# FilterModel1: Filtering algorithm for the type one model. In SMFilter: Filtering Algorithms for the State Space Models on the Stiefel Manifold

## Description

This function implements the filtering algorithm for the type one model. See Details part below.

## Usage

 1 2 FilterModel1(mY, mX, mZ, beta, mB = NULL, Omega, vD, U0, method = "max_1") 

## Arguments

 mY the matrix containing Y_t with dimension T \times p. mX the matrix containing X_t with dimension T \times q_1. mZ the matrix containing Z_t with dimension T \times q_2. beta the β matrix. mB the coefficient matrix \boldsymbol{B} before mZ with dimension p \times q_2. Omega covariance matrix of the errors. vD vector of the diagonals of D. U0 initial value of the alpha sequence. method a string representing the optimization method from c('max_1','max_2','max_3','min_1','min_2').

## Details

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

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.

## Value

an array aAlpha containing the modal orientations of alpha in the prediction step.

## Author(s)

Yukai Yang, yukai.yang@statistik.uu.se

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 iT = 50 ip = 2 ir = 1 iqx = 4 iqz=0 ik = 0 Omega = diag(ip)*.1 if(iqx==0) mX=NULL else mX = matrix(rnorm(iT*iqx),iT, iqx) if(iqz==0) mZ=NULL else mZ = matrix(rnorm(iT*iqz),iT, iqz) if(ik==0) mY=NULL else mY = matrix(0, ik, ip) alpha_0 = matrix(c(runif_sm(num=1,ip=ip,ir=ir)), ip, ir) beta = matrix(c(runif_sm(num=1,ip=ip*ik+iqx,ir=ir)), ip*ik+iqx, ir) mB=NULL vD = 100 ret = SimModel1(iT=iT, mX=mX, mZ=mZ, mY=mY, alpha_0=alpha_0, beta=beta, mB=mB, vD=vD, Omega=Omega) mYY=as.matrix(ret\$dData[,1:ip]) fil = FilterModel1(mY=mYY, mX=mX, mZ=mZ, beta=beta, mB=mB, Omega=Omega, vD=vD, U0=alpha_0) 

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