Mixtures & Gaussian Copulas

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Finite Gaussian mixture (EM)

A \(K\)-component mixture of TAM regressions is fitted by Expectation-Maximization [Dempster et al., 1977] whose M-step is exactly the weighted atom with the posterior responsibilities \(r_{ik}\) as per-observation weights:

\[ \text{E: } r_{ik} = \frac{\pi_k\,\mathcal N(t_i;\,\Phi_i\theta_k,\,\sigma_k)}{\sum_j \pi_j\,\mathcal N(t_i;\,\Phi_i\theta_j,\,\sigma_j)}, \qquad \text{M: } \theta_k = \text{atom}(x,\,t;\,W=r_k),\ \ \sigma_k,\pi_k \text{ from } r_k. \]

No standalone class: a mixture is StaticTAM on a different schedule (single-group).

Gaussian copula

Sklar’s theorem [Sklar, 1959] separates several correlated responses into independent margins (each a distributional StaticTAM) plus a copula for the dependence. Map each margin through its probability integral transform to a normal score \(z_j=\Phi^{-1}(F_j(y_j))\), estimate the score correlation \(R\), and obtain a joint anomaly score from the Mahalanobis distance:

\[ \text{joint p-value} = \chi^2_d.\mathrm{sf}\big(z^\top R^{-1} z\big). \]