# embedding

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Related to Embedding dimension: Correlation dimension

## embedding

[em′bed·iŋ]
(mathematics)
An injective homomorphism between two algebraic systems of the same type.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## embedding

(mathematics)
One instance of some mathematical object contained with in another instance, e.g. a group which is a subgroup.

## embedding

(theory)
(domain theory) A complete partial order F in [X -> Y] is an embedding if

(1) For all x1, x2 in X, x1 <= x2 <=> F x1 <= F x2 and

(2) For all y in Y, x | F x <= y is directed.

("<=" is written in LaTeX as \sqsubseteq).
References in periodicals archive ?
Given that system (1) is quadratic, an embedding dimension of 5 should be necessary to reconstruct this process.(11) That is, embedding dimensions greater than 5 should not increase the correlation dimension.
As mentioned above, the embedding dimension for the pulse waves is 4.
where t = 1, 2, ..., M; M = N - (m - 1)[tau]; and m and [[tau].sub.d] are the embedding dimension and delay time, respectively.
Given a one dimensional time series, x(t) = [[x.sub.1](t), [x.sub.2](t), ..., [x.sub.n](t)] for an appropriate embedding dimension m and embedding time delay [tau], the time series x(t) can be transform to the m-dimensional space.
Suppose that chaotic time series is {[x.sub.1], [x.sub.2],..., [x.sub.n]}, embedding dimension is m, and time delay is [tau]; then reconstruction phase space is as follows:
Before the phase space reconstruction, we determine the embedding dimension m and delay time [tau].
3 shows the averaged relative running time for computing synchronization indices according to the MCM-PCA and M-SSA algorithms versus the embedding dimension M, for a network of AR time series and a network of coupled chaotic Roessler oscillators of equivalent size.
When the same returns series is randomly scrambled (and thereby destroying the temporal order), one observes that the point estimates of the GP correlation dimension rises rather rapidly with embedding dimensions. This provides us with an indication that there might be some chaotic influences present in the observed returns series.
Before using IMPE, four parameters including the embedding dimension h, the length of signal N, the time delay q, and the scale factor [tau] need to be set.
The embedding dimension and embedding delay characterize the geometry structure of the pixel intensity series.
The reconstructed phase space is shown by Lopez-Mendez and Casas and Takens [28, 29] for the large enough m, which is a homeomorph m (embedding dimension) of the true dynamical system in the generated time series.
where m is the embedding dimension, [tau] is the delay time, and N is the number of points in the reconstructed phase space (N = K - (m - 1)[tau]).

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