SMFilter (version 1.0.3)

FilterModel1: Filtering algorithm for the type one model.

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 \(\beta\) 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').

Value

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

Details

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.

Examples

# NOT RUN {
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)

# }