fractal (version 2.0-4)

FNS: Estimation of the proper embedding dimension for a single-variable time series

Description

Invokes the method of False Nearest Strands (FNS) to estimate the minimal embedding dimension of a multivariate data set.

Usage

FNS(x, dimension=5, tlag=NULL, atol=1,
    image.tol=1, olag=1)

Arguments

x

a vector containing a uniformly-sampled real-valued time series.

atol

fNS statitstic threshold. Default: 1.

dimension

the maximal embedding dimension. Default: 5.

image.tol

an integer defining the so-called iterate tolerance. Nearest neighbor pairs (i,J(i)) are separated in time by a point index span dindex = |i-J(i)|, where J(i) represents the index of the nearest neighbor to point i. If a point near i, say k points away also has a nearest neighbor such that |k - J(k)| = dindex +/- M, where M is the iterate tolerance, then the pair (k, J(k)) is added to the current strand. Typically, M=0 or M=1. If M=0, then the difference in index must be exactly the same for each pair included in the strand. If M=1, the index difference is allowed to be 1 point off from the reference pair. Default: 1.

olag

orbital lag. The number of points along the trajectory (orbit) of the current point that must be exceeded in order for another point in the phase space to be considered a neighbor candidate. This argument is used to help attenuate temporal correlation in the the embedding which can lead to spuriously low minimal embedding dimension estimates. The orbital lag must be positive or zero. Default: 1.

tlag

the time delay between coordinates. Default: the decorrelation time of the autocorrelation function.

Value

an single-dimensional matrix containing the FNS percentage as a function of embedding dimension.

Details

The statistic used for determining a false nearest strand (FNS) is based on a Euclidean tolerance supplied by the user (atol). Let \(S(d)\) be the mean Euclidean distance in the projected \((d+1)\)th coordinate between strand pairs found to be nearest neighbors in embedding dimension $d$. If \(S(d) / A > \mbox{atol}\), where \(A\) is the estimated attractor size, then the strand is considered to be a false strand. \(A\) is typically calculated to be the sample standard deviation of the original time series. The \(S(d)\) statistic is a measure of the average additional Euclidean distance we gain by embedding the strand in the next dimension, and is used to assess when this extra distance has grown too large, indicating a false strand.

References

M. B. Kennel and Henry D.I. Abarbanel (2002), False neighbors and false strands: A reliable minimum embedding dimension algorithm, Physical Review E, 66, 026209, 1--19.

M. B. Kennel, R. Brown, and H. D. I. Abarbanel (1992), Determining embedding dimension for phase-space reconstruction using a geometrical construction, Physical Review A, 45(6), 3403--3411.

Fredkin, D. R., and Rice, J. A. (1995), Method of false nearest neighbors: A cautionary note, Physical Review E, 51(4), 2950--2954.

See Also

FNN, embedSeries, infoDim, corrDim, timeLag, determinism.

Examples

# NOT RUN {
## perform False Nearest Strands tests on chaotic 
## beam data for embedding dimensions 1 through 
## 10, using a time delay embedding with a time 
## lag of 10 and an orbital lag of 15 
x <- FNS(beamchaos, dim=10, tlag=10, olag=15)

## print the results 
print(x)
# }