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So I recently came across Poincaré plots (Wiki) which plots x(t) against x(t+1). Thereafter I found out about lag plots (Description)which seem to do the same thing just the other way around, i.e. x(t) against x(t-1). Now besides for the obvious offset difference, my question is, does it matter in terms of analysis.

The reason I ask is that pandas has a lag plot implemented already and don't see the point of implementing a special Poincaré plot if it is essentially the same thing.

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    $\begingroup$ You are right that (apart from labelling) this is the same thing! $\endgroup$ Feb 2, 2017 at 11:11

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The Poincaré plot that you specifically refer to (from wikipedia) is indeed the same as a lag plot.

However, as Fabrice Pautot notes in his answer a Poincaré map(/plot) more commonly refers to something different, something more general (also on wikipedia).


This ambiguity in the reference to Poincaré (map or plot) is probably due to the researchers in the field of cardiology, who at some point started to refer to a lag plot as Poincaré plot.

A search in literature for the term 'Poincaré map' shows that some early references to this use of Poincaré maps date at least from the early 90's of the last century (e.g. Raetz, S.L., Richard, C.A., Garfinkel, A. and Harper, R.M., 1991. Dynamic characteristics of cardiac RR intervals during sleep and waking states. Sleep, 14(6), pp.526-533). In that research the Poincaré map is used to refer to a lag plot. This is stated explicitly

Each cardiac interval of a particular epoch was plotted on the y-axis against the previous cardiac interval value on the x-axis resulting in a $RR_{n+l}$ vs. $RR_n$ Poincare plot (21)

The reference, numbered 21, relates to the book 'the geometry of behavior' by Abraham and Shaw. However they do not describe such thing as a lag plot. They speak about the 'Poincaré section' and the 'first return map'.

I would think that this might be the origin of the ambiguity, because there is a discrepancy between the description in the article and the reference in the article, and it is a very early reference to 'Poincaré plot' in that field. However, when you look for slightly different terms then you find some older references. The research on the dynamics and chaos of heart beat seems to be a rising trend around the end of the 80s (for instance a reference to the term 'Poincaré return map' is made in "Zbilut et al 1989 Chaotic Heart Rate Dynamics in Isolated Perfused Rat Hearts").

The ambiguity is not 'wrong'. A lag plot may be seen as a specific case of a Poincaré map. This is also mentioned in a paper on chaos theory in relation to cardiology (see: Denton, T. A., Diamond, G. A., Helfant, R. H., Khan, S., & Karagueuzian, H. (1990). Fascinating rhythm: a primer on chaos theory and its application to cardiology. American heart journal, 120(6), 1419-1440 where it is mentioned that there are two ways to make a Poincaré intersection. The lag plot relates to the second 'stroboscopic Poincaré section' case.).

See also ("Nonlinear Dynamics and Chaos" Thompson and Stewart)

By continuing to mark the trajectory stroboscopically at times that are integer multiples of the forcing period 2n, a sequence of strictly comparable points A, B, C, D, etc., is accumulated in Figure 1.4. Proceeding now to simplify the picture, we erase the continuous trajectory and show only the strobed points in the Poincare section.

https://books.google.com/books?id=80ChNIpUDVAC&pg=PA5&source=gbs_toc_r&cad=4#v=onepage&q&f=false


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    $\begingroup$ +1 This is a very nice example of why it is so much more constructive and informative to address the issues raised by a question rather than holding a singular opinion and insisting it is the only correct one. $\endgroup$
    – whuber
    Sep 5, 2019 at 14:59

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