The State Space Model

Ivan Jacob Agaloos Pesigan

2026-01-17

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 $t = 1000$ 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] 1000
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.   :-2.8405340   Min.   :-3.1732718  
#>  1st Qu.: 25.75   1st Qu.:249.8   1st Qu.:-0.4213725   1st Qu.:-0.4999526  
#>  Median : 50.50   Median :499.5   Median :-0.0007939   Median : 0.0031390  
#>  Mean   : 50.50   Mean   :499.5   Mean   : 0.0001801   Mean   : 0.0009951  
#>  3rd Qu.: 75.25   3rd Qu.:749.2   3rd Qu.: 0.4236361   3rd Qu.: 0.4990295  
#>  Max.   :100.00   Max.   :999.0   Max.   : 2.6029049   Max.   : 3.2748708  
#>        y3           
#>  Min.   :-3.029883  
#>  1st Qu.:-0.455867  
#>  Median : 0.003032  
#>  Mean   : 0.003844  
#>  3rd Qu.: 0.465214  
#>  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:  8.625165e-05 0.0004086102 0.001471604 0.7080791 
#> 0.0006374059 1.947947e-05 0.5065289 0.5921055 0.004164591 -0.1007597 0.3952978 
#> 0.5017908 -2.338339 -0.002554279 -0.003423727 -2.267178 -0.001514532 -2.321312 
#> -1.602984 -1.621672 -1.596002 -0.02756067 -0.08560135 0.01939701 -1.476967 
#> 0.575084 0.2698641 -1.120811 0.2716345 -1.630752 
#> 
#> Transformed fitted parameters:  8.625165e-05 0.0004086102 0.001471604 0.7080791 
#> 0.0006374059 1.947947e-05 0.5065289 0.5921055 0.004164591 -0.1007597 0.3952978 
#> 0.5017908 0.09648776 -0.0002464566 -0.0003303477 0.1036048 -0.000156068 
#> 0.09814614 0.201295 0.1975681 0.2027053 -0.02756067 -0.08560135 0.01939701 
#> 0.2283291 0.1313084 0.06161781 0.4015286 0.1239924 0.2364659 
#> 
#> Doing end processing
#> Successful trial
#> Total Time: 1.29726 
#> Backend Time: 1.297134

Summary

summary(results)
#> Coefficients:
#>              Estimate Std. Error t value   ci.lower   ci.upper Pr(>|t|)    
#> alpha_1_1   8.625e-05  1.067e-03   0.081 -2.006e-03  2.178e-03   0.4678    
#> alpha_2_1   4.086e-04  1.373e-03   0.298 -2.282e-03  3.099e-03   0.3830    
#> alpha_3_1   1.472e-03  1.348e-03   1.092 -1.171e-03  4.114e-03   0.1375    
#> beta_1_1    7.081e-01  9.515e-03  74.421  6.894e-01  7.267e-01   <2e-16 ***
#> beta_1_2    6.374e-04  5.903e-03   0.108 -1.093e-02  1.221e-02   0.4570    
#> beta_1_3    1.948e-05  4.364e-03   0.004 -8.534e-03  8.573e-03   0.4982    
#> beta_2_1    5.065e-01  1.028e-02  49.288  4.864e-01  5.267e-01   <2e-16 ***
#> beta_2_2    5.921e-01  7.952e-03  74.456  5.765e-01  6.077e-01   <2e-16 ***
#> beta_2_3    4.165e-03  5.651e-03   0.737 -6.911e-03  1.524e-02   0.2306    
#> beta_3_1   -1.008e-01  7.436e-03 -13.550 -1.153e-01 -8.619e-02   <2e-16 ***
#> beta_3_2    3.953e-01  7.282e-03  54.283  3.810e-01  4.096e-01   <2e-16 ***
#> beta_3_3    5.018e-01  6.985e-03  71.842  4.881e-01  5.155e-01   <2e-16 ***
#> psi_1_1     9.649e-02  3.009e-03  32.067  9.059e-02  1.024e-01   <2e-16 ***
#> psi_2_1    -2.465e-04  1.103e-03  -0.223 -2.408e-03  1.915e-03   0.4116    
#> psi_3_1    -3.303e-04  1.031e-03  -0.320 -2.351e-03  1.691e-03   0.3743    
#> psi_2_2     1.036e-01  2.610e-03  39.696  9.849e-02  1.087e-01   <2e-16 ***
#> psi_3_2    -1.561e-04  1.050e-03  -0.149 -2.214e-03  1.902e-03   0.4409    
#> psi_3_3     9.815e-02  3.016e-03  32.546  9.224e-02  1.041e-01   <2e-16 ***
#> theta_1_1   2.013e-01  2.546e-03  79.064  1.963e-01  2.063e-01   <2e-16 ***
#> theta_2_2   1.976e-01  2.558e-03  77.245  1.926e-01  2.026e-01   <2e-16 ***
#> theta_3_3   2.027e-01  3.050e-03  66.462  1.967e-01  2.087e-01   <2e-16 ***
#> mu0_1_1    -2.756e-02  6.018e-02  -0.458 -1.455e-01  9.038e-02   0.3235    
#> mu0_2_1    -8.560e-02  7.412e-02  -1.155 -2.309e-01  5.967e-02   0.1241    
#> mu0_3_1     1.940e-02  6.399e-02   0.303 -1.060e-01  1.448e-01   0.3809    
#> sigma0_1_1  2.283e-01  4.761e-02   4.796  1.350e-01  3.216e-01   <2e-16 ***
#> sigma0_2_1  1.313e-01  4.671e-02   2.811  3.975e-02  2.229e-01   0.0025 ** 
#> sigma0_3_1  6.162e-02  3.970e-02   1.552 -1.618e-02  1.394e-01   0.0603 .  
#> sigma0_2_2  4.015e-01  8.158e-02   4.922  2.416e-01  5.614e-01   <2e-16 ***
#> sigma0_3_2  1.240e-01  4.949e-02   2.505  2.700e-02  2.210e-01   0.0061 ** 
#> sigma0_3_3  2.365e-01  6.032e-02   3.920  1.183e-01  3.547e-01   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> -2 log-likelihood value at convergence = 530766.20
#> AIC = 530826.20
#> BIC = 531111.59

Parameter Estimates

alpha_hat
#> [1] 8.625165e-05 4.086102e-04 1.471604e-03
beta_hat
#>            [,1]         [,2]         [,3]
#> [1,]  0.7080791 0.0006374059 1.947947e-05
#> [2,]  0.5065289 0.5921054723 4.164591e-03
#> [3,] -0.1007597 0.3952978037 5.017908e-01
psi_hat
#>               [,1]          [,2]          [,3]
#> [1,]  0.0964877604 -0.0002464566 -0.0003303477
#> [2,] -0.0002464566  0.1036047755 -0.0001560680
#> [3,] -0.0003303477 -0.0001560680  0.0981461397
theta_hat
#>          [,1]      [,2]      [,3]
#> [1,] 0.201295 0.0000000 0.0000000
#> [2,] 0.000000 0.1975681 0.0000000
#> [3,] 0.000000 0.0000000 0.2027053
mu0_hat
#> [1] -0.02756067 -0.08560135  0.01939701
sigma0_hat
#>            [,1]      [,2]       [,3]
#> [1,] 0.22832905 0.1313084 0.06161781
#> [2,] 0.13130839 0.4015286 0.12399240
#> [3,] 0.06161781 0.1239924 0.23646593

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. (2025). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/