Fit the Continuous-Time Vector Autoregressive Model Using the OpenMx Package
Ivan Jacob Agaloos Pesigan
2025-08-23
Source:vignettes/fit-ct-var-mx.Rmd
fit-ct-var-mx.RmdData Generation Using the SimSSMOUFixed Function from
the simStateSpace Package
n
#> [1] 100
time
#> [1] 1000
delta_t
#> [1] 0.1
mu0
#> [1] 0 0 0
sigma0
#> [,1] [,2] [,3]
#> [1,] 1.0 0.2 0.2
#> [2,] 0.2 1.0 0.2
#> [3,] 0.2 0.2 1.0
sigma0_l # sigma0_l <- t(chol(sigma0))
#> [,1] [,2] [,3]
#> [1,] 1.0 0.0000000 0.0000000
#> [2,] 0.2 0.9797959 0.0000000
#> [3,] 0.2 0.1632993 0.9660918
mu
#> [1] 0 0 0
phi
#> [,1] [,2] [,3]
#> [1,] -0.357 0.000 0.000
#> [2,] 0.771 -0.511 0.000
#> [3,] -0.450 0.729 -0.693
sigma
#> [,1] [,2] [,3]
#> [1,] 0.24455556 0.02201587 -0.05004762
#> [2,] 0.02201587 0.07067800 0.01539456
#> [3,] -0.05004762 0.01539456 0.07553061
sigma_l # sigma_l <- t(chol(sigma))
#> [,1] [,2] [,3]
#> [1,] 0.49452559 0.0000000 0.000000
#> [2,] 0.04451917 0.2620993 0.000000
#> [3,] -0.10120330 0.0759256 0.243975
nu
#> [1] 0 0 0
lambda
#> [,1] [,2] [,3]
#> [1,] 1 0 0
#> [2,] 0 1 0
#> [3,] 0 0 1
theta
#> [,1] [,2] [,3]
#> [1,] 0.2 0.0 0.0
#> [2,] 0.0 0.2 0.0
#> [3,] 0.0 0.0 0.2
theta_l # theta_l <- t(chol(theta))
#> [,1] [,2] [,3]
#> [1,] 0.4472136 0.0000000 0.0000000
#> [2,] 0.0000000 0.4472136 0.0000000
#> [3,] 0.0000000 0.0000000 0.4472136
library(simStateSpace)
sim <- SimSSMOUFixed(
n = n,
time = time,
delta_t = delta_t,
mu0 = mu0,
sigma0_l = sigma0_l,
mu = mu,
phi = phi,
sigma_l = sigma_l,
nu = nu,
lambda = lambda,
theta_l = theta_l,
type = 0
)
data <- as.data.frame(sim)
colnames(data) <- c("id", "time", "x", "m", "y")
head(data)
#> id time x m y
#> 1 1 0.0 -0.3504435 0.41877429 2.611996
#> 2 1 0.1 -0.5920330 1.07433208 1.669272
#> 3 1 0.2 -0.7619855 1.21483834 2.369837
#> 4 1 0.3 -1.6964652 0.21209722 2.128531
#> 5 1 0.4 -1.2282686 0.09950326 1.891140
#> 6 1 0.5 0.1433985 0.66784226 2.036033Model Fitting
We use the OpenMx package to fit the continuous-time
vector autoregressive model.
Prepare the Initial Condition
mu0 <- mxMatrix(
type = "Full",
nrow = 3,
ncol = 1,
free = TRUE,
values = matrix(
data = c(
0, 0, 0
),
nrow = 3,
ncol = 1
),
labels = matrix(
data = c(
"mu0_1_1", "mu0_2_1", "mu0_3_1"
),
nrow = 3,
ncol = 1
),
lbound = NA,
ubound = NA,
byrow = FALSE,
dimnames = list(
c("eta_x", "eta_m", "eta_y"),
"mu0"
),
name = "mu0"
)
sigma0 <- mxMatrix(
type = "Symm",
nrow = 3,
ncol = 3,
free = TRUE,
values = matrix(
data = c(
1.0, 0.2, 0.2,
0.2, 1.0, 0.2,
0.2, 0.2, 1.0
),
nrow = 3,
ncol = 3
),
labels = matrix(
data = c(
"sigma0_1_1", "sigma0_2_1", "sigma0_3_1",
"sigma0_2_1", "sigma0_2_2", "sigma0_3_2",
"sigma0_3_1", "sigma0_3_2", "sigma0_3_3"
),
nrow = 3,
ncol = 3
),
lbound = matrix(
data = c(
0, NA, NA,
NA, 0, NA,
NA, NA, 0
),
nrow = 3,
ncol = 3
),
ubound = NA,
byrow = FALSE,
dimnames = list(
c("eta_x", "eta_m", "eta_y"),
c("eta_x", "eta_m", "eta_y")
),
name = "sigma0"
)Prepare the Dynamic Model
phi <- mxMatrix(
type = "Full",
nrow = 3,
ncol = 3,
free = TRUE,
values = matrix(
data = c(
-0.2, 0.0, 0.0,
0.0, -0.2, 0.0,
0.0, 0.0, -0.2
),
nrow = 3,
ncol = 3
),
labels = matrix(
data = c(
"phi_1_1", "phi_2_1", "phi_3_1",
"phi_1_2", "phi_2_2", "phi_3_2",
"phi_1_3", "phi_2_3", "phi_3_3"
),
nrow = 3,
ncol = 3
),
lbound = -1.5,
ubound = 1.5,
byrow = FALSE,
dimnames = list(
c("eta_x", "eta_m", "eta_y"),
c("eta_x", "eta_m", "eta_y")
),
name = "phi"
)Prepare the Noise Matrices
sigma <- mxMatrix(
type = "Symm",
nrow = 3,
ncol = 3,
free = TRUE,
values = 0.2 * diag(3),
labels = matrix(
data = c(
"sigma_1_1", "sigma_2_1", "sigma_3_1",
"sigma_2_1", "sigma_2_2", "sigma_3_2",
"sigma_3_1", "sigma_3_2", "sigma_3_3"
),
nrow = 3,
ncol = 3
),
lbound = matrix(
data = c(
0, NA, NA,
NA, 0, NA,
NA, NA, 0
),
nrow = 3,
ncol = 3
),
ubound = NA,
byrow = FALSE,
dimnames = list(
c("eta_x", "eta_m", "eta_y"),
c("eta_x", "eta_m", "eta_y")
),
name = "sigma"
)
theta <- mxMatrix(
type = "Diag",
nrow = 3,
ncol = 3,
free = TRUE,
values = 0.2 * diag(3),
labels = matrix(
data = c(
"theta_1_1", "fixed", "fixed",
"fixed", "theta_2_2", "fixed",
"fixed", "fixed", "theta_3_3"
),
nrow = 3,
ncol = 3
),
lbound = matrix(
data = c(
0, NA, NA,
NA, 0, NA,
NA, NA, 0
),
nrow = 3,
ncol = 3
),
ubound = NA,
byrow = FALSE,
dimnames = list(
c("x", "m", "y"),
c("x", "m", "y")
),
name = "theta"
)Prepare Miscellaneous Matrices
time <- mxMatrix(
type = "Full",
nrow = 1,
ncol = 1,
free = FALSE,
labels = "data.time",
name = "time"
)
gamma <- mxMatrix(
type = "Zero",
nrow = 3,
ncol = 1,
name = "gamma"
)
kappa <- mxMatrix(
type = "Zero",
nrow = 3,
ncol = 1,
name = "kappa"
)
covariate <- mxMatrix(
type = "Zero",
nrow = 1,
ncol = 1,
name = "covariate"
)Prepare the Model
In this parameterization, we fit the same model to all individuals
(id) using a multi-group framework, assuming that the
parameters remain fixed across individuals.
model <- mxModel(
model = "CTVAR",
phi,
gamma,
lambda,
kappa,
sigma,
theta,
mu0,
sigma0,
covariate,
time,
mxExpectationStateSpaceContinuousTime(
A = "phi",
B = "gamma",
C = "lambda",
D = "kappa",
Q = "sigma",
R = "theta",
x0 = "mu0",
P0 = "sigma0",
u = "covariate",
t = "time",
dimnames = c("x", "m", "y")
),
mxFitFunctionML(),
mxData(
observed = data,
type = "raw"
)
)
ids <- sort(
unique(data[, "id"])
)
model_id <- lapply(
X = ids,
FUN = function(i,
data,
model) {
return(
mxModel(
name = paste0("CTVAR", "_", i),
model = model,
mxData(
observed = data[
which(
data[, "id"] == i
), ,
drop = FALSE
],
type = "raw"
)
)
)
},
data = data,
model = model
)Fit the Model
fit <- mxTryHardctsem(
model = mxModel(
name = "CTVAR",
model_id,
mxFitFunctionMultigroup(
paste0(
"CTVAR",
"_",
ids
)
)
),
extraTries = 1000
)
#> Running CTVAR with 27 parameters
#>
#> Beginning initial fit attempt
#> Running CTVAR with 27 parameters
#>
#> Lowest minimum so far: 429365.485590996
#>
#> Solution found#>
#> Solution found! Final fit=429365.49 (started at 446038.15) (1 attempt(s): 1 valid, 0 errors)
#> Start values from best fit:
#> -0.351821137097335,0.744058613532521,-0.458865746117413,0.017325500704327,-0.488580124271284,0.726989949717352,-0.0238038665034885,-0.00993040056057583,-0.688416447176184,0.237552845809196,0.0317028936192271,0.0717950634740514,-0.0536134825530202,0.0123389991374601,0.0757517174981564,0.198871348346359,0.199532512118898,0.201182561465738,0.00635060323101044,-0.042423026750225,0.130144639885915,1.15044508486601,0.413785996635709,1.22186586248913,0.226070445628877,0.235391509788928,0.962738053951411
summary(fit)
#> Summary of CTVAR
#>
#> free parameters:
#> name matrix row col Estimate Std.Error A lbound
#> 1 phi_1_1 CTVAR_1.phi eta_x eta_x -0.351821137 0.035299440 -1.5
#> 2 phi_2_1 CTVAR_1.phi eta_m eta_x 0.744058614 0.021699954 -1.5
#> 3 phi_3_1 CTVAR_1.phi eta_y eta_x -0.458865746 0.022757419 -1.5
#> 4 phi_1_2 CTVAR_1.phi eta_x eta_m 0.017325501 0.030881024 -1.5
#> 5 phi_2_2 CTVAR_1.phi eta_m eta_m -0.488580124 0.019172291 -1.5
#> 6 phi_3_2 CTVAR_1.phi eta_y eta_m 0.726989950 0.020130685 -1.5
#> 7 phi_1_3 CTVAR_1.phi eta_x eta_y -0.023803867 0.023615965 -1.5
#> 8 phi_2_3 CTVAR_1.phi eta_m eta_y -0.009930401 0.014638885 -1.5
#> 9 phi_3_3 CTVAR_1.phi eta_y eta_y -0.688416447 0.015482401 -1.5
#> 10 sigma_1_1 CTVAR_1.sigma eta_x eta_x 0.237552846 0.006725262 0
#> 11 sigma_2_1 CTVAR_1.sigma eta_x eta_m 0.031702894 0.002522145
#> 12 sigma_2_2 CTVAR_1.sigma eta_m eta_m 0.071795063 0.001922595 0
#> 13 sigma_3_1 CTVAR_1.sigma eta_x eta_y -0.053613483 0.002639335
#> 14 sigma_3_2 CTVAR_1.sigma eta_m eta_y 0.012338999 0.001396768
#> 15 sigma_3_3 CTVAR_1.sigma eta_y eta_y 0.075751717 0.002137272 0
#> 16 theta_1_1 CTVAR_1.theta x x 0.198871348 0.001163754 0
#> 17 theta_2_2 CTVAR_1.theta m m 0.199532512 0.001000012 0
#> 18 theta_3_3 CTVAR_1.theta y y 0.201182561 0.001015357 0
#> 19 mu0_1_1 CTVAR_1.mu0 eta_x mu0 0.006350603 0.110265994 !
#> 20 mu0_2_1 CTVAR_1.mu0 eta_m mu0 -0.042423027 0.113052803 !
#> 21 mu0_3_1 CTVAR_1.mu0 eta_y mu0 0.130144640 0.100521830 !
#> 22 sigma0_1_1 CTVAR_1.sigma0 eta_x eta_x 1.150445085 0.173008126 ! 0
#> 23 sigma0_2_1 CTVAR_1.sigma0 eta_x eta_m 0.413785997 0.132331823 !
#> 24 sigma0_2_2 CTVAR_1.sigma0 eta_m eta_m 1.221865862 0.179918449 0
#> 25 sigma0_3_1 CTVAR_1.sigma0 eta_x eta_y 0.226070446 0.114458254 !
#> 26 sigma0_3_2 CTVAR_1.sigma0 eta_m eta_y 0.235391510 0.116842831 !
#> 27 sigma0_3_3 CTVAR_1.sigma0 eta_y eta_y 0.962738054 0.143039642 0
#> ubound
#> 1 1.5
#> 2 1.5
#> 3 1.5
#> 4 1.5
#> 5 1.5
#> 6 1.5
#> 7 1.5
#> 8 1.5
#> 9 1.5
#> 10
#> 11
#> 12
#> 13
#> 14
#> 15
#> 16
#> 17
#> 18
#> 19
#> 20
#> 21
#> 22
#> 23
#> 24
#> 25
#> 26
#> 27
#>
#> Model Statistics:
#> | Parameters | Degrees of Freedom | Fit (-2lnL units)
#> Model: 27 299973 429365.5
#> Saturated: NA NA NA
#> Independence: NA NA NA
#> Number of observations/statistics: 1e+05/3e+05
#>
#> Information Criteria:
#> | df Penalty | Parameters Penalty | Sample-Size Adjusted
#> AIC: -170580.5 429419.5 429419.5
#> BIC: -3024201.3 429676.3 429590.5
#> To get additional fit indices, see help(mxRefModels)
#> timestamp: 2025-08-23 06:10:23
#> Wall clock time: 4006.813 secs
#> optimizer: SLSQP
#> OpenMx version number: 2.22.7
#> Need help? See help(mxSummary)
coefs <- coef(fit)
vcovs <- vcov(fit)Extract Matrices from the Fitted Model to use in cTMed
phi_names <- c(
"phi_1_1", "phi_2_1", "phi_3_1",
"phi_1_2", "phi_2_2", "phi_3_2",
"phi_1_3", "phi_2_3", "phi_3_3"
)
sigma_names <- c(
"sigma_1_1", "sigma_2_1", "sigma_3_1",
"sigma_2_1", "sigma_2_2", "sigma_3_2",
"sigma_3_1", "sigma_3_2", "sigma_3_3"
)
sigma_vech_names <- c(
"sigma_1_1", "sigma_2_1", "sigma_3_1",
"sigma_2_2", "sigma_3_2",
"sigma_3_3"
)
theta_names <- c(
"phi_1_1", "phi_2_1", "phi_3_1",
"phi_1_2", "phi_2_2", "phi_3_2",
"phi_1_3", "phi_2_3", "phi_3_3",
"sigma_1_1", "sigma_2_1", "sigma_3_1",
"sigma_2_2", "sigma_3_2",
"sigma_3_3"
)
phi <- matrix(
data = coefs[phi_names],
nrow = 3,
ncol = 3
)
sigma <- matrix(
data = coefs[sigma_names],
nrow = 3,
ncol = 3
)
theta <- coefs[theta_names]
vcov_phi_vec <- vcovs[phi_names, phi_names]
vcov_sigma_vech <- vcovs[sigma_vech_names, sigma_vech_names]
vcov_theta <- vcovs[theta_names, theta_names]Estimated Drift Matrix with Corresponding Sampling Covariance Matrix
phi
#> [,1] [,2] [,3]
#> [1,] -0.3518211 0.0173255 -0.023803867
#> [2,] 0.7440586 -0.4885801 -0.009930401
#> [3,] -0.4588657 0.7269899 -0.688416447
vcov_phi_vec
#> phi_1_1 phi_2_1 phi_3_1 phi_1_2 phi_2_2
#> phi_1_1 1.246050e-03 6.150931e-05 -1.797393e-04 -1.043572e-03 -6.010544e-05
#> phi_2_1 6.150931e-05 4.708880e-04 9.205053e-06 -3.966508e-05 -3.984006e-04
#> phi_3_1 -1.797393e-04 9.205053e-06 5.179001e-04 1.447459e-04 -1.422049e-06
#> phi_1_2 -1.043572e-03 -3.966508e-05 1.447459e-04 9.536376e-04 4.629107e-05
#> phi_2_2 -6.010544e-05 -3.984006e-04 -1.422049e-06 4.629107e-05 3.675767e-04
#> phi_3_2 1.578125e-04 -1.533727e-05 -4.386718e-04 -1.420256e-04 7.731061e-06
#> phi_1_3 7.017864e-04 2.196281e-05 -8.787919e-05 -6.648937e-04 -2.731141e-05
#> phi_2_3 4.861771e-05 2.676394e-04 -5.042368e-06 -4.105661e-05 -2.565891e-04
#> phi_3_3 -1.135519e-04 1.576431e-05 2.958719e-04 1.067020e-04 -1.113070e-05
#> phi_3_2 phi_1_3 phi_2_3 phi_3_3
#> phi_1_1 1.578125e-04 7.017864e-04 4.861771e-05 -1.135519e-04
#> phi_2_1 -1.533727e-05 2.196281e-05 2.676394e-04 1.576431e-05
#> phi_3_1 -4.386718e-04 -8.787919e-05 -5.042368e-06 2.958719e-04
#> phi_1_2 -1.420256e-04 -6.648937e-04 -4.105661e-05 1.067020e-04
#> phi_2_2 7.731061e-06 -2.731141e-05 -2.565891e-04 -1.113070e-05
#> phi_3_2 4.052445e-04 9.151455e-05 1.485134e-06 -2.844768e-04
#> phi_1_3 9.151455e-05 5.577138e-04 3.150218e-05 -8.703403e-05
#> phi_2_3 1.485134e-06 3.150218e-05 2.142970e-04 3.442292e-06
#> phi_3_3 -2.844768e-04 -8.703403e-05 3.442292e-06 2.397048e-04Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix
sigma
#> [,1] [,2] [,3]
#> [1,] 0.23755285 0.03170289 -0.05361348
#> [2,] 0.03170289 0.07179506 0.01233900
#> [3,] -0.05361348 0.01233900 0.07575172
vcov_sigma_vech
#> sigma_1_1 sigma_2_1 sigma_3_1 sigma_2_2 sigma_3_2
#> sigma_1_1 4.522914e-05 -7.455500e-07 -4.778394e-06 -5.205253e-07 5.537197e-07
#> sigma_2_1 -7.455500e-07 6.361218e-06 -1.832842e-08 -3.428689e-08 -1.071143e-06
#> sigma_3_1 -4.778394e-06 -1.832842e-08 6.966087e-06 1.598234e-08 -3.733166e-08
#> sigma_2_2 -5.205253e-07 -3.428689e-08 1.598234e-08 3.696373e-06 -4.757941e-08
#> sigma_3_2 5.537197e-07 -1.071143e-06 -3.733166e-08 -4.757941e-08 1.950960e-06
#> sigma_3_3 -1.927519e-08 1.881402e-07 -1.975318e-06 -4.604885e-08 -9.723122e-08
#> sigma_3_3
#> sigma_1_1 -1.927519e-08
#> sigma_2_1 1.881402e-07
#> sigma_3_1 -1.975318e-06
#> sigma_2_2 -4.604885e-08
#> sigma_3_2 -9.723122e-08
#> sigma_3_3 4.567934e-06Estimated Drift Matrix and Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix
theta
#> phi_1_1 phi_2_1 phi_3_1 phi_1_2 phi_2_2 phi_3_2
#> -0.351821137 0.744058614 -0.458865746 0.017325501 -0.488580124 0.726989950
#> phi_1_3 phi_2_3 phi_3_3 sigma_1_1 sigma_2_1 sigma_3_1
#> -0.023803867 -0.009930401 -0.688416447 0.237552846 0.031702894 -0.053613483
#> sigma_2_2 sigma_3_2 sigma_3_3
#> 0.071795063 0.012338999 0.075751717
vcov_theta
#> phi_1_1 phi_2_1 phi_3_1 phi_1_2 phi_2_2
#> phi_1_1 1.246050e-03 6.150931e-05 -1.797393e-04 -1.043572e-03 -6.010544e-05
#> phi_2_1 6.150931e-05 4.708880e-04 9.205053e-06 -3.966508e-05 -3.984006e-04
#> phi_3_1 -1.797393e-04 9.205053e-06 5.179001e-04 1.447459e-04 -1.422049e-06
#> phi_1_2 -1.043572e-03 -3.966508e-05 1.447459e-04 9.536376e-04 4.629107e-05
#> phi_2_2 -6.010544e-05 -3.984006e-04 -1.422049e-06 4.629107e-05 3.675767e-04
#> phi_3_2 1.578125e-04 -1.533727e-05 -4.386718e-04 -1.420256e-04 7.731061e-06
#> phi_1_3 7.017864e-04 2.196281e-05 -8.787919e-05 -6.648937e-04 -2.731141e-05
#> phi_2_3 4.861771e-05 2.676394e-04 -5.042368e-06 -4.105661e-05 -2.565891e-04
#> phi_3_3 -1.135519e-04 1.576431e-05 2.958719e-04 1.067020e-04 -1.113070e-05
#> sigma_1_1 -1.915569e-04 -1.636915e-05 3.037297e-05 1.500395e-04 1.381028e-05
#> sigma_2_1 1.913283e-05 -3.237521e-05 -6.856165e-06 -2.061782e-05 2.408541e-05
#> sigma_3_1 1.056219e-05 1.986607e-06 -3.667489e-05 -4.214403e-06 -1.428526e-06
#> sigma_2_2 3.767400e-06 1.338905e-05 7.844918e-07 -3.434966e-06 -1.416929e-05
#> sigma_3_2 -5.792140e-06 1.827631e-06 7.743783e-06 5.787144e-06 2.357006e-07
#> sigma_3_3 3.815037e-06 -1.085609e-06 2.355086e-06 -4.528709e-06 7.624446e-07
#> phi_3_2 phi_1_3 phi_2_3 phi_3_3 sigma_1_1
#> phi_1_1 1.578125e-04 7.017864e-04 4.861771e-05 -1.135519e-04 -1.915569e-04
#> phi_2_1 -1.533727e-05 2.196281e-05 2.676394e-04 1.576431e-05 -1.636915e-05
#> phi_3_1 -4.386718e-04 -8.787919e-05 -5.042368e-06 2.958719e-04 3.037297e-05
#> phi_1_2 -1.420256e-04 -6.648937e-04 -4.105661e-05 1.067020e-04 1.500395e-04
#> phi_2_2 7.731061e-06 -2.731141e-05 -2.565891e-04 -1.113070e-05 1.381028e-05
#> phi_3_2 4.052445e-04 9.151455e-05 1.485134e-06 -2.844768e-04 -2.484618e-05
#> phi_1_3 9.151455e-05 5.577138e-04 3.150218e-05 -8.703403e-05 -9.292911e-05
#> phi_2_3 1.485134e-06 3.150218e-05 2.142970e-04 3.442292e-06 -9.517195e-06
#> phi_3_3 -2.844768e-04 -8.703403e-05 3.442292e-06 2.397048e-04 1.640337e-05
#> sigma_1_1 -2.484618e-05 -9.292911e-05 -9.517195e-06 1.640337e-05 4.522914e-05
#> sigma_2_1 6.846581e-06 1.428969e-05 -1.399999e-05 -4.891883e-06 -7.455500e-07
#> sigma_3_1 2.742907e-05 -4.235913e-06 4.114099e-07 -1.510750e-05 -4.778394e-06
#> sigma_2_2 -1.113749e-06 2.310964e-06 9.327113e-06 8.758013e-07 -5.205253e-07
#> sigma_3_2 -8.036564e-06 -4.220293e-06 -2.606510e-06 5.179229e-06 5.537197e-07
#> sigma_3_3 1.741461e-06 5.213264e-06 -1.014618e-07 -6.773320e-06 -1.927519e-08
#> sigma_2_1 sigma_3_1 sigma_2_2 sigma_3_2 sigma_3_3
#> phi_1_1 1.913283e-05 1.056219e-05 3.767400e-06 -5.792140e-06 3.815037e-06
#> phi_2_1 -3.237521e-05 1.986607e-06 1.338905e-05 1.827631e-06 -1.085609e-06
#> phi_3_1 -6.856165e-06 -3.667489e-05 7.844918e-07 7.743783e-06 2.355086e-06
#> phi_1_2 -2.061782e-05 -4.214403e-06 -3.434966e-06 5.787144e-06 -4.528709e-06
#> phi_2_2 2.408541e-05 -1.428526e-06 -1.416929e-05 2.357006e-07 7.624446e-07
#> phi_3_2 6.846581e-06 2.742907e-05 -1.113749e-06 -8.036564e-06 1.741461e-06
#> phi_1_3 1.428969e-05 -4.235913e-06 2.310964e-06 -4.220293e-06 5.213264e-06
#> phi_2_3 -1.399999e-05 4.114099e-07 9.327113e-06 -2.606510e-06 -1.014618e-07
#> phi_3_3 -4.891883e-06 -1.510750e-05 8.758013e-07 5.179229e-06 -6.773320e-06
#> sigma_1_1 -7.455500e-07 -4.778394e-06 -5.205253e-07 5.537197e-07 -1.927519e-08
#> sigma_2_1 6.361218e-06 -1.832842e-08 -3.428689e-08 -1.071143e-06 1.881402e-07
#> sigma_3_1 -1.832842e-08 6.966087e-06 1.598234e-08 -3.733166e-08 -1.975318e-06
#> sigma_2_2 -3.428689e-08 1.598234e-08 3.696373e-06 -4.757941e-08 -4.604885e-08
#> sigma_3_2 -1.071143e-06 -3.733166e-08 -4.757941e-08 1.950960e-06 -9.723122e-08
#> sigma_3_3 1.881402e-07 -1.975318e-06 -4.604885e-08 -9.723122e-08 4.567934e-06References
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