Fit the Continuous-Time Vector Autoregressive Model Using the dynr Package

Data Generation Using the `SimSSMOUFixed` Function from the `simStateSpace` Package

n
#> [1] 50
time
#> [1] 100
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 2.036033

Model Fitting

We use the dynr package to fit the continuous-time vector autoregressive model.

library(dynr)

Prepare the Data

dynr_data <- dynr.data(
  dataframe = data,
  id = "id",
  time = "time",
  observed = c("x", "m", "y")
)

Prepare the Initial Condition

dynr_initial <- prep.initial(
  values.inistate = c(
    0, 0, 0
  ),
  params.inistate = c(
    "mu0_1_1", "mu0_2_1", "mu0_3_1"
  ),
  values.inicov = matrix(
    data = c(
      1.0, 0.2, 0.2,
      0.2, 1.0, 0.2,
      0.2, 0.2, 1.0
    ),
    nrow = 3
  ),
  params.inicov = 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
  )
)

Prepare the Measurement Model

dynr_measurement <- prep.measurement(
  values.load = diag(3),
  params.load = matrix(
    data = "fixed",
    nrow = 3,
    ncol = 3
  ),
  state.names = c("eta_x", "eta_m", "eta_y"),
  obs.names = c("x", "m", "y")
)

Prepare the Dynamic Model

dynr_dynamics <- prep.formulaDynamics(
  formula = list(
    eta_x ~ phi_1_1 * eta_x + phi_1_2 * eta_m + phi_1_3 * eta_y,
    eta_m ~ phi_2_1 * eta_x + phi_2_2 * eta_m + phi_2_3 * eta_y,
    eta_y ~ phi_3_1 * eta_x + phi_3_2 * eta_m + phi_3_3 * eta_y
  ),
  startval = c(
    phi_1_1 = -0.2, phi_2_1 = 0.0, phi_3_1 = 0.0,
    phi_1_2 = 0.0, phi_2_2 = -0.2, phi_3_2 = 0.0,
    phi_1_3 = 0.0, phi_2_3 = 0.0, phi_3_3 = -0.2
  ),
  isContinuousTime = TRUE
)

Prepare the Noise Matrices

dynr_noise <- prep.noise(
  values.latent = matrix(
    data = 0.2 * diag(3),
    nrow = 3
  ),
  params.latent = 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
  ),
  values.observed = 0.2 * diag(3),
  params.observed = matrix(
    data = c(
      "theta_1_1", "fixed", "fixed",
      "fixed", "theta_2_2", "fixed",
      "fixed", "fixed", "theta_3_3"
    ),
    nrow = 3
  )
)

Prepare the Model

In this example, we increase the maximum evaluations in the optimization process and constrain the lower and upper bounds of the parameters for the drift matrix.

dynr_model <- dynr.model(
  data = dynr_data,
  initial = dynr_initial,
  measurement = dynr_measurement,
  dynamics = dynr_dynamics,
  noise = dynr_noise,
  outfile = tempfile(
    paste0(
      "src-",
      format(
        Sys.time(),
        "%Y-%m-%d-%H-%M-%OS3"
      )
    ),
    fileext = ".c"
  )
)
dynr_model@options$maxeval <- 100000
lb <- ub <- rep(NA, times = length(dynr_model$xstart))
names(ub) <- names(lb) <- names(dynr_model$xstart)
lb[
  c(
    "phi_1_1", "phi_2_1", "phi_3_1",
    "phi_1_2", "phi_2_2", "phi_3_2",
    "phi_1_3", "phi_2_3", "phi_3_3"
  )
] <- -1.5
ub[
  c(
    "phi_1_1", "phi_2_1", "phi_3_1",
    "phi_1_2", "phi_2_2", "phi_3_2",
    "phi_1_3", "phi_2_3", "phi_3_3"
  )
] <- 1.5
dynr_model$lb <- lb
dynr_model$ub <- ub

Fit the Model

fit <- dynr.cook(
  dynr_model,
  verbose = FALSE
)
#> [1] "Get ready!!!!"
#> using C compiler: ‘gcc (Ubuntu 13.3.0-6ubuntu2~24.04) 13.3.0’
#> Optimization function called.
#> Starting Hessian calculation ...
#> Finished Hessian calculation.
#> Original exit flag:  3 
#> Modified exit flag:  3 
#> Optimization terminated successfully: ftol_rel or ftol_abs was reached. 
#> Original fitted parameters:  -0.2938068 0.8243958 -0.4543161 -0.07191554 
#> -0.5689988 0.707647 0.02579061 0.08266777 -0.6875018 -1.444748 0.07916771 
#> -0.2743246 -2.582169 0.1772108 -2.804538 -1.604288 -1.652135 -1.622277 
#> -0.06854665 0.08457349 0.1157207 -0.05902917 0.1716259 0.1095353 0.379774 
#> -0.08084045 -0.07765938 
#> 
#> Transformed fitted parameters:  -0.2938068 0.8243958 -0.4543161 -0.07191554 
#> -0.5689988 0.707647 0.02579061 0.08266777 -0.6875018 0.2358056 0.01866819 
#> -0.06468727 0.07708775 0.008277734 0.08065449 0.2010327 0.1916402 0.1974485 
#> -0.06854665 0.08457349 0.1157207 0.9426793 0.1617881 0.1032567 1.489721 
#> -0.1004635 0.9461439 
#> 
#> Doing end processing
#> Successful trial
#> Total Time: 6.042105 
#> Backend Time: 6.031054
summary(fit)
#> Coefficients:
#>             Estimate Std. Error t value  ci.lower  ci.upper Pr(>|t|)    
#> phi_1_1    -0.293807   0.082574  -3.558 -0.455648 -0.131966   0.0002 ***
#> phi_2_1     0.824396   0.056914  14.485  0.712846  0.935946   <2e-16 ***
#> phi_3_1    -0.454316   0.057647  -7.881 -0.567303 -0.341329   <2e-16 ***
#> phi_1_2    -0.071916   0.068787  -1.045 -0.206735  0.062904   0.1479    
#> phi_2_2    -0.568999   0.048532 -11.724 -0.664119 -0.473878   <2e-16 ***
#> phi_3_2     0.707647   0.048636  14.550  0.612322  0.802972   <2e-16 ***
#> phi_1_3     0.025791   0.062589   0.412 -0.096882  0.148463   0.3402    
#> phi_2_3     0.082668   0.043529   1.899 -0.002647  0.167983   0.0288 *  
#> phi_3_3    -0.687502   0.044092 -15.593 -0.773920 -0.601084   <2e-16 ***
#> sigma_1_1   0.235806   0.022759  10.361  0.191198  0.280413   <2e-16 ***
#> sigma_2_1   0.018668   0.010045   1.858 -0.001020  0.038356   0.0316 *  
#> sigma_3_1  -0.064687   0.010768  -6.007 -0.085793 -0.043582   <2e-16 ***
#> sigma_2_2   0.077088   0.009447   8.160  0.058572  0.095603   <2e-16 ***
#> sigma_3_2   0.008278   0.006655   1.244 -0.004765  0.021321   0.1068    
#> sigma_3_3   0.080654   0.010196   7.910  0.060670  0.100639   <2e-16 ***
#> theta_1_1   0.201033   0.005107  39.363  0.191023  0.211043   <2e-16 ***
#> theta_2_2   0.191640   0.004373  43.823  0.183069  0.200211   <2e-16 ***
#> theta_3_3   0.197449   0.004521  43.677  0.188588  0.206309   <2e-16 ***
#> mu0_1_1    -0.068547   0.142210  -0.482 -0.347273  0.210180   0.3149    
#> mu0_2_1     0.084573   0.176685   0.479 -0.261722  0.430869   0.3161    
#> mu0_3_1     0.115721   0.141485   0.818 -0.161585  0.393026   0.2067    
#> sigma0_1_1  0.942679   0.205369   4.590  0.540164  1.345195   <2e-16 ***
#> sigma0_2_1  0.161788   0.179880   0.899 -0.190769  0.514346   0.1842    
#> sigma0_3_1  0.103257   0.143856   0.718 -0.178696  0.385209   0.2365    
#> sigma0_2_2  1.489721   0.311204   4.787  0.879772  2.099670   <2e-16 ***
#> sigma0_3_2 -0.100464   0.178078  -0.564 -0.449491  0.248563   0.2863    
#> sigma0_3_3  0.946144   0.204219   4.633  0.545882  1.346406   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> -2 log-likelihood value at convergence = 21626.57
#> AIC = 21680.57
#> BIC = 21856.53
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,] -0.2938068 -0.07191554  0.02579061
#> [2,]  0.8243958 -0.56899879  0.08266777
#> [3,] -0.4543161  0.70764697 -0.68750180
vcov_phi_vec
#>               phi_1_1       phi_2_1       phi_3_1       phi_1_2       phi_2_2
#> phi_1_1  0.0068183887  3.328110e-04 -0.0014352362 -0.0049087319 -2.786146e-04
#> phi_2_1  0.0003328110  3.239243e-03 -0.0001186669 -0.0001298602 -2.425089e-03
#> phi_3_1 -0.0014352362 -1.186669e-04  0.0033232239  0.0009753620  1.079346e-04
#> phi_1_2 -0.0049087319 -1.298602e-04  0.0009753620  0.0047316027  1.927033e-04
#> phi_2_2 -0.0002786146 -2.425089e-03  0.0001079346  0.0001927033  2.355321e-03
#> phi_3_2  0.0010343753  3.893230e-05 -0.0024481377 -0.0009945998 -8.261422e-05
#> phi_1_3  0.0033608473  4.732055e-05 -0.0006083507 -0.0034296101 -8.603025e-05
#> phi_2_3  0.0002256333  1.691774e-03 -0.0001006453 -0.0001924514 -1.731951e-03
#> phi_3_3 -0.0007319213 -7.091233e-06  0.0016871746  0.0007482711  3.091309e-05
#>               phi_3_2       phi_1_3       phi_2_3       phi_3_3
#> phi_1_1  1.034375e-03  3.360847e-03  2.256333e-04 -7.319213e-04
#> phi_2_1  3.893230e-05  4.732055e-05  1.691774e-03 -7.091233e-06
#> phi_3_1 -2.448138e-03 -6.083507e-04 -1.006453e-04  1.687175e-03
#> phi_1_2 -9.945998e-04 -3.429610e-03 -1.924514e-04  7.482711e-04
#> phi_2_2 -8.261422e-05 -8.603025e-05 -1.731951e-03  3.091309e-05
#> phi_3_2  2.365471e-03  6.672321e-04  1.034336e-04 -1.733689e-03
#> phi_1_3  6.672321e-04  3.917403e-03  1.804875e-04 -8.681116e-04
#> phi_2_3  1.034336e-04  1.804875e-04  1.894760e-03 -9.774066e-05
#> phi_3_3 -1.733689e-03 -8.681116e-04 -9.774066e-05  1.944073e-03

Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix

sigma
#>             [,1]        [,2]         [,3]
#> [1,]  0.23580555 0.018668186 -0.064687273
#> [2,]  0.01866819 0.077087752  0.008277734
#> [3,] -0.06468727 0.008277734  0.080654489
vcov_sigma_vech
#>               sigma_1_1     sigma_2_1     sigma_3_1     sigma_2_2     sigma_3_2
#> sigma_1_1  5.179936e-04  1.286404e-05 -9.425124e-05  9.869593e-07 -5.327209e-06
#> sigma_2_1  1.286404e-05  1.009005e-04 -3.493273e-06  4.327901e-06 -2.213247e-05
#> sigma_3_1 -9.425124e-05 -3.493273e-06  1.159554e-04 -4.600131e-07  3.190146e-06
#> sigma_2_2  9.869593e-07  4.327901e-06 -4.600131e-07  8.924541e-05 -2.785630e-06
#> sigma_3_2 -5.327209e-06 -2.213247e-05  3.190146e-06 -2.785630e-06  4.428372e-05
#> sigma_3_3  1.933227e-05  1.524740e-06 -4.759391e-05 -1.091407e-07 -2.565877e-06
#>               sigma_3_3
#> sigma_1_1  1.933227e-05
#> sigma_2_1  1.524740e-06
#> sigma_3_1 -4.759391e-05
#> sigma_2_2 -1.091407e-07
#> sigma_3_2 -2.565877e-06
#> sigma_3_3  1.039638e-04

Estimated 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 
#> -0.293806753  0.824395779 -0.454316070 -0.071915544 -0.568998789  0.707646969 
#>      phi_1_3      phi_2_3      phi_3_3    sigma_1_1    sigma_2_1    sigma_3_1 
#>  0.025790607  0.082667770 -0.687501801  0.235805550  0.018668186 -0.064687273 
#>    sigma_2_2    sigma_3_2    sigma_3_3 
#>  0.077087752  0.008277734  0.080654489
vcov_theta
#>                 phi_1_1       phi_2_1       phi_3_1       phi_1_2       phi_2_2
#> phi_1_1    6.818389e-03  3.328110e-04 -1.435236e-03 -4.908732e-03 -2.786146e-04
#> phi_2_1    3.328110e-04  3.239243e-03 -1.186669e-04 -1.298602e-04 -2.425089e-03
#> phi_3_1   -1.435236e-03 -1.186669e-04  3.323224e-03  9.753620e-04  1.079346e-04
#> phi_1_2   -4.908732e-03 -1.298602e-04  9.753620e-04  4.731603e-03  1.927033e-04
#> phi_2_2   -2.786146e-04 -2.425089e-03  1.079346e-04  1.927033e-04  2.355321e-03
#> phi_3_2    1.034375e-03  3.893230e-05 -2.448138e-03 -9.945998e-04 -8.261422e-05
#> phi_1_3    3.360847e-03  4.732055e-05 -6.083507e-04 -3.429610e-03 -8.603025e-05
#> phi_2_3    2.256333e-04  1.691774e-03 -1.006453e-04 -1.924514e-04 -1.731951e-03
#> phi_3_3   -7.319213e-04 -7.091233e-06  1.687175e-03  7.482711e-04  3.091309e-05
#> sigma_1_1 -1.106468e-03 -1.398287e-04  2.517698e-04  6.261763e-04  8.119764e-05
#> sigma_2_1  1.010203e-04 -2.425122e-04 -3.718890e-05 -1.273680e-04  1.291759e-04
#> sigma_3_1  1.334626e-04  2.947566e-05 -2.751090e-04 -3.525638e-05 -1.042500e-05
#> sigma_2_2 -4.057314e-06  1.151342e-04  1.195059e-05  5.574693e-06 -1.403873e-04
#> sigma_3_2 -5.525374e-06  2.791248e-05  5.403340e-05  1.195423e-05  8.511003e-06
#> sigma_3_3 -2.843026e-05 -1.384687e-05  5.809073e-05  7.685408e-06  5.321714e-06
#>                 phi_3_2       phi_1_3       phi_2_3       phi_3_3     sigma_1_1
#> phi_1_1    1.034375e-03  3.360847e-03  2.256333e-04 -7.319213e-04 -1.106468e-03
#> phi_2_1    3.893230e-05  4.732055e-05  1.691774e-03 -7.091233e-06 -1.398287e-04
#> phi_3_1   -2.448138e-03 -6.083507e-04 -1.006453e-04  1.687175e-03  2.517698e-04
#> phi_1_2   -9.945998e-04 -3.429610e-03 -1.924514e-04  7.482711e-04  6.261763e-04
#> phi_2_2   -8.261422e-05 -8.603025e-05 -1.731951e-03  3.091309e-05  8.119764e-05
#> phi_3_2    2.365471e-03  6.672321e-04  1.034336e-04 -1.733689e-03 -1.438765e-04
#> phi_1_3    6.672321e-04  3.917403e-03  1.804875e-04 -8.681116e-04 -2.973809e-04
#> phi_2_3    1.034336e-04  1.804875e-04  1.894760e-03 -9.774066e-05 -4.129221e-05
#> phi_3_3   -1.733689e-03 -8.681116e-04 -9.774066e-05  1.944073e-03  7.127045e-05
#> sigma_1_1 -1.438765e-04 -2.973809e-04 -4.129221e-05  7.127045e-05  5.179936e-04
#> sigma_2_1  3.881781e-05  7.657799e-05 -5.873909e-05 -2.257847e-05  1.286404e-05
#> sigma_3_1  1.436551e-04 -6.840144e-05 -7.334321e-06 -4.241756e-05 -9.425124e-05
#> sigma_2_2 -1.299325e-05 -3.273776e-06  8.554974e-05  8.082729e-06  9.869593e-07
#> sigma_3_2 -6.474756e-05 -4.432167e-06 -4.542482e-05  3.582377e-05 -5.327209e-06
#> sigma_3_3  1.565448e-05  2.001541e-05  5.982094e-06 -1.016672e-04  1.933227e-05
#>               sigma_2_1     sigma_3_1     sigma_2_2     sigma_3_2     sigma_3_3
#> phi_1_1    1.010203e-04  1.334626e-04 -4.057314e-06 -5.525374e-06 -2.843026e-05
#> phi_2_1   -2.425122e-04  2.947566e-05  1.151342e-04  2.791248e-05 -1.384687e-05
#> phi_3_1   -3.718890e-05 -2.751090e-04  1.195059e-05  5.403340e-05  5.809073e-05
#> phi_1_2   -1.273680e-04 -3.525638e-05  5.574693e-06  1.195423e-05  7.685408e-06
#> phi_2_2    1.291759e-04 -1.042500e-05 -1.403873e-04  8.511003e-06  5.321714e-06
#> phi_3_2    3.881781e-05  1.436551e-04 -1.299325e-05 -6.474756e-05  1.565448e-05
#> phi_1_3    7.657799e-05 -6.840144e-05 -3.273776e-06 -4.432167e-06  2.001541e-05
#> phi_2_3   -5.873909e-05 -7.334321e-06  8.554974e-05 -4.542482e-05  5.982094e-06
#> phi_3_3   -2.257847e-05 -4.241756e-05  8.082729e-06  3.582377e-05 -1.016672e-04
#> sigma_1_1  1.286404e-05 -9.425124e-05  9.869593e-07 -5.327209e-06  1.933227e-05
#> sigma_2_1  1.009005e-04 -3.493273e-06  4.327901e-06 -2.213247e-05  1.524740e-06
#> sigma_3_1 -3.493273e-06  1.159554e-04 -4.600131e-07  3.190146e-06 -4.759391e-05
#> sigma_2_2  4.327901e-06 -4.600131e-07  8.924541e-05 -2.785630e-06 -1.091407e-07
#> sigma_3_2 -2.213247e-05  3.190146e-06 -2.785630e-06  4.428372e-05 -2.565877e-06
#> sigma_3_3  1.524740e-06 -4.759391e-05 -1.091407e-07 -2.565877e-06  1.039638e-04

References

Ou, L., Hunter, M. D., & Chow, S.-M. (2019). What’s for dynr: A package for linear and nonlinear dynamic modeling in R. The R Journal, 11(1), 91. https://doi.org/10.32614/rj-2019-012

R Core Team. (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/

Ivan Jacob Agaloos Pesigan

2025-01-18

Data Generation Using the `SimSSMOUFixed` Function from the `simStateSpace` Package

Model Fitting

Prepare the Data

Prepare the Initial Condition

Prepare the Measurement Model

Prepare the Dynamic Model

Prepare the Noise Matrices

Prepare the Model

Fit the Model

Extract Matrices from the Fitted Model to use in cTMed

Estimated Drift Matrix with Corresponding Sampling Covariance Matrix

Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix

Estimated Drift Matrix and Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix

References

Fit the Continuous-Time Vector Autoregressive Model Using the dynr Package

Ivan Jacob Agaloos Pesigan

2025-01-18

Data Generation Using the SimSSMOUFixed Function from the simStateSpace Package

Model Fitting

Prepare the Data

Prepare the Initial Condition

Prepare the Measurement Model

Prepare the Dynamic Model

Prepare the Noise Matrices

Prepare the Model

Fit the Model

Extract Matrices from the Fitted Model to use in cTMed

Estimated Drift Matrix with Corresponding Sampling Covariance Matrix

Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix

Estimated Drift Matrix and Process Noise Covariance Matrix with Corresponding Sampling Covariance Matrix

References

Data Generation Using the `SimSSMOUFixed` Function from the `simStateSpace` Package