1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
|
#! /usr/bin/env python
import openturns as ot
from openturns.experimental import GaussianProcessRegression, GaussianProcessFitter
import openturns.testing as ott
ot.TESTPREAMBLE()
ot.ResourceMap.SetAsUnsignedInteger(
"OptimizationAlgorithm-DefaultMaximumCallsNumber", 20000
)
ot.ResourceMap.SetAsScalar("Cobyla-DefaultRhoBeg", 0.5)
ot.ResourceMap.SetAsScalar("OptimizationAlgorithm-DefaultMaximumAbsoluteError", 1e-8)
# Test 1
def test_one_input_one_output():
sampleSize = 6
dimension = 1
f = ot.SymbolicFunction(["x0"], ["x0 * sin(x0)"])
X = ot.Sample(sampleSize, dimension)
X2 = ot.Sample(sampleSize, dimension)
for i in range(sampleSize):
X[i, 0] = 3.0 + i
X2[i, 0] = 2.5 + i
X[0, 0] = 1.0
X[1, 0] = 3.0
X2[0, 0] = 2.0
X2[1, 0] = 4.0
Y = f(X)
Y2 = f(X2)
# create covariance model
basis = ot.ConstantBasisFactory(dimension).build()
covarianceModel = ot.SquaredExponential()
# create algorithm
fit_algo = GaussianProcessFitter(X, Y, covarianceModel, basis)
# set sensible optimization bounds and estimate hyper parameters
fit_algo.setOptimizationBounds(ot.Interval(X.getMin(), X.getMax()))
fit_algo.run()
# perform an evaluation
fit_result = fit_algo.getResult()
algo = GaussianProcessRegression(fit_result)
algo.run()
result = algo.getResult()
ott.assert_almost_equal(result.getMetaModel()(X), Y)
ott.assert_almost_equal(result.getResiduals(), [1.32804e-07], 1e-3, 1e-3)
ott.assert_almost_equal(result.getRelativeErrors(), [5.20873e-21])
# Prediction accuracy
ott.assert_almost_equal(Y2, result.getMetaModel()(X2), 0.3, 0.0)
# Test 2
def test_two_inputs_one_output():
inputDimension = 2
# Learning data
levels = [8, 5]
box = ot.Box(levels)
inputSample = box.generate()
# Scale each direction
inputSample *= 10.0
model = ot.SymbolicFunction(["x", "y"], ["cos(0.5*x) + sin(y)"])
outputSample = model(inputSample)
# Validation
sampleSize = 10
inputValidSample = ot.JointDistribution(2 * [ot.Uniform(0, 10.0)]).getSample(
sampleSize
)
outputValidSample = model(inputValidSample)
# 2) Definition of exponential model
# The parameters have been calibrated using TNC optimization
# and AbsoluteExponential models
scale = [5.33532, 2.61534]
amplitude = [1.61536]
covarianceModel = ot.SquaredExponential(scale, amplitude)
# 3) Basis definition
basis = ot.ConstantBasisFactory(inputDimension).build()
# 4) GPF algorithm
fit_algo = GaussianProcessFitter(inputSample, outputSample, covarianceModel, basis)
# set sensible optimization bounds and estimate hyper parameters
fit_algo.setOptimizationBounds(
ot.Interval(inputSample.getMin(), inputSample.getMax())
)
fit_algo.run()
# perform an evaluation
fit_result = fit_algo.getResult()
# Regression algorithm
algo = GaussianProcessRegression(fit_result)
algo.run()
result = algo.getResult()
# Get meta model
metaModel = result.getMetaModel()
outData = metaModel(inputValidSample)
# 5) Errors
# Interpolation
ott.assert_almost_equal(outputSample, metaModel(inputSample), 3.0e-5, 3.0e-5)
# Prediction
ott.assert_almost_equal(outputValidSample, outData, 1.0e-1, 1e-1)
def test_two_outputs():
f = ot.SymbolicFunction(["x"], ["x * sin(x)", "x * cos(x)"])
sampleX = ot.Sample([[1.0], [2.0], [3.0], [4.0], [5.0], [6.0], [7.0], [8.0]])
sampleY = f(sampleX)
# Build a basis phi from R --> R^2
# phi_{0,0} = phi_{0,1} = x
# phi_{1,0} = phi_{1,1} = x^2
phi0 = ot.AggregatedFunction(
[ot.SymbolicFunction(["x"], ["x"]), ot.SymbolicFunction(["x"], ["x"])]
)
phi1 = ot.AggregatedFunction(
[ot.SymbolicFunction(["x"], ["x^2"]), ot.SymbolicFunction(["x"], ["x^2"])]
)
basis = ot.Basis([phi0, phi1])
covarianceModel = ot.SquaredExponential([1.0])
covarianceModel.setActiveParameter([])
covarianceModel = ot.TensorizedCovarianceModel([covarianceModel] * 2)
fit_algo = GaussianProcessFitter(sampleX, sampleY, covarianceModel, basis)
# set sensible optimization bounds and estimate hyper parameters
fit_algo.run()
# perform an evaluation
fit_result = fit_algo.getResult()
algo = GaussianProcessRegression(fit_result)
algo.run()
result = algo.getResult()
mm = result.getMetaModel()
assert mm.getOutputDimension() == 2, "wrong output dim"
ott.assert_almost_equal(mm([5.5]), [-3.88368, 3.90286])
def test_stationary_fun():
# fix https://github.com/openturns/openturns/issues/1861
ot.RandomGenerator.SetSeed(0)
rho = ot.SymbolicFunction("tau", "exp(-abs(tau))*cos(2*pi_*abs(tau))")
model = ot.StationaryFunctionalCovarianceModel([1], [1], rho)
x = ot.Normal().getSample(20)
x.setDescription(["J0"])
y = x + ot.Normal(0, 0.1).getSample(20)
y.setDescription(["G0"])
fit_algo = GaussianProcessFitter(x, y, model, ot.LinearBasisFactory().build())
# set sensible optimization bounds and estimate hyper parameters
fit_algo.run()
# perform an evaluation
fit_result = fit_algo.getResult()
algo = GaussianProcessRegression(fit_result)
algo.run()
result = algo.getResult()
mm = result.getMetaModel()
ott.assert_almost_equal(mm([5.5]), [5.58283])
def test_gpr_no_opt():
sampleSize = 6
dimension = 1
f = ot.SymbolicFunction(["x0"], ["x0 * sin(x0)"])
X = ot.Sample(sampleSize, dimension)
X2 = ot.Sample(sampleSize, dimension)
for i in range(sampleSize):
X[i, 0] = 3.0 + i
X2[i, 0] = 2.5 + i
X[0, 0] = 1.0
X[1, 0] = 3.0
X2[0, 0] = 2.0
X2[1, 0] = 4.0
Y = f(X)
Y2 = f(X2)
# create covariance model
covarianceModel = ot.SquaredExponential([1.6326932047296538], [4.895995962015954])
trend_function = ot.SymbolicFunction("x", "1.49543")
# GPR (comparable with test_one_input_one_output)
algo = GaussianProcessRegression(X, Y, covarianceModel, trend_function)
algo.run()
result = algo.getResult()
ott.assert_almost_equal(result.getMetaModel()(X), Y)
ott.assert_almost_equal(result.getResiduals(), [1.32804e-07], 1e-3, 1e-3)
ott.assert_almost_equal(result.getRelativeErrors(), [5.20873e-21])
# Prediction accuracy
ott.assert_almost_equal(Y2, result.getMetaModel()(X2), 0.3, 0.0)
if __name__ == "__main__":
test_one_input_one_output()
test_two_inputs_one_output()
test_two_outputs()
test_stationary_fun()
test_gpr_no_opt()
|
