KarhunenLoeve#

class FDApy.simulation.KarhunenLoeve(n_functions=5, basis_name='fourier', argvals=None, basis=None, random_state=None, **kwargs_basis)[source]#

Simulate based on Karhunen-Loève decomposition.

This class is used to simulate functional data \(X_1, \dots, X_N\) based on a truncated Karhunen-Loève decomposition:

\[X_i(t) = \sum_{K = 1}^K c_{i, k}\phi_{k}(t), i = 1, \dots, N,\]

on one- or higher-dimensional domains. The eigenfunctions \(\phi_{k}(t)\) could be generated using different basis functions or be user-defined. The scores \(c_{i, k}\) are simulated independently from a normal distribution with zero mean and decreasing variance. For higher-dimensional domains, the eigenfunctions are constructed as tensors of marginal orthonormal function systems.

Parameters:

  • n_functions (List[Tuple[int] | int]) – Number of functions to use to generate the basis. See Basis and MultivariateBasis for more information.

  • basis_name (List[Tuple[str] | str] | None) – Name of the basis to use. See Basis and MultivariateBasis for more information.

  • argvals (List[DenseArgvals] | None) – The sampling points of the functional data.

  • basis (Basis | Sequence[Basis] | None) – Basis of functions as a Basis object. Used to have a user-defined basis of function.

  • random_state (int | None) – A seed to initialize the random number generator.

  • kwargs_basis (Any) – Basis parameters.

Attributes:

  • data (DenseFunctionalData | MultivariateFunctionalData) – An object that represents the simulated data.

  • noisy_data (DenseFunctionalData | MultivariateFunctionalData) – An object that represents a noisy version of the simulated data.

  • sparse_data (IrregularFunctionalData | MultivariateFunctionalData) – An object that represents a sparse version of the simulated data.

  • labels (npt.NDArray[np.float64], shape=(n_obs,)) – The integer labels for cluster membership of each sample.

  • basis (Basis | MultivariateBasis) – The eigenfunctions used to simulate the data.

  • eigenvalues (npt.NDArray[np.float64], shape=(n_functions,)) – The eigenvalues used to simulate the data.

Notes

In the case of multivariate functional data, \(X_i\) and \(\phi_{k}\) are vectors and according to the multivariate Karhunen-Loève theorem (see, e.g, [1]), the coefficients do not depend on the component \(p\).

If the basis is user-defined, the object has to be an element of the class Basis and not just DenseFunctionalData or MultivariateFunctionalData.

References

Methods

add_noise([noise_variance])

Add noise to functional data objects.

add_noise_and_sparsify([noise_variance, ...])

Generate a noisy and sparse version of functional data objects.

new(n_obs[, n_clusters, argvals])

Simulate realizations from Karhunen-Loève decomposition.

sparsify([percentage, epsilon])

Generate a sparse version of functional data objects.
add_noise(noise_variance=1.0)[source]#

Add noise to functional data objects.

This function generates an artificial noisy version of a functional data object of class DenseFunctionalData by adding realizations of Gaussian random variables \(\epsilon \sim \mathcal{N}(0, \sigma^2)\) to the observations. The variance \(\sigma^2\) can be supplied by the user. The generated data are given by

\[Y(t) = X(t) + \epsilon.\]
Parameters:

noise_variance (float) – The variance \(\sigma^2\) of the Gaussian noise that is added to the data.

Returns:

Create the class attribute noisy_data.

Return type:

None

add_noise_and_sparsify(noise_variance=1.0, percentage=0.9, epsilon=0.05)[source]#

Generate a noisy and sparse version of functional data objects.

This function generates an artificially noisy and sparse version of a functional datasets. From a functional dataset, it first generates the noisy version and then the sparse version based on the noisy one.

Parameters:

  • noise_variance (float) – The variance \(\sigma^2\) of the Gaussian noise that is added to the data.

  • percentage (float) – The percentage of observations to be retained.

  • epsilon (float) – The uncertainty around the percentage of observations to be retained.

Returns:

Create the class attributes noisy_data and sparse_data.

Return type:

None

new(n_obs, n_clusters=1, argvals=None, **kwargs)[source]#

Simulate realizations from Karhunen-Loève decomposition.

This function generates n_obs realizations of a Gaussian process using the Karhunen-Loève decomposition on a common grid argvals.

Parameters:

  • n_obs (int) – Number of observations to simulate.

  • n_clusters (int) – Number of clusters to generate.

  • argvals (ndarray[Any, dtype[float64]] | None) – Not used in this context. We will use the argvals from the Basis object as argvals of the simulation. Here to be compliant with the class Simulation.

  • kwargs – See below

Keyword Arguments:

  • centers (npt.NDArray[np.float64], shape=(n_features, n_clusters)) – The centers of the clusters to generate. The n_features correspond to the number of functions within the basis.

  • cluster_std (npt.NDArray[np.float64],shape=(n_features, n_clusters)) – The standard deviation of the clusters to generate. The n_features correspond to the number of functions within the basis.

Returns:

Create the class attributes data.

Return type:

None

sparsify(percentage=0.9, epsilon=0.05)[source]#

Generate a sparse version of functional data objects.

This function generates an artificially sparsified version of a functional data object of class DenseFunctionalData. The percentage (and the uncertainty around it) of the number of observation points retained can be supplied by the user. Let \(p\) be the defined percentage and \(\epsilon\) be the uncertainty value. The retained number of observations will be different for each curve and be between \(p - \epsilon\) and \(p + \epsilon\).

Parameters:

  • percentage (float) – The percentage of observations to be retained.

  • epsilon (float) – The uncertainty around the percentage of observations to be retained.

Returns:

Create the class attribute sparse_data.

Return type:

None

Examples using FDApy.simulation.KarhunenLoeve#

Smoothing of dense one-dimensional functional data

Smoothing of dense one-dimensional functional data

Smoothing of dense two-dimensional functional data

Smoothing of dense two-dimensional functional data

FPCA of 1-dimensional data

FPCA of 1-dimensional data

FPCA of 1-dimensional sparse data

FPCA of 1-dimensional sparse data

FPCA of 2-dimensional data

FPCA of 2-dimensional data

MFPCA of 1-dimensional data

MFPCA of 1-dimensional data

MFPCA of 2-dimensional data

MFPCA of 2-dimensional data

MFPCA of 1- and 2-dimensional data

MFPCA of 1- and 2-dimensional data

MFPCA of 1-dimensional sparse data

MFPCA of 1-dimensional sparse data

Simulation using Karhunen-Loève decomposition

Simulation using Karhunen-Loève decomposition

Simulation using multivariate Karhunen-Loève decomposition

Simulation using multivariate Karhunen-Loève decomposition

Simulation of functional data

Simulation of functional data

Simulation of clusters of univariate functional data

Simulation of clusters of univariate functional data

Simulation of clusters of multivariate functional data

Simulation of clusters of multivariate functional data