fit.fmou,fmou-method: The fast EM algorithm of multivariate Ornstein-Uhlenbeck...

In FastGaSP: Fast and Exact Computation of Gaussian Stochastic Process

The fast EM algorithm of multivariate Ornstein-Uhlenbeck processes

Description

This function implements an efficient EM algorithm to estimate the parameters in the FMOU model, a latent factor model with a fixed or estimated orthogonal factor loading matrix, where each latent factor is modeled as an O-U (Ornstein-Uhlenbeck) process.

Usage

## S4 method for signature 'fmou'
fit.fmou(object, M=50, threshold=1e-4,
         track_iterations=FALSE,track_neg_log_lik=FALSE,
         U_init=NULL, rho_init=NULL, sigma2_init=NULL, d_ub=NULL)

Arguments

object	an objecft of class fmou.
M	number of iterations in the EM algorithm, default is 50.
threshold	stopping criteria with respect to predictive mean of observations, default is 1e-4.
track_iterations	a bool value, default is FALSE. If TRUE, the estimations in each EM iteration will be recorded and returned.
track_neg_log_lik	a bool value, default is FALSE. If TRUE, the negative log marginal likelihood of the output in each EM iteration will be recorded and returned.
U_init	user-specified initial factor loading matrix in the EM algorithm. Default is NULL. The dimension is k*d, where k is the length of observations at each time step and d is the number of latent factors.
rho_init	user-specified initial correlation parameters in the EM algorithm. Default is NULL. The length is equal to the number of latent factors.
sigma2_init	user-specified initial variance parameters in the EM algorithm. Default is NULL. The length is equal to the number of latent factors.
d_ub	upper bound of d when d is estimated. Default is null.

Value

output	the observation matrix.
U	the estimated (or fixed) factor loading matrix.
post_z_mean	the posterior mean of latent factors.
post_z_var	the posterior variance of latent factors.
post_z_cov	the posterior covariance between two consecutive time steps of a latent process.
mean_obs	the predictive mean of the observations.
mean_obs_95lb	the lower bound of the 95% posterior credible intervals of predictive mean.
mean_obs_95ub	the upper bound of the 95% posterior credible intervals of predictive mean.
sigma0_2	estimated variance of noise.
rho	estimated correlation parameters.
sigma2	estimated variance parameters
num_iterations	number of iterations in the EM algorithm.
d	the estimated (or fixed) number of latent factors.
record_sigma0_2	estimated variance of noise in each iteration.
record_rho	estimated correlation parameters in each iteration.
record_sigma2	estimation variance parameters in each iteration.

Author(s)

Mengyang Gu [aut, cre], Xinyi Fang [aut], Yizi Lin [aut]

Maintainer: Mengyang Gu <[e-mail]>

References

Lin, Y., Liu, X., Segall, P., & Gu, M. (2025). Fast data inversion for high-dimensional dynamical systems from noisy measurements. arXiv preprint arXiv:2501.01324.

Examples

## generate simulated data
library(FastGaSP)
library(rstiefel)

d = 5  # number of latent factors
k = 20 # length of observation at each time step
n = 500 # number time step
noise_level = 1 # variance of noise

U = rustiefel(k, k) # factor loading matrix
z = matrix(NA, d, n)
sigma_2 = runif(d, 0.5, 1)
rho = runif(d, 0.95, 1)
for(l in 1:d){
  R = matrix(NA, n, n)
  diag(R) = 1
  for(ir in 1:n){
    for(ic in 1:n){
      R[ir, ic] = rho[l]^(abs(ir-ic)) * R[ir, ir]
    }
  }
  R = (sigma_2[l]/(1-rho[l]^2) )* R
  z[l, ] = t(chol(R)) %*% rnorm(n)
}

signal = U[,1:d] %*% z
y = signal + matrix(rnorm(n*k,mean=0,sd=sqrt(noise_level)),k,n)

##constucting the fmou.model
fmou.model=fmou(output=y, d=d, est_U0=TRUE, est_sigma0_2=TRUE)

## estimate the parameters
em_alg <- fit.fmou(fmou.model, M=500)

## root mean square error (RMSE) of predictive mean of observations
sqrt(mean((em_alg$mean_obs-signal)^2))

## standard deviation of (truth) mean of observations
sd(signal)

## estimated variance of noise
em_alg$sigma0_2

FastGaSP documentation built on April 4, 2025, 5:16 a.m.