*2.6. Decision Boundaries for Two-Group Clustering*

In this subsection, we examine decision boundaries produced by MCLUST and MCLUST-ME for partitioning a sample into *G* = 2 clusters.

## 2.6.1. MCLUST Boundary

Suppose we would like to separate a *<sup>d</sup>*-dimensional i.i.d. random sample *<sup>S</sup>* = {*yi*}*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> into two clusters with MCLUST. Let (*τ*ˆ*k*,*μ*<sup>ˆ</sup> *<sup>k</sup>*,Σ<sup>ˆ</sup> *<sup>k</sup>*) denote MLEs for (*τk*,*μk*,Σ*k*) upon convergence. If we assign each point to the more probable cluster, then the two clusters can be expressed as follows.

$$E\_1 = \{ \mathbf{y}\_i \in \mathbb{S} : \vec{\pi}\_1 f\_1(\mathbf{y}\_i; \hat{\mathfrak{p}}\_1, \hat{\mathfrak{T}}\_1) - \vec{\pi}\_2 f\_2(\mathbf{y}\_i; \hat{\mathfrak{p}}\_2, \hat{\mathfrak{T}}\_2) > 0 \}; \quad E\_2 = S \searrow E\_1 \tag{22}$$

and the decision boundary separating *E*<sup>1</sup> and *E*<sup>2</sup> is

$$B = \{ \mathbf{t} \in \mathbb{R}^d : \mathbf{t}\_1 f\_1(\mathbf{t}; \mathbf{\hat{p}}\_1, \mathbf{\hat{t}}\_1) - \mathbf{t}\_2 f\_2(\mathbf{t}; \mathbf{\hat{p}}\_2, \mathbf{\hat{t}}\_2) = 0 \},\tag{23}$$

where *fk*, *<sup>k</sup>* = 1, 2, is defined in (2). Equivalently, the boundary *<sup>B</sup>* is the set of all points in R*<sup>d</sup>* with classification uncertainty equal to 0.5. Notice that since the solution set *B* does not depend on *i*, a common boundary is shared by *all* observations. When *d* = 2, under the model assumption of MCLUST, the boundary *<sup>B</sup>* is a straight line when <sup>Σ</sup><sup>ˆ</sup> <sup>1</sup> <sup>=</sup> <sup>Σ</sup><sup>ˆ</sup> 2, and a conic section when <sup>Σ</sup><sup>ˆ</sup> <sup>1</sup> <sup>=</sup> <sup>Σ</sup><sup>ˆ</sup> 2, with its shape and position determined by the values of the MLEs. This can be shown by simplifying the equality in (23) (see [12] for more details).
