[R] [OT] good reference for mixed models and EM algorithm

[(Ted Harding)]

On 11-Feb-08 15:07:37, Douglas Bates wrote:
> [...]
> Except that Doug Bates doesn't use the EM algorithm for fitting mixed
> models any more.  The lme4 package previously had an option for
> starting with EM (actually ECME, which is a variant of EM) iterations
> but I have since removed it.  For large data sets and especially for
> models with non-nested random effects, the EM iterations just slowed
> things down relative to direct optimisation of the log-likelihood.
> [...]

The "raw" EM Algorithm can be slow. I have had good success
using Aitken Acceleration for it.

The basic principle is that, once an interative algorithm
gets to a stage where (approximately)

[A]  (X[n+1] - X) = k*(X[n] -X)

where X[n] is the result at the n-th iteration, -1 < k < 1, and
X is the limit, then you can use recent results to predict the
limit. Taking the above equation literally, along with its
analogue for the next step:

[B]  (X[n+2] - X) = k*(X[n+1] -X)

from which k = (X[n+2] - X[[n+1])/(X[n+1] - X[n])

and then

[C] X = (X[n+1] - X[n])/(1 - k).

If X is multidimensional (say dimension = p), then k is a
pxp matrix, and you want all its eigenvalues to be less than 1
in modulus. Then you use the matrix analogues of the above
equations, based it on (p+1) successive iterations
X[n], X[n+1], ... , X[n+p+1]), i.e. on the p-vector

  c(X[n+1]-X[n], X[n+1]-X[n+1], ... , X[n+p+1]-X[n+p])

I have had good experience with this too!

The best method of proceeding is:

Stage 1: Monitor the sequence {X[n]} until it seems that
  equation [A] is beginning to be approximately true;

Stage 2: Apply equations [A], [B], [C] to estimate X.

Stage 3: Starting at this X, run a few more iterations
  so that you get a better (later) estimate of k, and
  then apply [A], [B], [C] aqain to re-estimate X.

Repeat stage 3 until happy (or bored).

The EM Algorithm, in most cases, falls into the class
of procedures to which Aitken Acceleration is applicable.

Best wishes to all,
Ted.

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E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 11-Feb-08                                       Time: 16:18:45
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