Package 'roahd'

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)

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)

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 )

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 )

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 ) ) )

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.

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.

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 )

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 )

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)

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' )

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' )

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")

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 )

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)

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 )

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')

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' ) )

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 )

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 )

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 ))

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 ) )

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 )

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 )

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 )

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 )

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 )

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 $\tilde{\lambda(A)} = \lambda( A ) / ( b - a )$.