SMFilter: SMFilter: a package implementing the 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.

Arguments

Author and Maintainer

Yukai Yang

Department of Statistics, Uppsala University

yukai.yang@statistik.uu.se

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.

Details

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

The type one model on Stiefel manifold takes the form: $$\boldsymbol{y}_t \quad = \quad \boldsymbol{\alpha}_t \boldsymbol{\beta} ' \boldsymbol{x}_t + \boldsymbol{B} \boldsymbol{z}_t + \boldsymbol{\varepsilon}_t$$ $$\boldsymbol{\alpha}_{t+1} | \boldsymbol{\alpha}_{t} \quad \sim \quad ML (p, r, \boldsymbol{\alpha}_{t} \boldsymbol{D})$$ 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 $\boldsymbol{\alpha}_t$ and $\boldsymbol{\beta}$ have dimensions $p \times r$ and $q_1 \times r$, respectively. Note that $r$ is strictly smaller than both $p$ and $q_1$. $\boldsymbol{\alpha}_t$ and $\boldsymbol{\beta}$ are both non-singular matrices. $\boldsymbol{\alpha}_t$ is time-varying while $\boldsymbol{\beta}$ is time-invariant.

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

$ML (p, r, \boldsymbol{\alpha}_{t} \boldsymbol{D})$ denotes the Matrix Langevin distribution or matrix von Mises-Fisher distribution on the Stiefel manifold. Its density function takes the form $$f(\boldsymbol{\alpha_{t+1}} ) = \frac{ \mathrm{etr} \left\{ \boldsymbol{D} \boldsymbol{\alpha}_{t}' \boldsymbol{\alpha_{t+1}} \right\} }{ _{0}F_1 (\frac{p}{2}; \frac{1}{4}\boldsymbol{D}^2 ) }$$ 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: $$\boldsymbol{y}_t \quad = \quad \boldsymbol{\alpha} \boldsymbol{\beta}_t ' \boldsymbol{x}_t + \boldsymbol{B}' \boldsymbol{z}_t + \boldsymbol{\varepsilon}_t$$ $$\boldsymbol{\beta}_{t+1} | \boldsymbol{\beta}_{t} \quad \sim \quad ML (q_1, r, \boldsymbol{\beta}_{t} \boldsymbol{D})$$ 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 $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}_t$ have dimensions $p \times r$ and $q_1 \times r$, respectively. Note that $r$ is strictly smaller than both $p$ and $q_1$. $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}_t$ are both non-singular matrices. $\boldsymbol{\beta}_t$ is time-varying while $\boldsymbol{\alpha}$ is time-invariant.

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

$ML (p, r, \boldsymbol{\beta}_t \boldsymbol{D})$ denotes the Matrix Langevin distribution or matrix von Mises-Fisher distribution on the Stiefel manifold. Its density function takes the form $$f(\boldsymbol{\beta_{t+1}} ) = \frac{ \mathrm{etr} \left\{ \boldsymbol{D} \boldsymbol{\beta}_{t}' \boldsymbol{\beta_{t+1}} \right\} }{ _{0}F_1 (\frac{p}{2}; \frac{1}{4}\boldsymbol{D}^2 ) }$$ 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.

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.