class GaussianKernel

class deeptime.kernels.GaussianKernel(sigma, impl='cdist')

Implementation of the Gaussian kernel

$$\kappa (x,y) = \exp \left(-\sum_k (x_k - y_k)^2 / (2\sigma^2)\right),$$

where $\sigma$ is the bandwidth of the kernel.

Parameters:

sigma (float) – The bandwidth.

Attributes

impl
sigma	Bandwidth of the Gaussian kernel.
valid_impls	Valid implementation modes.

Methods

apply(data_1, data_2)	Applies the kernel to data arrays.
gram(data)	Computes the corresponding Gram matrix, see also apply().

__call__(x, y)

Computes the value of the kernel at two specific points.

Parameters:

• x ((d,) ndarray) – X point.

• y ((d,) ndarray) – Y point.

Returns:

kxy – The kernel evaluation $\kappa (x, y)$.

Return type:

float

Notes

This dispatches to _evaluate() which is an interface method intended to be overridden by a specific kernel implementation.

apply(data_1: ndarray, data_2: ndarray) → ndarray

Applies the kernel to data arrays.

Given $x\in\mathbb{R}^{T_1 \times d}$ and $y\in\mathbb{R}^{T_2\times d}$, it yields a kernel matrix

$$K = (\kappa(x_i, y_j))_{i,j}\in\mathbb{R}^{T_1\times T_2}.$$

Note that this corresponds to the kernel Gramian in case $x = y$.

Parameters:

• data_1 ((T_1, d) ndarray) – Data array.

• data_2 ((T_2, d) ndarray) – Data array.

Returns:

K – The kernel matrix.

Return type:

(T_1, T_2) ndarray

gram(data: ndarray) → ndarray

Computes the corresponding Gram matrix, see also apply().

Parameters:

data ((T, d) ndarray) – The data array.

Returns:

G – The kernel Gramian with G[i, j] = k(x[i], x[j]).

Return type:

(T, T) ndarray

property sigma: ndarray

Bandwidth of the Gaussian kernel.

Getter:

Yields the bandwidth.

Type:

(1,) ndarray

valid_impls = ('cdist', 'binomial')

Valid implementation modes.