Causal-State Splitting Reconstruction Ziba Rostamian CS 590 – Winter 2008

Probabilistic Automaton A probabilistic automaton is a finite automaton with transition probabilities which represents a distribution over the set of all strings defined over a finite alphabet. They are used in a variety of applications, including text and speech processing, image processing, and computational biology

Notations Consider, a discrete-time,discrete-valued stochastic process where value of ∈ A and A is a finite alphabet of k symbols At any time t, is future and is history. For all events A in future : does not depend on t. Problem : knowing, what is P( )?

Causal States Definition: The causal states of a process are the members of the range of a function that maps from histories to sets of histories. If μ( ) is the collection of all measurable future events, then

Recurrent, Transient and Synchronization states Recurrent: Recurrent states are returned to infinitely. Transient: Transient states visited only finitely. Synchronization: Synchronization states can never be returned to once a recurrent state has been visited.

Properties of causal states The causal states are homogeneous for future events. The causal states themselves form Markov process. The –machine is deterministic.

Causal-State Spiting Reconstruction A string w is suffix of a history if for some L. A state has been represented as a set of suffixes. The function maps a finite history to that which contains a suffix of One suffix is the child of another suffix if w = av where a is a single symbol. A suffix is a descendent of its ancestor if w = uv where u in any non-null string.

State’s Morph Each is associated with a distribution for the next observable ;i.e., is defined for each and each. They call this distribution the state’s morph.

CSSR Algorithm There are three steps in CSSR algorithm: Initialize, Homogenize and Determinize. Initialize : Start L = 0 and where.They regard suffix of any history, so that initially maps all histories to one causal state. The morph of that state is defined as:

CSSR Algorithm Homogenize: Generate states whose member histories have no significant difference in their individual states. 1.For each calculate the future distribution from that state given the data sequence. 2.For each test the null hypothesis. For each length-L history. and each, generate the suffix of length L Increment L by one. 4.Repeat steps 1-3 until reaching the maximum history length.

CSSR Algorithm Determinize: 1.Eliminate all transient states from the current state-transition, leaving only recurrent states. 2.Convert it to deterministic machine. 3.From new, deterministic states eliminate those which are transient.