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Custom delay reconstruction

To create delay embeddings from sequential data, use customembed. This function returns allows for more flexibility than embed in DynamicalSystems.jl¹, but also returns a Dataset, so these embedding methods may be used interchangeably.

Delay reconstruction

# CausalityToolsBase.customembed — Function.

customembed(pts, positions::Positions, lags::Lags)

Do custom state space reconstructions with customembed(pts, positions::Positions, lags::Lags). This function acts almost as DynamicalSystems.reconstruct, but allows for more flexibility in the ordering of dynamical variables and allows for negative lags. The positions variable indicates which dynamical variables are mapped to which variables in the final reconstruction, while lags indicates the lags for each of the embedding variables.

Note: customembed expects an array of state vectors, i.e. pts[k] must refer to the kth point of the dataset, not the kth dynamical variable/column.*. To embed a vector of time series, load DynamicalSystems and wrap the time series in a Dataset first, e.g. if x = rand(100); y = rand(100) are two time series, then customembed(Dataset(x, y), Positions(1, 2, 2), Lags(0, 0, 1) will create the embedding with state vectors (x(t), y(t), y(t + 1)).

Pre-embedded points may be wrapped in a CustomReconstruction instance by simply calling customembed(preembedded_pts) without any position/lag instructions.

Examples

Example 1

customembed([rand(3) for i = 1:50], Positions(1, 2, 1, 3), Lags(0, 0, 1, -2)

gives a 4-dimensional embedding with state vectors (x1(t), x2(t), x1(t + 1), x3(t - 2)).

Example 2

Say we want to construct an appropriate delay reconstruction for transfer entropy (TE) analysis

$$E = \{S_{pp}, T_{pp}, T_f \}= \{x_t, (y_t, y_{t-\tau}), y_{t+\eta} \}``),$$

so that we're computing the following TE

$$TE_{x \to y} = \int_E P(x_t, y_{t-\tau} y_t, y_{t + \eta}) \log{\left( \dfrac{P(y_{t + \eta} | (y_t, y_{t - \tau}, x_t)}{P(y_{t + \eta} | y_t, y_{t-\tau})} \right)}.$$

We'll use a prediction lag $\eta = 2$ and use first minima of the lagged mutual information function for the embedding delay $\tau$.

using CausalityToolsBase, DynamicalSystems

x, y = rand(100), rand(100)
D = Dataset(x, y)
embedlag = optimal_delay(y)
CustomReconstruction(D, Positions(2, 2, 2, 1), Lags(2, 0, embedlag, 0))

Variable mapping

# CausalityToolsBase.Positions — Type.

Positions

Type specifying the positions the different dynamical variables appear in when constructing a custom state space reconstruction. Used in combination with Lags to specify how a CustomReconstruction should be constructed. Each of the positions must refer to a dynamical variable (column) actually present in the dataset.

Examples

• Positions(1, 2, 1, 5) indicates a 4-dimensional state space reconstruction where the 1^st coordinate axis of the reconstruction should be formed from the first variable/column of the input data, the 2^nd coordinate axis of the reconstruction should be formed from the 2^nd variable/column of the input data, the 3^rd coordinate axis of the reconstruction should be formed from the 1^st variable/column of the input data, the 4^th coordinate axis of the reconstruction should be formed from the 5^th variable/column of the input data.
• Positions(-1, 2) indicates a 2-dimensional reconstruction, but will not work, because each position must refer to the index of a dynamical variable (column) of a dataset (indexed from 1 and up, so negative values will not work).

Reconstruction lag mapping

# CausalityToolsBase.Lags — Type.

Lags

Wrapper type for lags used when performing custom state space reconstructions. Used in combination with Positions to specify how a CustomReconstruction should be constructed.

Examples

• Lags(2, 0, -3, 1) indicates a 4-dimensional state space reconstruction where the first variable has a positive lag of 2, the second variable is not lagged, the third variable has a lag of -3, and the fourth variable has a positive lag of 1.
• Lags(0, 0) indicates a 2-dimensional state space reconstruction where both variables are not lagged.

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1. Datseris, George. "DynamicalSystems. jl: A Julia software library for chaos and nonlinear dynamics." J. Open Source Software 3.23 (2018): 598. ↩