Package 'adamethods'

Title: Archetypoid Algorithms and Anomaly Detection
Description: Collection of several algorithms to obtain archetypoids with small and large databases, and with both classical multivariate data and functional data (univariate and multivariate). Some of these algorithms also allow to detect anomalies (outliers). Please see Vinue and Epifanio (2020) <doi:10.1007/s11634-020-00412-9>.
Authors: Guillermo Vinue, Irene Epifanio
Maintainer: Guillermo Vinue <[email protected]>
License: GPL (>= 2)
Version: 1.2.1
Built: 2024-11-23 06:48:11 UTC
Source: CRAN

Help Index

Multivariate parallel archetypoid algorithm for large applications (ADALARA)

Description

The ADALARA algorithm is based on the CLARA clustering algorithm. This is the parallel version of the algorithm to try to get faster results. It allows to detect anomalies (outliers). There are two different methods to detect them: the adjusted boxplot (default and most reliable option) and tolerance intervals. If needed, tolerance intervals allow to define a degree of outlierness.

Usage

adalara(data, N, m, numArchoid, numRep, huge, prob, type_alg = "ada", 
        compare = FALSE, vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
        outl_degree = c("outl_strong", "outl_semi_strong", "outl_moderate"), 
        method = "adjbox", frame)

Arguments

data

Data matrix. Each row corresponds to an observation and each column corresponds to a variable. All variables are numeric. The data must have row names so that the algorithm can identify the archetypoids in every sample.
N

Number of samples.
m

Sample size of each sample.
numArchoid

Number of archetypes/archetypoids.
numRep

For each numArchoid, run the archetype algorithm numRep times.
huge

Penalization added to solve the convex least squares problems.
prob

Probability with values in [0,1].
type_alg

String. Options are 'ada' for the non-robust adalara algorithm and 'ada_rob' for the robust adalara algorithm.
compare

Boolean argument to compute the robust residual sum of squares if type_alg = "ada" and the non-robust if type_alg = "ada_rob".
vect_tol

Vector with the tolerance values. Default c(0.95, 0.9, 0.85). Needed if method='toler'.
alpha

Significance level. Default 0.05. Needed if method='toler'.
outl_degree

Type of outlier to identify the degree of outlierness. Default c("outl_strong", "outl_semi_strong", "outl_moderate"). Needed if method='toler'.
method

Method to compute the outliers. Options allowed are 'adjbox' for using adjusted boxplots for skewed distributions, and 'toler' for using tolerance intervals.
frame

Boolean value to indicate whether the frame is computed (Mair et al., 2017) or not. The frame is made up of a subset of extreme points, so the archetypoids are only computed on the frame. Low frame densities are obtained when only small portions of the data were extreme. However, high frame densities reduce this speed-up.

Value

A list with the following elements:

  • cases Optimal vector of archetypoids.

  • rss Optimal residual sum of squares.

  • outliers: Outliers.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Eugster, M.J.A. and Leisch, F., From Spider-Man to Hero - Archetypal Analysis in R, 2009. Journal of Statistical Software 30(8), 1-23, https://doi.org/10.18637/jss.v030.i08

Hubert, M. and Vandervieren, E., An adjusted boxplot for skewed distributions, 2008. Computational Statistics and Data Analysis 52(12), 5186-5201, https://doi.org/10.1016/j.csda.2007.11.008

Kaufman, L. and Rousseeuw, P.J., Clustering Large Data Sets, 1986. Pattern Recognition in Practice, 425-437.

Mair, S., Boubekki, A. and Brefeld, U., Frame-based Data Factorizations, 2017. Proceedings of the 34th International Conference on Machine Learning, Sydney, Australia, 1-9.

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Vinue, G., Anthropometry: An R Package for Analysis of Anthropometric Data, 2017. Journal of Statistical Software 77(6), 1-39, https://doi.org/10.18637/jss.v077.i06

See Also

do_ada, do_ada_robust, adalara_no_paral

Examples

## Not run: 
library(Anthropometry)
library(doParallel)

# Prepare parallelization (including the seed for reproducibility):
no_cores <- detectCores() - 1
cl <- makeCluster(no_cores)
registerDoParallel(cl)
clusterSetRNGStream(cl, iseed = 1)

# Load data:
data(mtcars)
data <- mtcars
n <- nrow(data)

# Arguments for the archetype/archetypoid algorithm:
# Number of archetypoids:
k <- 3 
numRep <- 2
huge <- 200

# Size of the random sample of observations:
m <- 10
# Number of samples:
N <- floor(1 + (n - m)/(m - k))
N
           
prob <- 0.75            

# ADALARA algorithm:
preproc <- preprocessing(data, stand = TRUE, percAccomm = 1)
data1 <- as.data.frame(preproc$data)

adalara_aux <- adalara(data1, N, m, k, numRep, huge, prob, 
                       "ada_rob", FALSE, method = "adjbox", frame = FALSE)

#adalara_aux <- adalara(data1, N, m, k, numRep, huge, prob, 
#                       "ada_rob", FALSE, vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
#                       outl_degree = c("outl_strong", "outl_semi_strong", "outl_moderate"),
#                       method = "toler", frame = FALSE)

# Take the minimum RSS, which is in the second position of every sublist:
adalara <- adalara_aux[which.min(unlist(sapply(adalara_aux, function(x) x[2])))][[1]]
adalara

# End parallelization:
stopCluster(cl)

## End(Not run)

Multivariate non-parallel archetypoid algorithm for large applications (ADALARA)

Description

The ADALARA algorithm is based on the CLARA clustering algorithm. This is the non-parallel version of the algorithm. It allows to detect anomalies (outliers). There are two different methods to detect them: the adjusted boxplot (default and most reliable option) and tolerance intervals. If needed, tolerance intervals allow to define a degree of outlierness.

Usage

adalara_no_paral(data, seed, N, m, numArchoid, numRep, huge, prob, type_alg = "ada", 
                 compare = FALSE, verbose = TRUE, vect_tol = c(0.95, 0.9, 0.85), 
                 alpha = 0.05, outl_degree = c("outl_strong", "outl_semi_strong", 
                 "outl_moderate"), method = "adjbox", frame)

Arguments

data

Data matrix. Each row corresponds to an observation and each column corresponds to a variable. All variables are numeric. The data must have row names so that the algorithm can identify the archetypoids in every sample.
seed

Integer value to set the seed. This ensures reproducibility.
N

Number of samples.
m

Sample size of each sample.
numArchoid

Number of archetypes/archetypoids.
numRep

For each numArchoid, run the archetype algorithm numRep times.
huge

Penalization added to solve the convex least squares problems.
prob

Probability with values in [0,1].
type_alg

String. Options are 'ada' for the non-robust adalara algorithm and 'ada_rob' for the robust adalara algorithm.
compare

Boolean argument to compute the robust residual sum of squares if type_alg = "ada" and the non-robust if type_alg = "ada_rob".
verbose

Display progress? Default TRUE.
vect_tol

Vector the tolerance values. Default c(0.95, 0.9, 0.85). Needed if method='toler'.
alpha

Significance level. Default 0.05. Needed if method='toler'.
outl_degree

Type of outlier to identify the degree of outlierness. Default c("outl_strong", "outl_semi_strong", "outl_moderate"). Needed if method='toler'.
method

Method to compute the outliers. Options allowed are 'adjbox' for using adjusted boxplots for skewed distributions, and 'toler' for using tolerance intervals.
frame

Boolean value to indicate whether the frame is computed (Mair et al., 2017) or not. The frame is made up of a subset of extreme points, so the archetypoids are only computed on the frame. Low frame densities are obtained when only small portions of the data were extreme. However, high frame densities reduce this speed-up.

Value

A list with the following elements:

  • cases Optimal vector of archetypoids.

  • rss Optimal residual sum of squares.

  • outliers: Outliers.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Eugster, M.J.A. and Leisch, F., From Spider-Man to Hero - Archetypal Analysis in R, 2009. Journal of Statistical Software 30(8), 1-23, https://doi.org/10.18637/jss.v030.i08

Hubert, M. and Vandervieren, E., An adjusted boxplot for skewed distributions, 2008. Computational Statistics and Data Analysis 52(12), 5186-5201, https://doi.org/10.1016/j.csda.2007.11.008

Kaufman, L. and Rousseeuw, P.J., Clustering Large Data Sets, 1986. Pattern Recognition in Practice, 425-437.

Mair, S., Boubekki, A. and Brefeld, U., Frame-based Data Factorizations, 2017. Proceedings of the 34th International Conference on Machine Learning, Sydney, Australia, 1-9.

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Vinue, G., Anthropometry: An R Package for Analysis of Anthropometric Data, 2017. Journal of Statistical Software 77(6), 1-39, https://doi.org/10.18637/jss.v077.i06

See Also

do_ada, do_ada_robust, adalara

Examples

## Not run: 
library(Anthropometry)

# Load data:
data(mtcars)
data <- mtcars
n <- nrow(data)

# Arguments for the archetype/archetypoid algorithm:
# Number of archetypoids:
k <- 3 
numRep <- 2
huge <- 200

# Size of the random sample of observations:
m <- 10
# Number of samples:
N <- floor(1 + (n - m)/(m - k))
N
           
prob <- 0.75            

# ADALARA algorithm:
preproc <- preprocessing(data, stand = TRUE, percAccomm = 1)
data1 <- as.data.frame(preproc$data)
res_adalara <- adalara_no_paral(data1, 1, N, m, k, 
                                numRep, huge, prob, "ada_rob", FALSE, TRUE, 
                                method = "adjbox", frame = FALSE)

# Examine the results:
res_adalara

res_adalara1 <- adalara_no_paral(data1, 1, N, m, k, 
                                numRep, huge, prob, "ada_rob", FALSE, TRUE, 
                                vect_tol = c(0.95, 0.9, 0.85), 
                                alpha = 0.05, outl_degree = c("outl_strong", "outl_semi_strong", 
                                                              "outl_moderate"),
                                method = "toler", frame = FALSE)
res_adalara1                                 
                                 

## End(Not run)

Archetypoid algorithm with the functional Frobenius norm

Description

Archetypoid algorithm with the functional Frobenius norm to be used with functional data.

Usage

archetypoids_funct(numArchoid, data, huge = 200, ArchObj, PM)

Arguments

numArchoid

Number of archetypoids.
data

Data matrix. Each row corresponds to an observation and each column corresponds to a variable. All variables are numeric.
huge

Penalization added to solve the convex least squares problems.
ArchObj

The list object returned by the stepArchetypesRawData_funct function.
PM

Penalty matrix obtained with eval.penalty.

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • archet_ini: Vector of initial archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • resid: Matrix with the residuals.

Author(s)

Irene Epifanio

References

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

See Also

archetypoids

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- t(growth$hgtm)
# Create basis:
basis_fd <- create.bspline.basis(c(1,ncol(hgtm)), 10)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:ncol(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = growth$hgtm, basisobj = basis_fd)
data_archs <- t(temp_fd$coefs)

lass <- stepArchetypesRawData_funct(data = data_archs, numArch = 3, 
                                    numRep = 5, verbose = FALSE, 
                                    saveHistory = FALSE, PM)

af <- archetypoids_funct(3, data_archs, huge = 200, ArchObj = lass, PM) 
str(af)                                

## End(Not run)

Archetypoid algorithm with the functional multivariate Frobenius norm

Description

Archetypoid algorithm with the functional multivariate Frobenius norm to be used with functional data.

Usage

archetypoids_funct_multiv(numArchoid, data, huge = 200, ArchObj, PM)

Arguments

numArchoid

Number of archetypoids.
data

Data matrix. Each row corresponds to an observation and each column corresponds to a variable. All variables are numeric.
huge

Penalization added to solve the convex least squares problems.
ArchObj

The list object returned by the stepArchetypesRawData_funct function.
PM

Penalty matrix obtained with eval.penalty.

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • archet_ini: Vector of initial archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • resid: Matrix with the residuals.

Author(s)

Irene Epifanio

References

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

See Also

archetypoids

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- growth$hgtm
hgtf <- growth$hgtf[,1:39]

# Create array:
nvars <- 2
data.array <- array(0, dim = c(dim(hgtm), nvars))
data.array[,,1] <- as.matrix(hgtm)
data.array[,,2] <- as.matrix(hgtf)
rownames(data.array) <- 1:nrow(hgtm)
colnames(data.array) <- colnames(hgtm)
str(data.array)

# Create basis:
nbasis <- 10
basis_fd <- create.bspline.basis(c(1,nrow(hgtm)), nbasis)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:nrow(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = data.array, basisobj = basis_fd)

X <- array(0, dim = c(dim(t(temp_fd$coefs[,,1])), nvars))
X[,,1] <- t(temp_fd$coef[,,1]) 
X[,,2] <- t(temp_fd$coef[,,2])

# Standardize the variables:
Xs <- X
Xs[,,1] <- scale(X[,,1])
Xs[,,2] <- scale(X[,,2])

lass <- stepArchetypesRawData_funct_multiv(data = Xs, numArch = 3, 
                                           numRep = 5, verbose = FALSE, 
                                           saveHistory = FALSE, PM)
                                           
afm <- archetypoids_funct_multiv(3, Xs, huge = 200, ArchObj = lass, PM)
str(afm)

## End(Not run)

Archetypoid algorithm with the functional multivariate robust Frobenius norm

Description

Archetypoid algorithm with the functional multivariate robust Frobenius norm to be used with functional data.

Usage

archetypoids_funct_multiv_robust(numArchoid, data, huge = 200, ArchObj, PM, prob)

Arguments

numArchoid

Number of archetypoids.
data

Data matrix. Each row corresponds to an observation and each column corresponds to a variable. All variables are numeric.
huge

Penalization added to solve the convex least squares problems.
ArchObj

The list object returned by the stepArchetypesRawData_funct function.
PM

Penalty matrix obtained with eval.penalty.
prob

Probability with values in [0,1].

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • archet_ini: Vector of initial archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • resid: Matrix with the residuals.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

See Also

archetypoids

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- growth$hgtm
hgtf <- growth$hgtf[,1:39]

# Create array:
nvars <- 2
data.array <- array(0, dim = c(dim(hgtm), nvars))
data.array[,,1] <- as.matrix(hgtm)
data.array[,,2] <- as.matrix(hgtf)
rownames(data.array) <- 1:nrow(hgtm)
colnames(data.array) <- colnames(hgtm)
str(data.array)

# Create basis:
nbasis <- 10
basis_fd <- create.bspline.basis(c(1,nrow(hgtm)), nbasis)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:nrow(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = data.array, basisobj = basis_fd)

X <- array(0, dim = c(dim(t(temp_fd$coefs[,,1])), nvars))
X[,,1] <- t(temp_fd$coef[,,1]) 
X[,,2] <- t(temp_fd$coef[,,2])

# Standardize the variables:
Xs <- X
Xs[,,1] <- scale(X[,,1])
Xs[,,2] <- scale(X[,,2])

lass <- stepArchetypesRawData_funct_multiv_robust(data = Xs, numArch = 3, 
                                                  numRep = 5, verbose = FALSE, 
                                                  saveHistory = FALSE, PM, prob = 0.8, 
                                                  nbasis, nvars)

afmr <- archetypoids_funct_multiv_robust(3, Xs, huge = 200, ArchObj = lass, PM, 0.8)
str(afmr)

## End(Not run)

Archetypoid algorithm with the functional robust Frobenius norm

Description

Archetypoid algorithm with the functional robust Frobenius norm to be used with functional data.

Usage

archetypoids_funct_robust(numArchoid, data, huge = 200, ArchObj, PM, prob)

Arguments

numArchoid

Number of archetypoids.
data

Data matrix. Each row corresponds to an observation and each column corresponds to a variable. All variables are numeric.
huge

Penalization added to solve the convex least squares problems.
ArchObj

The list object returned by the stepArchetypesRawData_funct_robust function.
PM

Penalty matrix obtained with eval.penalty.
prob

Probability with values in [0,1].

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • archet_ini: Vector of initial archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • resid: Matrix with the residuals.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

See Also

archetypoids

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- t(growth$hgtm)
# Create basis:
basis_fd <- create.bspline.basis(c(1,ncol(hgtm)), 10)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:ncol(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = growth$hgtm, basisobj = basis_fd)
data_archs <- t(temp_fd$coefs)

lass <- stepArchetypesRawData_funct_robust(data = data_archs, numArch = 3, 
                                           numRep = 5, verbose = FALSE, 
                                           saveHistory = FALSE, PM, prob = 0.8)

afr <- archetypoids_funct_robust(3, data_archs, huge = 200, ArchObj = lass, PM, 0.8)
str(afr)

## End(Not run)

Archetypoid algorithm with the Frobenius norm

Description

This function is the same as archetypoids but the 2-norm is replaced by the Frobenius norm. Thus, the comparison with the robust archetypoids can be directly made.

Usage

archetypoids_norm_frob(numArchoid, data, huge = 200, ArchObj)

Arguments

numArchoid

Number of archetypoids.
data

Data matrix. Each row corresponds to an observation and each column corresponds to a variable. All variables are numeric.
huge

Penalization added to solve the convex least squares problems.
ArchObj

The list object returned by the stepArchetypesRawData_norm_frob function.

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • archet_ini: Vector of initial archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • resid: Matrix with the residuals.

Author(s)

Irene Epifanio

References

Eugster, M.J.A. and Leisch, F., From Spider-Man to Hero - Archetypal Analysis in R, 2009. Journal of Statistical Software 30(8), 1-23, https://doi.org/10.18637/jss.v030.i08

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Vinue, G., Epifanio, I., and Alemany, S., Archetypoids: a new approach to define representative archetypal data, 2015. Computational Statistics and Data Analysis 87, 102-115, https://doi.org/10.1016/j.csda.2015.01.018

Vinue, G., Anthropometry: An R Package for Analysis of Anthropometric Data, 2017. Journal of Statistical Software 77(6), 1-39, https://doi.org/10.18637/jss.v077.i06

See Also

archetypoids

Examples

data(mtcars)
data <- mtcars

k <- 3
numRep <- 2
huge <- 200

lass <- stepArchetypesRawData_norm_frob(data = data, numArch = k, 
                                        numRep = numRep, verbose = FALSE)

res <- archetypoids_norm_frob(k, data, huge, ArchObj = lass)
str(res)  
res$cases
res$rss

Archetypoid algorithm with the robust Frobenius norm

Description

Robust version of the archetypoid algorithm with the Frobenius form.

Usage

archetypoids_robust(numArchoid, data, huge = 200, ArchObj, prob)

Arguments

numArchoid

Number of archetypoids.
data

Data matrix. Each row corresponds to an observation and each column corresponds to a variable. All variables are numeric.
huge

Penalization added to solve the convex least squares problems.
ArchObj

The list object returned by the stepArchetypesRawData_robust function.
prob

Probability with values in [0,1].

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • archet_ini: Vector of initial archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • resid: Matrix with the residuals.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

See Also

archetypoids_norm_frob

Examples

data(mtcars)
data <- mtcars

k <- 3
numRep <- 2
huge <- 200

lass <- stepArchetypesRawData_robust(data = data, numArch = k, 
                                     numRep = numRep, verbose = FALSE, 
                                     saveHistory = FALSE, prob = 0.8)

res <- archetypoids_robust(k, data, huge, ArchObj = lass, 0.8)
str(res)    
res$cases
res$rss

Bisquare function

Description

This function belongs to the bisquare family of loss functions. The bisquare family can better cope with extreme outliers.

Usage

bisquare_function(resid, prob, ...)

Arguments

resid

Vector of residuals, computed from the m×nm \times n residuals data matrix.
prob

Probability with values in [0,1].
...

Additional possible arguments.

Value

Vector of real numbers.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Examples

resid <- c(2.47, 11.85)  
bisquare_function(resid, 0.8)

Run the whole classical archetypoid analysis with the Frobenius norm

Description

This function executes the entire procedure involved in the archetypoid analysis. Firstly, the initial vector of archetypoids is obtained using the archetypal algorithm and finally, the optimal vector of archetypoids is returned.

Usage

do_ada(subset, numArchoid, numRep, huge, compare = FALSE,
              vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
              outl_degree = c("outl_strong", "outl_semi_strong", "outl_moderate"),
              method = "adjbox", prob)

Arguments

subset

Data to obtain archetypes. In ADALARA this is a subset of the entire data frame.
numArchoid

Number of archetypes/archetypoids.
numRep

For each numArch, run the archetype algorithm numRep times.
huge

Penalization added to solve the convex least squares problems.
compare

Boolean argument to compute the robust residual sum of squares to compare these results with the ones provided by do_ada_robust.
vect_tol

Vector the tolerance values. Default c(0.95, 0.9, 0.85). Needed if method='toler'.
alpha

Significance level. Default 0.05. Needed if method='toler'.
outl_degree

Type of outlier to identify the degree of outlierness. Default c("outl_strong", "outl_semi_strong", "outl_moderate"). Needed if method='toler'.
method

Method to compute the outliers. Options allowed are 'adjbox' for using adjusted boxplots for skewed distributions, and 'toler' for using tolerance intervals.
prob

If compare=TRUE, probability with values in [0,1].

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • rss_rob: If compare=TRUE, this is the residual sum of squares using the robust Frobenius norm. Otherwise, NULL.

  • resid: Vector with the residuals.

  • outliers: Outliers.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Eugster, M.J.A. and Leisch, F., From Spider-Man to Hero - Archetypal Analysis in R, 2009. Journal of Statistical Software 30(8), 1-23, https://doi.org/10.18637/jss.v030.i08

Vinue, G., Epifanio, I., and Alemany, S., Archetypoids: a new approach to define representative archetypal data, 2015. Computational Statistics and Data Analysis 87, 102-115, https://doi.org/10.1016/j.csda.2015.01.018

Vinue, G., Anthropometry: An R Package for Analysis of Anthropometric Data, 2017. Journal of Statistical Software 77(6), 1-39, https://doi.org/10.18637/jss.v077.i06

See Also

stepArchetypesRawData_norm_frob, archetypoids_norm_frob

Examples

library(Anthropometry)
data(mtcars)
#data <- as.matrix(mtcars)
data <- mtcars

k <- 3
numRep <- 2
huge <- 200

preproc <- preprocessing(data, stand = TRUE, percAccomm = 1)
suppressWarnings(RNGversion("3.5.0"))
set.seed(2018)
res_ada <- do_ada(preproc$data, k, numRep, huge, FALSE, method = "adjbox")
str(res_ada)     

res_ada1 <- do_ada(preproc$data, k, numRep, huge, FALSE, 
                   vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
                   outl_degree = c("outl_strong", "outl_semi_strong", 
                                   "outl_moderate"), method = "toler")
str(res_ada1) 

res_ada2 <- do_ada(preproc$data, k, numRep, huge, TRUE, method = "adjbox", prob = 0.8)
str(res_ada2)

Run the whole robust archetypoid analysis with the robust Frobenius norm

Description

This function executes the entire procedure involved in the robust archetypoid analysis. Firstly, the initial vector of archetypoids is obtained using the robust archetypal algorithm and finally, the optimal vector of robust archetypoids is returned.

Usage

do_ada_robust(subset, numArchoid, numRep, huge, prob, compare = FALSE,
              vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
              outl_degree = c("outl_strong", "outl_semi_strong", "outl_moderate"),
              method = "adjbox")

Arguments

subset

Data to obtain archetypes. In ADALARA this is a subset of the entire data frame.
numArchoid

Number of archetypes/archetypoids.
numRep

For each numArch, run the archetype algorithm numRep times.
huge

Penalization added to solve the convex least squares problems.
prob

Probability with values in [0,1].
compare

Boolean argument to compute the non-robust residual sum of squares to compare these results with the ones provided by do_ada.
vect_tol

Vector the tolerance values. Default c(0.95, 0.9, 0.85). Needed if method='toler'.
alpha

Significance level. Default 0.05. Needed if method='toler'.
outl_degree

Type of outlier to identify the degree of outlierness. Default c("outl_strong", "outl_semi_strong", "outl_moderate"). Needed if method='toler'.
method

Method to compute the outliers. Options allowed are 'adjbox' for using adjusted boxplots for skewed distributions, and 'toler' for using tolerance intervals.

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • rss_non_rob: If compare=TRUE, this is the residual sum of squares using the non-robust Frobenius norm. Otherwise, NULL.

  • resid Vector of residuals.

  • outliers: Outliers.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

See Also

stepArchetypesRawData_robust, archetypoids_robust

Examples

## Not run: 
library(Anthropometry)
data(mtcars)
#data <- as.matrix(mtcars)
data <- mtcars

k <- 3
numRep <- 2
huge <- 200

preproc <- preprocessing(data, stand = TRUE, percAccomm = 1)
suppressWarnings(RNGversion("3.5.0"))
set.seed(2018)
res_ada_rob <- do_ada_robust(preproc$data, k, numRep, huge, 0.8,
                             FALSE, method = "adjbox")
str(res_ada_rob)    

res_ada_rob1 <- do_ada_robust(preproc$data, k, numRep, huge, 0.8,
                             FALSE, vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
                             outl_degree = c("outl_strong", "outl_semi_strong", 
                                             "outl_moderate"),
                             method = "toler")
str(res_ada_rob1)  

## End(Not run)

Alphas and RSS of every set of archetypoids

Description

In the ADALARA algorithm, every time that a set of archetypoids is computed using a sample of the data, the alpha coefficients and the associated residual sum of squares (RSS) for the entire data set must be computed.

Usage

do_alphas_rss(data, subset, huge, k_subset, rand_obs, alphas_subset, 
              type_alg = "ada", PM, prob)

Arguments

data

Data matrix with all the observations.
subset

Data matrix with a sample of the data observations.
huge

Penalization added to solve the convex least squares problems.
k_subset

Archetypoids obtained from subset.
rand_obs

Sample observations that form subset.
alphas_subset

Alpha coefficients related to k_subset.
type_alg

String. Options are 'ada' for the non-robust multivariate adalara algorithm, 'ada_rob' for the robust multivariate adalara algorithm, 'fada' for the non-robust fda fadalara algorithm and 'fada_rob' for the robust fda fadalara algorithm.
PM

Penalty matrix obtained with eval.penalty. Needed when type_alg = 'fada' or type_alg = 'fada_rob'.
prob

Probability with values in [0,1]. Needed when type_alg = 'ada_rob' or type_alg = 'fada_rob'.

Value

A list with the following elements:

  • rss Real number of the residual sum of squares.

  • resid_rss Matrix with the residuals.

  • alphas Matrix with the alpha values.

Author(s)

Guillermo Vinue

See Also

archetypoids_norm_frob

Examples

data(mtcars)
data <- mtcars
n <- nrow(data)
m <- 10

k <- 3 
numRep <- 2
huge <- 200

suppressWarnings(RNGversion("3.5.0"))
set.seed(1)
rand_obs_si <- sample(1:n, size = m) 

si <- data[rand_obs_si,]
ada_si <- do_ada(si, k, numRep, huge, FALSE) 

k_si <- ada_si$cases
alphas_si <- ada_si$alphas
colnames(alphas_si) <- rownames(si)     

rss_si <- do_alphas_rss(data, si, huge, k_si, rand_obs_si, alphas_si, "ada")
str(rss_si)

Alphas and RSS of every set of multivariate archetypoids

Description

In the ADALARA algorithm, every time that a set of archetypoids is computed using a sample of the data, the alpha coefficients and the associated residual sum of squares (RSS) for the entire data set must be computed.

Usage

do_alphas_rss_multiv(data, subset, huge, k_subset, rand_obs, alphas_subset, 
                  type_alg = "ada", PM, prob, nbasis, nvars)

Arguments

data

Data matrix with all the observations.
subset

Data matrix with a sample of the data observations.
huge

Penalization added to solve the convex least squares problems.
k_subset

Archetypoids obtained from subset.
rand_obs

Sample observations that form subset.
alphas_subset

Alpha coefficients related to k_subset.
type_alg

String. Options are 'ada' for the non-robust multivariate adalara algorithm, 'ada_rob' for the robust multivariate adalara algorithm, 'fada' for the non-robust fda fadalara algorithm and 'fada_rob' for the robust fda fadalara algorithm.
PM

Penalty matrix obtained with eval.penalty. Needed when type_alg = 'fada' or type_alg = 'fada_rob'.
prob

Probability with values in [0,1]. Needed when type_alg = 'ada_rob' or type_alg = 'fada_rob'.
nbasis

Number of basis.
nvars

Number of variables.

Value

A list with the following elements:

  • rss Real number of the residual sum of squares.

  • resid_rss Matrix with the residuals.

  • alphas Matrix with the alpha values.

Author(s)

Guillermo Vinue

See Also

archetypoids_norm_frob

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- growth$hgtm
hgtf <- growth$hgtf[,1:39]

# Create array:
nvars <- 2
data.array <- array(0, dim = c(dim(hgtm), nvars))
data.array[,,1] <- as.matrix(hgtm)
data.array[,,2] <- as.matrix(hgtf)
rownames(data.array) <- 1:nrow(hgtm)
colnames(data.array) <- colnames(hgtm)
str(data.array)

# Create basis:
nbasis <- 10
basis_fd <- create.bspline.basis(c(1,nrow(hgtm)), nbasis)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:nrow(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = data.array, basisobj = basis_fd)

X <- array(0, dim = c(dim(t(temp_fd$coefs[,,1])), nvars))
X[,,1] <- t(temp_fd$coef[,,1]) 
X[,,2] <- t(temp_fd$coef[,,2])

# Standardize the variables:
Xs <- X
Xs[,,1] <- scale(X[,,1])
Xs[,,2] <- scale(X[,,2])
# We have to give names to the dimensions to know the 
# observations that were identified as archetypoids.
dimnames(Xs) <- list(paste("Obs", 1:dim(hgtm)[2], sep = ""), 
                     1:nbasis,
                     c("boys", "girls"))

n <- dim(Xs)[1] 
# Number of archetypoids:
k <- 3 
numRep <- 20
huge <- 200

# Size of the random sample of observations:
m <- 15
# Number of samples:
N <- floor(1 + (n - m)/(m - k))
N
prob <- 0.75
data_alg <- Xs

nbasis <- dim(data_alg)[2] # number of basis.
nvars <- dim(data_alg)[3] # number of variables.
n <- nrow(data_alg)

suppressWarnings(RNGversion("3.5.0"))
set.seed(1) 
rand_obs_si <- sample(1:n, size = m)  
si <- apply(data_alg, 2:3, function(x) x[rand_obs_si])  

fada_si <- do_fada_multiv_robust(si, k, numRep, huge, 0.8, FALSE, PM)

k_si <- fada_si$cases
alphas_si <- fada_si$alphas
colnames(alphas_si) <- rownames(si)

rss_si <- do_alphas_rss_multiv(data_alg, si, huge, k_si, rand_obs_si, alphas_si, 
                               "fada_rob", PM, 0.8, nbasis, nvars)
str(rss_si)                                

## End(Not run)

Cleaning outliers

Description

Cleaning of the most remarkable outliers. This improves the performance of the archetypoid algorithm since it is not affected by spurious points.

Usage

do_clean(data, num_pts, range = 1.5, out_perc = 80)

Arguments

data

Data frame with (temporal) points in the rows and observations in the columns.
num_pts

Number of temporal points.
range

Same parameter as in function boxplot. A value of 1.5 is enough to detect amplitude and shift outliers, while a value of 3 is needed to detect isolated outliers.
out_perc

Minimum number of temporal points (in percentage) to consider the observation as an outlier. Needed when range=1.5.

Value

Numeric vector with the outliers.

Author(s)

Irene Epifanio

See Also

boxplot

Examples

data(mtcars)
data <- mtcars
num_pts <- ncol(data)
do_clean(t(data), num_pts, 1.5, 80)

Cleaning multivariate functional outliers

Description

Cleaning of the most remarkable multivariate functional outliers. This improves the performance of the archetypoid algorithm since it is not affected by spurious points.

Usage

do_clean_multiv(data, num_pts, range = 1.5, out_perc = 80, nbasis, nvars)

Arguments

data

Data frame with (temporal) points in the rows and observations in the columns.
num_pts

Number of temporal points.
range

Same parameter as in function boxplot. A value of 1.5 is enough to detect amplitude and shift outliers, while a value of 3 is needed to detect isolated outliers.
out_perc

Minimum number of temporal points (in percentage) to consider the observation as an outlier. Needed when range=1.5.
nbasis

Number of basis.
nvars

Number of variables.

Value

List with the outliers for each variable.

Author(s)

Irene Epifanio

See Also

boxplot

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- growth$hgtm
hgtf <- growth$hgtf[,1:39]

# Create array:
nvars <- 2
data.array <- array(0, dim = c(dim(hgtm), nvars))
data.array[,,1] <- as.matrix(hgtm)
data.array[,,2] <- as.matrix(hgtf)
rownames(data.array) <- 1:nrow(hgtm)
colnames(data.array) <- colnames(hgtm)
str(data.array)

# Create basis:
nbasis <- 10
basis_fd <- create.bspline.basis(c(1,nrow(hgtm)), nbasis)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:nrow(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = data.array, basisobj = basis_fd)

X <- array(0, dim = c(dim(t(temp_fd$coefs[,,1])), nvars))
X[,,1] <- t(temp_fd$coef[,,1]) 
X[,,2] <- t(temp_fd$coef[,,2])

# Standardize the variables:
Xs <- X
Xs[,,1] <- scale(X[,,1])
Xs[,,2] <- scale(X[,,2])

x1 <- t(Xs[,,1]) 
for (i in 2:nvars) { 
 x12 <- t(Xs[,,i]) 
 x1 <- rbind(x1, x12) 
}
data_all <- t(x1) 

num_pts <- ncol(data_all) / nvars
range <- 3 
outl <- do_clean_multiv(t(data_all), num_pts, range, out_perc, nbasis, nvars)
outl

## End(Not run)

Run the whole functional archetypoid analysis with the Frobenius norm

Description

This function executes the entire procedure involved in the functional archetypoid analysis. Firstly, the initial vector of archetypoids is obtained using the functional archetypal algorithm and finally, the optimal vector of archetypoids is returned.

Usage

do_fada(subset, numArchoid, numRep, huge, compare = FALSE, PM,
              vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
              outl_degree = c("outl_strong", "outl_semi_strong", "outl_moderate"),
              method = "adjbox", prob)

Arguments

subset

Data to obtain archetypes. In fadalara this is a subset of the entire data frame.
numArchoid

Number of archetypes/archetypoids.
numRep

For each numArch, run the archetype algorithm numRep times.
huge

Penalization added to solve the convex least squares problems.
compare

Boolean argument to compute the robust residual sum of squares to compare these results with the ones provided by do_fada_robust.
PM

Penalty matrix obtained with eval.penalty.
vect_tol

Vector the tolerance values. Default c(0.95, 0.9, 0.85). Needed if method='toler'.
alpha

Significance level. Default 0.05. Needed if method='toler'.
outl_degree

Type of outlier to identify the degree of outlierness. Default c("outl_strong", "outl_semi_strong", "outl_moderate"). Needed if method='toler'.
method

Method to compute the outliers. Options allowed are 'adjbox' for using adjusted boxplots for skewed distributions, and 'toler' for using tolerance intervals.
prob

If compare=TRUE, probability with values in [0,1].

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • rss_rob: If compare_robust=TRUE, this is the residual sum of squares using the robust Frobenius norm. Otherwise, NULL.

  • resid: Vector of residuals.

  • outliers: Outliers.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

See Also

stepArchetypesRawData_funct, archetypoids_funct

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- t(growth$hgtm)

# Create basis:
basis_fd <- create.bspline.basis(c(1,ncol(hgtm)), 10)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:ncol(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = growth$hgtm, basisobj = basis_fd)
data_archs <- t(temp_fd$coefs)

suppressWarnings(RNGversion("3.5.0"))
set.seed(2018)
res_fada <- do_fada(subset = data_archs, numArchoid = 3, numRep = 5, huge = 200, 
                    compare = FALSE, PM = PM, method = "adjbox")
str(res_fada)      

suppressWarnings(RNGversion("3.5.0"))
set.seed(2018)
res_fada1 <- do_fada(subset = data_archs, numArchoid = 3, numRep = 5, huge = 200, 
                     compare = FALSE, PM = PM, 
                     vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
                     outl_degree = c("outl_strong", "outl_semi_strong", "outl_moderate"),
                     method = "toler")
str(res_fada1)                               

res_fada2 <- do_fada(subset = data_archs, numArchoid = 3, numRep = 5, huge = 200, 
                    compare = TRUE, PM = PM, method = "adjbox", prob = 0.8)
str(res_fada2)  


## End(Not run)

Run the whole archetypoid analysis with the functional multivariate Frobenius norm

Description

This function executes the entire procedure involved in the functional archetypoid analysis. Firstly, the initial vector of archetypoids is obtained using the functional archetypal algorithm and finally, the optimal vector of archetypoids is returned.

Usage

do_fada_multiv(subset, numArchoid, numRep, huge, compare = FALSE, PM,
               method = "adjbox", prob)

Arguments

subset

Data to obtain archetypes. In fadalara this is a subset of the entire data frame.
numArchoid

Number of archetypes/archetypoids.
numRep

For each numArch, run the archetype algorithm numRep times.
huge

Penalization added to solve the convex least squares problems.
compare

Boolean argument to compute the robust residual sum of squares to compare these results with the ones provided by do_fada_robust.
PM

Penalty matrix obtained with eval.penalty.
method

Method to compute the outliers. So far the only option allowed is 'adjbox' for using adjusted boxplots for skewed distributions. The use of tolerance intervals might also be explored in the future for the multivariate case.
prob

If compare=TRUE, probability with values in [0,1].

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • rss_rob: If compare_robust=TRUE, this is the residual sum of squares using the robust Frobenius norm. Otherwise, NULL.

  • resid: Vector of residuals.

  • outliers: Outliers.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

See Also

stepArchetypesRawData_funct_multiv, archetypoids_funct_multiv

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- growth$hgtm
hgtf <- growth$hgtf[,1:39]

# Create array:
nvars <- 2
data.array <- array(0, dim = c(dim(hgtm), nvars))
data.array[,,1] <- as.matrix(hgtm)
data.array[,,2] <- as.matrix(hgtf)
rownames(data.array) <- 1:nrow(hgtm)
colnames(data.array) <- colnames(hgtm)
str(data.array)

# Create basis:
nbasis <- 10
basis_fd <- create.bspline.basis(c(1,nrow(hgtm)), nbasis)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:nrow(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = data.array, basisobj = basis_fd)

X <- array(0, dim = c(dim(t(temp_fd$coefs[,,1])), nvars))
X[,,1] <- t(temp_fd$coef[,,1]) 
X[,,2] <- t(temp_fd$coef[,,2])

# Standardize the variables:
Xs <- X
Xs[,,1] <- scale(X[,,1])
Xs[,,2] <- scale(X[,,2])

suppressWarnings(RNGversion("3.5.0"))
set.seed(2018)
res_fada <- do_fada_multiv(subset = Xs, numArchoid = 3, numRep = 5, huge = 200, 
                           compare = FALSE, PM = PM, method = "adjbox")
str(res_fada)     

## End(Not run)

Run the whole archetypoid analysis with the functional multivariate robust Frobenius norm

Description

This function executes the entire procedure involved in the functional archetypoid analysis. Firstly, the initial vector of archetypoids is obtained using the functional archetypal algorithm and finally, the optimal vector of archetypoids is returned.

Usage

do_fada_multiv_robust(subset, numArchoid, numRep, huge, prob, compare = FALSE, PM,
                      method = "adjbox")

Arguments

subset

Data to obtain archetypes. In fadalara this is a subset of the entire data frame.
numArchoid

Number of archetypes/archetypoids.
numRep

For each numArch, run the archetype algorithm numRep times.
huge

Penalization to solve the convex least squares problem, see archetypoids.
prob

Probability with values in [0,1].
compare

Boolean argument to compute the non-robust residual sum of squares to compare these results with the ones provided by do_fada.
PM

Penalty matrix obtained with eval.penalty.
method

Method to compute the outliers. So far the only option allowed is 'adjbox' for using adjusted boxplots for skewed distributions. The use of tolerance intervals might also be explored in the future for the multivariate case.

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • rss_non_rob: If compare=TRUE, this is the residual sum of squares using the non-robust Frobenius norm. Otherwise, NULL.

  • resid Vector of residuals.

  • outliers: Outliers.

  • local_rel_imp Matrix with the local (casewise) relative importance (in percentage) of each variable for the outlier identification. Only for the multivariate case. It is relative to the outlier observation itself. The other observations are not considered for computing this importance. This procedure works because the functional variables are in the same scale, after standardizing. Otherwise, it couldn't be interpreted like that.

  • margi_rel_imp Matrix with the marginal relative importance of each variable (in percentage) for the outlier identification. Only for the multivariate case. In this case, the other points are considered, since the value of the outlier observation is compared with the remaining points.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

See Also

stepArchetypesRawData_funct_multiv_robust, archetypoids_funct_multiv_robust

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- growth$hgtm
hgtf <- growth$hgtf[,1:39]

# Create array:
nvars <- 2
data.array <- array(0, dim = c(dim(hgtm), nvars))
data.array[,,1] <- as.matrix(hgtm)
data.array[,,2] <- as.matrix(hgtf)
rownames(data.array) <- 1:nrow(hgtm)
colnames(data.array) <- colnames(hgtm)
str(data.array)

# Create basis:
nbasis <- 10
basis_fd <- create.bspline.basis(c(1,nrow(hgtm)), nbasis)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:nrow(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = data.array, basisobj = basis_fd)

X <- array(0, dim = c(dim(t(temp_fd$coefs[,,1])), nvars))
X[,,1] <- t(temp_fd$coef[,,1]) 
X[,,2] <- t(temp_fd$coef[,,2])

# Standardize the variables:
Xs <- X
Xs[,,1] <- scale(X[,,1])
Xs[,,2] <- scale(X[,,2])

suppressWarnings(RNGversion("3.5.0"))
set.seed(2018)
res_fada <- do_fada_multiv_robust(subset = Xs, numArchoid = 3, numRep = 5, huge = 200, 
                                  prob = 0.75, compare = FALSE, PM = PM, method = "adjbox")
str(res_fada)
res_fada$cases
#[1]  8 24 29
res_fada$rss
#[1] 2.301741

## End(Not run)

Run the whole archetypoid analysis with the functional robust Frobenius norm

Description

This function executes the entire procedure involved in the functional archetypoid analysis. Firstly, the initial vector of archetypoids is obtained using the functional archetypal algorithm and finally, the optimal vector of archetypoids is returned.

Usage

do_fada_robust(subset, numArchoid, numRep, huge, prob, compare = FALSE, PM,
              vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
              outl_degree = c("outl_strong", "outl_semi_strong", "outl_moderate"),
              method = "adjbox")

Arguments

subset

Data to obtain archetypes. In fadalara this is a subset of the entire data frame.
numArchoid

Number of archetypes/archetypoids.
numRep

For each numArch, run the archetype algorithm numRep times.
huge

Penalization added to solve the convex least squares problems.
prob

Probability with values in [0,1].
compare

Boolean argument to compute the non-robust residual sum of squares to compare these results with the ones provided by do_fada.
PM

Penalty matrix obtained with eval.penalty.
vect_tol

Vector the tolerance values. Default c(0.95, 0.9, 0.85). Needed if method='toler'.
alpha

Significance level. Default 0.05. Needed if method='toler'.
outl_degree

Type of outlier to identify the degree of outlierness. Default c("outl_strong", "outl_semi_strong", "outl_moderate"). Needed if method='toler'.
method

Method to compute the outliers. Options allowed are 'adjbox' for using adjusted boxplots for skewed distributions, and 'toler' for using tolerance intervals.

Value

A list with the following elements:

  • cases: Final vector of archetypoids.

  • alphas: Alpha coefficients for the final vector of archetypoids.

  • rss: Residual sum of squares corresponding to the final vector of archetypoids.

  • rss_non_rob: If compare=TRUE, this is the residual sum of squares using the non-robust Frobenius norm. Otherwise, NULL.

  • resid: Vector of residuals.

  • outliers: Outliers.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

See Also

stepArchetypesRawData_funct_robust, archetypoids_funct_robust

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- t(growth$hgtm)

# Create basis:
basis_fd <- create.bspline.basis(c(1,ncol(hgtm)), 10)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:ncol(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = growth$hgtm, basisobj = basis_fd)
data_archs <- t(temp_fd$coefs)

suppressWarnings(RNGversion("3.5.0"))
set.seed(2018)
res_fada_rob <- do_fada_robust(subset = data_archs, numArchoid = 3, numRep = 5, huge = 200,
                               prob = 0.75, compare = FALSE, PM = PM, method = "adjbox")
str(res_fada_rob)  

suppressWarnings(RNGversion("3.5.0"))
set.seed(2018)
res_fada_rob1 <- do_fada_robust(subset = data_archs, numArchoid = 3, numRep = 5, huge = 200,
                                prob = 0.75, compare = FALSE, PM = PM, 
                                vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
                                outl_degree = c("outl_strong", "outl_semi_strong", "outl_moderate"),
                                method = "toler")
str(res_fada_rob1) 

## End(Not run)

kNN for outlier detection

Description

Ramaswamy et al. proposed the k-nearest neighbors outlier detection method (kNNo). Each point's anomaly score is the distance to its kth nearest neighbor in the data set. Then, all points are ranked based on this distance. The higher an example's score is, the more anomalous it is.

Usage

do_knno(data, k, top_n)

Arguments

data

Data observations.
k

Number of neighbors of a point that we are interested in.
top_n

Total number of outliers we are interested in.

Value

Vector of outliers.

Author(s)

Guillermo Vinue

References

Ramaswamy, S., Rastogi, R. and Shim, K. Efficient Algorithms for Mining Outliers from Large Data Sets. SIGMOD'00 Proceedings of the 2000 ACM SIGMOD international conference on Management of data, 2000, 427-438.

Examples

data(mtcars)
data <- as.matrix(mtcars)
outl <- do_knno(data, 3, 2)
outl
data[outl,]

Degree of outlierness

Description

Classification of outliers according to their degree of outlierness. They are classified using the tolerance proportion. For instance, outliers from a 95

Usage

do_outl_degree(vect_tol = c(0.95, 0.9, 0.85), resid_vect, alpha = 0.05, 
               outl_degree = c("outl_strong", "outl_semi_strong", "outl_moderate"))

Arguments

vect_tol

Vector the tolerance values. Default c(0.95, 0.9, 0.85).
resid_vect

Vector of n residuals, where n was the number of rows of the data matrix.
alpha

Significance level. Default 0.05.
outl_degree

Type of outlier to identify the degree of outlierness. Default c("outl_strong", "outl_semi_strong", "outl_moderate").

Value

List with the type outliers.

Author(s)

Guillermo Vinue

See Also

outl_toler

Examples

do_outl_degree(0.95, 1:100, 0.05, "outl_strong")

Functional parallel archetypoid algorithm for large applications (FADALARA)

Description

The FADALARA algorithm is based on the CLARA clustering algorithm. This is the parallel version of the algorithm. It allows to detect anomalies (outliers). In the univariate case, there are two different methods to detect them: the adjusted boxplot (default and most reliable option) and tolerance intervals. In the multivariate case, only adjusted boxplots are used. If needed, tolerance intervals allow to define a degree of outlierness.

Usage

fadalara(data, N, m, numArchoid, numRep, huge, prob, type_alg = "fada", 
         compare = FALSE, PM, vect_tol = c(0.95, 0.9, 0.85), alpha = 0.05, 
         outl_degree = c("outl_strong", "outl_semi_strong", "outl_moderate"),
         method = "adjbox", multiv, frame)

Arguments

data

Data matrix. Each row corresponds to an observation and each column corresponds to a variable. All variables are numeric. The data must have row names so that the algorithm can identify the archetypoids in every sample.
N

Number of samples.
m

Sample size of each sample.
numArchoid

Number of archetypes/archetypoids.
numRep

For each numArch, run the archetype algorithm numRep times.
huge

Penalization added to solve the convex least squares problems.
prob

Probability with values in [0,1].
type_alg

String. Options are 'fada' for the non-robust fadalara algorithm, whereas 'fada_rob' is for the robust fadalara algorithm.
compare

Boolean argument to compute the robust residual sum of squares if type_alg = "fada" and the non-robust if type_alg = "fada_rob".
PM

Penalty matrix obtained with eval.penalty.
vect_tol

Vector the tolerance values. Default c(0.95, 0.9, 0.85). Needed if method='toler'.
alpha

Significance level. Default 0.05. Needed if method='toler'.
outl_degree

Type of outlier to identify the degree of outlierness. Default c("outl_strong", "outl_semi_strong", "outl_moderate"). Needed if method='toler'.
method

Method to compute the outliers. Options allowed are 'adjbox' for using adjusted boxplots for skewed distributions, and 'toler' for using tolerance intervals. The tolerance intervals are only computed in the univariate case, i.e., method='toler' only valid if multiv=FALSE.
multiv

Multivariate (TRUE) or univariate (FALSE) algorithm.
frame

Boolean value to indicate whether the frame is computed (Mair et al., 2017) or not. The frame is made up of a subset of extreme points, so the archetypoids are only computed on the frame. Low frame densities are obtained when only small portions of the data were extreme. However, high frame densities reduce this speed-up.

Value

A list with the following elements:

  • cases Vector of archetypoids.

  • rss Optimal residual sum of squares.

  • outliers: Outliers.

  • alphas: Matrix with the alpha coefficients.

  • local_rel_imp Matrix with the local (casewise) relative importance (in percentage) of each variable for the outlier identification. Only for the multivariate case. It is relative to the outlier observation itself. The other observations are not considered for computing this importance. This procedure works because the functional variables are in the same scale, after standardizing. Otherwise, it couldn't be interpreted like that.

  • margi_rel_imp Matrix with the marginal relative importance of each variable (in percentage) for the outlier identification. Only for the multivariate case. In this case, the other points are considered, since the value of the outlier observation is compared with the remaining points.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

Hubert, M. and Vandervieren, E., An adjusted boxplot for skewed distributions, 2008. Computational Statistics and Data Analysis 52(12), 5186-5201, https://doi.org/10.1016/j.csda.2007.11.008

Kaufman, L. and Rousseeuw, P.J., Clustering Large Data Sets, 1986. Pattern Recognition in Practice, 425-437.

Mair, S., Boubekki, A. and Brefeld, U., Frame-based Data Factorizations, 2017. Proceedings of the 34th International Conference on Machine Learning, Sydney, Australia, 1-9.

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

See Also

do_fada, do_fada_robust

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- growth$hgtm
hgtf <- growth$hgtf[,1:39]

# Create array:
nvars <- 2
data.array <- array(0, dim = c(dim(hgtm), nvars))
data.array[,,1] <- as.matrix(hgtm)
data.array[,,2] <- as.matrix(hgtf)
rownames(data.array) <- 1:nrow(hgtm)
colnames(data.array) <- colnames(hgtm)
str(data.array)

# Create basis:
nbasis <- 10
basis_fd <- create.bspline.basis(c(1,nrow(hgtm)), nbasis)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:nrow(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = data.array, basisobj = basis_fd)

X <- array(0, dim = c(dim(t(temp_fd$coefs[,,1])), nvars))
X[,,1] <- t(temp_fd$coef[,,1]) 
X[,,2] <- t(temp_fd$coef[,,2])

# Standardize the variables:
Xs <- X
Xs[,,1] <- scale(X[,,1])
Xs[,,2] <- scale(X[,,2])
# We have to give names to the dimensions to know the 
# observations that were identified as archetypoids.
dimnames(Xs) <- list(paste("Obs", 1:dim(hgtm)[2], sep = ""), 
                     1:nbasis,
                     c("boys", "girls"))

n <- dim(Xs)[1] 
# Number of archetypoids:
k <- 3 
numRep <- 20
huge <- 200

# Size of the random sample of observations:
m <- 15
# Number of samples:
N <- floor(1 + (n - m)/(m - k))
N
prob <- 0.75
data_alg <- Xs

# Parallel:
# Prepare parallelization (including the seed for reproducibility):
library(doParallel)
no_cores <- detectCores() - 1
no_cores 
cl <- makeCluster(no_cores)
registerDoParallel(cl)
clusterSetRNGStream(cl, iseed = 2018)
res_fl <- fadalara(data = data_alg, N = N, m = m, numArchoid = k, numRep = numRep, 
                   huge = huge, prob = prob, type_alg = "fada_rob", compare = FALSE, 
                   PM = PM, method = "adjbox", multiv = TRUE, frame = FALSE) # frame = TRUE
stopCluster(cl)

res_fl_copy <- res_fl
res_fl <- res_fl[which.min(unlist(sapply(res_fl, function(x) x[2])))][[1]]
str(res_fl)
res_fl$cases
res_fl$rss
as.vector(res_fl$outliers)

## End(Not run)

Functional non-parallel archetypoid algorithm for large applications (FADALARA)

Description

The FADALARA algorithm is based on the CLARA clustering algorithm. This is the non-parallel version of the algorithm. It allows to detect anomalies (outliers). In the univariate case, there are two different methods to detect them: the adjusted boxplot (default and most reliable option) and tolerance intervals. In the multivariate case, only adjusted boxplots are used. If needed, tolerance intervals allow to define a degree of outlierness.

Usage

fadalara_no_paral(data, seed, N, m, numArchoid, numRep, huge, prob, type_alg = "fada", 
                 compare = FALSE, verbose = TRUE, PM, vect_tol = c(0.95, 0.9, 0.85), 
                 alpha = 0.05, outl_degree = c("outl_strong", "outl_semi_strong", 
                 "outl_moderate"), method = "adjbox", multiv, frame)

Arguments

data

Data matrix. Each row corresponds to an observation and each column corresponds to a variable (temporal point). All variables are numeric. The data must have row names so that the algorithm can identify the archetypoids in every sample.
seed

Integer value to set the seed. This ensures reproducibility.
N

Number of samples.
m

Sample size of each sample.
numArchoid

Number of archetypes/archetypoids.
numRep

For each numArch, run the archetype algorithm numRep times.
huge

Penalization added to solve the convex least squares problems.
prob

Probability with values in [0,1].
type_alg

String. Options are 'fada' for the non-robust fadalara algorithm, whereas 'fada_rob' is for the robust fadalara algorithm.
compare

Boolean argument to compute the robust residual sum of squares if type_alg = "fada" and the non-robust if type_alg = "fada_rob".
verbose

Display progress? Default TRUE.
PM

Penalty matrix obtained with eval.penalty.
vect_tol

Vector the tolerance values. Default c(0.95, 0.9, 0.85). Needed if method='toler'.
alpha

Significance level. Default 0.05. Needed if method='toler'.
outl_degree

Type of outlier to identify the degree of outlierness. Default c("outl_strong", "outl_semi_strong", "outl_moderate"). Needed if method='toler'.
method

Method to compute the outliers. Options allowed are 'adjbox' for using adjusted boxplots for skewed distributions, and 'toler' for using tolerance intervals. The tolerance intervals are only computed in the univariate case, i.e., method='toler' only valid if multiv = FALSE.
multiv

Multivariate (TRUE) or univariate (FALSE) algorithm.
frame

Boolean value to indicate whether the frame is computed (Mair et al., 2017) or not. The frame is made up of a subset of extreme points, so the archetypoids are only computed on the frame. Low frame densities are obtained when only small portions of the data were extreme. However, high frame densities reduce this speed-up.

Value

A list with the following elements:

  • cases Vector of archetypoids.

  • rss Optimal residual sum of squares.

  • outliers: Vector of outliers.

  • alphas: Matrix with the alpha coefficients.

  • local_rel_imp Matrix with the local (casewise) relative importance (in percentage) of each variable for the outlier identification. Only for the multivariate case. It is relative to the outlier observation itself. The other observations are not considered for computing this importance. This procedure works because the functional variables are in the same scale, after standardizing. Otherwise, it couldn't be interpreted like that.

  • margi_rel_imp Matrix with the marginal relative importance of each variable (in percentage) for the outlier identification. Only for the multivariate case. In this case, the other points are considered, since the value of the outlier observation is compared with the remaining points.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

Hubert, M. and Vandervieren, E., An adjusted boxplot for skewed distributions, 2008. Computational Statistics and Data Analysis 52(12), 5186-5201, https://doi.org/10.1016/j.csda.2007.11.008

Kaufman, L. and Rousseeuw, P.J., Clustering Large Data Sets, 1986. Pattern Recognition in Practice, 425-437.

Mair, S., Boubekki, A. and Brefeld, U., Frame-based Data Factorizations, 2017. Proceedings of the 34th International Conference on Machine Learning, Sydney, Australia, 1-9.

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

See Also

fadalara

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- growth$hgtm
hgtf <- growth$hgtf[,1:39]

# Create array:
nvars <- 2
data.array <- array(0, dim = c(dim(hgtm), nvars))
data.array[,,1] <- as.matrix(hgtm)
data.array[,,2] <- as.matrix(hgtf)
rownames(data.array) <- 1:nrow(hgtm)
colnames(data.array) <- colnames(hgtm)
str(data.array)

# Create basis:
nbasis <- 10
basis_fd <- create.bspline.basis(c(1,nrow(hgtm)), nbasis)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:nrow(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = data.array, basisobj = basis_fd)

X <- array(0, dim = c(dim(t(temp_fd$coefs[,,1])), nvars))
X[,,1] <- t(temp_fd$coef[,,1]) 
X[,,2] <- t(temp_fd$coef[,,2])

# Standardize the variables:
Xs <- X
Xs[,,1] <- scale(X[,,1])
Xs[,,2] <- scale(X[,,2])
# We have to give names to the dimensions to know the 
# observations that were identified as archetypoids.
dimnames(Xs) <- list(paste("Obs", 1:dim(hgtm)[2], sep = ""), 
                     1:nbasis,
                     c("boys", "girls"))

n <- dim(Xs)[1] 
# Number of archetypoids:
k <- 3 
numRep <- 20
huge <- 200

# Size of the random sample of observations:
m <- 15
# Number of samples:
N <- floor(1 + (n - m)/(m - k))
N
prob <- 0.75
data_alg <- Xs

seed <- 2018
res_fl <- fadalara_no_paral(data = data_alg, seed = seed, N = N, m = m, 
                            numArchoid = k, numRep = numRep, huge = huge, 
                            prob = prob, type_alg = "fada_rob", compare = FALSE, 
                            verbose = TRUE, PM = PM, method = "adjbox", multiv = TRUE,
                            frame = FALSE) # frame = TRUE
                   
str(res_fl)
res_fl$cases
res_fl$rss
as.vector(res_fl$outliers)

## End(Not run)

Compute archetypes frame

Description

Computing the frame with the approach by Mair et al. (2017).

Usage

frame_in_r(X)

Arguments

X

Data frame.

Value

Vector with the observations that belong to the frame.

Author(s)

Sebastian Mair, code kindly provided by him.

References

Mair, S., Boubekki, A. and Brefeld, U., Frame-based Data Factorizations, 2017. Proceedings of the 34th International Conference on Machine Learning, Sydney, Australia, 1-9.

Examples

## Not run: 
X <- mtcars
q <- frame_in_r(X)
H <- X[q,]
q

## End(Not run)

Frobenius norm

Description

Computes the Frobenius norm.

Usage

frobenius_norm(m)

Arguments

m

Data matrix with the residuals. This matrix has the same dimensions as the original data matrix.

Details

Residuals are vectors. If there are p variables (columns), for every observation there is a residual that there is a p-dimensional vector. If there are n observations, the residuals are an n times p matrix.

Value

Real number.

Author(s)

Guillermo Vinue, Irene Epifanio

References

Eugster, M.J.A. and Leisch, F., From Spider-Man to Hero - Archetypal Analysis in R, 2009. Journal of Statistical Software 30(8), 1-23, https://doi.org/10.18637/jss.v030.i08

Vinue, G., Epifanio, I., and Alemany, S.,Archetypoids: a new approach to define representative archetypal data, 2015. Computational Statistics and Data Analysis 87, 102-115, https://doi.org/10.1016/j.csda.2015.01.018

Vinue, G., Anthropometry: An R Package for Analysis of Anthropometric Data, 2017. Journal of Statistical Software 77(6), 1-39, https://doi.org/10.18637/jss.v077.i06

Examples

mat <- matrix(1:4, nrow = 2)
frobenius_norm(mat)

Functional Frobenius norm

Description

Computes the functional Frobenius norm.

Usage

frobenius_norm_funct(m, PM)

Arguments

m

Data matrix with the residuals. This matrix has the same dimensions as the original data matrix.
PM

Penalty matrix obtained with eval.penalty.

Details

Residuals are vectors. If there are p variables (columns), for every observation there is a residual that there is a p-dimensional vector. If there are n observations, the residuals are an n times p matrix.

Value

Real number.

Author(s)

Irene Epifanio

References

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

Examples

library(fda)
mat <- matrix(1:9, nrow = 3)
fbasis <- create.fourier.basis(rangeval = c(1, 32), nbasis = 3)
PM <- eval.penalty(fbasis)
frobenius_norm_funct(mat, PM)

Functional multivariate Frobenius norm

Description

Computes the functional multivariate Frobenius norm.

Usage

frobenius_norm_funct_multiv(m, PM)

Arguments

m

Data matrix with the residuals. This matrix has the same dimensions as the original data matrix.
PM

Penalty matrix obtained with eval.penalty.

Details

Residuals are vectors. If there are p variables (columns), for every observation there is a residual that there is a p-dimensional vector. If there are n observations, the residuals are an n times p matrix.

Value

Real number.

Author(s)

Irene Epifanio

References

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

Examples

mat <- matrix(1:400, ncol = 20)
PM <- matrix(1:100, ncol = 10)
frobenius_norm_funct_multiv(mat, PM)

Functional multivariate robust Frobenius norm

Description

Computes the functional multivariate robust Frobenius norm.

Usage

frobenius_norm_funct_multiv_robust(m, PM, prob, nbasis, nvars)

Arguments

m

Data matrix with the residuals. This matrix has the same dimensions as the original data matrix.
PM

Penalty matrix obtained with eval.penalty.
prob

Probability with values in [0,1].
nbasis

Number of basis.
nvars

Number of variables.

Details

Residuals are vectors. If there are p variables (columns), for every observation there is a residual that there is a p-dimensional vector. If there are n observations, the residuals are an n times p matrix.

Value

Real number.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Examples

mat <- matrix(1:400, ncol = 20)
PM <- matrix(1:100, ncol = 10)
frobenius_norm_funct_multiv_robust(mat, PM, 0.8, 10, 2)

Functional robust Frobenius norm

Description

Computes the functional robust Frobenius norm.

Usage

frobenius_norm_funct_robust(m, PM, prob)

Arguments

m

Data matrix with the residuals. This matrix has the same dimensions as the original data matrix.
PM

Penalty matrix obtained with eval.penalty.
prob

Probability with values in [0,1].

Details

Residuals are vectors. If there are p variables (columns), for every observation there is a residual that there is a p-dimensional vector. If there are n observations, the residuals are an n times p matrix.

Value

Real number.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Examples

library(fda)
mat <- matrix(1:9, nrow = 3)
fbasis <- create.fourier.basis(rangeval = c(1, 32), nbasis = 3)
PM <- eval.penalty(fbasis)
frobenius_norm_funct_robust(mat, PM, 0.8)

Robust Frobenius norm

Description

Computes the robust Frobenius norm.

Usage

frobenius_norm_robust(m, prob)

Arguments

m

Data matrix with the residuals. This matrix has the same dimensions as the original data matrix.
prob

Probability with values in [0,1].

Details

Residuals are vectors. If there are p variables (columns), for every observation there is a residual that there is a p-dimensional vector. If there are n observations, the residuals are an n times p matrix.

Value

Real number.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Examples

mat <- matrix(1:4, nrow = 2)
frobenius_norm_robust(mat, 0.8)

Interior product between matrices

Description

Helper function to compute the Frobenius norm.

Usage

int_prod_mat(m)

Arguments

m

Data matrix.

Value

Data matrix.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Examples

mat <- matrix(1:4, nrow = 2)
int_prod_mat(mat)

Interior product between matrices for FDA

Description

Helper function to compute the Frobenius norm in the functional data analysis (FDA) scenario.

Usage

int_prod_mat_funct(m, PM)

Arguments

m

Data matrix.
PM

Penalty matrix obtained with eval.penalty.

Value

Data matrix.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Examples

library(fda)
mat <- matrix(1:9, nrow = 3)
fbasis <- create.fourier.basis(rangeval = c(1, 32), nbasis = 3)
PM <- eval.penalty(fbasis)  
int_prod_mat_funct(mat, PM)

Squared interior product between matrices

Description

Helper function to compute the robust Frobenius norm.

Usage

int_prod_mat_sq(m)

Arguments

m

Data matrix.

Value

Data matrix.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Examples

mat <- matrix(1:4, nrow = 2)
int_prod_mat_sq(mat)

Squared interior product between matrices for FDA

Description

Helper function to compute the robust Frobenius norm in the functional data analysis (FDA) scenario.

Usage

int_prod_mat_sq_funct(m, PM)

Arguments

m

Data matrix.
PM

Penalty matrix obtained with eval.penalty.

Value

Data matrix.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Examples

library(fda)
mat <- matrix(1:9, nrow = 3)
fbasis <- create.fourier.basis(rangeval = c(1, 32), nbasis = 3)
PM <- eval.penalty(fbasis)  
int_prod_mat_sq_funct(mat, PM)

Tolerance outliers

Description

Outliers according to a tolerance interval. This function is used by the archetypoid algorithms to identify the outliers. See the function nptol.int in package tolerance.

Usage

outl_toler(p_tol = 0.95, resid_vect, alpha = 0.05)

Arguments

p_tol

The proportion of observations to be covered by this tolerance interval.
resid_vect

Vector of n residuals, where n was the number of rows of the data matrix.
alpha

Significance level.

Value

Vector with the outliers.

Author(s)

Guillermo Vinue

References

Young, D., tolerance: An R package for estimating tolerance intervals, 2010. Journal of Statistical Software, 36(5), 1-39, https://doi.org/10.18637/jss.v036.i05

See Also

adalara, fadalara, do_outl_degree

Examples

outl_toler(0.95, 1:100, 0.05)

Archetype algorithm to raw data with the functional Frobenius norm

Description

This is a slight modification of stepArchetypesRawData to use the functional archetype algorithm with the Frobenius norm.

Usage

stepArchetypesRawData_funct(data, numArch, numRep = 3, 
                            verbose = TRUE, saveHistory = FALSE, PM)

Arguments

data

Data to obtain archetypes.
numArch

Number of archetypes to compute, from 1 to numArch.
numRep

For each numArch, run the archetype algorithm numRep times.
verbose

If TRUE, the progress during execution is shown.
saveHistory

Save execution steps.
PM

Penalty matrix obtained with eval.penalty.

Value

A list with the archetypes.

Author(s)

Irene Epifanio

References

Cutler, A. and Breiman, L., Archetypal Analysis. Technometrics, 1994, 36(4), 338-347, https://doi.org/10.2307/1269949

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

Eugster, M.J.A. and Leisch, F., From Spider-Man to Hero - Archetypal Analysis in R, 2009. Journal of Statistical Software 30(8), 1-23, https://doi.org/10.18637/jss.v030.i08

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- t(growth$hgtm)
# Create basis:
basis_fd <- create.bspline.basis(c(1,ncol(hgtm)), 10)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:ncol(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = growth$hgtm, basisobj = basis_fd)
data_archs <- t(temp_fd$coefs)

lass <- stepArchetypesRawData_funct(data = data_archs, numArch = 3, 
                                    numRep = 5, verbose = FALSE, 
                                    saveHistory = FALSE, PM)
str(lass)   
length(lass[[1]])
class(lass[[1]])  
class(lass[[1]][[5]])                                 

## End(Not run)

Archetype algorithm to raw data with the functional multivariate Frobenius norm

Description

This is a slight modification of stepArchetypesRawData to use the functional archetype algorithm with the multivariate Frobenius norm.

Usage

stepArchetypesRawData_funct_multiv(data, numArch, numRep = 3, 
                                   verbose = TRUE, saveHistory = FALSE, PM)

Arguments

data

Data to obtain archetypes.
numArch

Number of archetypes to compute, from 1 to numArch.
numRep

For each numArch, run the archetype algorithm numRep times.
verbose

If TRUE, the progress during execution is shown.
saveHistory

Save execution steps.
PM

Penalty matrix obtained with eval.penalty.

Value

A list with the archetypes.

Author(s)

Irene Epifanio

References

Cutler, A. and Breiman, L., Archetypal Analysis. Technometrics, 1994, 36(4), 338-347, https://doi.org/10.2307/1269949

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

Eugster, M.J.A. and Leisch, F., From Spider-Man to Hero - Archetypal Analysis in R, 2009. Journal of Statistical Software 30(8), 1-23, https://doi.org/10.18637/jss.v030.i08

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- growth$hgtm
hgtf <- growth$hgtf[,1:39]

# Create array:
nvars <- 2
data.array <- array(0, dim = c(dim(hgtm), nvars))
data.array[,,1] <- as.matrix(hgtm)
data.array[,,2] <- as.matrix(hgtf)
rownames(data.array) <- 1:nrow(hgtm)
colnames(data.array) <- colnames(hgtm)
str(data.array)

# Create basis:
nbasis <- 10
basis_fd <- create.bspline.basis(c(1,nrow(hgtm)), nbasis)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:nrow(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = data.array, basisobj = basis_fd)

X <- array(0, dim = c(dim(t(temp_fd$coefs[,,1])), nvars))
X[,,1] <- t(temp_fd$coef[,,1]) 
X[,,2] <- t(temp_fd$coef[,,2])

# Standardize the variables:
Xs <- X
Xs[,,1] <- scale(X[,,1])
Xs[,,2] <- scale(X[,,2])

lass <- stepArchetypesRawData_funct_multiv(data = Xs, numArch = 3, 
                                           numRep = 5, verbose = FALSE, 
                                           saveHistory = FALSE, PM)
                                           
str(lass)   
length(lass[[1]])
class(lass[[1]])  
class(lass[[1]][[5]])                                             

## End(Not run)

Archetype algorithm to raw data with the functional multivariate robust Frobenius norm

Description

This is a slight modification of stepArchetypesRawData to use the functional archetype algorithm with the multivariate Frobenius norm.

Usage

stepArchetypesRawData_funct_multiv_robust(data, numArch, numRep = 3, 
                            verbose = TRUE, saveHistory = FALSE, PM, prob, nbasis, nvars)

Arguments

data

Data to obtain archetypes.
numArch

Number of archetypes to compute, from 1 to numArch.
numRep

For each numArch, run the archetype algorithm numRep times.
verbose

If TRUE, the progress during execution is shown.
saveHistory

Save execution steps.
PM

Penalty matrix obtained with eval.penalty.
prob

Probability with values in [0,1].
nbasis

Number of basis.
nvars

Number of variables.

Value

A list with the archetypes.

Author(s)

Irene Epifanio

References

Cutler, A. and Breiman, L., Archetypal Analysis. Technometrics, 1994, 36(4), 338-347, https://doi.org/10.2307/1269949

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

Eugster, M.J.A. and Leisch, F., From Spider-Man to Hero - Archetypal Analysis in R, 2009. Journal of Statistical Software 30(8), 1-23, https://doi.org/10.18637/jss.v030.i08

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- growth$hgtm
hgtf <- growth$hgtf[,1:39]

# Create array:
nvars <- 2
data.array <- array(0, dim = c(dim(hgtm), nvars))
data.array[,,1] <- as.matrix(hgtm)
data.array[,,2] <- as.matrix(hgtf)
rownames(data.array) <- 1:nrow(hgtm)
colnames(data.array) <- colnames(hgtm)
str(data.array)

# Create basis:
nbasis <- 10
basis_fd <- create.bspline.basis(c(1,nrow(hgtm)), nbasis)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:nrow(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = data.array, basisobj = basis_fd)

X <- array(0, dim = c(dim(t(temp_fd$coefs[,,1])), nvars))
X[,,1] <- t(temp_fd$coef[,,1]) 
X[,,2] <- t(temp_fd$coef[,,2])

# Standardize the variables:
Xs <- X
Xs[,,1] <- scale(X[,,1])
Xs[,,2] <- scale(X[,,2])

lass <- stepArchetypesRawData_funct_multiv_robust(data = Xs, numArch = 3, 
                                                  numRep = 5, verbose = FALSE, 
                                                  saveHistory = FALSE, PM, prob = 0.8, 
                                                  nbasis, nvars)
str(lass)   
length(lass[[1]])
class(lass[[1]])  
class(lass[[1]][[5]]) 

## End(Not run)

Archetype algorithm to raw data with the functional robust Frobenius norm

Description

This is a slight modification of stepArchetypesRawData to use the functional archetype algorithm with the functional robust Frobenius norm.

Usage

stepArchetypesRawData_funct_robust(data, numArch, numRep = 3, 
                                verbose = TRUE, saveHistory = FALSE, PM, prob)

Arguments

data

Data to obtain archetypes.
numArch

Number of archetypes to compute, from 1 to numArch.
numRep

For each numArch, run the archetype algorithm numRep times.
verbose

If TRUE, the progress during execution is shown.
saveHistory

Save execution steps.
PM

Penalty matrix obtained with eval.penalty.
prob

Probability with values in [0,1].

Value

A list with the archetypes.

Author(s)

Irene Epifanio

References

Cutler, A. and Breiman, L., Archetypal Analysis. Technometrics, 1994, 36(4), 338-347, https://doi.org/10.2307/1269949

Epifanio, I., Functional archetype and archetypoid analysis, 2016. Computational Statistics and Data Analysis 104, 24-34, https://doi.org/10.1016/j.csda.2016.06.007

Eugster, M.J.A. and Leisch, F., From Spider-Man to Hero - Archetypal Analysis in R, 2009. Journal of Statistical Software 30(8), 1-23, https://doi.org/10.18637/jss.v030.i08

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Examples

## Not run: 
library(fda)
?growth
str(growth)
hgtm <- t(growth$hgtm)
# Create basis:
basis_fd <- create.bspline.basis(c(1,ncol(hgtm)), 10)
PM <- eval.penalty(basis_fd)
# Make fd object:
temp_points <- 1:ncol(hgtm)
temp_fd <- Data2fd(argvals = temp_points, y = growth$hgtm, basisobj = basis_fd)
data_archs <- t(temp_fd$coefs)

lass <- stepArchetypesRawData_funct_robust(data = data_archs, numArch = 3, 
                                           numRep = 5, verbose = FALSE, 
                                           saveHistory = FALSE, PM, prob = 0.8)
str(lass)   
length(lass[[1]])
class(lass[[1]])  
class(lass[[1]][[5]]) 

## End(Not run)

Archetype algorithm to raw data with the Frobenius norm

Description

This is a slight modification of stepArchetypesRawData to use the archetype algorithm with the Frobenius norm.

Usage

stepArchetypesRawData_norm_frob(data, numArch, numRep = 3, 
                                verbose = TRUE, saveHistory = FALSE)

Arguments

data

Data to obtain archetypes.
numArch

Number of archetypes to compute, from 1 to numArch.
numRep

For each numArch, run the archetype algorithm numRep times.
verbose

If TRUE, the progress during execution is shown.
saveHistory

Save execution steps.

Value

A list with the archetypes.

Author(s)

Irene Epifanio

References

Eugster, M.J.A. and Leisch, F., From Spider-Man to Hero - Archetypal Analysis in R, 2009. Journal of Statistical Software 30(8), 1-23, https://doi.org/10.18637/jss.v030.i08

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

Vinue, G., Epifanio, I., and Alemany, S., Archetypoids: a new approach to define representative archetypal data, 2015. Computational Statistics and Data Analysis 87, 102-115, https://doi.org/10.1016/j.csda.2015.01.018

Vinue, G., Anthropometry: An R Package for Analysis of Anthropometric Data, 2017. Journal of Statistical Software 77(6), 1-39, https://doi.org/10.18637/jss.v077.i06

See Also

stepArchetypesRawData, stepArchetypes

Examples

data(mtcars)
data <- as.matrix(mtcars)

numArch <- 5 
numRep <- 2

lass <- stepArchetypesRawData_norm_frob(data = data, numArch = 1:numArch, 
                                        numRep = numRep, verbose = FALSE)
                                        
str(lass)   
length(lass[[1]])
class(lass[[1]])

Archetype algorithm to raw data with the robust Frobenius norm

Description

This is a slight modification of stepArchetypesRawData to use the archetype algorithm with the robust Frobenius norm.

Usage

stepArchetypesRawData_robust(data, numArch, numRep = 3, 
                             verbose = TRUE, saveHistory = FALSE, prob)

Arguments

data

Data to obtain archetypes.
numArch

Number of archetypes to compute, from 1 to numArch.
numRep

For each numArch, run the archetype algorithm numRep times.
verbose

If TRUE, the progress during execution is shown.
saveHistory

Save execution steps.
prob

Probability with values in [0,1].

Value

A list with the archetypes.

Author(s)

Irene Epifanio

References

Moliner, J. and Epifanio, I., Robust multivariate and functional archetypal analysis with application to financial time series analysis, 2019. Physica A: Statistical Mechanics and its Applications 519, 195-208. https://doi.org/10.1016/j.physa.2018.12.036

See Also

stepArchetypesRawData_norm_frob

Examples

data(mtcars)
data <- as.matrix(mtcars)

numArch <- 5 
numRep <- 2

lass <- stepArchetypesRawData_robust(data = data, numArch = 1:numArch, 
                                     numRep = numRep, verbose = FALSE,
                                     saveHistory = FALSE, prob = 0.8)
str(lass)   
length(lass[[1]])
class(lass[[1]])