Fit the Continuous-Time Vector Autoregressive Model Using the OpenMx Package
Ivan Jacob Agaloos Pesigan
2026-02-04
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: 429365.485591036
#>
#> Solution found#>
#> Solution found! Final fit=429365.49 (started at 446038.15) (1 attempt(s): 1 valid, 0 errors)
#> Start values from best fit:
#> -0.351820812571319,0.744058348126096,-0.458865690430299,0.0173253212869458,-0.488579953983734,0.726990071828104,-0.023803730310335,-0.00993023993483604,-0.688416480027775,0.237552827206534,0.0317029444801824,0.0717951385747256,-0.053613558460382,0.012338997867509,0.0757517888198224,0.198871372142446,0.199532519044795,0.201182587340908,0.00635388230183679,-0.0424239172867786,0.13014992344374,1.150441191294,0.413787071269287,1.22186739084888,0.226069493859024,0.235392674509846,0.962739024156933
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 -0.351820813 0.035321153 -1.5
#> 2 phi_2_1 CTVAR_1.phi eta_m eta_x 0.744058348 0.021718167 -1.5
#> 3 phi_3_1 CTVAR_1.phi eta_y eta_x -0.458865690 0.022742841 -1.5
#> 4 phi_1_2 CTVAR_1.phi eta_x eta_m 0.017325321 0.030891811 -1.5
#> 5 phi_2_2 CTVAR_1.phi eta_m eta_m -0.488579954 0.019189836 -1.5
#> 6 phi_3_2 CTVAR_1.phi eta_y eta_m 0.726990072 0.020119692 -1.5
#> 7 phi_1_3 CTVAR_1.phi eta_x eta_y -0.023803730 0.023614255 -1.5
#> 8 phi_2_3 CTVAR_1.phi eta_m eta_y -0.009930240 0.014649985 -1.5
#> 9 phi_3_3 CTVAR_1.phi eta_y eta_y -0.688416480 0.015475497 -1.5
#> 10 sigma_1_1 CTVAR_1.sigma eta_x eta_x 0.237552827 0.006729458 0
#> 11 sigma_2_1 CTVAR_1.sigma eta_x eta_m 0.031702944 0.002522389
#> 12 sigma_2_2 CTVAR_1.sigma eta_m eta_m 0.071795139 0.001922894 0
#> 13 sigma_3_1 CTVAR_1.sigma eta_x eta_y -0.053613558 0.002638619
#> 14 sigma_3_2 CTVAR_1.sigma eta_m eta_y 0.012338998 0.001396694
#> 15 sigma_3_3 CTVAR_1.sigma eta_y eta_y 0.075751789 0.002137223 0
#> 16 theta_1_1 CTVAR_1.theta x x 0.198871372 0.001163882 0
#> 17 theta_2_2 CTVAR_1.theta m m 0.199532519 0.001000026 0
#> 18 theta_3_3 CTVAR_1.theta y y 0.201182587 0.001015359 0
#> 19 mu0_1_1 CTVAR_1.mu0 eta_x mu0 0.006353882 0.109652563
#> 20 mu0_2_1 CTVAR_1.mu0 eta_m mu0 -0.042423917 0.112217050 !
#> 21 mu0_3_1 CTVAR_1.mu0 eta_y mu0 0.130149923 0.100486548 !
#> 22 sigma0_1_1 CTVAR_1.sigma0 eta_x eta_x 1.150441191 0.171254549 ! 0
#> 23 sigma0_2_1 CTVAR_1.sigma0 eta_x eta_m 0.413787071 0.132444266 !
#> 24 sigma0_2_2 CTVAR_1.sigma0 eta_m eta_m 1.221867391 0.180735288 0
#> 25 sigma0_3_1 CTVAR_1.sigma0 eta_x eta_y 0.226069494 0.114217258 !
#> 26 sigma0_3_2 CTVAR_1.sigma0 eta_m eta_y 0.235392675 0.116761937 !
#> 27 sigma0_3_3 CTVAR_1.sigma0 eta_y eta_y 0.962739024 0.143527285 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: 2026-02-04 03:28:05
#> Wall clock time: 1748.603 secs
#> optimizer: SLSQP
#> OpenMx version number: 2.22.10
#> 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,] -0.3518208 0.01732532 -0.02380373
#> [2,] 0.7440583 -0.48857995 -0.00993024
#> [3,] -0.4588657 0.72699007 -0.68841648
vcov_phi_vec
#> phi_1_1 phi_2_1 phi_3_1 phi_1_2 phi_2_2
#> phi_1_1 1.247584e-03 6.204680e-05 -1.797110e-04 -1.044633e-03 -6.054430e-05
#> phi_2_1 6.204680e-05 4.716788e-04 9.538355e-06 -4.016560e-05 -3.991326e-04
#> phi_3_1 -1.797110e-04 9.538355e-06 5.172368e-04 1.447003e-04 -1.731015e-06
#> phi_1_2 -1.044633e-03 -4.016560e-05 1.447003e-04 9.543040e-04 4.669631e-05
#> phi_2_2 -6.054430e-05 -3.991326e-04 -1.731015e-06 4.669631e-05 3.682498e-04
#> phi_3_2 1.577580e-04 -1.561336e-05 -4.381310e-04 -1.419511e-04 7.988600e-06
#> phi_1_3 7.022339e-04 2.238038e-05 -8.777993e-05 -6.650863e-04 -2.765071e-05
#> phi_2_3 4.887109e-05 2.681547e-04 -4.792982e-06 -4.128968e-05 -2.570596e-04
#> phi_3_3 -1.134849e-04 1.591634e-05 2.955074e-04 1.066127e-04 -1.127583e-05
#> phi_3_2 phi_1_3 phi_2_3 phi_3_3
#> phi_1_1 1.577580e-04 7.022339e-04 4.887109e-05 -1.134849e-04
#> phi_2_1 -1.561336e-05 2.238038e-05 2.681547e-04 1.591634e-05
#> phi_3_1 -4.381310e-04 -8.777993e-05 -4.792982e-06 2.955074e-04
#> phi_1_2 -1.419511e-04 -6.650863e-04 -4.128968e-05 1.066127e-04
#> phi_2_2 7.988600e-06 -2.765071e-05 -2.570596e-04 -1.127583e-05
#> phi_3_2 4.048020e-04 9.139347e-05 1.275627e-06 -2.841744e-04
#> phi_1_3 9.139347e-05 5.576330e-04 3.170146e-05 -8.690637e-05
#> phi_2_3 1.275627e-06 3.170146e-05 2.146221e-04 3.565454e-06
#> phi_3_3 -2.841744e-04 -8.690637e-05 3.565454e-06 2.394910e-04Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix
sigma
#> [,1] [,2] [,3]
#> [1,] 0.23755283 0.03170294 -0.05361356
#> [2,] 0.03170294 0.07179514 0.01233900
#> [3,] -0.05361356 0.01233900 0.07575179
vcov_sigma_vech
#> sigma_1_1 sigma_2_1 sigma_3_1 sigma_2_2 sigma_3_2
#> sigma_1_1 4.528560e-05 -7.425746e-07 -4.781448e-06 -5.223248e-07 5.533432e-07
#> sigma_2_1 -7.425746e-07 6.362446e-06 -1.639466e-08 -3.618622e-08 -1.070976e-06
#> sigma_3_1 -4.781448e-06 -1.639466e-08 6.962308e-06 1.520680e-08 -3.684501e-08
#> sigma_2_2 -5.223248e-07 -3.618622e-08 1.520680e-08 3.697522e-06 -4.742080e-08
#> sigma_3_2 5.533432e-07 -1.070976e-06 -3.684501e-08 -4.742080e-08 1.950754e-06
#> sigma_3_3 -2.040071e-08 1.873694e-07 -1.974478e-06 -4.580901e-08 -9.723002e-08
#> sigma_3_3
#> sigma_1_1 -2.040071e-08
#> sigma_2_1 1.873694e-07
#> sigma_3_1 -1.974478e-06
#> sigma_2_2 -4.580901e-08
#> sigma_3_2 -9.723002e-08
#> sigma_3_3 4.567723e-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
#> -0.35182081 0.74405835 -0.45886569 0.01732532 -0.48857995 0.72699007
#> phi_1_3 phi_2_3 phi_3_3 sigma_1_1 sigma_2_1 sigma_3_1
#> -0.02380373 -0.00993024 -0.68841648 0.23755283 0.03170294 -0.05361356
#> sigma_2_2 sigma_3_2 sigma_3_3
#> 0.07179514 0.01233900 0.07575179
vcov_theta
#> phi_1_1 phi_2_1 phi_3_1 phi_1_2 phi_2_2
#> phi_1_1 1.247584e-03 6.204680e-05 -1.797110e-04 -1.044633e-03 -6.054430e-05
#> phi_2_1 6.204680e-05 4.716788e-04 9.538355e-06 -4.016560e-05 -3.991326e-04
#> phi_3_1 -1.797110e-04 9.538355e-06 5.172368e-04 1.447003e-04 -1.731015e-06
#> phi_1_2 -1.044633e-03 -4.016560e-05 1.447003e-04 9.543040e-04 4.669631e-05
#> phi_2_2 -6.054430e-05 -3.991326e-04 -1.731015e-06 4.669631e-05 3.682498e-04
#> phi_3_2 1.577580e-04 -1.561336e-05 -4.381310e-04 -1.419511e-04 7.988600e-06
#> phi_1_3 7.022339e-04 2.238038e-05 -8.777993e-05 -6.650863e-04 -2.765071e-05
#> phi_2_3 4.887109e-05 2.681547e-04 -4.792982e-06 -4.128968e-05 -2.570596e-04
#> phi_3_3 -1.134849e-04 1.591634e-05 2.955074e-04 1.066127e-04 -1.127583e-05
#> sigma_1_1 -1.918630e-04 -1.645251e-05 3.037485e-05 1.502686e-04 1.387956e-05
#> sigma_2_1 1.910338e-05 -3.241454e-05 -6.868620e-06 -2.058373e-05 2.412367e-05
#> sigma_3_1 1.058652e-05 1.958428e-06 -3.663026e-05 -4.237886e-06 -1.403297e-06
#> sigma_2_2 3.777621e-06 1.342082e-05 7.952239e-07 -3.444068e-06 -1.419808e-05
#> sigma_3_2 -5.787097e-06 1.830464e-06 7.733913e-06 5.781118e-06 2.325849e-07
#> sigma_3_3 3.816940e-06 -1.075800e-06 2.349552e-06 -4.527518e-06 7.540172e-07
#> phi_3_2 phi_1_3 phi_2_3 phi_3_3 sigma_1_1
#> phi_1_1 1.577580e-04 7.022339e-04 4.887109e-05 -1.134849e-04 -1.918630e-04
#> phi_2_1 -1.561336e-05 2.238038e-05 2.681547e-04 1.591634e-05 -1.645251e-05
#> phi_3_1 -4.381310e-04 -8.777993e-05 -4.792982e-06 2.955074e-04 3.037485e-05
#> phi_1_2 -1.419511e-04 -6.650863e-04 -4.128968e-05 1.066127e-04 1.502686e-04
#> phi_2_2 7.988600e-06 -2.765071e-05 -2.570596e-04 -1.127583e-05 1.387956e-05
#> phi_3_2 4.048020e-04 9.139347e-05 1.275627e-06 -2.841744e-04 -2.484591e-05
#> phi_1_3 9.139347e-05 5.576330e-04 3.170146e-05 -8.690637e-05 -9.304952e-05
#> phi_2_3 1.275627e-06 3.170146e-05 2.146221e-04 3.565454e-06 -9.558393e-06
#> phi_3_3 -2.841744e-04 -8.690637e-05 3.565454e-06 2.394910e-04 1.640237e-05
#> sigma_1_1 -2.484591e-05 -9.304952e-05 -9.558393e-06 1.640237e-05 4.528560e-05
#> sigma_2_1 6.855446e-06 1.425503e-05 -1.402873e-05 -4.894158e-06 -7.425746e-07
#> sigma_3_1 2.739421e-05 -4.219143e-06 3.921594e-07 -1.508627e-05 -4.781448e-06
#> sigma_2_2 -1.122974e-06 2.318940e-06 9.347026e-06 8.813382e-07 -5.223248e-07
#> sigma_3_2 -8.028180e-06 -4.213819e-06 -2.603417e-06 5.173124e-06 5.533432e-07
#> sigma_3_3 1.745357e-06 5.208954e-06 -9.585209e-08 -6.774855e-06 -2.040071e-08
#> sigma_2_1 sigma_3_1 sigma_2_2 sigma_3_2 sigma_3_3
#> phi_1_1 1.910338e-05 1.058652e-05 3.777621e-06 -5.787097e-06 3.816940e-06
#> phi_2_1 -3.241454e-05 1.958428e-06 1.342082e-05 1.830464e-06 -1.075800e-06
#> phi_3_1 -6.868620e-06 -3.663026e-05 7.952239e-07 7.733913e-06 2.349552e-06
#> phi_1_2 -2.058373e-05 -4.237886e-06 -3.444068e-06 5.781118e-06 -4.527518e-06
#> phi_2_2 2.412367e-05 -1.403297e-06 -1.419808e-05 2.325849e-07 7.540172e-07
#> phi_3_2 6.855446e-06 2.739421e-05 -1.122974e-06 -8.028180e-06 1.745357e-06
#> phi_1_3 1.425503e-05 -4.219143e-06 2.318940e-06 -4.213819e-06 5.208954e-06
#> phi_2_3 -1.402873e-05 3.921594e-07 9.347026e-06 -2.603417e-06 -9.585209e-08
#> phi_3_3 -4.894158e-06 -1.508627e-05 8.813382e-07 5.173124e-06 -6.774855e-06
#> sigma_1_1 -7.425746e-07 -4.781448e-06 -5.223248e-07 5.533432e-07 -2.040071e-08
#> sigma_2_1 6.362446e-06 -1.639466e-08 -3.618622e-08 -1.070976e-06 1.873694e-07
#> sigma_3_1 -1.639466e-08 6.962308e-06 1.520680e-08 -3.684501e-08 -1.974478e-06
#> sigma_2_2 -3.618622e-08 1.520680e-08 3.697522e-06 -4.742080e-08 -4.580901e-08
#> sigma_3_2 -1.070976e-06 -3.684501e-08 -4.742080e-08 1.950754e-06 -9.723002e-08
#> sigma_3_3 1.873694e-07 -1.974478e-06 -4.580901e-08 -9.723002e-08 4.567723e-06References
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