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Cross map over multiple time series lengths

To perform a convergent cross map analysis as in 1 one can apply the crossmap functions on time series of increasing length. The convergentcrossmap(driver, response, timeserieslengths; kwargs...) function does so by applying crossmap for each time series length in timeserieslengths, where time windows always start at the first data point.

Documentation

# CrossMappings.convergentcrossmap — Function.

convergentcrossmap(driver,
        response,
        timeseries_lengths;
        summarise::Bool = true,
        average_measure::Symbol = :median,
        uncertainty_measure::Symbol = :quantile,
        quantiles = [0.327, 0.673],
        kwargs...)

Algorithm

Compute the cross mapping between a driver series and a response series over different timeseries_lengths. If summarise = true, then call ccm_with_summary. If summarise = false, then call ccm (returns raw crossmap skills).

Arguments

  • driver: The data series representing the putative driver process.
  • response: The data series representing the putative response process.
  • timeseries_lengths: Time series length(s) for which to compute the cross mapping(s).

Summary keyword arguments

  • summarise: Should cross map skills be summarised for each time series length? Default is summarise = true.
  • average_measure: Either :median or :mean. Default is :median.
  • uncertainty_measure: Either :quantile or :std. Default is :quantile.
  • quantiles: Compute uncertainty over quantile(s) if uncertainty_measure is :quantile. Default is [0.327, 0.673], roughly corresponding to 1s for normally distributed data.

Keyword arguments to crossmap

  • dim: The dimension of the state space reconstruction (delay embedding) constructed from the response series. Default is dim = 3.
  • τ: The embedding lag for the delay embedding constructed from response. Default is τ = 1.
  • ν: The prediction lag to use when predicting scalar values of driver fromthe delay embedding of response. ν > 0 are forward lags (causal; driver's past influences response's future), and ν < 0 are backwards lags (non-causal; driver's' future influences response's past). Adjust the prediction lag if you want to performed lagged ccm (Ye et al., 2015). Default is ν = 0, as in Sugihara et al. (2012). Note: The sign of the lag ν is organized to conform with the conventions in TransferEntropy.jl, and is opposite to the convention used in the rEDM package (Ye et al., 2016).
  • libsize: Among how many delay embedding points should we sample time indices and look for nearest neighbours at each cross mapping realization (of which there are n_reps)?
  • n_reps: The number of times we draw a library of libsize points from the delay embedding of response and try to predict driver values. Equivalently, how many times do we cross map for this value of libsize? Default is n_reps = 100.
  • replace: Sample delay embedding points with replacement? Default is replace = true.
  • exclusion_radius: How many temporal neighbors of the delay embedding point response_embedding(t) to exclude when searching for neighbors to determine weights for predicting the scalar point driver(t + ν). Default is exclusion_radius = 0.
  • which_is_surr: Which data series should be replaced by a surrogate realization of the type given by surr_type? Must be one of the following: :response, :driver, :none, :both. Default is :none.
  • surr_func: A valid surrogate function from TimeseriesSurrogates.jl.
  • tree_type: The type of tree to build when looking for nearest neighbors. Must be a tree type from NearestNeighbors.jl. For now, this is either BruteTree, KDTree or BallTree.
  • distance_metric: An instance of a Metric from Distances.jl. BallTree and BruteTree work with any Metric. KDTree only works with the axis aligned metrics Euclidean, Chebyshev, Minkowski and Cityblock. Default is metric = Euclidean() (note the instantiation of the metric).
  • correspondence_measure: The function that computes the correspondence between actual values of driver and predicted values. Can be any function returning a similarity measure between two vectors of values. Default is correspondence_measure = StatsBase.cor, which returns values on [-1, 1]. In this case, any negative values are usually filtered out (interpreted as zero coupling) and a value of 1 means perfect prediction. Sugihara et al. (2012) also proposes to use the root mean square deviation, for which a value of 0 would be perfect prediction.

References

Sugihara, George, et al. "Detecting causality in complex ecosystems." Science (2012): 1227079. http://science.sciencemag.org/content/early/2012/09/19/science.1227079

Ye, Hao, et al. "Distinguishing time-delayed causal interactions using convergent cross mapping." Scientific Reports 5 (2015): 14750. https://www.nature.com/articles/srep14750

Ye, H., et al. "rEDM: Applications of empirical dynamic modeling from time series." R Package Version 0.4 7 (2016). https://cran.r-project.org/web/packages/rEDM/index.html

  1. Sugihara, G., May, R., Ye, H., Hsieh, C. H., Deyle, E., Fogarty, M., & Munch, S. (2012). Detecting causality in complex ecosystems. Science. https://doi.org/10.1126/science.1227079 ↩