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

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

\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.

Value

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

Author(s)

Yukai Yang, [e-mail]

Examples

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.