The State Space Model

Ivan Jacob Agaloos Pesigan

2025-05-09

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 $n = 1000$ 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] 1000
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.   :   1.0   Min.   :  0.0   Min.   :-2.880700   Min.   :-3.395670  
#>  1st Qu.: 250.8   1st Qu.:249.8   1st Qu.:-0.422772   1st Qu.:-0.496613  
#>  Median : 500.5   Median :499.5   Median : 0.001104   Median : 0.001196  
#>  Mean   : 500.5   Mean   :499.5   Mean   : 0.001169   Mean   : 0.001253  
#>  3rd Qu.: 750.2   3rd Qu.:749.2   3rd Qu.: 0.425543   3rd Qu.: 0.498803  
#>  Max.   :1000.0   Max.   :999.0   Max.   : 2.997266   Max.   : 3.752081  
#>        y3            
#>  Min.   :-3.2399955  
#>  1st Qu.:-0.4605500  
#>  Median :-0.0001519  
#>  Mean   :-0.0002244  
#>  3rd Qu.: 0.4602780  
#>  Max.   : 3.1835079
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:  0.0003183676 -7.720703e-05 -0.0004995016 0.6980319 
#> 0.001729944 -0.0005521516 0.4960978 0.6021552 -0.001368254 -0.100953 0.4013302 
#> 0.4991841 -2.298619 0.004124438 0.0003224472 -2.297285 -0.0002552043 -2.288587 
#> -1.613556 -1.61031 -1.617563 -0.001094629 -0.01500538 -0.03377115 -1.669574 
#> 0.6009867 0.1203478 -1.326947 0.4397236 -1.710145 
#> 
#> Transformed fitted parameters:  0.0003183676 -7.720703e-05 -0.0004995016 
#> 0.6980319 0.001729944 -0.0005521516 0.4960978 0.6021552 -0.001368254 -0.100953 
#> 0.4013302 0.4991841 0.1003973 0.0004140826 3.237285e-05 0.1005331 -2.552254e-05 
#> 0.1014097 0.199178 0.1998256 0.1983815 -0.001094629 -0.01500538 -0.03377115 
#> 0.1883273 0.1131822 0.02266478 0.3333069 0.1302737 0.2348621 
#> 
#> Doing end processing
#> Successful trial
#> Total Time: 15.41867 
#> Backend Time: 15.41852

Summary

summary(results)
#> Coefficients:
#>              Estimate Std. Error t value   ci.lower   ci.upper Pr(>|t|)    
#> alpha_1_1   3.184e-04  3.446e-04   0.924 -3.571e-04  9.939e-04   0.1778    
#> alpha_2_1  -7.721e-05  4.260e-04  -0.181 -9.121e-04  7.576e-04   0.4281    
#> alpha_3_1  -4.995e-04  4.309e-04  -1.159 -1.344e-03  3.450e-04   0.1232    
#> beta_1_1    6.980e-01  2.958e-03 235.997  6.922e-01  7.038e-01   <2e-16 ***
#> beta_1_2    1.730e-03  1.830e-03   0.945 -1.857e-03  5.317e-03   0.1723    
#> beta_1_3   -5.521e-04  1.354e-03  -0.408 -3.206e-03  2.101e-03   0.3417    
#> beta_2_1    4.961e-01  3.168e-03 156.576  4.899e-01  5.023e-01   <2e-16 ***
#> beta_2_2    6.022e-01  2.437e-03 247.068  5.974e-01  6.069e-01   <2e-16 ***
#> beta_2_3   -1.368e-03  1.719e-03  -0.796 -4.738e-03  2.001e-03   0.2131    
#> beta_3_1   -1.010e-01  2.241e-03 -45.042 -1.053e-01 -9.656e-02   <2e-16 ***
#> beta_3_2    4.013e-01  2.172e-03 184.739  3.971e-01  4.056e-01   <2e-16 ***
#> beta_3_3    4.992e-01  2.087e-03 239.242  4.951e-01  5.033e-01   <2e-16 ***
#> psi_1_1     1.004e-01  9.646e-04 104.085  9.851e-02  1.023e-01   <2e-16 ***
#> psi_2_1     4.141e-04  3.445e-04   1.202 -2.612e-04  1.089e-03   0.1147    
#> psi_3_1     3.237e-05  3.231e-04   0.100 -6.008e-04  6.655e-04   0.4601    
#> psi_2_2     1.005e-01  7.772e-04 129.344  9.901e-02  1.021e-01   <2e-16 ***
#> psi_3_2    -2.552e-05  3.257e-04  -0.078 -6.640e-04  6.129e-04   0.4688    
#> psi_3_3     1.014e-01  9.419e-04 107.666  9.956e-02  1.033e-01   <2e-16 ***
#> theta_1_1   1.992e-01  8.096e-04 246.036  1.976e-01  2.008e-01   <2e-16 ***
#> theta_2_2   1.998e-01  7.685e-04 260.031  1.983e-01  2.013e-01   <2e-16 ***
#> theta_3_3   1.984e-01  9.450e-04 209.936  1.965e-01  2.002e-01   <2e-16 ***
#> mu0_1_1    -1.095e-03  1.780e-02  -0.062 -3.597e-02  3.379e-02   0.4755    
#> mu0_2_1    -1.501e-02  2.200e-02  -0.682 -5.813e-02  2.812e-02   0.2476    
#> mu0_3_1    -3.377e-02  2.030e-02  -1.664 -7.355e-02  6.010e-03   0.0481 *  
#> sigma0_1_1  1.883e-01  1.323e-02  14.230  1.624e-01  2.143e-01   <2e-16 ***
#> sigma0_2_1  1.132e-01  1.295e-02   8.738  8.779e-02  1.386e-01   <2e-16 ***
#> sigma0_3_1  2.266e-02  1.256e-02   1.805 -1.952e-03  4.728e-02   0.0356 *  
#> sigma0_2_2  3.333e-01  2.090e-02  15.949  2.923e-01  3.743e-01   <2e-16 ***
#> sigma0_3_2  1.303e-01  1.745e-02   7.467  9.608e-02  1.645e-01   <2e-16 ***
#> sigma0_3_3  2.349e-01  2.044e-02  11.492  1.948e-01  2.749e-01   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> -2 log-likelihood value at convergence = 5306384.65
#> AIC = 5306444.65
#> BIC = 5306799.11

Parameter Estimates

alpha_hat
#> [1]  3.183676e-04 -7.720703e-05 -4.995016e-04
beta_hat
#>            [,1]        [,2]          [,3]
#> [1,]  0.6980319 0.001729944 -0.0005521516
#> [2,]  0.4960978 0.602155205 -0.0013682542
#> [3,] -0.1009530 0.401330175  0.4991841350
psi_hat
#>              [,1]          [,2]          [,3]
#> [1,] 1.003973e-01  4.140826e-04  3.237285e-05
#> [2,] 4.140826e-04  1.005331e-01 -2.552254e-05
#> [3,] 3.237285e-05 -2.552254e-05  1.014097e-01
theta_hat
#>          [,1]      [,2]      [,3]
#> [1,] 0.199178 0.0000000 0.0000000
#> [2,] 0.000000 0.1998256 0.0000000
#> [3,] 0.000000 0.0000000 0.1983815
mu0_hat
#> [1] -0.001094629 -0.015005381 -0.033771148
sigma0_hat
#>            [,1]      [,2]       [,3]
#> [1,] 0.18832727 0.1131822 0.02266478
#> [2,] 0.11318219 0.3333069 0.13027371
#> [3,] 0.02266478 0.1302737 0.23486214

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

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