LMC

The linear model of co-regionalization (LMC) distinctly models the covariances between the $var element = document.getElementById("moose-equation-a2ecd724-5467-4a8e-9f61-9e0e48683470");katex.render("N", element, {displayMode:false,throwOnError:false});$ inputs and the $var element = document.getElementById("moose-equation-9dfba597-42c2-4832-be95-4b12f3a83e1e");katex.render("M", element, {displayMode:false,throwOnError:false});$ outputs. Mathematically, the LMC is defined as (Liu et al., 2018; Cheng et al., 2020):

$(1)var element = document.getElementById("moose-equation-7be5e0ce-8162-40db-9e01-4e752efa2628");katex.render(" \\bar{\\pmb{K}} = \\sum_{q=1}^Q \\bar{\\pmb{B}}_q \\otimes \\pmb{K}_q", element, {displayMode:true,throwOnError:false});$

where, $var element = document.getElementById("moose-equation-79e5183c-2642-4116-b9c3-380193895636");katex.render("q", element, {displayMode:false,throwOnError:false});$ denotes the latent basis index, $var element = document.getElementById("moose-equation-57fd4d7e-1338-4165-b61b-19283bf224a4");katex.render("\\bar{\\pmb{B}}_q", element, {displayMode:false,throwOnError:false});$ output covariance matrix of size $var element = document.getElementById("moose-equation-6c4b5bad-b4be-46f9-a551-b2e44bcf44ae");katex.render("M \\times M", element, {displayMode:false,throwOnError:false});$ for the $var element = document.getElementById("moose-equation-4c02819c-a614-41d2-b899-4751c4c32ed1");katex.render("q", element, {displayMode:false,throwOnError:false});$ -th covariate, $var element = document.getElementById("moose-equation-2c627c0f-a626-4608-afe9-ad309f6e8e86");katex.render("\\pmb{K}_q", element, {displayMode:false,throwOnError:false});$ is the input covariance matrix of size $var element = document.getElementById("moose-equation-582987e3-21f0-4234-ac8f-3941d804368b");katex.render("N \\times N", element, {displayMode:false,throwOnError:false});$ for the $var element = document.getElementById("moose-equation-23813dd1-76d6-4686-b44a-eac44f37e729");katex.render("q", element, {displayMode:false,throwOnError:false});$ -th covariate, $var element = document.getElementById("moose-equation-62e6950e-2dbc-42fc-9923-d469ac2a0116");katex.render("Q", element, {displayMode:false,throwOnError:false});$ is the total number of basis functions, and $var element = document.getElementById("moose-equation-67a17b4f-0189-4461-bd2b-ba2229e0b52a");katex.render("\\otimes", element, {displayMode:false,throwOnError:false});$ denotes the Kronecker product. $var element = document.getElementById("moose-equation-924dd256-d0b4-4dec-aac0-79fd1567338d");katex.render("\\bar{\\pmb{B}}_q", element, {displayMode:false,throwOnError:false});$ is further defined as the sum of two matrices of weights (Cheng et al., 2020):

$(2)var element = document.getElementById("moose-equation-e917e165-8c86-43cb-8c68-3536c3c57cf2");katex.render(" \\bar{\\pmb{B}}_q = \\pmb{A}_q \\pmb{A}_q^\\intercal + \\textrm{diag}\\Big(\\pmb{\\lambda}_q\\Big)", element, {displayMode:true,throwOnError:false});$

where, $var element = document.getElementById("moose-equation-d7504068-c236-42a3-a256-2d14368ddff7");katex.render("\\pmb{A}_q", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-aea01315-2c8d-46a2-97d5-a377dc8bb6b9");katex.render("\\pmb{\\lambda}_q", element, {displayMode:false,throwOnError:false});$ are vectors (size $var element = document.getElementById("moose-equation-0a28ab6d-bd72-453a-a9e2-3a316ace8c95");katex.render("M\\times 1", element, {displayMode:false,throwOnError:false});$ ) of hyper-parameters, both for the $var element = document.getElementById("moose-equation-2d2eb808-66f9-4768-9ce0-69eae7b0fe44");katex.render("q", element, {displayMode:false,throwOnError:false});$ -th basis. The size $var element = document.getElementById("moose-equation-e57bf04f-5a4f-40a7-b7fa-924cab602e5f");katex.render("Q", element, {displayMode:false,throwOnError:false});$ is user-defined and it can be greater than or equal to 1. The larger the $var element = document.getElementById("moose-equation-cbdb0d62-27ec-434f-a255-c628d7788b3a");katex.render("Q", element, {displayMode:false,throwOnError:false});$ , the more sophisticated the multi-output Gaussian Process in modeling complex outputs.

If $var element = document.getElementById("moose-equation-5b9db081-b5b5-47cb-8e1f-dfe4fe5264e8");katex.render("Q=1", element, {displayMode:false,throwOnError:false});$ , the LMC reduces to the intrinsic co-regionalization model (ICM).

Example Input File Syntax

[Covariance]
  [covar]
    type = SquaredExponentialCovariance
    signal_variance = 0.711441
    noise_variance = 0.0
    length_factor = '53.5829 6.92861'
  []
  [lmc]
    type = LMC
    covariance_functions = covar
    num_outputs = 2
    num_latent_funcs = 1
  []
[]

warning

The detailed documentation of this object is only available when Moose is compiled with Libtorch. For instructions on how to compile Moose with Libtorch, visit the general installation webpage or click here.

References

L. F. Cheng, B. Dumitrascu, G. Darnell, C. Chivers, M. Draugelis, K. Li, and B. E. Engelhardt. Sparse multi-output gaussian processes for online medical time series prediction. BMC medical informatics and decision making, 20(1):1–23, 2020.

BibTeX

@article{Cheng2020gp,
    author = "Cheng, L. F. and Dumitrascu, B. and Darnell, G. and Chivers, C. and Draugelis, M. and Li, K. and Engelhardt, B. E.",
    title = "Sparse multi-output Gaussian processes for online medical time series prediction",
    journal = "BMC medical informatics and decision making",
    volume = "20",
    number = "1",
    pages = "1--23",
    year = "2020"
}

RIS

TY  - JOUR
AU  - Cheng, L. F.
AU  - Dumitrascu, B.
AU  - Darnell, G.
AU  - Chivers, C.
AU  - Draugelis, M.
AU  - Li, K.
AU  - Engelhardt, B. E.
TI  - Sparse multi-output Gaussian processes for online medical time series prediction
JO  - BMC medical informatics and decision making
PY  - 2020
VL  - 20
IS  - 1
SP  - 1
EP  - 23
ER  -

Plain Text

L. F. Cheng, B. Dumitrascu, G. Darnell, C. Chivers, M. Draugelis, K. Li, and B. E. Engelhardt. Sparse multi-output gaussian processes for online medical time series prediction. BMC medical informatics and decision making, 20(1):1–23, 2020.

H. Liu, J. Cai, and Y. S. Ong. Remarks on multi-output gaussian process regression. Knowledge-Based Systems, 144:102–112, 2018.

BibTeX

@article{Liu2018gp,
    author = "Liu, H. and Cai, J. and Ong, Y. S.",
    title = "Remarks on multi-output Gaussian process regression",
    journal = "Knowledge-Based Systems",
    volume = "144",
    pages = "102--112",
    year = "2018"
}

RIS

TY  - JOUR
AU  - Liu, H.
AU  - Cai, J.
AU  - Ong, Y. S.
TI  - Remarks on multi-output Gaussian process regression
JO  - Knowledge-Based Systems
PY  - 2018
VL  - 144
SP  - 102
EP  - 112
ER  -

Plain Text

H. Liu, J. Cai, and Y. S. Ong. Remarks on multi-output gaussian process regression. Knowledge-Based Systems, 144:102–112, 2018.

(moose/modules/stochastic_tools/test/tests/surrogates/multioutput_gp/mogp_lmc.i)

[StochasticTools]
[]

[Distributions]
  [k_dist]
    type = Normal
    mean = 15.0
    standard_deviation = 2.0
  []
  [bc_dist]
    type = Normal
    mean = 1000.0
    standard_deviation = 100.0
  []
[]

[Samplers]
  [train]
    type = LatinHypercube
    num_rows = 10
    distributions = 'k_dist bc_dist'
    execute_on = PRE_MULTIAPP_SETUP
    seed = 100
  []
  [test]
    type = LatinHypercube
    num_rows = 5
    distributions = 'k_dist bc_dist'
    seed = 101
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    mode = batch-reset
    sampler = train
  []
[]

[Controls]
  [cmdline]
    type = MultiAppSamplerControl
    multi_app = sub
    sampler = train
    param_names = 'Materials/conductivity/prop_values BCs/right/value'
  []
[]

[Transfers]
  [data]
    type = SamplerReporterTransfer
    from_multi_app = sub
    sampler = train
    stochastic_reporter = results
    from_reporter = 'T_vec/T'
  []
[]

[Reporters]
  [results]
    type = StochasticReporter
    outputs = none
  []
  [eval_test]
    type = EvaluateSurrogate
    model = mogp
    response_type = vector_real
    parallel_type = ROOT
    execute_on = timestep_end
    sampler = test
    evaluate_std = true
  []
[]

[Trainers]
  [mogp_trainer]
    type = GaussianProcessTrainer
    execute_on = timestep_end
    covariance_function = 'lmc'
    standardize_params = 'true'
    standardize_data = 'true'
    sampler = train
    response_type = vector_real
    response = results/data:T_vec:T
  []
[]

[Covariance]
  [covar]
    type = SquaredExponentialCovariance
    signal_variance = 0.711441
    noise_variance = 0.0
    length_factor = '53.5829 6.92861'
  []
  [lmc]
    type = LMC
    covariance_functions = covar
    num_outputs = 2
    num_latent_funcs = 1
  []
[]

[Surrogates]
  [mogp]
    type = GaussianProcessSurrogate
    trainer = mogp_trainer
  []
[]

[VectorPostprocessors]
  [train_params]
    type = SamplerData
    sampler = train
    execute_on = final
  []
  [test_params]
    type = SamplerData
    sampler = test
    execute_on = final
  []
  [hyperparams]
    type = GaussianProcessData
    gp_name = mogp
    execute_on = final
  []
[]

[Outputs]
  [out]
    type = JSON
    execute_on = final
    vectorpostprocessors_as_reporters = true
    execute_system_information_on = NONE
  []
[]

Example Input File Syntax
References