fit.fmou | R Documentation |
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.
## 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)
object |
an objecft of class |
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 |
track_neg_log_lik |
a bool value, default is |
U_init |
user-specified initial factor loading matrix in the EM algorithm. Default is |
rho_init |
user-specified initial correlation parameters in the EM algorithm. Default is |
sigma2_init |
user-specified initial variance parameters in the EM algorithm. Default is |
d_ub |
upper bound of d when d is estimated. Default is null. |
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. |
Mengyang Gu [aut, cre], Xinyi Fang [aut], Yizi Lin [aut]
Maintainer: Mengyang Gu <mengyang@pstat.ucsb.edu>
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.
## 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
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