Algorithm 1: Outline of algorithm for determining outliers via Eigenvalue method

Initialize: Determine initial cluster membership for all observations;

Return: Number of clusters: G and

Cluster variance estimates: ${\widehat{V}}_g$ for g = 1,…, G;

for Observations i = 1,…, N do

For observation i assigned to cluster g, hold other N − 1 cluster memberships constant;

(A) Determine variance structure when removing observation i;

With N-1 observations, use the M-step (from the EM algorithm) to calculate parameter estimates for all G groups under all specified variance structures Select best variance structure through maximization of the BIC;

Return: Cluster variance estimate: ${\widehat{V}}_{g,-i}$;

if Selected variance structure differs from original variance structure,

then re-estimate cluster variance ${\widehat{V}}_g$ for g = 1,…, G with all N observations under the best variance structure selected for N − 1 observations ;

(B) Calculate the minimum eigenvalue $E_i^g$ of R−i, where $R_{-i}={\widehat{V}}_{g,-i}{\widehat{V}}_g^{-1}$

Define: the # of observations in cluster g as Ng, trimming value $T=\lceil \sqrt{N/G}\rceil$ and tj for j = 1,…, 5 to create ∼ 4 equally spaced intervals from 1,…, T;

for g = 1,…, G do

if Ng > T then

Compute thresholds with trimmed cluster-specific eigenvalues $E^{g,t_j}$

for j = 1,…, 5, do Mt,g = Mean( $E^{g,t_j}$) − 5 SD( $E^{g,t_j}$)

for j = 2,…, 5, do $M_{j,g}^{\ast }=M_{t_j,g}/M_{t_{j-1},g}$ = Mean

If $\exists (M{.}_{,g}^{\ast })>(1+1/N_g)$

Then $k=\underset{j=2,\dots ,5}{\text{argmax}j}:M_{t_{j,g}}^{\ast }>(1+1/N_g)$ and $C_g=M_{t_k,g}$

Else Cg = M1,g

else Cluster g denoted as “outlier cluster”, $C_g=E_{(N_g)}^g$

Return: Observations with Eg ≤ Cg, for g = 1,…, G