Linear Regression Model#

class pylit.models.lrm.LinearRegressionModel(name, tau, params)#

Bases: object

Abstract base class for parametric linear regression models of the form

\[f(\omega) = \sum_{\boldsymbol{\alpha} \in \mathcal{I}} c_\boldsymbol{\alpha} \, K(\omega; \boldsymbol{\theta}_\boldsymbol{\alpha}),\]
where

  • \(\omega \in \mathbb{R}\) is the input variable,

  • \(\mathcal{I}\) is a multi–index set determined by the parameter configurations,

  • \(c_\boldsymbol{\alpha} \in \mathbb{R}\) are the model coefficients,

  • \(K(\cdot; \boldsymbol{\theta}_\boldsymbol{\alpha})\) is a kernel function depending on a parameter tuple \(\boldsymbol{\theta}_{\boldsymbol{\alpha}} = (\theta_{\alpha_1}, \ldots, \theta_{\alpha_p})\),

  • \(p = \mathrm{pdim}\) is the number of parameter groups.

Given

  • a finite set of nodes \(\tau = (\tau_1, \ldots, \tau_m) \in \mathbb{R}^m\),

  • discrete parameter sets \(\{ \theta_{j,k} \}_{k=1}^{n_j}\) for each parameter group \(j = 1,\ldots,p\),

the model constructs the Cartesian multi–index set

\[\mathcal{I} = \{ (i_1, \ldots, i_p) \mid 1 \leq i_j \leq n_j \},\]

with degree \(|\mathcal{I}| = \prod_{j=1}^p n_j\).

The regression matrix \(R \in \mathbb{R}^{m \times |\mathcal{I}|}\) is defined entrywise by

\[R_{r,\boldsymbol{\alpha}} = \mathcal{L}\big[K(\cdot; \boldsymbol{\theta}_\boldsymbol{\alpha}) \big](\tau_r),\]

where \(\mathcal{L}\) denotes a Laplace-type transformation specified in ltransform().

The forward model at the nodes is then

\[f(\tau) = R \boldsymbol{c},\]

and evaluation at arbitrary inputs \(\omega\) uses the kernel representation.

Notes

  • This class is abstract and should not be instantiated directly.

  • Concrete subclasses must override:

    • kernel() — defines \(K(\omega; \boldsymbol{\theta})\),

    • ltransform() — defines \(\mathcal{L}[K](\tau; \boldsymbol{\theta})\).

  • Coefficients \(\boldsymbol{c}\) are free variables, whereas \(R\) depends only on the chosen parameters and nodes.

__call__(omega, matrix=False)#

Evaluate the linear regression model at given inputs.

For input vector \(\omega = (\omega_1,\ldots,\omega_n) \in \mathbb{R}^n\), the method constructs the evaluation matrix

\[E_{i,\boldsymbol{\alpha}} = K(\omega_i; \boldsymbol{\theta}_{\boldsymbol{\alpha}}),\]

where \(\boldsymbol{\alpha} \in \mathcal{I}\) indexes the multi–index set of parameter combinations. The model evaluation is then

\[f(\omega) = E \boldsymbol{c},\]

with coefficient vector \(\boldsymbol{c} \in \mathbb{R}^d\), where \(d = |\mathcal{I}|\).

Parameters:

  • omega (ndarray) – One-dimensional array of inputs \(\omega\) at which the model should be evaluated.

  • matrix (bool) – If True, return the evaluation matrix \(E \in \mathbb{R}^{n \times d}\) instead of the model output. Defaults to False.

Returns:

  • If matrix=False: one-dimensional array of length n containing \(f(\omega)\).

  • If matrix=True: two-dimensional array of shape (n, d) containing the evaluation matrix \(E\).

Return type:

np.ndarray

Raises:

ValueError – If omega is not one-dimensional.

__init__(name, tau, params)#

Initializes the linear regression model.

Parameters:

  • name (str) – The name of the linear regression model.

  • tau (ndarray) – The discrete time axis of the linear regression model.

  • params (List[ndarray]) – The parameters of the linear regression model.

compute_regression_matrix()#

Compute the regression matrix.

The regression matrix \(R \in \mathbb{R}^{m \times d}\) encodes the evaluation of the Laplace-transformed kernels at the discrete time axis \(\tau\) with the given parameter grid \(\{ \boldsymbol{\theta}_{\alpha} \}\).

Returns:

Two-dimensional array of shape (m, d) containing the regression matrix \(R\).

Return type:

np.ndarray

Notes

The regression matrix depends only on tau and params, but not on coeffs.

forward()#

Evaluation of the Laplace-transformed model at the discrete time axis.

Computes the model values at the discrete time axis \(\tau\) using the regression matrix \(R\) and the coefficient vector \(\mathbf{c}\):

\[R \mathbf{c},\]
where

  • \(R\) is the regression matrix of shape (m, d),

  • \(\mathbf{c}\) is the coefficient vector of length d,

  • \(m\) is the number of nodes in the discrete time axis.

Returns:

One-dimensional array of length m containing the model values at the discrete time axis \(\tau\).

Return type:

np.ndarray

kernel(omega, param)#

Evaluate a single kernel function for a given set of parameters.

The kernel function \(K(\omega; \boldsymbol{\theta})\) defines the model contribution for a specific parameter tuple \(\boldsymbol{\theta} \in \Theta\). Concrete subclasses must implement this method.

Parameters:

  • omega (ndarray) – One-dimensional array representing the discrete frequency axis at which the kernel is evaluated.

  • param (List[ndarray]) – List of parameter arrays corresponding to the parameter tuple \(\boldsymbol{\theta}\) for this kernel.

Return type:

ndarray[float64]

Returns:

Values of the kernel function \(K(\omega; \boldsymbol{\theta})\) at the input frequencies.

Notes

This method must be overridden in the concrete child class to define the actual kernel function.

ltransform(tau, param)#

Evaluate the Laplace-transformed kernel function at the discrete time axis.

For a given parameter tuple \(\boldsymbol{\theta}\), this method computes the Laplace transform of the kernel function:

\[\mathcal{L}[K(\cdot; \boldsymbol{\theta})](\tau),\]

where \(\tau\) is the discrete time axis. Concrete subclasses must implement this method to define the Laplace-transformed model function.

Parameters:

  • tau (ndarray) – One-dimensional array representing the discrete time axis at which the Laplace transform is evaluated.

  • param (List[ndarray]) – List of parameter arrays corresponding to the parameter tuple \(\boldsymbol{\theta}\).

Return type:

ndarray

Returns:

Values of the Laplace-transformed kernel function at the discrete time axis.

Notes

This method must be overridden in the concrete child class to define the Laplace-transformed model.

property coeffs: ndarray#
property degree: int64#
property multi_index_set: ndarray[int64]#
property params: List[ndarray[float64]]#
property pdim: int64#
property regression_matrix: ndarray#
property tau: ndarray#