• ^{Hamaker, E. L., & Grasman, R. P. P. P. (2014). To center or not to center? Investigating inertia with a multilevel autoregressive model. Frontiers in Psychology, 5, 1492. http://doi.org/10.3389/fpsyg.2014.01492}

• Psychological constructs can be conceptualized as dynamical systems, featuring complex emergent behavior:
  • Correlated responses
  • Stable "traits"
  • Phase transitions
  • Individual Differences
• These systems can be portrayed as networks

• Introduction Vector Auto-regression
• Problem 1: \(n=1\) and limited observations
  • Graphical VAR
• Problem 2: \(n>1\)
  • Multi-level VAR
• Problem 3: Measurement error
  • State-space models
• Even more problems

• Regress a vector of variables from a single subject on the previous time point
• Assume multivariate normality
• Assume errors are correlated
• This leads to three things to estimate:
  • A vector of intercepts
  • A matrix encoding a temporal network
  • A matrix encoding a contemporaneous network
• Can be estimated using lm() or any least squares regression method

For a single subject: \[ \begin{aligned} \pmb{y}_t &= \pmb{\tau} + \pmb{B} \pmb{y}_{t-1} + \pmb{\varepsilon}_t \\ \pmb{\varepsilon}_t &\sim N\left( \pmb{0}, \pmb{\Theta} \right). \end{aligned} \] \(\pmb{B}\) encodes a directed temporal network and \(\pmb{\Theta}\) and undirected contemporaneous network

With \(\pmb{K} = \pmb{\Sigma}^{-1}\) encoding partial correlations coefficients

• Measure a patient over a short time, estimate network structures and use these in clinical practice
• Naturally \(n=1\) problem
  • Different estimation periods
  • Different nodes
• Naturally a limited data problem
  • You can't measure a patient 10 times per day
  • You can't measure a patient for months

• Only model temporal effects between consecutive measurements
  • Lag-1
• Assume both the temporal and contemporaneous effects are sparse
  • Only a relatively little number of edges in both networks
• To do this, we use the graphical VAR model (Wild et al. 2010)
  • Estimation via LASSO regularization, using extended BIC to select optimal tuning parameter (Rothman, Levina, and Zhu 2010; Abegaz and Wit 2013).
• We implemented these methods in the R package graphicalVAR (cran.r-project.org/package=graphicalVAR)

Data collected by Date C. Van der Veen, in collaboration with Harriette Riese en Renske Kroeze.

• Patient suffering from panic disorder and depressive symptoms
  • Perfectionist
• Measured over a period of two weeks
• Five times per day
• Items were chosen after intake together with therapist