particles.hmm¶
BaumWelch filter/smoother for hidden Markov models.
Overview¶
A hidden Markov model (HMM) is a statespace model with a finite statespace, {1, …, K}. The BaumWelch algorithm allows to compute (exactly) the filtering and smoothing distributions of such a model; i.e. the probabilities that X_t=k given data Y_{0:t} (filtering) or the complete data Y_{0:T} (smoothing). In addition, one may sample trajectories from the complete smoothing distribution (the distribution of all the states, X_{0:T}, given all the data).
Hidden Markov models and the BaumWelch algorithm are covered in Chapter 6 of the book.
Running the BaumWelch algorithm¶
Class BaumWelch
is instantiated as follows:
bw = BaumWelch(hmm=my_hmm, data=y)
To actually run the algorithm, one must invoke the appropriate methods, e.g.:
bw.forward() # computes the filtering probs
bw.backward() # computes the marginal smoothing probs
bw.sample(N=30) # generate 30 smoothing trajectories
If you invoke either backward
or sample
directly, the forward pass will be
run first. The output of the forward and backward passes are attributes of
object bw
, which are lists of Klength numpy arrays. For instance,
self.filt
is a list of arrays containing the filtering probabilities; see the
documentation of BaumWelch
for more details.
Running the forward pass step by step¶
A BaumWelch
object is an iterator; each iteration performs a single step of
the forward pass. It is thus possible for the user to run the forward pass step
by step:
next(bw) # performs one step of the forward pass
This may be useful in a variety of scenarios, such as when data are acquired on
the fly (in that case, modify attribute self.data
directly), or when one
wants to perform the smoothing pass at different times; in particular:
bw = BaumWelch(hmm=mh_hmm, data=y)
for t, _ in enumerate(y):
bw.step()
bw.backward()
## save the results in bw.smth somewhere
would compute all the intermediate smoothing distributions (for data $Y_0$, then $Y_{0:1}$, and so on). This is expensive, of course (cost is O(T^2)).
Classes

BaumWelch filtering/smoothing algorithm. 

Gaussian HMM: \(Y_tX_t=k \sim N(\mu_k, \sigma_k^2)\) 

Base class for hidden Markov models. 