continuous_discrete_linear_gaussian_ssm

Continuous Discrete Linear Gaussian State Space Models

• Implementation of Continuous-discrete Linear Gaussian State Space Models

  • We provide a model class definition

    • NB: t0 and t1 refer to $t_k$ and $t_{k+1}$, not necessarily regularly sampled

  • We provide a set of filtering and smoothing algorithms described below.

Implemented algorithms

• The codebase is based on algorithms as defined in

  • [1] Särkkä, Simo. Recursive Bayesian inference on stochastic differential equations. Helsinki University of Technology, 2006.

• The Continuous Discrete Kalman Filter implementation

  • i.e., Algorithm 3.15 in [1]

• Continuous Discrete Kalman Smoothers:

  • The Continuous Discrete Kalman Smoother type I implementation

    • i.e., Algorithm 3.17 in [1]

  • The Continuous Discrete Kalman Smoother type II implementation

    • i.e., Algorithm 3.18 in [1]

Parameter inference

• Parameter (point)-estimation is possible via stochastic gradient descent based MLE

  • See fit_sgd() in ssm_temissions.py
    • Which leverages the marginal log-likelihood computed in closed form
    • The analytical marginalized log-likelihood is possible given the Gaussian nature of the (recursive) transition and emission distributions

• We do not provide a parameter (point)-estimation via EM

  • The m-step requires MLE for continuous time parameters

Pending

• Note that the codebase currently only supports inputs at measurement times, i.e., $u$ is observed at times $t_k$ as given in t_emissions.

• Note that even if the model definition seems to allow for time-varying emission weights, the implementation is not ready to do so (dynamax wasn't either)