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In paper:

A. Vlad, A. Luca, and M. Frunzete, "Computational measurements of the transient time and of the sampling distance that enables statistical independence in the logistic map," in Proc. Int. Conf. Computational Science and its applications, Berlin, 2009, pp. 703-718

it is mentioned that for a time series $X = \{x_k,x_{k+1},\ldots,x_{k+n},\ldots\}$, $x_n$ and $x_{k+d}$ are statistically independent when $d \rightarrow \infty$ where $d$ is an integer denoting the sampling distance. Let's say $d = 5$, then for a time series that is generated by

$x_{k+1} = Rx_k(1-x_k) \tag{1}$

where the the parameter $R$ enables chaotic behavior. The distribution is not Gaussian and the Authors performed Kolmogorov-Smirnov test based for showing statistical independence. Quote : pg: 705

This test method is here required because the investigated random variables are not normally distributed, they are even of unknown statistical law. For each and every investigated $R$ value we used Smirnov tests to verify if two experimental data sets are coming out from the same probability law or not.

EDIT: Sampling distance as defined in paper

Vlad, Adriana, Adrian Luca, and Madalin Frunzete. "Computational measurements of the transient time and of the sampling distance that enables statistical independence in the logistic map." Computational Science and Its Applications–ICCSA 2009. Springer Berlin Heidelberg, 2009. 703-718.

(direct Quote)

The two random variables $X$ and $Y$ submitted to the statistical test are sampled from the random process assigned to (1) at two iterations $k_1$ and $k_1 + d$. We attempted to measure the minimum sampling distance $d$ which enables the statistical independence between $X$ and $Y$. We wanted to establish which is the minimum $d$ sampling distance which enables to accept that two random variables, $X$ and $Y$ extracted from the random process (1) at iterations are statistically independent. Our measurements bring into evidence that the minimum $d$ statistical independence distance is affected by $R$ parameter. The overall experimental study performed on all mentioned values of $R$ parameter (4; 3.9999; 3.999; 3.99; 3.95; 3.9; 3.8; 3.78) indicates that d minimum sampling distance is about: 15 iterations for R = 4; 20 iterations for R =3.9999 and R = 3.999; 30 iterations for R =3.99 (etc). The statistical independence distance is important in practice mainly for the stationarity region of the chaotic map, so that the quantitative results are not dependent on the initial distribution law of the chaotic process. Referring to the overall results, we can say that for initial conditions $x_0$ (for generating the random process in (1)) is uniformly distributed in (0,1) interval, a value of about 40-50 iterations is quite enough to ensure both the transient time (the time interval elapsed from an initial condition of the chaotic signal up to its entrance in stationarity) and the minimum statistical independence sampling distance for all R investigated values.

In Paper:

Yu, Lei, et al. "Compressive sensing with chaotic sequence." Signal Processing Letters, IEEE 17.8 (2010): 731-734.

download link, under Section C and Remark 2.2: it is mentioned that the output of (1) is not independent but its independence can be measured through higher order correlations which is determined by the sampling distance. If the sampling distance is chosen large enough, for distance $d$ = 15, then for any positive integer $m_0$, $m_1 < 2^d$, $X$ has $E(x_k^{m_0} x_{k+d}^{m_1}) = E(x_k^{m_0}) E(x_{k+d}^{m_1})$

Based on this, I have the following question:

Q1: Can somebody please show with an example time series vector what is meant by sampling distance? Will a sampling distance of say $d = 5$ mean that the resulting sampled time series, say $Z$ will have data points taken after every 5 samples? For example, if $X$ is represented as $X = \{x_n,x_{n+1},\ldots,x_{n+k},\ldots\}$ and $Z = \{z_n,z_{n+d},\ldots,z_{n+kd}\}$.

Let's say that from equation 1 $$X = [x_1,\, x_2,\, x_3,\, x_4,\, x_5,\, x_6,\, x_7,\, x_8,\, x_9,\, x_{10},\, x_{11},\, x_{12},\, x_{13},\, x_{14},\, x_{15},\, x_{16},\, x_{17},\, x_{18},\, x_{19},\, x_{20}]$$ then will $Z$ be:

$$Z = [x_6,\, x_7,\, x_8,\, x_9,\, x_{10},\, x_{16},\, x_{17},\, x_{18},\, x_{19},\, x_{20}]$$

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  • $\begingroup$ Can you define what the paper means by "statistically independent"? $\endgroup$ – Jim Clay Sep 10 '14 at 14:32
  • $\begingroup$ @JimClay: I have updated the Question and provided the quotes where sampling distance and statistical independence is mentioned; also added a reference which can be downloaded. The paper "Computational measurements of the transient time and of the sampling distance that enables statistical independence in the logistic map" is mentioned under Ref[18] of the link. Thank you for your suggestions. $\endgroup$ – Srishti M Sep 10 '14 at 16:38
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The authors you mention don't talk about a "resulting time series", but only about the correlation of two random variables (or their powers) of the original sequence $X$. They consider the expectation

$$E\{x^{m_0}_nx^{m_1}_{n+d}\}\tag{1}$$

and discuss for which values of $d$ (the sampling distance) (and of $m_0$ and $m_1$) the expectation in (1) equals

$$E\{x^{m_0}_n\}E\{x^{m_1}_{n+d}\}$$

i.e., for which values of $d$ and of $m_0$ and $m_1$ the variables (not sequences!) $x_n^{m_0}$ and $x_{n+d}^{m_1}$ are uncorrelated.

So I think your confusion arose because you thought that $x_n$ and $x_{n+d}$ refer to sequences, when in fact they refer to the $n^{th}$ and $(n+d)^{th}$ values of the sequence $X$.

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  • $\begingroup$ So, $x_n$ and $x_{n+d}$ refer to data points (numbers) OR the values of $X$ are variables and not data points? Then how come the convention is $x_{k+1}$ which according to my understanding means the data point at iteration $k+1$; $x_{k+2}$ the value of data point at iteration $k+2$ etc since the model in Eq(1) is two dimensional discrete and the paper mentions that $k$ is the iteration?Could you plz explain how the time series $Z$ will look like with an example for any sampling distance since the rest of the paper works with the sampled time series $Z$ instead of $X$. $\endgroup$ – Srishti M Sep 10 '14 at 22:58
  • $\begingroup$ @SrishtiM: The $x_n$ is simply the $n^{th}$ value of $X$. Call it a data point or a variable or whatever you like, but it is a single value. The sequence $Z(d,k,z_0)$ is subsampled from $Z$ with a subsampling factor $d$, so it contains every $d^{th}$ value of $Z$. $\endgroup$ – Matt L. Sep 11 '14 at 6:08
  • $\begingroup$ Right, thank you for explaining it once again. But there is the other problem which I mentioned before that I am having difficulty to follow, which is if $d$ = 4 then in general any sampled time series will contain values after every $d$ number of sample? For ex: if original vector A = [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16],then will the result of sampling for $d = 4$ be B = [5 6 7 8 13 14 15 16]?Could you kindly clarify this? $\endgroup$ – Srishti M Sep 11 '14 at 20:13
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    $\begingroup$ @SrishtiM: No, you need to take every $d^{th}$ value, i.e. take one, skip $d-1$, take one, etc. E.g. if $d=4$ and $A$ is as in your example above, then $B=[1, 5,9,13,\ldots]$. $\endgroup$ – Matt L. Sep 12 '14 at 6:55

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