Vector-valued skewness for model-based clustering

MaxSkew and MultiSkew: Two R Packages for Detecting, Measuring and Removing Multivariate Skewness

“…Statistical applications of directional skewness include normality testing ( [28]), point estimation ( [36]), independent component analysis ( [6,29]) and cluster analysis ( [4,[9][10][11][12]18]).…”

“…It coincides with the third standardized cumulant of a random variable in the univariate case, and with the null d-dimensional vector when the underlying distribution is centrally symmetric. [11] applied it to model-based clustering.…”

“…Similarly, the third moment (cumulant) of a random vector is a matrix containing all moments (cumulants) of order three which can be obtained from the random vector itself. Statistical applications of the third moment include, but are not limited to: factor analysis ( [1,2]), density approximation ( [3][4][5]), independent component analysis ( [6]), financial econometrics ( [7,8]), cluster analysis ( [4,[9][10][11][12]), Edgeworth expansions ( [13], page 189), portfolio theory ( [14]), linear models ( [15,16]), likelihood inference ( [17]), projection pursuit ( [18]), time series ( [7,19]), spatial statistics ( [20][21][22]) and nonrandom sampling ( [23]).…”

Some remarks on theR2for clustering

“…For example, if indeed there are real, distinct groups in the data and the means of these groups are collinear, as in an allometric extension model , then “stretching” the distribution sufficiently in the direction of cluster differences will force the estimated cluster means from the k ‐means algorithm to lie in this direction . Also, the existence of distinct symmetric groups in a population (eg, a normal mixture) may cause the overall distribution to appear skewed; by estimating the direction of skewness and stretching the distribution in this direction via projection pursuit prior to clustering will yield cluster means that better align with this direction of skewness. Note also that cluster results based on standardizing the variables to unit variance first (similar to PCA performed on the correlation matrix) will not necessarily improve cluster results.…”

Third cumulant for multivariate aggregate claim models

“…Malkovich and Afifi (1973) defined the multivariate skewness of a random vector X as the maximum value β is the maximum attainable skewness by a projection of the random vector X onto a direction. Statistical applications of directional skewness include normality testing (see Malkovich and Afifi (1973)), point estimation ( see Loperfido (2010)), projection pursuit and cluster analysis (see Loperfido (2015b)). …”