Toy (Quark) Model

ToyQuark([configs])

Implementation of a toy quark model, from the paper [1].

create_ToyQuark_model([Nq, x_grid])

This module provides functionality to create and test very basic toy models for debugging, performance evaluation, etc.

class ToyQuarkConfigs(*, debug=False, verbose=False, output_dir='./_PYOED_RESULTS_', model_name='Toy-Quark', screen_output_iter=1, file_output_iter=1, x_grid=<factory>, Nq=2)

Bases: TimeIndependentModelConfigs

Configuration class for the ToyQuark model

Parameters:

• verbose (bool) – a boolean flag to control verbosity of the object.

• debug (bool) – a boolean flag that enables adding extra functionlity in a debug mode

• output_dir (str | Path) – the base directory where the output files will be saved.

• screen_output_iter (int) – iteration interval for screen output. Default is 1. Note that this should be a positive integer to enforce proper effect.

• file_out_iter – iteration interval for file output. Default is 1. Note that this should be a positive integer to enforce proper effect.

• x_grid (ndarray) – discretization of the state space. Default is np.linspace(0, 1, 10). Must be a 1D numpy array.

• Nq (int) – number of quarks. Default is 2. Must be a postiive integer.

• model_name (str) – name of the model. Default is “Toy-Quark”

x_grid: ndarray

Nq: int

validate()

__init__(*, debug=False, verbose=False, output_dir='./_PYOED_RESULTS_', model_name='Toy-Quark', screen_output_iter=1, file_output_iter=1, x_grid=<factory>, Nq=2)

class ToyQuark(configs=None)

Bases: TimeIndependentModel

Implementation of a toy quark model, from the paper [1]. It is given by the expression

\[\begin{align*} \sigma(x) &= \sum_{i=1}^{N_q} q_i(x) \ q_i(x) &= N_i x^{\alpha_i} (1-x)^{\beta_i} \end{align*}\]

where \(N_i, \alpha_i, \beta_i\) are parameters to be estimated. Hence, this is a toy model with 3*N_q parameters. The model is defined on the interval \(x \in [0, 1]\) and the discretization of the state is left variable (i.e., the user can choose the number of discretization points).

[1] (1,2)

Hunt-Smith, N. T., Accardi, A., Melnitchouk, W., Sato, N., Thomas, A. W., & White, M. J. (2022). Determination of uncertainties in parton densities. In Physical Review D (Vol. 106, Issue 3). American Physical Society (APS). https://doi.org/10.1103/physrevd.106.036003

__init__(configs=None)

Create an instance of the toy quark model.

Parameters:

configs (ToyQuarkConfigs | dict | None) – configurations of the model. See ToyQuarkConfigs for the possible configurations. If None, default configurations are used.

validate_configurations(configs, raise_for_invalid=True)

Check the passed configuratios and make sure they are conformable with each other, and with current configurations once combined. This guarantees that any key-value pair passed in configs can be properly used

Note

Here only the locally-defined configurations in ToyQuarkConfigs are validated. Finally, super classes validators are called.

Parameters:

• configs (dict | ToyQuarkConfigs) – full or partial (subset) configurations to be validated

• raise_for_invalid (bool) – if True raise TypeError for invalid configrations type/key. Default True

Returns:

flag indicating whether passed configurations dictionary is valid or not

Raises:

• AttributeError – if any (or a group) of the configurations does not exist in the model configurations ToyQuarkConfigs.

• PyOEDConfigsValidationError – if the configurations are invalid and raise_for_invalid is set to True.

Return type:

bool

parameter_vector(init_val=0)

Create an instance of model parameter vector. This is a vector of size \(3N_q\), and corresponds to parameters in the order \([N_1, \alpha_1, \beta_1, \dots, N_{N_q}, \alpha_{N_q}, \beta_{N_q}]\).

Parameters:

init_val (float) – (optional) value assigned to entries of the state vector upon initialization

is_parameter_vector(parameter)

Test whether the passed parameter vector is valid or not

state_vector(init_val=0)

Create an instance of model state vector

Parameters:

init_val (float) – (optional) value assigned to entries of the state vector upon initialization

is_state_vector(state)

Test whether the passed state vector is valid or not

q_i(parameter, i)

Helper function to retrieve the i-th quark distribution function from the parameter vector. The parameter is assumed to be a vector of size \(3N_q\), and corresponds to parameters in the order \([N_1, \alpha_1, \beta_1, \dots, N_{N_q}, \alpha_{N_q}, \beta_{N_q}]\).

Parameters:

• parameter – parameter from which the i-th quark distribution function is retrieved.

• i (int) – index of the quark distribution function to retrieve. Note, this is 1-based indexing.

Returns:

\(q_i(x; p)\) where \(p\) is the parameter vector. This is of size len(self.x_grid).

solve_forward(parameter, verbose=False)

Apply (solve the forward model) to the given parameter, and return the resulting state. The parameter is assumed to be a vector of size \(3N_q\), and corresponds to parameters in the order \([N_1, \alpha_1, \beta_1, \dots, N_{N_q}, \alpha_{N_q}, \beta_{N_q}]\).

Parameters:

parameter – parameter to which the forward model is applied.

Returns:

\(\sigma(x; p)\) where \(p\) is the parameter vector. This is of size len(self.x_grid).

Jacobian_T_matvec(state, eval_at=None, verbose=False)

Multiply the Jacobian of the model (derivative of model equation w.r.t parameter) by the passed state

Parameters:

state – state to which the forward sensitivities are applied to.

Returns:

result resulting from applying the forward sensitivities to model to the passed state. This is of size \(3N_q\).

get_model_grid()

retrieve the model grid

property x_grid

retrieve the model grid

property Nq

retrieve the number of quarks

create_ToyQuark_model(Nq=2, x_grid=array([0.1, 0.18888889, 0.27777778, 0.36666667, 0.45555556, 0.54444444, 0.63333333, 0.72222222, 0.81111111, 0.9]))