SILVar: Single Index Latent Variable Models

How real is this relationship? This is a ubiquitous question that presents itself not only in judging interpersonal connections but also in evaluating correlations and causality throughout science and engineering. Two reasons for reaching incorrect conclusions based on observed relationships in collected data are chance and outside influences. For example, we can flip two coins that both show heads, or observe that today's temperature measurements on the west coast of the continental USA seem to correlate with tomorrow's on the east coast throughout the year. In the first case, we might not immediately conclude that coins are related, since the number of flips we observe is not very large relative to the possible variance of the process, and the apparent link we observed is up to chance. In the second case, we still may hesitate to use west coast weather to understand and predict east coast weather, since in reality both are closely following a seasonal trend. Establishing interpretable relationships between entities while mitigating the effects of chance can be achieved via sparse optimization methods, such as regression (Lasso) [1] and inverse covariance estimation [2]. In addition, the extension to time series via vector autoregression [3], [4] yields interpretations related to Granger causality [5]. In each of these settings, estimated nonzero values correspond to actual relations, while zeros correspond to absence of relations.