# smoothing¶

This module implements:

1. particle history classes, which store the full or partial history of a SMC algorithm.
2. off-line smoothing algorithms as methods of these classes.

For on-line smoothing, see instead the collectors module.

## History classes¶

A SMC object has a hist attribute, which is used to record at certain times t:

• the N current particles $$X_t^n$$;
• their weights;
• (optionally, see below), the ancestor variables $$A_t^n$$.

The frequency at which history is recorded depends on option store_history of class SMC. Possible options are:

• True: records full history (at every time t);
• False: no history (attribute hist set to None);
• callable f: history is recorded at time t if f(t) returns True
• int k: records a rolling window history of length k (may be used to perform fixed-lag smoothing)

This module implements different classes that correspond to the different cases:

All these classes provide a similar interface. If smc is a SMC object, then:

• smc.hist.X[t] returns the N particles at time t
• smc.hist.wgts[t] returns the N weights at time t (see resampling.weights)
• smc.hist.A[t] returns the N ancestor variables at time t

## Partial History¶

Here are some examples on one may record history only at certain times:

# store every other 10 iterations
smc = SMC(fk=fk, N=100, store_history=lambda t: (t % 10) == 0)

# store at certain times given by a list
times = [10, 30, 84]
smc = SMC(fk=fk, N=100, store_history=lambda t: t in times)


Once the algorithm is run, smc.hist.X and smc.hist.wgts are dictionaries, the keys of which are the times where history was recorded. The ancestor variables are not recorded in that case:

smc.run()
smc.hist.X[10]  # the N particles at time 10
smc.hist.A[10]  # raises an error


## Full history, off-line smoothing algorithms¶

For a given state-space model, off-line smoothing amounts to approximate the distribution of the complete trajectory $$X_{0:T}$$, given data $$y_{0:T}$$, at some fixed time horizon T. The corresponding algorithms take as an input the complete history of a particle filter, run until time T (forward pass). Say:

# forward pass
fk = ssm.Bootstrap(ssm=my_ssm, data=y)
pf = particles.SMC(fk=fk, N=100, store_history=True)
pf.run()


Then, pf.hist is an instance of class particles.smoothing.ParticleHistory. It implements two types of approaches:

• two-filter smoothing: uses two particle filters (one forward, one backward) to estimate marginal expectations; see two_filter_smoothing.
• FFBS (forward filtering backward sampling): uses one particle filter, then generates trajectories from its history, using different methods (exact, rejection, MCMC, QMC). See backward_sampling_mcmc, backward_sampling_ON2, backward_sampling_reject, and backward_sampling_qmc. Recommended method is backward_sampling_mcmc, see discussion in Dang & Chopin (2022).

For more details, see the documentation of particles.smoothing.ParticleHistory, the ipython notebook on smoothing, Chapter 12 of the book, and Dang & Chopin (2022).

Warning

the complete history of a particle filter may take a lot of memory.

## Rolling history, Fixed-lag smoothing¶

To obtain a rolling window (fixed-length) history:

smc = SMC(fk=fk, N=100, store_history=10)
smc.run()


In that case, fields smc.hist.X, smc.hist.wgts and smc.hist.A are deques of max length 10. Using negative indices:

smc.hist.X[-1]  # the particles at final time T
smc.hist.X[-2]  # the particles at time T - 1
# ...
smc.hist.X[-10] # the N particles at time T - 9
smc.hist.X[-11] # raises an error


Note that this type of history makes it possible to perform fixed-lag smoothing as follows:

B = smc.hist.compute_trajectories()
# B[t, n] is index of ancestor of X_T^n at time t
phi = lambda x: x  # any test function
est = np.average(phi(smc.hist.X[-10][B[-10, :]]), weights=smc.W)
# est is an estimate of E[ phi(X_{T-9}) | Y_{0:T}]


Note

recall that it is possible to run SMC algorithms step by step, since they are iterators. Hence it is possible to do fixed-lag smoothing step-by-step as well.

## Module summary¶

 ParticleHistory(fk, qmc) Particle history. PartialParticleHistory(func) Partial history. RollingParticleHistory(length) Rolling window history. smoothing_worker([method, N, fk, fk_info, …]) Generic worker for off-line smoothing algorithms.