Visualizing the Relationship between Two Time Series by Hierarchical Smoothing Models

Abstract

Let x(t_i), y(t_i) be two time series such that y(t_i) = μ(t_i, x) + ε_i, where μ is a smooth function and ε_i is a zero mean stationary process. Which model may be assumed for μ depends on the subject specific context. This article was motivated by questions raised in the context of musical performance theory. The general problem is to understand the relationship between the symbolic structure of a music score and its performance. Musical structure typically consists of a hierarchy of global and local structures. This motivates the definition of hierarchical smoothing models (or HISMOOTH models) that are characterized by a hierarchy of bandwidths b ₁ > b ₂ > … > b_M and a vector of coefficients β ∈ R^M. The expected value μ(t_i x) = Ey(t_i)‖x] is equal to a weighted sum of smoothed versions of x. The “errors” ε_i are modeled by a Gaussian process that may exhibit long memory. More generally, we may observe a collection of time series y_r (r = 1, …, N) that are related to a common time series x by y_r(t_i) = μ _r(t_i, x) + ε_{r, i} where ε _r are independent error processes. For repeated time series, HISMOOTH models lead to a visual and formal classification into clusters that can be interpreted in terms of the relationship to x. An analysis of tempo curves from 28 performances of Schumann's “Träumerei” op. 15/7 illustrates the method. In particular, similarities and differences of “melodic styles” can be identified.