Title:	Methodologies for Functional Data Based on the Epigraph and Hypograph Indices
Version:	0.1.1
Description:	Implements methods for functional data analysis based on the epigraph and hypograph indices. These methods transform functional datasets, whether in one or multiple dimensions, into multivariate datasets. The transformation involves applying the epigraph, hypograph, and their modified versions to both the original curves and their first and second derivatives. The calculation of these indices is tailored to the dimensionality of the functional dataset, with special considerations for dependencies between dimensions in multidimensional cases. This approach extends traditional multivariate data analysis techniques to the functional data setting. A key application of this package is the EHyClus method, which enhances clustering analysis for functional data across one or multiple dimensions using the epigraph and hypograph indices. See Pulido et al. (2023) <doi:10.1007/s11222-023-10213-7> and Pulido et al. (2024) <doi:10.48550/arXiv.2307.16720>.
License:	MIT + file LICENSE
Encoding:	UTF-8
URL:	https://github.com/bpulidob/ehymet, https://bpulidob.github.io/ehymet/
BugReports:	https://github.com/bpulidob/ehymet/issues
Depends:	R (≥ 4.1)
Imports:	clusterCrit, kernlab, stats, tf
Suggests:	ggplot2, knitr, MASS, parallel, rmarkdown, testthat (≥ 3.0.0), tidyr
RoxygenNote:	7.3.2
Config/testthat/edition:	3
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2024-11-26 12:11:07 UTC; belen
Author:	Belen Pulido [aut, cre], Jose Ignacio Diez [ctr]
Maintainer:	Belen Pulido <bpulidob4@gmail.com>
Repository:	CRAN
Date/Publication:	2024-11-26 23:00:13 UTC

Clustering using Epigraph and Hypograph indices

Description

It creates a multivariate dataset containing the epigraph, hypograph and/or its modified versions on the curves and derivatives and then perform hierarchical clustering, kmeans, kernel kmeans, and spectral clustering

Usage

EHyClus(
  curves,
  vars_combinations,
  k = 30,
  n_clusters = 2,
  bs = "cr",
  clustering_methods = c("hierarch", "kmeans", "kkmeans", "spc"),
  l_method_hierarch = c("single", "complete", "average", "centroid", "ward.D2"),
  l_dist_hierarch = c("euclidean", "manhattan"),
  l_dist_kmeans = c("euclidean", "mahalanobis"),
  l_kernel = c("rbfdot", "polydot"),
  true_labels = NULL,
  only_best = FALSE,
  verbose = FALSE,
  n_cores = 1,
  ...
)

Arguments

Value

A list containing the clustering partition for each method and indices combination and, if true_labels is provided a data frame containing the time elapsed for obtaining a clustering partition of the indices dataset for each methodology. Also, the number of generated clusters and the combinations of variables used can be seen as attributes of this object.

Examples

# univarariate data without labels
curves <- sim_model_ex1(n = 10)
vars_combinations <- list(c("dtaEI", "dtaMEI"), c("dtaHI", "dtaMHI"))
EHyClus(curves, vars_combinations = vars_combinations)

# multivariate data with labels
curves <- sim_model_ex2(n = 5)
true_labels <- c(rep(1, 5), rep(2, 5))
vars_combinations <- list(c("dtaMEI", "ddtaMEI"), c("dtaMEI", "d2dtaMEI"))
res <- EHyClus(curves, vars_combinations = vars_combinations, true_labels = true_labels)
res$cluster # clustering results

# multivariate data and generic (default) vars_combinations
curves <- sim_model_ex2(n = 5)
EHyClus(curves)

Epigraph Index (EI) for a functional dataset

Description

The Epigraph Index of a curve x is one minus the proportion of curves in the sample that are above x.

Usage

EI(curves, ...)

Arguments

Value

numeric vector containing the EI for each curve.

Examples

x <- matrix(c(1, 2, 3, 3, 2, 1, 5, 2, 3, 9, 8, 7), ncol = 3, nrow = 4)
EI(x)

y <- array(c(1, 2, 3, 3, 2, 1, 5, 2, 3, 9, 8, 7, -1, -5, -6, 2, 3, 0, -1, 0, 2, -1, -2, 0),
  dim = c(3, 4, 2)
)
EI(y)

Hypograph Index (HI) for a functional dataset

Description

The Hypograph Index of a curve x is the proportion of curves in the sample that are below x.

Usage

HI(curves, ...)

Arguments

Value

numeric vector containing the HI for each curve.

Examples

x <- matrix(c(1, 2, 3, 3, 2, 1, 5, 2, 3, 9, 8, 7), ncol = 3, nrow = 4)
HI(x)

y <- array(c(1, 2, 3, 3, 2, 1, 5, 2, 3, 9, 8, 7, -1, -5, -6, 2, 3, 0, -1, 0, 2, -1, -2, 0),
  dim = c(3, 4, 2)
)
HI(y)

Modified Epigraph Index (MEI) for functional dataset.

Description

The Modified Epigraph Index of a curve x is one minus the proportion of "time" the curves in the sample are above x.

Usage

MEI(curves, ...)

Arguments

Value

numeric vector containing the MEI for each curve.

Examples

x <- matrix(c(1, 2, 3, 3, 2, 1, 5, 2, 3, 9, 8, 7), ncol = 3, nrow = 4)
MEI(x)
y <- array(c(1, 2, 3, 3, 2, 1, 5, 2, 3, 9, 8, 7, -1, -5, -6, 2, 3, 0, -1, 0, 2, -1, -2, 0),
  dim = c(3, 4, 2)
)
MEI(y)

Modified Hypograph Index (MHI) for a functional dataset

Description

The Modified Hypograph Index of a curve x is the proportion of "time" the curves in the sample are below x.

Usage

MHI(curves, ...)

Arguments

Value

numeric vector containing the MHI for each curve.

Examples

x <- matrix(c(1, 2, 3, 3, 2, 1, 5, 2, 3, 9, 8, 7), ncol = 3, nrow = 4)
MHI(x)
y <- array(c(1, 2, 3, 3, 2, 1, 5, 2, 3, 9, 8, 7, -1, -5, -6, 2, 3, 0, -1, 0, 2, -1, -2, 0),
  dim = c(3, 4, 2)
)
MHI(y)

Perform hierarchical clustering for a different combinations of indices, method and distance

Description

Perform hierarchical clustering for a different combinations of indices, method and distance

Usage

clustInd_hierarch(
  ind_data,
  vars_combinations,
  method_list = c("single", "complete", "average", "centroid", "ward.D2"),
  dist_vector = c("euclidean", "manhattan"),
  n_cluster = 2,
  true_labels = NULL,
  n_cores = 1
)

Arguments

Value

A list containing hierarchical clustering results for each configuration.

Examples

vars1 <- c("dtaEI", "dtaMEI")
vars2 <- c("dtaHI", "dtaMHI")
data <- ehymet::sim_model_ex1()
data_ind <- generate_indices(data)
clustInd_hierarch(data_ind, list(vars1, vars2))

Kernel k-means clustering using indices

Description

Perform kernel kmeans clustering for a different combinations of indices and kernel

Usage

clustInd_kkmeans(
  ind_data,
  vars_combinations,
  kernel_list = c("rbfdot", "polydot"),
  n_cluster = 2,
  true_labels = NULL,
  n_cores = 1
)

Arguments

Value

A list containing kernel-kmeans clustering results for each configuration.

Examples

vars1 <- c("dtaEI", "dtaMEI")
vars2 <- c("dtaHI", "dtaMHI")
data <- ehymet::sim_model_ex1()
data_ind <- generate_indices(data)
clustInd_kkmeans(data_ind, list(vars1, vars2))

K-means clustering with indices

Description

Perform k-means clustering for a different combinations of indices and distances.

Usage

clustInd_kmeans(
  ind_data,
  vars_combinations,
  dist_vector = c("euclidean", "mahalanobis"),
  n_cluster = 2,
  init = "random",
  true_labels = NULL,
  n_cores = 1
)

Arguments

Value

A list containing hierarchical clustering results for each configuration

A list containing kmeans clustering results for each configuration

Examples

vars1 <- c("dtaEI", "dtaMEI")
vars2 <- c("dtaHI", "dtaMHI")
data <- ehymet::sim_model_ex1()
data_ind <- generate_indices(data)
clustInd_kmeans(data_ind, list(vars1, vars2))

Spectral clustering using indices

Description

Perform spectral clustering for a different combinations of indices and kernels

Usage

clustInd_spc(
  ind_data,
  vars_combinations,
  kernel_list = c("rbfdot", "polydot"),
  n_cluster = 2,
  true_labels = NULL,
  n_cores = 1
)

Arguments

Value

A list containing kkmeans clustering results for each configuration

Examples

vars1 <- c("dtaEI", "dtaMEI")
vars2 <- c("dtaHI", "dtaMHI")
data <- ehymet::sim_model_ex1()
data_ind <- generate_indices(data)
clustInd_spc(data_ind, list(vars1, vars2))

Create a table containing four validation metrics for clustering: Purity, F-measure and Rand Index (RI) and Adjusted Rand Index (ARI). This function considers pairs of points

Description

Create a table containing four validation metrics for clustering: Purity, F-measure and Rand Index (RI) and Adjusted Rand Index (ARI). This function considers pairs of points

Usage

clustering_validation(clusters, true_labels, digits = 4)

Arguments

Value

A list containing values for Purity, F-measure, RI and ARI.

Examples

set.seed(1221)
vars <- list(c("dtaEI", "dtaMEI"))
data <- sim_model_ex1()
true_labels <- c(rep(1, 50), rep(2, 50))
data_ind <- generate_indices(data)
clus_kmeans <- clustInd_kmeans(data_ind, vars)
cluskmeans_mahalanobis_dtaEIdtaMEI <- clus_kmeans$kmeans_mahalanobis_dtaEIdtaMEI$cluster
clustering_validation(cluskmeans_mahalanobis_dtaEIdtaMEI, true_labels)

Create a dataset with indices from a functional dataset in one or multiple dimensions

Description

Create a dataset with indices from a functional dataset in one or multiple dimensions

Usage

generate_indices(
  curves,
  k,
  bs = "cr",
  indices = c("EI", "HI", "MEI", "MHI"),
  n_cores = 1,
  ...
)

Arguments

Value

A dataframe containing the indices provided in indices for original data, first and second derivatives

Examples

# 3-dimensional array
x1 <- array(c(1, 2, 3, 3, 2, 1, 5, 2, 3, 9, 8, 7, -1, -5, -6, 2, 3, 0, -1, 0, 2, -1, -2, 0),
  dim = c(3, 4, 2)
)
generate_indices(x1, k = 4)

# matrix
x2 <- matrix(c(1, 2, 3, 3, 2, 1, 5, 2, 3, 9, 8, 7), nrow = 3, ncol = 4)
generate_indices(x2, k = 4)

# using additional parameter for tf::tfb
curves <- sim_model_ex1(n = 10)
generate_indices(
  curves = curves,
  k = 20,
  bs = "bs",
  m = c(3,2),        # additional parameter for tfb
  penalized = FALSE  # additional parameter for tfb
)

Function for generating functional data in one dimension

Description

Each dataset has 2 groups with n curves each, defined in the interval t=[0, 1] with p equidistant points. The first n curves are generated fron the following model X_1(t)=E_1(t)+e(t) where E_1(t)=E_1(X(t))=30t^{ \frac{3}{2}}(1-t) is the mean function and e(t) is a centered Gaussian process with covariance matrix Cov(e(t_i),e(t_j))=0.3 \exp(-\frac{\lvert t_i-t_j \rvert}{0.3}) The remaining 50 functions are generated from model i_sim with i_sim \in \{1, \ldots, 8\}. The first three models contain changes in the mean, while the covariance matrix does not change. Model 4 and 5 are obtained by multiplying the covariance matrix by a constant. Model 6 is obtained from adding to E_1(t) a centered Gaussian process h(t) whose covariance matrix is given by Cov(e(t_i),e(t_j))=0.5 \exp (-\frac{\lvert t_i-t_j\rvert}{0.2}). Model 7 and 8 are obtained by a different mean function.

Model 1.

X_1(t)=30t^{\frac{3}{2}}(1-t)+0.5+e(t).

Model 2.

X_2(t)=30t^{\frac{3}{2}}(1-t)+0.75+e(t).

Model 3.

X_3(t)=30t^{\frac{3}{2}}(1-t)+1+e(t).

Model 4.

X_4(t)=30t^{\frac{3}{2}}(1-t)+2 e(t).

Model 5.

X_5(t)=30t^{\frac{3}{2}}(1-t)+0.25 e(t).

Model 6.

X_6(t)=30t^{\frac{3}{2}}(1-t)+ h(t).

Model 7.

X_7(t)=30t{(1-t)}^2+ h(t).

Model 8.

X_8(t)=30t{(1-t)}^2+ e(t).

Usage

sim_model_ex1(n = 50, p = 30, i_sim = 1)

Arguments

Value

data matrix of size 2n \times p.

Examples

sm1 <- sim_model_ex1()
dim(sm1)

Function for generating functional data in one or multiple dimension

Description

The function can generate one-dimensional or multi-dimensional curves. For i_sim 1 or 2, one-dimensional curves are generated. For i_sim 3 or 4, multi-dimensional curves are generated.

Usage

sim_model_ex2(n = 50, p = 150, i_sim = 1)

Arguments

Value

data matrix of size 2n \times p if i\_sim \in {1,2} or an array of dimensions 2n \times p \times 2 if i\_sim \in {3, 4}.

Examples

sm1 <- sim_model_ex2()
dim(sm1) # This should output (100, 150) by default, since n = 50 and p = 150

sm4 <- sim_model_ex2(i_sim = 4)
dim(sm4) # This should output (100, 150, 2) by default, since n = 50 and p = 150