class Monomials

class deeptime.basis.Monomials(p: int, d: int)

Monomial basis observable which transforms a number of d-dimensional datapoints \(\mathbf{x}\in\mathbb{R}^d\) into (unique) monomials of at most degree \(p\).

This means, that

\[\mathbf{x} \mapsto \left\{ \prod_{d=1}^n \mathbf{x}_d^{k_d} : \sum k_d \leq p \right\}.\]

The set is returned as a numpy ndarray of shape (n_test_points, n_monomials), where n_monomials is the size of the set.

Parameters:

• p (int) – Maximum degree of the monomial basis. Must be positive.

• d (int) – The dimension of the input.

Examples

Given three test points in one dimension

>>> import numpy as np
>>> X = np.random.normal(size=(3, 1))

Evaluating the monomial basis up to degree two yields \(x^0, x^1, x^2\), i.e., the expected shape is (3, 3)

>>> Y = Monomials(p=2, d=1)(X)
>>> Y.shape
(3, 3)

and, e.g., the second monomial of the third test point is the third test point itself:

>>> np.testing.assert_almost_equal(Y[2, 1], X[2, 0])

Methods

get_feature_names([input_features])	See get_feature_names_out().
get_feature_names_out([input_features])	Yields a list of feature names, optionally given input feature names.
transform(data, **kwargs)	Transforms the input data.

__call__(x: ndarray)

Evaluation of the observable.

Parameters:

x ((N, d) np.ndarray) – Evaluates the observable for N d-dimensional data points.

Returns:

out – Result of the evaluation for each data point.

Return type:

(N, p) np.ndarray

get_feature_names(input_features: Optional[List[str]] = None) → List[str]

See get_feature_names_out().

get_feature_names_out(input_features: Optional[List[str]] = None) → List[str]

Yields a list of feature names, optionally given input feature names.

Parameters:

input_features (list of str, optional, default=None) – If not None, replaces the input feature names.

Returns:

feature_names – Feature names corresponding to each monomial.

Return type:

list of str

transform(data, **kwargs)

Transforms the input data.

Parameters:

data (array_like) – Input data.

Returns:

transformed – The transformed data

Return type:

array_like