Different investigators have been using dissimilar values of embedding dimensions for reconstructing stat-space of normal human walking gaits such as some researchers have selected low embedding dimensions as: m = 3 [1], m = 6 [2], and m = 5 [3]. While ref. [4] used reasonably large embedding dimension; m = 10 for reconstructing state-space.
[1] B. A. Smith, N. Stergiou, and B. D. Ulrich, "Lyapunov exponent and surrogation analysis of patterns of variability: profiles in new walkers with and without down syndrome," Motor control, vol. 14, pp. 126.
[2] J. A. Nessler, C. J. De Leone, and S. Gilliland, "Nonlinear time series analysis of knee and ankle kinematics during side by side treadmill walking," Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 19, pp. 02610401-02610411, 2009.
[3] J. B. Dingwell and J. P. Cusumano, "Nonlinear time series analysis of normal and pathological human walking," Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 10, pp. 848-863, 2000.
[4] M. Perc, "The dynamics of human gait," European journal of physics, vol. 26, pp. 525-534, 2005.
Not my strongest area and I should probably research a question before answering, but... I would think that there is no 'correct' answer; it depends on the goal. In machine learning I would optimize the embedding space based on some prediction/performance criterion. More generally, I might ask which mapping produces dimensions that can be related to physical/biological concepts, and does this relationship help push the research forward by suggesting new hypotheses? Of course the 'it depends' answer is almost always right and almost always useless :)
If your data come from a system whose dimension is known, such as m coupled ordinary differential equations, the Takens condition ensures that an embedding of 2m+1 is sufficient. Sauer relaxed that condition slightly under appropriate conditions. However, for real-world data where you do not have a good model and only limited data, there is no one correct answer. In the early days, the method of choice was to calculate the correlation dimension in various embeddings and look for a saturation in its value as the embedding dimension increases. However, a saturation will always occur when you no longer have enough data to adequately fill your high-dimensional space. More recently the method of choice has been false nearest neighbors, although that suffers from the same problem when the neighborhood does not contain sufficiently many points. As a rule of thumb, you might demand that each dimension have at least ten points, so that if you only have a million data points, it is senseless to embed in a dimension higher than about six. An approach I worked on a few years ago is to model the data with an artificial neural network, and then to look at the sensitivity of each data point to its preceding points using the model equations. Freeware for doing this is at http://sprott.physics.wisc.edu/chaos/maus/lagspace.htm. Good luck!
In addition to what has been said already, let me remind you that, in practice i.e. when using data that are limited in length and quality, the embedding dimension might depend on other factors such as the sampling and delay times and also which variable is used. I found the following recent study quite interesting: Journal of Complex Systems 2015 10.1155/2015/932750.
As a final remark, the best embedding dimension will also depend slightly on the intended aplication and, in many cases, the choice of a precise value is not critical.
my favorite test is the False Nearest Neighbor. See Abarbanel's nice book if you are unfamiliar with this simple and revealing test. Extremely powerful. If you really span the attractor, you will know it. Conversely, if you didn't you will know it too!
I agree that the False Nearest Neighbors method is helpful for determining the embedding dimension, as well as the saturation of the slope of the correlation integral - provided that the time series are sufficiently long in both cases. You can check and compare these methods using the freely available software package Dataplore (http://dataplore.com). It also contains further methods of nonlinear analysis, like mutual information, recurrence plots, or Lyapunov exponents.
Hi all. In fact for human data there is not "one correct answer" on the exact number and method to calculate embedding dimensions. Typically, in biological systems, the number of embedding dimensions stay between 10 and maximum of 20, no higher.
False Nearest Neighbors is probably the most common method to make such a decision. You should take in mind that this method aims to find an optimal number of embedding dimensions such that there is no gain in adding more dimensions.However, I consider that your call should also take in consideration other parameters such as delay, sampling rate and length of the time series.
Maybe it might be interesting for you to consider the extensive body of work by the Cincinnati group (e.g., M.A. Riley, R. Balasubramaniam, M.T. Turvey, Recurrence quantification analysis of postural fluctuations, Gait and Posture 9 (1999) 65–78). There is also an interesting book you might consider: Riley, M. A., & Van Orden, G. C. (2005). Tutorials in contemporary nonlinea methods for the behavioral sciences.
Visual Recurrence Analysis implements false nearest neighbors method for the estimation of embedding dimension in a time series. The software link is: https://visual-recurrence-analysis.software.informer.com/5.0/
Dear Sajid,
False Nearest Neighbors (FNN) is one of the essential methods used in estimating the minimally sufficient embedding dimension in delay-coordinate embedding concept of deterministic time series. It can identify random time series as not having a finite embedding dimension and provide information about the deterministic part of correlated stochastic processes.
Refer site for further researching
Article Reliable Estimation of Minimum Embedding Dimension Through S...
Ashish
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