Advanced kriging

In this example we will build a metamodel using gaussian process regression of the function.

We will choose the number of learning points, the basis and the covariance model.

import openturns as ot
from openturns.viewer import View
import numpy as np
import matplotlib.pyplot as plt
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)

Generate design of experiment

We create training samples from the function . We can change their number and distribution in the range. If the with_error boolean is True, then the data is computed by adding a gaussian noise to the function values.

dim = 1
xmin = 0
xmax = 10
n_pt = 20 # number of initial points
with_error = True  # whether to use generation with error

ref_func_with_error = ot.SymbolicFunction(['x', 'eps'], ['x * sin(x) + eps'])
ref_func = ot.ParametricFunction(ref_func_with_error, [1], [0.0])
x = np.vstack(np.linspace(xmin, xmax, n_pt))
ot.RandomGenerator.SetSeed(1235)
eps = ot.Normal(0, 1.5).getSample(n_pt)
X = ot.Sample(n_pt, 2)
X[:, 0] = x
X[:, 1] = eps
if with_error:
    y = np.array(ref_func_with_error(X))
else:
    y = np.array(ref_func(x))

graph = ref_func.draw(xmin, xmax, 200)
cloud = ot.Cloud(x, y)
cloud.setColor('red')
cloud.setPointStyle('bullet')
graph.add(cloud)
graph.setLegends(["Function","Data"])
graph.setLegendPosition("topleft")
graph.setTitle("Sample size = %d" % (n_pt))
view = viewer.View(graph)

Create the kriging algorithm

# 1. basis
ot.ResourceMap.SetAsBool('GeneralLinearModelAlgorithm-UseAnalyticalAmplitudeEstimate', True)
basis = ot.ConstantBasisFactory(dim).build()
print(basis)

# 2. covariance model
cov = ot.MaternModel([1.], [2.5], 1.5)
print(cov)

# 3. kriging algorithm
algokriging = ot.KrigingAlgorithm(x, y, cov, basis)

## error measure
#algokriging.setNoise([5*1e-1]*n_pt)

# 4. Optimization
# algokriging.setOptimizationAlgorithm(ot.NLopt('GN_DIRECT'))
startingPoint = ot.LHSExperiment(ot.Uniform(1e-1, 1e2), 50).generate()
algokriging.setOptimizationAlgorithm(ot.MultiStart(ot.TNC(), startingPoint))
algokriging.setOptimizationBounds(ot.Interval([0.1], [1e2]))

# if we choose not to optimize parameters
#algokriging.setOptimizeParameters(False)

# 5. run the algorithm
algokriging.run()

Out:

Basis( [class=LinearEvaluation name=Unnamed center=[0] constant=[1] linear=[[ 0 ]]] )
MaternModel(scale=[1], amplitude=[2.5], nu=1.5)

Results

get some results

krigingResult = algokriging.getResult()
print('residual = ', krigingResult.getResiduals())
print('R2 = ', krigingResult.getRelativeErrors())
print('Optimal scale= {}'.format(krigingResult.getCovarianceModel().getScale()))
print('Optimal amplitude = {}'.format(krigingResult.getCovarianceModel().getAmplitude()))
print('Optimal trend coefficients = {}'.format(krigingResult.getTrendCoefficients()))

Out:

residual =  [0.0147631]
R2 =  [0.000224304]
Optimal scale= [0.849535]
Optimal amplitude = [4.54402]
Optimal trend coefficients = [[-0.141313]]

get the metamodel

krigingMeta = krigingResult.getMetaModel()

n_pts_plot = 1000
x_plot = np.vstack(np.linspace(xmin, xmax, n_pts_plot))
fig, [ax1, ax2] = plt.subplots(1, 2, figsize=(12, 6))

# On the left, the function
graph = ref_func.draw(xmin, xmax, n_pts_plot)
graph.setLegends(["Function"])
graphKriging = krigingMeta.draw(xmin, xmax, n_pts_plot)
graphKriging.setColors(["green"])
graphKriging.setLegends(["Kriging"])
graph.add(graphKriging)
cloud = ot.Cloud(x,y)
cloud.setColor("red")
cloud.setLegend("Data")
graph.add(cloud)
graph.setLegendPosition("topleft")
View(graph, axes=[ax1])

# On the right, the conditional kriging variance
graph = ot.Graph("", "x", "Conditional kriging variance", True, '')
# Sample for the data
sample = ot.Sample(n_pt,2)
sample[:,0] = x
cloud = ot.Cloud(sample)
cloud.setColor("red")
graph.add(cloud)
# Sample for the variance
sample = ot.Sample(n_pts_plot,2)
sample[:,0] = x_plot
variance = [[krigingResult.getConditionalCovariance(xx)[0, 0]] for xx in x_plot]
sample[:,1] = variance
curve = ot.Curve(sample)
curve.setColor("green")
graph.add(curve)
View(graph, axes=[ax2])

fig.suptitle("Kriging result");

Out:

Text(0.5, 0.98, 'Kriging result')

Display the confidence interval

level = 0.95
quantile = ot.Normal().computeQuantile((1-level)/2)[0]
borne_sup = krigingMeta(x_plot) + quantile * np.sqrt(variance)
borne_inf = krigingMeta(x_plot) - quantile * np.sqrt(variance)

fig, ax = plt.subplots(figsize=(8, 8))
ax.plot(x, y, ('ro'))
ax.plot(x_plot, borne_sup, '--', color='orange', label='Confidence interval')
ax.plot(x_plot, borne_inf, '--', color='orange')
View(ref_func.draw(xmin, xmax, n_pts_plot), axes=[ax], plot_kw={'label':'$x sin(x)$'})
View(krigingMeta.draw(xmin, xmax, n_pts_plot), plot_kw={'color':'green', 'label':'prediction'}, axes=[ax])
legend = ax.legend()
ax.autoscale()

Generate conditional trajectories

support for trajectories with training samples removed

values = np.linspace(0, 10, 500)
for xx in x:
    if len(np.argwhere(values==xx)) == 1:
        values = np.delete(values, np.argwhere(values==xx)[0, 0])

Conditional Gaussian process

krv = ot.KrigingRandomVector(krigingResult, np.vstack(values))
krv_sample = krv.getSample(5)

x_plot = np.vstack(np.linspace(xmin, xmax, n_pts_plot))
fig, ax = plt.subplots(figsize=(8, 6))
ax.plot(x, y, ('ro'))
for i in range(krv_sample.getSize()):
    if i == 0:
        ax.plot(values, krv_sample[i, :], '--', alpha=0.8, label='Conditional trajectories')
    else:
        ax.plot(values, krv_sample[i, :], '--', alpha=0.8)
View(ref_func.draw(xmin, xmax, n_pts_plot), axes=[ax],
     plot_kw={'color':'black', 'label':'$x*sin(x)$'})
View(krigingMeta.draw(xmin, xmax, n_pts_plot), axes=[ax],
     plot_kw={'color':'green', 'label':'prediction'})
legend = ax.legend()
ax.autoscale()

Validation

n_valid = 10
x_valid = ot.Uniform(xmin, xmax).getSample(n_valid)
if with_error:
    X_valid = ot.Sample(x_valid)
    X_valid.stack(ot.Normal(0.0, 1.5).getSample(n_valid))
    y_valid = np.array(ref_func_with_error(X_valid))
else:
    y_valid = np.array(ref_func(X_valid))

validation = ot.MetaModelValidation(x_valid, y_valid, krigingMeta)
validation.computePredictivityFactor()

[0.863752]

graph = validation.drawValidation()
view = viewer.View(graph)

graph =validation.getResidualDistribution().drawPDF()
graph.setXTitle("Residuals")
view = viewer.View(graph)
plt.show()