ParticleSmoother Class Template Reference

Inheritance diagram for ParticleSmoother:

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List of all members.

Detailed Description

template<class T = unsigned int>
class indii::ml::filter::ParticleSmoother< T >

Forward-backward particle smoother.

Author:: Lawrence Murray <lawrence@indii.org>

Version:

Rev

Date:

Date

ParticleSmoother is suitable for models with nonlinear transition and measurement functions, approximating state and noise with indii::ml::aux::DiracMixturePdf distributions. It is particularly suitable in situations where an appropriate backwards transition function cannot be defined, such as for transition functions defined as convergent differential equations that become divergent when reversed.

See also:: indii::ml::filter for general usage guidelines.

Details

The implementation here is of the second algorithm described in Isard & Blake (1998). This reweights particles obtained during the forwards pass rather than performing a backwards filtering pass to obtain new particles. While this means a backwards transition function need not be defined for the model, the drawback is that the approach is computationally and spatially expensive -- $O(P^2)$

in the number of particles $P$

in both cases.

The forwards pass provides a weighted sample set $\{(\mathbf{s}_t^{(i)},\pi_t^{(i)})\}$ at each time point $t = t_1,\ldots,t_T$ for $i = 1,\ldots,P$ . Initialising with $\psi_{t_T} = \pi_{t_T}$ , the backwards step to calculate weights at time $t_n$ is as follows:

$\begin{eqnarray*} \alpha_{t_n}^{(i,j)} &=& p(\mathbf{x}({t_{n+1}}) = \mathbf{s}_{t_{n+1}}^{(i)}\,|\,\mathbf{x}({t_n}) = \mathbf{s}_{t_n}^{(j)}) \\ \mathbf{\gamma}_{t_n} &=& \alpha_{t_n} \mathbf{\pi}_{t_n} \\ \mathbf{\delta}_{t_n} &=& \alpha_{t_n}^T(\mathbf{\psi}_{t_{n+1}} \oslash \mathbf{\gamma}_{t_n})\\ \mathbf{\psi}_{t_n} &=& \mathbf{\pi}_{t_n} \otimes \mathbf{\delta}_{t_n} \end{eqnarray*}$

where $\oslash$ is element-wise division and $\otimes$ element-wise multiplication.

These are then normalised so that $\sum \psi_{t_n}^{(i)} = 1$ and the smoothed result $\{(\mathbf{s}_{t_n}^{(i)},\psi_{t_n}^{(i)})\}$ for $i = 1,\ldots,P$ is stored.

This implementation performs calculations in the above matrix form to take advantage of low-level optimisations in the matrix library. It also uses a sparse matrix implementation of $\alpha$ to alleviate spatial complexity somewhat.

References

Isard, M. & Blake, A. A smoothing filter for Condensation. Proceedings of the 5th European Conference on Computer Vision, 1998, 1, 767-781.

Definition at line 78 of file ParticleSmoother.hpp.


Public Member Functions
	ParticleSmoother (ParticleSmootherModel< T > *model, const T tT, const indii::ml::aux::DiracMixturePdf &p_xT)
	Create new particle smoother.
virtual	~ParticleSmoother ()
	Destructor.
virtual ParticleSmootherModel < T > *	getModel ()
	Get the model being estimated.
virtual void	smooth (const T tn, const indii::ml::aux::DiracMixturePdf &p_xtn_ytn)
	Rewind system to time of previous measurement and smooth.
virtual indii::ml::aux::DiracMixturePdf	smoothedMeasure ()
	Apply the measurement function to the current smoothed state to obtain an estimated measurement.
void	smoothedResample (ParticleResampler *resampler)
	Resample the smoothed state.

Constructor & Destructor Documentation

ParticleSmoother	(	ParticleSmootherModel< T > *	model,
		const T	tT,
		const indii::ml::aux::DiracMixturePdf &	p_xT
	)			`[inline]`

Create new particle smoother.

Parameters:

	model	Model to estimate.
	tT	; starting time.
	p_xT	$p(\mathbf{x}_T)$ ; prior over the state at time .

Definition at line 145 of file ParticleSmoother.hpp.

~ParticleSmoother ( ) [inline, virtual]

Destructor.

Definition at line 153 of file ParticleSmoother.hpp.

Member Function Documentation

indii::ml::filter::ParticleSmootherModel< T > * getModel ( ) [inline, virtual]

Get the model being estimated.

Returns:: The model being estimated.

Definition at line 159 of file ParticleSmoother.hpp.

void smooth	(	const T	tn,
		const indii::ml::aux::DiracMixturePdf &	p_xtn_ytn
	)			`[inline, virtual]`

Rewind system to time of previous measurement and smooth.

Parameters:

	tn	; the time to which to rewind the system. This must be less than the current time $t_{n+1}$ .
	p_xtn_ytn	$p(\mathbf{x}_n\,\|\,\mathbf{y}_{1:n})$ ; filter density at time .