SMFilter: SMFilter: a package implementing the filtering algorithms for...

In SMFilter: Filtering Algorithms for the State Space Models on the Stiefel Manifold

Description

The package implements the filtering algorithms for the state-space models on the Stiefel manifold. It also implements sampling algorithms for uniform, vector Langevin-Bingham and matrix Langevin-Bingham distributions on the Stiefel manifold.

Details

Two types of the state-space models on the Stiefel manifold are considered.

The type one model on Stiefel manifold takes the form:

\boldsymbol{y}_t \quad = \quad \boldsymbol{α}_t \boldsymbol{β} ' \boldsymbol{x}_t + \boldsymbol{B} \boldsymbol{z}_t + \boldsymbol{\varepsilon}_t

\boldsymbol{α}_{t+1} | \boldsymbol{α}_{t} \quad \sim \quad ML (p, r, \boldsymbol{α}_{t} \boldsymbol{D})

where \boldsymbol{y}_t is a p-vector of the dependent variable, \boldsymbol{x}_t and \boldsymbol{z}_t are explanatory variables wit dimension q_1 and q_2, \boldsymbol{x}_t and \boldsymbol{z}_t have no overlap, matrix \boldsymbol{B} is the coefficients for \boldsymbol{z}_t, \boldsymbol{\varepsilon}_t is the error vector.

The matrices \boldsymbol{α}_t and \boldsymbol{β} have dimensions p \times r and q_1 \times r, respectively. Note that r is strictly smaller than both p and q_1. \boldsymbol{α}_t and \boldsymbol{β} are both non-singular matrices. \boldsymbol{α}_t is time-varying while \boldsymbol{β} is time-invariant.

Furthermore, \boldsymbol{α}_t fulfills the condition \boldsymbol{α}_t' \boldsymbol{α}_t = \boldsymbol{I}_r, and therefor it evolves on the Stiefel manifold.

ML (p, r, \boldsymbol{α}_{t} \boldsymbol{D}) denotes the Matrix Langevin distribution or matrix von Mises-Fisher distribution on the Stiefel manifold. Its density function takes the form

f(\boldsymbol{α_{t+1}} ) = \frac{ \mathrm{etr} ≤ft\{ \boldsymbol{D} \boldsymbol{α}_{t}' \boldsymbol{α_{t+1}} \right\} }{ _{0}F_1 (\frac{p}{2}; \frac{1}{4}\boldsymbol{D}^2 ) }

where \mathrm{etr} denotes \mathrm{exp}(\mathrm{tr}()), and _{0}F_1 (\frac{p}{2}; \frac{1}{4}\boldsymbol{D}^2 ) is the (0,1)-type hypergeometric function for matrix.

The type two model on Stiefel manifold takes the form:

\boldsymbol{y}_t \quad = \quad \boldsymbol{α} \boldsymbol{β}_t ' \boldsymbol{x}_t + \boldsymbol{B}' \boldsymbol{z}_t + \boldsymbol{\varepsilon}_t

\boldsymbol{β}_{t+1} | \boldsymbol{β}_{t} \quad \sim \quad ML (q_1, r, \boldsymbol{β}_{t} \boldsymbol{D})

where \boldsymbol{y}_t is a p-vector of the dependent variable, \boldsymbol{x}_t and \boldsymbol{z}_t are explanatory variables wit dimension q_1 and q_2, \boldsymbol{x}_t and \boldsymbol{z}_t have no overlap, matrix \boldsymbol{B} is the coefficients for \boldsymbol{z}_t, \boldsymbol{\varepsilon}_t is the error vector.

The matrices \boldsymbol{α} and \boldsymbol{β}_t have dimensions p \times r and q_1 \times r, respectively. Note that r is strictly smaller than both p and q_1. \boldsymbol{α} and \boldsymbol{β}_t are both non-singular matrices. \boldsymbol{β}_t is time-varying while \boldsymbol{α} is time-invariant.

Furthermore, \boldsymbol{β}_t fulfills the condition \boldsymbol{β}_t' \boldsymbol{β}_t = \boldsymbol{I}_r, and therefor it evolves on the Stiefel manifold.

ML (p, r, \boldsymbol{β}_t \boldsymbol{D}) denotes the Matrix Langevin distribution or matrix von Mises-Fisher distribution on the Stiefel manifold. Its density function takes the form

f(\boldsymbol{β_{t+1}} ) = \frac{ \mathrm{etr} ≤ft\{ \boldsymbol{D} \boldsymbol{β}_{t}' \boldsymbol{β_{t+1}} \right\} }{ _{0}F_1 (\frac{p}{2}; \frac{1}{4}\boldsymbol{D}^2 ) }

where \mathrm{etr} denotes \mathrm{exp}(\mathrm{tr}()), and _{0}F_1 (\frac{p}{2}; \frac{1}{4}\boldsymbol{D}^2 ) is the (0,1)-type hypergeometric function for matrix.

Author and Maintainer

Yukai Yang

Department of Statistics, Uppsala University

[e-mail]

References

Yang, Yukai and Bauwens, Luc. (2018) "State-Space Models on the Stiefel Manifold with a New Approach to Nonlinear Filtering", Econometrics, 6(4), 48.

Simulation

SimModel1 simulate from the type one state-space model on the Stiefel manifold.

SimModel2 simulate from the type two state-space model on the Stiefel manifold.

Filtering

FilterModel1 filtering algorithm for the type one model.

FilterModel2 filtering algorithm for the type two model.

Sampling

runif_sm sample from the uniform distribution on the Stiefel manifold.

rvlb_sm sample from the vector Langevin-Bingham distribution on the Stiefel manifold.

rmLB_sm sample from the matrix Langevin-Bingham distribution on the Stiefel manifold.

Other Functions

version shows the version number and some information of the package.

SMFilter documentation built on May 1, 2019, 8:01 p.m.