Package 'Kurt'

Title: Performs Kurtosis-Based Statistical Analyses
Description: Computes measures of multivariate kurtosis, matrices of fourth-order moments and cumulants, kurtosis-based projection pursuit. Franceschini, C. and Loperfido, N. (2018, ISBN:978-3-319-73905-2). "An Algorithm for Finding Projections with Extreme Kurtosis". Loperfido, N. (2017,ISSN:0024-3795). "A New Kurtosis Matrix, with Statistical Applications".
Authors: Cinzia Franceschini [aut, cre], Nicola Loperfido [aut]
Maintainer: Cinzia Franceschini <[email protected]>
License: GPL (>= 2)
Version: 1.1
Built: 2024-11-27 06:31:16 UTC
Source: CRAN

Help Index

Kurt: Performs kurtosis-based statistical analyses

Description

Computes measures of multivariate kurtosis, matrices of fourth-order moments and cumulants, kurtosis-based projection pursuit

Details

Index of help topics: 

Cum4                    Cum4: fourth multivariate cumulant
ExtKur                  ExtKur: kurtosis based projection pursuit
ExtKurBiv               ExtKurBiv: kurtosis-based projection pursuit
                        for bivariate random vectors
Fourth                  Fourth: matrices of fourth moments or fourth
                        cumulants
Fourth4                 Fourth4: fourth moment of a data matrix
Kurt-package            Kurt: Performs kurtosis-based statistical
                        analyses
NoKurt                  NoKurt: data projections whose excess kurtosis
                        is as close to zero as possible
ScalarKurt              ScalarKurt
optik                   optik

ScalarKurt(), ExtKurt(), ExtKurtBiv(), optik(), NoKurt(), Cum4(), Fourth(), Fourth4()

Author(s)

Cinzia Franceschini and Nicola Loperfido

Maintainer: Cinzia Franceschini [email protected]

References

Franceschini, C. and Loperfido, N. (2012). On some Inequalities between Measures of Multivariate Kurtosis, with Application to Financial Returns. In "Mathematical and Statistical Methods for Actuarial Sciences and Finance", Perna, C. and Sibillo, M. (Eds.), Springer, 211-218.

Franceschini, C. and Loperfido, N. (2018). An Algorithm for Finding Projections with Extreme Kurtosis. In "Studies in Theoretical and Applied Statistics: SIS2016-48th Meeting of the Italian Statistical Society, Salerno 8-10 June 2016", Perna C., Pratesi M. and Ruiz-Gazen A. (Eds.), Springer.

Henze, N. (1994). On Mardia's kurtosis test for multivariate normality. Communications in statistics-Theory and Methods 23:4, 1031-1045.

Kollo, T. (2008). Multivariate skewness and kurtosis measures with an application in ICA. Journal of Multivariate Analysis 99, 2328-2338.

Kollo, T. and Srivastava, M.S. (2005). Estimation and testing of parameters in multivariate Laplace distribution. Comm. Statist. 33, 2363–2687.

Koziol, J.A. (1987). An alternative formulation of Neyman's smooth goodness of fit tests under composite alternatives. Metrika 34, 17-24.

Koziol, J.A. (1989). A note on measures of multivariate kurtosis. Biometrical Journal 31, 619-624.

Loperfido, N. (2011). Spectral Analysis of the Fourth Moment Matrix. Linear Algebra and its Applications 435, 1837-1844.

Loperfido, N. (2014). A Note on the Fourth Cumulant of a Finite Mixture Distribution. Journal of Multivariate Analysis 123, 386-394.

Loperfido, N. (2017). A New Kurtosis Matrix, with Statistical Applications. Linear Algebra and its Applications 512, 1-17.

Loperfido N. (2019). Kurtosis-Based Projection Pursuit for Outlier Detection in Financial Time Series. The European Journal of Finance, to appear.

Loperfido, N. (2020). Some Remarks on Koziol's Kurtosis. Journal of Multivariate Analysis 175, to appear.

Malkovich, J.F. and Afifi, A.A. (1973). On Tests for Multivariate Normality. J. Amer. Statist. Ass. 68, 176-179.

Mardia, K.V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 519-530.

Mardia, K. V. and Kent, J. T. (1991). Rao Score Tests for Goodness of Fit and Independence. Biometrika 78, 355-36.

Miettinen J., Taskinen S., Nordhausen K. and Oja H. (2015). Fourth Moments and Independent Component Analysis. Statistical Science 30, 372-390.

Mori T.F., Rohatgi V.K. and Szekely G.J. (1993). On multivariate skewness and kurtosis. Theory Probab. Appl. 38, 547-551.

Cum4: fourth multivariate cumulant

Description

returns a matrix containing the fourth cumulants of the given data

Usage

Cum4(data, type, shape)

Arguments

data

data matrix

type

type=0 uses original data, type=1 uses centered data, type=2 uses standardized data

shape

if shape="square" the output is a d^2 x d^2 matrix. If shape="rectangular", the output is a d x d^3 matrix. Where d is the number of variables

Value

K4 is the matrix containing the fourth cumulants of the given data

Author(s)

Cinzia Franceschini and Nicola Loperfido

Examples

data(iris)
iris<-data.matrix(iris)#returns the matrix obtained by converting the data frame to numeric mode
Cum4(iris[,1:4], 1, "square") # returns a matrix containing the fourth cumulants of the given data

ExtKur: kurtosis based projection pursuit

Description

Returns a data projection with either maximal or minimal kurtosis.

Usage

ExtKur(data, iterations, maxmin)

Arguments

data

data matrix

iterations

number of required iterations

maxmin

is the choice to either maximise ("MAX") or minimise ("MIN") kurtosis

Value

linear

vector of coefficients
projection

vector of projected data
kurt

extreme kurtosis attainable by a data projection

Author(s)

Cinzia Franceschini and Nicola Loperfido

Examples

data(iris)
iris<-data.matrix(iris)#returns the matrix obtained by converting the data frame to numeric mode
ExtKur(iris[,1:4],10,"MAX") #returns a data projection with maximal kurtosis

ExtKurBiv: kurtosis-based projection pursuit for bivariate random vectors

Description

Returns a projection of bivariate data with either maximal or minimal kurtosis.

Usage

ExtKurBiv(data, maxmin)

Arguments

data

data matrix

maxmin

choice between maximal ("MAX") and minimal ("MIN") kurtosis

Value

linearMAX

coefficients of the projections maximising kurtosis
projectionMAX

projection with maximal kurtosis
kurtMAX

maximal kurtosis
linearMIN

coefficients of the projections minimising kurtosis
projectionMIN

projection with minimal kurtosis
kurtMIN

minimal kurtosis

Author(s)

Cinzia Franceschini and Nicola Loperfido

Examples

data(iris)
iris<-data.matrix(iris)#returns the matrix obtained by converting the data frame to numeric mode

ExtKurBiv(iris[,1:2],"MAX")# returns a projection of bivariate data with maximal kurtosis

Fourth: matrices of fourth moments or fourth cumulants

Description

Returns a matrix containing either the fourth moments or the fourth cumulants. It recalls the functions Fourth4 and Cum4.

Usage

Fourth(data, type, shape, feature)

Arguments

data

data matrix

type

type =0 is the ordinary fourth moment / cumulant; type =1 is the centered fourth moment / cumulant; type =2 is the standardized fourth moment / cumulant

shape

"square" or "rectangular"

feature

"moment" or "cumulant". If feature is "moment", the function computes the fourth moment of a data matrix. The function recalls the function Fourth4. If feature is "cumulant", the function computes the fourth multivariate cumulant. The function recalls the function Cum4.

Value

M

Fourth square moment matrix
MM

Fourth rectangular moment matrix
K4

Fourth cumulants of the given data

Author(s)

Cinzia Franceschini and Nicola Loperfido

Examples

data(iris)
iris<-data.matrix(iris)#returns the matrix obtained by converting the data frame to numeric mode
Fourth(iris[,1:4], 1,"square", "moment")#returns a matrix containing the fourth moments

Fourth4: fourth moment of a data matrix

Description

Returns a matrix containing the fourth moments.

Usage

Fourth4(data, type, shape)

Arguments

data

data matrix

type

type=0 is the ordinary fourth moment, type=1 is the centered fourth moment, type=2 is the standardized fourh moment

shape

"square" or "rectangular"

Value

M

Fourth square moment matrix
MM

Fourth rectangular moment matrix

Author(s)

Cinzia Franceschini and Nicola Loperfido

Examples

data(iris)
iris<-data.matrix(iris)#returns the matrix obtained by converting the data frame to numeric mode
Fourth4(iris[,1:4], 1,"square") #returns a matrix containing the fourth moments

NoKurt: data projections whose excess kurtosis is as close to zero as possible

Description

Data projections whose excess kurtosis is as close to zero as possible. Excess kurtosis is the fourth standardized cumulant , that is the fourth standardized moment minus three.

Usage

NoKurt(data, number)

Arguments

data

data matrix

number

number of required projections. It must be greater than one and less than the number of variables

Value

Nkurtoses

kurtoses of Nprojections
Nprojections

data projections ordered according to the absolute values of their excess kurtoses
MATRIX

matrix characterizing the projection

Author(s)

Cinzia Franceschini and Nicola Loperfido

Examples

data(iris)
iris<-data.matrix(iris[,1:4])
NoKurt(iris[,1:4],3)#returns data projections whose excess kurtosis is as close to zero as possible

optik

Description

It computes the matrix containing the smallest and largest kurtoses of data projections as well as the corresponding directions.

Usage

optik(data)

Arguments

data

data matrix

Value

kurMAX

kurtosis of the projection maximizing kurtosis
pMAX

projection maximizing kurtosis
dMAX

direction maximizing kurtosis
kurMINbis

kurtosis of the projection minimizing kurtosis
pMINbis

projection minimizing kurtosis
dMINbis

direction minimizing kurtosis

Author(s)

Cinzia Franceschini and Nicola Loperfido

Examples

data(iris)
iris<-data.matrix(iris)#returns the matrix obtained by converting the data frame to numeric mode
optik(iris[,1:4])#starting values of the projections with extreme kurtosis

ScalarKurt

Description

Returns the statistic and the p-value of either Mardia's kurtosis or Koziol's kurtosis tests for normality .

Usage

ScalarKurt(data, feature, type, prob)

Arguments

data

data matrix

feature

"moment" or "cumulant"

type

"Mardia" or "Koziol"

prob

"lower" if probability is P[X <= x], "upper" if probability is P[X > x], "twoside" if probability is computed on both tails

Details

For Koziol kurtosis only the upper tail is meaningful

Value

statistic
pvalue

Author(s)

Cinzia Franceschini and Nicola Loperfido

Examples

data(iris)
iris<-data.matrix(iris)#returns the matrix obtained by converting the data frame to numeric mode
ScalarKurt(iris[,1:4],"moment","Mardia","upper")#returns the statistic and the p-value
#of Mardia's kurtosis test for normality