MaternHalfIntCovariance

Matern half-integer covariance function.

Overview

A special case of the Matern class of covariance functions, in which the $var element = document.getElementById("moose-equation-aafb1cfb-14c3-48e1-9a34-ab6310a60e2c");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ hyperparameter takes on a half integer value. Substituting a positive integer $var element = document.getElementById("moose-equation-73f466f2-763a-42a7-bf2b-1fc10b64147b");katex.render("p", element, {displayMode:false,throwOnError:false});$ for $var element = document.getElementById("moose-equation-2fb14450-f50f-4e51-aa62-3872e355c180");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ using $var element = document.getElementById("moose-equation-bd2af87e-ee51-46b4-ae49-40f8083a8319");katex.render("\\nu = p + 1/2", element, {displayMode:false,throwOnError:false});$ the covariance function is given as

var element = document.getElementById("moose-equation-ba0b9cb7-b9ac-4466-bdbf-bca58416d6e4");katex.render("k(x,x^\\prime) = \\sigma_f^2 \\, exp \\left(- \\sqrt{2p+1} r_\\ell(x,x^\\prime) \\right) \\frac{p!}{(2p)!} \\sum_{i=0}^p \\left[ \\frac{(p+i)!}{i! (2p-i)!} \\left( 2 \\sqrt{2p+1} r_\\ell(x,x^\\prime) \\right)^{p-i} \\right] + \\sigma_n^2 \\, \\delta_{x,x^\\prime}.", element, {displayMode:true,throwOnError:false});

$var element = document.getElementById("moose-equation-62aae311-62d5-40bd-8c3c-850b03e2984f");katex.render("r_\\ell(x,x^\\prime)", element, {displayMode:false,throwOnError:false});$ is a scaled distance based on the length factor $var element = document.getElementById("moose-equation-eab6548d-cb69-460b-bbe9-125289120d32");katex.render("\\vec{\\ell}", element, {displayMode:false,throwOnError:false});$ , defined as

var element = document.getElementById("moose-equation-286eccb5-037e-4e35-99c7-110eebc359c7");katex.render("r_\\ell(x,x^\\prime) = \\sqrt{ \\sum_n \\left( \\frac{x_i - x^\\prime_i}{\\ell_i} \\right)^2}.", element, {displayMode:true,throwOnError:false});

Hyperparameters

Table 1: Hyperparameters for Matern Covariance Function

Variable	Domain	Description
$var element = document.getElementById("moose-equation-377f531a-a9ca-4902-9d16-5f0c4f580fd3");katex.render("\\vec{\\ell}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-d9f5e482-b853-4376-b4fd-ba6eb3a6df4d");katex.render("\\mathbb{R}_{>0}^n", element, {displayMode:false,throwOnError:false});$	Length factors corresponding to input parameters*
$var element = document.getElementById("moose-equation-2f2f24f7-3190-4ef6-b7e2-1dca43f9317e");katex.render("\\sigma_f", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-2764bf15-e16e-4e72-b56c-02a1e8d44a8b");katex.render("\\mathbb{R}_{\\geq 0}", element, {displayMode:false,throwOnError:false});$	Signal variance*
$var element = document.getElementById("moose-equation-d75274d1-42dc-4640-8dcb-c08375b8ceb2");katex.render("\\sigma_n", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-d649a1a4-db5e-45c8-892f-e992a230078c");katex.render("\\mathbb{R}_{\\geq 0}", element, {displayMode:false,throwOnError:false});$	Noise variance*
$var element = document.getElementById("moose-equation-545827ad-484d-4615-b747-0de3c6dbbdc1");katex.render("p", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-1f3dd63c-1a57-4dc1-8e79-c7f5cf3bb975");katex.render("\\mathbb{N}", element, {displayMode:false,throwOnError:false});$

*See the Gaussian Process Trainer documentation for more in depth explanation of $var element = document.getElementById("moose-equation-c285c838-4c8e-47c5-8b99-22fa68f92899");katex.render("\\vec{\\ell}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-4300f2e8-a469-45f0-9714-469071aee747");katex.render("\\sigma_f", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-f0c30342-5a5e-4e0c-a2ab-b546c390e7e9");katex.render("\\sigma_n", element, {displayMode:false,throwOnError:false});$ hyperparameters.

Example Input File Syntax

[Covariance<<<{"href": "../../syntax/Covariance/index.html"}>>>]
  [covar]
    type = MaternHalfIntCovariance<<<{"description": "Matern half-integer covariance function.", "href": "MaternHalfIntCovariance.html"}>>>
    p<<<{"description": "Integer p to use for Matern Half Integer Covariance Kernel"}>>> = 2 #Define the exponential factor
    signal_variance<<<{"description": "Signal Variance ($\\sigma_f^2$) to use for kernel calculation."}>>> = 1 #Use a signal variance of 1 in the kernel
    noise_variance<<<{"description": "Noise Variance ($\\sigma_n^2$) to use for kernel calculation."}>>> = 1e-6 #A small amount of noise can help with numerical stability
    length_factor<<<{"description": "Length factors to use for Covariance Kernel"}>>> = '0.551133 0.551133' #Select a length factor for each parameter (k and q)
  []
[]

Input Parameters

length_factorLength factors to use for Covariance Kernel
C++ Type:std::vector<double>
Unit:(no unit assumed)
Controllable:No
Description:Length factors to use for Covariance Kernel
pInteger p to use for Matern Half Integer Covariance Kernel
C++ Type:unsigned int
Controllable:No
Description:Integer p to use for Matern Half Integer Covariance Kernel
signal_varianceSignal Variance ($\sigma_f^2$) to use for kernel calculation.
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Signal Variance ($\sigma_f^2$) to use for kernel calculation.

Required Parameters

covariance_functionsCovariance functions that this covariance function depends on.
C++ Type:std::vector<UserObjectName>
Controllable:No
Description:Covariance functions that this covariance function depends on.
noise_variance0Noise Variance ($\sigma_n^2$) to use for kernel calculation.
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Noise Variance ($\sigma_n^2$) to use for kernel calculation.
num_outputs1The number of outputs expected for this covariance function.
Default:1
C++ Type:unsigned int
Controllable:No
Description:The number of outputs expected for this covariance function.

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:No
Description:Set the enabled status of the MooseObject.

Advanced Parameters

(moose/modules/stochastic_tools/test/tests/surrogates/gaussian_process/GP_Matern_half_int.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Uniform
    lower_bound = 1
    upper_bound = 10
  []
  [q_dist]
    type = Uniform
    lower_bound = 9000
    upper_bound = 11000
  []
[]

[Samplers]
  [train_sample]
    type = MonteCarlo
    num_rows = 10
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
  [test_sample]
    type = MonteCarlo
    num_rows = 100
    distributions = 'k_dist q_dist'
    execute_on = PRE_MULTIAPP_SETUP
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = train_sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppSamplerControl
    multi_app = sub
    sampler = train_sample
    param_names = 'Materials/conductivity/prop_values Kernels/source/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = train_sample
    stochastic_reporter = results
    from_reporter = 'avg/value'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
    parallel_type = ROOT
  []
  [samp_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = test_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
  [train_avg]
    type = EvaluateSurrogate
    model = GP_avg
    sampler = train_sample
    evaluate_std = 'true'
    parallel_type = ROOT
    execute_on = final
  []
[]

[VectorPostprocessors]
  [hyperparams]
    type = GaussianProcessData
    gp_name = 'GP_avg'
    execute_on = final
  []
[]

[Trainers]
  [GP_avg_trainer]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    covariance_function = 'covar' #Choose a Matern with half-integer argument for the kernel
    standardize_params = 'true' #Center and scale the training params
    standardize_data = 'true' #Center and scale the training data
    sampler = train_sample
    response = results/data:avg:value
  []
[]

[Surrogates]
  [GP_avg]
    type = GaussianProcessSurrogate
    trainer = GP_avg_trainer
  []
[]

[Covariance]
  [covar]
    type = MaternHalfIntCovariance
    p = 2 #Define the exponential factor
    signal_variance = 1 #Use a signal variance of 1 in the kernel
    noise_variance = 1e-6 #A small amount of noise can help with numerical stability
    length_factor = '0.551133 0.551133' #Select a length factor for each parameter (k and q)
  []
[]

[Outputs]
  [out]
    type = CSV
    execute_on = FINAL
  []
[]

Overview
Hyperparameters
Example Input File Syntax
Input Parameters