TransportMaps.Algorithms.SequentialInference.SequentialInferenceBase

Module Contents

Classes

Filter	Perform the on-line filtering of a sequential Hidded Markov chain.
Smoother	Perform the on-line smoothing and filtering of a sequential Hidded Markov chain.

class TransportMaps.Algorithms.SequentialInference.SequentialInferenceBase.Filter(pi_hyper=None, **kwargs)

Bases: TransportMaps.ObjectBase.TMO

Perform the on-line filtering of a sequential Hidded Markov chain.

Given the prior distribution on the hyper-parameters \(\pi(\Theta)\), provides the functions neccessary to assimilate new pieces of data or missing data (defined in terms of transition densities \(\pi\left({\bf Z}_{k+1} \middle\vert {\bf Z}_k, \Theta \right)\) and log-likelihoods \(\log \mathcal{L}\left({\bf y}_{k+1}\middle\vert {\bf Z}_{k+1}, \Theta\right)\)), to return the maps pushing forward \(\mathcal{N}(0,{\bf I})\) to the filtering/forecast distributions \(\{\pi\left(\Theta, {\bf Z}_k \middle\vert {\bf y}_{0:k} \right)\}_k\).

For more details see also [TM3] and the tutorial.

Parameters:

pi_hyper (Distribution) – prior distribution on the hyper-parameters \(\pi(\Theta)\)

Note

This is a super-class. Part of its methods need to be implemented by sub-classes.

property pi

property nsteps

property filtering_map_list

Returns the maps \(\{ \widetilde{\mathfrak{M}}_k({\bf x}_\theta, {\bf x}_{k+1}) \}_{i=0}^{k-1}\) pushing forward \(\mathcal{N}(0,{\bf I})\) to the filtering/forecast distributions \(\{\pi\left(\Theta, {\bf Z}_k \middle\vert {\bf y}_{0:k} \right)\}_k\).

The maps \(\widetilde{\mathfrak{M}}_k({\bf x}_\theta, {\bf x}_{k+1})\) are defined as follows:

\[\begin{split}\widetilde{\mathfrak{M}}_k({\bf x}_\theta, {\bf x}_{k+1}) = \left[\begin{array}{l} \mathfrak{M}_0^\Theta \circ \cdots \circ \mathfrak{M}_{k}^\Theta ({\bf x}_\theta) \\ \mathfrak{M}_k^1\left({\bf x}_\theta, {\bf x}_{k+1}\right) \end{array}\right] = \left[\begin{array}{l} \mathfrak{H}_{k}({\bf x}_\theta) \\ \mathfrak{M}_k^1\left({\bf x}_\theta, {\bf x}_{k+1}\right) \end{array}\right]\end{split}\]

Returns:

(list of TransportMap) – list of transport maps \(\widetilde{\mathfrak{M}}_k({\bf x}_\theta, {\bf x}_{k+1})\)

assimilate(pi, ll, *args, **kwargs)

Assimilate one piece of data \(\left( \pi\left({\bf Z}_{k+1} \middle\vert {\bf Z}_k, \Theta \right), \log \mathcal{L}\left({\bf y}_{k+1}\middle\vert {\bf Z}_{k+1}, \Theta\right) \right)\).

Given the new piece of data \(\left( \pi\left({\bf Z}_{k+1} \middle\vert {\bf Z}_k, \Theta \right), \log \mathcal{L}\left({\bf y}_{k+1}\middle\vert {\bf Z}_{k+1}, \Theta\right) \right)\), determine the maps pushing forward \(\mathcal{N}(0,{\bf I})\) to the filtering/forecast distributions \(\{\pi\left(\Theta, {\bf Z}_k \middle\vert {\bf y}_{0:k} \right)\}_k\).

Parameters:

• pi (Distribution) – transition distribution \(\pi\left({\bf Z}_{k+1} \middle\vert {\bf Z}_k, \Theta \right)\)

• ll (LogLikelihood) – log-likelihood \(\log \mathcal{L}\left({\bf y}_{k+1}\middle\vert {\bf Z}_{k+1}, \Theta\right)\). The value None stands for missing observation.

• *args – arguments required by the particular sub-classes implementations of _assimilation_step().

• **kwargs –

  arguments required by the particular sub-classes implementations of _assimilation_step().

Note

This method requires the implementation of the function _assimilation_step() in sub-classes

abstract _assimilation_step(*args, **kwargs)

[Abstract] Implements the map approximation for one step in the sequential inference.

get_filtering_map_list()

Deprecated since version Use: filtering_map_list instead

class TransportMaps.Algorithms.SequentialInference.SequentialInferenceBase.Smoother(pi_hyper=None, **kwargs)

Bases: Filter

Perform the on-line smoothing and filtering of a sequential Hidded Markov chain.

Given the prior distribution on the hyper-parameters \(\pi(\Theta)\), provides the functions neccessary to assimilate new pieces of data or missing data (defined in terms of transition densities \(\pi\left({\bf Z}_{k+1} \middle\vert {\bf Z}_k, \Theta \right)\) and log-likelihoods \(\log \mathcal{L}\left({\bf y}_{k+1}\middle\vert {\bf Z}_{k+1}, \Theta\right)\)), to return the map pushing forward \(\mathcal{N}(0,{\bf I})\) to the smoothing distribution \(\pi\left(\Theta, {\bf Z}_\Lambda \middle\vert {\bf y}_\Xi \right)\) and to return the maps pushing forward \(\mathcal{N}(0,{\bf I})\) to the filtering/forecast distributions \(\{\pi\left(\Theta, {\bf Z}_k \middle\vert {\bf y}_{0:k} \right)\}_k\).

For more details see also [TM3] and the tutorial.

Parameters:

pi_hyper (Distribution) – prior distribution on the hyper-parameters \(\pi(\Theta)\)

Note

This is a super-class. Part of its methods need to be implemented by sub-classes.

property smoothing_map

Returns the map \(\mathfrak{T}\) pushing forward \(\mathcal{N}(0,{\bf I})\) to the smoothing distribution \(\pi\left(\Theta, {\bf Z}_\Lambda \middle\vert {\bf y}_\Xi\right)\).

The map \(\mathfrak{T}\) is given by the composition \(T_0 \circ \cdots \circ T_{k-1}\) maps constructed in \(k\) assimilation steps.

Returns:

(TransportMap) – the map \(\mathfrak{T}\)

assimilate(pi, ll, *args, **kwargs)

Assimilate one piece of data \(\left( \pi\left({\bf Z}_{k+1} \middle\vert {\bf Z}_k, \Theta \right), \log \mathcal{L}\left({\bf y}_{k+1}\middle\vert {\bf Z}_{k+1}, \Theta\right) \right)\).

Given the new piece of data \(\left( \pi\left({\bf Z}_{k+1} \middle\vert {\bf Z}_k, \Theta \right), \log \mathcal{L}\left({\bf y}_{k+1}\middle\vert {\bf Z}_{k+1}, \Theta\right) \right)\), retrieve the \(k\)-th Markov component \(\pi^k\) of \(\pi\), determine the transport map

\[\begin{split}\mathfrak{M}_k({\boldsymbol \theta}, {\bf z}_k, {\bf z}_{k+1}) = \left[ \begin{array}{l} \mathfrak{M}^\Theta_k({\boldsymbol \theta}) \\ \mathfrak{M}^0_k({\boldsymbol \theta}, {\bf z}_k, {\bf z}_{k+1}) \\ \mathfrak{M}^1_k({\boldsymbol \theta}, {\bf z}_{k+1}) \end{array} \right] = Q \circ R_k \circ Q\end{split}\]

that pushes forward \(\mathcal{N}(0,{\bf I})\) to \(\pi^k\), and embed it into the linear map which will remove the desired conditional dependencies from \(\pi\).

Parameters:

• pi (Distribution) – transition distribution \(\pi\left({\bf Z}_{k+1} \middle\vert {\bf Z}_k, \Theta \right)\)

• ll (LogLikelihood) – log-likelihood \(\log \mathcal{L}\left({\bf y}_{k+1}\middle\vert {\bf Z}_{k+1}, \Theta\right)\). The value None stands for missing observation.

• *args – arguments required by the particular sub-classes implementations of _assimilation_step().

• **kwargs –

  arguments required by the particular sub-classes implementations of _assimilation_step().

Note

This method requires the implementation of the function _assimilation_step() in sub-classes

get_smoothing_map()

Deprecated since version Use: filtering_map_list instead