Representation of functional data

The first step is the analysis of functional data is to represent them in a suitable way. The package provides several classes to represent functional data, depending on the type of data to represent. The package provides classes to represent univariate functional data, irregular functional data, multivariate functional data, and functional data in a basis representation.

Generic representation

Representations of functional data are based on three abstract classes. The first one, Argvals, represents the arguments of the functions. The second one, Values, represents the values of the functions. The third one, FunctionalData, represents the functional data object, which is a pair of Argvals and Values. The package provides several implementations of these classes, depending on the type of functional data to represent. These classes cannot be instantiated directly, but they are used as base classes for the specific implementations.

FDApy.representation.Argvals([dict])	Define the structure of Argvals.
FDApy.representation.Values()	Define the structure of Values.
FDApy.representation.FunctionalData()	Define the structure of FunctionalData.

Representing Argvals and Values

Functional data representations are based on two main components: the arguments of the functions and the values of the functions. The package provides several classes to represent these components, depending on the type of data to represent.

FDApy.representation.DenseArgvals([dict])	Represent the argvals of dense functional data.
FDApy.representation.DenseValues(input_array)	Represent the values of dense functional data.

FDApy.representation.IrregularArgvals([dict])	Represent the argvals of irregular functional data.
FDApy.representation.IrregularValues([dict])	Represent the values of irregular functional data.

Univariate Functional Data

Univariate functional data are realizations of a random process:

\[X: \mathcal{T} \subset \mathbb{R}^d \rightarrow \mathbb{R}.\]

The package provides two representations of univariate functional data: grid representation and basis representation.

Grid representation

Univariate functional data can be represented as a set of values on a grid. The GridFunctionalData class is the abstract class to represent univariate functional data on a grid. The package provides two implementations of this class: DenseFunctionalData and IrregularFunctionalData. The class DenseFunctionalData represents functional data of arbitrary dimension (one for curves, two for images, etc.) on a common set of sampling points, while the class IrregularFunctionalData represents functional data of arbitrary dimension sampled on different sets of points (the number and location of the sampling points vary between functional observations).

FDApy.representation.GridFunctionalData(...)	Represent discretised functional data.
FDApy.representation.DenseFunctionalData(...)	Represent densely sampled functional data.
FDApy.representation.IrregularFunctionalData(...)	Represent irregularly sampled functional data.

Basis representation

The basis representation of univariate functional data consists of a linear combination of basis functions.

FDApy.representation.BasisFunctionalData(...)

Represent functional data with a basis.

Multivariate Functional Data

Multivariate functional data are realizations of a multivariate random process. Multivariate functional data objects are vectors of univariate functional data objects, eventually defined on different domains. The class MultivariateFunctionalData allows for the combination of different types of functional data objects (DenseFunctionalData, IrregularFunctionalData, and BasisFunctionalData). It is also possible to mix unidimensional data (curves) with multidimensional data (images, surfaces, etc.).

FDApy.representation.MultivariateFunctionalData(...)

Represent multivariate functional data.

Basis

The package provides two classes to represent basis of functions. The class Basis represents a basis of functions, while the class MultivariateBasis represents a multivariate basis of functions. Currently, the available bases are: Fourier basis, B-spline basis, Legendre basis and Wiener basis. The user may also define custom bases.

FDApy.representation.Basis([name, ...])	Define univariate orthonormal basis.
FDApy.representation.MultivariateBasis([...])	Define multivariate orthonormal basis.

Iterators

The package provides several iterators to handle functional data objects. These iterators allow for the iteration over the functional data objects (e.g. for loops, list comprehensions, etc.). These are defined in the classes: DenseFunctionalDataIterator, IrregularFunctionalDataIterator, and BasisFunctionalDataIterator.