• ^{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

• ^{Scheffer, M., Bascompte, J., Brock, W. A., Brovkin, V., Carpenter, S. R., Dakos, V., … & Sugihara, G. (2009). Early-warning signals for critical transitions. Nature, 461(7260), 53-59.}

• CSD seems to signal the vicinity of a tipping point
  • Climate change
  • Collapse of Ecosystems
  • Proven in fold bifurcations
• CSD should lead to an increase in auto-correlation
• Thus, build up of auto-correlation can be an early warning signal for a critical phase transition

• Data from:
  • The general population (females; n = 535)
  • Depressed patients eligible for treatment (n = 93)
• Experience Sampling Method
  • Wristwatch and booklet
  • 10 measurements per day
  • 5 days of measuring
• Follow up course of depression measured 6-8 weeks later

• One patient: 57 year old male
  • History of depression
  • Has been using antidepressants for 8.5 years
• 10 measurements per day for 239 days
  • irritated, content, lonely, anxious, enthusiastic, cheerful, guilty, indecisive, strong, restless and agitated.
  • 3 principal components: Negative affect, positive affect and mental unrest
  • In addition: Suspicious and worrying
• During this time: antidepressant dose reduced in blind scheme

Movie!

• Critical transition at day 127
  • Change point analysis
  • Clinically significant: patient returned to taking medications
• Preceded by critical slowing down

• Depression can be characterized as a complex system
• Increasing auto correlation can function as an early warning signal for the onset of depression

• Only model temporal effects between consecutive measurements
  • Lag-1
• Assume both the temporal and contemporaneous effects are sparse
  • Only a relatively little amount of edges in both networks
• To do this, we use the graphical VAR model (Wild et al. 2010)
  • Estimation via LASSO regularization, using 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

Feeling worthless interacts with feeling helpless

Feeling stressed interacts with feeling the need to do things

Central node: Feeling sad

Cycle of enjoyment, feeling sad, feeling worthless and being active

Having to had to do things leads to letting important things pass

Let \(\pmb{y}_t\) represent the \(P\) length random vector of responses at time point \(t\) in the ESM design. We assume that \(\pmb{y}_t\) follows a stationary centered normal distribution: \[ \pmb{y}_t \sim N\left( \pmb{0}, \pmb{\Sigma} \right) \quad \forall t. \]

We use a lag-1 autoregression model: \[ \begin{aligned} \pmb{y}_{t} &= \pmb{B} \pmb{y}_{t-1} + \pmb{\varepsilon}_{t} \\ \pmb{\varepsilon}_{t} &\sim N\left( \pmb{0}, \pmb{K}^{-1} \right) \end{aligned} \] \(\pmb{B}\) is a asymmetric matrix encoding a directed network and \(\pmb{K}\) is a symmetric matrix encoding an undirected network.

The stationary covariance matrix \(\pmb{\Sigma}\) can be obtained after jointly estimating \(\pmb{K}\) and \(\pmb{B}\):

\[ \mathrm{Vec}\left(\pmb{\Sigma}\right) = \left( \pmb{I} - \pmb{B} \otimes \pmb{B}\right)^{-1} \mathrm{Vec}\left(\pmb{\Theta}\right) \]

The joint distribution of \(\pmb{y}_{t}\) and \(\pmb{y}_{t+1}\) can now be formulated as follows: \[ \begin{bmatrix} \pmb{y}_{t} \\ \pmb{y}_{t+1} \end{bmatrix} \sim N \left( \pmb{0}, \pmb{\Sigma}_{\mathrm{TP}} \right), \] in which \(\pmb{\Sigma}_{\mathrm{TP}}\) is the Toeplitz matrix: \[ \pmb{\Sigma}_{\mathrm{TP}} = \begin{bmatrix} \pmb{\Sigma} & \pmb{\Sigma}\pmb{B}^\top\\ \pmb{B}\pmb{\Sigma} & \pmb{\Sigma} \end{bmatrix}, \]

the stationary differential entropy of the system can be derived to be: \[ h\left( \pmb{y}_t \right) = \frac{1}{2} \log_2 \left( (2\pi e)^P \mid \pmb{\Sigma} \mid \right) \quad \forall t, \]

The joint differential entropy of \(\pmb{y}_t\) and \(\pmb{y}_{t+1}\) is identical except that it uses the Toeplitz matrix: \[ \begin{aligned} h\left( \pmb{y}_t , \pmb{y}_{t+1} \right) &= \frac{1}{2} \log_2 \left( (2\pi e)^{2P} \mid \pmb{\Sigma}_{\mathrm{TP}} \mid \right) \end{aligned} \]

Using these expressions, we can obtain the mutual information between \(\pmb{y}_t\) and \(\pmb{y}_{t+1}\), making use of \(h\left( \pmb{y}_t \right) = h\left( \pmb{y}_{t+1} \right)\): \[ \begin{aligned} I\left(\pmb{y}_t; \pmb{y}_{t+1} \right) &= 2 h\left( \pmb{y}_t \right) - h\left( \pmb{y}_t , \pmb{y}_{t+1} \right) \\ &= \frac{1}{2} \log_2 \left( \frac{ \mid \pmb{\Sigma} \mid^2 }{ \mid \pmb{\Sigma}_{\mathrm{TP}} \mid } \right) \end{aligned} \]

This measure encodes the stability of the system, the amount of information retained in consecutive time points.

The differential entropy of variable \(j\) at time point \(t\) equals: \[ h\left( y_{t,j} \right) = \frac{1}{2} \log_2\left( 2 \pi e \sigma_{jj} \right). \] To compute the joint differential entropy between between \(y_{t,j}\) and \(\pmb{y}_{t+1}\), we need to form the block matrix \(\pmb{\Sigma}^{(j)}_{\mathrm{TP}}\): \[ \pmb{\Sigma}^{(j)}_{\mathrm{TP}} = \begin{bmatrix} \sigma_{jj} & \left(\pmb{\Sigma}\pmb{B}^\top\right)_{j,+} \\ \left(\pmb{B}\pmb{\Sigma}\right)_{+,j} & \pmb{\Sigma} \end{bmatrix}, \] in which \(\left(\pmb{\Sigma}\pmb{B}^\top\right)_{j,+}\) indicates the \(j\)th row-vector of \(\pmb{\Sigma}\pmb{B}^\top\) and \(\left(\pmb{B}\pmb{\Sigma}\right)_{+,j}\) the \(j\)th column-vector of \(\pmb{B}\pmb{\Sigma}\)

the mutual information between \(y_{t,j}\) and \(\pmb{y}_{t+1}\) becomes: \[ \begin{aligned} I\left(y_{t,j}; \pmb{y}_{t+1} \right) &= h\left( y_{t,j} \right) + h\left( \pmb{y}_{t+1} \right) - h\left( y_{t,j} , \pmb{y}_{t+1} \right) \\ &= \frac{1}{2} \log_2\left( \frac{ \sigma_{jj} \mid \pmb{\Sigma} \mid }{ \mid \pmb{\Sigma}^{(j)}_{\mathrm{TP}} \mid } \right). \end{aligned} \]

This measure gives a comparable measure between variables that indicates the relative influence of each variable on the rest of the system at the next time point.

To investigate the unique influence of a variable on the system, we can investigate the mutual information between one variable and all other variables on the next time point conditioned on all other variables on the current time point; \(I\left(y_{t,j}; \pmb{y}_{t+1} \mid \pmb{y}_{t+1}^{-(j)} \right)\):

\[ \begin{aligned} I\left(y_{j,t} ; \pmb{y}_{t+1} \mid \pmb{y}^{-(j)}_{t} \right) &= I\left(\pmb{y}_{t}; \pmb{y}_{t+1} \right) - I\left(\pmb{y}^{-(j)}_{t} ; \pmb{y}_{t+1} \right) \\ &= \frac{1}{2} \log_2\left( \frac{ \mid \pmb{\Sigma} \mid \mid \pmb{\Sigma}^{-(j)}_{\mathrm{TP}} \mid}{ \mid \pmb{\Sigma}_{\mathrm{TP}} \mid \mid \pmb{\Sigma}^{-(j)} \mid} \right) \end{aligned} \]

Lag-1 model

Level 1: \[ \pmb{y}_t^{(p)} = \pmb{B}^{(p)} \pmb{y}_ {t-1}^{(p)} + \pmb{\varepsilon}_t^{(p)} \]

Level 2: \[ \begin{aligned} \pmb{\beta}_ {ij}^{(p)} &= b_{ij} + u^{(p)}_{ij} \\ u^{(p)}_{ij} &\sim N(0, \sigma_{ij}) \end{aligned} \]

Lag-2 model

Level 1: \[ \pmb{y}_t^{(p)} = \pmb{B}_1^{(p)} \pmb{y}_ {t-1}^{(p)} + \pmb{B}_2^{(p)} \pmb{y}_ {t-2}^{(p)} + \pmb{\varepsilon}_t^{(p)} \]

Level 2: \[ \begin{aligned} \pmb{\beta}_ {lij}^{(p)} &= b_{lij} + u^{(p)}_{lij} \\ u^{(p)}_{lij} &\sim N(0, \sigma_{lij}) \end{aligned} \]

C = cheerful, E = pleasant event, W = worry, F = fearful, S = sad and R = relaxed

^{Bringmann, L. F., Vissers, N., Wichers, M., Geschwind, N., Kuppens, P., Peeters, F., … & Tuerlinckx, F. (2013). A network approach to psychopathology: new insights into clinical longitudinal data. PloS one, 8(4), e60188.}

• Multi level VAR only possible up to 6 nodes
• For more nodes, stepwise estimation:
  • For each node:
  • Start with model containing only auto-regression (fixed + random)
  • Remove or re-add edges (fixed + random) as long as it improves AIC/BIC