| SMFilter {SMFilter} | R Documentation |
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.
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.
Author and Maintainer
Yukai Yang
Department of Statistics, Uppsala University
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.