# Numerical Data

Trajectory and Timeseries

The word "timeseries" can be confusing, because it can mean a univariate (also called scalar or one-dimensional) timeseries or a multivariate (also called multi-dimensional) timeseries. To resolve this confusion, in DynamicalSystems.jl we have the following convention: "timeseries" always refers to a one-dimensional vector of numbers, which exists with respect to some other one-dimensional vector of numbers that corresponds to a time-vector. On the other hand, the word "trajectory" is used to refer to a multi-dimensional timeseries, which is of course simply a group/set of one-dimensional timeseries. A trajectory is represented by a Dataset and is the return type of trajectory.

Trajectories (and in general sets in state space) in DynamicalSystems.jl are represented by a structure called Dataset (while timeseries are standard Julia Vectors).

DelayEmbeddings.DatasetType
Dataset{D, T} <: AbstractDataset{D,T}

A dedicated interface for datasets. It contains equally-sized datapoints of length D, represented by SVector{D, T}.

When indexed with 1 index, a dataset is like a vector of datapoints. When indexed with 2 indices it behaves like a matrix that has each of the columns be the timeseries of each of the variables.

Dataset also supports most sensible operations like append!, push!, hcat, eachrow, among others.

Description of indexing

In the following let i, j be integers, typeof(data) <: AbstractDataset and v1, v2 be <: AbstractVector{Int} (v1, v2 could also be ranges).

• data[i] gives the ith datapoint (returns an SVector)
• data[v1] will return a vector of datapoints
• data[v1, :] using a Colon as a second index will return a Dataset of these points
• data[:, j] gives the jth variable timeseries, as Vector
• data[v1, v2] returns a Dataset with the appropriate entries (first indices being "time"/point index, while second being variables)
• data[i, j] value of the jth variable, at the ith timepoint

Use Matrix(dataset) or Dataset(matrix) to convert. It is assumed that each column of the matrix is one variable. If you have various timeseries vectors x, y, z, ... pass them like Dataset(x, y, z, ...). You can use columns(dataset) to obtain the reverse, i.e. all columns of the dataset in a tuple.

In essence a Dataset is simply a wrapper for a Vector of SVectors. However, it is visually represented as a matrix, similarly to how numerical data would be printed on a spreadsheet (with time being the column direction). It also offers a lot more functionality than just pretty-printing. Besides the examples in the documentation string, you can also do:

using DynamicalSystems
hen = Systems.henon()
data = trajectory(hen, 10000) # this returns a dataset
for point in data
# do stuff with each data-point
# (vector with as many elements as system dimension)
end

Most functions from DynamicalSystems.jl that manipulate and use data are expecting a Dataset. This allows us to define efficient methods that coordinate well with each other, like e.g. embed.

## Dataset Functions

Functions that operate on datasets.

DelayEmbeddings.minimaFunction
minima(dataset)

Return an SVector that contains the minimum elements of each timeseries of the dataset.

DelayEmbeddings.maximaFunction
maxima(dataset)

Return an SVector that contains the maximum elements of each timeseries of the dataset.

## Dataset I/O

Input/output functionality for an AbstractDataset is already achieved using base Julia, specifically writedlm and readdlm. To write and read a dataset, simply do:

using DelimitedFiles

data = Dataset(rand(1000, 2))

# I will write and read using delimiter ','
writedlm("data.txt", data, ',')

# Don't forget to convert the matrix to a Dataset when reading
data = Dataset(readdlm("data.txt", ',', Float64))

## Neighborhoods

Combining the excellent performance of NearestNeighbors.jl with the AbstractDataset allows us to define a function that calculates a "neighborhood" of a given point, i.e. finds other points near it. The different "types" of the neighborhoods are subtypes of AbstractNeighborhood.

DelayEmbeddings.neighborhoodFunction
neighborhood(point, tree, ntype)
neighborhood(point, tree, ntype, n::Int, w::Int = 1)

Return a vector of indices which are the neighborhood of point in some data, where the tree was created using tree = KDTree(data [, metric]). The ntype is the type of neighborhood and can be any subtype of AbstractNeighborhood.

Use the second method when the point belongs in the data, i.e. point = data[n]. Then w stands for the Theiler window (positive integer). Only points that have index abs(i - n) ≥ w are returned as a neighborhood, to exclude close temporal neighbors. The default w=1 is the case of excluding the point itself.

References

neighborhood simply interfaces the functions NearestNeighbors.knn and inrange from NearestNeighbors.jl by using the argument ntype.

DelayEmbeddings.AbstractNeighborhoodType
AbstractNeighborhood

Supertype of methods for deciding the neighborhood of points for a given point.

Concrete subtypes:

• FixedMassNeighborhood(K::Int) : The neighborhood of a point consists of the K nearest neighbors of the point.
• FixedSizeNeighborhood(ε::Real) : The neighborhood of a point consists of all neighbors that have distance < ε from the point.

See neighborhood for more.