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I am trying to implement Takens' correlation dimension for noisy time series formula given in Eq (8) in (Estimation of the dimension of a noisy attractor. Schouten JC, Takens F, van den Bleek CM Physical Review E, vol 50, number 3, 1994) . The actual derivation and explanation is given in Takens' maximum likelihood estimation (F. Takens, On the numerical determination of the dimension of an attractor, in: D. Rand, L.S. Yong (Eds.), Dynamical Systems and Turbulence, Warwick, 1980, Lecture Notes in Mathematics, Vol. 898, Springer, Berlin, 1981.) and summarized below according to my understanding. My time series are noisy with additive white Gaussian noise, Z = x+noise

Step1 = Perform phase space embedding of the 1D time series. This becomes U.

Step2: Find euclidean distances or maximum norm distances of the higher dimensional space. Find a scaling distance, $l_0$ (or $r_0$) called as the maximum scaling distance.

Step3: Then find those distances, $l_z$, which are $\le l_0$. This step converts distances into $l_z$ that now belongs to range $[0,1]$. Sometimes, the maximum scaling length $l_0$ is taken as the $l_0$ = average absolute deviation of the time series by the formula $l_0 = 1/{\rm total\ elements}[ {\sum (u_i - {\tt mean}(u)}]$.

Step 4: In Takens' paper, it is mentioned under Section2, that the distances can be made independent of $l_0$ by dividing all distances $l_z$ by $l_0$.

Step5: Take the logarithm of each element so found by Step 4.

${\rm Takens\ dimension} = -(\frac{2}{n(n-1)} \sum [\log (||X_i-X_j||/r_o)]^{-1}$

(Formula for Takens) $n$ = number of samples.

Questions:

(A) It is not clear to me what should be $l_0$ - the upper bound of the scaling distances.

Due to the term "distance", I initially thought them to be the average standard deviation of the euclidean distances calculated. But, then this doe snot follow what is mentioned under Eq (12).

When I took $l_0$ as the average absolute deviation of the time series and divided the distances by it, the result of correlation dimension is incorrect. So, if I follow what is mentioned in the paper1, then the value of the dimension is coming out to be incorrect. What should be $l_0$ ?

(B) How do I formulate the dimension for noisy case , I mean will the final formula change and do I have to include the noise effects as done by Ellner http://rullf2.xs4all.nl/msct/node28.html

I really do not understand what should be $l_0$, the upper bound of scaling distances. Should be the maximum distance, or the maximum standard deviation of the distance vector or the average standard deviation of the time series ?

Please help, I am not sure if I understood correctly.

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Hi @Srishti M: Minor comment to the post (v3): Please consider to mention explicitly author, etc. of links, so it is possible to reconstruct links in case of link rot. The 1st link does not work for me. Also please link to abstract pages rather than pdf files. –  Qmechanic Nov 27 '13 at 19:30
    
@Qmechanic: I have added the paper description –  Srishti M Nov 27 '13 at 20:48

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