Parametric Bootstrap for the Ornstein–Uhlenbeck Model using a State Space Model Parameterization (Fixed Parameters)
Source:R/bootStateSpace-pb-ssm-ou-fixed.R
PBSSMOUFixed.RdThis function simulates data from
a Ornstein–Uhlenbeck (OU) model
using a state-space model parameterization
and fits the model using the dynr package.
The process is repeated R times.
It assumes that the parameters remain constant
across individuals and over time.
At the moment, the function only supports
type = 0.
Usage
PBSSMOUFixed(
R,
path,
prefix,
n,
time,
delta_t = 0.1,
mu0,
sigma0_l,
mu,
phi,
sigma_l,
nu,
lambda,
theta_l,
type = 0,
x = NULL,
gamma = NULL,
kappa = NULL,
mu0_fixed = FALSE,
sigma0_fixed = FALSE,
alpha_level = 0.05,
optimization_flag = TRUE,
hessian_flag = FALSE,
verbose = FALSE,
weight_flag = FALSE,
debug_flag = FALSE,
perturb_flag = FALSE,
xtol_rel = 1e-07,
stopval = -9999,
ftol_rel = -1,
ftol_abs = -1,
maxeval = as.integer(-1),
maxtime = -1,
ncores = NULL,
seed = NULL,
clean = TRUE
)Arguments
- R
Positive integer. Number of bootstrap samples.
- path
Path to a directory to store bootstrap samples and estimates.
- prefix
Character string. Prefix used for the file names for the bootstrap samples and estimates.
- n
Positive integer. Number of individuals.
- time
Positive integer. Number of time points.
- delta_t
Numeric. Time interval (\(\Delta_t\)).
- mu0
Numeric vector. Mean of initial latent variable values (\(\boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0}\)).
- sigma0_l
Numeric matrix. Cholesky factorization (
t(chol(sigma0))) of the covariance matrix of initial latent variable values (\(\boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0}\)).- mu
Numeric vector. The long-term mean or equilibrium level (\(\boldsymbol{\mu}\)).
- phi
Numeric matrix. The drift matrix which represents the rate of change of the solution in the absence of any random fluctuations (\(\boldsymbol{\Phi}\)). It also represents the rate of mean reversion, determining how quickly the variable returns to its mean.
- sigma_l
Numeric matrix. Cholesky factorization (
t(chol(sigma))) of the covariance matrix of volatility or randomness in the process (\(\boldsymbol{\Sigma}\)).- nu
Numeric vector. Vector of intercept values for the measurement model (\(\boldsymbol{\nu}\)).
- lambda
Numeric matrix. Factor loading matrix linking the latent variables to the observed variables (\(\boldsymbol{\Lambda}\)).
- theta_l
Numeric matrix. Cholesky factorization (
t(chol(theta))) of the covariance matrix of the measurement error (\(\boldsymbol{\Theta}\)).- type
Integer. State space model type. See Details for more information.
- x
List. Each element of the list is a matrix of covariates for each individual
iinn. The number of columns in each matrix should be equal totime.- gamma
Numeric matrix. Matrix linking the covariates to the latent variables at current time point (\(\boldsymbol{\Gamma}\)).
- kappa
Numeric matrix. Matrix linking the covariates to the observed variables at current time point (\(\boldsymbol{\kappa}\)).
- mu0_fixed
Logical. If
mu0_fixed = TRUE, fix the initial mean vector tomu0. Ifmu0_fixed = FALSE,mu0is estimated.- sigma0_fixed
Logical. If
sigma0_fixed = TRUE, fix the initial covariance matrix totcrossprod(sigma0_l). Ifsigma0_fixed = FALSE,sigma0is estimated.- alpha_level
Numeric vector. Significance level \(\alpha\).
- optimization_flag
a flag (TRUE/FALSE) indicating whether optimization is to be done.
- hessian_flag
a flag (TRUE/FALSE) indicating whether the Hessian matrix is to be calculated.
- verbose
a flag (TRUE/FALSE) indicating whether more detailed intermediate output during the estimation process should be printed
- weight_flag
a flag (TRUE/FALSE) indicating whether the negative log likelihood function should be weighted by the length of the time series for each individual
- debug_flag
a flag (TRUE/FALSE) indicating whether users want additional dynr output that can be used for diagnostic purposes
- perturb_flag
a flag (TRUE/FLASE) indicating whether to perturb the latent states during estimation. Only useful for ensemble forecasting.
- xtol_rel
Stopping criteria option for parameter optimization. See
dynr::dynr.model()for more details.- stopval
Stopping criteria option for parameter optimization. See
dynr::dynr.model()for more details.- ftol_rel
Stopping criteria option for parameter optimization. See
dynr::dynr.model()for more details.- ftol_abs
Stopping criteria option for parameter optimization. See
dynr::dynr.model()for more details.- maxeval
Stopping criteria option for parameter optimization. See
dynr::dynr.model()for more details.- maxtime
Stopping criteria option for parameter optimization. See
dynr::dynr.model()for more details.- ncores
Positive integer. Number of cores to use. If
ncores = NULL, use a single core. Consider using multiple cores when number of bootstrap samplesRis a large value.- seed
Random seed.
- clean
Logical. If
clean = TRUE, delete intermediate files generated by the function.
Value
Returns an object
of class bootstatespace which is a list with the following elements:
- call
Function call.
- args
Function arguments.
- thetahatstar
Sampling distribution of \(\boldsymbol{\hat{\theta}}\).
- vcov
Sampling variance-covariance matrix of \(\boldsymbol{\hat{\theta}}\).
- est
Vector of estimated \(\boldsymbol{\hat{\theta}}\).
- fun
Function used ("PBSSMOUFixed").
- method
Bootstrap method used ("parametric").
Details
Type 0
The measurement model is given by $$ \mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right) $$ where \(\mathbf{y}_{i, t}\), \(\boldsymbol{\eta}_{i, t}\), and \(\boldsymbol{\varepsilon}_{i, t}\) are random variables and \(\boldsymbol{\nu}\), \(\boldsymbol{\Lambda}\), and \(\boldsymbol{\Theta}\) are model parameters. \(\mathbf{y}_{i, t}\) represents a vector of observed random variables, \(\boldsymbol{\eta}_{i, t}\) a vector of latent random variables, and \(\boldsymbol{\varepsilon}_{i, t}\) a vector of random measurement errors, at time \(t\) and individual \(i\). \(\boldsymbol{\nu}\) denotes a vector of intercepts, \(\boldsymbol{\Lambda}\) a matrix of factor loadings, and \(\boldsymbol{\Theta}\) the covariance matrix of \(\boldsymbol{\varepsilon}\).
An alternative representation of the measurement error is given by $$ \boldsymbol{\varepsilon}_{i, t} = \boldsymbol{\Theta}^{\frac{1}{2}} \mathbf{z}_{i, t}, \quad \mathrm{with} \quad \mathbf{z}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \mathbf{I} \right) $$ where \(\mathbf{z}_{i, t}\) is a vector of independent standard normal random variables and \( \left( \boldsymbol{\Theta}^{\frac{1}{2}} \right) \left( \boldsymbol{\Theta}^{\frac{1}{2}} \right)^{\prime} = \boldsymbol{\Theta} . \)
The dynamic structure is given by $$ \mathrm{d} \boldsymbol{\eta}_{i, t} = \boldsymbol{\Phi} \left( \boldsymbol{\eta}_{i, t} - \boldsymbol{\mu} \right) \mathrm{d}t + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t} $$ where \(\boldsymbol{\mu}\) is the long-term mean or equilibrium level, \(\boldsymbol{\Phi}\) is the rate of mean reversion, determining how quickly the variable returns to its mean, \(\boldsymbol{\Sigma}\) is the matrix of volatility or randomness in the process, and \(\mathrm{d}\boldsymbol{W}\) is a Wiener process or Brownian motion, which represents random fluctuations.
Type 1
The measurement model is given by $$ \mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right) . $$
The dynamic structure is given by $$ \mathrm{d} \boldsymbol{\eta}_{i, t} = \boldsymbol{\Phi} \left( \boldsymbol{\eta}_{i, t} - \boldsymbol{\mu} \right) \mathrm{d}t + \boldsymbol{\Gamma} \mathbf{x}_{i, t} + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t} $$ where \(\mathbf{x}_{i, t}\) represents a vector of covariates at time \(t\) and individual \(i\), and \(\boldsymbol{\Gamma}\) the coefficient matrix linking the covariates to the latent variables.
Type 2
The measurement model is given by $$ \mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\kappa} \mathbf{x}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right) $$ where \(\boldsymbol{\kappa}\) represents the coefficient matrix linking the covariates to the observed variables.
The dynamic structure is given by $$ \mathrm{d} \boldsymbol{\eta}_{i, t} = \boldsymbol{\Phi} \left( \boldsymbol{\eta}_{i, t} - \boldsymbol{\mu} \right) \mathrm{d}t + \boldsymbol{\Gamma} \mathbf{x}_{i, t} + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t} . $$
The OU model as a linear stochastic differential equation model
The OU model is a first-order linear stochastic differential equation model in the form of
$$ \mathrm{d} \boldsymbol{\eta}_{i, t} = \left( \boldsymbol{\iota} + \boldsymbol{\Phi} \boldsymbol{\eta}_{i, t} \right) \mathrm{d}t + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t} $$ where \(\boldsymbol{\mu} = - \boldsymbol{\Phi}^{-1} \boldsymbol{\iota}\) and, equivalently \(\boldsymbol{\iota} = - \boldsymbol{\Phi} \boldsymbol{\mu}\).
References
Chow, S.-M., Ho, M. R., Hamaker, E. L., & Dolan, C. V. (2010). Equivalence and differences between structural equation modeling and state-space modeling techniques. Structural Equation Modeling: A Multidisciplinary Journal, 17(2), 303–332. doi:10.1080/10705511003661553
See also
Other Bootstrap for State Space Models Functions:
PBSSMFixed(),
PBSSMLinSDEFixed(),
PBSSMVARFixed()
Examples
# \donttest{
# prepare parameters
## number of individuals
n <- 5
## time points
time <- 50
delta_t <- 0.10
## dynamic structure
p <- 2
mu0 <- c(-3.0, 1.5)
sigma0 <- 0.001 * diag(p)
sigma0_l <- t(chol(sigma0))
mu <- c(5.76, 5.18)
phi <- matrix(
data = c(
-0.10,
0.05,
0.05,
-0.10
),
nrow = p
)
sigma <- matrix(
data = c(
2.79,
0.06,
0.06,
3.27
),
nrow = p
)
sigma_l <- t(chol(sigma))
## measurement model
k <- 2
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.001 * diag(k)
theta_l <- t(chol(theta))
path <- tempdir()
pb <- PBSSMOUFixed(
R = 10L, # use at least 1000 in actual research
path = path,
prefix = "ou",
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,
ncores = 1, # consider using multiple cores
seed = 42
)
print(pb)
#> Call:
#> PBSSMOUFixed(R = 10L, path = path, prefix = "ou", 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, ncores = 1, seed = 42)
#>
#> Parametric bootstrap confidence intervals.
#> type = "pc"
#> est se R 2.5% 97.5%
#> phi_1_1 -0.100 0.1121 10 -0.4299 -0.0979
#> phi_2_1 0.050 0.2090 10 -0.1890 0.3585
#> phi_1_2 0.050 0.2549 10 -0.1524 0.5515
#> phi_2_2 -0.100 0.2801 10 -0.7518 0.0733
#> mu_1_1 5.760 17.5412 10 -0.1145 46.4755
#> mu_2_1 5.180 16.5698 10 -18.3931 35.8675
#> sigma_1_1 2.790 0.2068 10 2.4150 2.9959
#> sigma_2_1 0.060 0.1520 10 -0.2386 0.2020
#> sigma_2_2 3.270 0.1900 10 2.9998 3.4951
#> theta_1_1 0.001 0.0008 10 0.0000 0.0022
#> theta_2_2 0.001 0.0006 10 0.0000 0.0013
#> mu0_1_1 -3.000 0.0110 10 -3.0157 -2.9836
#> mu0_2_1 1.500 0.0250 10 1.4730 1.5375
#> sigma0_1_1 0.001 0.0009 10 0.0000 0.0021
#> sigma0_2_2 0.001 0.0020 10 0.0000 0.0047
summary(pb)
#> est se R 2.5% 97.5%
#> phi_1_1 -0.100 0.1121 10 -0.4299 -0.0979
#> phi_2_1 0.050 0.2090 10 -0.1890 0.3585
#> phi_1_2 0.050 0.2549 10 -0.1524 0.5515
#> phi_2_2 -0.100 0.2801 10 -0.7518 0.0733
#> mu_1_1 5.760 17.5412 10 -0.1145 46.4755
#> mu_2_1 5.180 16.5698 10 -18.3931 35.8675
#> sigma_1_1 2.790 0.2068 10 2.4150 2.9959
#> sigma_2_1 0.060 0.1520 10 -0.2386 0.2020
#> sigma_2_2 3.270 0.1900 10 2.9998 3.4951
#> theta_1_1 0.001 0.0008 10 0.0000 0.0022
#> theta_2_2 0.001 0.0006 10 0.0000 0.0013
#> mu0_1_1 -3.000 0.0110 10 -3.0157 -2.9836
#> mu0_2_1 1.500 0.0250 10 1.4730 1.5375
#> sigma0_1_1 0.001 0.0009 10 0.0000 0.0021
#> sigma0_2_2 0.001 0.0020 10 0.0000 0.0047
confint(pb)
#> 2.5 % 97.5 %
#> phi_1_1 -4.299497e-01 -0.097851049
#> phi_2_1 -1.889892e-01 0.358461692
#> phi_1_2 -1.524410e-01 0.551539224
#> phi_2_2 -7.518428e-01 0.073305346
#> mu_1_1 -1.145149e-01 46.475539280
#> mu_2_1 -1.839307e+01 35.867508640
#> sigma_1_1 2.415048e+00 2.995855339
#> sigma_2_1 -2.386130e-01 0.201978103
#> sigma_2_2 2.999791e+00 3.495110318
#> theta_1_1 2.047164e-20 0.002190152
#> theta_2_2 3.993967e-15 0.001347848
#> mu0_1_1 -3.015728e+00 -2.983648643
#> mu0_2_1 1.473001e+00 1.537463816
#> sigma0_1_1 3.849675e-14 0.002105113
#> sigma0_2_2 1.389864e-16 0.004749876
vcov(pb)
#> phi_1_1 phi_2_1 phi_1_2 phi_2_2
#> phi_1_1 1.257624e-02 -8.568968e-03 -2.080176e-02 -2.195136e-03
#> phi_2_1 -8.568968e-03 4.366183e-02 3.067842e-02 -3.749987e-02
#> phi_1_2 -2.080176e-02 3.067842e-02 6.497826e-02 -1.138220e-02
#> phi_2_2 -2.195136e-03 -3.749987e-02 -1.138220e-02 7.847953e-02
#> mu_1_1 -8.324203e-02 -4.898370e-02 7.909102e-01 -3.925981e-02
#> mu_2_1 -1.035602e+00 1.352011e+00 2.767269e+00 1.016360e+00
#> sigma_1_1 8.728227e-03 1.424105e-02 -2.119593e-02 -4.376327e-02
#> sigma_2_1 9.301139e-03 -5.212394e-03 -1.044109e-02 -2.135751e-02
#> sigma_2_2 -5.057923e-03 1.686322e-02 2.169282e-02 -5.087701e-03
#> theta_1_1 -1.385641e-05 4.777764e-05 1.993931e-05 1.478306e-05
#> theta_2_2 1.233515e-05 -2.789183e-05 -1.025475e-05 6.049828e-05
#> mu0_1_1 6.252507e-04 -1.024925e-04 -1.331778e-03 -1.346019e-03
#> mu0_2_1 5.561032e-04 3.732341e-04 -1.662070e-03 1.751828e-03
#> sigma0_1_1 3.145341e-05 -1.190031e-05 -6.703418e-05 -1.276839e-04
#> sigma0_2_2 -3.144040e-05 1.523709e-04 -2.131209e-06 -2.277832e-04
#> mu_1_1 mu_2_1 sigma_1_1 sigma_2_1
#> phi_1_1 -8.324203e-02 -1.035601618 8.728227e-03 9.301139e-03
#> phi_2_1 -4.898370e-02 1.352010523 1.424105e-02 -5.212394e-03
#> phi_1_2 7.909102e-01 2.767268977 -2.119593e-02 -1.044109e-02
#> phi_2_2 -3.925981e-02 1.016360401 -4.376327e-02 -2.135751e-02
#> mu_1_1 3.076943e+02 59.666953508 4.050331e-01 5.925189e-01
#> mu_2_1 5.966695e+01 274.559371058 -1.731122e+00 -1.703492e+00
#> sigma_1_1 4.050331e-01 -1.731122284 4.275492e-02 1.547843e-02
#> sigma_2_1 5.925189e-01 -1.703491566 1.547843e-02 2.311192e-02
#> sigma_2_2 9.443886e-01 0.710359847 4.128853e-03 -6.081077e-03
#> theta_1_1 -3.258928e-03 0.002960687 -3.815606e-06 -7.913358e-05
#> theta_2_2 9.241816e-03 0.002693798 -9.676339e-06 -1.764450e-06
#> mu0_1_1 -7.213094e-02 -0.122234807 1.434826e-03 9.251777e-04
#> mu0_2_1 1.384580e-01 -0.055059759 7.537726e-04 -5.032274e-04
#> sigma0_1_1 -9.229027e-05 -0.006786605 1.031865e-04 9.906573e-05
#> sigma0_2_2 -1.823011e-02 -0.004003851 1.470355e-04 3.178681e-05
#> sigma_2_2 theta_1_1 theta_2_2 mu0_1_1
#> phi_1_1 -5.057923e-03 -1.385641e-05 1.233515e-05 6.252507e-04
#> phi_2_1 1.686322e-02 4.777764e-05 -2.789183e-05 -1.024925e-04
#> phi_1_2 2.169282e-02 1.993931e-05 -1.025475e-05 -1.331778e-03
#> phi_2_2 -5.087701e-03 1.478306e-05 6.049828e-05 -1.346019e-03
#> mu_1_1 9.443886e-01 -3.258928e-03 9.241816e-03 -7.213094e-02
#> mu_2_1 7.103598e-01 2.960687e-03 2.693798e-03 -1.222348e-01
#> sigma_1_1 4.128853e-03 -3.815606e-06 -9.676339e-06 1.434826e-03
#> sigma_2_1 -6.081077e-03 -7.913358e-05 -1.764450e-06 9.251777e-04
#> sigma_2_2 3.610609e-02 4.817568e-05 2.643107e-05 1.449178e-04
#> theta_1_1 4.817568e-05 7.166276e-07 -3.861286e-08 -2.755199e-06
#> theta_2_2 2.643107e-05 -3.861286e-08 3.721091e-07 -2.953469e-06
#> mu0_1_1 1.449178e-04 -2.755199e-06 -2.953469e-06 1.210159e-04
#> mu0_2_1 1.024676e-03 7.565769e-06 7.652731e-06 -6.654761e-05
#> sigma0_1_1 -2.864729e-05 -5.908546e-07 -1.256564e-07 7.670170e-06
#> sigma0_2_2 1.582927e-05 -4.663689e-07 -8.219677e-07 1.484417e-05
#> mu0_2_1 sigma0_1_1 sigma0_2_2
#> phi_1_1 5.561032e-04 3.145341e-05 -3.144040e-05
#> phi_2_1 3.732341e-04 -1.190031e-05 1.523709e-04
#> phi_1_2 -1.662070e-03 -6.703418e-05 -2.131209e-06
#> phi_2_2 1.751828e-03 -1.276839e-04 -2.277832e-04
#> mu_1_1 1.384580e-01 -9.229027e-05 -1.823011e-02
#> mu_2_1 -5.505976e-02 -6.786605e-03 -4.003851e-03
#> sigma_1_1 7.537726e-04 1.031865e-04 1.470355e-04
#> sigma_2_1 -5.032274e-04 9.906573e-05 3.178681e-05
#> sigma_2_2 1.024676e-03 -2.864729e-05 1.582927e-05
#> theta_1_1 7.565769e-06 -5.908546e-07 -4.663689e-07
#> theta_2_2 7.652731e-06 -1.256564e-07 -8.219677e-07
#> mu0_1_1 -6.654761e-05 7.670170e-06 1.484417e-05
#> mu0_2_1 6.231900e-04 -1.082977e-05 -2.047552e-05
#> sigma0_1_1 -1.082977e-05 8.350329e-07 1.134016e-06
#> sigma0_2_2 -2.047552e-05 1.134016e-06 4.043062e-06
coef(pb)
#> phi_1_1 phi_2_1 phi_1_2 phi_2_2 mu_1_1 mu_2_1 sigma_1_1
#> -0.100 0.050 0.050 -0.100 5.760 5.180 2.790
#> sigma_2_1 sigma_2_2 theta_1_1 theta_2_2 mu0_1_1 mu0_2_1 sigma0_1_1
#> 0.060 3.270 0.001 0.001 -3.000 1.500 0.001
#> sigma0_2_2
#> 0.001
print(pb, type = "bc") # bias-corrected
#> Call:
#> PBSSMOUFixed(R = 10L, path = path, prefix = "ou", 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, ncores = 1, seed = 42)
#>
#> Parametric bootstrap confidence intervals.
#> type = "bc"
#> est se R 2.5% 97.5%
#> phi_1_1 -0.100 0.1121 10 -0.1763 -0.0972
#> phi_2_1 0.050 0.2090 10 -0.1890 0.3585
#> phi_1_2 0.050 0.2549 10 -0.1532 0.4014
#> phi_2_2 -0.100 0.2801 10 -0.3337 0.0817
#> mu_1_1 5.760 17.5412 10 -0.3628 44.8531
#> mu_2_1 5.180 16.5698 10 -18.3931 35.8675
#> sigma_1_1 2.790 0.2068 10 2.4150 2.9959
#> sigma_2_1 0.060 0.1520 10 -0.1439 0.2382
#> sigma_2_2 3.270 0.1900 10 2.9948 3.4753
#> theta_1_1 0.001 0.0008 10 0.0000 0.0021
#> theta_2_2 0.001 0.0006 10 0.0000 0.0014
#> mu0_1_1 -3.000 0.0110 10 -3.0158 -2.9862
#> mu0_2_1 1.500 0.0250 10 1.4723 1.5333
#> sigma0_1_1 0.001 0.0009 10 0.0000 0.0021
#> sigma0_2_2 0.001 0.0020 10 0.0000 0.0048
summary(pb, type = "bc")
#> est se R 2.5% 97.5%
#> phi_1_1 -0.100 0.1121 10 -0.1763 -0.0972
#> phi_2_1 0.050 0.2090 10 -0.1890 0.3585
#> phi_1_2 0.050 0.2549 10 -0.1532 0.4014
#> phi_2_2 -0.100 0.2801 10 -0.3337 0.0817
#> mu_1_1 5.760 17.5412 10 -0.3628 44.8531
#> mu_2_1 5.180 16.5698 10 -18.3931 35.8675
#> sigma_1_1 2.790 0.2068 10 2.4150 2.9959
#> sigma_2_1 0.060 0.1520 10 -0.1439 0.2382
#> sigma_2_2 3.270 0.1900 10 2.9948 3.4753
#> theta_1_1 0.001 0.0008 10 0.0000 0.0021
#> theta_2_2 0.001 0.0006 10 0.0000 0.0014
#> mu0_1_1 -3.000 0.0110 10 -3.0158 -2.9862
#> mu0_2_1 1.500 0.0250 10 1.4723 1.5333
#> sigma0_1_1 0.001 0.0009 10 0.0000 0.0021
#> sigma0_2_2 0.001 0.0020 10 0.0000 0.0048
confint(pb, type = "bc")
#> 2.5 % 97.5 %
#> phi_1_1 -1.763001e-01 -0.097215403
#> phi_2_1 -1.889892e-01 0.358461692
#> phi_1_2 -1.532111e-01 0.401366134
#> phi_2_2 -3.336856e-01 0.081651472
#> mu_1_1 -3.627641e-01 44.853142707
#> mu_2_1 -1.839307e+01 35.867508640
#> sigma_1_1 2.415048e+00 2.995855339
#> sigma_2_1 -1.438877e-01 0.238220036
#> sigma_2_2 2.994832e+00 3.475302415
#> theta_1_1 1.799215e-20 0.002077927
#> theta_2_2 3.579658e-12 0.001354334
#> mu0_1_1 -3.015769e+00 -2.986220287
#> mu0_2_1 1.472278e+00 1.533275935
#> sigma0_1_1 6.300405e-11 0.002145621
#> sigma0_2_2 8.786878e-13 0.004833874
# }