
3.3.3 Comovement Coefficient
Stefan Leigh Statistical Engineering Division
S. Perlman National Security Agency
Andrew Rukhin University of Maryland/Statistical Engineering Division
Time series problems encountered in numerous scientific
disciplines  engineering, geophysical, biological,
economic etc.  often involve the matching of two sequences
for common geometric features, implying some causal or
other relationship.
Often the matching in the
scientific literature
is done numerically
by the computation of a correlation
coefficient. While informative to some degree, the correlation
does not quantify features that the human eye readily detects
as indicative of ``comovement.'' A statistic
close to the correlation of derivatives (first differences) is proposed as a comovement coefficient. The statistic is much more relevant to comovement assay, and yet as a normalized inner product retains many of the desirable properties of the classic correlation: symmetry, translationinvariance, positive homogeneity, and so forth.
In order to estimate sampling moments/distribution of the
comovement between two arbitrary time sequences, a procedure
was originally proposed involving ARMA modeling the two
individual sequences, followed by innovations bootstrapping
of the models in parallel, recomputing the comovement at
each iteration of the bootstrap.
Direct closedform asymptotic results for the first
and second moments of the comovement computed between loworder
MA or AR processes have been obtained.
For two AR(1) processes
of the form
with zeromean i.i.d. random error vectors, with arbitrary covariance structure, it can be shown that the limiting distribution is Gaussian. More precisely and limiting variance that can be explicitly calculated. These new results, of utility and interest on their own merits, lead to modifications of the original resampling specification. We were originally introduced to this problem during a review discussion of surface profile matching in a tribology application here at NIST.
Figure 20: The limit of the sample comovement coefficient for two AR(1) processes.
Date created: 7/20/2001 