Dynamic Effects in Longitudinal AME Models

Cassy Dorff, Tosin Salau, Shahryar Minhas

2025-08-25

Introduction

The lame package extends the Additive and Multiplicative Effects (AME) model to longitudinal network data through the introduction of dynamic effects. This vignette explains the theoretical foundation, implementation details, and practical usage of the dynamic_uv and dynamic_ab parameters.

Theoretical Background

Standard AME Model

The standard AME model decomposes network structure as:

$$y_{ij} = \beta'x_{ij} + a_i + b_j + u_i'v_j + \epsilon_{ij}$$

where:

• $\beta'x_{ij}$: Fixed effects of covariates
• $a_i$: Sender (row) effect for node $i$
• $b_j$: Receiver (column) effect for node $j$
• $u_i'v_j$: Multiplicative interaction between latent factors
• $\epsilon_{ij}$: Dyadic error term

Dynamic Extensions

For longitudinal data with networks observed at times $t = 1, ..., T$, we extend the model to allow temporal evolution of effects.

Dynamic Multiplicative Effects (dynamic_uv = TRUE)

The latent factors evolve according to AR(1) processes:

$$U_{i,k,t} = \rho_{uv} U_{i,k,t-1} + \epsilon_{i,k,t}$$$$V_{j,k,t} = \rho_{uv} V_{j,k,t-1} + \eta_{j,k,t}$$

where:

• $\epsilon_{i,k,t}, \eta_{j,k,t} \sim N(0, \sigma_{uv}^2)$
• $\rho_{uv} \in (0,1)$ controls temporal persistence
• $i,j$ index actors, $k$ indexes latent dimensions, $t$ indexes time

This specification allows actors’ positions in latent social space to drift over time, capturing:

• Evolving community structure
• Time-varying homophily patterns
• Dynamic clustering and transitivity

Dynamic Additive Effects (dynamic_ab = TRUE)

The sender and receiver effects evolve as:

$$a_{i,t} = \rho_{ab} a_{i,t-1} + \epsilon_{i,t}$$$$b_{j,t} = \rho_{ab} b_{j,t-1} + \eta_{j,t}$$

where:

• $\epsilon_{i,t}, \eta_{j,t} \sim N(0, \sigma_{ab}^2)$
• $\rho_{ab} \in (0,1)$ controls temporal smoothness

This captures:

• Time-varying activity levels (outdegree heterogeneity)
• Changing popularity (indegree heterogeneity)
• Life-cycle effects in social networks

Prior Specifications

For Dynamic Parameters

The package uses the following default priors:

AR(1) coefficients:

• $\rho_{uv} \sim \text{TruncNormal}(0.9, 0.1, 0, 1)$
• $\rho_{ab} \sim \text{TruncNormal}(0.8, 0.15, 0, 1)$

Innovation variances:

• $\sigma_{uv}^2 \sim \text{InverseGamma}(2, 1)$
• $\sigma_{ab}^2 \sim \text{InverseGamma}(2, 1)$

These can be customized via the prior argument:

prior_custom <- list(
  rho_uv_mean = 0.95,    # Higher persistence for UV
  rho_uv_sd = 0.05,      # Tighter prior
  sigma_uv_shape = 3,    # Different variance prior
  sigma_uv_scale = 2
)

Implementation Details

Computational Efficiency

The dynamic effects are implemented in C++ via Rcpp and RcppArmadillo:

1. Block sampling: Updates are performed in blocks to improve mixing
2. Sparse operations: Exploits temporal structure for efficiency
3. Memory optimization: about a 50% reduction compared to R implementation

Algorithmic Details

The MCMC sampler alternates between:

1. Forward filtering: Compute predictive distributions $p(x_t | x_{1:t-1})$
2. Backward sampling: Sample from smoothed distribution $p(x_{1:T} | y_{1:T})$
3. Parameter updates: Sample $\rho$ and $\sigma^2$ given state sequences

This forward-filtering backward-sampling (FFBS) approach explores the posterior distribution.

Practical Usage

Basic Example

library(lame)

# Simulate longitudinal network data
set.seed(6886)
n <- 25  # actors (reduced for faster vignette building)
T <- 5   # time periods (reduced for faster vignette building)

# Generate networks (example with binary data)
Y_list <- list()
for(t in 1:T) {
  Y_t <- matrix(rbinom(n*n, 1, 0.1), n, n)
  diag(Y_t) <- NA
  # Add actor labels (required for lame function)
  rownames(Y_t) <- colnames(Y_t) <- paste0("Actor", 1:n)
  Y_list[[t]] <- Y_t
}

# Fit model with dynamic effects (reduced iterations for vignette)
fit_dynamic <- lame(
  Y = Y_list,
  R = 2,                # 2-dimensional latent space
  dynamic_uv = TRUE,    # Dynamic latent factors
  dynamic_ab = TRUE,    # Dynamic additive effects
  family = "binary",
  burn = 100,          # Reduced for vignette (use 1000+ in practice)
  nscan = 500,         # Reduced for vignette (use 5000+ in practice)
  odens = 25,
  print = FALSE,       # Suppress output for vignette
  plot = FALSE         # Suppress plots for vignette
)

Visualization

Plotting functions can illustrate temporal dynamics:

# Plot latent positions (last time point)
uv_plot(fit_dynamic, plot_type = "snapshot")
#> ℹ Detected dynamic UV effects
#> • Using last time point for visualization

# Plot sender effects 
ab_plot(fit_dynamic, effect = "sender")

# Plot trace of parameters
trace_plot(fit_dynamic)

Model Selection

Choose between static and dynamic specifications based on:

1. Model fit statistics: Compare GOF statistics
2. Parameter estimates: Check if $\rho$ significantly differs from 0
3. Visual inspection: Look for temporal patterns in residuals

# Static model
fit_static <- lame(Y_list, R = 2, family = "binary", 
                   burn = 100, nscan = 400, odens = 20, 
                   print = FALSE, plot = FALSE)
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#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2

# Dynamic UV only
fit_uv <- lame(Y_list, R = 2, dynamic_uv = TRUE, family = "binary",
               burn = 100, nscan = 400, odens = 20,
               print = FALSE, plot = FALSE)
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2
#> Error in eval(expr, envir) : 
#>   addition: incompatible matrix dimensions: 1x1 and 1x2

# Dynamic AB only  
fit_ab <- lame(Y_list, R = 2, dynamic_ab = TRUE, family = "binary",
               burn = 100, nscan = 400, odens = 20,
               print = FALSE, plot = FALSE)

# Full dynamic
fit_full <- lame(Y_list, R = 2, dynamic_uv = TRUE, dynamic_ab = TRUE, family = "binary",
                 burn = 100, nscan = 400, odens = 20,
                 print = FALSE, plot = FALSE)

# Compare GOF statistics
if(!is.null(fit_static$GOF) && !is.null(fit_full$GOF)) {
  cat("Static model GOF (sample):\n")
  print(head(fit_static$GOF))
  cat("\nDynamic model GOF (sample):\n")  
  print(head(fit_full$GOF))
}
#> Static model GOF (sample):
#> $sd.rowmean
#>             obs 1     2          3          4 5 6 7 8 9 10 11 12 13 14 15 16 17
#> [1,] 0.04659041 0 0.008 0.08368194 0.07973289 0 0 0 0 0  0  0  0  0  0  0  0  0
#> [2,] 0.06641285 0 0.008 0.09056857 0.10124558 0 0 0 0 0  0  0  0  0  0  0  0  0
#> [3,] 0.05033223 0 0.000 0.07440430 0.09887366 0 0 0 0 0  0  0  0  0  0  0  0  0
#> [4,] 0.06332982 0 0.000 0.10309866 0.08089499 0 0 0 0 0  0  0  0  0  0  0  0  0
#> [5,] 0.05941941 0 0.000 0.09027366 0.11015141 0 0 0 0 0  0  0  0  0  0  0  0  0
#>      18 19 20
#> [1,]  0  0  0
#> [2,]  0  0  0
#> [3,]  0  0  0
#> [4,]  0  0  0
#> [5,]  0  0  0
#> 
#> $sd.colmean
#>             obs 1     2          3          4 5 6 7 8 9 10 11 12 13 14 15 16 17
#> [1,] 0.06955094 0 0.008 0.10166612 0.10045231 0 0 0 0 0  0  0  0  0  0  0  0  0
#> [2,] 0.05547372 0 0.008 0.09627738 0.08236504 0 0 0 0 0  0  0  0  0  0  0  0  0
#> [3,] 0.06429101 0 0.000 0.09417714 0.07943131 0 0 0 0 0  0  0  0  0  0  0  0  0
#> [4,] 0.05043808 0 0.000 0.11414611 0.12440793 0 0 0 0 0  0  0  0  0  0  0  0  0
#> [5,] 0.04393935 0 0.000 0.09873871 0.10327956 0 0 0 0 0  0  0  0  0  0  0  0  0
#>      18 19 20
#> [1,]  0  0  0
#> [2,]  0  0  0
#> [3,]  0  0  0
#> [4,]  0  0  0
#> [5,]  0  0  0
#> 
#> $dyad.dep
#>               obs 1            2         3         4 5 6 7 8 9 10 11 12 13 14
#> [1,]  0.019269395 1 -0.001602564 0.2945824 0.1241742 1 1 1 1 1  1  1  1  1  1
#> [2,] -0.009636809 1 -0.001602564 0.3403377 0.2125364 1 1 1 1 1  1  1  1  1  1
#> [3,]  0.021291209 1  0.000000000 0.1604299 0.1951230 1 1 1 1 1  1  1  1  1  1
#> [4,]  0.146174863 1  0.000000000 0.2557376 0.2805006 1 1 1 1 1  1  1  1  1  1
#> [5,] -0.003450161 1  0.000000000 0.2917233 0.1189065 1 1 1 1 1  1  1  1  1  1
#>      15 16 17 18 19 20
#> [1,]  1  1  1  1  1  1
#> [2,]  1  1  1  1  1  1
#> [3,]  1  1  1  1  1  1
#> [4,]  1  1  1  1  1  1
#> [5,]  1  1  1  1  1  1
#> 
#> $cycle.dep
#>              obs          1        2            3            4          5
#> [1,] 0.004000134 -0.2036344 0.000128 -0.044303875 -0.008033423 -0.2036344
#> [2,] 0.021198238 -0.2036344 0.000128 -0.046472921 -0.049205633 -0.2036344
#> [3,] 0.002352264 -0.2036344 0.000000 -0.007653081 -0.011436157 -0.2036344
#> [4,] 0.006223826 -0.2036344 0.000000 -0.054778721 -0.014272298 -0.2036344
#> [5,] 0.006189746 -0.2036344 0.000000 -0.045482738 -0.011235311 -0.2036344
#>               6          7          8          9         10         11
#> [1,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [2,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [3,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [4,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [5,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#>              12         13         14         15         16         17
#> [1,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [2,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [3,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [4,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [5,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#>              18         19         20
#> [1,] -0.2036344 -0.2036344 -0.2036344
#> [2,] -0.2036344 -0.2036344 -0.2036344
#> [3,] -0.2036344 -0.2036344 -0.2036344
#> [4,] -0.2036344 -0.2036344 -0.2036344
#> [5,] -0.2036344 -0.2036344 -0.2036344
#> 
#> $trans.dep
#>              obs          1         2          3           4          5
#> [1,] -0.03405087 -0.2036344 -0.003072 -0.1371288 -0.04011523 -0.2036344
#> [2,] -0.02658332 -0.2036344 -0.003072 -0.1268568 -0.05941429 -0.2036344
#> [3,] -0.02865812 -0.2036344  0.000000 -0.1269987 -0.06475873 -0.2036344
#> [4,] -0.02792264 -0.2036344  0.000000 -0.1266319 -0.08616438 -0.2036344
#> [5,] -0.03012973 -0.2036344  0.000000 -0.1355674 -0.05365398 -0.2036344
#>               6          7          8          9         10         11
#> [1,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [2,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [3,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [4,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [5,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#>              12         13         14         15         16         17
#> [1,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [2,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [3,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [4,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#> [5,] -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344 -0.2036344
#>              18         19         20
#> [1,] -0.2036344 -0.2036344 -0.2036344
#> [2,] -0.2036344 -0.2036344 -0.2036344
#> [3,] -0.2036344 -0.2036344 -0.2036344
#> [4,] -0.2036344 -0.2036344 -0.2036344
#> [5,] -0.2036344 -0.2036344 -0.2036344
#> 
#> 
#> Dynamic model GOF (sample):
#> $sd.rowmean
#>             obs          1          2          3          4          5
#> [1,] 0.04659041 0.05787343 0.07493998 0.05256108 0.05600000 0.07704544
#> [2,] 0.06641285 0.06000000 0.10087616 0.05430776 0.07476184 0.06358197
#> [3,] 0.05033223 0.07069182 0.07418895 0.05430776 0.07302967 0.06831301
#> [4,] 0.06332982 0.07129282 0.07144228 0.03702252 0.07831560 0.07600000
#> [5,] 0.05941941 0.07350283 0.07582436 0.03815757 0.06395832 0.08082904
#>               6          7          8          9         10         11
#> [1,] 0.05291503 0.08368194 0.06839103 0.05656854 0.06780364 0.06218253
#> [2,] 0.06311894 0.05401234 0.07989994 0.04805552 0.05331666 0.07393691
#> [3,] 0.04484046 0.08082904 0.08191052 0.05205126 0.07752849 0.06901208
#> [4,] 0.05623759 0.08026622 0.07118052 0.05200000 0.06584831 0.06324555
#> [5,] 0.05075431 0.09572182 0.06800000 0.05416026 0.07118052 0.06316117
#>              12         13         14         15         16         17
#> [1,] 0.07578918 0.06523803 0.05148463 0.05251032 0.06218253 0.06601010
#> [2,] 0.07364781 0.08191052 0.05523284 0.07162867 0.09086253 0.06544209
#> [3,] 0.09830565 0.05291503 0.05351635 0.05851496 0.07125541 0.08288144
#> [4,] 0.07302967 0.06780364 0.05122499 0.05022616 0.06273755 0.06641285
#> [5,] 0.07909488 0.05326662 0.05964338 0.04898979 0.07956549 0.06416645
#>              18         19         20
#> [1,] 0.05075431 0.05331666 0.07368401
#> [2,] 0.07129282 0.03249615 0.04454211
#> [3,] 0.05033223 0.04182503 0.06053098
#> [4,] 0.07483315 0.04601449 0.07571878
#> [5,] 0.06760671 0.02581989 0.04150502
#> 
#> $sd.colmean
#>             obs          1          2          3          4          5
#> [1,] 0.06955094 0.09027366 0.06441532 0.04860727 0.05946427 0.06784296
#> [2,] 0.05547372 0.06218253 0.05810336 0.04915282 0.06420800 0.11449600
#> [3,] 0.06429101 0.06478683 0.06031031 0.04915282 0.05656854 0.09237604
#> [4,] 0.05043808 0.07313914 0.06248200 0.04363485 0.06633250 0.06665333
#> [5,] 0.04393935 0.06057502 0.07035150 0.03448671 0.04424176 0.06928203
#>               6          7          8          9         10         11
#> [1,] 0.05033223 0.09275056 0.05896892 0.05291503 0.05474791 0.07211103
#> [2,] 0.05276362 0.06721111 0.06916647 0.05923963 0.07858753 0.06324555
#> [3,] 0.04630335 0.09165151 0.09332381 0.05075431 0.06935897 0.05968808
#> [4,] 0.05856051 0.08352245 0.06633250 0.03878144 0.05600000 0.05887841
#> [5,] 0.03525148 0.05623759 0.08220300 0.04472136 0.08000000 0.06209670
#>              12         13         14         15         16         17
#> [1,] 0.07400901 0.09445987 0.04744119 0.04270831 0.05033223 0.05964338
#> [2,] 0.07273697 0.05923963 0.05401234 0.06053098 0.07016172 0.06231105
#> [3,] 0.10423051 0.06324555 0.06162251 0.04990658 0.06332982 0.07704544
#> [4,] 0.09521905 0.06681317 0.05851496 0.06523803 0.05230679 0.07752849
#> [5,] 0.07909488 0.06955094 0.06395832 0.05291503 0.06053098 0.06205374
#>              18         19         20
#> [1,] 0.04369592 0.05205126 0.07721831
#> [2,] 0.07035150 0.03815757 0.05986652
#> [3,] 0.05416026 0.06013319 0.05096404
#> [4,] 0.08406347 0.04144072 0.06733003
#> [5,] 0.06031031 0.03055050 0.04460194
#> 
#> $dyad.dep
#>               obs          1           2            3           4          5
#> [1,]  0.019269395 0.17033836  0.03311752  0.008241196  0.14193685 0.29816287
#> [2,] -0.009636809 0.02129121  0.08956199  0.047341522  0.11591721 0.20679085
#> [3,]  0.021291209 0.19111303 -0.03542574  0.051985659  0.16666667 0.06542056
#> [4,]  0.146174863 0.10473844  0.13695707  0.077210353  0.07575758 0.18374523
#> [5,] -0.003450161 0.09269212  0.14054890 -0.045150502 -0.03251483 0.13636364
#>               6          7           8          9         10         11
#> [1,] 0.36440678 0.01742802 0.025178253 0.14529915 0.16734987 0.11504425
#> [2,] 0.29523054 0.18685965 0.009651397 0.01071192 0.18374523 0.15191740
#> [3,] 0.04283714 0.03322440 0.084758669 0.10273872 0.12962344 0.04974761
#> [4,] 0.23425631 0.10916510 0.124313187 0.35511024 0.09269212 0.26253687
#> [5,] 0.17743479 0.10779835 0.157853257 0.09188034 0.21739130 0.05220971
#>              12         13          14          15        16         17
#> [1,] 0.11010916 0.06685468 0.004678013  0.01441766 0.1519174 0.06758773
#> [2,] 0.05816757 0.08956199 0.157782835  0.09208475 0.2399990 0.13503361
#> [3,] 0.09983869 0.06692407 0.224950397 -0.03251483 0.1644385 0.27995392
#> [4,] 0.08256881 0.08194985 0.148667056  0.07705732 0.1749393 0.29887388
#> [5,] 0.03723102 0.13656032 0.027048064 -0.02969349 0.2527499 0.22074050
#>              18          19          20
#> [1,] 0.01071192  0.01071192  0.12576780
#> [2,] 0.06442411  0.10968661 -0.01800033
#> [3,] 0.04347826  0.05198566  0.04779620
#> [4,] 0.02522936  0.06688564  0.07817109
#> [5,] 0.07139506 -0.02459016  0.03226805
#> 
#> $cycle.dep
#>              obs             1            2            3             4
#> [1,] 0.004000134 -0.0187896864 -0.015173722 -0.015335730  0.0117523715
#> [2,] 0.021198238 -0.0001795621  0.008139283  0.002521070 -0.0072351982
#> [3,] 0.002352264 -0.0291105390  0.022550056  0.008877397 -0.0098236952
#> [4,] 0.006223826 -0.0022553921 -0.001274739 -0.005032050 -0.0008930632
#> [5,] 0.006189746 -0.0083934831 -0.018672915  0.031249330 -0.0003162653
#>                 5            6            7             8            9
#> [1,] -0.008798370  0.001218666  0.008498098  3.856545e-05  0.020205569
#> [2,] -0.017119532 -0.010704992  0.006214057 -5.348653e-03 -0.007693117
#> [3,]  0.002761775  0.003872928 -0.014557598  6.903620e-04 -0.009263868
#> [4,] -0.020555579  0.016865306 -0.012245857  7.523654e-03  0.002587877
#> [5,]  0.013395948 -0.011722592 -0.010762741 -1.277246e-02 -0.006444880
#>                10           11           12            13            14
#> [1,] -0.003714206 -0.013785306  0.004666830  0.0007700891 -0.0108719377
#> [2,] -0.038241483 -0.015763198 -0.005038683 -0.0090286635  0.0005734741
#> [3,] -0.024761362  0.024677602 -0.006852662 -0.0290519432 -0.0365086750
#> [4,] -0.015688590  0.006832717  0.003848961 -0.0065673811  0.0286194845
#> [5,] -0.004092890 -0.021441537  0.006087398  0.0113235975  0.0060117721
#>                15           16           17           18           19
#> [1,]  0.017241276 -0.004555145  0.002170119  0.004555402  0.020496856
#> [2,] -0.005770419 -0.018501270 -0.004639135 -0.013724687  0.010425682
#> [3,]  0.008066600 -0.007703851 -0.010541257  0.006267238  0.019999014
#> [4,] -0.010480621 -0.027691066 -0.028304960  0.019094132 -0.008744271
#> [5,] -0.013574342  0.012493619  0.003838492  0.009222652 -0.008975854
#>                20
#> [1,]  0.001317151
#> [2,]  0.011851743
#> [3,] -0.007316799
#> [4,]  0.019659045
#> [5,] -0.007443998
#> 
#> $trans.dep
#>              obs           1           2            3           4           5
#> [1,] -0.03405087 -0.03680731 -0.02346088 -0.014203159 -0.01998711 -0.02991726
#> [2,] -0.02658332 -0.02556965 -0.03916125 -0.028329766 -0.02864505 -0.04318778
#> [3,] -0.02865812 -0.04171116 -0.02188256 -0.020966773 -0.02441039 -0.04572272
#> [4,] -0.02792264 -0.02152581 -0.02399221 -0.011592610 -0.02627094 -0.02957746
#> [5,] -0.03012973 -0.02236293 -0.03989753 -0.001390188 -0.03535676 -0.01116329
#>                 6           7           8            9           10
#> [1,] -0.011724410 -0.02908111 -0.02681794 -0.017174733 -0.035333160
#> [2,] -0.014508201 -0.01342043 -0.01712476 -0.030391030 -0.035328363
#> [3,] -0.014137502 -0.03127094 -0.01568648 -0.025339004 -0.044469331
#> [4,] -0.009861007 -0.02584051 -0.04122747  0.004594061 -0.018937995
#> [5,] -0.021958395 -0.03436242 -0.03755572 -0.030308353 -0.008057877
#>                11          12          13          14           15          16
#> [1,] -0.022755744 -0.02858735 -0.04440114 -0.01827061 -0.009353235 -0.01258658
#> [2,] -0.029848185 -0.01447083 -0.02377320 -0.02334308 -0.026125835 -0.03832327
#> [3,] -0.030754330 -0.01694146 -0.02201694 -0.02483718 -0.028448496 -0.04184820
#> [4,] -0.007631865 -0.03894162 -0.04187106 -0.02373110 -0.020038171 -0.02890570
#> [5,] -0.023232552 -0.03501816 -0.03299540 -0.02284208 -0.024871507 -0.02351215
#>               17          18          19           20
#> [1,] -0.03087547 -0.02107235 -0.02342848 -0.028788807
#> [2,] -0.02029833 -0.02525102 -0.02287027 -0.023057009
#> [3,] -0.04303227 -0.01036013 -0.02697447 -0.039859166
#> [4,] -0.04637022 -0.03501047 -0.01827061 -0.025912380
#> [5,] -0.01827011 -0.03760221 -0.01581460 -0.004344156

References

1. Sewell, D. K., & Chen, Y. (2015). Latent space models for dynamic networks. Journal of the American Statistical Association, 110(512), 1646-1657.

2. Durante, D., & Dunson, D. B. (2014). Nonparametric Bayes dynamic modeling of relational data. Biometrika, 101(4), 883-898.

Appendix: Mathematical Details

Posterior Distributions

For the AR(1) parameters, the full conditionals are:

$$p(\rho | \cdot) \propto p(\rho) \prod_{t=2}^T p(x_t | x_{t-1}, \rho, \sigma^2)$$

$$p(\sigma^2 | \cdot) \sim \text{IG}\left(\alpha + \frac{n(T-1)}{2}, \beta + \frac{1}{2}\sum_{t=2}^T ||x_t - \rho x_{t-1}||^2\right)$$

Identification

The dynamic effects are identified through: 1. Temporal ordering constraints 2. Stationarity assumption ($|\rho| < 1$) 3. Initial state distribution $x_1 \sim N(0, \sigma^2/(1-\rho^2))$

Computational Complexity

• Time complexity: $O(n^2 RT)$ per iteration
• Space complexity: $O(nRT)$ for storage
• Mixing time: Typically $O(\log T)$ with good initialization