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Nicola LoperfidoUniversità degli Studi di Urbino "Carlo Bo“,
Dipartimento di Economia, Società e PoliticaVia Saffi 42, Urbino (PU), ITALY
e-mail: [email protected]
Wirtschaftsuniversität Wien, 03-24-20171/42
Nicola Loperfido, Urbino University
OutlineFinite mixtures
Third moment
Multivariate skewness
Decathlon data
Nicola Loperfido, Urbino UniversityWirtschaftsuniversität Wien,
03-24-20172/42
Mixtures: problem
Are the data skewed because they come fromdifferent populations?
Nicola Loperfido, Urbino University3/42Wirtschaftsuniversität Wien,
03-24-2017
Mixtures: quotation"I have at present been unable to find any generalcondition among the moments, which would beimpossible for a skew curve and possible for acompound, and so indicate compoundness. I donot, however, despair of one being found".(Pearson, 1895)
Nicola Loperfido, Urbino University4/42Wirtschaftsuniversität Wien,
03-24-2017
Mixtures: blood pressureBlood pressure data are skewed.
Platt (1963): hypertension is an illness present in agenetically defined subpopulation.
Pickering (1968): hypertension is a labeling forthose in the upper tail of the population.
Nicola Loperfido, Urbino University5/42Wirtschaftsuniversität Wien,
03-24-2017
Mixtures: definition
A probability density function f(∙) is a finitemixture if it can be represented as a weightedaverage of several probability densityfunctions:
g
i
ii xfxf1
Nicola Loperfido, Urbino University6/42Wirtschaftsuniversität Wien,
03-24-2017
Mixtures: special cases
Normal: each component is normally distributed.
Two-component: there are only two components.
Location: components only differ in location.
Nicola Loperfido, Urbino University7/42Wirtschaftsuniversität Wien,
03-24-2017
Mixtures: options
Components
Densities
Parameters
Nicola Loperfido, Urbino University8/42Wirtschaftsuniversität Wien,
03-24-2017
Mixtures: advantages
Flexibility
Interpretability
Tractability
Nicola Loperfido, Urbino University9/42Wirtschaftsuniversität Wien,
03-24-2017
Mixtures: inferenceMaximum likelihood overestimates the number ofcomponents when they are erroneously assumedto be symmetric.
How can we choose between different models?
Nicola Loperfido, Urbino University10/42Wirtschaftsuniversität Wien,
03-24-2017
Kullback-Leibler divergenceLet p and q be two probability density functions.The Kullback-Leibler divergence of p from q is
xdxq
xpxpqpJ ln,
Nicola Loperfido, Urbino University11/42Wirtschaftsuniversität Wien,
03-24-2017
Kronecker product
Nicola Loperfido, Urbino University12/42Wirtschaftsuniversität Wien,
03-24-2017
Third moment: definition
Nicola Loperfido, Urbino University13/42Wirtschaftsuniversität Wien,
03-24-2017
Kullback-Leibler approximationLet p and q be two densities with identical means,identical covariances and possibly different thirdcumulants K3,p and K3,q. Then their Kullback-Lieblerdivergence is approximately ||K3,p -K3,q||2/12, where||.|| denotes the Euclidean distance (Lin et al, 1999).
Nicola Loperfido, Urbino University14/42Wirtschaftsuniversität Wien,
03-24-2017
Third moment: standardization
The third standardized moment (cumulant) of x isthe third moment of z=-½(x-μ).
Nicola Loperfido, Urbino University15/42Wirtschaftsuniversität Wien,
03-24-2017
TensorsA real tensor is a multidimensional array ofreal values identified by a vector ofsubscripts:
Nicola Loperfido, Urbino University16/42Wirtschaftsuniversität Wien,
03-24-2017
Third-order tensors
Nicola Loperfido, Urbino University17/42Wirtschaftsuniversität Wien,
03-24-2017
Tensor rankThe rank of a n1 n2 n3 third-order tensorA is the smallest number r satisfying
Nicola Loperfido, Urbino University18/42Wirtschaftsuniversität Wien,
03-24-2017
Symmetric tensor rankA p²×p third moment M3,x has symmetric tensorrank k if there are k p-dimensional real vectors v1,..., vk satisfying
.111,3 k
T
kk
T
x vvvvvvM
Nicola Loperfido, Urbino University19/42Wirtschaftsuniversität Wien,
03-24-2017
Third moment: special cases
The symmetric tensor rank of the third moment is
easily recovered when the underlying distribution
is either a shape mixture of skew-normals or a
location normal mixture.
Nicola Loperfido, Urbino University20/42Wirtschaftsuniversität Wien,
03-24-2017
Mardia’s skewness: definition
x, y P E(x) = V(x) = x, y i.i.d.
b1,pM = E[(x- )T -1(y- )]3
Nicola Loperfido, Urbino University21/42Wirtschaftsuniversität Wien,
03-24-2017
Mardia’s skewness: standardizationMardia’s skewness is the trace of the product of
the third standardized moment and its transpose
Nicola Loperfido, Urbino University22/42Wirtschaftsuniversität Wien,
03-24-2017
Mardias skewness: application
The Kullback-Liebler divergence between asymmetric, standardized random vector andanother standardized random vector isapproximately proportional to the Mardia’sskewness of the latter.
Nicola Loperfido, Urbino University23/42Wirtschaftsuniversität Wien,
03-24-2017
Vectorial skewness: definition
22
1
22
11
pd
p
T
V
ZZZ
ZZZ
zzz
Nicola Loperfido, Urbino University24/42Wirtschaftsuniversität Wien,
03-24-2017
Vectorial skewness: standardization
p
T
zV IvecM ,3
Nicola Loperfido, Urbino University25/42Wirtschaftsuniversität Wien,
03-24-2017
Vectorial skewness: clustering
Nicola Loperfido, Urbino University26/42
When data come from a mixture of two weakly symmetricdistributions with different means and proportionalcovariances, the projection of the standardized data onto thedirection of a vectorial skewness consistently estimates thebest linear discriminant projection.
Wirtschaftsuniversität Wien, 03-24-2017
Directional skewness: definition
Nicola Loperfido, Urbino University27/42Wirtschaftsuniversität Wien,
03-24-2017
Directional skewness: standardization
Directional skewness is a function
of the third standardized moment
Nicola Loperfido, Urbino University28/42Wirtschaftsuniversität Wien,
03-24-2017
Directional skewness: fit Let be the total and directionalskewness of the p-dimensional density q. Then thesmallest Kullback-Liebler divergence of q from alocation mixture of two symmetric densities is
approximately .
⇓The fit to the data X of a location mixture of two symmetric densities might be assessed by the difference .
and ,2,2 qq D
p
M
p bb
- ,2,2 qq D
p
M
p bb
- ,1,2 XbXb D
p
M
p
Nicola Loperfido, Urbino University29/42Wirtschaftsuniversität Wien,
03-24-2017
Decathlon data: description Units. The 23 athletes who scored points in all 10
events of the Olympic decathlon in Rio 2016.
Variables. Performances in each event, convertedinto decathlon points using IAAF scoring tables.
Source. The official website of the InternationalAssociation of Athletics Federations (IAAF).
Nicola Loperfido, Urbino University30/42Wirtschaftsuniversität Wien,
03-24-2017
Decathlon data: original variables
Nicola Loperfido, Urbino University31/42Wirtschaftsuniversität Wien,
03-24-2017
Decathlon data: summaries Data are slightly nonnormal, with low to moderate
levels of skewness and kurtosis.
The multiple scatterplot does not show anyparticular features, as for example outliers.
The number of variables is quite large with respectto the number of units.
Nicola Loperfido, Urbino University32/42Wirtschaftsuniversität Wien,
03-24-2017
Decathlon data: principal components
Nicola Loperfido, Urbino University33/42Wirtschaftsuniversität Wien,
03-24-2017
Decathlon data: summary PCA The first two principal component account for
about 55% of the total variation.
Their joint distribution is approximately normal.
They do not clearly suggest the presence ofoutliers.
Nicola Loperfido, Urbino University34/42Wirtschaftsuniversität Wien,
03-24-2017
Decathlon data: Skewed components
We computed the two most skewed and mutuallyorthogonal projections of the Decathlon datausing the R package MaxSkew.
Nicola Loperfido, Urbino University35/42Wirtschaftsuniversität Wien,
03-24-2017
Decathlon data: skewed components
Nicola Loperfido, Urbino University36/42Wirtschaftsuniversität Wien,
03-24-2017
Decathlon data: outliersKarl Robert Saluri (EST). He scored lowest
due to lower-than-average performances innearly all events.
Jeremi Taiwo (USA). He obtained an aboutaverage score due to a very unusual patternof performances.
Nicola Loperfido, Urbino University37/42Wirtschaftsuniversität Wien,
03-24-2017
Decathlon data : conclusion
The scatterplot of the first two most skewedcomponents clearly hints for the presence of twooutliers.
Nicola Loperfido, Urbino University38/42Wirtschaftsuniversität Wien,
03-24-2017
Skewness &
Mixtures
Finite Mixtures
Definition
Properties
Problem
Third MomentsDefinition
Approximation
Tensor
Skewness Measures
Total
Vectorial
Directional
Decathlon dataOriginal variables:
no outliers
Principal components: one outliers
Skewed components: two outliers
39/42Wirtschaftsuniversität Wien, 03-24-2017Nicola Loperfido, Urbino University
Future research
Package Multiskew
Fourth moment
Tensor approach
Nicola Loperfido, Urbino University40/42Wirtschaftsuniversität Wien,
03-24-2017
Essential references
Loperfido, N. (2015). Singular value decomposition of the thirdmultivariate moment. Lin. Alg. Appl. 473, 202-216.
Malkovich, J.F. and Afifi, A.A.(1973). On tests formultivariate normality. J. Amer. Statist. Ass. 68, 176-179.
Mardia, K.V. (1970). Measures of multivariate skewness andkurtosis with applications. Biometrika 57, 519-530.
• McLachlan, G. and Peel, D. (2000). Finite Mixture Models. JohnWiley and Sons Inc, New York.
Mori T.F., Rohatgi V.K. and Székely G.J. (1993). On multivariateskewness and kurtosis. Theory Probab. Appl. 38, 547-551.
Nicola Loperfido, Urbino University41/42Wirtschaftsuniversität Wien,
03-24-2017
Thank you for your attention
Nicola Loperfido, Urbino University42/42Wirtschaftsuniversität Wien,
03-24-2017