Often we are interested in finding patterns which appear over a
space of time. These patterns occur in many areas; the pattern
of commands someone uses in instructing a computer, sequences of
words in sentences, the sequence of phonemes in spoken words -
any area where a sequence of events occurs could produce useful
Consider the simple example of someone trying to deduce the
weather from a piece of seaweed - folklore tells us that `soggy'
seaweed means wet weather, while `dry' seaweed means sun. If it
is in an intermediate state (`damp'), then we cannot be sure.
However, the state of the weather is not restricted to the state
of the seaweed, so we may say on the basis of an examination
that the weather is probably raining or sunny. A second useful
clue would be the state of the weather on the preceding day (or,
at least, its probable state) - by combining knowledge about
what happened yesterday with the observed seaweed state, we
might come to a better forecast for today.
This is typical of the type of system we will consider in this
- First we will introduce systems which generate probabalistic
patterns in time, such as the weather fluctuating between sunny
- We then look at systems where what we wish to predict is not
what we observe - the underlying system is hidden. In the above
example, the observed sequence would be the seaweed and the
hidden system would be the actual weather.
- We then look at some problems that can be solved once the system
has been modeled. For the above example, we may want to know
- What the weather was for a week given each day's seaweed
- Given a sequence of seaweed observations, is it winter or
summer? Intuitively, if the seaweed has been dry for a while
it may be summer, if it has been soggy for a while it might