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#! /usr/bin/env python
import openturns as ot
from openturns.usecases import ishigami_function
from openturns.testing import assert_almost_equal
import openturns.experimental as otexp
def computeMSENaiveLOO(
inputSample,
outputSample,
inputDistribution,
adaptiveStrategy,
projectionStrategy,
):
"""
Compute mean squared error by (naive) LOO.
Parameters
----------
inputSample : Sample(size, input_dimension)
The inputSample dataset.
outputSample : Sample(size, output_dimension)
The outputSample dataset.
inputDistribution : ot.Distribution.
The distribution of the input variable.
adaptiveStrategy : ot.AdaptiveStrategy
The method to select relevant coefficients.
projectionStrategy : ot.ProjectionStrategy
The method to compute the coefficients.
Returns
-------
mse : Point(output_dimension)
The mean squared error.
"""
#
sampleSize = inputSample.getSize()
outputDimension = outputSample.getDimension()
splitter = ot.LeaveOneOutSplitter(sampleSize)
residualsLOO = ot.Sample(sampleSize, outputDimension)
indexLOO = 0
for indicesTrain, indicesTest in splitter:
inputSampleTrain, inputSampleTest = (
inputSample[indicesTrain],
inputSample[indicesTest],
)
outputSampleTrain, outputSampleTest = (
outputSample[indicesTrain],
outputSample[indicesTest],
)
algoLOO = ot.FunctionalChaosAlgorithm(
inputSampleTrain,
outputSampleTrain,
inputDistribution,
adaptiveStrategy,
projectionStrategy,
)
algoLOO.run()
chaosResultLOO = algoLOO.getResult()
metamodelLOO = chaosResultLOO.getMetaModel()
outputPrediction = metamodelLOO(inputSampleTest)
for j in range(outputDimension):
residualsLOO[indexLOO, j] = outputSampleTest[0, j] - outputPrediction[0, j]
indexLOO += 1
mse = ot.Point(outputDimension)
for j in range(outputDimension):
marginalResidualsLOO = residualsLOO.getMarginal(j).asPoint()
mse[j] = marginalResidualsLOO.normSquare() / sampleSize
return mse
def computeMSENaiveKFold(
inputSample,
outputSample,
inputDistribution,
adaptiveStrategy,
projectionStrategy,
kParameter=5,
):
"""
Compute mean squared error by (naive) KFold.
Parameters
----------
inputSample : Sample(size, input_dimension)
The inputSample dataset.
outputSample : Sample(size, output_dimension)
The outputSample dataset.
inputDistribution : ot.Distribution.
The distribution of the input variable.
adaptiveStrategy : ot.AdaptiveStrategy
The method to select relevant coefficients.
projectionStrategy : ot.ProjectionStrategy
The method to compute the coefficients.
kParameter : int, in (2, sampleSize)
The parameter K.
Returns
-------
mse : Point(output_dimension)
The mean squared error.
"""
#
sampleSize = inputSample.getSize()
outputDimension = outputSample.getDimension()
splitter = ot.KFoldSplitter(sampleSize, kParameter)
squaredResiduals = ot.Sample(sampleSize, outputDimension)
for indicesTrain, indicesTest in splitter:
inputSampleTrain, inputSampleTest = (
inputSample[indicesTrain],
inputSample[indicesTest],
)
outputSampleTrain, outputSampleTest = (
outputSample[indicesTrain],
outputSample[indicesTest],
)
algoKFold = ot.FunctionalChaosAlgorithm(
inputSampleTrain,
outputSampleTrain,
inputDistribution,
adaptiveStrategy,
projectionStrategy,
)
algoKFold.run()
chaosResultKFold = algoKFold.getResult()
metamodelKFold = chaosResultKFold.getMetaModel()
predictionsKFold = metamodelKFold(inputSampleTest)
residualsKFold = outputSampleTest - predictionsKFold
foldSize = indicesTest.getSize()
for j in range(outputDimension):
for i in range(foldSize):
squaredResiduals[indicesTest[i], j] = residualsKFold[i, j] ** 2
mse = squaredResiduals.computeMean()
return mse
ot.TESTPREAMBLE()
# Problem parameters
im = ishigami_function.IshigamiModel()
dimension = im.inputDistribution.getDimension()
# Compute the sample size from number of folds to guarantee a non constant integer
# number of points per fold
kFoldParameter = 10
foldSampleSize = 20
sampleSize = foldSampleSize * kFoldParameter + 1
degree = 5
enumerateFunction = ot.LinearEnumerateFunction(dimension)
basisSize = enumerateFunction.getBasisSizeFromTotalDegree(degree)
print("basisSize = ", basisSize)
productBasis = ot.OrthogonalProductPolynomialFactory(
[ot.LegendreFactory()] * dimension, enumerateFunction
)
adaptiveStrategy = ot.FixedStrategy(productBasis, basisSize)
selectionAlgorithm = (
ot.PenalizedLeastSquaresAlgorithmFactory()
) # Get a full PCE: do not use model selection.
projectionStrategy = ot.LeastSquaresStrategy(selectionAlgorithm)
inputSample = im.inputDistribution.getSample(sampleSize)
outputSample = im.model(inputSample)
algo = ot.FunctionalChaosAlgorithm(
inputSample,
outputSample,
im.inputDistribution,
adaptiveStrategy,
projectionStrategy,
)
algo.run()
chaosResult = algo.getResult()
#
print("1. Analytical leave-one-out")
splitterLOO = ot.LeaveOneOutSplitter(sampleSize)
validationLOO = otexp.FunctionalChaosValidation(chaosResult, splitterLOO)
mseLOOAnalytical = validationLOO.computeMeanSquaredError()
print("Analytical LOO MSE = ", mseLOOAnalytical)
assert validationLOO.getSplitter().getN() == sampleSize
# Naive leave-one-out
mseLOOnaive = computeMSENaiveLOO(
inputSample,
outputSample,
im.inputDistribution,
adaptiveStrategy,
projectionStrategy,
)
print("Naive LOO MSE = ", mseLOOnaive)
# Test
rtolLOO = 1.0e-8
atolLOO = 0.0
assert_almost_equal(mseLOOAnalytical, mseLOOnaive, rtolLOO, atolLOO)
# Check LOO R2
r2ScoreLOO = validationLOO.computeR2Score()
print("Analytical LOO R2 score = ", r2ScoreLOO)
sampleVariance = outputSample.computeCentralMoment(2)
print("sampleVariance = ", sampleVariance)
r2ScoreReference = 1.0 - mseLOOAnalytical[0] / sampleVariance[0]
print("Computed R2 score = ", r2ScoreReference)
rtolLOO = 1.0e-12
atolLOO = 0.0
assert_almost_equal(r2ScoreReference, r2ScoreLOO[0], rtolLOO, atolLOO)
#
print("2. Analytical K-Fold")
splitterKF = ot.KFoldSplitter(sampleSize, kFoldParameter)
validationKFold = otexp.FunctionalChaosValidation(chaosResult, splitterKF)
print("KFold with K = ", kFoldParameter)
assert validationKFold.getSplitter().getN() == sampleSize
# Compute mean squared error
mseKFoldAnalytical = validationKFold.computeMeanSquaredError()
print("Analytical KFold MSE = ", mseKFoldAnalytical)
# Naive KFold
mseKFoldnaive = computeMSENaiveKFold(
inputSample,
outputSample,
im.inputDistribution,
adaptiveStrategy,
projectionStrategy,
kFoldParameter,
)
print("Naive KFold MSE = ", mseKFoldnaive)
# Test
rtolKFold = 1.0e-5
atolKFold = 0.0
assert_almost_equal(mseKFoldAnalytical, mseKFoldnaive, rtolKFold, atolKFold)
# Check K-Fold R2
r2ScoreKFold = validationKFold.computeR2Score()
print("Analytical K-Fold R2 score = ", r2ScoreKFold)
r2ScoreReference = 1.0 - mseKFoldAnalytical[0] / sampleVariance[0]
print("Computed R2 score = ", r2ScoreReference)
rtolKFold = 1.0e-12
atolKFold = 0.0
assert_almost_equal(r2ScoreReference, r2ScoreKFold[0], rtolKFold, atolKFold)
#
print("3. Setting FunctionalChaosValidation-ModelSelection to true")
# enables to do LOO CV on a sparse model.
ot.ResourceMap.SetAsBool("FunctionalChaosValidation-ModelSelection", True)
selectionAlgorithm = (
ot.LeastSquaresMetaModelSelectionFactory()
) # Get a sparse PCE (i.e. with model selection).
projectionStrategy = ot.LeastSquaresStrategy(selectionAlgorithm)
inputSample = im.inputDistribution.getSample(sampleSize)
outputSample = im.model(inputSample)
algo = ot.FunctionalChaosAlgorithm(
inputSample,
outputSample,
im.inputDistribution,
adaptiveStrategy,
projectionStrategy,
)
algo.run()
chaosResult = algo.getResult()
# Analytical leave-one-out
splitterLOO = ot.LeaveOneOutSplitter(sampleSize)
validationLOO = otexp.FunctionalChaosValidation(chaosResult, splitterLOO)
mseLOOAnalytical = validationLOO.computeMeanSquaredError()
print("Analytical LOO MSE = ", mseLOOAnalytical)
# Naive leave-one-out
mseLOOnaive = computeMSENaiveLOO(
inputSample,
outputSample,
im.inputDistribution,
adaptiveStrategy,
projectionStrategy,
)
print("Naive LOO MSE = ", mseLOOnaive)
# Test
rtolLOO = 1.0e-1 # We cannot have more accuracy, as the MSE estimator is then biased
atolLOO = 0.0
assert_almost_equal(mseLOOAnalytical, mseLOOnaive, rtolLOO, atolLOO)
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