| Title: | Filtering Algorithms for the State Space Models on the Stiefel Manifold |
| Version: | 1.0.3 |
| Description: | Provides the filtering algorithms for the state space models on the Stiefel manifold as well as the corresponding sampling algorithms for uniform, vector Langevin-Bingham and matrix Langevin-Bingham distributions on the Stiefel manifold. |
| Depends: | R (≥ 3.0.0) |
| License: | GPL-3 |
| Encoding: | UTF-8 |
| LazyData: | true |
| RoxygenNote: | 6.1.1 |
| URL: | https://github.com/yukai-yang/SMFilter |
| BugReports: | https://github.com/yukai-yang/SMFilter/issues |
| Suggests: | knitr, rmarkdown, ggplot2 |
| VignetteBuilder: | knitr |
| NeedsCompilation: | no |
| Packaged: | 2018-12-12 22:10:57 UTC; yyang |
| Author: | Yukai Yang [aut, cre] |
| Maintainer: | Yukai Yang <yukai.yang@statistik.uu.se> |
| Repository: | CRAN |
| Date/Publication: | 2018-12-12 22:20:03 UTC |
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.
Compute the squared Frobenius distance between two matrices.
Description
This function Compute the squared Frobenius distance between two matrices.
Usage
FDist2(mX, mY)
Arguments
mX |
a |
mY |
another |
Details
The Frobenius distance between two matrices is defined to be
d(X, Y) = \sqrt{ \mathrm{tr} \{ A' A \} }
where A = X - Y.
The Frobenius distance is a possible measure of the distance between two points on the Stiefel manifold.
Value
the Frobenius distance.
Author(s)
Yukai Yang, yukai.yang@statistik.uu.se
Examples
FDist2(runif_sm(1,4,2)[1,,], runif_sm(1,4,2)[1,,])
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 |
mX |
the matrix containing X_t with dimension |
mZ |
the matrix containing Z_t with dimension |
beta |
the |
mB |
the coefficient matrix |
Omega |
covariance matrix of the errors. |
vD |
vector of the diagonals of |
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{\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.
Value
an array aAlpha containing the modal orientations of alpha in the prediction step.
Author(s)
Yukai Yang, yukai.yang@statistik.uu.se
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)
Filtering algorithm for the type two model.
Description
This function implements the filtering algorithm for the type two model. See Details part below.
Usage
FilterModel2(mY, mX, mZ, alpha, mB = NULL, Omega, vD, U0,
method = "max_1")
Arguments
mY |
the matrix containing Y_t with dimension |
mX |
the matrix containing X_t with dimension |
mZ |
the matrix containing Z_t with dimension |
alpha |
the |
mB |
the coefficient matrix |
Omega |
covariance matrix of the errors. |
vD |
vector of the diagonals of |
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 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.
Value
an array aAlpha containing the modal orientations of alpha in the prediction step.
Author(s)
Yukai Yang, yukai.yang@statistik.uu.se
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 = matrix(c(runif_sm(num=1,ip=ip,ir=ir)), ip, ir)
beta_0 = matrix(c(runif_sm(num=1,ip=ip*ik+iqx,ir=ir)), ip*ik+iqx, ir)
mB=NULL
vD = 100
ret = SimModel2(iT=iT, mX=mX, mZ=mZ, mY=mY, alpha=alpha, beta_0=beta_0, mB=mB, vD=vD)
mYY=as.matrix(ret$dData[,1:ip])
fil = FilterModel2(mY=mYY, mX=mX, mZ=mZ, alpha=alpha, mB=mB, Omega=Omega, vD=vD, U0=beta_0)
Simulate from the type one state-space Model on Stiefel manifold.
Description
This function simulates from the type one model on Stiefel manifold. See Details part below.
Usage
SimModel1(iT, mX = NULL, mZ = NULL, mY = NULL, alpha_0, beta,
mB = NULL, Omega = NULL, vD, burnin = 100)
Arguments
iT |
the sample size. |
mX |
the matrix containing X_t with dimension |
mZ |
the matrix containing Z_t with dimension |
mY |
initial values of the dependent variable for |
alpha_0 |
the initial alpha, |
beta |
the |
mB |
the coefficient matrix |
Omega |
covariance matrix of the errors. |
vD |
vector of the diagonals of |
burnin |
burn-in sample size (matrix Langevin). |
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.
Note that the function does not add intercept automatically.
Value
A list containing the sampled data and the dynamics of alpha.
The object is a list containing the following components:
dData |
a data.frame of the sampled data |
aAlpha |
an array of the |
Author(s)
Yukai Yang, yukai.yang@statistik.uu.se
Examples
iT = 50 # sample size
ip = 2 # dimension of the dependent variable
ir = 1 # rank number
iqx=2 # number of variables in X
iqz=2 # number of variables in Z
ik = 1 # lag length
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)
if(ip*ik+iqz==0) mB=NULL else mB = matrix(c(runif_sm(num=1,ip=(ip*ik+iqz)*ip,ir=1)), ip, ip*ik+iqz)
vD = 50
ret = SimModel1(iT=iT, mX=mX, mZ=mZ, mY=mY, alpha_0=alpha_0, beta=beta, mB=mB, vD=vD)
Simulate from the type two state-space Model on Stiefel manifold.
Description
This function simulates from the type two model on Stiefel manifold. See Details part below.
Usage
SimModel2(iT, mX = NULL, mZ = NULL, mY = NULL, beta_0, alpha,
mB = NULL, Omega = NULL, vD, burnin = 100)
Arguments
iT |
the sample size. |
mX |
the matrix containing X_t with dimension |
mZ |
the matrix containing Z_t with dimension |
mY |
initial values of the dependent variable for |
beta_0 |
the initial beta, iqx+ip*ik, y_1,t-1,y_1,t-2,...,y_2,t-1,y_2,t-2,.... |
alpha |
the |
mB |
the coefficient matrix |
Omega |
covariance matrix of the errors. |
vD |
vector of the diagonals of |
burnin |
burn-in sample size (matrix Langevin). |
Details
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.
Note that the function does not add intercept automatically.
Value
A list containing the sampled data and the dynamics of beta.
The object is a list containing the following components:
dData |
a data.frame of the sampled data |
aBeta |
an array of the |
Author(s)
Yukai Yang, yukai.yang@statistik.uu.se
Examples
iT = 50
ip = 2
ir = 1
iqx =3
iqz=2
ik = 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 = matrix(c(runif_sm(num=1,ip=ip,ir=ir)), ip, ir)
beta_0 = matrix(c(runif_sm(num=1,ip=ip*ik+iqx,ir=ir)), ip*ik+iqx, ir)
if(ip*ik+iqz==0) mB=NULL else mB = matrix(c(runif_sm(num=1,ip=(ip*ik+iqz)*ip,ir=1)), ip, ip*ik+iqz)
vD = 50
ret = SimModel2(iT=iT, mX=mX, mZ=mZ, mY=mY, alpha=alpha, beta_0=beta_0, mB=mB, vD=vD)
Sample from the matrix Langevin-Bingham on the Stiefel manifold.
Description
This function draws a sample from the matrix Langevin-Bingham on the Stiefel manifold.
Usage
rmLB_sm(num, mJ, mH, mC, mX, ir)
Arguments
num |
number of observations or sample size. |
mJ |
symmetric ip*ip matrix |
mH |
symmetric ir*ir matrix |
mC |
ip*ir matrix |
mX |
ip*ir matrix, the initial value |
ir |
ir |
Details
The matrix Langevin-Bingham distribution on the Stiefel manifold has the density kernel:
f(X) \propto \mathrm{etr}\{ H X' J X + C' X \}
where X satisfies X'X = I_r, and H and J are symmetric matrices.
Value
an array containing a sample of draws from the matrix Langevin-Bingham on the Stiefel manifold.
Author(s)
Yukai Yang, yukai.yang@statistik.uu.se
#' @section References: Hoff, P. D. (2009) "Simulation of the Matrix Bingham—von Mises—Fisher Distribution, With Applications to Multivariate and Relational Data", Journal of Computational and Graphical Statistics, Vol. 18, pp. 438-456.
Sample from the uniform distribution on the Stiefel manifold.
Description
This function draws a sample from the uniform distribution on the Stiefel manifold.
Usage
runif_sm(num, ip, ir)
Arguments
num |
number of observations or sample size. |
ip |
the first dimension |
ir |
the second dimension |
Details
The Stiefel manifold with dimension p and r (p \geq r) is a space whose points are r-frames in R^p.
A set of r orthonormal vectors in R^p is called an r-frame in R^p.
The Stiefel manifold is a collection of p \times r full rank matrices X such that X'X = I_r.
Value
an array with dimension num, ip and ir containing a sample of draws from the uniform distribution on the Stiefel manifold.
Author(s)
Yukai Yang, yukai.yang@statistik.uu.se
Examples
runif_sm(10,4,2)
Sample from the vector Langevin-Bingham on the Stiefel manifold.
Description
This function draws a sample from the vector Langevin-Bingham on the Stiefel manifold.
Usage
rvlb_sm(num, mA, vc, vx)
Arguments
num |
number of observations or sample size. |
mA |
the matrix A which is symmetric ip*ip matrix. |
vc |
the vector c with dimension ip. |
vx |
the vector x, the initial value. |
Details
The vector Langevin-Bingham distribution on the Stiefel manifold has the density kernel:
f(X) \propto \mathrm{etr}\{ x' A x + c' x \}
where x satisfies x'x = 1, and A is a symmetric matrix.
Value
an array containing a sample of draws from the vector Langevin-Bingham on the Stiefel manifold.
References
Hoff, P. D. (2009) "Simulation of the Matrix Bingham—von Mises—Fisher Distribution, With Applications to Multivariate and Relational Data", Journal of Computational and Graphical Statistics, Vol. 18, pp. 438-456.
Author(s)
Yukai Yang, yukai.yang@statistik.uu.se
Show the version number of some information.
Description
This function shows the version number and some information of the package.
Usage
version()
Author(s)
Yukai Yang, yukai.yang@statistik.uu.se