This constructs a series of vectors with overlapping entries called an embedding, with M called the
embedding dimension. If the "true" (but unknown) dimension of the system that produced the observations is n, if M [greater than or equal to] 2n + 1, then the M-histories can recreate the dynamics of the underlying system.
Regarding the calculation of LLE, the indicator of trajectory instability of the chaotic time series data, the optimal choice of the
embedding dimension is 4 [9, 10].
When reconstructing the attractor from the scalar time series, the most critical issue is the selection of the delay time and
embedding dimension [20].
The
embedding dimension d is computed by false nearest neighbors [22].
wherein d is the
embedding dimension, N is the number of data points, r is the embedding delay, [x.sub.i+d[tau]] and [x.sub.n(i d)+d[tau]] are the ith vector in the data sets and its nearest neighbors of d-dimensional phase space [18].
Therefore, we determined the
embedding dimension of EEG signals by using the false nearest neighbors algorithm [31].
Single variable time series can be reconstructed into a phase space by Takens' embedding theorem in phase space reconstruction [24, 25]; that is, the original dynamical system can be restored in the sense of topological equivalence as long as the
embedding dimension is sufficiently high.
Each channel can be reconstructed to an M-dimensional phase space by selecting the
embedding dimension M and time delay [tau].
When plotted against m, the number of
embedding dimensions, the point estimate of v (i.e., [v.sub.m]) will converge to a constant beyond a certain m if the data series is indeed chaotic.
Given a time series Y = {[y.sub.1] ,[y.sub.2], ...,[y.sub.N]} with the length of N, h dimensional delay embedding vector at the moment t can be constructed as [Y.sup.h,q.sub.t] = {[y.sub.t], [y.sub.t+q], ..., [y.sub.t+(h-1)q]} (t = 1, 2, ..., N - (h - l)q), where h represents the
embedding dimension and q is the time delay.
The underlying idea is that, for a sufficiently large
embedding dimension m and embedding delay [tau], the vector x' (t) = [[x.sub.i] (t), [x.sub.i+[tau]](t), ..., [x.sub.i+m[tau]](t)] performs the same functions as the original variables of the system.
Paladin and Vulpiani [23] presented the embedding trajectory dimension, which was similar to reconstruct the
embedding dimension of the phase spaces of the dynamic system.
