polynomial_exponents¶

sofia_redux.toolkit.resampling.polynomial_exponents(order, ndim=None, use_max_order=False)[source]¶

Define a set of polynomial exponents.

The resampling algorithm uses defines a set of polynomial exponents as an array of shape (dimensions, terms) for an equation of the form:

\[f( \Phi ) = \sum_{m=1}^{M}{c_m \Phi_m}\]

for \(M\) terms. Here, \(\Phi_m\) represents the product of independent variables, each raised to an appropriate power as defined by exponents. For example, consider the equation for 2-dimensional data with independent variables \(x\) and \(y\):

\[f(x, y) = c_1 + c_2 x + c_3 x^2 + c_4 y + c_5 x y + c_6 y^2\]

In this case:

exponents = [[0, 0],  # represents a constant or x^0 y^0
             [1, 0],  # represents x
             [2, 0],  # represents x^2
             [0, 1],  # represents y
             [1, 1],  # represents xy
             [0, 2]]  # represents y^2

The resampling algorithm solves for the coefficients (\(c\)) by converting \(f(X) \rightarrow f(\Phi)\) for \(K-\text{dimensional}\) independent variables (\(X\)) and exponents (\(p\)) by setting:

\[\Phi_m = \prod_{k=1}^{K}{X_{k}^{p_{m, k}}}\]

In most of the code, the \(\Phi\) terms are interchangable with “polynomial terms”, and in the above example \(\Phi_5 = xy\) since exponents[4] = [1, 1] representing \(x^1 y^1\).

Note that for all terms (\(m\)) in each dimension \(k\), \(\sum_{k=1}^{K}{p_{m, k}} \leq max(\text{order})\). In addition, if use_max_order is False (default), \(p_{m,k} \leq \text{order}[k]\).

Parameters:

orderint or array_like of int: Polynomial order for which to generate exponents. If an array will create full polynomial exponents over all len(order) dimensions.
ndimint, optional: If set, return Taylor expansion for ndim dimensions for the given order if order is not an array.
use_max_orderbool, optional: This keyword is only applicable for multi-dimensional data when orders are unequal across dimensions. When True, the maximum exponent for each dimension is equal to max(order). If False, the maximum available exponent for dimension k is equal to order[k].

Returns:

exponentsnumpy.ndarray: (n_terms, n_dimensions) array of polynomial exponents.

Examples

>>> polynomial_exponents(3)
array([[0],
       [1],
       [2],
       [3]])

>>> polynomial_exponents([1, 2])
array([[0, 0],
       [1, 0],
       [0, 1],
       [1, 1],
       [0, 2]])

>>> polynomial_exponents(3, ndim=2)
array([[0, 0],
       [1, 0],
       [2, 0],
       [3, 0],
       [0, 1],
       [1, 1],
       [2, 1],
       [0, 2],
       [1, 2],
       [0, 3]])

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polynomial_exponents¶