queens.models package#

Models.

Modules for multi-query mapping of inputs to outputs, such as parameter samples to model evaluations.

Subpackages#

Submodules#

queens.models.adjoint module#

Adjoint model.

class Adjoint(scheduler, driver, gradient_driver, adjoint_file='adjoint_grad_objective.csv')[source]#

Bases: Simulation

Adjoint model.

adjoint_file#

Name of the adjoint file that contains the evaluated derivative of the functional w.r.t. to the simulation output.

Type:

str

gradient_driver#

Driver object for the adjoint simulation run.

Type:

Driver

grad(samples, upstream_gradient)[source]#

Evaluate gradient of model w.r.t. current set of input samples.

Consider current model f(x) with input samples x, and upstream function g(f). The provided upstream gradient is \(\frac{\partial g}{\partial f}\) and the method returns \(\frac{\partial g}{\partial f} \frac{df}{dx}\).

Parameters:

  • samples (np.array) – Input samples

  • upstream_gradient (np.array) – Upstream gradient function evaluated at input samples \(\frac{\partial g}{\partial f}\)

Returns:

gradient (np.array) – Gradient w.r.t. current set of input samples \(\frac{\partial g}{\partial f} \frac{df}{dx}\)

queens.models.bmfmc module#

Bayesian multi-fidelity Monte-Carlo model.

class BMFMC(parameters, global_settings, probabilistic_mapping, features_config, predictive_var, BMFMC_reference, y_pdf_support_max, y_pdf_support_min, X_cols=None, num_features=None, hf_model=None, path_to_lf_mc_data=None, path_to_hf_mc_reference_data=None)[source]#

Bases: Model

Bayesian multi-fidelity Monte-Carlo model.

Bayesian multi-fidelity Monte-Carlo model for uncertainty quantification, which is a probabilistic mapping between a high-fidelity simulation model (HF) and one or more low fidelity simulation models (LFs), respectively informative features \((\gamma)\) from the input space. Based on this mapping and the LF samples \(\mathcal{D}_{LF}^*=\\{Z^*,Y_{LF}^*\\}\), the BMFMC model computes the posterior statistics:

\(\mathbb{E}_{f^*}\left[p(y_{HF}^*|f^*,D_f)\right]\), equation (14) in [1] and

\(\mathbb{V}_{f^*}\left[p(y_{HF}^*|f^*,D_f)\right]\), equation (15) in [1]

of the HF model’s output uncertainty.

The BMFMC model is designed to be constructed upon the sampling data of a LF model \(\mathcal{D}_{LF}^*=\\{Z^*,Y_{LF}^*\\}\), that is provided by pickle or csv-files, and offers then different options to obtain the HF data:

  1. Provide HF training data in a file (Attention: The user needs to make sure that this training set is representative and its input \(Z\) is a subset of the LF model sampling set: \(Z\subset Z^*\)).

  2. Run optimal HF simulations based on LF data. This requires a suitable simulation sub-model and sub-iterator for the HF model.

  3. Provide HF sampling data (as a file), calculated with same \(Z^*\) as the LF simulation runs and select optimal HF training set from this data. This is just helpful for scientific benchmarking when a ground-truth solution for the HF output uncertainty has been sampled before and this data exists anyway.

parameters#

Parameters object

Type:

obj

interface#

Interface object.

Type:

obj

features_config#

strategy that will be used to calculate the low-fidelity features \(Z_{\text{LF}}\): opt_features, no_features or man_features

Type:

str

X_cols#

for man_features, columns of X-matrix that should be used as an informative feature

Type:

list

num_features#

for opt_features, number of features to be used

Type:

int

high_fidelity_model#

HF (simulation) model to run simulations that yield the HF training set \(\mathcal{D}_{HF}=\\{Z, Y_{HF}\\}\).

Type:

obj

X_train#

Matrix of simulation inputs corresponding to the training data-set of the multi-fidelity mapping.

Type:

np.array

Y_HF_train#

Vector or matrix of HF output that correspond to training input according to \(Y_{HF} = y_{HF}(X)\).

Type:

np.array

Y_LFs_train#

Output vector/matrix of one or multiple LF models that correspond to the training input according to \(Y_{LF,i}=y_{LF,i}(X)\).

Type:

np.array

X_mc#

Matrix of simulation inputs that were used in the Monte-Carlo sampling of the LF models. Each row is one input set for a simulation. Columns refer to different field_realizations of the same variable.

Type:

np.array

Y_LFs_mc#

Output vector/matrix for the LF models that correspond to the X_mc according to \(Y_{LF,i}^*=y_{LF,i}(X^*)\). At the moment Y_LF_mc contains in one row scalar results for different LF models. (In the future we will change the format to pandas dataframes to handle vectorized/functional outputs for different models more elegantly).

Type:

np.array

Y_HF_mc#

(optional for benchmarking) Output vector/matrix for the HF model that corresponds to the X_mc according to \(Y_{HF}^*=y_{HF}(X^*)\).

Type:

np.array

gammas_ext_mc#

Matrix of extended low-fidelity informative features \(\boldsymbol{\Gamma}^*\) corresponding to Monte-Carlo input \(X^*\).

Type:

np.array

gammas_ext_train#

Matrix of extended low-fidelity informative features \(\boldsymbol{\Gamma}\) corresponding to the training input \(X\).

Type:

np.array

Z_train#

Training matrix of low-fidelity features according to \(Z=\left[y_{LF,i}(X),\Gamma\right]\).

Type:

np.array

Z_mc#

Monte-Carlo matrix of low-fidelity features according to \(Z^*=\left[y_{LF,i}(X^*),\Gamma^*\right]\).

Type:

np.array

m_f_mc#

Vector of posterior mean values of multi-fidelity mapping corresponding to the Monte-Carlo input Z_mc according to \(\mathrm{m}_{f^*}(Z^*)\).

Type:

np.array

var_y_mc#

Vector of posterior variance of multi-fidelity mapping corresponding to the Monte-Carlo input Z_mc according to \(\mathrm{m}_{f^*}(Z^*)\).

Type:

np.array

p_yhf_mean#

Vector that contains the mean approximation of the HF output density defined on y_hf_support. The vector p_yhf_mean is defined as: \(\mathbb{E}_{f^*}\left[p(y_{HF}^*|f^*,D_f)\right]\) according to eq. (14) in [1].

Type:

np.array

p_yhf_var#

Vector that contains the variance approximation of the HF output density defined on y_hf_support. The vector p_yhf_var is defined as: \(\mathbb{V}_{f^*}\left[p(y_{HF}^*|f^*,D_f)\right]\) according to eq. (15) in [1].

Type:

np.array

predictive_var_bool#

Flag that determines whether p_yhf_var should be computed.

Type:

bool

p_yhf_mc#

(optional) Monte-Carlo based kernel-density estimate of the HF output.

Type:

np.array

p_ylf_mc#

(optional) Kernel density estimate for LF model output. Note: For BMFMC the explicit density is never required, only the \(\mathcal{D}_{LF}\) is used in the algorithm.

Type:

np.array

no_features_comparison_bool#

If flag is True, the result will be compared to a prediction that used no LF input features.

Type:

bool

eigenfunc_random_fields#

Matrix containing the discretized eigenfunctions of a underlying random field. Note: This is an intermediate solution and should be moved to the variables module! The current solution works so far only for one random field!

Type:

np.array

eigenvals#

Eigenvalues corresponding to the eigenfunctions.

f_mean_train#

Vector of predicted mean values of multi-fidelity mapping corresponding to the training input Z_train according to \(\mathrm{m}_{f^*}(Z)\).

Type:

np.array

y_pdf_support#

Support grid for HF output density \(p(y_{HF})\).

Type:

np.array

lf_data_iterators#

Data iterators to load sampling data of low-fidelity models from a file.

Type:

obj

hf_data_iterator#

Data iterator to load the benchmark sampling data from a HF model from a file (optional and only for scientific benchmark).

Type:

obj

training_indices#

Vector with indices to select the training data subset from the larger data set of Monte-Carlo data.

Type:

np.array

uncertain_parameters#

UncertainParameters object.

Type:

obj

visualization#

BMFMC visualization object.

Type:

BMFMCVisualization

References

[1] Nitzler, J., Biehler, J., Fehn, N., Koutsourelakis, P.-S. and Wall, W.A. (2020),

“A Generalized Probabilistic Learning Approach for Multi-Fidelity Uncertainty Propagation in Complex Physical Simulations”, arXiv:2001.02892

build_approximation(approx_case=True)[source]#

Train surrogate model.

Construct the probabilistic surrogate/mapping based on the provided training-data and optimize the hyper-parameters by maximizing the data’s evidence or its lower bound (ELBO).

Parameters:

approx_case (bool) – Boolean that switches input features \(\boldsymbol{\gamma}\) off if set to False. If not specified or set to True informative input features will be used in the BMFMC framework

calculate_extended_gammas()[source]#

Calculate extended input features.

Given the low-fidelity sampling data, calculate the extended input features \(\boldsymbol{\gamma_{\text{LF,ext}}}\). The informative input features \(\boldsymbol{\gamma}\) are calculated so that they would maximize the Pearson correlation coefficient between \(\boldsymbol{\gamma_i^*}\) and \(Y_{\text{LF}}^*\). Afterwards \(z_{\text{LF}}\) is composed by \(y_{\text{LF}}\) and \(\boldsymbol{\gamma_{LF}}\).

calculate_p_yhf_mean()[source]#

Calculate the posterior mean estimate for the HF density.

calculate_p_yhf_var()[source]#

Calculate the posterior variance of the HF density prediction.

compute_pyhf_statistics()[source]#

Compute high-fidelity output density prediction.

Calculate the high-fidelity output density prediction p_yhf_mean and its credible bounds p_yhf_var on the support y_pdf_support according to equation (14) and (15) in [1].

compute_pymc_reference()[source]#

Compute reference kernel density estimate.

Given a high-fidelity Monte-Carlo benchmark dataset, compute the reference kernel density estimate for the quantity of interest and optimize the bandwidth of the kde.

evaluate(samples)[source]#

Evaluate the Bayesian Multi-Fidelity Monte-Carlo (BMFMC) model.

This method evaluates the BMFMC routine by performing two main tasks:

  1. Probabilistic Mapping Evaluation: Evaluates the probabilistic mapping for both the LF

    Monte Carlo points and the LF training points.

  2. BMFMC Posterior Statistics Evaluation: Uses the results from the probabilistic

    mapping to compute the BMFMC posterior statistics.

Parameters:

samples (any type) – Currently not used.

Returns:

output (dict) –

A dictionary containing the results of the evaluation. The dictionary

includes:

  • Z_mc: (np.array) Low-fidelity features Monte-Carlo data used in the

    evaluation.

  • m_f_mc: (np.array) Posterior mean values of the probabilistic mapping

    (denoted f) for LF Monte-Carlo inputs. These values represent the mean predictions based on the Monte-Carlo simulations.

  • var_y_mc: (np.array) Posterior variance of the probabilistic mapping

    (denoted f) for LF Monte-Carlo inputs. These values indicate the uncertainty associated with the mean predictions.

  • y_pdf_support: (np.array) Support grid for the probability density function

    (PDF) of the HF output used in this analysis. This provides the range over which the PDF is evaluated.

  • p_yhf_mean: (np.array) Posterior mean prediction of the HF output PDF,

    representing the expected values of the HF output given the model and data.

  • p_yhf_var: (np.array) Posterior variance prediction of the HF output PDF,

    indicating the uncertainty in the HF output predictions.

  • p_yhf_mean_BMFMC: (np.array or None) Reference posterior mean prediction of

    the HF output PDF without using informative features, for comparison purposes. This value is calculated if self.no_features_comparison_bool is True.

  • p_yhf_var_BMFMC: (np.array or None) Reference posterior variance prediction of

    the HF output PDF without using informative features, for comparison purposes. This value is calculated if self.no_features_comparison_bool is True.

  • p_ylf_mc: (np.array) Kernel density estimate of the LF model output PDF, used

    for illustration purposes.

  • p_yhf_mc: (np.array) Kernel density estimate of the HF model output PDF based

    on a full Monte-Carlo simulation. Used for benchmarking and comparison.

  • Z_train: (np.array) LF feature vector used for training the probabilistic

    mapping. This contains the features that were used to fit the model.

  • X_train: (np.array) Input matrix for simulations used to train the

    probabilistic mapping, corresponding to the training data.

  • Y_HF_train: (np.array) Outputs of the high-fidelity model corresponding to the

    training inputs X_train, representing the HF output values used for model fitting.

get_hf_training_data()[source]#

Get high-fidelity training data.

Given the low-fidelity sampling data and the optimal training input \(X\), either simulate the high-fidelity response for \(X\) or load the corresponding high-fidelity response from the high-fidelity benchmark data provided by a pickle file.

get_random_fields_and_truncated_basis(explained_var=95.0)[source]#

Get random fields and their truncated basis.

Get the random fields and their description from the data files (pickle-files) and return their truncated basis. The truncation is determined based on the explained variance threshold (explained_var).

Parameters:

explained_var (float) – Threshold for truncation in percent.

Returns:

  • random_fields_trunc_dict (dict) – Dictionary containing samples of the random fields as well as their truncated basis

  • x_uncorr (np.array) – Samples of remaining uncorrelated random variables.

grad(samples, upstream_gradient)[source]#

Evaluate gradient of model w.r.t. current set of input samples.

Consider current model f(x) with input samples x, and upstream function g(f). The provided upstream gradient is \(\frac{\partial g}{\partial f}\) and the method returns \(\frac{\partial g}{\partial f} \frac{df}{dx}\).

Parameters:

  • samples (np.array) – Input samples

  • upstream_gradient (np.array) – Upstream gradient function evaluated at input samples \(\frac{\partial g}{\partial f}\)

Returns:

gradient (np.array) – Gradient w.r.t. current set of input samples \(\frac{\partial g}{\partial f} \frac{df}{dx}\)

input_dim_red()[source]#

Reduce dimensionality of the input space.

Unsupervised dimensionality reduction of the input space. The random are first expressed via a truncated Karhunen-Loeve expansion that still contains, e.g. 95 % of the field’s variance. Afterwards, input samples of the random fields get projected on the reduced basis and the coefficients of the projection sever as the new reduced encoding for the latter. Eventually the uncorrelated input samples and the reduced representation of random field samples get assembled to a new reduced input vector which is also standardized along each of the remaining dimensions.

Returns:

  • X_red_test_stdizd (np.array) – Dimensionality reduced input matrix corresponding to

  • testing/sampling data for the probabilistic mapping

load_sampling_data()[source]#

Load sampling data.

Load the low-fidelity sampling data from a pickle file into QUEENS. Check if high-fidelity benchmark data is available and load this as well.

run_BMFMC()[source]#

Run BMFMC.

Run with additional informative features.

run_BMFMC_without_features()[source]#

Run BMFMC without further informative LF features.

Returns:

  • p_yhf_mean_BMFMC (np.array) – Posterior mean function of the HF output density

  • p_yhf_var_BMFMC (np.array) – Posterior variance function of the HF output density

set_feature_strategy()[source]#

Set feature strategy.

Depending on the method specified in the input file, set the strategy that will be used to calculate the low-fidelity features \(Z_{\text{LF}}\).

update_probabilistic_mapping_with_features()[source]#

Update probabilistic mapping.

Given the number of additional informative features of the input and the extended feature matrix \(\Gamma_{LF,ext}\), assemble first the LF feature matrix \(Z_{LF}\). In a next step, update the probabilistic mapping with the LF-features. The former steps includes a training and prediction step. The determination of optimal training points is outsourced to the BMFMC iterator and the results get only called at this place.

class BmfmcInterface(probabilistic_mapping)[source]#

Bases: object

Interface for grouping outputs with inputs.

Interface for grouping the outputs of several simulations with identical input to one data point. The BmfmcInterface is basically a version of the approximation_interface class, that allows vectorized mapping and implicit function relationships.

probabilistic_mapping#

Instance of the probabilistic mapping, which models the probabilistic dependency between high-fidelity model, low-fidelity models and informative input features.

Type:

obj

Returns:

BMFMCInterface (obj) – Instance of the BMFMCInterface

build_approximation(Z_LF_train, Y_HF_train)[source]#

Build and train the probabilistic mapping.

Based on the training inputs \(\mathcal{D}_f={Y_{HF},Z_{LF}}\).

Parameters:

  • Z_LF_train (np.array) – Training inputs for probabilistic mapping

  • Y_HF_train (np.array) – Training outputs for probabilistic mapping

evaluate(samples, support='y', full_cov=False, gradient_bool=False)[source]#

Predict on probabilistic mapping.

Call the probabilistic mapping and predict the mean and variance for the high-fidelity model, given the inputs z_lf (called samples here).

Parameters:

  • samples (np.array) – Low-fidelity feature vector z_lf that contains the corresponding Monte-Carlo points on which the probabilistic mapping should be evaluated

  • support (str) – Support/variable for which we predict the mean and (co)variance. For support=f the Gaussian process predicts w.r.t. the latent function f. For the choice of support=y we predict w.r.t. the simulation/experimental output y

  • full_cov (bool) – Boolean that specifies whether the entire posterior covariance matrix should be returned or only the posterior variance

  • gradient_bool (bool) – Flag to determine whether the gradient of the function at the evaluation point is expected (True) or not (False)

Returns:

  • mean_Y_HF_given_Z_LF (np.array) – Vector of mean predictions \(\mathbb{E}_{f^*}[p(y_{HF}^*|f^*,z_{LF}^*, \mathcal{D}_{f})]\) for the HF model given the low-fidelity feature input

  • var_Y_HF_given_Z_LF (np.array) – Vector of variance predictions \(\mathbb{V}_{f^*}[p(y_{HF}^*|f^*,z_{LF}^*,\mathcal{D}_{f})]\) for the HF model given the low-fidelity feature input

assemble_x_red_stdizd(x_uncorr, coef_mat)[source]#

Assemble and standardize the dimension-reduced input x_red.

Parameters:

  • x_uncorr (np.array) – Samples of remaining uncorrelated random variables.

  • coef_mat (np.array) – Optional coefficient matrix to concatenate.

Returns:

X_red_test_stdizd (np.array) – Standardized dimension-reduced input.

linear_scale_a_to_b(data_a, data_b)[source]#

Scale linearly.

Scale a data vector ‘data_a’ linearly to the range of data vector ‘data_b’.

Parameters:

  • data_a (np.array) – Data vector that should be scaled.

  • data_b (np.array) – Reference data vector that provides the range for scaling.

Returns:

scaled_a (np.array) – Scaled data_a vector.

project_samples_on_truncated_basis(truncated_basis_dict, num_samples)[source]#

Project samples on truncated basis.

Project the high-dimensional samples of the random field on the truncated bases to yield the projection coefficients of the series expansion that serve as a new reduced representation of the random fields.

Parameters:

  • truncated_basis_dict (dic) – Dictionary containing random field samples and truncated bases.

  • num_samples (int) – Number of Monte-Carlo samples.

Returns:

coefs_mat (np.array) – Matrix containing the reduced representation of all random fields stacked together along the columns.

queens.models.finite_difference module#

Finite difference model.

class FiniteDifference(scheduler, driver, finite_difference_method, step_size=1e-05, bounds=None)[source]#

Bases: Simulation

Finite difference model.

finite_difference_method#

Method to calculate a finite difference based approximation of the Jacobian matrix:

  • ‘2-point’: a one-sided scheme by definition

  • ‘3-point’: more exact but needs twice as many function

    evaluations

Type:

str

step_size#

Step size for the finite difference approximation

Type:

float

bounds#

Lower and upper bounds on independent variables. Defaults to no bounds meaning: [-inf, inf] Each bound must match the size of x0 or be a scalar, in the latter case the bound will be the same for all variables. Use it to limit the range of function evaluation.

Type:

np.array

evaluate(samples)[source]#

Evaluate model with current set of input samples.

Parameters:

samples (np.ndarray) – Input samples

Returns:

response (dict) – Response of the underlying model at input samples

evaluate_finite_differences(samples)[source]#

Evaluate model gradient based on FDs.

Parameters:

samples (np.array) – Current samples at which model should be evaluated.

Returns:

  • response (np.array) – Array with model response for given input samples

  • gradient_response (np.array) – Array with row-wise model/objective fun gradients for given samples.

grad(samples, upstream_gradient)[source]#

Evaluate gradient of model w.r.t. current set of input samples.

Consider current model f(x) with input samples x, and upstream function g(f). The provided upstream gradient is \(\frac{\partial g}{\partial f}\) and the method returns \(\frac{\partial g}{\partial f} \frac{df}{dx}\).

Parameters:

  • samples (np.array) – Input samples

  • upstream_gradient (np.array) – Upstream gradient function evaluated at input samples \(\frac{\partial g}{\partial f}\)

Returns:

gradient (np.array) – Gradient w.r.t. current set of input samples \(\frac{\partial g}{\partial f} \frac{df}{dx}\)

queens.models.logpdf_gp module#

GPLogpdf model.

class LogpdfGP(approx_type, num_hyper=100, num_optimizations=3, hmc_burn_in=1000, hmc_steps=2000, prior_rate=None, prior_gp_mean=-1.0, upper_bound=None, quantile=0.9, jitter=1e-16)[source]#

Bases: Model

LogpdfGP Class.

approx_type#

Approximation type (GPMAP-I’, ‘CGPMAP-II’ or ‘CFBGP’)

Type:

str

num_hyper#

Number of hyperparameter samples (if CFBGP)

Type:

int

num_optimizations#

Number of hyperparameter optimization restarts

Type:

int

hmc_burn_in#

Number of HMC burn-in steps (if CFBGP)

Type:

int

hmc_steps#

Number of HMC steps (if CFBGP)

Type:

int

prior_rate#

Rates of exponential priors for hyperparameters [lengthscales, signal_std, noise_var]

Type:

np.ndarray

prior_gp_mean#

Transformed prior GP mean. Range: [-1, 0]

Type:

float

upper_bound#

Transformed upper bound for Gaussian process. Range: [-1, 0]. If None provided, it is derived from the number of observations.

Type:

float

quantile#

Confidence quantile

Type:

float

jitter#

Nugget term for numerical stability of Cholesky decomposition

Type:

float

x_train#

Training input samples

Type:

np.ndarray

y_train#

Training likelihood output samples

Type:

np.ndarray

scaler_x#

Scaler for training input samples

Type:

float

scaler_y#

Scaler for training likelihood output samples

Type:

float

num_dim#

number of dimensions

Type:

np.ndarray

hyperparameters#

Hyperparameter (samples)

Type:

np.ndarray

chol_k_train_train#

Cholesky decomposition of Gram matrix evaluated at the training samples

Type:

np.ndarray

v_train#

Matrix product of inverse of Gram matrix evaluated at training samples and training output samples

Type:

np.ndarray

jit_func_generate_output#

Jitted partial ‘generate_output_*’ function

Type:

obj

partial_hyperparameter_log_prob#

Jitted partial function of hyperparameter log posterior probability

Type:

obj

batch_size#

Batch size for concurrent prediction evaluations

Type:

int

calc_train_factor(hyperparameters)[source]#

Calculate training factors.

Parameters:

hyperparameters (np.ndarray) – Hyperparameters

Returns:

  • chol_k_train_train (np.ndarray) – Cholesky decomposition of Gram matrix evaluated at the training samples

  • v_train (np.ndarray) – Matrix product of inverse of Gram matrix evaluated at training samples and training output samples

evaluate(samples)[source]#

Evaluate model with current set of input samples.

Parameters:

samples (np.ndarray) – Input samples

Returns:

log_likelihood (np.ndarray) – Approximation of log-likelihood values at input samples

static evaluate_mean_and_std(dists_test_train, hyperparameters, v_train, chol_k_train_train, prior_gp_mean, scaler_y)[source]#

Mean and standard deviation of unconstrained GP at test samples.

Parameters:

  • dists_test_train (np.ndarray) – Distance Matrix between test and train samples

  • hyperparameters (np.ndarray) – Hyperparameters

  • v_train (np.ndarray) – Matrix product of inverse of Gram matrix evaluated at training samples and training output samples

  • chol_k_train_train (np.ndarray) – Cholesky decomposition of Gram matrix evaluated at the training samples

  • prior_gp_mean (float, opt) – Transformed prior GP mean. Range: [-1, 0]

  • scaler_y (float) – Scaler for training likelihood output samples

Returns:

  • mean (np.ndarray) – Mean of unconstrained GP at test samples

  • std (np.ndarray) – Standard deviation of unconstrained GP at test samples.

static generate_output_cfbgp(x_test, x_train, hyperparameters, v_train, chol_k_train_train, eval_mean_and_std, upper_bound, quantile)[source]#

Approximation of log-likelihood using CGPMAP-II approach.

Parameters:

  • x_test (np.ndarray) – Input test samples

  • x_train (np.ndarray) – Training input samples

  • hyperparameters (np.ndarray) – Hyperparameters

  • v_train (np.ndarray) – Matrix product of inverse of Gram matrix evaluated at training samples and training output samples

  • chol_k_train_train (np.ndarray) – Cholesky decomposition of Gram matrix evaluated at the training samples

  • eval_mean_and_std (obj) – eval_mean_and_std function

  • upper_bound (float) – Transformed upper bound for Gaussian process

  • quantile (float) – Confidence quantile

Returns:

log_likelihood (np.ndarray) – Approximation of log-likelihood values at x_test

static generate_output_cgpmap_2(x_test, x_train, hyperparameters, v_train, chol_k_train_train, eval_mean_and_std, upper_bound, quantile)[source]#

Approximation of log-likelihood using CGPMAP-II approach.

Parameters:

  • x_test (np.ndarray) – Input test samples

  • x_train (np.ndarray) – Training input samples

  • hyperparameters (np.ndarray) – Hyperparameters

  • v_train (np.ndarray) – Matrix product of inverse of Gram matrix evaluated at training samples and training output samples

  • chol_k_train_train (np.ndarray) – Cholesky decomposition of Gram matrix evaluated at the training samples

  • eval_mean_and_std (obj) – eval_mean_and_std function

  • upper_bound (float) – Transformed upper bound for Gaussian process

  • quantile (float) – Confidence quantile

Returns:

log_likelihood (np.ndarray) – Approximation of log-likelihood values at x_test

static generate_output_gpmap_1(x_test, x_train, hyperparameters, v_train, prior_gp_mean, scaler_y)[source]#

Approximation of log-likelihood using CGPMAP-II approach.

Parameters:

  • x_test (np.ndarray) – Input test samples

  • x_train (np.ndarray) – Training input samples

  • hyperparameters (np.ndarray) – Hyperparameters

  • v_train (np.ndarray) – Matrix product of inverse of Gram matrix evaluated at training samples and training output samples

  • prior_gp_mean (float, opt) – Transformed prior GP mean

  • scaler_y (float) – Scaler for training likelihood output samples

Returns:

log_likelihood (np.ndarray) – Approximation of log-likelihood values at x_test

grad(samples, upstream_gradient)[source]#

Evaluate gradient of model w.r.t.

current set of input samples.

static hyperparameter_log_likelihood(hyperparameters, x_train, y_train, jitter)[source]#

Log likelihood function for hyperparameters.

Parameters:

  • hyperparameters (np.ndarray) – Hyperparameters

  • x_train (np.ndarray) – Training input samples

  • y_train (np.ndarray) – Training output samples

  • jitter (float) – Nugget term for numerical stability of Cholesky decomposition

Returns:

log_likelihood (np.ndarray) – Log likelihood of data given hyperparameters

static hyperparameter_log_prior(hyperparameters, prior_rate)[source]#

Log prior function for hyperparameters.

Parameters:

  • hyperparameters (np.ndarray) – Hyperparameters

  • prior_rate (np.ndarray) – prior_rate (np.ndarray): Rates of exponential priors for hyperparameters

Returns:

joint_log_prior (np.ndarray) – Joint log prior of hyperparameters

static hyperparameter_log_prob(transformed_hyperparameters, x_train, y_train, jitter, log_likelihood_func, log_prior_func, prior_rate)[source]#

Log joint probability function for hyperparameters.

Parameters:

  • transformed_hyperparameters (np.ndarray) – Transformed hyperparameters

  • x_train (np.ndarray) – Training input samples

  • y_train (np.ndarray) – Training output samples

  • jitter (float) – Nugget term for numerical stability of Cholesky decomposition

  • log_likelihood_func (obj) – Log likelihood function of hyperparameters

  • log_prior_func (obj) – log prior function of hyperparameters

  • prior_rate (np.ndarray) – prior_rate (np.ndarray): Rates of exponential priors for hyperparameters

Returns:

np.ndarray – Log joint probability of hyperparameters

initialize(x_train, y_train, num_observations)[source]#

Initialize Gaussian process model.

Parameters:

  • x_train (np.ndarray) – Training input samples

  • y_train (np.ndarray) – Training likelihood output samples

  • num_observations (int) – Number of observations

optimize_hyperparameters()[source]#

Optimize hyperparameters.

Returns:

hyperparameters (np.ndarray) – Optimized hyperparameters

sample_hyperparameters()[source]#

Draw samples from hyperparameter posterior.

Returns:

hyperparameters (np.ndarray) – Samples of hyperparameter posterior

distances(x1, x2)[source]#

Distance Matrix between two sample sets.

Parameters:

  • x1 (np.ndarray) – input samples

  • x2 (np.ndarray) – input samples

Returns:

dists (np.ndarray) – Distance Matrix

rbf(x1, x2, hyperparameters)[source]#

Gram Matrix of RBF Kernel.

Parameters:

  • x1 (np.ndarray) – input samples

  • x2 (np.ndarray) – input samples

  • hyperparameters (np.ndarray) – hyperparameters of kernel

Returns:

k_x1_x2 (np.ndarray) – Gram Matrix of RBF Kernel

rbf_by_dists(dists, hyperparameters)[source]#

Gram Matrix of RBF Kernel.

Parameters:

  • dists (np.ndarray) – Distance Matrix

  • hyperparameters (np.ndarray) – hyperparameters of kernel

Returns:

k_x1_x2 (np.ndarray) – Gram Matrix of RBF Kernel

queens.models.simulation module#

Simulation model class.

class Simulation(scheduler, driver)[source]#

Bases: Model

Simulation model class.

scheduler#

Scheduler for the simulations

Type:

Scheduler

driver#

Driver for the simulations

Type:

Driver

evaluate(samples)[source]#

Evaluate model with current set of input samples.

Parameters:

samples (np.ndarray) – Input samples

Returns:

response (dict) – Response of the underlying model at input samples

grad(samples, upstream_gradient)[source]#

Evaluate gradient of model w.r.t. current set of input samples.

Consider current model f(x) with input samples x, and upstream function g(f). The provided upstream gradient is \(\frac{\partial g}{\partial f}\) and the method returns \(\frac{\partial g}{\partial f} \frac{df}{dx}\).

Parameters:

  • samples (np.array) – Input samples

  • upstream_gradient (np.array) – Upstream gradient function evaluated at input samples \(\frac{\partial g}{\partial f}\)

Returns:

gradient (np.array) – Gradient w.r.t. current set of input samples \(\frac{\partial g}{\partial f} \frac{df}{dx}\)

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