A new mixture model on the simplex

This paper is meant to introduce a significant extension of the flexible Dirichlet (FD) distribution, which is a quite tractable special mixture model for compositional data, i.e. data representing vectors of proportions of a whole. The FD model displays several theoretical properties which make it suitable for inference, and fairly easy to handle from a computational viewpoint. However, the rigid type of mixture structure implied by the FD makes it unsuitable to describe many compositional datasets. Furthermore, the FD only allows for negative correlations. The new extended model, by considerably relaxing the strict constraints among clusters entailed by the FD, allows for a more general dependence structure (including positive correlations) and greatly expands its applicative potential. At the same time, it retains, to a large extent, its good properties. EM-type estimation procedures can be developed for this more complex model, including ad hoc reliable initialization methods, which permit to keep the computational issues at a rather uncomplicated level. Accurate evaluation of standard error estimates can be provided as well.

This study was partially funded by Università degli Studi di Milano-Bicocca (Grant No. FA 2018).

Authors and Affiliations

1. DEMS - Dept of Economics, Management and Statistics, University of Milano-Bicocca, Milano, Italy

   Andrea Ongaro, Sonia Migliorati & Roberto Ascari

Corresponding author

Correspondence to Sonia Migliorati.

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Appendix

1.1 Proof of Proposition 3

The conditional distribution function of \({\varvec{S}}_1\mid {\varvec{X}}_{2}={\varvec{x}}_{2}\) can be derived most easily by conditioning on \({\varvec{Z}}\):

Given that \({\varvec{X}}\mid {\varvec{Z}}={\varvec{e}}_i\sim \mathcal{D}({\dot{\varvec{\alpha }}}_i)\), by using well-known Dirichlet independence properties we have that:

Recalling that the Dirichlet distribution is closed under the operation of subcomposition, it follows that:

and

The probabilities \(P({\varvec{Z}}={\varvec{e}}_i\mid {\varvec{X}}_{2}={\varvec{x}}_{2})\) can be computed by the Bayes theorem. In particular, the distribution of \(({\varvec{X}}_{2},1-X_2^+)^\intercal | {\varvec{Z}}={\varvec{e}}_i\) can be obtained by resorting to closure of the Dirichlet under marginalization; it takes the form

if \(i\le k\) and

if \(i> k\). From the Bayes formula, some algebraic manipulations show that the probabilities \(P({\varvec{Z}}={\varvec{e}}_i\mid {\varvec{X}}_{2}={\varvec{x}}_{2})\) are proportional to the \(p_{i}^{'}\)’s provided by (14). By plugging all the computed quantities into (35), the result is obtained.

1.2 Proof of Proposition 5

It is obvious that if \(\varvec{\theta }=\varvec{\theta }^\prime \), then \(\mathbf{X } \sim \mathbf{X }^\prime \). In order to show the converse, one can focus on the marginal distribution of \(X_i\). By virtue of Proposition 3, we can write its density function \(g(x_i; \varvec{\theta })\) as:

If \(\mathbf{X } \sim \mathbf{X }^\prime \), then \(X_i \sim X_i^\prime \) and therefore, \(g(x_i; \varvec{\theta }) = g(x_i; \varvec{\theta }^\prime )\)\(\forall \)\(x_i\)\(\in \) (0, 1), as these density functions are continuous. It follows that \(\displaystyle \lim \limits _{x \rightarrow 0^+} \frac{g(x_i; \varvec{\theta })}{x_i^{\alpha _i - 1}} = \lim \limits _{x \rightarrow 0^+} \frac{g(x_i; \varvec{\theta }^\prime )}{x_i^{\alpha _i - 1}}\). We have:

and

In order to satisfy the equality of these two limits, the quantity \(\displaystyle \left( \lim _{x_i \rightarrow 0^+} \frac{x_i^{\alpha _i^\prime - 1}}{x^{\alpha _i - 1}}\right) \) must be finite and different from 0.

This implies that \(\varvec{\alpha }=\varvec{\alpha }^\prime \). As a consequence, the equality \(g(x_i; \varvec{\theta }) = g(x_i; \varvec{\theta }^\prime )\) can be rewritten as:

By taking the limits as \(x_i \rightarrow 1^-\) on both sides, one obtains:

Equation (38) implies that \(p_i\) and \( p_i^\prime \) are either both null or both strictly positive. In the former case, because of the parameter space definition, \( \tau _i=\tau _i^\prime =1\). In the latter case, plugging (38) into equality (37) and deriving both sides, the following equality must hold \(\forall \)\(x_i\)\(\in \) (0, 1):

Taking the limits as \(x_i \rightarrow 1^-\) on both sides, we have:

It follows that \(\tau _i=\tau _i^\prime \) for any i such that \(p_i>0\) and hence for all i. Finally, substituting this constraint in (38), it is possible to conclude that \(\mathbf{p }= \mathbf{p }^\prime \).

1.3 Proof of Proposition 8

Recall that \(\mathbf{X }| Y^+ = y^+ \sim EFD(\varvec{\alpha },\mathbf{p }^*(y^+),\varvec{\tau }, \beta )\), where \(\mathbf{p }^*(y^+)\) are defined as in (23). Then, if \(\tau _i=\tau \)\(\forall i\), it can be seen immediately that the \(p_i^*(y^+)\)’s are independent of \(y^+\) (and coincide with the \(p_i\)’s). Conversely, if the basis is compositional invariant, then \(p_i^*(y^+)\) does not depend on \(y^+\), and therefore, neither does the ratio \(p_i^*(y^+)/p_l^*(y^+)\)\(\forall i\ne l\). Because this ratio is proportional to \({(y^+)}^{\tau _i-\tau _l}\), \(\tau _i=\tau _l\)\(\forall i\ne l\).

1.4 Partial derivatives

In this section we show the partial derivatives of the complete-data log-likelihood (25). In particular, for \(i=1, \ldots ,D\), the first-order partial derivatives are:

where \(z_{\cdot i}=\sum _{j=1}^n z_{ji}\).

Histograms and estimated densities of the eight two-part compositions

Full size image

The second-order partial derivatives are:

where \(\mathbb {1}_{i=h}\) is the indicator function that is equal to 1 if \(i = h\) and 0 otherwise.

where \(\psi ^\prime (\cdot )\) is the trigamma function.

1.5 Results of the univariate case of the olive oil dataset

In this section we report the AIC and BIC criteria for the considered models (Table 7), and the fitted density curves (Fig. 9).

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