TY - JOUR
T1 - Visualizing the relationship between two time series by hierarchical smoothing models?
AU - Beran, Jan
AU - Mazzola, Guerino
PY - 1999/6
Y1 - 1999/6
N2 - Let x(ti), y(ti) be two time series such that y(ti) = μ(ti, x) + εi, where μ is a smooth function and εi 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 > … > bM and a vector of coefficients β ∈ RM. The expected value μ(ti x) = E[y(ti)‖x] is equal to a weighted sum of smoothed versions of x. The “errors” εi are modeled by a Gaussian process that may exhibit long memory. More generally, we may observe a collection of time series yr (r = 1, …, N) that are related to a common time series x by yr(ti) = μ r(ti, x) + εr, 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.
AB - Let x(ti), y(ti) be two time series such that y(ti) = μ(ti, x) + εi, where μ is a smooth function and εi 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 > … > bM and a vector of coefficients β ∈ RM. The expected value μ(ti x) = E[y(ti)‖x] is equal to a weighted sum of smoothed versions of x. The “errors” εi are modeled by a Gaussian process that may exhibit long memory. More generally, we may observe a collection of time series yr (r = 1, …, N) that are related to a common time series x by yr(ti) = μ r(ti, x) + εr, 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.
KW - Fractional autoregressive model
KW - Hierarchical smoothing
KW - Kernel smoothing
KW - Long-range dependence
KW - Musical performance
KW - Musical structure
KW - Time series
UR - http://www.scopus.com/inward/record.url?scp=0033433090&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0033433090&partnerID=8YFLogxK
U2 - 10.1080/10618600.1999.10474811
DO - 10.1080/10618600.1999.10474811
M3 - Article
AN - SCOPUS:0033433090
VL - 8
SP - 213
EP - 238
JO - Journal of Computational and Graphical Statistics
JF - Journal of Computational and Graphical Statistics
SN - 1061-8600
IS - 2
ER -