dataset of elements Y_1(t), Y_2(t) \ldots, Y_M(t), defined over the same compact interval I=[a,b], this function computes the MBD of elements of the former with respect to elements of the latter, i.e.:

MBD( X_i( t ) ) = {M \choose 2 }^{-1} \sum_{1 \leq i_1 < i_2 \leq M} \tilde{\lambda}\big( {t : \min( Y_{i_1}(t), Y_{i_2}(t) ) \leq X_i(t) \leq \max( Y_{i_1}(t), Y_{i_2}(t) ) } \big),

\forall i = 1, \ldots, N, where \tilde{\lambda}(\cdot) is the normalized Lebesgue measure over I=[a,b], that is \tilde{\lambda(A)} = \lambda( A ) / ( b - a ).

Value

The function returns a vector containing the MBD of elements in Data_target with respect to elements in Data_reference.

See Also

MBD, BD, BD_relative, fData

Examples

grid = seq( 0, 1, length.out = 1e2 )

Data_ref = matrix( c( 0  + sin( 2 * pi * grid ),
                      1  + sin( 2 * pi * grid ),
                      -1 + sin( 2 * pi * grid ) ),
                   nrow = 3, ncol = length( grid ), byrow = TRUE )

Data_test_1 = matrix( c( 0.6 + sin( 2 * pi * grid ) ),
                      nrow = 1, ncol = length( grid ), byrow = TRUE )

Data_test_2 = matrix( c( 0.6 + sin( 2 * pi * grid ) ),
                      nrow = length( grid ), ncol = 1, byrow = TRUE )

Data_test_3 = 0.6 + sin( 2 * pi * grid )

Data_test_4 = array( 0.6 + sin( 2 * pi * grid ), dim = length( grid ) )

Data_test_5 = array( 0.6 + sin( 2 * pi * grid ), dim = c( 1, length( grid ) ) )

Data_test_6 = array( 0.6 + sin( 2 * pi * grid ), dim = c( length( grid ), 1 ) )

Data_test_7 = matrix( c( 0.5  + sin( 2 * pi * grid ),
                         -0.5 + sin( 2 * pi * grid ),
                         1.1 + sin( 2 * pi * grid ) ),
                      nrow = 3, ncol = length( grid ), byrow = TRUE )

fD_ref = fData( grid, Data_ref )
fD_test_1 = fData( grid, Data_test_1 )
fD_test_2 = fData( grid, Data_test_2 )
fD_test_3 = fData( grid, Data_test_3 )
fD_test_4 = fData( grid, Data_test_4 )
fD_test_5 = fData( grid, Data_test_5 )
fD_test_6 = fData( grid, Data_test_6 )
fD_test_7 = fData( grid, Data_test_7 )

MBD_relative( fD_test_1, fD_ref )
MBD_relative( Data_test_1, Data_ref )

MBD_relative( fD_test_2, fD_ref )
MBD_relative( Data_test_2, Data_ref )

MBD_relative( fD_test_3, fD_ref )
MBD_relative( Data_test_3, Data_ref )

MBD_relative( fD_test_4, fD_ref )
MBD_relative( Data_test_4, Data_ref )

MBD_relative( fD_test_5, fD_ref )
MBD_relative( Data_test_5, Data_ref )

MBD_relative( fD_test_6, fD_ref )
MBD_relative( Data_test_6, Data_ref )

MBD_relative( fD_test_7, fD_ref )
MBD_relative( Data_test_7, Data_ref )

Modified Epigraph Index of univariate functional dataset

Description

This function computes the Modified Epigraphic Index (MEI) of elements of a univariate functional dataset.

Usage

MEI(Data)

## S3 method for class 'fData'
MEI(Data)

## Default S3 method:
MEI(Data)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t), defined over a compact interval I=[a,b], this function computes the MEI, i.e.:

MEI( X(t) ) = \frac{1}{N} \sum_{i=1}^N \tilde{\lambda}( X(t) \leq X_i(t) ),

where \tilde{\lambda}(\cdot) is the normalized Lebesgue measure over I=[a,b], that is \tilde{\lambda(A)} = \lambda( A ) / ( b - a ).

Value

The function returns a vector containing the values of MEI for each element of the functional dataset provided in Data.

References

Lopez-Pintado, S. and Romo, J. (2012). A half-region depth for functional data, Computational Statistics and Data Analysis, 55, 1679-1695.

Arribas-Gil, A., and Romo, J. (2014). Shape outlier detection and visualization for functional data: the outliergram, Biostatistics, 15(4), 603-619.

See Also

EI, MHI, HI, fData

Examples

N = 20
P = 1e2

grid = seq( 0, 1, length.out = P )

C = exp_cov_function( grid, alpha = 0.2, beta = 0.3 )

Data = generate_gauss_fdata( N,
                             centerline = sin( 2 * pi * grid ),
                             C )

fD = fData( grid, Data )

MEI( fD )

MEI( Data )

Modified Hypograph Index of univariate functional dataset

Description

This function computes the Modified Hypograph Index (MEI) of elements of a univariate functional dataset.

Usage

MHI(Data)

## S3 method for class 'fData'
MHI(Data)

## Default S3 method:
MHI(Data)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t), defined over a compact interval I=[a,b], this function computes the MHI, i.e.:

MHI( X(t) ) = \frac{1}{N} \sum_{i=1}^N \tilde{\lambda}( X(t) \geq X_i(t) ),

where \tilde{\lambda}(\cdot) is the normalized Lebesgue measure over I=[a,b], that is \tilde{\lambda(A)} = \lambda( A ) / ( b - a ).

Value

The function returns a vector containing the values of MHI for each element of the functional dataset provided in Data.

References

Lopez-Pintado, S. and Romo, J. (2012). A half-region depth for functional data, Computational Statistics and Data Analysis, 55, 1679-1695.

Arribas-Gil, A., and Romo, J. (2014). Shape outlier detection and visualization for functional data: the outliergram, Biostatistics, 15(4), 603-619.

See Also

HI, MEI, EI, fData

Examples

N = 20
P = 1e2

grid = seq( 0, 1, length.out = P )

C = exp_cov_function( grid, alpha = 0.2, beta = 0.3 )

Data = generate_gauss_fdata( N,
                             centerline = sin( 2 * pi * grid ),
                             C )
fD = fData( grid, Data )

MHI( fD )

MHI( Data )

Modified Half-Region Depth for univariate functional data

Description

This function computes the Modified Half-Region Depth (MHRD) of elements of a univariate functional dataset.

Usage

MHRD(Data)

## S3 method for class 'fData'
MHRD(Data)

## Default S3 method:
MHRD(Data)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t), defined over a compact interval I=[a,b], this function computes the MHRD of its elements, i.e.:

MHRD(X(t)) = \min( MEI( X(t) ), MHI(X(t)) ),

where MEI(X(t)) indicates the Modified Epigraph Index (MEI) of X(t) with respect to the dataset, and MHI(X(t)) indicates the Modified Hypograph Index of X(t) with respect to the dataset.

Value

The function returns a vector containing the values of MHRD for each element of the functional dataset provided in Data.

References

Lopez-Pintado, S. and Romo, J. (2012). A half-region depth for functional data, Computational Statistics and Data Analysis, 55, 1679-1695.

Arribas-Gil, A., and Romo, J. (2014). Shape outlier detection and visualization for functional data: the outliergram, Biostatistics, 15(4), 603-619.

See Also

HRD, MEI, MHI

Examples

N = 20
P = 1e2

grid = seq( 0, 1, length.out = P )

C = exp_cov_function( grid, alpha = 0.2, beta = 0.3 )

Data = generate_gauss_fdata( N,
                             centerline = sin( 2 * pi * grid ),
                             C )
fD = fData( grid, Data )

MHRD( fD )

MHRD( Data )

Append two compatible univariate functional datasets

Description

This is a convenience function that simplifies the task of appending univariate functional observations of two datasets to a unique univariate functional dataset.

Usage

append_fData(fD1, fD2)

Arguments

Details

The two original datasets must be compatible, i.e. must be defined on the same grid. If we denote with X_1, \ldots, X_n the first dataset, defined over the grid I = t_1, \ldots, t_P, and with Y_1, \ldots, Y_m the second functional dataset, defined on the same grid, the method returns the union dataset obtained by taking all the n + m observations together.

Value

The function returns an fData object containing the union of fD1 and fD2

See Also

append_mfData, fData

Examples

# Creating two simple univariate datasets

grid = seq(0, 2 * pi, length.out = 100)

values1 = matrix( c(sin(grid),
                    sin(2 * grid)), nrow = 2, ncol = length(grid),
                   byrow=TRUE)

values2 = matrix( c(cos(grid),
                    cos(2 * grid)), nrow = 2, ncol = length(grid),
                   byrow=TRUE)

fD1 = fData( grid, values1 )
fD2 = fData( grid, values2 )

# Appending them to a unique dataset
append_fData(fD1, fD2)

Append two compatible multivariate functional datasets

Description

This is a convenience function that simplifies the task of appending multivariate functional observations of two datasets to a unique multivariate functional dataset.

Usage

append_mfData(mfD1, mfD2)

Arguments

Details

The two original datasets must be compatible, i.e. must have same number of components (dimensions) and must be defined on the same grid. If we denote with X_1^(i), \ldots, X_n^(i), i=0, \ldots, L the first dataset, defined over the grid I = t_1, \ldots, t_P, and with Y_1^(i), \ldots, Y_m^(i), i=0, \ldots, L the second functional dataset, the method returns the union dataset obtained by taking all the n + m observations together.

Value

The function returns a mfData object containing the union of mfD1 and mfD2

See Also

append_fData, mfData

Examples

# Creating two simple bivariate datasets

grid = seq(0, 2 * pi, length.out = 100)

values11 = matrix( c(sin(grid),
                     sin(2 * grid)), nrow = 2, ncol = length(grid),
                   byrow=TRUE)
values12 = matrix( c(sin(3 * grid),
                     sin(4 * grid)), nrow = 2, ncol = length(grid),
                   byrow=TRUE)
values21 = matrix( c(cos(grid),
                     cos(2 * grid)), nrow = 2, ncol = length(grid),
                   byrow=TRUE)
values22 = matrix( c(cos(3 * grid),
                     cos(4 * grid)), nrow = 2, ncol = length(grid),
                   byrow=TRUE)

mfD1 = mfData( grid, list(values11, values12) )
mfD2 = mfData( grid, list(values21, values22) )

# Appending them to a unique dataset
append_mfData(mfD1, mfD2)

Area-under-curve order relation between univariate functional data

Description

This function implements an order relation between univariate functional data based on the area-under-curve relation, that is to say a pre-order relation obtained by comparing the area-under-curve of two different functional data.

Usage

area_ordered(fData, gData)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t) and another functional dataset Y_1(t), Y_2(t), \ldots, Y_M(t) defined over the same compact interval I=[a,b], the function computes the area-under-curve (namely, the integral) in both the datasets, and checks whether the first ones are lower or equal than the second ones.

By default the function tries to compare each X_i(t) with the corresponding Y_i(t), thus assuming N=M, but when either N=1 or M=1, the comparison is carried out cycling over the dataset with fewer elements. In all the other cases (N\neq M, and either N \neq 1 or M \neq 1) the function stops.

Value

The function returns a logical vector of length \max(N,M) containing the value of the predicate for all the corresponding elements.

References

Valencia, D., Romo, J. and Lillo, R. (2015). A Kendall correlation coefficient for functional dependence, Universidad Carlos III de Madrid technical report, http://EconPapers.repec.org/RePEc:cte:wsrepe:ws133228.

See Also

area_under_curve, fData

Examples

P = 1e3

grid = seq( 0, 1, length.out = P )

Data_1 = matrix( c( 1 * grid,
                    2 *  grid ),
                 nrow = 2, ncol = P, byrow = TRUE )

Data_2 = matrix( 3 * ( 0.5 - abs( grid - 0.5 ) ),
                 nrow = 1, byrow = TRUE )

Data_3 = rbind( Data_1, Data_1 )


fD_1 = fData( grid, Data_1 )
fD_2 = fData( grid, Data_2 )
fD_3 = fData( grid, Data_3 )

area_ordered( fD_1, fD_2 )

area_ordered( fD_2, fD_3 )

Area under curve of elements of univariate functional data

Description

This method computes the (signed) area under the curve of elements of a univariate functional dataset, namely, their integral.

Usage

area_under_curve(fData)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t), defined over a compact interval I=[a,b] and observed on an evenly spaced 1D grid [a = t_0, t_1, \ldots, t_{P-1} = b] \subset I, the function computes:

\sum_{i=1}^{P-2} \frac{X(t_{i+1}) - X(t_{i-1})}{2} h \approx \int_a^b X(t) dt,

where h = t_1 - t_0.

Value

The function returns a numeric vector containing the values of areas under the curve for all the elements of the functional dataset fData.

See Also

area_ordered, fData

Examples

P = 1e3
grid = seq( 0, 1, length.out = P )

fD = fData( grid,
            matrix( c( sin( 2 * pi * grid ),
                       cos( 2 * pi * grid ),
                       4 * grid * ( 1 - grid ) ),
                    nrow = 3, ncol = P, byrow = TRUE ) )
plot( fD )

area_under_curve( fD )

Converting object to mfData class

Description

This S3 method provides a way to convert some objects to the class mfData, thus obtaining a multivariate functional dataset.

Usage

as.mfData(x, ...)

## S3 method for class 'list'
as.mfData(x, ...)

Arguments

Value

The function returns a mfData object, obtained starting from argument x.

Examples

grid = seq( 0, 1, length.out = 100 )

fD_1 = fData( grid, sin( 2 * pi * grid ) )
fD_2 = fData( grid, cos( 2 * pi * grid ) )

plot( as.mfData( list( fD_1, fD_2 ) ) )

Kendall's tau correlation coefficient for bivariate functional data

Description

This function computes the Kendall's tau correlation coefficient for a bivariate functional dataset, with either a max or area-under-curve order order relation between univariate functional elements (components).

Usage

cor_kendall(mfD, ordering = "max")

Arguments

Details

Given a bivariate functional dataset, with first components X_1(t), X_2(t), \ldots, X_N(t) and second components Y_1(t), Y_2(t), \ldots, Y_N(t), the function exploits either the order relation based on the maxima or the area-under-curve relation to compare data and produce concordances and discordances, that are then used to compute the tau coefficient.

See the references for more details.

Value

The function returns the Kendall's tau correlation coefficient for the bivariate dataset provided with mfData.

References

Valencia, D., Romo, J. and Lillo, R. (2015). A Kendall correlation coefficient for functional dependence, Universidad Carlos III de Madrid technical report, http://EconPapers.repec.org/RePEc:cte:wsrepe:ws133228.

See Also

mfData, area_ordered, max_ordered

Examples

#### TOTALLY INDEPENDENT COMPONENTS
N = 2e2
P = 1e3

grid = seq( 0, 1, length.out = P )

# Creating an exponential covariance function to simulate guassian data
Cov = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )

# Simulating (independent) gaussian functional data with given center and
# covariance function
Data_1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = Cov )
Data_2 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = Cov )

# Using the simulated data as (independent) components of a bivariate functional
# dataset
mfD = mfData( grid, list( Data_1, Data_2 ) )

# Correlation approx. zero (components were created independently)
cor_kendall( mfD, ordering = 'max' )

# Correlation approx. zero (components were created independently)
cor_kendall( mfD, ordering = 'area' )

#### TOTALLY DEPENDENT COMPONENTS

# Nonlinear transform of first component
Data_3 = t( apply( Data_1, 1, exp ) )

# Creating bivariate dataset starting from nonlinearly-dependent components
mfD = mfData( grid, list( Data_1, Data_3 ) )

# Correlation very high (components are nonlinearly dependent)
cor_kendall( mfD, ordering = 'max' )

# Correlation very high (components are nonlinearly dependent)
cor_kendall( mfD, ordering = 'area' )

Spearman's correlation coefficient for multivariate functional data

Description

This function computes the Spearman's correlation coefficient for a multivariate functional dataset, with either a Modified Epigraph Index (MEI) or Modified Hypograph Index (MHI) ranking of univariate elements of data components.

Usage

cor_spearman(mfD, ordering = "MEI")

Arguments

Details

Given a multivariate functional dataset, with first components X^1_1(t), X^1_2(t), \ldots, X^1_N(t), second components X^2_1(t), X^2_2(t), \ldots, X^2_N(t), etc., the function exploits either the MEI or MHI to compute the matrix of Spearman's correlation coefficients. Such matrix is symmetrical and has ones on the diagonal. The entry (i, j) represents the Spearman correlation coefficient between curves of component i and j.

See the references for more details.

Value

If the original dataset is bivariate, the function returns only the scalar value of the correlation coefficient for the two components. When the number of components is L >2, it returns the whole matrix of Spearman's correlation coefficients for all the components.

References

Valencia, D., Romo, J. and Lillo, R. (2015). Spearman coefficient for functions, Universidad Carlos III de Madrid technical report, http://EconPapers.repec.org/RePEc:cte:wsrepe:ws133329.

See Also

mfData, MEI, MHI

Examples

#### TOTALLY INDEPENDENT COMPONENTS

N = 2e2
P = 1e3

grid = seq( 0, 1, length.out = P )

# Creating an exponential covariance function to simulate guassian data
Cov = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )

# Simulating (independent) gaussian functional data with given center and
# covariance function
Data_1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = Cov )
Data_2 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = Cov )

# Using the simulated data as (independent) components of a bivariate functional
# dataset
mfD = mfData( grid, list( Data_1, Data_2 ) )

# Correlation approx. zero (components were created independently)
cor_spearman( mfD, ordering = 'MEI' )

# Correlation approx. zero (components were created independently)
cor_spearman( mfD, ordering = 'MHI' )

#### TOTALLY DEPENDENT COMPONENTS

# Nonlinear transform of first component
Data_3 = t( apply( Data_1, 1, exp ) )

# Creating bivariate dataset starting from nonlinearly-dependent components
mfD = mfData( grid, list( Data_1, Data_3 ) )

# Correlation very high (components are nonlinearly dependent)
cor_spearman( mfD, ordering = 'MEI' )

# Correlation very high (components are nonlinearly dependent)
cor_spearman( mfD, ordering = 'MHI' )

Bootstrap Spearman's correlation coefficient for multivariate functional data

Description

This function computes the bootstrap estimates of standard error and bias of the Spearman's correlation coefficient for a multivariate functional dataset.

Usage

cor_spearman_accuracy(
  mfD,
  ordering = "MEI",
  bootstrap_iterations = 1000,
  verbose = FALSE
)

Arguments

Details

Given a multivariate functional dataset X_1^(i), \ldots, X_n^(i), i=0, \ldots, L defined over the grid I = t_0, \ldots, t_P, having components i=1, \ldots, L, and a chosen ordering strategy (MEI or MHI), the function computes the matrix of Spearman's correlation indices of the dataset components, as well as their bias and standard deviation estimates through a specified number of bootstrap iterations (bias and standard error are updated with on-line formulas).

Value

a list of three elements: mean, the mean of the matrix of correlation coefficients; bias, a matrix containing the estimated bias (mean - point estimate of correlation coefficients); sd, a matrix containing the estimated standard deviation of the coefficients' matrix. In case the multivariate functional dataset has only two components, the return type is scalar and not matrix.

See Also

cor_spearman, mfData

Examples

N <- 200
P <- 100

grid <- seq(0, 1, length.out = P)

# Creating an exponential covariance function to simulate Gaussian data
Cov <- exp_cov_function(grid, alpha = 0.3, beta = 0.4)

# Simulating (independent) Gaussian functional data with given center and covariance function

Data_1 <- generate_gauss_fdata(
  N = N,
  centerline = sin(2 * pi * grid),
  Cov = Cov
)

Data_2 <- generate_gauss_fdata(
  N = N,
  centerline = sin(2 * pi * grid),
  Cov = Cov
)

# Using the simulated data as (independent) components of a bivariate functional dataset
mfD <- mfData(grid, list(Data_1, Data_2))


# Computes bootstrap estimate of Spearman correlation
cor_spearman_accuracy(mfD, ordering = "MEI")
cor_spearman_accuracy(mfD, ordering = "MHI")

Covariance function for functional data

Description

S3 method to compute the sample covariance and cross-covariance functions for a set of functional data.

Usage

cov_fun(X, Y = NULL)

## S3 method for class 'fData'
cov_fun(X, Y = NULL)

## S3 method for class 'mfData'
cov_fun(X, Y = NULL)

Arguments

Details

Given a univariate random function X, defined over the grid I = [a,b], the covariance function is defined as:

C(s,t) = Cov( X(s), X(t) ), \qquad s,t \in I.

Given another random function, Y, defined over the same grid as X, the cross- covariance function of X and Y is:

C^{X,Y}( s,t ) = Cov( X(s), Y(t) ), \qquad s, t \in I.

For a generic L-dimensional random function X, i.e. an L-dimensional multivariate functional datum, the covariance function is defined as the set of blocks:

C_{i,j}(s,t) = Cov( X_i(s), X_j(t)), i,j = 1, ...,L, s,t \in I,

while the cross-covariance function is defined by the blocks:

C^{X,Y}_{i,j}(s,t) = Cov( X_i(s), Y_j(t))

The method cov_fun provides the sample estimator of the covariance or cross-covariance functions for univariate or multivariate functional datasets.

The class of X (fData or mfData) is used to dispatch the correct implementation of the method.

Value

The following cases are given:

• if X is of class fData and Y is NULL, then the covariance function of X is returned;

• if X is of class fData and Y is of class fData, the cross-covariance function of the two datasets is returned;

• if X is of class mfData and Y is NULL, the upper-triangular blocks of the covariance function of X are returned (in form of list and by row, i.e. in the sequence 1_1, 1_2, ..., 1_L, 2_2, ... - have a look at the labels of the list with str);

• if X is of class mfData and Y is of class fData, the cross-covariances of X's components and Y are returned (in form of list);

• if X is of class mfData and Y is of class mfData, the upper-triangular blocks of the cross-covariance of X's and Y's components are returned (in form of list and by row, i.e. in the sequence 1_1, 1_2, ..., 1_L, 2_2, ... - have a look at the labels of the list with str));

In any case, the return type is either an instance of the S3 class Cov or a list of instances of such class (for the case of multivariate functional data).

See Also

fData, mfData, plot.Cov

Examples

# Generating a univariate functional dataset
N = 1e2

P = 1e2
t0 = 0
t1 = 1

time_grid = seq( t0, t1, length.out = P )

Cov = exp_cov_function( time_grid, alpha = 0.3, beta = 0.4 )

D1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * time_grid ), Cov = Cov )
D2 = generate_gauss_fdata( N, centerline = sin( 2 * pi * time_grid ), Cov = Cov )

fD1 = fData( time_grid, D1 )
fD2 = fData( time_grid, D2 )

# Computing the covariance function of fD1

C = cov_fun( fD1 )
str( C )

# Computing the cross-covariance function of fD1 and fD2
CC = cov_fun( fD1, fD2 )
str( CC )

# Generating a multivariate functional dataset
L = 3

C1 = exp_cov_function( time_grid, alpha = 0.1, beta = 0.2 )
C2 = exp_cov_function( time_grid, alpha = 0.2, beta = 0.5 )
C3 = exp_cov_function( time_grid, alpha = 0.3, beta = 1 )

centerline = matrix( c( sin( 2 * pi * time_grid ),
                        sqrt( time_grid ),
                        10 * ( time_grid - 0.5 ) * time_grid ),
                     nrow = 3, byrow = TRUE )

D3 = generate_gauss_mfdata( N, L, centerline,
                       correlations = c( 0.5, 0.5, 0.5 ),
                       listCov = list( C1, C2, C3 ) )

# adding names for better readability of BC3's labels
names( D3 ) = c( 'comp1', 'comp2', 'comp3' )
mfD3 = mfData( time_grid, D3 )

D1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * time_grid ),
                              Cov = Cov )
fD1 = fData( time_grid, D1 )

# Computing the block covariance function of mfD3
BC3 = cov_fun( mfD3 )
str( BC3 )

# computing cross-covariance between mfData and fData objects
CC = cov_fun( mfD3, fD1 )
str( CC )

Depthgram for univariate and multivariate functional data sets

Description

This function computes the three 'DepthGram' representations from a p-variate functional data set.

Usage

depthgram(
  Data,
  marginal_outliers = FALSE,
  boxplot_factor = 1.5,
  outliergram_factor = 1.5,
  ids = NULL
)

## Default S3 method:
depthgram(
  Data,
  marginal_outliers = FALSE,
  boxplot_factor = 1.5,
  outliergram_factor = 1.5,
  ids = NULL
)

## S3 method for class 'fData'
depthgram(
  Data,
  marginal_outliers = FALSE,
  boxplot_factor = 1.5,
  outliergram_factor = 1.5,
  ids = NULL
)

## S3 method for class 'mfData'
depthgram(
  Data,
  marginal_outliers = FALSE,
  boxplot_factor = 1.5,
  outliergram_factor = 1.5,
  ids = NULL
)

Arguments

Value

An object of class depthgram which is a list with the following items:

• mbd.mei.d: vector MBD of the MEI dimension-wise.

• mei.mbd.d: vector MEI of the MBD dimension-wise.

• mbd.mei.t: vector MBD of the MEI time-wise.

• mei.mbd.t: vector MEI of the MEI time-wise.

• mbd.mei.t2: vector MBD of the MEI time/correlation-wise.

• mei.mbd.t2: vector MEI of the MBD time/correlation-wise.

• shp.out.det: detected shape outliers by dimension.

• mag.out.det: detected magnitude outliers by dimension.

• mbd.d: matrix ⁠n x p⁠ of MBD dimension-wise.

• mei.d: matrix ⁠n x p⁠ of MEI dimension-wise.

• mbd.t: matrix ⁠n x p⁠ of MBD time-wise.

• mei.t: matrix ⁠n x p⁠ of MEI time-wise.

• mbd.t2: matrix ⁠n x p⁠ of MBD time/correlation-wise

• mei.t2: matrix ⁠n x p⁠ of MBD time/correlation-wise.

References

Aleman-Gomez, Y., Arribas-Gil, A., Desco, M. Elias-Fernandez, A., and Romo, J. (2021). "Depthgram: Visualizing Outliers in High Dimensional Functional Data with application to Task fMRI data exploration".

Examples

N <- 2e2
P <- 1e3
grid <- seq(0, 1, length.out = P)
Cov <- exp_cov_function(grid, alpha = 0.3, beta = 0.4)

Data <- list()
Data[[1]] <- generate_gauss_fdata(
  N,
  centerline = sin(2 * pi * grid),
  Cov = Cov
)
Data[[2]] <- generate_gauss_fdata(
  N,
  centerline = sin(2 * pi * grid),
  Cov = Cov
)
names <- paste0("id_", 1:nrow(Data[[1]]))

DG1 <- depthgram(Data, marginal_outliers = TRUE, ids = names)

fD <- fData(grid, Data[[1]])
DG2 <- depthgram(fD, marginal_outliers = TRUE, ids = names)

mfD <- mfData(grid, Data)
DG3 <- depthgram(mfD, marginal_outliers = TRUE, ids = names)

Exponential covariance function over a grid

Description

This function computes the discretization of an exponential covariance function of the form:

C( s, t ) = \alpha e^{ - \beta | s - t | }

over a 1D grid [t_0, t_1, \ldots, t_{P-1}], thus obtaining the P \times P matrix of values:

C_{i,j} = C( t_i, t_j ) = \alpha e^{ - \beta | t_i - t_j | } .

Usage

exp_cov_function(grid, alpha, beta)

Arguments

See Also

generate_gauss_fdata, generate_gauss_mfdata

Examples

grid = seq( 0, 1, length.out = 5e2 )

alpha = 0.2
beta = 0.3

dev.new()
image( exp_cov_function( grid, alpha, beta ),
       main = 'Exponential covariance function',
       xlab = 'grid', ylab = 'grid')

A set of fancy color to plot functional datasets

Description

This function can be used to generate a palette of colors useful to plot functional datasets with the plot methods.

Usage

fDColorPalette(N, hue_range = c(0, 360), alpha = 0.8, ...)

Arguments

Details

The function, built around scales::hue_pal, allows to set up the HCL parameters of the set of colors desired, and besides to set up the alpha channel value.

See Also

plot.fData, plot.mfData

Examples

N = 1e2
angular_grid = seq( 0, 359, length.out = N )

dev.new()
plot( angular_grid, angular_grid,
      col = fDColorPalette( N, hue_range = c( 0, 359 ), alpha = 1 ),
      pch = 16, cex = 3 )

S3 Class for univariate functional datasets.

Description

This function implements a constructor for elements of S3 class fData, aimed at implementing a representation of a functional dataset.

Usage

fData(grid, values)

Arguments

Details

The functional dataset is represented as a collection of measurement of the observations on an evenly spaced, 1D grid of discrete points (representing, e.g. time), namely, for functional data defined over a grid [t_0, t_1, \ldots, t_{P-1}]:

f_{i,j} = f_i( t_0 + j h ), \quad h = \frac{t_P - t_0}{N}, \quad \forall j = 1, \ldots, P, \quad \forall i = 1, \ldots N.

Value

The function returns a S3 object of class fData, containing the following elements:

• "N": the number of elements in the dataset;

• "P": the number of points in the 1D grid over which elements are measured;

• "t0": the starting point of the 1D grid;

• "tP": the ending point of the 1D grid;

• "values": the matrix of measurements of the functional observations on the 1D grid provided with grid.

See Also

generate_gauss_fdata, sub-.fData

Examples

# Defining parameters
N = 20
P = 1e2

# One dimensional grid
grid = seq( 0, 1, length.out = P )

# Generating an exponential covariance function (see related help for more
# information )
C = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )

# Generating a synthetic dataset with a gaussian distribution and
# required mean and covariance function:
values = generate_gauss_fdata( N,
                               centerline = sin( 2 * pi * grid ),
                               Cov = C )

fD = fData( grid, values )

Functional boxplot of univariate and multivariate functional data

Description

This function can be used to perform the functional boxplot of univariate or multivariate functional data.

Usage

fbplot(
  Data,
  Depths = "MBD",
  Fvalue = 1.5,
  adjust = FALSE,
  display = TRUE,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  ...
)

## S3 method for class 'fData'
fbplot(
  Data,
  Depths = "MBD",
  Fvalue = 1.5,
  adjust = FALSE,
  display = TRUE,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  ...
)

## S3 method for class 'mfData'
fbplot(
  Data,
  Depths = list(def = "MBD", weights = "uniform"),
  Fvalue = 1.5,
  adjust = FALSE,
  display = TRUE,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  ...
)

Arguments

Value

Even when used in graphical way to plot the functional boxplot, the function returns a list of three elements:

• Depths: contains the depths of each element of the functional dataset.

• Fvalue: is the value of F used to obtain the outliers.

• ID_out: contains the vector of indices of dataset elements flagged as outliers (if any).

Adjustment

In the univariate functional case, when the adjustment option is selected, the value of F is optimized for the univariate functional dataset provided with Data.

In practice, a number adjust$N_trials of times a synthetic population (of size adjust$tiral_size with the same covariance (robustly estimated from data) and centerline as fData is simulated without outliers and each time an optimized value F_i is computed so that a given proportion (adjust$TPR) of observations is flagged as outliers. The final value of F for the functional boxplot is determined as an average of F_1, F_2, \dots, F_{N_{trials}}. At each time step the optimization problem is solved using stats::uniroot (Brent's method).

References

1. Sun, Y., & Genton, M. G. (2012). Functional boxplots. Journal of Computational and Graphical Statistics.

2. Sun, Y., & Genton, M. G. (2012). Adjusted functional boxplots for spatio-temporal data visualization and outlier detection. Environmetrics, 23(1), 54-64.

See Also

fData, MBD, BD, mfData, multiMBD, multiBD

Examples

# UNIVARIATE FUNCTIONAL BOXPLOT - NO ADJUSTMENT

set.seed(1)

N = 2 * 100 + 1
P = 2e2

grid = seq( 0, 1, length.out = P )

D = 10 * matrix( sin( 2 * pi * grid ), nrow = N, ncol = P, byrow = TRUE )

D = D + rexp(N, rate = 0.05)


# c( 0, 1 : (( N - 1 )/2), -( ( ( N - 1 ) / 2 ) : 1 ) )^4


fD = fData( grid, D )

dev.new()
oldpar <- par(mfrow = c(1, 1))
par(mfrow = c(1, 3))

plot( fD, lwd = 2, main = 'Functional dataset',
      xlab = 'time', ylab = 'values' )

fbplot( fD, main = 'Functional boxplot', xlab = 'time', ylab = 'values', Fvalue = 1.5 )

boxplot(fD$values[,1], ylim = range(fD$values), main = 'Boxplot of functional dataset at t_0 ' )

par(oldpar)

# UNIVARIATE FUNCTIONAL BOXPLOT - WITH ADJUSTMENT


set.seed( 161803 )

P = 2e2
grid = seq( 0, 1, length.out = P )

N = 1e2

# Generating a univariate synthetic gaussian dataset
Data = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ),
                             Cov = exp_cov_function( grid,
                                                     alpha = 0.3,
                                                     beta  = 0.4 ) )
fD = fData( grid, Data )

dev.new()

fbplot( fD, adjust = list( N_trials = 10,
                           trial_size = 5 * N,
                           VERBOSE = TRUE ),
                     xlab = 'time', ylab = 'Values',
                     main = 'My adjusted functional boxplot' )


# MULTIVARIATE FUNCTIONAL BOXPLOT - NO ADJUSTMENT

set.seed( 1618033 )

P = 1e2
N = 1e2
L = 2

grid = seq( 0, 1, length.out = 1e2 )

C1 = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )
C2 = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )

# Generating a bivariate functional dataset of gaussian data with partially
# correlated components
Data = generate_gauss_mfdata( N, L,
                              centerline = matrix( sin( 2 * pi * grid ),
                                                   nrow = 2, ncol = P,
                                                   byrow = TRUE ),
                              correlations = rep( 0.5, 1 ),
                              listCov = list( C1, C2 ) )

mfD = mfData( grid, Data )

dev.new()
fbplot( mfD, Fvalue = 2.5, xlab = 'time', ylab = list( 'Values 1',
                                                       'Values 2' ),
        main = list( 'First component', 'Second component' ) )

Generation of gaussian univariate functional data

Description

generate_gauss_fdata generates a dataset of univariate functional data with a desired mean and covariance function.

Usage

generate_gauss_fdata(N, centerline, Cov = NULL, CholCov = NULL)

Arguments

Details

In particular, the following model is considered for the generation of data:

X(t) = m( t ) + \epsilon( t ), \quad t \in I = [a, b]

where m(t) is the center and \epsilon(t) is a centered gaussian process with covariance function C_i. That is to say:

Cov( \epsilon(s), \epsilon(t) ) = C( s, t ), \quad \forall s, t \in I

All the functions are supposed to be observed on an evenly-spaced, one- dimensional grid of P points: [a = t_0, t_1, \ldots, t_{P-1} = b] \subset I .

Value

The function returns a matrix containing the discretized values of the generated observations (in form of an N \times P matrix).

See Also

exp_cov_function, fData, generate_gauss_mfdata

Examples

N = 30
P = 1e2

t0 = 0
tP = 1

time_grid = seq( t0, tP, length.out = P )

C = exp_cov_function( time_grid, alpha = 0.1, beta = 0.2 )

CholC = chol( C )

centerline = sin( 2 * pi * time_grid )

invisible(generate_gauss_fdata( N, centerline, Cov = C ))

invisible(generate_gauss_fdata( N, centerline, CholCov = CholC ))

Generation of gaussian multivariate functional data

Description

generate_gauss_mfdata generates a dataset of multivariate functional data with a desired mean and covariance function in each dimension and a desired correlation structure among components.

Usage

generate_gauss_mfdata(
  N,
  L,
  centerline,
  correlations,
  listCov = NULL,
  listCholCov = NULL
)

Arguments

Details

In particular, the following model is considered for the generation of data:

X(t) = ( m_1( t ) + \epsilon_1( t ), \ldots, m_L(t) + \epsilon_L(t)), \quad t \in I = [a, b]

where L is the number of components of the multivariate functional random variable, m_i(t) is the i-th component of the center and \epsilon_i(t) is a centered gaussian process with covariance function C_i. That is to say:

Cov( \epsilon_{i}(s), \epsilon_{i}(t) ) = C( s, t ), \quad \forall i = 1, \ldots, L, \quad \forall s, t \in I

A correlation structure among \epsilon_1(t),\ldots,\epsilon_L(t) is allowed in the following way:

Cor( \epsilon_i(t), \epsilon_j(t)) = \rho_{i,j}, \quad \forall i \neq j, \quad \forall t \in I.

All the functions are supposed to be observed on an evenly-spaced, one- dimensional grid of P points: [ a = t_0, t_1, \ldots, t_{P-1} = b] \subset I .

Value

The function returns a list of L matrices, one for each component of the multivariate functional random variable, containing the discretized values of the generated observations (in form of N \times P matrices).

See Also

exp_cov_function, mfData, generate_gauss_fdata

Examples

N = 30
P = 1e2
L = 3

time_grid = seq( 0, 1, length.out = P )

C1 = exp_cov_function( time_grid, alpha = 0.1, beta = 0.2 )
C2 = exp_cov_function( time_grid, alpha = 0.2, beta = 0.5 )
C3 = exp_cov_function( time_grid, alpha = 0.3, beta = 1 )


centerline = matrix( c( sin( 2 * pi * time_grid ),
                        sqrt( time_grid ),
                        10 * ( time_grid - 0.5 ) * time_grid ),
                     nrow = 3, byrow = TRUE )

generate_gauss_mfdata( N, L, centerline,
                       correlations = c( 0.5, 0.5, 0.5 ),
                       listCov = list( C1, C2, C3 ) )

CholC1 = chol( C1 )
CholC2 = chol( C2 )
CholC3 = chol( C3 )

generate_gauss_mfdata( N, L, centerline,
                       correlations = c( 0.5, 0.5, 0.5 ),
                      listCholCov = list( CholC1, CholC2, CholC3 ) )

Maximum order relation between univariate functional data

Description

This function implements an order relation between univariate functional data based on the maximum relation, that is to say a pre-order relation obtained by comparing the maxima of two different functional data.

Usage

max_ordered(fData, gData)

Arguments

Details

Given a univariate functional dataset, X_1(t), X_2(t), \ldots, X_N(t) and another functional dataset Y_1(t), Y_2(t), \ldots, Y_M(t) defined over the same compact interval I=[a,b], the function computes the maxima in both the datasets, and checks whether the first ones are lower or equal than the second ones.

By default the function tries to compare each X_i(t) with the corresponding Y_i(t), thus assuming N=M, but when either N=1 or M=1, the comparison is carried out cycling over the dataset with fewer elements. In all the other cases (N\neq M, and either N \neq 1 or M \neq 1) the function stops.

Value

The function returns a logical vector of length \max(N,M) containing the value of the predicate for all the corresponding elements.

References

Valencia, D., Romo, J. and Lillo, R. (2015). A Kendall correlation coefficient for functional dependence, Universidad Carlos III de Madrid technical report, http://EconPapers.repec.org/RePEc:cte:wsrepe:ws133228.

See Also

maxima, minima, fData, area_ordered

Examples

P = 1e2

grid = seq( 0, 1, length.out = P )

Data_1 = matrix( c( 1 * grid,
                    2 *  grid ),
                 nrow = 2, ncol = P, byrow = TRUE )

Data_2 = matrix( 3 * ( 0.5 - abs( grid - 0.5 ) ),
                 nrow = 1, byrow = TRUE )

Data_3 = rbind( Data_1, Data_1 )


fD_1 = fData( grid, Data_1 )
fD_2 = fData( grid, Data_2 )
fD_3 = fData( grid, Data_3 )

max_ordered( fD_1, fD_2 )

max_ordered( fD_2, fD_3 )

Maxima of a univariate functional dataset

Description

This function computes the maximum value of each element of a univariate functional dataset, optionally returning also the value of the grid where they are fulfilled.

Usage

maxima(fData, ..., which = FALSE)

Arguments

Value

If which = FALSE, the function returns a vector containing the maxima for each element of the functional dataset; if which = TRUE, the function returns a data.frame whose field value contains the values of maxima, and grid contains the grid points where maxima are reached.

See Also

minima

Examples

P = 1e3

grid = seq( 0, 1, length.out = P )

Data = matrix( c( 1 * grid,
                  2 *  grid,
                  3 * ( 0.5 - abs( grid - 0.5 ) ) ),
               nrow = 3, ncol = P, byrow = TRUE )

fD = fData( grid, Data )

maxima( fD, which = TRUE )

Cross-sectional mean of of a fData object.

Description

This S3 method implements the cross-sectional mean of a univariate functional dataset stored in a fData object, i.e. the mean computed point-by-point along the grid over which the dataset is defined.

Usage

## S3 method for class 'fData'
mean(x, ...)

Arguments

Value

The function returns a fData object with one observation defined on the same grid as the argument x's representing the desired cross-sectional mean.

See Also

fData

Examples

N = 1e2
P = 1e2
grid = seq( 0, 1, length.out = P )

# Generating a gaussian functional sample with desired mean
target_mean = sin( 2 * pi * grid )
C = exp_cov_function( grid, alpha = 0.2, beta = 0.2 )
fD = fData( grid, generate_gauss_fdata( N,
                                      centerline = target_mean,
                                       Cov = C ) )

# Graphical representation of the mean
plot( fD )
plot( mean( fD ), col = 'black', lwd = 2, lty = 2, add = TRUE )

Cross-sectional mean of of a mfData object.

Description

This S3 method implements the cross-sectional mean of a multivariate functional dataset stored in a mfData object, i.e. the mean computed point-by-point along the grid over which the dataset is defined.

Usage

## S3 method for class 'mfData'
mean(x, ...)

Arguments

Value

The function returns a mfData object with one observation defined on the same grid as the argument x's representing the desired cross-sectional mean.

See Also

mfData

Examples

N = 1e2
L = 3
P = 1e2
grid = seq( 0, 1, length.out = P )

# Generating a gaussian functional sample with desired mean
target_mean = sin( 2 * pi * grid )
C = exp_cov_function( grid, alpha = 0.2, beta = 0.2 )
# Independent components
correlations = c( 0, 0, 0 )
mfD = mfData( grid,
              generate_gauss_mfdata( N, L,
                                     correlations = correlations,
                                     centerline = matrix( target_mean,
                                                          nrow = 3,
                                                          ncol = P,
                                                          byrow = TRUE ),
                                     listCov = list( C, C, C ) )
)

# Graphical representation of the mean
oldpar <- par(mfrow = c(1, 1))
par(mfrow = c(1, L))

for(iL in 1:L)
{
  plot(mfD$fDList[[iL]])
  plot(
    mean(mfD)$fDList[[iL]],
    col = 'black',
    lwd = 2,
    lty = 2,
    add = TRUE
  )
}

par(oldpar)

Median of a univariate functional dataset

Description

This method computes the sample median of a univariate functional dataset based on a definition of depth for univariate functional data.

Usage

median_fData(fData, type = "MBD", ...)

Arguments

Details

Provided a definition of functional depth for univariate data, the corresponding median (i.e. the deepest element of the sample) is returned as the desired median. This method does not coincide with the computation of the cross-sectional median of the sample of the point-by-point measurements on the grid. Hence, the sample median is a member of the dataset provided.

Value

The function returns a fData object containing the desired sample median.

See Also

fData, mean.fData, median_mfData

Examples

N = 1e2
P = 1e2
grid = seq( 0, 1, length.out = P )

# Generating a gaussian functional sample with desired mean
# Being the distribution symmetric, the sample mean and median are coincident
target_median = sin( 2 * pi * grid )
C = exp_cov_function( grid, alpha = 0.2, beta = 0.2 )
fD = fData( grid, generate_gauss_fdata( N,
                                      centerline = target_median,
                                       Cov = C ) )

# Graphical representation of the mean
plot( fD )
plot( median_fData( fD ), col = 'black', lwd = 2, lty = 2, add = TRUE )

Median of a multivariate functional dataset

Description

This method computes the sample median of a multivariate functional dataset based on a definition of depth for multivariate functional data.

Usage

median_mfData(mfData, type = "multiMBD", ...)

Arguments

Details

Provided a definition of functional depth for multivariate data, the corresponding median (i.e. the deepest element of the sample) is returned as the desired median. This method does not coincide with the computation of the cross-sectional median of the sample of the point-by-point measurements on the grid. Hence, the sample median is a member of the dataset provided.

Value

The function returns a mfData object containing the desired sample median.

See Also

mfData, mean.mfData, median_fData

Examples

N = 1e2
L = 3
P = 1e2
grid = seq( 0, 1, length.out = P )

# Generating a gaussian functional sample with desired mean
# Being the distribution symmetric, the sample mean and median are coincident
target_median = sin( 2 * pi * grid )
C = exp_cov_function( grid, alpha = 0.2, beta = 0.2 )

# Strongly dependent components
correlations = c( 0.9, 0.9, 0.9 )
mfD = mfData( grid,
              generate_gauss_mfdata( N, L,
                                     correlations = correlations,
                                     centerline = matrix( target_median,
                                                          nrow = 3,
                                                          ncol = P,
                                                          byrow = TRUE ),
                                     listCov = list( C, C, C ) )
)

med_mfD = median_mfData( mfD, type = 'multiMBD', weights = 'uniform' )

# Graphical representation of the mean
oldpar <- par(mfrow = c(1, 1))
par(mfrow = c(1, L))

for(iL in 1:L)
{
  plot(mfD$fDList[[iL]])
  plot(
    med_mfD$fDList[[iL]],
    col = 'black',
    lwd = 2,
    lty = 2,
    add = TRUE
  )
}

par(oldpar)

ECG trace of subjects suffering from Left-Bundle-Branch-Block (LBBB)

Description

A dataset containing the 8-Lead ECG traces of 50 subjects suffering from Left-Bundle-Branch-Block (LBBB), a cardiac pathology affecting the conduction process and resulting in some peculiar distortions of the ECG.

Usage

mfD_LBBB

Format

A mfData object.

Details

The 8 leads are, in order, V1, V2, V3, V4, V5, D1 and D2. The signals have been registered and smoothed over an evenly spaced grid of 1024 time points at 1kHz.

ECG trace of healthy subjects

Description

A dataset containing the 8-Lead ECG traces of 50 healthy subjects. They can be used to compare the signals of pathological subjects stored in mfD_LBBB and mfD_RBBB objects.

Usage

mfD_healthy

Format

A mfData object.

Details

The 8 leads are, in order, V1, V2, V3, V4, V5, D1 and D2. The signals have been registered and smoothed over an evenly spaced grid of 1024 time points at 1kHz.

S3 class for multivariate functional datasets

Description

This function implements a constructor for elements of S3 class mfData, aimed at implementing a representation of a multivariate functional dataset.

Usage

mfData(grid, Data_list)

Arguments

Details

The functional dataset is represented as a collection of L components, each one an object of class fData. Each component must contain elements defined on the same grid as the others, and must contain the same number of elements (N).

Value

The function returns a S3 object of class mfData, containing the following elements:

• "N": the number of elements in the dataset;

• "L": the number of components of the functional dataset;

• "P": the number of points in the 1D grid over which elements are measured;

• "t0": the starting point of the 1D grid;

• "tP": the ending point of the 1D grid;

• "fDList": the list of fData objects representing the L components as corresponding univariate functional datasets.

See Also

fData, generate_gauss_fdata, generate_gauss_mfdata

Examples

# Defining parameters
N = 1e2

P = 1e3

t0 = 0
t1 = 1

# Defining the measurement grid
grid = seq( t0, t1, length.out = P )

# Generating an exponential covariance matrix to be used in the simulation of
# the functional datasets (see the related help for details)
C = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )

# Simulating the measurements of two univariate functional datasets with
# required center and covariance function
Data_1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = C )
Data_2 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = C )

# Building the mfData object
mfData( grid, list( Data_1, Data_2 ) )

Minima of a univariate functional dataset

Description

This function computes computes the minimum value of each element of a univariate functional dataset, optionally returning also the value of the grid where they are fulfilled.

Usage

minima(fData, ..., which = FALSE)

Arguments

Value

If which = FALSE, the function returns a vector containing the minima for each element of the functional dataset; if which = TRUE, the function returns a data.frame whose field value contains the values of minima, and grid contains the grid points where minima are reached.

See Also

maxima

Examples

P = 1e3

grid = seq( 0, 1, length.out = P )

Data = matrix( c( 1 * grid,
                  2 *  grid,
                  3 * ( 0.5 - abs( grid - 0.5 ) ) ),
               nrow = 3, ncol = P, byrow = TRUE )

fD = fData( grid, Data )

minima( fD, which = TRUE )

(Modified) Band Depth for multivariate functional data

Description

These functions compute the Band Depth (BD) and Modified Band Depth (MBD) of elements of a multivariate functional dataset.

Usage

multiMBD(Data, weights = "uniform", manage_ties = FALSE)

## S3 method for class 'mfData'
multiMBD(Data, weights = "uniform", manage_ties = FALSE)

## Default S3 method:
multiMBD(Data, weights = "uniform", manage_ties = FALSE)

multiBD(Data, weights = "uniform")

## S3 method for class 'mfData'
multiBD(Data, weights = "uniform")

## Default S3 method:
multiBD(Data, weights = "uniform")

Arguments

Details

Given a multivariate functional dataset composed of N elements with L components each, \mathbf{X_1} =( X^1_1(t), X^2_1(t), \ldots, X^L_1(t)), and a set of L non-negative weights,

w_1, w_2, \ldots, w_L, \qquad \sum_{i=1}^L w_i = 1,

these functions compute the BD and MBD of each element of the functional dataset, namely:

BD( \mathbf{X_j} ) = \sum_{i=1}^{L} w_i BD( X^i_j ), \quad \forall j = 1, \ldots N.

MBD( \mathbf{X_j} ) = \sum_{i=1}^{L} w_i MBD( X^i_j ), \quad \forall j = 1, \ldots N.

Value

The function returns a vector containing the depths of each element of the multivariate functional dataset.

References

Ieva, F. and Paganoni, A. M. (2013). Depth measures for multivariate functional data, Communications in Statistics: Theory and Methods, 41, 1265-1276.

Tarabelloni, N., Ieva, F., Biasi, R. and Paganoni, A. M. (2015). Use of Depth Measure for Multivariate Functional Data in Disease Prediction: An Application to Electrocardiograph Signals, International Journal of Biostatistics, 11.2, 189-201.

See Also

MBD, BD, toListOfValues, mfData

Examples

N = 20
P = 1e3

grid = seq( 0, 10, length.out = P )

# Generating an exponential covariance function to be used to simulate gaussian
# functional data
Cov = exp_cov_function( grid, alpha = 0.2, beta = 0.8 )

# First component of the multivariate guassian functional dataset
Data_1 = generate_gauss_fdata( N, centerline = rep( 0, P ), Cov = Cov )

# First component of the multivariate guassian functional dataset
Data_2 = generate_gauss_fdata( N, centerline = rep( 0, P ), Cov = Cov )

mfD = mfData( grid, list( Data_1, Data_2 ) )

multiBD( mfD, weights = 'uniform' )
multiMBD( mfD, weights = 'uniform', manage_ties = TRUE )

multiBD( mfD, weights = c( 1/3, 2/3 ))
multiMBD( mfD, weights = c( 1/3, 2/3 ), manage_ties = FALSE )

multiBD( list( Data_1, Data_2 ), weights = 'uniform')
multiMBD( list( Data_1, Data_2 ), weights = 'uniform', manage_ties = TRUE )

multiBD( list( Data_1, Data_2 ), weights = c( 1/3, 2/3 ))
multiMBD( list( Data_1, Data_2 ), weights = c( 1/3, 2/3 ), manage_ties = FALSE )

Modified Epigraph Index for multivariate functional data

Description

These functions compute the Modified Epigraph Index of elements of a multivariate functional dataset.

Usage

multiMEI(Data, weights = "uniform")

## S3 method for class 'mfData'
multiMEI(Data, weights = "uniform")

## Default S3 method:
multiMEI(Data, weights = "uniform")

Arguments

Details

Given a multivariate functional dataset composed of N elements with L components each, \mathbf{X_1} =( X^1_1(t), X^2_1(t), \ldots, X^L_1(t)), and a set of L non-negative weights,

w_1, w_2, \ldots, w_L, \qquad \sum_{i=1}^L w_i = 1,

these functions compute the MEI of each element of the functional dataset, namely:

MEI( \mathbf{X_j} ) = \sum_{i=1}^{L} w_i MEI( X^i_j ), \quad \forall j = 1, \ldots N.

Value

The function returns a vector containing the values of MEI of each element of the multivariate functional dataset.

See Also

mfData, MEI, MHI, multiMHI

Examples

N = 20
P = 1e3

grid = seq( 0, 10, length.out = P )

# Generating an exponential covariance function to be used to simulate gaussian
# functional data
Cov = exp_cov_function( grid, alpha = 0.2, beta = 0.8 )

# First component of the multivariate guassian functional dataset
Data_1 = generate_gauss_fdata( N, centerline = rep( 0, P ), Cov = Cov )

# First component of the multivariate guassian functional dataset
Data_2 = generate_gauss_fdata( N, centerline = rep( 0, P ), Cov = Cov )

mfD = mfData( grid, list( Data_1, Data_2 ) )

# Uniform weights
multiMEI( mfD, weights = 'uniform' )

# Non-uniform, custom weights
multiMEI( mfD, weights = c(2/3, 1/3) )

Modified Hypograph Index for multivariate functional data

Description

These functions compute the Modified Hypograph Index of elements of a multivariate functional dataset.

Usage

multiMHI(Data, weights = "uniform")

## S3 method for class 'mfData'
multiMHI(Data, weights = "uniform")

## Default S3 method:
multiMHI(Data, weights = "uniform")

Arguments

Details

Given a multivariate functional dataset composed of N elements with L components each, \mathbf{X_1} =( X^1_1(t), X^2_1(t), \ldots, X^L_1(t)), and a set of L non-negative weights,

w_1, w_2, \ldots, w_L, \qquad \sum_{i=1}^L w_i = 1,

these functions compute the MHI of each element of the functional dataset, namely:

MHI( \mathbf{X_j} ) = \sum_{i=1}^{L} w_i MHI( X^i_j ), \quad \forall j = 1, \ldots N.

Value

The function returns a vector containing the values of MHI of each element of the multivariate functional dataset.

See Also

mfData, MHI, MEI, multiMEI

Examples

N = 20
P = 1e3

grid = seq( 0, 10, length.out = P )

# Generating an exponential covariance function to be used to simulate gaussian
# functional data
Cov = exp_cov_function( grid, alpha = 0.2, beta = 0.8 )

# First component of the multivariate guassian functional dataset
Data_1 = generate_gauss_fdata( N, centerline = rep( 0, P ), Cov = Cov )

# First component of the multivariate guassian functional dataset
Data_2 = generate_gauss_fdata( N, centerline = rep( 0, P ), Cov = Cov )

mfD = mfData( grid, list( Data_1, Data_2 ) )

# Uniform weights
multiMHI( mfD, weights = 'uniform' )

# Non-uniform, custom weights
multiMHI( mfD, weights = c(2/3, 1/3) )

Outliergram for multivariate functional datasets

Description

This function performs the outliergram of a multivariate functional dataset.

Usage

multivariate_outliergram(
  mfData,
  MBD_data = NULL,
  MEI_data = NULL,
  weights = "uniform",
  p_check = 0.05,
  Fvalue = 1.5,
  shift = TRUE,
  display = TRUE,
  xlab = NULL,
  ylab = NULL,
  main = NULL
)

Arguments

Details

The method applies the extension of the univariate outliergram to the case of multivariate functional datasets. Differently from the function for the univariate case, only the outliergram plot is displayed.

Adjustment

Differently from the case of univariate functional data, in this case the function does not apply an automatic tuning of the F parameter, since the related procedure would become computationally too heavy for general datasets. If a good value of F is sought, it is recommended to run several trials of the outliergram and manually select the best value.

References

Ieva, F. & Paganoni, A.M. Stat Papers (2017). https://doi.org/10.1007/s00362-017-0953-1.

See Also

outliergram, mfData, MBD, MEI

Examples

N = 2e2
P = 1e2

t0 = 0
t1 = 1

set.seed(1)

# Defining the measurement grid
grid = seq( t0, t1, length.out = P )

# Generating an exponential covariance matrix to be used in the simulation of
# the functional datasets (see the related help for details)
C = exp_cov_function( grid, alpha = 0.3, beta = 0.2)

# Simulating the measurements of two univariate functional datasets with
# required center and covariance function
f1 = function(x) x * ( 1 - x )
f2 = function(x) x^3
Data = generate_gauss_mfdata( N, L = 2,
                              centerline = matrix(c(sin(2 * pi * grid),
                                                    cos(2 * pi * grid)), nrow=2, byrow=TRUE),
                              listCov = list(C, C), correlations = 0.1 )

# Building the mfData object
mfD = mfData( grid, Data )


dev.new()
out = multivariate_outliergram(mfD, Fvalue = 2., shift=TRUE)
col_non_outlying = scales::hue_pal( h = c( 180, 270 ),
                                    l = 60 )( N - length( out$ID_outliers ) )
col_non_outlying = set_alpha( col_non_outlying, 0.5 )
col_outlying = scales::hue_pal( h = c( - 90, 180  ),
                                c = 150 )( length( out$ID_outliers ) )
colors = rep('black', N)
colors[out$ID_outliers] = col_outlying
colors[colors == 'black'] = col_non_outlying

lwd = rep(1, N)
lwd[out$ID_outliers] = 2

dev.new()
plot(mfD, col=colors, lwd=lwd)

Outliergram for univariate functional data sets

Description

This function performs the outliergram of a univariate functional data set, possibly with an adjustment of the true positive rate of outliers discovered under assumption of gaussianity.

Usage

outliergram(
  fData,
  MBD_data = NULL,
  MEI_data = NULL,
  p_check = 0.05,
  Fvalue = 1.5,
  adjust = FALSE,
  display = TRUE,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  ...
)

Arguments

Value

Even when used graphically to plot the outliergram, the function returns a list containing:

• Fvalue: the value of the parameter F used;

• d: the vector of values of the parameter d for each observation (distance to the parabolic border of the outliergram);

• ID_outliers: the vector of observations id corresponding to outliers.

Adjustment

When the adjustment option is selected, the value of F is optimized for the univariate functional dataset provided with fData. In practice, a number adjust$N_trials of times a synthetic population (of size adjust$trial_size with the same covariance (robustly estimated from data) and centerline as fData is simulated without outliers and each time an optimized value F_i is computed so that a given proportion (adjust$TPR) of observations is flagged as outliers. The final value of F for the outliergram is determined as an average of F_1, F_2, \ldots, F_{N_{trials}}. At each time step the optimization problem is solved using stats::uniroot (Brent's method).

References

Arribas-Gil, A., and Romo, J. (2014). Shape outlier detection and visualization for functional data: the outliergram, Biostatistics, 15(4), 603-619.

See Also

fData, MEI, MBD, fbplot

Examples

set.seed(1618)

N <- 200
P <- 200
N_extra <- 4

grid <- seq(0, 1, length.out = P)

Cov <- exp_cov_function(grid, alpha = 0.2, beta = 0.8)

Data <- generate_gauss_fdata(
  N = N,
  centerline = sin(4 * pi * grid),
  Cov = Cov
)

Data_extra <- array(0, dim = c(N_extra, P))

Data_extra[1, ] <- generate_gauss_fdata(
  N = 1,
  centerline = sin(4 * pi * grid + pi / 2),
  Cov = Cov
)

Data_extra[2, ] <- generate_gauss_fdata(
  N = 1,
  centerline = sin(4 * pi * grid - pi / 2),
  Cov = Cov
)

Data_extra[3, ] <- generate_gauss_fdata(
  N = 1,
  centerline = sin(4 * pi * grid + pi / 3),
  Cov = Cov
)

Data_extra[4, ] <- generate_gauss_fdata(
  N = 1,
  centerline = sin(4 * pi * grid - pi / 3),
  Cov = Cov
)

Data <- rbind(Data, Data_extra)

fD <- fData(grid, Data)

# Outliergram with default Fvalue = 1.5
outliergram(fD, display = TRUE)

# Outliergram with Fvalue enforced to 2.5
outliergram(fD, Fvalue = 2.5, display = TRUE)


# Outliergram with estimated Fvalue to ensure TPR of 1%
outliergram(
  fData = fD,
  adjust = list(
    N_trials = 10,
    trial_size = 5 * nrow(Data),
    TPR = 0.01,
    VERBOSE = FALSE
  ),
  display = TRUE
)

Specialized method to plot Cov objects

Description

This function performs the plot of an object of class Cov, i.e. a covariance or cross-covariance function.

Usage

## S3 method for class 'Cov'
plot(x, ...)

Arguments

Details

It builds above the function graphics::image, therefore any additional parameter suitable for graphics::image will also be suitable as ... argument to plot.Cov.

See Also

cov_fun

Examples

# Generating a univariate functional dataset
N = 1e2

P = 1e2
t0 = 0
t1 = 1

time_grid = seq( t0, t1, length.out = P )

Cov = exp_cov_function( time_grid, alpha = 0.3, beta = 0.4 )

D1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * time_grid ), Cov = Cov )

fD1 = fData( time_grid, D1 )

# Computing the covariance function of fD1

plot( cov_fun( fD1 ), main = 'Covariance function', xlab = 'time', ylab = 'time' )

Specialized method to plot 'depthgram' objects

Description

This function plots the three 'DepthGram' representations from the output of the depthgram function.

Usage

## S3 method for class 'depthgram'
plot(
  x,
  limits = FALSE,
  ids = NULL,
  print = FALSE,
  plot_title = "",
  shorten = TRUE,
  col = NULL,
  pch = 19,
  sp = 2,
  st = 4,
  sa = 10,
  text_labels = "",
  ...
)

Arguments

Value

A list with the following items:

• p: list with all the interactive (plotly) depthGram plots;

• out: outliers detected;

• colors: used colors for plotting.

References

Aleman-Gomez, Y., Arribas-Gil, A., Desco, M. Elias-Fernandez, A., and Romo, J. (2021). "Depthgram: Visualizing Outliers in High Dimensional Functional Data with application to Task fMRI data exploration".

Examples

N <- 50
P <- 50
grid <- seq(0, 1, length.out = P)
Cov <- exp_cov_function(grid, alpha = 0.3, beta = 0.4)

Data <- list()
Data[[1]] <- generate_gauss_fdata(
  N,
  centerline = sin(2 * pi * grid),
  Cov = Cov
)
Data[[2]] <- generate_gauss_fdata(
  N,
  centerline = sin(2 * pi * grid),
  Cov = Cov
)
names <- paste0("id_", 1:nrow(Data[[1]]))
DG <- depthgram(Data, marginal_outliers = TRUE, ids = names)
plot(DG)

Specialized method to plot fData objects

Description

This function performs the plot of a functional univariate dataset stored in an object of class fData. It is able to accept all the usual customizable graphical parameters, otherwise it will use the default ones.

Usage

## S3 method for class 'fData'
plot(x, ...)

Arguments

See Also

fData

Examples

N = 20
P = 1e2

# One dimensional grid
grid = seq( 0, 1, length.out = P )

# Generating an exponential covariance function (see related help for more
# information )
C = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )

# Generating a synthetic dataset with a gaussian distribution and
# required mean and covariance function:
values = generate_gauss_fdata( N,
                               centerline = sin( 2 * pi * grid ),
                               Cov = C )

fD = fData( grid, values )

plot( fD )

Specialized method to plot mfData objects

Description

This function performs the plot of a functional multivariate dataset stored in an object of class mfData. It is able to accept all the usual customizable graphical parameters, otherwise it will use the default ones.

Usage

## S3 method for class 'mfData'
plot(x, ...)

Arguments

Details

The current active graphical device is split into a number of sub-figures, each one meant to contain the plot of the corresponding dimension of the mfData object. In particular, they are arranged in a rectangular lattice with a number of rows equal to \lfloor \sqrt{ L } \rfloor and a number of columns equal to \lceil L / \lfloor \sqrt{L} \rfloor \rceil .

A special use of the graphical parameters allows to set up y-labels and titles for all the sub-figures in the graphical window. In particular, parameters ylab and main can take as argument either a single string, that are repeatedly used for all the sub-graphics, or a list of different strings (one for each of the L dimensions) that have to be used in the corresponding graphic.

See Also

mfData, fData, plot.fData

Examples

N = 1e2

P = 1e3

t0 = 0
t1 = 1

# Defining the measurement grid
grid = seq( t0, t1, length.out = P )

# Generating an exponential covariance matrix to be used in the simulation of
# the functional datasets (see the related help for details)
C = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )

# Simulating the measurements of two univariate functional datasets with
# required center and covariance function
Data_1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = C )
Data_2 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = C )

# Building the mfData object and plotting tt
plot( mfData( grid, list( Data_1, Data_2 ) ),
      xlab = 'time', ylab = list( '1st dim.', '2nd dim.' ),
      main = list( 'An important plot here', 'And another one here' ) )

Operator + and - for fData objects

Description

These methods provide operators + and - to perform sums or differences between an fData object and either another fData object or other compliant data structures, like matrices or vectors or arrays, representing the pointwise measurements of the second term of the sum.

Usage

## S3 method for class 'fData'
fD + A

## S3 method for class 'fData'
fD - A

Arguments

Details

If the second term of the operation is an fData object, it must be defined over the same grid as the first.

Value

The function returns an fData object, whose function values have undergone the sum/difference.

Examples

fD = fData( seq( 0, 1, length.out = 10 ),
            values = matrix( seq( 1, 10 ),
                             nrow = 21, ncol = 10, byrow = TRUE ) )
fD + 1 : 10

fD + array( 1, dim = c( 1, 10 ) )

fD + fD

fD = fData( seq( 0, 1, length.out = 10 ),
            values = matrix( seq( 1, 10 ),
                             nrow = 21, ncol = 10, byrow = TRUE ) )
fD - 2 : 11

fD - array( 1, dim = c( 1, 10 ) )

fD - fD

Function to setup alpha value for a set of colors

Description

set_alpha manipulates a vector of color representations in order to setup the alpha value, and get the desired transparency level.

Usage

set_alpha(col, alpha)

Arguments

See Also

fDColorPalette

Examples

original_col = c( 'blue', 'red', 'green', 'yellow' )

alpha_col = set_alpha( original_col, 0.5 )

alpha_col = set_alpha( original_col, c(0.5, 0.5, 0.2, 0.1 ) )

dev.new()
oldpar <- par(mfrow = c(1, 1))
par(mfrow = c(1, 2))

plot( seq_along( original_col ),
      seq_along( original_col ),
      col = original_col,
      pch = 16,
      cex = 2,
      main = 'Original colors' )

plot( seq_along( alpha_col ),
      seq_along( alpha_col ),
      col = alpha_col,
      pch = 16,
      cex = 2,
      main = 'Alpha colors' )

par(oldpar)

Operator sub-.fData to subset fData objects

Description

This method provides an easy and natural way to subset a functional dataset stored in a fData object, without having to deal with the inner representation of fData class.

Usage

## S3 method for class 'fData'
fD[i, j, as_fData = TRUE]

Arguments

Value

The method returns either an fData object ( if as_fData = TRUE ) or a matrix ( if as_fData = FALSE ) containing the required subset ( both in terms of observations and measurement points ) of the univariate functional dataset.

See Also

fData

Examples

N = 20
P = 1e2

# One dimensional grid
grid = seq( 0, 1, length.out = P )

# Generating an exponential covariance function (see related help for more
# information )
C = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )

# Generating a synthetic dataset with a gaussian distribution and
# required mean and covariance function:
fD = fData( grid,
            generate_gauss_fdata( N,
                                  centerline = sin( 2 * pi * grid ),
                                  Cov = C ) )

dev.new()
oldpar <- par(mfrow = c(1, 1))
par(mfrow = c(2, 2))

# Original data
plot(fD)

# Subsetting observations
plot(fD[c(1, 2, 3), , as_fData = TRUE])

# Subsetting measurements
plot(fD[, 1:30])

# Subsetting both observations and measurements
plot(fD[1:10, 50:P])

par(oldpar)

# Subsetting both observations and measurements but returning a matrix
fD[1:10, 50:P, as_fData = FALSE]

Operator sub-.mfData to subset mfData objects

Description

This method provides an easy and natural way to subset a multivariate functional dataset stored in a mfData object, without having to deal with the inner representation of mfData class.

Usage

## S3 method for class 'mfData'
mfD[i, j]

Arguments

Value

The method returns and mfData object containing the required subset ( both in terms of observations and measurement points ) of the multivariate functional dataset.

See Also

mfData

Examples

# Defining parameters
N = 1e2

P = 1e3

t0 = 0
t1 = 1

# Defining the measurement grid
grid = seq( t0, t1, length.out = P )

# Generating an exponential covariance matrix to be used in the simulation of
# the functional datasets (see the related help for details)
C = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )

# Simulating the measurements of two univariate functional datasets with
# required center and covariance function
Data_1 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = C )
Data_2 = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ), Cov = C )

# Building the mfData object
mfD = mfData( grid, list( Data_1, Data_2 ) )

# Subsetting the first 10 elements and 10 time points
mfD[1:10, 1:10]

# Subsetting only observations
mfD[1:10,]

# Subsetting only time points (contiguously)
mfD[,1:10]

Operator * and / for fData objects

Description

These methods provide operators * and / to perform products or divisions between an fData object and either a number or a compliant 1D data structure, like numeric vector, array or matrix. The operation is computed by performing the element-wise product or division between fD's observations and the provided value(s).

Usage

## S3 method for class 'fData'
fD * a

## S3 method for class 'fData'
fD / a

Arguments

Details

If the second argument is a 1D data structure, it must have length N equal to the number of observations in fD.

Value

The function returns an fData object, whose function values have undergone the product/division.

Examples

N = 11
fD = fData( seq( 0, 1, length.out = 10 ),
            values = matrix( seq( 1, 10 ),
                             nrow = N, ncol = 10, byrow = TRUE ) )
fD * 2

fD * seq( 1, N )


N = 11
fD = fData( seq( 0, 1, length.out = 10 ),
            values = matrix( seq( 1, 10 ),
                             nrow = N, ncol = 10, byrow = TRUE ) )
fD / 2

fD / rep( 10, N )

Manipulation of mfData list of values

Description

This utility function manipulates a mfData object in order to extract from the list of its fData objects ( namely, mfData$fDList ) the measurement values of each component and stores them into a list.

Usage

toListOfValues(mfData)

Arguments

Details

Given a mfData of L components, the function is equivalent to list( mfData$fDList[[ 1 ]]$values, ..., mfData$fDList[[ L ]]$values ) .

Value

The function returns the list of values of each fData object representing the components of mfData.

See Also

mfData

Examples

grid = seq( 0, 1, length.out = 5 )

D_1 = matrix( 1 : 5, nrow = 10, ncol = 5, byrow = TRUE )
D_2 = 2 * D_1
D_3 = 3 * D_1

mfD = mfData( grid, list( D_1, D_2, D_3 ) )
mfD

toListOfValues( mfD )

Conversion of vector/array/matrix to row-matrix form

Description

This function manipulates a numeric data structure of vector/array/matrix type in order to obtain a matrix representation. For 1D data structures and column/row arrays and matrices the output is turned in a matrix format with just one row. If the input structure is rectangular, instead, it is only converted in matrix format.

Usage

toRowMatrixForm(D)

Arguments

Warning

The function is not supposed to work with arbitrary N-dimensional arrays.

Examples

toRowMatrixForm( 1 : 10 )

toRowMatrixForm( array( 1 : 10, dim = c(1,10 ) ) )

toRowMatrixForm( array( 1 : 10, dim = c( 10, 1 ) ) )

toRowMatrixForm( matrix( 1 : 10, ncol = 10, nrow = 1 ) )

toRowMatrixForm( matrix( 1 : 10, ncol = 1, nrow = 10 ) )

toRowMatrixForm( matrix( 1 : 12, ncol = 3, nrow = 4 ) )

toRowMatrixForm( matrix( 1 : 12, ncol = 4, nrow = 3 ) )

Unfolding a univariate functional dataset

Description

This function operates on a univariate functional dataset and transforms its observations unfolding their values and turning them into monotone functions.

Usage

unfold(fData)

Arguments

Details

Each function of the fData object is transformed into a non-monotone function into a monotone function by “unfolding” it at any of its maxima. For more details about the definition of the transform, see the reference.

Value

The function returns an fData object whose observations are the unfolded version of the corresponding observations in the argument fData.

References

Arribas-Gil, A. and Romo, J. (2012). Robust depth-based estimation in the time warping model, Biostatistics, 13 (3), 398–414.

See Also

fData, warp

Examples

P = 1e3

time_grid = seq( 0, 1, length.out = P )

D = matrix( c( sin( 2 * pi * time_grid ),
               cos( 2 * pi * time_grid ),
               sin( 10 * pi * time_grid ) * time_grid + 2 ),
            ncol = P, nrow = 3, byrow = TRUE )

# Functional dataset
fD = fData( time_grid, D )

# Unfolded version
fD_unfold = unfold( fD )

dev.new()
oldpar <- par(mfrow = c(1, 1))
par(mfrow = c(1, 2))

plot(fD, main = 'Original data')
plot(fD_unfold, main = 'Unfolded data')

par(oldpar)

Warp elements of a univariate functional dataset

Description

This function carries out the warping of elements of a univariate functional dataset by using a set of pre-computed warping functions.

Usage

warp(fData, warpings)

Arguments

Details

Given a univariate functional dataset X_1(t), \ldots, X_N(t) and a set of warping functions H_1(t), \ldots, H_N(t), such that:

H_i : s \longrightarrow t = H_i(s), \quad \forall i = 1, \ldots, N,

where s spans the warped (or registered) grid and t spans the original grid, the function computes the warping given by the following composition:

X_1 \circ H_1(t), \ldots, X_N \circ H_N(t).

Value

The function returns the univariate functional dataset of warped functions, in form of fData object.

See Also

fData

Examples

set.seed( 1618033 )

N = 30

t0 = 0
t1 = 1
P = 1e3 + 1

time_grid = seq( t0, t1, length.out = P )

means = round( runif( N,
                      t0 + (t1 - t0) / 8,
                      t1 - (t1 - t0) / 8  ), 3 )

Data = matrix( sapply( means,
                       function( m )( dnorm( time_grid, mean = m, sd = 0.05 ) ) ),
               ncol = P, nrow = N, byrow = TRUE )

fD = fData( time_grid, Data )

# Piecewise linear warpings
template_warping = function( m )( c( time_grid[ time_grid <= 0.5 ] * m / 0.5,
                                     ( time_grid[ time_grid > 0.5 ]
                                       - 0.5 ) * (1 - m ) / 0.5 + m ) )


warpings = matrix( sapply( means, template_warping ),
                   ncol = P,
                   nrow = N, byrow = TRUE )

wfD = fData( time_grid, warpings )

fD_warped = warp( fD, wfD )

dev.new()
oldpar <- par(mfrow = c(1, 1))
par(mfrow = c(1, 3))

plot( fD,
     main = 'Unregistered functions', xlab = 'actual grid', ylab = 'values'  )
plot( wfD,
     main = 'Warping functions', xlab = 'registered grid',
     ylab = 'actual grid' )
plot( fD_warped,
     main = 'Warped functions', xlab = 'registered grid',
     ylab = 'values' )

par(oldpar)