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Windowed RQA

In some cases, specially with very long time series, it may be suitable to perform the analysis at different points, considering only a limited window of data around each observation. The macro @windowed modifies the behaviour of the basic functions to calculate RQA parameters in that fashion. For instance, if rmat is a 104×104 recurrence matrix, then

@windowed determinism(rmat, theiler=2, lmin=3) width=1000 step=100

will return a 91-element vector, such that each value is the determinism associated to a 1000-point fragment, starting at every 100 points (i.e. at 1, 101, … 9001).

The general syntax of that macro is:

@windowed expr w                 #1
@windowed expr width=w step=s    #2

where:

  • expr is an expression used to calculate RQA parameters
  • w is the width of the window for relevant data around each point.
  • s is the step or distance between points where the calculations are done (starting in the first point).

To prevent syntax failures in the expansion of the macro, identify the RQA function (rqa, recurrencerate, determinism,...) directly by its name (avoid aliases), and use simple variable names (not complex expressions) for the arguments. On the other hand, the windowing options width and step can be given in any order. If step is ommitted, the calculations are done at every point, and the keyword width may be ommitted. (However, using step=1 may be computationally very expensive, and that will provide just overly redundant results around each point, so it is advisable to set step a relatively big fraction of the window width.)

The value returned by the macro will normally be a vector with the same type of numbers as expected by expr. In the case of @windowed rqa(...) ..., it will return a NamedTuple with a similar structure as in the default rqa function, but replacing scalar values by vectors.

The macro @windowed can also be applied to the functions that calculate recurrence matrices (RecurrenceMatrix, CrossRecurrenceMatrix, JointRecurrenceMatrix). That creates a sparse matrix with the same size as if the macro was not used, but only containing valid values for pairs of points that belong to the w first main diagonals (i.e. the separation in time from one point to the other is w or smaller). The ‘step’ parameter s has no effect on those functions. Such ‘windowed’ matrices can be used as the input arguments to calculate windowed RQA parameters, obtaining the same results as if the complete matrix was used (under certain conditions, see below). For instance, the following calculations are equivalent:

# Using complete matrix
rmat = RecurrenceMatrix(x, 1.5)
d = @windowed determinism(rmat) width=1000 step=250

# Using windowed matrix
rmatw = @windowed RecurrenceMatrix(x, 1.5) 1000
d = @windowed determinism(rmatw) width=1000 step=250

The main difference between the two alternatives is that the second one will be faster and consume less memory. To ensure the equivalence between both approaches, the window width used to create the matrix must be greater than the one used to calculate the RQA parameters. Otherwise, the computation of RQA parameters might involve data points whose value is not well defined. Besides, the threshold to identify recurrences should be referred to a fixed scale. For instance:

rmat  =           RecurrenceMatrix(x, 0.1, scale=maximum)
rmatw = @windowed RecurrenceMatrix(x, 0.1, scale=maximum) 1000
rmat[1:1000,1:1000] == rmatw[1:1000,1:1000] # FALSE!!!

In this example, the 1000×1000 blocks of both matrices differ, because the threshold 0.1 is scaled with respect to the maximum distance between all points of x in rmat, but in the case of rmatw the scale changes between subsets of points. Something similar may happen if the recurrence matrix is calculated for a fixed recurrence rate (with the option fixedrate=true).

Docstring

#RecurrenceAnalysis.@windowed — Macro.

@windowed(f(x,...), width)
@windowed(f(x,...); width, step=1)

Calculate windowed RQA parameters with a given window width.

f(x,...) may be any call to RQA functions (e.g. recurrencerate, determinism, etc.), with x being a named variable that designates the recurrence matrix (do not use in-place calculations of the recurrence matrix). The results are returned in a vector with one value for each position of the window. By default the window moves at one-point intervals, but a longer step length may be specified, together with the window width, by declaring those options as keyword arguments.

This macro may be also used with recurrence matrix constructors (RecurrenceMatrix, CrossRecurrenceMatrix, JointRecurrenceMatrix), to create 'incomplete' matrices that are suitable for such windowed RQA. The values of the resulting matrix in the diagonals within the window width will be equal to those obtained without the @windowed macro, if the distances are not scaled (using the option scale=1, see RecurrenceMatrix). Outside the window width, the values of the recurrence matrix will be undefined (mostly zero).

Alternative syntax for @windowed

The following ways of using the macro @windowed are equivalent:

y = @windowed f(x,...) w
@windowed y=f(x,...) w
y = @windowed(f(x,...), w)
@windowed(y=f(x,...), w)

In all four cases, the width parameter w might have been qualified with a keyword as width=w. If the step parameter is added, the keyword qualification is mandatory.