Factor Analysis: Evolutionary

Public debates are characterized by a complex structure that is constantly changing. As such, they pose severe challenges to traditional methodologies of analysis: techniques focusing on detecting latent patterns and structures typically require time-invariant data, while dynamic analyses are mostly limited to investigating the behavior of a few known processes over time. The Evolutionary Factor Analysis (EFA) of frames, a technique for analyzing the dynamic structure of high-dimensional semantic network data with time-varying latent structure, allows tracing subtle changes in the latent organization of a debate over time, identifying and describing the main underlying processes.

To understand what EFA is, we need to introduce the main ideas behind factor analysis for time series. The main idea behind a factor model is that a large number N of series can be explained by a small number q of factors. In a factor model, a vector of N observations at time t, say YN(t), is decomposed into a common component XN(t) and an idiosyncratic component ZN(t):

The common components are a linear combination of the latent common factors f(t), with weights given by the so-called factor loadings LN:

In this notation, the matrix LN is N×q, whereas the vector of factors is q×1. Both X and Z are unobservable. The covariance matrix of the observations can be decomposed accordingly into

In public debates, for example, the covariance matrix CX captures the correlations between the semantic concepts and the common factors, whereas the covariance matrix CZ explains the covariation between specific semantic meanings in the debates which cannot be explained by the factors.

Factor models are appealing for two reasons: they are a dimension-reduction tool and, at the same time, a meaningful representation of the principal components of the covariance matrix of the observations. While the first feature (dimension reduction) is common to all factor models, the second property is achieved only when the number of series N is large. The traditional or strict approach assumes that N is finite and that the covariance matrix of the errors CZ is diagonal (for identification, some other conditions need to be imposed on the loadings). Then the parameters in the models can be estimated by Maximum Likelihood. The most recent literature of approximate factor models differs from the traditional one in that the covariance matrix of the idiosyncratic components CZ is allowed to be non-diagonal and the cross-section size N is large. With sufficiently large N, the parameters in the model can be estimated by Principal Components. Allowing for a non-diagonal CZ is important: it means that we allow for non-zero correlation among specific concepts in our debate analysis.

The traditional or stationary approach in factor analysis is based on the assumption that the loadings (which are weighting the factors) are time-invariant. As a consequence, the covariance matrix of the observations does not change over time. However, many empirical applications in [Page 511]communication science show that the process underlying the data may be non-stationary. The EFA considers a non-stationarity, which is explained by smooth evolutions of the dynamics and modeled with parameters which are slowly-varying (or slowly evolving) over time.

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