

VisitationFrequencyTest

# CausalityTools.CausalityTests.VisitationFrequencyTest — Type.

VisitationFrequencyTest(k::Int = 1, l::Int = 1, m::Int = 1, n::Int = 1, 
    τ::Int = 1, b = 2, estimator::VisitationFrequency = VisitationFrequency(), 
    binning_summary_statistic::Function = StatsBase.mean,
    binning::RectangularBinning, ηs)

The parameters for a transfer entropy test using the VisitationFrequency estimator. This is the original transfer entropy estimator from [1], as implemented in [2].

Mandatory keyword arguments

binning::RectangularBinning: An instance of a RectangularBinning that dictates how the delay embedding is discretized.
ηs: The prediction lags (that go into the $T_{f}$ component of the embedding).

Optional keyword arguments

k::Int: The dimension of the $T_{f}$ component of the embedding.
l::Int: The dimension of the $T_{pp}$ component of the embedding.
m::Int: The dimension of the $S_{pp}$ component of the embedding.
n::Int: The dimension of the $C_{pp}$ component of the embedding.
τ::Int: The embedding lag. Default is τ = 1.
b: Base of the logarithm. The default (b = 2) gives the TE in bits.
estimator::VisitationFrequency: A VisitationFrequency estimator instance.
binning_summary_statistic::Function: A summary statistic to summarise the transfer entropy values if multiple binnings are provided.

Estimation of the invariant measure

With a VisitationFrequencyTest, the invariant distribution from which transfer entropy is computed is estimated using simple counting (hence, visitation frequency). Counting is done over the partition elements of a discretization of an appropriate delay reconstruction of the time series to be analysed.

About the delay reconstruction for transfer entropy analysis

Denote the time series for the source process $S$ as $S(t)$ , and the time series for the target process $T$ as $T(t)$ , and $C_i(t)$ as the time series for any conditional processes $C_i$ that also may influence $T$ . To compute (conditional) TE, we need a generalised embedding [3, 4] incorporating all of these processes.

For convenience, define the state vectors

$\begin{align} T_f^{(k)} &= \{(T(t+\eta_k), \ldots, T(t+\eta_2), T(t+\eta_1))\}, \label{eq:Tf} \\ T_{pp}^{(l)} &= \{ (T(t), T(t-\tau_1), T(t-\tau_2), \ldots, T(t - \tau_{l - 1})) \\ S_{pp}^{(m)} &= \{(S(t), S(t-\tau_1), S(t-\tau_2), \ldots, S(t-\tau_{m - 1}))\},\\ C_{pp}^{(n)} &= \{ (C_1(t), C_1(t-\tau_1), \ldots, C_2(t), C_2(t-\tau_1) \}, \end{align}$

where the state vectors $T_f^{(k)}$ contain $k$ future values of the target variable, $T_{pp}^{(l)}$ contain $l$ present and past values of the target variable, $S_{pp}^{(m)}$ contain $m$ present and past values of the source variable, $C_{pp}^{(n)}$ contain a total of $n$ present and past values of any conditional variable(s). Combining all variables, we have the generalised embedding

$\begin{align} \mathbb{E} = (T_f^{(k)}, T_{pp}^{(l)}, S_{pp}^{(m)}, C_{pp}^{(n)}) \end{align}$

with a total embedding dimension of $k + l + m + n$ . Here, $\tau$ indicates the embedding lag (in the VisitationFrequencyTest, we set $\tau_1 = \tau_2 = \tau_3 = \ldots$ ), and $\eta$ indicates the prediction lag (the lag of the influence the source has on the target).

Hence, in the generalised embedding, only $T_f$ depends on the prediction lag $\eta$ , which is to be determined by the analyst. For transfer entropy analysis, $\eta$ is chosen to be some positive integer, while for predictive asymmetry analysis, symmetric negative and positive $\eta$ s are used for computing $\mathbb{A}$ .

In terms of this generalised embedding, transfer entropy from a source variable $S$ to a target variable $T$ with conditioning on variable(s) $C$ is thus defined as

$\begin{align} TE_{S \rightarrow T|C} = \int_{\mathbb{E}} P(T_f, T_{pp}, S_{pp}, C_{pp}) \log_{2}{\left( \frac{P(T_f | T_{pp}, S_{pp}, C_{pp})}{P(T_f | T_{pp}, C_{pp})}\right)} \end{align}$

Without conditioning, we have

$\begin{align} TE_{S \rightarrow T} = \int_{\mathbb{E}} P(T_f, T_{pp}, S_{pp}) \log_{2}{\left(\frac{P(T_f | T_{pp}, S_{pp})}{P(T_f | T_{pp})}\right)} \end{align}$

Low-level estimator

This test uses the VisitationFrequency estimator on the following low-level method under the hood.

transferentropy(::Any, ::TEVars, ::RectangularBinning, ::TransferEntropyEstimator)

In this estimator, the mapping between variables of the generalised embedding and the marginals during transfer entropy computation is controlled using a TEVars instance. It is highly recommended that you check the documentation for this method, because it describes the transfer entropy estimation procedure in detail.

Notes:

Use causality(source, target, params::VisitationFrequencyTest) for regular transfer entropy analysis. This method uses only the k, l, m and ignores n when constructing the delay reconstruction.
Use causality(source, target, cond, params::VisitationFrequencyTest) for conditional transfer entropy analysis. This method uses the k, l, m and n when constructing the delay reconstruction.

Example

# Prediction lags
ηs = 1:10
binning = RectangularBinning(10)

# Use defaults, binning and prediction lags are required. 
# Note that `binning` and `ηs` are *mandatory* keyword arguments.
VisitationFrequencyTest(binning = binning, ηs = ηs)

# The other keywords can also be adjusted
VisitationFrequencyTest(k = 1, l = 2, binning = binning, ηs = ηs)

References

Schreiber, Thomas. "Measuring information transfer." Physical review letters 85.2 (2000): 461. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.85.461
Diego, David, Kristian Agasøster Haaga, and Bjarte Hannisdal. "Transfer entropy computation using the Perron-Frobenius operator." Physical Review E 99.4 (2019): 042212. https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.042212
Sauer, Tim, James A. Yorke, and Martin Casdagli. "Embedology." Journal of statistical Physics 65.3-4 (1991): 579-616.
Deyle, Ethan R., and George Sugihara. "Generalized theorems for nonlinear state space reconstruction." PLoS One 6.3 (2011): e18295.

source