Synchronization analysis

Once the goodness-of-fit for an ELSA model has been established the model’s output can be analyzed for synchronization of emotional responding. In the following sections, we present four types of measures that we developed for this purpose. We categorized these measures according to whether they were partial or complete, and whether they were transient or intransient. We discuss each of the possible combinations but focus mainly on complete transient synchronization, since this corresponds most closely to the type of synchronization that emotion theorists—and the CPM in particular—refer to when speaking of emotional synchronization. We propose a measure epsilon, ε, that quantifies this type of synchronization. In Section 4.2.3, we noted that the concept of emotional synchronization can

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be broken down into two (potentially) independent meanings, which are component correlation and component patterning (Bulteel et al., 2013). Our measure epsilon focuses primarily on the former type, although it has aspects of the latter, due to the involvement of all emotion components simultaneously and the heavily interactional data transform inside ELSA’s liquid layers. Nevertheless, we emphasize that the epsilon measure was not developed to identify patterns of componential responding that are unique to certain emotion episodes. Moreover, the epsilon measure also does not measure attractors or the likelihood of passing one. It is merely assumed that, the higher the value of epsilon, the more likely that ELSA’s activity is passing an attractor trajectory.²⁶

Partial synchronization. One obvious measure of emotional synchronization in ELSA would be the strength of the (Pearson) correlation between the observed and the fitted values—or its squared counterpart—of each response time series. ELSA already outputs these values by default—as part of its fit statistics—and they provide direct information about the strength of the association between different emotion components. For example, if a motivation time series achieves a correlation, r = 0.70, then that series must depend on some combination of appraisal, physiology, or expression time series.²⁷ Observed-fitted correlations can be considered a measure of partial synchronization, since they quantify the fit of only one response at a time (cfr. univariate regression) and do not necessarily imply complete system synchronization.²⁸ Observed-fitted correlations should also be considered a measure of intransient synchronization because they are calculated over a whole time series, without taking into account qualitatively different phases such as baseline activity, emotional episodes, or recovery. A transient version of partial synchronization that we propose is the moving correlation between observed and fitted time series. To calculate this measure, one chooses a certain window size (e.g., 60 time steps) and then computes the observed-fitted correlation for consecutive segments of this size, each time shifting the window for a fixed step (see supplementary material, Figure 4.S6). Usually, the moving window is weighted such that the middle observations impact the correlation measure more strongly (see supplementary material, Figure 4.S6). Weighting is often convenient because it enables one to interpret the moving statistic on the time scale of the original series rather than the window time scale (e.g., per 60 steps).

Complete synchronization. Pearson correlation is a measure of bivariate coherence. For more than two time series, a relatively straightforward index of multivariate coherence can be obtained from principal component analysis (PCA). The more correlated a set of variables, the larger the eigenvalue of the first principal component. This eigenvalue is also called the spectral radius. It is known mathematically that, for a dataset with Q variables, the spectral radius of its covariance matrix can be at most equal to the sum of the total variance. This sets a useful upper limit on the degree of coherence

26 Measuring attractors is a considerably more challenging problem that measuring (nonlinear) correlation.

27 Which components account for more variability can be investigated with a deletion analysis

28 ELSA allows for the “paradox” that one emotion component can be predicted using another but not the other way around.

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in multivariate data. When we divide the spectral radius simply by its maximal possible value, we obtain a (relatively) normalized measure of multivariate association bounded between 0 and 1.²⁹ A value closer to 0 would mean that the multivariate data are less coherent, whereas a value close to 1 would mean that the multivariate data are more coherent. Earlier, we presented a moving version of Pearson correlation. Likewise, one could calculate a moving version of the standardized spectral radius. To do this, we would again choose a window size (e.g., 60 steps), a weighting function (e.g., triangular weighting), and then compute successive spectral radii along a multivariate time series.

Moving measures of association deal with the problem of transience in time series but they do not address the problem of delay. The latter could be solved by calculating lagged cross-correlation (e.g., cor[Xt–s,Yt]), or its moving counterpart. Lagged cross-correlation requires additional choices of the analyst, such as the amount of lag to impose and which time series to lag (e.g., X or Y). This problem becomes even more complicated for a multivariate time series. For a multivariate time series, it is not clear which variables to lag, and for how many steps. Computing the standardized spectral radius for all possible combinations of lags among the variables would be impractical if not impossible for large datasets. Rather than lagging variables manually, it would be more convenient if the time series themselves already encoded information about their past history. This is exactly what a liquid state transformation can accomplish. Our synchronization measure epsilon, ε, is thus defined as:

ε = the weighted moving standardized spectral radius of ELSA’s weighted motivation, physiology,and expression eigenliquid states.

We have already explained the concept of the standardized spectral radius. In addition, we know that ELSA’s eigenliquid states represent a reduced factorization of liquid wavelet-transformed states.

The definition of epsilon also specifies a double weighting. The first weighting refers to the standard window weighting for calculating moving statistics. For epsilon, we use normalized triangular weighting (supplementary material, Figure 4.S6). The second weighting in the epsilon definition refers to predictive weighting of the eigenliquid time series. That is, not all eigenliquid time series will be relevant for predicting the response time series. We can assign more weight to the highly predictive time series by multiplying each with the sum of their absolute coefficients of the penalized regression, thus magnifying their contribution to the final ε estimate.

The epsilon measure can be interpreted as a moving PCA that focuses only the first principal component. This procedure is itself relatively straightforward. What makes the epsilon measure special (and flexible) is that it is calculated on ELSA’s eigenliquid time series, which encode delay (i.e., fading memory), nonlinear transformations, and wavelet transformations. Because epsilon is a moving measure of association, it can quantify transient synchronization. Finally, epsilon is called a

29 See supplementary material for more details on the lower bound of the spectral radius, which we do not discuss here.

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measure of complete synchronization because it is calculated on the combined (i.e., concatenated) eigenliquid time series of the motivation, physiology, and expression submodels. For these reasons, we feel that epsilon captures most fully the concept of emotional synchronization proposed by emotion theorists. Note that the idea of applying a moving PCA to detect multivariate coherence in emotion time series has also been used by Bulteel and colleagues (2013), whose DeCon tool applies robust PCA on a moving window to identify points of change in multivariate time series.

Calculating epsilon requires a few practical choices which we discuss in the supplementary material. Here, we note briefly that we recommend to calculate epsilon on covariance matrices (not correlation matrices) and to calculate it for both training and validation data. Finally, the most important choice for conducting a synchronization analysis—and perhaps the only real control parameter—is the width of the moving window. This choice cannot easily be automated and depends both on reasonable theoretical limits and data-specific knowledge on the expected length of synchronization. In general, the window size will function as a smoothing parameter, with small values producing very noisy synchronization data and large values producing very smooth synchronization data.