Convergent cross mapping algorithm



Convergent cross mapping algorithmConsider time series of hypothetical variables X and Y. Convergent cross-mapping (CCM) employs state space reconstruction (SSR), thereby using time-lagged coordinates of each of these variables to produce shadow versions of their respective source manifolds. We will refer to these projection manifolds as Mx and My. To test whether X causes Y, CCM applies the following logic: Because manifold reconstruction preserves the Lyapunov exponents of the original system ADDIN CSL_CITATION { "citationItems" : [ { "id" : "ITEM-1", "itemData" : { "DOI" : "10.1016/0167-2789(91)90222-U", "ISSN" : "01672789", "abstract" : "Takens' theorem demonstrates that in the absence of noise a multidimensional state space can be reconstructed from a scalar time series. This theorem gives little guidance, however, about practical considerations for reconstructing a good state space. We extend Takens' treatment, applying statistical methods to incorporate the effects of observational noise and estimation error. We define the distortion matrix, which is proportional to the conditional covariance of a state, given a series of noisy measurements, and the noise amplification, which is proportional to root-square time series prediction errors with an ideal model. We derive explicit formulae for these quantities, and we prove that in the low noise limit minimizing the distortion is equivalent to minimizing the noise amplification. We identify several different scaling regimes for distortion and noise amplification, and derive asymptotic scaling laws. When the dimension and Lyapunov exponents are sufficiently large these scaling laws show that, no matter how the state space is reconstructed, there is an explosion in the noise amplification - from a practical point of view determinism is lost, and the time series is effectively a random process. In the low noise, large data limit we show that the technique of local singular value decomposition is an optimal coordinate transformation, in the sense that it achieves the minimum distortion in a state space of the lowest possible dimension. However, in numerical experiments we find that estimation error complicates this issue. For local approximation methods, we analyze the effect of reconstruction on estimation error, derive a scaling law, and suggest an algorithm for reducing estimation errors.", "author" : [ { "dropping-particle" : "", "family" : "Casdagli", "given" : "Martin", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Eubank", "given" : "Stephen", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Farmer", "given" : "J.Doyne", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" }, { "dropping-particle" : "", "family" : "Gibson", "given" : "John", "non-dropping-particle" : "", "parse-names" : false, "suffix" : "" } ], "container-title" : "Physica D: Nonlinear Phenomena", "id" : "ITEM-1", "issue" : "1-3", "issued" : { "date-parts" : [ [ "1991", "8" ] ] }, "page" : "52-98", "title" : "State space reconstruction in the presence of noise", "type" : "article-journal", "volume" : "51" }, "uris" : [ "" ] } ], "mendeley" : { "previouslyFormattedCitation" : "[18]" }, "properties" : { "noteIndex" : 0 }, "schema" : "" }[18], if X causes Y, then time points that are close in My should also be close in Mx. Since Mx is constructed from lags of the observations of X, the points that are close in Mx will also have similar values in the corresponding time series. Therefore, if X causes Y, then My can tell us which observations of X should best predict a given point from X. Furthermore, predictability should increase with the number of manifold points that are considered. To test whether X causes Y, My is used to infer the points in X that will best predict a given held-out point from X. We measure this performance using predictive skill, quantified by ρccm. Intuitively, this procedure works as follows: A point is held out from X. We then use My to infer the points in Mx that will be closest to that point of interest. Using exponential weights derived from the relative pairwise distances of corresponding points in My, we predict the held-out point using other observations from X. Finally, ρccm is calculated as the Pearson correlation between observed and predicted points, and so is a cross-validated measure. To examine whether the signal converges as expected for a causal relationship, these steps are repeated using increasing time series length (L).Paramecium-Didinium system514351945640One-third thinnedOne-half thinnedFull datasetOne-third thinnedOne-half thinnedFull datasetDidinium is a free-living unicellular carnivore. Paramecium is its prey. More information about this system, as well as interactive graphs of time series and manifold constructions, can be found at: Figure S1. The maximal predictive skill as a function of E, tau p, and the number of included points.Fourier transform analysisWe calculated the characteristic frequencies of the paramecium and didinium time series by performing fourier transform analysis using the rfft function in the python module scipy. ................
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