Mixtures of Skew-t Factor Analyzers
In this paper, we introduce a mixture of skew- factor analyzers as well as a family of mixture models based thereon. The mixture of skew- distributions model that we use arises as a limiting case of the mixture of generalized hyperbolic distributions. Like their Gaussian and -distribution analogues, our mixture of skew- factor analyzers are very well-suited to the model-based clustering of high-dimensional data. Imposing constraints on components of the decomposed covariance parameter results in the development of eight flexible models. The alternating expectation-conditional maximization algorithm is used for model parameter estimation and the Bayesian information criterion is used for model selection. The models are applied to both real and simulated data, giving superior clustering results compared to a well-established family of Gaussian mixture models.
Model-based clustering employs finite mixture models to estimate the group memberships of a given set of unlabelled observations. A finite mixture model has density of the form , where , with , are the mixing proportions, is the vector of parameters, and is the th component density. Traditionally, model-based clustering has been performed with the component densities taken to be multivariate Gaussian. However, there has recently been significant interest in non-Gaussian approaches to mixture model-based clustering (e.g., McLachlan and Peel, 1998; McLachlan et al., 2007; Karlis and Meligkotsidou, 2007; Lin, 2009; Browne et al., 2012; Lee and McLachlan, 2012; Franczak et al., 2012; Vrbik and McNicholas, 2012; Morris and McNicholas, 2013; Morris et al., 2013).
Browne and McNicholas (2013) introduced a mixture of generalized hyperbolic distributions with model density , where
is the density of the generalized hyperbolic distribution, is a vector of parameters, and is the squared Mahalanobis distance between and . As we will discuss herein (Section 3), the skew- distribution can be obtained as a special case of the generalized hyperbolic distribution (cf. Barndorff-Nielsen and Shephard, 2001).
Several alternative forms of the multivariate skew- distribution have appeared in the literature (e.g., Sahu et al., 2003; Branco and Dey, 2001; Jones and Faddy, 2003; Ma and Genton, 2004). However, the form of the distribution arising from the generalized hyperbolic distribution is chosen for this work because of its particularly attractive form, i.e., because its form is particularly conducive to the development of skew- factors (cf. Section 3.1). Furthermore, this form of the multivariate skew- distribution has computational advantages (cf. Section 3.2) and it has not previously been used for model-based clustering.
In this paper, we use this version of the skew- distribution to introduce a skew- analogue of the mixture of factor analyzers model, as well as a family of parsimonious models based thereon. The remainder of this paper is laid out as follows. In Section 2, we go through some important background material. Then we introduce our methodology, and illustrate our approach on real and simulated data (Section 4). The paper concludes with a discussion and suggestions for future work (Section 5).
2.1 Mixtures of Factor Analyzers
The factor analysis model (Spearman, 1904) assumes that a -dimensional vector of observed variables can be modelled by a -dimensional vector of latent factors where , making it useful in the analysis of high-dimensional data. We can write the factor analysis model as , where is a matrix of factor loadings, is the vector of factors, and with diag. Under this model, the marginal distribution of is .
McNicholas and Murphy (2008) introduced a family of eight parsimonious Gaussian mixture models (PGMM) that are suitable for clustering high dimensional data. These models include both mixtures of probabilistic principal component analyzers (Tipping and Bishop, 1999) as well as mixtures of factor analyzers models (Ghahramani and Hinton, 1997; McLachlan and Peel, 2000) as special cases. The most general such model is a Gaussian mixture model component covariance structure . By allowing the constraints , , and the isotropic constraint , McNicholas and Murphy (2008) introduced the eight models which comprise the PGMM family. In addition to introducing a mixture of skew- distributions model herein, we also implement a family of mixtures of skew- factor analyzers that is analogous to the PGMM family (Table 1). We refer to this family as parsimonious mixtures of skew- factor analyzers (PMSTFA).
|ID||Loading Matrix||Error Variance||Isotropic||Free Covariance Parameters|
2.2 Generalized Inverse Gaussian Distribution
We write to indicate that the random variable follows a generalized inverse Gaussian (GIG) distribution (Good, 1953; Barndorff-Nielsen and Halgreen, 1977; Blæsild, 1978; Halgreen, 1979; Jørgensen, 1982) with parameters . The density of is given by
for , where , , and is the modified Bessel function of the third kind with index . The generalized inverse Gaussian distribution has some attractive features, including the tractability of the following expected values:
where and is the modified Bessel function of the third kind with index .
2.3 AECM Algorithm
The expectation-maximization (EM) algorithm (Dempster et al., 1977) is commonly used for parameter estimation in model-based clustering. This iterative algorithm facilitates maximum likelihood estimation in the presence of incomplete data, making it well-suited for clustering problems. The E-step of the algorithm consists of computing the expected value of the complete-data log-likelihood based on the current model parameters and the M-step maximizes this expected value with respect to the model parameters. The EM algorithm iterates between these two steps until some convergence criterion is met.
The expectation-conditional maximization (ECM) algorithm (Meng and Rubin, 1993) is a variant of the EM algorithm that replaces the M-step by multiple conditional-maximization (CM) steps and is useful when the complete-data log likelihood is relatively complicated. The alternating expectation-conditional maximization (AECM) algorithm (Meng and van Dyk, 1997) is a further extension that allows the source of incomplete data to change between the CM steps. An AECM algorithm is used for parameter estimation for members of the PGMM family because there are two sources of missing data: the component membership labels and the latent factors . Note that we define such that if observation is in component and otherwise, for and . Similarly, we will use the AECM algorithms for parameter estimation for the models used herein (Section 3.2).
3.1 A Mixture of Skew- Factor Analyzers
The density of a skew- distribution can obtained as a limiting case of the generalized hyperbolic distribution by setting and , and letting . A -dimensional skew- random variable arising in this way has density
where is the location, is the scale matrix, is the skewness, and is the value for degrees of freedom. We write to denote that the random variable follows the skew- distribution such that it has the density given in (3). Now, can be obtained through the relationship
where and , where denotes the inverse Gamma distribution. We have and so, from Bayes’ theorem, .
Our skew- factor analysis model arises by setting
The marginal distribution of arising from the model in (6) is . Accordingly, our mixture of skew- factor analyzers model has density
where again denotes all model parameters.
3.2 Parameter Estimation
Parameter estimation is carried out within the AECM algorithm framework. In addition to the GIG expected value formulae given in Section 2.2, we need the expected value of the component labels, i.e.,
In the first stage of our AECM algorithm, the complete-data consist of the data , the group membership labels , and the latent , for , . Hence, the complete-data log-likelihood at this stage is
By maximizing the expected value of the complete-data log-likelihood, we obtain the parameter updates:
where , , , , , and . Note that these are analogous to the updates given by Browne and McNicholas (2013). The update for degrees of freedom does not exist in closed form and the equation
must be solved numerically to obtain the updated value of .
On the second CM-step, the complete-data consist of the data , the group membership labels , the latent , and the latent factors , for and . The expected value of the resulting complete-data log-likelihood is maximized to find updates for the parameters and . The exact nature of this complete-data log-likelihood and the resulting parameter estimates will depend on which of the eight models in the PMSTFA family is under consideration. These parameter updates are analogous to those given by McNicholas and Murphy (2008) in the Gaussian case but now the ‘sample covariance’ matrix takes the more complicated form
Note that the form of the skew- distribution introduced in Section 3.1 has a computational simplicity not present in other versions of the skew- distribution that have been used for model-based clustering (cf. Lee and McLachlan, 2013). The elegant parameter estimation for our skew- distribution arises because of the properties of the GIG distribution (Section 2.2). Note that, in the same way described by (McNicholas et al., 2010), the Woodbury identity (Woodbury, 1950) can be used to avoid inverting any non-diagonal matrices. This latter trick is particularly useful with dealing with higher dimensional data.
3.3 Model Selection
The Bayesian information criterion (BIC; Schwarz, 1978) is used to select the best model in terms of number of mixture components, number of latent factors, and the covariance structure for each member of the PMSTFA family. The BIC is defined as BIC2 log , where is the maximized log-likelihood, is the maximum likelihood estimate of the model parameters , is the number of free parameters in the model, and is the number of observations. Support for the use of the BIC in mixture model selection is given by Campbell et al. (1997) and Dasgupta and Raftery (1998), while Lopes and West (2004) provide support for its use in selecting the number of latent factors.
4.1 Performance Assessment
Although all of our illustrations are carried out as bona fide cluster analysis, i.e., no knowledge of labels is assumed, the true labels are known and can be used to asses the classification performance of our PMSTFA models. The adjusted Rand index (ARI; Hubert and Arabie, 1985) was used to assess model performance in this study. The ARI indicates class agreement between the true and estimated group memberships and in doing so, accounts for the fact that random classification would almost certainly classify some observations correctly by chance. An ARI value of 1 indicates perfect classification, a value of 0 would be expected under random classification, and a negative ARI value indicates classification which is in some sense worse than one would expect under random classification.
For all analyses performed herein, -means clustering was performed on the data and the resulting cluster memberships were used to initialize the . The degrees of freedom were initialized at and the skewness parameters were initialized to be close to zero. The estimated group means and the sample covariance matrices were initialized based on the initial values, and the matrices and were initialized following McNicholas and Murphy (2008).
4.3 Simulation Study
A thirteen-dimensional toy data set was simulated from a two-component skew- mixture model, with and . The two components are very well separated but are non-elliptical (cf. Figure 1). The PMSTFA models were applied to these simulated data for groups and factors using -means starting values. The model with the highest BIC () was the CCC model with groups and latent factors. This model obtained perfect clustering results (AR1=1.00).
For comparison, the PGMM family was also fit to the simulated data using the pgmm package (McNicholas et al., 2011) for the R software (R Core Team, 2013). The model with the highest BIC () was the CUC model with components and latent factor (ARI=0.749). Note that the classification results obtained by this model (Table 2) could be considered correct if the second and third components were merged (cf. Figure 1). This nicely illustrates that Gaussian mixtures — in this case, a special case of a mixture of factor analyzers model — can sometimes capture non-elliptical components via a posteriori merging of components (cf. Figure 2).
4.4 Australian Institute of Sport Data
The Australian Institute of Sport (AIS) data contains measured variables on 102 male and 100 female athletes. We consider body fat percentage and body mass index (BMI), taking gender to be unknown. The PMSTFA models were fit to the AIS data for with the number of latent factors fixed at . The model with the highest BIC () was the component CCC model (ARI=0.811). The classification results for this model are given in Table 3.
The PGMM models were also fit to the AIS data. The model with the highest value of the BIC () was the component UUC model (ARI=0.685). Comparing the true and predicted classifications (Table 3), we see that merging components still results in inferior classification performance when compared to the results from the best PMSTFA model (Table 3). The scatter plots in Figure 3 reinforce this point; here, we can see that the best PGMM model has effectively fitted a noise component (i.e., the green component) to pick up points that do not neatly fit into one of the other components. While we have previously demonstrated (Figure 2) that Gaussian mixtures can sensibly use multiple components to model a single asymmetric cluster, a different situation has emerged here. The PGMM solution on our AIS example illustrates that the imposition of a Gaussian component density is ipso facto a stringent constraint that can lead to classification results that are not sensible in any practical sense.
Note that this real data example was chosen to illustrate our PMSFTA models outperforming their Gaussian analogues in a fashion that cannot be overcome by merging Gaussian components. Our chosen model, using latent factor, has same number of misclassifications obtained by Lee and McLachlan (2012) using mixtures of skew- distributions. As reported by Vrbik and McNicholas (2012), this number may decrease if deterministic annealing (Zhou and Lange, 2010) is used.
4.5 Yeast Data
The PMSTFA models were applied to data containing information on cellular localization sites of 1,484 yeast proteins. The data contains measured variables including the McGeoch’s method for signal sequence recognition (mcg), the score of the ALOM membrane spanning region prediction program (alm), and the score of discriminant analysis of amino acid content of vacuolar and extracellular proteins (vac). The data are available in the UCI Machine Learning Repository with a detailed description of the data given by Nakai and Kanehisa (1991, 1992). For this analysis, we considered two localization sites, CYT and ME3, and clustered the data based on the three variables described above.
The PMSTFA models were applied for groups with latent factors from -means starting values. The best model in terms of BIC () was the component model (). The PGMM models were also fit to the data. The model with the highest BIC () was the =3 component UUC model (). Cross-tabulations of the true and estimated classifications for both the PMSTFA and PGMM models are given in Table 4. Note that for the yeast data analysis, as for the AIS data analysis, the best possible merging of Gaussian components still fails to outperform our PMSTFA models (cf. Table 4 and Figure 4).
We have introduced a mixture of skew- factor analyzers model as well as a family of mixture models based thereon, i.e., the PMSTFA family. This is the first use of a mixture of skewed factor analyzers within the literature. These models were developed using a form of the skew- distribution that had not previously been used for model-based clustering. This form of the skew- distribution arises as a special case of the generalized hyperbolic distribution and offers a natural representation for skew- factor analyzers. Furthermore, borrowing attractive features of the GIG distribution, the form of the skew- distribution we use lends itself to elegant parameter estimation via an AECM algorithm.
We illustrated our PMSTFA models on real and simulated data, comparing them to their Gaussian analogous. The real data examples illustrate our PMSTFA family outperforming the PGMM family in ways that cannot be mitigated by merging Gaussian components. Note that the data we used to illustrate our PMSFTA models were low dimensional; however, our models are well suited to high dimensional applications by dint of the fact that the number of covariance parameters is linear in data dimensionality for all eight models. Developing a mixtures of skew- distributions approach to the analysis of longitudinal data, after the fashion of McNicholas and Murphy (2010) and McNicholas and Subedi (2012), is a subject of ongoing work. Finally, although used for clustering herein, our PMSTFA models can also be used for model-based classification (see McNicholas, 2010, for the PGMM analogue) or discriminant analysis (Hastie and Tibshirani, 1996).
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