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Computational Models for Affect Dynamics

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Affect Dynamics

Abstract

Computational models of affect dynamics are ubiquitous. These models are appropriate for either exploring intensive longitudinal data or testing theories about affect dynamics. In this chapter, we give a brief overview of some of the computational models that have been applied in the field of affect dynamics, focusing on both discrete-time and continuous-time models. The emphasis of this chapter lies on describing the core ideas of the models and how they can be interpreted. At the end, we provide references to other important topics for the interested reader.

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Notes

  1. 1.

    Because of this, we present network models in the section on autoregressive models. This was a practical choice, and we do not mean to imply that network models are always autoregressive in nature. In fact, most networks are not (e.g., Ising models; Kruis & Maris, 2016).

  2. 2.

    Note that we specifically talk about reinforcement learning in the context of emotion dynamics: This class of model is applicable to many more subjects, like decision-making, conditioning, and learned behavior (see Sutton & Barto, 2018).

  3. 3.

    To remain in line with the mathematical notations of this chapter, we changed the notation of this model (see Rutledge et al., 2014).

  4. 4.

    Importantly, this implies that vector fields are deterministic—they show what the model would expect if there were no perturbations to the system.

  5. 5.

    In the application that we do not discuss, the OU model was only part of a series of equations (see Pellert et al., 2020; Schweitzer & Garcia, 2010).

  6. 6.

    The damped linear oscillator is an example of a second-order differential equation, where speed and location of a variable y at time t are both related to changes in speed over time (i.e., acceleration; speeding up or slowing down over time).

  7. 7.

    If this is the case, we are able to predict the emotional behavior of an individual for eternity, as the system is deterministic.

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Author notes

  1. Niels Vanhasbroeck and Sigert Ariens contributed equally to this work.

Authors and Affiliations

  1. KU Leuven, Leuven, Belgium

    Niels Vanhasbroeck, Sigert Ariens, Francis Tuerlinckx & Tim Loossens

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  1. Niels Vanhasbroeck
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Correspondence to Niels Vanhasbroeck .

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  1. Wake Forest University, Winston Salem, NC, USA

    Christian E. Waugh

  2. KU Leuven, Leuven, Belgium

    Peter Kuppens

Appendices

Appendix 1: Properties of the VAR

1.1 Properties of the AR Model

Given the AR model (repeated here):

$$ {y}_j=\delta +\varphi {y}_{j-1}+{\varepsilon}_j. $$

we can define some properties of the process. These properties are defined and mathematically derived below.

Predictions.Given a first observation y0 collected at time t0, we are able to predict the next measurement y1, as:

$$ {\displaystyle \begin{array}{cc}\left\langle {y}_1\right.\left|{y}_0\right\rangle & =\delta +\varphi {y}_0+{\varepsilon}_1\\ {}& =\delta +\varphi {y}_0\end{array}} $$

where denotes the time-dependent expected value (i.e., E[.]). Note that the innovations do not play a role in the expectation of y1, given that their expected value is equal to 0.

Using the same principle, we can also make predictions about observations further in the future. For instance, the expectation of y2 conditional on the observation y0 is given by:

$$ {\displaystyle \begin{array}{cc}\left\langle {y}_2\right.\left|{y}_0\right\rangle & =\delta +\varphi {y}_1\\ {}& =\delta +\varphi \left(\delta +\varphi {y}_0\right)\\ {}& =\left(1+\varphi \right)\delta +{\varphi}^2{y}_0\end{array}} $$

In general, the prediction of a future observation yj conditional on y0 is:

$$ \left\langle {y}_j\right.\left|{y}_0\right\rangle =\left(\sum \limits_{k=0}^{j-1}{\varphi}^k\right)\delta +{\varphi}^j{y}_0 $$
(10.7)

Baseline. Since the magnitude of φj shrinks as j increases, in the long-time limit, it holds that:

$$ li{m}_{j\to \infty}\left(\sum \limits_{j-1}^{k=0}{\varphi}^k\right)=\frac{1}{1-\varphi} $$

As a result, the predictions ⟨yj| y0⟩ converge towards a fixed point:

$$ {\displaystyle \begin{array}{cc} li{m}_{j\to \infty}\left(\left\langle {y}_1\right.\left|{y}_0\right\rangle \right)& = li{m}_{j\to \infty}\left(\left(\sum \limits_{k=0}^{j-1}{\varphi}^k\right)\delta +{\varphi}^j{y}_0\right)\\ {}& =\frac{1}{1-\varphi}\delta +0\\ {}& =\frac{\delta}{1-\varphi}\\ {}& =\mu \end{array}} $$
(10.8)

This fixed point μ can be considered the emotional baseline (i.e., the dotted line in Fig. 10.1) and represents the emotional state to which the emotional state is expected to evolve to. As such coincides with the end state of a regulation process (provided nothing happens to disrupt the regulation process).

It also represents the state that will be visited the most by the individual over longer periods of time. For that reason, it coincides with the mean of the distribution of observations {yj | j,…,N} for sufficiently large N. Because of this, the baseline is also referred to as the stationary mean. The term stationary is used to stress that the baseline is independent of time.

When an AR(1) process is only observed during a short period of time during which the emotional state is still relaxing (i.e., converging) towards the baseline, then the mean of the observations will differ from the stationary mean. Only when measurements have been collected for a sufficiently long period of time will the mean of the data distribution coincide with the stationary mean.

Uncertainty. Until now, we were only concerned with point-predictions of future observations. However, we can also compute the uncertainty that is associated with these predictions. For this, we realize that the observation y1 is normally distributed with mean δ + φy0(the prediction) and variance σε2:

$$ {y}_1\mid {y}_0\sim N\left(\delta +\varphi {y}_0,{\sigma}_{\varepsilon}^2\right) $$

Because of stochasticity, uncertainty about predictions typically grows the further in the future you go. It can be shown that the future observation yj, given observation y0, is normally distributed with the mean being the point-prediction in Eq. (10.7) and variance given by:

$$ {\sigma}_j^2=\sum \limits_{j-1}^{k=0}{\varphi}^{2k}{\sigma}_{\varepsilon}^2 $$
(10.9)

where in the long-time limit:

$$ li{m}_{j\to \infty}\left(\sum \limits_{j-1}^{k=0}{\varphi}^{2k}\right)=\frac{1}{1-{\varphi}^2} $$

so that the variance of the uncertainty distribution in the long-time limit converges to

$$ {\sigma}^2=\frac{\sigma_{\varepsilon}^2}{1-{\varphi}^2} $$
(10.10)

Like the stationary mean, this variance is time-independent and thus called the stationary variance.

Autocovariance. An AR model relies on the assumption that measurements yj at time tj are related to measurements yj − 1 at time tj − 1, i.e. that there is a time-dependence between measurements. The extent to which this relationship holds is expressed by the autocovariance. The autocovariance at lag-p σp is defined as:

$$ {\sigma}_p=\left\langle \left({y}_{j+p}-\mu \right)\right.\left.\left({y}_j-\mu \right)\right\rangle $$

To compute the autocovariance of the AR process, we first reformulate the model in terms of the baseline μ. To do so, we substitute δ for (1 − φ)μ (see Eq. (10.8)) to obtain:

$$ {\displaystyle \begin{array}{c}{y}_j=\left(1-\varphi \right)\mu +\varphi {y}_{j-1}+{\varepsilon}_j\\ {}=\mu -\varphi \mu +\varphi {y}_{j-1}+{\varepsilon}_j\end{array}} $$

Then, by rearranging the terms, we can write

$$ {y}_j-\mu =\varphi \left({y}_{j-1}-\mu \right)+{\varepsilon}_j. $$

Setting the innovations to zero (they do not correlate with anything), we find (see Eq. (10.7))

$$ {\displaystyle \begin{array}{cc}{\sigma}_p& =\left\langle \left({y}_{j+p}-\mu \right)\right.\left.\left({y}_j-\mu \right)\right\rangle \left(\mathrm{Def}.\mathrm{autocovariance}\right)\\ {}& =\left\langle {\varphi}^p\right.\left({y}_j-\mu \right)\left.\left({y}_j-\mu \right)\right\rangle \left(\mathrm{Generalization}\kern0.17em \mathrm{previous}\kern0.17em \mathrm{property}\right)\\ {}& =\left\langle {\varphi}^p{\left({y}_j-\mu \right)}^2\right\rangle \\ {}& ={\varphi}^p{\sigma}^2\end{array}} $$
(10.11)

Here we have used the fact that the centered variable yj − μ have the same stationary variance (Eq. (10.10)) as the variable yj themselves. If we standardize the measurements so that σ2 = 1, we obtain the autocorrelation:

$$ \rho (p)={\varphi}^p $$

From this expression, it can be seen that the autoregressive coefficient φ of the AR model corresponds to the autocorrelation between measurements at lag 1.

1.2 Properties of the VAR Model

We can generalize the properties of the AR model to fit the d-dimensional VAR model (repeated here):

$$ {\displaystyle \begin{array}{c}{\boldsymbol{y}}_j=\boldsymbol{\delta} +\varPhi {\boldsymbol{y}}_{j-1}+{\boldsymbol{\varepsilon}}_j\\ {}{\boldsymbol{\varepsilon}}_j\sim N\left(\mathbf{0},{\varSigma}_{\varepsilon}\right)\end{array}} $$

Predictions. Just like for the AR model, the prediction of a future observation conditional on the observation y0 is given by (see Eq. (10.7))

$$ \left\langle {y}_j|{\mathrm{y}}_0\right\rangle =\left(\sum \limits_{k=0}^{j-1}{\varPhi}^k\right)\boldsymbol{\delta} +{\varPhi}^j{\boldsymbol{y}}_0 $$
(10.12)

Importantly, this equation results in a vector that contains all expectation values for all d variables of the model.

Baseline. Using a similar reasoning as for the AR model (see Eq. (10.8)), but this time using matrices instead of scalars, it can be shown that the predictions of the VAR model Eq. (10.12) converge to the baseline:

$$ \boldsymbol{\mu} ={\left({I}_d-\varPhi \right)}^{-1}\boldsymbol{\delta} $$

where Id is the d-dimensional identity matrix.

Uncertainty. An expression similar to Eq. (10.9) can be obtained for the growing uncertainty of the VAR model:

$$ {\varSigma}_j=\sum \limits_{j-1}^{k=0}{\varPhi}^k{\varSigma}_{\varepsilon}{\left({\varPhi}^k\right)}^T $$

For stable transition matrices Φ, this covariance matrix becomes constant in the long-time limit. This stationary covariance is given by

$$ \varSigma =\sum \limits_{\infty}^{k=0}{\varPhi}^k{\varSigma}_{\varepsilon}{\left({\varPhi}^k\right)}^T $$

and is a solution of the discrete-time Lyapunov equation

$$ \varSigma -\varPhi \varSigma {\varPhi}^T={\varSigma}_{\varepsilon} $$

Given the transition matrix Φ and the covariance Σε of the innovations, this Lyapunov equation enables us to compute the stationary covariance without having to compute an infinite sum.

Autocovariance. The autocovariance of the VAR model is similar to the autocovariance of the AR (see Eq. (10.11), namely

$$ {\displaystyle \begin{array}{cc}{\varSigma}_p& =\left\langle \left({y}_{j+p}-\mu \right){\left({y}_j-\mu \right)}^T\right\rangle \\ {}& ={\varPhi}^p\varSigma \end{array}} $$

Appendix 2: Autocorrelation of Bivariate VAR

If we take a bivariate VAR model with the intercepts δ = 0, then we can compute the autocovariance as:

$$ {\displaystyle \begin{array}{cc}{\sigma}_{tt-1}& =\left({y}_{1t}{y}_{1t-1}\right)\\ {}& =\left({\delta}_1+{\varphi}_{11}{y}_{1t-1}+{\varphi}_{12}{y}_{2t-1}+{\varepsilon}_{1t}\right){y}_{1t-1}\\ {}& ={\delta}_1{y}_{1t-1}+{\varphi}_{11}{y}_{1t-1}^2+{\varphi}_{12}{y}_{2t-1}{y}_{1t-1}+{\varepsilon}_{1t}\\ {}& ={\delta}_1{y}_{1t-1}+{\varphi}_{11}{y}_{1t-1}^2+{\varphi}_{12}{y}_{2t-1}{y}_{1t-1}+{\varepsilon}_{1t}\\ {}& =0+{\varphi}_{11}{\sigma}_1^2+{\varphi}_{12}{\sigma}_{12}+0\\ {}& ={\varphi}_{11}{\sigma}_1^2+{\varphi}_{12}{\sigma}_{12}\end{array}} $$

We can compute the autocorrelation as:

$$ {\displaystyle \begin{array}{cc}{\rho}_{tt-1}& =\frac{\sigma_{tt-1}}{\sigma_{1t}{\sigma}_{1t-1}}\\ {}& =\frac{\sigma_{tt-1}}{\sigma_1^2}\\ {}& =\frac{\varphi_{11}{\sigma}_1^2+{\varphi}_{12}{\sigma}_{12}}{\sigma_1^2}\\ {}& ={\varphi}_{11}+\frac{\varphi_{12}{\sigma}_{12}}{\sigma_1^2}\end{array}} $$

More generally, it holds that for a variable yi:

$$ {\displaystyle \begin{array}{cc}{\sigma}_{tt-1}& ={y}_{it}{y}_{it-1}\\ {}& =\left({\delta}_i+{\varphi}_{ii}{y}_{it-1}+\sum \limits_{\begin{array}{c}{i}^{\prime}=1\\ {}{i}^{\prime}\ne i\end{array}}^d{\varphi}_{i{i}^{\prime}}{y}_{i^{\prime}t-1}+{\varepsilon}_{it}\right){y}_{it-1}\\ {}& ={\varphi}_{ii}{\sigma}_i^2+\sum \limits_{\begin{array}{c}{i}^{\prime}=1\\ {}{i}^{\prime}\ne i\end{array}}^d{\varphi}_{i{i}^{\prime}}{\sigma}_{i{i}^{\prime}}\end{array}} $$

and:

$$ {\rho}_{tt-1}={\varphi}_{ii}+\sum \limits_d^{\begin{array}{c}{i}^{\prime}=1\\ {}{i}^{\prime}\ne i\end{array}}\frac{\varphi_{i{i}^{\prime}}{\sigma}_{i{i}^{\prime}}}{\sigma_i^2} $$

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Vanhasbroeck, N., Ariens, S., Tuerlinckx, F., Loossens, T. (2021). Computational Models for Affect Dynamics. In: Waugh, C.E., Kuppens, P. (eds) Affect Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-030-82965-0_10

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