**smoothing**¶

This module implements:

- particle history classes, which store the full or partial history of a SMC algorithm.
- 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:

- particles.smoothing.ParticleHistory: full history (based on lists)
`PartialParticleHistory`

: partial history (based on dictionaries)`RollingParticleHistory`

: rolling window history (based on deques)

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. |