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3.3.1 Comovement Coefficient

Stefan Leigh

Statistical Engineering Division, ITL

S. Perlman

National Security Agency

Andrew Rukhin

University of Maryland and Statistical Engineering Division, ITL

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

\begin{displaymath}cm(u,v) \ = \
\frac{\sum \Delta u \cdot \Delta v}
{(\sum (\Delta u)^{2} \cdot \sum (\Delta v)^{2})^{\frac{1}{2}}}
\end{displaymath}

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, translation-invariance, 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 of the two individual sequences, followed by innovations bootstrapping of the models in parallel, recomputing the comovement at each iteration of the bootstrap. Direct closed-form asymptotic results for the first and second moments of the comovement computed between low-order MA or AR processes have been obtained. For two AR(1) processes of the form

\begin{displaymath}X_t =\phi_1 X_{t-1} + \epsilon_t^{(1)},\\
Y_t =\phi_2 Y_{t-1} + \epsilon_t^{(2)},
\end{displaymath}

with zero-mean i.i.d. random error vectors, with arbitrary covariance structure, it can be shown that the limiting distribution is Gaussian, with mean

\begin{displaymath}\gamma = \frac
{\rho(2-\phi_1-\phi_2)\sqrt{(1+\phi_1)(1+\phi_2)}}
{2(1-\phi_1\phi_2)}
\end{displaymath}

and a 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.




\begin{figure}
\epsfig{file=/proj/sedshare/panelbk/97/data/projects/inf/step.ps,width=6.0in}\end{figure}

Figure 22: The limit of the sample comovement coefficient for two AR(1) processes.



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Date created: 7/20/2001
Last updated: 7/20/2001
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