Nonlinear Timeseries Analysis

broomhead_king(s::AbstractVector, d::Int) -> U, S, Vtr

Return the Broomhead-King coordinates of a timeseries s by performing svd on high-dimensional embedding if s with dimension d with minimum delay.

Description

Broomhead and King coordinates is an approach proposed in ^{[Broomhead1987]} that applies the Karhunen–Loève theorem to delay coordinates embedding with smallest possible delay.

The function performs singular value decomposition on the d-dimensional matrix $X$ of $s$,

\[X = \frac{1}{\sqrt{N}}\left( \begin{array}{cccc} x_1 & x_2 & \ldots & x_d \\ x_2 & x_3 & \ldots & x_{d+1}\\ \vdots & \vdots & \vdots & \vdots \\ x_{N-d+1} & x_{N-d+2} &\ldots & x_N \end{array} \right) = U\cdot S \cdot V^{tr}.\]

where $x := s - \bar{s}$. The columns of $U$ can then be used as a new coordinate system, and by considering the values of the singular values $S$ you can decide how many columns of $U$ are "important". See the documentation page for example application.

dyca(data, eig_thresold) -> eigenvalues, proj_mat, projected_data

Compute the Dynamical Component analysis (DyCA) of the given data ^[Uhl2018] used for dimensionality reduction.

Return the eigenvalues, projection matrix, and reduced-dimension data (which are just data*proj_mat).

Keyword Arguments

• norm_eigenvectors=false : if true, normalize the eigenvectors

Description

Dynamical Component Analysis (DyCA) is a method to detect projection vectors to reduce the dimensionality of multi-variate, high-dimensional deterministic datasets. Unlike methods like PCA or ICA that make a stochasticity assumption, DyCA relies on a determinacy assumption on the time-series and is based on the solution of a generalized eigenvalue problem. After choosing an appropriate eigenvalue threshold and solving the eigenvalue problem, the obtained eigenvectors are used to project the high-dimensional dataset onto a lower dimension. The obtained eigenvalues measure the quality of the assumption of linear determinism for the investigated data. Furthermore, the number of the generalized eigenvalues with a value of approximately 1.0 are a measure of the number of linear equations contained in the dataset. This property is useful in detecting regions with highly deterministic parts in the time-series and also as a preprocessing step for reservoir computing of high-dimensional spatio-temporal data.

The generalised eigenvalue problem we solve is:

\[C_1 C_0^{-1} C_1^{\top} \bar{u} = \lambda C_2 \bar{u} \]

where $C_0$ is the correlation matrix of the data with itself, $C_1$ the correlation matrix of the data with its derivative, and $C_2$ the correlation matrix of the derivative of the data with itself. The eigenvectors $\bar{u}$ with eigenvalues approximately 1 and their $C_1^{-1} C_2 u$ counterpart, form the space where the data is projected onto.

mean_return_times(ds::DynamicalSystem, u₀, εs, T; kwargs...) → τ, c

Return the mean return times τ, as well as the amount of returns c, for subsets of the state space of ds defined by u₀, εs. The ds is evolved for a maximum of T time. This function behaves similarly to exit_entry_times and thus see that one for the meaning of u₀ and εs.

This function supports both discrete and continuous systems, however the optimizations done in discrete systems (where all nested ε-sets are checked at the same time), are not done here yet, which leads to disproportionally lower performance since each ε-related set is checked individually from start.

Continuous systems allow for the following keywords:

• i=10 How many points to interpolate the trajectory in-between steps to find candidate crossing regions.
• dmin If the trajectory is at least dmin distance away from u0, the algorithm that checks for crossings of the ε-set is not initiated. By default obtains the a value 4 times as large as the radius of the maximum ε-set.
• diffeq... All extra keywords are propagated to solvers of DifferentialEquations.jl, see trajectory. It is strongly recommended to use high accuracy keywords here, e.g. alg = Vern9(), reltol = 1e-12, abstol = 1e-12, maxiters = typemax(Int).

For continuous systems T, i, dmin can be vectors with same size as εs, to help increase accuracy of small ε.

exit_entry_times(ds, u₀, εs, T; diffeq...) → exits, entries

Collect exit and entry times for a ball/box centered at u₀ with radii εs (see below), in the state space of the given discrete dynamical system (function not yet available for continuous systems). Return the exit and (re-)entry return times to the set(s), where each of these is a vector containing all collected times for the respective ε-radius set, for ε ∈ εs.

Use transit_return(exits, entries) to transform the output into transit and return times, and see also mean_return_times for both continuous and discrete systems.

Description

Transit time statistics are important for the transport properties of dynamical systems^[Meiss1997] and can even be connected with the fractal dimension of chaotic sets^[Boev2014].

The current algorithm collects exit and re-entry times to given sets in the state space, which are centered at u₀ (algorithm always starts at u₀ and the initial state of ds is irrelevant). εs is always a Vector.

The sets around u₀ are nested hyper-spheres of radius ε ∈ εs, if each entry of εs is a real number. The sets can also be hyper-rectangles (boxes), if each entry of εs is a vector itself. Then, the i-th box is defined by the space covered by u0 .± εs[i] (thus the actual box size is 2εs[i]!).

The reason to input multiple εs at once is purely for performance.

For discrete systems, exit time is recorded immediatelly after exitting of the set, and re-entry is recorded immediatelly on re-entry. This means that if an orbit needs 1 step to leave the set and then it re-enters immediatelly on the next step, the return time is 1. For continuous systems high-order interpolation is done to accurately record the time of exactly crossing the ε-ball/box.