.. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_meta_modeling_kriging_metamodel_plot_kriging_sequential.py: Sequentially adding new points to a kriging =========================================== In this example, we show how to sequentially add new points to a kriging in order to improve the predictivity of the metamodel. In order to create simple graphics, we consider a 1D function. Create the function and the design of experiments ------------------------------------------------- .. code-block:: default import openturns as ot from openturns.viewer import View import numpy as np import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) .. code-block:: default sampleSize = 4 dimension = 1 Define the function. .. code-block:: default g = ot.SymbolicFunction(['x'], ['0.5*x^2 + sin(2.5*x)']) Create the design of experiments. .. code-block:: default xMin = -0.9 xMax = 1.9 X_distr = ot.Uniform(xMin, xMax) X = ot.LHSExperiment(X_distr, sampleSize, False, False).generate() Y = g(X) .. code-block:: default graph = g.draw(xMin, xMax) data = ot.Cloud(X,Y) data.setColor("red") graph.add(data) view = viewer.View(graph) .. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_sequential_001.png :alt: y0 as a function of x :class: sphx-glr-single-img Create the algorithms --------------------- .. code-block:: default def createMyBasicKriging(X,Y): ''' Create a kriging from a pair of X and Y samples. We use a 3/2 Matérn covariance model and a constant trend. ''' basis = ot.ConstantBasisFactory(dimension).build() covarianceModel = ot.MaternModel([1.0], 1.5) algo = ot.KrigingAlgorithm(X, Y, covarianceModel, basis) algo.run() krigResult = algo.getResult() return krigResult .. code-block:: default def linearSample(xmin,xmax,npoints): '''Returns a sample created from a regular grid from xmin to xmax with npoints points.''' step = (xmax-xmin)/(npoints-1) rg = ot.RegularGrid(xmin, step, npoints) vertices = rg.getVertices() return vertices .. code-block:: default def plot_kriging_bounds(vLow,vUp,n_test): ''' From two lists containing the lower and upper bounds of the region, create a PolygonArray. ''' palette = ot.Drawable.BuildDefaultPalette(2) myPaletteColor = palette[1] polyData = [[vLow[i], vLow[i+1], vUp[i+1], vUp[i]] for i in range(n_test-1)] polygonList = [ot.Polygon(polyData[i], myPaletteColor, myPaletteColor) for i in range(n_test-1)] boundsPoly = ot.PolygonArray(polygonList) boundsPoly.setLegend("95% bounds") return boundsPoly The following `sqrt` function will be used later to compute the standard deviation from the variance. .. code-block:: default sqrt = ot.SymbolicFunction(["x"],["sqrt(x)"]) .. code-block:: default def plotMyBasicKriging(krigResult, xMin, xMax, X, Y, level = 0.95): ''' Given a kriging result, plot the data, the kriging metamodel and a confidence interval. ''' samplesize = X.getSize() meta = krigResult.getMetaModel() graphKriging = meta.draw(xMin, xMax) graphKriging.setLegends(["Kriging"]) # Create a grid of points and evaluate the function and the kriging nbpoints = 50 xGrid = linearSample(xMin,xMax,nbpoints) yFunction = g(xGrid) yKrig = meta(xGrid) # Compute the conditional covariance epsilon = ot.Point(nbpoints,1.e-8) conditionalVariance = krigResult.getConditionalMarginalVariance(xGrid)+epsilon conditionalVarianceSample = ot.Sample([[cv] for cv in conditionalVariance]) conditionalSigma = sqrt(conditionalVarianceSample) # Compute the quantile of the Normal distribution alpha = 1-(1-level)/2 quantileAlpha = ot.DistFunc.qNormal(alpha) # Graphics of the bounds epsilon = 1.e-8 dataLower = [yKrig[i,0] - quantileAlpha * conditionalSigma[i,0] for i in range(nbpoints)] dataUpper = [yKrig[i,0] + quantileAlpha * conditionalSigma[i,0] for i in range(nbpoints)] # Coordinates of the vertices of the Polygons vLow = [[xGrid[i,0],dataLower[i]] for i in range(nbpoints)] vUp = [[xGrid[i,0],dataUpper[i]] for i in range(nbpoints)] # Compute the Polygon graphics boundsPoly = plot_kriging_bounds(vLow,vUp,nbpoints) boundsPoly.setLegend("95% bounds") # Validate the kriging metamodel mmv = ot.MetaModelValidation(xGrid, yFunction, meta) Q2 = mmv.computePredictivityFactor()[0] # Plot the function graphFonction = ot.Curve(xGrid,yFunction) graphFonction.setLineStyle("dashed") graphFonction.setColor("magenta") graphFonction.setLineWidth(2) graphFonction.setLegend("Function") # Draw the X and Y observed cloudDOE = ot.Cloud(X, Y) cloudDOE.setPointStyle("circle") cloudDOE.setColor("red") cloudDOE.setLegend("Data") # Assemble the graphics graph = ot.Graph() graph.add(boundsPoly) graph.add(graphFonction) graph.add(cloudDOE) graph.add(graphKriging) graph.setLegendPosition("bottomright") graph.setAxes(True) graph.setGrid(True) graph.setTitle("Size = %d, Q2=%.2f%%" % (samplesize,100*Q2)) graph.setXTitle("X") graph.setYTitle("Y") return graph We start by creating the initial kriging metamodel on the 4 points in the design of experiments. .. code-block:: default krigResult = createMyBasicKriging(X,Y) graph = plotMyBasicKriging(krigResult,xMin,xMax, X, Y) view = viewer.View(graph) .. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_sequential_002.png :alt: Size = 4, Q2=96.36% :class: sphx-glr-single-img Sequentially add new points --------------------------- The following function is the building block of the algorithm. It returns a new point which maximizes the conditional variance. .. code-block:: default def getNewPoint(xMin, xMax,krigResult): ''' Returns a new point to be added to the design of experiments. This point maximizes the conditional variance of the kriging. ''' nbpoints = 50 xGrid = linearSample(xMin,xMax,nbpoints) conditionalVariance = krigResult.getConditionalMarginalVariance(xGrid) iMaxVar = int(np.argmax(conditionalVariance)) xNew = xGrid[iMaxVar,0] xNew = ot.Point([xNew]) return xNew .. code-block:: default krigingStep = 0 We first call `getNewPoint` to get a point to add to the design of experiments. .. code-block:: default xNew = getNewPoint(xMin, xMax,krigResult) xNew .. raw:: html

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Then we evaluate the function on the new point and add it to the training design of experiments. .. code-block:: default yNew = g(xNew) X.add(xNew) Y.add(yNew) We now plot the updated kriging. .. code-block:: default krigResult = createMyBasicKriging(X,Y) krigingStep += 1 myTitle = "Krigeage #%d" % (krigingStep+1) graph = plotMyBasicKriging(krigResult,xMin,xMax, X, Y) view = viewer.View(graph) .. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_sequential_003.png :alt: Size = 5, Q2=97.26% :class: sphx-glr-single-img The algorithm added a point to the right bound of the domain. .. code-block:: default for krigingStep in range(5): xNew = getNewPoint(xMin, xMax,krigResult) yNew = g(xNew) X.add(xNew) Y.add(yNew) krigResult = createMyBasicKriging(X,Y) krigingStep += 1 myTitle = "Krigeage #%d" % (krigingStep+1) graph = plotMyBasicKriging(krigResult,xMin,xMax, X, Y) View(graph) .. rst-class:: sphx-glr-horizontal * .. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_sequential_004.png :alt: Size = 6, Q2=98.76% :class: sphx-glr-multi-img * .. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_sequential_005.png :alt: Size = 7, Q2=99.71% :class: sphx-glr-multi-img * .. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_sequential_006.png :alt: Size = 8, Q2=99.80% :class: sphx-glr-multi-img * .. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_sequential_007.png :alt: Size = 9, Q2=99.86% :class: sphx-glr-multi-img * .. image:: /auto_meta_modeling/kriging_metamodel/images/sphx_glr_plot_kriging_sequential_008.png :alt: Size = 10, Q2=99.85% :class: sphx-glr-multi-img We observe that the second added point is the left bound of the domain. The remaining points were added strictly inside the domain where the accuracy was drastically improved. With only 10 points, the metamodel accuracy is already very good with a Q2 which is equal to 99.9%. Conclusion ---------- The current example presents the naive implementation on the creation of a sequential design of experiments based on kriging. More pratical algorithms are presented in the following references. * Mona Abtini. Plans prédictifs à taille fixe et séquentiels pour le krigeage (2008). Thèse de doctorat de l'Université de Lyon. * Céline Scheidt. Analyse statistique d’expériences simulées : Modélisation adaptative de réponses non régulières par krigeage et plans d’expériences (2007). Thèse présentée pour obtenir le grade de Docteur de l’Université Louis Pasteur. * David Ginsbourger. Sequential Design of Computer Experiments. Wiley StatsRef: Statistics Reference Online, Wiley (2018 ) .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.724 seconds) .. _sphx_glr_download_auto_meta_modeling_kriging_metamodel_plot_kriging_sequential.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_kriging_sequential.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_kriging_sequential.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_