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Let $x_t = F(x_{t-1})$ be a discrete-time dynamical system in the chaotic regime. Starting from an initial condition $x_0$, we can generate a time series $(x_t)$ where $t =1,2,...,T$ indicates the time index.
From this, we can generate symbols as follows: $s_t = 1$ if $F^t(x_0) > c$, and $s_t = 0$ otherwise, with $c$ being the critical point of the one-dimensional map. So each iterate of the map $F$ gives a new symbol. Putting the sequences of 0s and 1s into a vector of symbols, we get $\mathbf{s} = (s_0, s_1, s_2, …).$ (Please refer to this question about the symbolic dynamics of one-dimensional maps for a more mathematical explanation on how symbols are generated.)

Now, suppose we have a two-dimensional system or say, the above univariate (one-dimensional) system is transformed to two-dimensional system using Takens’ delay-embedding technique in the following way: Given a one-dimensional time series $(x_t)$, a delay $\tau$, and some embedding dimension $m$, one considers an embedding map $\mathbf{z}_t = (x_t,x_{t+\tau},...,x_{t+(m-1)\tau}) \in R^m$ which generates the phase-space vectors $\mathbf{z}_1, \mathbf{z}_2, ….$

As an example, let $m=2$, $\tau=1$, let the first co-ordinate obtained be called $x$ and the second coordinate $y$. $(x,y)$ forms the new multidimensional system. My problem is: How do I obtain the symbolic dynamics for this case? Example:

$$x = 0.10, 0.45, 0.60, 0.42, …$$ $$y = 0.00, 0.10, 0.45, 0.60, …$$

Will there be a symbolic sequence for each dimension or will a symbol be assigned to a point $(x,y)$? An explanation will be very helpful to clear the concept.

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Will there be a symbolic sequence for each dimension or will a symbol be assigned to a point $(x,y)$?

This depends what you eventually want to do with your symbol sequence, but for typical applications, such as determining the entropy or modelling, you want to assign one symbol to the point. The general reason behind this is that (for a proper reconstruction), you care about the location in phase space and not individual components – that’s more or less the entire point of the phase-space in general.

The straightforward way to do this is symbolising each component and then making a compound symbol (or “word”). For example, if you have two possible symbols, $0$ and $1$ for each component, then you have four possible compound symbols $(0,0); (0,1); (1,0); (1,1)$. For further reading, see, e.g., Daw et al. – A review of symbolic analysis of experimental data.

Another way is using permutation symbols as suggested in Bandt and Pompe – Permutation Entropy: A Natural Complexity Measure for Time Series. Here you consider the ordering of the components and assign a different symbol to each possible order. From another point of view, you look at which permutation you would have to apply to the components for them to be in ascending order and assign a symbol to each of the possible $m!$ permutations. For example, for a delay embedding with dimension $m=3$, you have six possible symbols, one for each of the following cases:

  1. $x_{t+2τ} > x_{t+τ\hphantom{2}} > x_{t\hphantom{τ+2}}$;
  2. $x_{t+2τ} > x_{t\hphantom{τ+2}} ≥ x_{t+τ\hphantom{2}}$;
  3. $x_{t+τ\hphantom{2}} ≥ x_{t+2τ} > x_{t\hphantom{τ+2}}$;
  4. $x_{t+τ\hphantom{2}} > x_{t\hphantom{τ+2}} ≥ x_{t+2τ}$;
  5. $x_{t\hphantom{τ+2}} ≥ x_{t+2τ} > x_{t+τ\hphantom{2}}$;
  6. $x_{t\hphantom{τ+2}} ≥ x_{t+τ\hphantom{2}} ≥ x_{t+2τ}$.

Using the numbers as symbols, with $τ=1$, and with no overlap between symbols, you would translate the following time series to symbols as follows:

$$ \underbrace{1, 7, 6}_{3}, \underbrace{4, 6, 5}_{3}, \underbrace{3, 0, 3}_{5}, \underbrace{6, 1, 0}_{6}, \underbrace{2, 2, 4}_{6}, \underbrace{7, 3, 5}_{5}, \underbrace{7, 6, 9}_{2}.$$

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  • $\begingroup$ Thank you for your reply. Based on my understanding, you suggest that each coordinate should be symbolized independently of the other other coordinate using the critical point (or mean of the time series for that coordinate) for each coordinate. So in essence we can say that each coordinate is iterated from an initial condition and the the symbolic representation of that coordinate is the binary expansion of the real valued initial condition. Please correct me if wrong. $\endgroup$ – SKM Apr 11 '16 at 17:42
  • $\begingroup$ Another way of symbolizing when the system is delay embedded is using the delay point instead of the critical point. This technique of symbolizing is new for me, thank you for introducing/mentioning it. Could you please present an example for $m=3$ how the symbolic representation for delay embedding would look like? Thank you very much for your effort and time. $\endgroup$ – SKM Apr 11 '16 at 17:44
  • $\begingroup$ I have asked another Question which is in continuation to this post here physics.stackexchange.com/q/248951 Could you please have a look and maybe you can provide a better explanation to it. $\endgroup$ – SKM Apr 11 '16 at 17:49
  • $\begingroup$ @SKM: So in essence we can say that each coordinate is iterated from an initial condition and the the symbolic representation of that coordinate is the binary expansion of the real valued initial condition. – The initial condition is not real-valued but vector-valued. Also do not forget that the components interact with each other. Finally, I don’t see how this is relevant to the problem at hand. $\endgroup$ – Wrzlprmft Apr 11 '16 at 18:15
  • $\begingroup$ @SKM: Another way of symbolizing when the system is delay embedded is using the delay point instead of the critical point. – I am not exactly sure what you mean by delay point.Could you please present an example for $m=3$ how the symbolic representation for delay embedding would look like? – See my edit. $\endgroup$ – Wrzlprmft Apr 11 '16 at 18:39

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