The State Space Model

Ivan Jacob Agaloos Pesigan

2025-10-19

Model

The measurement model is given by $$\begin{equation} \mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right) \end{equation}$$ 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 $$\begin{equation} \boldsymbol{\varepsilon}_{i, t} = \boldsymbol{\Theta}^{\frac{1}{2}} \mathbf{z}_{i, t}, \quad \mathrm{with} \quad \mathbf{z}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \mathbf{I} \right) \end{equation}$$ 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 $$\begin{equation} \boldsymbol{\eta}_{i, t} = \boldsymbol{\alpha} + \boldsymbol{\beta} \boldsymbol{\eta}_{i, t - 1} + \boldsymbol{\zeta}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\zeta}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Psi} \right) \end{equation}$$ where $\boldsymbol{\eta}_{i, t}$, $\boldsymbol{\eta}_{i, t - 1}$, and $\boldsymbol{\zeta}_{i, t}$ are random variables, and $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$, and $\boldsymbol{\Psi}$ are model parameters. Here, $\boldsymbol{\eta}_{i, t}$ is a vector of latent variables at time $t$ and individual $i$, $\boldsymbol{\eta}_{i, t - 1}$ represents a vector of latent variables at time $t - 1$ and individual $i$, and $\boldsymbol{\zeta}_{i, t}$ represents a vector of dynamic noise at time $t$ and individual $i$. $\boldsymbol{\alpha}$ denotes a vector of intercepts, $\boldsymbol{\beta}$ a matrix of autoregression and cross regression coefficients, and $\boldsymbol{\Psi}$ the covariance matrix of $\boldsymbol{\zeta}_{i, t}$.

An alternative representation of the dynamic noise is given by $$\begin{equation} \boldsymbol{\zeta}_{i, t} = \boldsymbol{\Psi}^{\frac{1}{2}} \mathbf{z}_{i, t}, \quad \mathrm{with} \quad \mathbf{z}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \mathbf{I} \right) \end{equation}$$ where $\left( \boldsymbol{\Psi}^{\frac{1}{2}} \right) \left( \boldsymbol{\Psi}^{\frac{1}{2}} \right)^{\prime} = \boldsymbol{\Psi}$ .

Data Generation

Notation

Let $t = 500$ be the number of time points and $n = 100$ be the number of individuals.

Let the measurement model intecept vector $\boldsymbol{\nu}$ be given by

$$\begin{equation} \boldsymbol{\nu} = \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ \end{array} \right) . \end{equation}$$

Let the factor loadings matrix $\boldsymbol{\Lambda}$ be given by

$$\begin{equation} \boldsymbol{\Lambda} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) . \end{equation}$$

Let the measurement error covariance matrix $\boldsymbol{\Theta}$ be given by

$$\begin{equation} \boldsymbol{\Theta} = \left( \begin{array}{ccc} 0.2 & 0 & 0 \\ 0 & 0.2 & 0 \\ 0 & 0 & 0.2 \\ \end{array} \right) . \end{equation}$$

Let the initial condition $\boldsymbol{\eta}_{0}$ be given by

$$\begin{equation} \boldsymbol{\eta}_{0} \sim \mathcal{N} \left( \boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0}, \boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0} \right) \end{equation}$$

$$\begin{equation} \boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ \end{array} \right) \end{equation}$$

$$\begin{equation} \boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{ccc} 0.1960784 & 0.1183232 & 0.0298539 \\ 0.1183232 & 0.3437711 & 0.1381855 \\ 0.0298539 & 0.1381855 & 0.2663828 \\ \end{array} \right) . \end{equation}$$

Let the constant vector $\boldsymbol{\alpha}$ be given by

$$\begin{equation} \boldsymbol{\alpha} = \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ \end{array} \right) . \end{equation}$$

Let the transition matrix $\boldsymbol{\beta}$ be given by

$$\begin{equation} \boldsymbol{\beta} = \left( \begin{array}{ccc} 0.7 & 0 & 0 \\ 0.5 & 0.6 & 0 \\ -0.1 & 0.4 & 0.5 \\ \end{array} \right) . \end{equation}$$

Let the dynamic process noise $\boldsymbol{\Psi}$ be given by

$$\begin{equation} \boldsymbol{\Psi} = \left( \begin{array}{ccc} 0.1 & 0 & 0 \\ 0 & 0.1 & 0 \\ 0 & 0 & 0.1 \\ \end{array} \right) . \end{equation}$$

R Function Arguments

n
#> [1] 100
time
#> [1] 500
mu0
#> [1] 0 0 0
sigma0
#>            [,1]      [,2]       [,3]
#> [1,] 0.19607843 0.1183232 0.02985385
#> [2,] 0.11832319 0.3437711 0.13818551
#> [3,] 0.02985385 0.1381855 0.26638284
sigma0_l # sigma0_l <- t(chol(sigma0))
#>            [,1]      [,2]     [,3]
#> [1,] 0.44280744 0.0000000 0.000000
#> [2,] 0.26721139 0.5218900 0.000000
#> [3,] 0.06741949 0.2302597 0.456966
alpha
#> [1] 0 0 0
beta
#>      [,1] [,2] [,3]
#> [1,]  0.7  0.0  0.0
#> [2,]  0.5  0.6  0.0
#> [3,] -0.1  0.4  0.5
psi
#>      [,1] [,2] [,3]
#> [1,]  0.1  0.0  0.0
#> [2,]  0.0  0.1  0.0
#> [3,]  0.0  0.0  0.1
psi_l # psi_l <- t(chol(psi))
#>           [,1]      [,2]      [,3]
#> [1,] 0.3162278 0.0000000 0.0000000
#> [2,] 0.0000000 0.3162278 0.0000000
#> [3,] 0.0000000 0.0000000 0.3162278
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

Visualizing the Dynamics Without Process Noise (n = 5 with Different Initial Condition)

Using the SimSSMFixed Function from the simStateSpace Package to Simulate Data

library(simStateSpace)
sim <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)
data <- as.data.frame(sim)
head(data)
#>   id time          y1         y2         y3
#> 1  1    0 -0.66908807 0.16178434  0.2955693
#> 2  1    1 -0.23021811 0.22873073 -0.2540210
#> 3  1    2  0.93689389 0.09496693 -0.9267506
#> 4  1    3  0.04445794 0.67521289 -0.1792168
#> 5  1    4  0.15413707 0.82591676  0.8976536
#> 6  1    5 -0.09943698 0.67154173  0.2853507
summary(data)
#>        id              time             y1                  y2           
#>  Min.   :  1.00   Min.   :  0.0   Min.   :-2.840534   Min.   :-2.671337  
#>  1st Qu.: 25.75   1st Qu.:124.8   1st Qu.:-0.421845   1st Qu.:-0.498760  
#>  Median : 50.50   Median :249.5   Median :-0.001813   Median : 0.004066  
#>  Mean   : 50.50   Mean   :249.5   Mean   :-0.000370   Mean   : 0.001287  
#>  3rd Qu.: 75.25   3rd Qu.:374.2   3rd Qu.: 0.421838   3rd Qu.: 0.498712  
#>  Max.   :100.00   Max.   :499.0   Max.   : 2.602905   Max.   : 3.274871  
#>        y3           
#>  Min.   :-2.675596  
#>  1st Qu.:-0.452601  
#>  Median : 0.001744  
#>  Mean   : 0.003782  
#>  3rd Qu.: 0.462944  
#>  Max.   : 3.016221
plot(sim)

Model Fitting

Prepare Data

dynr_data <- dynr::dynr.data(
  data = data,
  id = "id",
  time = "time",
  observed = c("y1", "y2", "y3")
)

Prepare Initial Condition

dynr_initial <- dynr::prep.initial(
  values.inistate = mu0,
  params.inistate = c("mu0_1_1", "mu0_2_1", "mu0_3_1"),
  values.inicov = sigma0,
  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 Measurement Model

dynr_measurement <- dynr::prep.measurement(
  values.load = diag(3),
  params.load = matrix(data = "fixed", nrow = 3, ncol = 3),
  state.names = c("eta_1", "eta_2", "eta_3"),
  obs.names = c("y1", "y2", "y3")
)

Prepare Dynamic Process

dynr_dynamics <- dynr::prep.formulaDynamics(
  formula = list(
    eta_1 ~ alpha_1_1 * 1 + beta_1_1 * eta_1 + beta_1_2 * eta_2 + beta_1_3 * eta_3,
    eta_2 ~ alpha_2_1 * 1 + beta_2_1 * eta_1 + beta_2_2 * eta_2 + beta_2_3 * eta_3,
    eta_3 ~ alpha_3_1 * 1 + beta_3_1 * eta_1 + beta_3_2 * eta_2 + beta_3_3 * eta_3
  ),
  startval = c(
    alpha_1_1 = alpha[1], alpha_2_1 = alpha[2], alpha_3_1 = alpha[3],
    beta_1_1 = beta[1, 1], beta_1_2 = beta[1, 2], beta_1_3 = beta[1, 3],
    beta_2_1 = beta[2, 1], beta_2_2 = beta[2, 2], beta_2_3 = beta[2, 3],
    beta_3_1 = beta[3, 1], beta_3_2 = beta[3, 2], beta_3_3 = beta[3, 3]
  ),
  isContinuousTime = FALSE
)

Prepare Process Noise

dynr_noise <- dynr::prep.noise(
  values.latent = psi,
  params.latent = matrix(
    data = c(
      "psi_1_1", "psi_2_1", "psi_3_1",
      "psi_2_1", "psi_2_2", "psi_3_2",
      "psi_3_1", "psi_3_2", "psi_3_3"
    ),
    nrow = 3
  ),
  values.observed = theta,
  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

model <- dynr::dynr.model(
  data = dynr_data,
  initial = dynr_initial,
  measurement = dynr_measurement,
  dynamics = dynr_dynamics,
  noise = dynr_noise,
  outfile = "ssm.c"
)

Fit the Model

results <- dynr::dynr.cook(
  model,
  debug_flag = TRUE,
  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:  -7.482091e-05 0.0009771779 0.001185204 0.6978014 
#> 0.006914791 -0.007611952 0.5060786 0.5923553 0.003602732 -0.08436625 0.3818816 
#> 0.5098833 -2.303145 -0.006457767 -0.006052534 -2.261517 0.008481604 -2.381506 
#> -1.605916 -1.6393 -1.571531 -0.01046285 -0.1363822 0.07357103 -1.927046 
#> 0.4206043 0.1606324 -1.003452 0.3994318 -1.785151 
#> 
#> Transformed fitted parameters:  -7.482091e-05 0.0009771779 0.001185204 
#> 0.6978014 0.006914791 -0.007611952 0.5060786 0.5923553 0.003602732 -0.08436625 
#> 0.3818816 0.5098833 0.09994399 -0.0006454149 -0.0006049144 0.1041965 
#> 0.0008876247 0.0924225 0.2007056 0.1941158 0.2077269 -0.01046285 -0.1363822 
#> 0.07357103 0.1455775 0.06123054 0.02338447 0.3923654 0.1562719 0.2300193 
#> 
#> Doing end processing
#> Successful trial
#> Total Time: 44.89606 
#> Backend Time: 44.8868

Summary

summary(results)
#> Coefficients:
#>              Estimate Std. Error t value   ci.lower   ci.upper Pr(>|t|)    
#> alpha_1_1  -7.482e-05  1.542e-03  -0.049 -3.096e-03  2.947e-03   0.4806    
#> alpha_2_1   9.772e-04  1.942e-03   0.503 -2.828e-03  4.783e-03   0.3074    
#> alpha_3_1   1.185e-03  1.858e-03   0.638 -2.457e-03  4.828e-03   0.2618    
#> beta_1_1    6.978e-01  1.378e-02  50.644  6.708e-01  7.248e-01   <2e-16 ***
#> beta_1_2    6.915e-03  8.563e-03   0.808 -9.868e-03  2.370e-02   0.2097    
#> beta_1_3   -7.612e-03  6.424e-03  -1.185 -2.020e-02  4.979e-03   0.1180    
#> beta_2_1    5.061e-01  1.473e-02  34.362  4.772e-01  5.349e-01   <2e-16 ***
#> beta_2_2    5.924e-01  1.149e-02  51.567  5.698e-01  6.149e-01   <2e-16 ***
#> beta_2_3    3.603e-03  8.271e-03   0.436 -1.261e-02  1.981e-02   0.3316    
#> beta_3_1   -8.437e-02  1.027e-02  -8.215 -1.045e-01 -6.424e-02   <2e-16 ***
#> beta_3_2    3.819e-01  1.025e-02  37.273  3.618e-01  4.020e-01   <2e-16 ***
#> beta_3_3    5.099e-01  9.995e-03  51.015  4.903e-01  5.295e-01   <2e-16 ***
#> psi_1_1     9.994e-02  4.469e-03  22.365  9.119e-02  1.087e-01   <2e-16 ***
#> psi_2_1    -6.454e-04  1.578e-03  -0.409 -3.739e-03  2.448e-03   0.3413    
#> psi_3_1    -6.049e-04  1.458e-03  -0.415 -3.462e-03  2.253e-03   0.3391    
#> psi_2_2     1.042e-01  3.753e-03  27.767  9.684e-02  1.116e-01   <2e-16 ***
#> psi_3_2     8.876e-04  1.462e-03   0.607 -1.977e-03  3.753e-03   0.2718    
#> psi_3_3     9.242e-02  4.085e-03  22.626  8.442e-02  1.004e-01   <2e-16 ***
#> theta_1_1   2.007e-01  3.747e-03  53.562  1.934e-01  2.080e-01   <2e-16 ***
#> theta_2_2   1.941e-01  3.659e-03  53.052  1.869e-01  2.013e-01   <2e-16 ***
#> theta_3_3   2.077e-01  4.181e-03  49.678  1.995e-01  2.159e-01   <2e-16 ***
#> mu0_1_1    -1.046e-02  5.176e-02  -0.202 -1.119e-01  9.098e-02   0.4199    
#> mu0_2_1    -1.364e-01  7.315e-02  -1.864 -2.798e-01  6.999e-03   0.0311 *  
#> mu0_3_1     7.357e-02  6.367e-02   1.155 -5.123e-02  1.984e-01   0.1240    
#> sigma0_1_1  1.456e-01  3.749e-02   3.883  7.209e-02  2.191e-01   0.0001 ***
#> sigma0_2_1  6.123e-02  3.805e-02   1.609 -1.336e-02  1.358e-01   0.0538 .  
#> sigma0_3_1  2.338e-02  3.212e-02   0.728 -3.957e-02  8.634e-02   0.2333    
#> sigma0_2_2  3.924e-01  7.628e-02   5.144  2.429e-01  5.419e-01   <2e-16 ***
#> sigma0_3_2  1.563e-01  4.900e-02   3.190  6.024e-02  2.523e-01   0.0007 ***
#> sigma0_3_3  2.300e-01  5.756e-02   3.996  1.172e-01  3.428e-01   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> -2 log-likelihood value at convergence = 265103.99
#> AIC = 265163.99
#> BIC = 265428.59

Parameter Estimates

alpha_hat
#> [1] -7.482091e-05  9.771779e-04  1.185204e-03
beta_hat
#>             [,1]        [,2]         [,3]
#> [1,]  0.69780139 0.006914791 -0.007611952
#> [2,]  0.50607855 0.592355283  0.003602732
#> [3,] -0.08436625 0.381881593  0.509883330
psi_hat
#>               [,1]          [,2]          [,3]
#> [1,]  0.0999439866 -0.0006454149 -0.0006049144
#> [2,] -0.0006454149  0.1041965235  0.0008876247
#> [3,] -0.0006049144  0.0008876247  0.0924224996
theta_hat
#>           [,1]      [,2]      [,3]
#> [1,] 0.2007056 0.0000000 0.0000000
#> [2,] 0.0000000 0.1941158 0.0000000
#> [3,] 0.0000000 0.0000000 0.2077269
mu0_hat
#> [1] -0.01046285 -0.13638224  0.07357103
sigma0_hat
#>            [,1]       [,2]       [,3]
#> [1,] 0.14557753 0.06123054 0.02338447
#> [2,] 0.06123054 0.39236542 0.15627195
#> [3,] 0.02338447 0.15627195 0.23001929

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. https://doi.org/10.1080/10705511003661553

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

Pesigan, I. J. A., Russell, M. A., & Chow, S.-M. (2025). Inferences and effect sizes for direct, indirect, and total effects in continuous-time mediation models. Psychological Methods. https://doi.org/10.1037/met0000779

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