Circumplex Models with Multivariate Time Series: An Idiographic Approach

Abstract

The circumplex model posits a circular representation of affect and some personality traits. There is an increasing need to examine the viability of the circumplex model with multivariate time series data collected on the same individuals due to the development of new data collection methods such as smartphone applications and wearable sensors. Estimating the circumplex model with time series data is more complex than with cross-sectional data because scores at nearby time points tend to be correlated. We adapt Browne’s circumplex model to accommodate time series data. We illustrate the proposed method with an empirical data set of daily affect ratings of an individual over 70 days. We conducted a simulation study to explore the statistical properties of the proposed method. The results show that the method provides more satisfactory confidence intervals and test statistics than a method that treats time series data as if they were cross-sectional data.

Keywords:

• Circumplex model
• multivariate time series
• time series

Notes

1 An idiographic approach is defined to “involve the thorough, intensive study of a single person or case in order to obtain an in-depth understanding of that person or case, as contrasted with a study of the universal aspects of groups of people or cases.” (APA Dictionary of Psychology, n.d.)

2 Molenaar (2004) defined ergodic process as “a process in which the structures of intraindividual variation and interindividual variation are (asymptotically) equivalent.”

3 Because one variable is chosen as the reference variable, its angle is fixed as $0^{°}.$ Thus, the model involves only p − 1 angles. Because ${\theta }_j-{\theta }_i=0$ implies a correlation of 1, ${\beta }_0+{\sum }_{i=1}^m{\beta }_i=1.$ We can compute β₀ from other weights.

4 We present a sketch of the proof for the adaptation in Appendix B.

5 Details of the derivatives were described by Lee and Zhang (2022).

6 We present a sketch of the proof for the adaptation in Appendix B.

7 We thank David Watson for sharing the data.

8 Watson et al. (1999, p. 824) originally designed the 60 items to measure 8 affects, but “disengagement” was not assessed in the within-subject situations. Indicators of high positive affect are enthusiastic, interested, determined, excited, inspired, alert, active, strong, proud, and attentive; indicators of high negative affect are scared, afraid, upset, distressed, jittery, nervous, ashamed, guilty, irritable, and hostile; indicators of low positive affect are sleepy, tired, sluggish, and drowsy; indicators of low negative affect are calm, relaxed, and at ease; indicators of pleasantness are happy, joyful, cheerful, and delighted; indicators of unpleasantness are sad, blue, downhearted, alone, and lonely; and indicators of engagement are surprised, amazed, and astonished.

9 The appendix contains R code for the illustration.

10 We present common score correlations (${\mathit{P}}_c$) of both models in an online support file (Figures A1 and A2).

11 We assume that the time series is weakly stationary (Brockwell & Davis, 1991, Definition (1.3.3)). Thus, the within-subject correlations are invariant across different time points. More sophisticated methods (Hamilton, 2010) are needed if the stationarity assumption seems inappropriate. The vector AR process is a simple way to simulate a stationary time series. Because the proposed method is valid for any stationary process, we use the simulation study to confirm a theoretical expectation. We expect that the general results will hold if we simulate stationary time series with other methods (e.g. more complex AR weight matrices, higher AR orders, with a moving average process).