The dissimilarity between time series x
and y
is given by:
$$ d(x,y) = \Phi[CORT(x,y)] \delta(x,y) $$
where:
CORT(x,y) measures the proximity between the dynamic behaviors of x and y by means of the first order temporal correlation coefficient defined by:
$$ CORT(x,y) = \frac{ \sum_{t=1} (x_{t+1} - x_t) ( y_{t+1} - y_t) }{ \sqrt{ \sum_{t=1} (x_{t+1} - x_t)^2} \sqrt{ \sum_{t=1} (y_{t+1} - y_t)^2 } } $$
\(\Phi[u]\) is an adaptive tuning function taking the form:
$$ \frac{2}{1+e^{ku}} $$ with \(k \geq 0\) so that both \(\Phi\) and k
modulate the weight that CORT(x,y) has on d(x,y).
\(\delta(x,y)\) denotes a dissimilarity measure between the raw values of series x
and y
, such as the Euclidean distance, the Frechet distance or the Dynamic Time Warping distance. Note that \( d(x,y) = \delta(x,y)\) if k=0
.
More details of the procedure can be seen in Chouakria-Douzal and Nagabhushan (2007).
deltamethod
(\(\delta\)) can be either Euclidean (deltamethod = "Euclid"
), Frechet ( deltamethod = "Frechet"
) or Dynamic Time Warping (deltamethod ="DTW"
) distances. When calling from dis.CORT
, DTW uses Manhattan as local distance.