aitken function

Aitken Acceleration

Calculates the Aitken acceleration estimate of the final converged maximised log-likelihood under the EM/CEM framework.

aitken(loglik)

Arguments

• loglik: A vector of three consecutive log-likelihood values. These three values should be in ascending order, though this is not checked.

Returns

A list with the following named components: - ll: The most current estimate of the log-likelihood, i.e. loglik[3].

• linf: The most current estimate of the final converged maximised log-likelihood.

• a: The Aitken acceleration value where typically 0 <= a <= 1. When a < 0, a numerical issue or bug has occurred; when a > 1, the algorithm is accelerating and should not be stopped.

• ldiff: The difference between linf and the previous estimate of the log-likelihood, i.e. loglik[2], in accordance with McNicholas et al. (2010).

When the "aitken" method is employed within MoE_clust (via MoE_control), ll at convergence gives the log-likelihood achieved by the estimated parameters, while linf at convergence estimates the log-likelihood that would be achieved after an infinite number of EM/CEM iterations.

Details

The final converged maximised log-likelihood can be used to determine convergence of the EM/CEM algorithm within MoE_clust, i.e. by checking whether the absolute difference between the previous log-likelihood estimate and the final converged maximised log-likelihood estimate is less than some tolerance.

Note

Within MoE_clust, as specified by the stopping argument of MoE_control, "aitken" is the default method used to assess convergence. The other option monitors the "relative" change in log-likelihood against some tolerance. See MoE_control.

Examples

(a1 <- aitken(-c(449.61534, 442.84221, 436.58999)))
a1$ldiff < 1e-05 # FALSE
(a2 <- aitken(-c(442.84221, 436.58999, 436.58998)))
a2$ldiff < 1e-05 # FALSE
(a3 <- aitken(-c(436.58999, 436.58998, 436.58998)))
a3$ldiff < 1e-05 # TRUE

References

Boehning, D., Dietz, E., Schaub, R., Schlattmann, P. and Lindsay, B. G. (1994). The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family. Annals of the Institute of Statistical Mathematics, 46(2): 373-388.

McNicholas, P. D., Murphy, T. B., McDaid, A. F. and Frost, D. (2010). Serial and parallel implementations of model-based clustering via parsimonious Gaussian mixture models. Computational Statistics & Data Analysis, 54(3): 711-723.

See Also

MoE_control

Author(s)

Keefe Murphy - <keefe.murphy@mu.ie >