Deussen, Philipp

Galvanin, Federico

English

UCL Discovery

On the effect of parametric uncertainty in Gaussian Processesapplied to geostatistical estimation: an in-silico case studyPhilipp Deussen, Federico GalvaninINTRODUCTIONDepartment of Chemical Engineering, University College London, Torrington Place, London, WC1E 7JE, UKCorresponding Author: f.galvanin@ucl.ac.ukMETHODOLOGYCASE STUDY[1]Dominy, S., O’Connor, L., Parbhakar-Fox, A., Glass, H., Purevgerel, S. (2018) Geometallurgy – aroute to more resilient mine operations, Minerals, 8(12).[2]Sahimi, M. (2011) Flow and Transport in Porous Media and Fractured Rock, 2nd Ed., Wiley.[3] Rossi, M. and Deutsch, C. (2014) Mineral Resource Estimation, 1st Ed, Springer.• In mining exploration and production, models of the subsurface are constructed usingGaussian Processes (GPs) to interpolate between samples of the ground (drill core).• Drilling the ground for samples is expensive and time-consuming. In value-densecommodities, sampling is a production bottleneck. Optimal sampling is thus desirable.• Commonly in industry, samples target the location with the highest Kriging variance.• However, this variance implicitly depends on estimates of the kernel parameters.• A methodology is proposed that fits multiple GPs with sets of parameter estimates fromMonte Carlo Simulations to account for parametric and model prediction uncertainty.• The goal is to produce maps to make quantitative uncertainty assessments possible.These shall be used as one of the optimality criteria for optimal experimental designs.Approach for Quantifying Parametric UncertaintyCONCLUSIONS & FUTURE WORK• Kernels with structurally unidentifiable parameters, e.g. the Gaussian kernel inSahimi[2], should be avoided. Alternative formulations that fix the numerical parametera = 1/3, e.g. in Rossi and Deutsch[3], should be used instead.• In industry practice, kernel parameter uncertainty, local anisotropies and qualitativefactors are often accounted for by fitting the kernel by hand[3]. Instead, regressionshould be used and the variance-covariance matrix should be computed so that theeffect of parametric uncertainty can be quantified and propagated to the GP.• Future Model-based optimal experiment designs will include a criterion to capturethe effect of parametric uncertainty on the GP. This will be based on the Fisher Infor-mation Matrix and constitute second design objective in model discrimination work.REFERENCESPhilippDeussenProf.FedericoGalvaninFormulation of Models• The kernel of the GP consists of a correlation function R(h), relating the semivarianceγ(h) of concentration at distance h and the distribution variance ௓ߪଶ of concentrations.• Simple (biased) Kriging is a form of GP which gives the best linear estimate of themean expected concentration at an unsampled location Z using the estimator መܼ, andthe variance of estimates ஻௄ߪ)ଶ ) based on the expected square error to the mean.• The estimators are based on the samples ௜ܼand their relative importance weights .௜ݓ• The optimal weights can࢏࢝ be found from samples i and j by substituting (3) into (2):• And then minimising the Kriging variance: ∂ ஻௄ߪଶ / ∂ wi = 0:• For all unsampled locations this results in a weights matrix W and correlation matricesbetween sampled points A, and sampled and unsampled locations P. This can besolved as it is a linear system of equations with no degrees of freedom.[2]1. Generating Ground Truth & Sampling: A known kernel, which describes the spatialcorrelation structure of concentration likelihood at different points, is used to generateground truth concentration data. Some points are sampled; the rest are to be modelled.2. Estimation of Kernel Parameters: Models based on different kernels are suggestedand the kernel parameters are estimated based on magnitude and position of samples.3. Estimation of Kriging Weights: Kriging weights determine the relative importance ofsamples for the GP. They are estimated for all proposed kernels; once with the optimalkernel parameters and once with the range of likely kernel parameters (MC simulation).4. Parameter identifiability: Compare predicted concentrations and variances from thesingle-pass and the Monte Carlo GP. If the difference is significant, reducing parametricuncertainty must be prioritised in experiment designs or the model rejected altogether.Spherical Kernel [2] :Gaussian Kernel [2] :Where s is the sill, n the nugget effect, r the characteristic range of correlations and a is constant.2. Estimation of Kernel Parameters from Sampled Locations3. Estimation of Kriging Weights & Fitting Gaussian ProcessesGenerate groundtruth concentrationdata in-silicoSelect kernel that isused in true modelGenerate candidatekernelsmodels withdifferent types ofEstimate kernelparameters usingsampled locationsEstimate Kriging weightsusing a range of kernelparameter estimates(Monte Carlo)Estimate Krigingweights using onlyoptimal kernelparameter estimatesPlot concentration andKriging Variance mapsParametricuncertaintysignificant?Accept parameterestimates forcandidate model(s)ߛ ݄ ൌ ௓ߪଶ െ ܴ ℎ (1)஻௄ߪଶ ൌ ௓ߪଶ − 2 ∑ ௜ܴݓ ܼǡܼ ௜ +ே௜ୀଵ ∑ ∑ ௝ܴݓ௜ݓ ௜ܼǡܼ ௝ே௝ୀଵே௜ୀଵ (4)መܼ= ∑ ௜ܼݓ ௜ே௜ୀଵ (3)஻௄ߪଶ ൌ ܧ ܼെ መܼଶ(2)∑ ܴ ௜ܼ ௝ܼே௜ୀଵ ௜ൌݓ ܴ ܼ ௝ܼ (5)YesNoGenerate randomsampling locationsif h < r ߛ ℎ = ݏଷ௛ଶ௥ −  ଵଶ௛௥ଷ+ n ; if h > r ߛ ݄ ൌ ൅ݏ ݊ (6)if h < r ߛ ℎ = ݏ ͳ െ ݁ି௔೓మೝమ + n ; if h > r ߛ ݄ ൌ ൅ݏ ݊ (7)Spherical Kernel: Variance-Covariance Matrix for Parameters1 2 31 4.142 -0.39 0.203 0.54 -0.18 0.18Figure 3 – True distribution of metal grade with 80random sampling locations used in analysisGaussian Kernel: Variance-Covariance Matrix for Parameters1 2 3 41 3.95 x 10152 3.98 x 106 0.783 -4.18 x 106 -0.77 0.774 5.50 x 1016 5.54 x 107 -5.83 x 107 7.66 x 1017TrueModelPlot concentration andKriging Variance maps12341. Generation of Candidate Models & Ground TruthKernel ParametersParameter True Values Spherical Kernel Estimates Gaussian Kernel Estimatesr 17 8.68 8.68n 3 2.31 2.31s 20 4.23 4.23a - - 251.62Figure 1: 3D model of a mineral deposit [1] with different concentrations in the blue, yellowand orange clouds. Drilled samples are in dark blue and a schematic of a drill is shown.Drill PipeRockCoreDrill BitMineralisation ZonesDrillholesSpherical Kernel Gaussian KernelSingle-PassMonteCarlo(100sims)Spherical Kernel Gaussian KernelSingle-PassMonteCarlo(100sims)RadicalincreaseinpredictionvarianceNon-physicalmodelpredictionsSlightincrease inpredictionvarianceSimilarpredictionmeanSurfacePrediction Variance(Kriging Variance):Prediction Mean(MetalConcentration):Figure 2 – Semivariance (squared concentration difference)between sampled point pairs vs. distance between themCoordinatescorrespondto those inFigure 2True Concentration Values (g/t)Semivariance(g/t)²Distance (m)

On the effect of parametric uncertainty in Gaussian Processes applied to geostatistical estimation: an in-silico case study

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On the effect of parametric uncertainty in Gaussian Processes applied to geostatistical estimation: an in-silico case study

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