Acceleration of the Universe as Acceleration of Probable Influence 11: Mean-Shifting

>>From Osher Doctorow mdoctorow@xxxxxxxxxxx

Gopal K. Basak and Zhan-Qian Lu, respectively of U. Bristol Math. Dept.
U.K. and Indian Statistical Institute India vs Statistical Engineering
Div., ITL, NIST, Gaithersburg Maryland USA, in "Stationarity of
switching VAR and other related models," math.ST/0507267 v1 13 Jul
2005, discuss switching ARMA (autoregressive moving averages time
series) models which allow a different ARMA model in a different
regime.

An especially interesting model is the mean shifting model, which they
discuss on p. 18 of their paper, given by:

1) Xn = Mn + AnX_(n-1) + En

where Mn is the shifting mean, namely mean ui or u_i when In = i for i
= 1 to r or Mn = sum ui 1_(In = i) where 1_ is the indicator function.
Here the In are taken to be a Markov chain taking values in a set S =
{1, 2, ..., r}, so that the mean shifts depending on the value that the
Markov chain takes, and there are r "potential regimes" namely 1, 2,
...., r which is related to r phases in a generalized sense. The
quantities An and Bi are defined as:

2) Bi = unknown or partially unknown p x p matrices
3) An = sum Bi I_(In = i), sum over i = 1 to r

The En are like An except Bi is replaced by epsilon_(ni) where the
latter are independent processes with each subsequence independent
identically distributed within itself and each having mean 0 and
covariance matrix the identity and the In's independent of the
epsilon_(ni) quantities among other things.

The use of the Markov chain, as I've indicated before in this thread,
is because Mainstream mathematical probability-statistics doesn't know
Probable Influence (PI), and Markov chains are in general a rather poor
approximation but at least tend to go in the same direction as PI away
from small neighborhoods of zero probability events.

More generally, the regimes could be any different phases or types of
events, as for example on p. 19 of their paper an example of 3 regimes:
normal (0), rising (1), or falling (2) riverflow.

Osher Doctorow

.

Relevant Pages