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#
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"""Oakley O'Hagan 15D function."""
import numpy as np
[docs]
def oakley_ohagan04(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15):
r"""Oakley O'Hagan 2004 function, 15 dimensional benchmark function in [1].
:math:`f({\bf x})=
{\bf a_1}^T{\bf x}+{\bf a_2}^T\sin({\bf x})+{\bf a_3}^T\cos({\bf x})+{\bf x^T M x}`
The *a*-coefficients are chosen so that 5 of the input variables contribute
significantly to the output variance, 5 have a much smaller effect, and the
remaining 5 have almost no effect on the output variance. Values of the
coefficient vectors *a1*, *a2* and *a3*, and the matrix *M*, can be found at:
http://www.jeremy-oakley.staff.shef.ac.uk/psa_example.txt.
In [1] :math:`x_i \sim N(\mu=0, \sigma=1), \textrm{for all i = 1, ..., 15}.`
Args:
x1 (float): Input parameter 1
x2 (float): Input parameter 2
x3 (float): Input parameter 3
x4 (float): Input parameter 4
x5 (float): Input parameter 5
x6 (float): Input parameter 6
x7 (float): Input parameter 7
x8 (float): Input parameter 8
x9 (float): Input parameter 9
x10 (float): Input parameter 10
x11 (float): Input parameter 11
x12 (float): Input parameter 12
x13 (float): Input parameter 13
x14 (float): Input parameter 14
x15 (float): Input parameter 15
Returns:
float: Value of the function at the parameters
References:
[1] Oakley, J. E., & O'Hagan, A. (2004). Probabilistic sensitivity analysis
of complex models: a Bayesian approach. Journal of the Royal Statistical
Society: Series B (Statistical Methodology), 66(3), 751-769.
"""
xx = np.array(
[
[x1],
[x2],
[x3],
[x4],
[x5],
[x6],
[x7],
[x8],
[x9],
[x10],
[x11],
[x12],
[x13],
[x14],
[x15],
]
)
a1 = np.array(
[
0.0118,
0.0456,
0.2297,
0.0393,
0.1177,
0.3865,
0.3897,
0.6061,
0.6159,
0.4005,
1.0741,
1.1474,
0.7880,
1.1242,
1.1982,
]
)
a2 = np.array(
[
0.4341,
0.0887,
0.0512,
0.3233,
0.1489,
1.0360,
0.9892,
0.9672,
0.8977,
0.8083,
1.8426,
2.4712,
2.3946,
2.0045,
2.2621,
]
)
a3 = np.array(
[
0.1044,
0.2057,
0.0774,
0.2730,
0.1253,
0.7526,
0.8570,
1.0331,
0.8388,
0.7970,
2.2145,
2.0382,
2.4004,
2.0541,
1.9845,
]
)
m = np.array(
[
[
-0.022482886,
-0.18501666,
0.13418263,
0.36867264,
0.17172785,
0.13651143,
-0.44034404,
-0.081422854,
0.71321025,
-0.44361072,
0.50383394,
-0.024101458,
-0.045939684,
0.21666181,
0.055887417,
],
[
0.25659630,
0.053792287,
0.25800381,
0.23795905,
-0.59125756,
-0.081627077,
-0.28749073,
0.41581639,
0.49752241,
0.083893165,
-0.11056683,
0.033222351,
-0.13979497,
-0.031020556,
-0.22318721,
],
[
-0.055999811,
0.19542252,
0.095529005,
-0.28626530,
-0.14441303,
0.22369356,
0.14527412,
0.28998481,
0.23105010,
-0.31929879,
-0.29039128,
-0.20956898,
0.43139047,
0.024429152,
0.044904409,
],
[
0.66448103,
0.43069872,
0.29924645,
-0.16202441,
-0.31479544,
-0.39026802,
0.17679822,
0.057952663,
0.17230342,
0.13466011,
-0.35275240,
0.25146896,
-0.018810529,
0.36482392,
-0.32504618,
],
[
-0.12127800,
0.12463327,
0.10656519,
0.046562296,
-0.21678617,
0.19492172,
-0.065521126,
0.024404669,
-0.096828860,
0.19366196,
0.33354757,
0.31295994,
-0.083615456,
-0.25342082,
0.37325717,
],
[
-0.28376230,
-0.32820154,
-0.10496068,
-0.22073452,
-0.13708154,
-0.14426375,
-0.11503319,
0.22424151,
-0.030395022,
-0.51505615,
0.017254978,
0.038957118,
0.36069184,
0.30902452,
0.050030193,
],
[
-0.077875893,
0.0037456560,
0.88685604,
-0.26590028,
-0.079325357,
-0.042734919,
-0.18653782,
-0.35604718,
-0.17497421,
0.088699956,
0.40025886,
-0.055979693,
0.13724479,
0.21485613,
-0.011265799,
],
[
-0.092294730,
0.59209563,
0.031338285,
-0.033080861,
-0.24308858,
-0.099798547,
0.034460195,
0.095119813,
-0.33801620,
0.0063860024,
-0.61207299,
0.081325416,
0.88683114,
0.14254905,
0.14776204,
],
[
-0.13189434,
0.52878496,
0.12652391,
0.045113625,
0.58373514,
0.37291503,
0.11395325,
-0.29479222,
-0.57014085,
0.46291592,
-0.094050179,
0.13959097,
-0.38607402,
-0.44897060,
-0.14602419,
],
[
0.058107658,
-0.32289338,
0.093139162,
0.072427234,
-0.56919401,
0.52554237,
0.23656926,
-0.011782016,
0.071820601,
0.078277291,
-0.13355752,
0.22722721,
0.14369455,
-0.45198935,
-0.55574794,
],
[
0.66145875,
0.34633299,
0.14098019,
0.51882591,
-0.28019898,
-0.16032260,
-0.068413337,
-0.20428242,
0.069672173,
0.23112577,
-0.044368579,
-0.16455425,
0.21620977,
0.0042702105,
-0.087399014,
],
[
0.31599556,
-0.027551859,
0.13434254,
0.13497371,
0.054005680,
-0.17374789,
0.17525393,
0.060258929,
-0.17914162,
-0.31056619,
-0.25358691,
0.025847535,
-0.43006001,
-0.62266361,
-0.033996882,
],
[
-0.29038151,
0.034101270,
0.034903413,
-0.12121764,
0.026030714,
-0.33546274,
-0.41424111,
0.053248380,
-0.27099455,
-0.026251302,
0.41024137,
0.26636349,
0.15582891,
-0.18666254,
0.019895831,
],
[
-0.24388652,
-0.44098852,
0.012618825,
0.24945112,
0.071101888,
0.24623792,
0.17484502,
0.0085286769,
0.25147070,
-0.14659862,
-0.084625150,
0.36931333,
-0.29955293,
0.11044360,
-0.75690139,
],
[
0.041494323,
-0.25980564,
0.46402128,
-0.36112127,
-0.94980789,
-0.16504063,
0.0030943325,
0.052792942,
0.22523648,
0.38390366,
0.45562427,
-0.18631744,
0.0082333995,
0.16670803,
0.16045688,
],
]
)
term1 = np.dot(a1, xx)
term2 = np.dot(a2, np.sin(xx))
term3 = np.dot(a3, np.cos(xx))
term4 = np.dot(np.dot(np.transpose(xx), m), xx)
y = term1 + term2 + term3 + term4
return y[0, 0]
