The optimization can be applied both to uni-dimensional
time-series (method = crqa), or multi-dimensional (method = mdcrqa)
The procedure is identical in both cases:
1) Identify a delay that accommodates both time-series
by finding the local minimum where mutual information
between them drops, and starts to level off.
When one ts has a considerably longer delay indicated
than the another, the function selects the longer delay
of the two to ensure that new information is gained for
both. When the delays are close to each other, the
function computes the mean of the two delays.
2) Determine embedding dimensions by using false nearest
neighbors and checking when it bottoms out (i.e., there
is no gain in adding more dimensions). If the embedding
dimension for the two ts are different the algorithm
selects the higher embedding dimension of the two to
make sure that both time series are sufficiently unfolded.
3) Determine radius yielding a recurrence rate between 2-5
To do so, we first determine a starting radius that yields
approximately 25
We generate a sampled sequence of equally spaced possible
radi from such radius till 0, using as unit for the sequence
step, the standard deviation of the distance matrix divided
by a scaling parameter (radiusspan). The larger this parameter,
the finer the unit.
For uni-dimensional time-series, the user has to decide how to
choose the value of average mutual information (i.e., typeami = mindip,
the lag at which minimal information is observed, or typeami = maxlag,
the maximum lag at which minimal information is observed) and the relative
percentage of information gained in FNN, relative to the first embedding
dimension, when higher embeddings are considered (i.e., fnnpercent).
Then, as crqa is integrated in the optimizeParam to estimate the
radius, most of the arguments are the same (e.g., mindiagline
or tw).
For multidimensional series, the user needs to specify the right RQA method
(i.e., method = "mdcrqa"). Then, for the estimation of the delay via
AMI: (1) nbins the number of bins to compute the two-dimensional
histogram of the original and delayed time series and (2) the criterion
to select the delay (firstBelow to use the lowest delay at which the AMI
function drops below the value set by the threshold argument, and
localMin to use the position of the first local AMI minimum).
The estimation of the embedding dimensions instead needs the following arguments:
(1) maxEmb, which is the maximum number of embedding dimensions considered,
(2) noSamples, which is the number of randomly drawn coordinates
from phase-space used to estimate the percentage of false-nearest neighbors,
(3) Rtol, which is the first distance criterion for separating false
neighbors, and (4) Atol, which is the second distance criterion for
separating false neighbors. The radius is estimated as before.