Skewness PP has already been studied for specific models [
18,
19,
48]. Here, we assume that
; its scaled version is
, where
and
are the covariance matrix and mean vector of
, respectively. PP is aimed at finding the direction
for which the variable
attains the maximum skewness, which implies solving the following problem:
with
the standard skewness measure:
, and
the set of all non-null
p-dimensional vectors.
In order to ensure a unique solution, taking into account the scale invariance of
, the problem can be stated by the following equivalent formulations:
where
and
with
. The maximal skewness
, with the superscript
D standing for directional skewness, is a well-known index of asymmetry [
24].
Proposition 1. Let be a vector such that . Provided that the first three moments, , of the mixing variable exist, the moments of up to the third order are given by
- 1.
.
- 2.
.
- 3.
where the symbols vec and ⊗ denote the vectorization and Kronecker operators and γ is a shape asymmetry vector given by.
The next proposition establishes the relationship between cumulants and moments of multivariate SMSN distributions.
Proposition 2. Let be a vector such that . Provided that the first three moments, , of the mixing variable exist, the cumulants of up to the third order are given by
- 1.
.
- 2.
.
- 3.
.
3.1. Projection Pursuit from the Third Cumulant Matrix
Now, skewness PP is revisited using an eigenvector approach from the third cumulant matrix of the scaled input vector
. The approach is motivated by the formulation of directional skewness as the maximum of an homogeneous third-order polynomial obtained from the third cumulant matrix as follows [
11]:
This formulation inspires the eigenvector approach, based on the third-cumulant tensor, and the use of the higher-order power method (HOPM) to implement the computation [
14]. The problem admits a simplification for skew-t vectors, which in turn provides the theoretical grounds and interpretation for the choice of the initial direction in the HOPM algorithm ([
11], Proposition 2). Theorem 1 below enriches the interpretation by extending the result to the family of SMSN distributions.
Firstly, we recall the definitions of left and right singular vectors of the third cumulant matrix of a vector : We say that the vector is a right singular vector of the third cumulant matrix if is proportional to . On the other hand, the vec operator can be used to define a left singular vector of as the vectorized matrix, with a matrix, such that gives a vector proportional to .
Theorem 1. Let be a vector such that , with mixing variable having finite second order moment satisfying the condition , and let be the scaled version of the input vector having third cumulant matrix . Then, it holds that the right dominant eigenvector of is given by .
Proof. Putting together the results from Propositions 1 and 2, we can write:
Taking into account the expression for the covariance matrix provided in the second item of Proposition 2, we obtain that
When these quantities are inserted into the previous expression for
, we obtain
where
and
. Now, taking into account that
[
50], an analogous argument as the one used in [
11] leads to
On the other hand, using the standard properties of the kronecker product, the following mathematical relationships can be derived after simple calculations:
from which we obtain
and as a result:
, where
. This finding implies that
is an eigenvector of
with
its associated eigenvalue. Hence,
is a right singular vector of
.
From the well-known moment inequality
we conclude that
. Thus, taking into account the condition for the moments of the mixing variable of the statement, we can assert that
which implies that
.
It can also be shown that is the right dominant eigenvector of using the following simple argument: If is another right eigenvector orthogonal to then with . Therefore, is the right dominant eigenvector of . □
The result given by Theorem 1 gives rise to the following meaningful remarks:
First of all, we known that the direction attaining the maximal skewness projection for the scaled vector
is proportional to
[
22]; henceforth, the vector driving the maximal skewness projection for
is
which in turn lies on the direction of
, being also an eigenvector of
([
22], Lemma 1). This fact enhances the role of
as a shape vector that regulates the asymmetry of the multivariate model in a directional fashion. Furthermore, the previous disquisition also provides a grounded theoretical basis for the choice of the right dominant eigenvector of
to set the initial direction of the HOPM algorithm, an issue advocated by De Lathauwer et al. [
14] without theoretical justification.
Secondly, we can also note that the expression
from which the right dominant singular vector of
is derived, reduces to that one found by (Loperfido [
11], Proposition 2) for the case of the multivariate skew-t distribution when the moments of the skew-t are inserted into the quantities
r and
s above. Actually, our result is an extension of (Loperfido [
11], Theorem 2) to the SMSN family.
Thirdly, coming back to expression Equation (
6), recall that a unit length vector
satisfies
if and only if
with
the largest eigenvalue [
11,
51]. Thus, we can put
, with
from Theorem 1, to show that
Therefore,
which, taking into account that
with
, reduces to the expression derived by [
22] using another approach. Moreover, for the case of a SN vector we know that
S is degenerate at
; so
,
and
, which gives
as previously stated by [
18].
Finally, it is worthwhile noting that the moment condition given in the statement of Theorem 1 is fulfilled by the most popular subfamilies of SMSN distributions such as those described in
Section 2.2.
3.2. Projection Pursuit from Scatter Matrices
In the previous section, projection pursuit under SMSN multivariate distributions is formulated as an eigenvector problem which stems from the third order cumulant matrix. In this section, the approach is formulated through an eigenvector problem concerned with the simultaneous diagonalization of the scatter matrices
and
; in fact, such formulation is related to the canonical transformation of an SMSN vector [
23]. The next theorem provides the theoretical foundation for the approach.
Theorem 2. Let be a vector such that with mixing variable having finite first and second order moments. Then, η is the dominant eigenvector of .
Proof. The proof follows pieces of previous work [
22,
23], so we sketch it. From Lemma 1 (Arevalillo and Navarro [
22], Lemma 1), we know that the shape vector
is an eigenvector of
; it is straightforward to show that it has eigenvalue
Now, let us consider another eigenvector
of
. Then, it holds that
since the vector
is orthogonal to
[
23]. As a result,
has associated eigenvalue equal to
which satisfies that
; hence, we conclude that
is the dominant eigenvector of
. □
Theorem 2 establishes that the dominant eigenvector of
lies on the direction of
, which in turn gives the direction at which the maximal skewness projection is attained [
22]; this finding points out the parametric interpretation of the dominant eigenvector of
under SMSN distributions as a shape parameter accounting for the directional asymmetry of the model. On the other hand, it is well-known that the canonical transformation of a SMSN vector can be derived by the simultaneous diagonalization of the pair of scatter matrices
, with the only skewed canonical variate being obtained by the projection onto the direction of the shape vector
[
23]; therefore, the computation of the skewness-based PP under the SMSN family would only require the estimation of both scatter matrices. Another alternative to derive the canonical transformation relies on the pair of scatter matrices
, where
is the scatter kurtosis matrix given by
[
52]; such alternative has appealing properties for SMSN distributions [
23]. Hence, we can work with either
or
, in order to derive the canonical form of a SMSN vector, and compute the corresponding dominant eigenvectors in their simultaneous diagonalizations to get the vector driving the projection pursuit direction.
It is worthwhile noting that the ideas behind the canonical form of SMSN vectors are inspired in the ICS method for representing multivariate data through the comparison of two scatter matrices [
53]. Although the choice of the scatter pair is of key relevance,
has become a standard choice which works well in the statistical practice. The idea behind the ICS method is the search of a new representation coordinate system of the data by sequential maximization of the quotient
. For SMSN distributions, a natural alternative is the scatter pair
, given by the covariance and scale matrices of the model; therefore, the projection index for maximization would be
. The study and comparison of projection pursuit and the ICS method has been previously discussed in the literature ascertaining the problem of which method is better suited to uncover structure in multivariate data [
54,
55,
56]; the SMSN family is a flexible and wide class of distributions that allows to settle the problem within a parametric framework so that we can deep dive into the investigation by means of a simulation study at controlled parametric scenarios.
The next section tackles several issues concerned with the estimation of the dominant eigenvector from the formulations we have studied so far: on the one hand, the PP from the third cumulant matrix and, on the other hand, the projection pursuit from scatter matrices with the indices and .
3.3. Estimation and Computational Issues
The previous theoretical material has outstanding practical implications, as they point to different estimation methods for computing the skewness PP direction for SMSN vectors. In this section, we discuss the computational side of the issue.
Let us denote by the data matrix and let be the regular covariance matrix. Furthermore, let be the scaled data matrix, where is a vector, is the sample mean vector and the symmetric positive definite square root of . Then, the third cumulant matrix can be estimated by , where is the i-th row of .
The result given by Theorem 1 gives theoretical support for the empirical conjecture that proposes the right dominant eigenvector of the cumulant matrix
as the starting direction of the HOPM iterative method [
14], an issue also ascertained by Loperfido [
11, Proposition 1] under the rank-one assumption for the left dominant singular vector of the third cumulant matrix, as well as in the context of the multivariate skew-t distribution with a reference to Arevalillo and Navarro [
19]. When the input vector
follows a SMSN distribution, we know that the vector
, with
the right dominant eigenvector of
, is proportional to the shape vector
that yields the maximal skewness projection [
22]. Then, it stands to reason that the skewness PP direction can be estimated by
, where
is the right dominant eigenvector of
; therefore, the first method we propose to obtain the maximal skewness direction for SMSN vectors is non-parametric in nature and relies on the computation of the right dominant eigenvector of the third cumulant sample matrix.
Method 1 (M1). Estimate the skewness PP direction by .
Another approach to compute the PP direction arises from the scatter matrices
and
which can also be estimated from data. In this case the PP direction can be derived by computing the dominant eigenvector of the matrix
, where
is a sample estimate of the population kurtosis matrix
. The ICS R package [
57] has functionalities for the estimation of
and the computation of the dominant eigenvector as a result. Hence, the second approach could be stated as follows.
Method 2 (M2). Estimate the skewness PP direction by the dominant eigenvector, , of .
Theorem 2 suggests an alternative approach to the problem, which brings it to the parametric arena: Let us denote by and the maximum likelihood estimators of and ; then the dominant eigenvector of is the estimator of the shape vector derived from the maximum likelihood (ML) method. The ML approach has obvious practical limitations as it requires knowing in advance the multivariate distribution of the underlying model that generated the data. In order to overcome this limitation, we propose an approach aimed at balancing the accuracy and the computational cost of the estimation procedure; a varied bunch of simulation assays—not reported here for the sake of space—suggests, as a good tradeoff, the estimation of the dominant eigenvector of by the dominant eigenvector of , where is the regular covariance matrix and is the estimation of the scale matrix , calculated by the ML method under the multivariate SN model. The proposal can be stated as follows:
Method 3 (M3). Estimate the skewness PP direction by which is the dominant eigenvector of .
In the next section, a simulation study with artificial data is carried out in order to investigate the performance of the proposed methods.