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Merge pull request #428 from damar-wicaksono/dev-332
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Add highly-curved function from Dette & Pepelyshev (2010)
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damar-wicaksono authored Nov 29, 2024
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4 changes: 3 additions & 1 deletion CHANGELOG.md
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Expand Up @@ -9,7 +9,9 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0

### Added

- The tree-dimensional exponential function from Dette and Pepelyshev (2010)
- The three-dimensional highly-curved function from Dette and Pepelyshev (2010)
for metamodeling exercises.
- The three-dimensional exponential function from Dette and Pepelyshev (2010)
for metamodeling exercises.
- The one-dimensional sine function from Higdon (2002) featuring behaviors
that are different in two scales; for metamodeling (multi-resolution)
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2 changes: 2 additions & 0 deletions docs/_toc.yml
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title: Damped Oscillator
- file: test-functions/damped-oscillator-reliability
title: Damped Oscillator Reliability
- file: test-functions/dette-curved
title: Dette & Pepelyshev (2010) Curved
- file: test-functions/dette-exp
title: Dette & Pepelyshev (2010) Exponential
- file: test-functions/flood
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87 changes: 44 additions & 43 deletions docs/fundamentals/metamodeling.md
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The table below listed the available test functions typically used
in the comparison of metamodeling approaches.

| Name | Input Dimension | Constructor |
|:------------------------------------------------------------------------:|:---------------:|:----------------------:|
| {ref}`Ackley <test-functions:ackley>` | M | `Ackley()` |
| {ref}`Alemazkoor & Meidani (2018) 2D <test-functions:alemazkoor-2d>` | 2 | `Alemazkoor2D()` |
| {ref}`Alemazkoor & Meidani (2018) 20D <test-functions:alemazkoor-20d>` | 20 | `Alemazkoor20D()` |
| {ref}`Borehole <test-functions:borehole>` | 8 | `Borehole()` |
| {ref}`Cheng and Sandu (2010) 2D <test-functions:cheng2d>` | 2 | `Cheng2D` |
| {ref}`Coffee Cup Model <test-functions:coffee-cup>` | 2 | `CoffeeCup()` |
| {ref}`Currin et al. (1988) Sine <test-functions:currin-sine>` | 1 | `CurrinSine()` |
| {ref}`Damped Cosine <test-functions:damped-cosine>` | 1 | `DampedCosine()` |
| {ref}`Damped Oscillator <test-functions:damped-oscillator>` | 7 | `DampedOscillator()` |
| {ref}`Dette & Pepelyshev (2010) Exponential <test-functions:dette-exp>` | 3 | `DetteExp()` |
| {ref}`Flood <test-functions:flood>` | 8 | `Flood()` |
| {ref}`Forrester et al. (2008) <test-functions:forrester>` | 1 | `Forrester2008()` |
| {ref}`(1st) Franke <test-functions:franke-1>` | 2 | `Franke1()` |
| {ref}`(2nd) Franke <test-functions:franke-2>` | 2 | `Franke2()` |
| {ref}`(3rd) Franke <test-functions:franke-3>` | 2 | `Franke3()` |
| {ref}`(4th) Franke <test-functions:franke-4>` | 2 | `Franke4()` |
| {ref}`(5th) Franke <test-functions:franke-5>` | 2 | `Franke5()` |
| {ref}`(6th) Franke <test-functions:franke-6>` | 2 | `Franke6()` |
| {ref}`Friedman (6D) <test-functions:friedman-6d>` | 6 | `Friedman6D()` |
| {ref}`Friedman (10D) <test-functions:friedman-10d>` | 10 | `Friedman10D()` |
| {ref}`Genz (Corner Peak) <test-functions:genz-corner-peak>` | M | `GenzCornerPeak()` |
| {ref}`Gramacy (2007) 1D Sine <test-functions:gramacy-1d-sine>` | 1 | `Gramacy1DSine()` |
| {ref}`Higdon (2002) Sine <test-functions:higdon-sine>` | 1 | `HigdonSine()` |
| {ref}`Holsclaw et al. (2013) Sine <test-functions:holsclaw-sine>` | 1 | `HolsclawSine()` |
| {ref}`Lim et al. (2002) Non-Polynomial <test-functions:lim-non-poly>` | 2 | `LimNonPoly()` |
| {ref}`Lim et al. (2002) Polynomial <test-functions:lim-poly>` | 2 | `LimPoly()` |
| {ref}`McLain S1 <test-functions:mclain-s1>` | 2 | `McLainS1()` |
| {ref}`McLain S2 <test-functions:mclain-s2>` | 2 | `McLainS2()` |
| {ref}`McLain S3 <test-functions:mclain-s3>` | 2 | `McLainS3()` |
| {ref}`McLain S4 <test-functions:mclain-s4>` | 2 | `McLainS4()` |
| {ref}`McLain S5 <test-functions:mclain-s5>` | 2 | `McLainS5()` |
| {ref}`Oakley & O'Hagan (2002) 1D <test-functions:oakley-1d>` | 1 | `Oakley1D()` |
| {ref}`OTL Circuit <test-functions:otl-circuit>` | 6 / 20 | `OTLCircuit()` |
| {ref}`Piston Simulation <test-functions:piston>` | 7 / 20 | `Piston()` |
| {ref}`Robot Arm <test-functions:robot-arm>` | 8 | `RobotArm()` |
| {ref}`Solar Cell Model <test-functions:solar-cell>` | 5 | `SolarCell()` |
| {ref}`Sulfur <test-functions:sulfur>` | 9 | `Sulfur()` |
| {ref}`Undamped Oscillator <test-functions:undamped-oscillator>` | 6 | `UndampedOscillator()` |
| {ref}`Webster et al. (1996) 2D <test-functions:webster-2d>` | 2 | `Webster2D()` |
| {ref}`Welch et al. (1992) <test-functions:welch1992>` | 20 | `Welch1992()` |
| {ref}`Wing Weight <test-functions:wing-weight>` | 10 | `WingWeight()` |
| Name | Input Dimension | Constructor |
|:-----------------------------------------------------------------------:|:---------------:|:----------------------:|
| {ref}`Ackley <test-functions:ackley>` | M | `Ackley()` |
| {ref}`Alemazkoor & Meidani (2018) 2D <test-functions:alemazkoor-2d>` | 2 | `Alemazkoor2D()` |
| {ref}`Alemazkoor & Meidani (2018) 20D <test-functions:alemazkoor-20d>` | 20 | `Alemazkoor20D()` |
| {ref}`Borehole <test-functions:borehole>` | 8 | `Borehole()` |
| {ref}`Cheng and Sandu (2010) 2D <test-functions:cheng2d>` | 2 | `Cheng2D` |
| {ref}`Coffee Cup Model <test-functions:coffee-cup>` | 2 | `CoffeeCup()` |
| {ref}`Currin et al. (1988) Sine <test-functions:currin-sine>` | 1 | `CurrinSine()` |
| {ref}`Damped Cosine <test-functions:damped-cosine>` | 1 | `DampedCosine()` |
| {ref}`Damped Oscillator <test-functions:damped-oscillator>` | 7 | `DampedOscillator()` |
| {ref}`Dette & Pepelyshev (2010) Curved <test-functions:dette-curved>` | 3 | `DetteCurved()` |
| {ref}`Dette & Pepelyshev (2010) Exponential <test-functions:dette-exp>` | 3 | `DetteExp()` |
| {ref}`Flood <test-functions:flood>` | 8 | `Flood()` |
| {ref}`Forrester et al. (2008) <test-functions:forrester>` | 1 | `Forrester2008()` |
| {ref}`(1st) Franke <test-functions:franke-1>` | 2 | `Franke1()` |
| {ref}`(2nd) Franke <test-functions:franke-2>` | 2 | `Franke2()` |
| {ref}`(3rd) Franke <test-functions:franke-3>` | 2 | `Franke3()` |
| {ref}`(4th) Franke <test-functions:franke-4>` | 2 | `Franke4()` |
| {ref}`(5th) Franke <test-functions:franke-5>` | 2 | `Franke5()` |
| {ref}`(6th) Franke <test-functions:franke-6>` | 2 | `Franke6()` |
| {ref}`Friedman (6D) <test-functions:friedman-6d>` | 6 | `Friedman6D()` |
| {ref}`Friedman (10D) <test-functions:friedman-10d>` | 10 | `Friedman10D()` |
| {ref}`Genz (Corner Peak) <test-functions:genz-corner-peak>` | M | `GenzCornerPeak()` |
| {ref}`Gramacy (2007) 1D Sine <test-functions:gramacy-1d-sine>` | 1 | `Gramacy1DSine()` |
| {ref}`Higdon (2002) Sine <test-functions:higdon-sine>` | 1 | `HigdonSine()` |
| {ref}`Holsclaw et al. (2013) Sine <test-functions:holsclaw-sine>` | 1 | `HolsclawSine()` |
| {ref}`Lim et al. (2002) Non-Polynomial <test-functions:lim-non-poly>` | 2 | `LimNonPoly()` |
| {ref}`Lim et al. (2002) Polynomial <test-functions:lim-poly>` | 2 | `LimPoly()` |
| {ref}`McLain S1 <test-functions:mclain-s1>` | 2 | `McLainS1()` |
| {ref}`McLain S2 <test-functions:mclain-s2>` | 2 | `McLainS2()` |
| {ref}`McLain S3 <test-functions:mclain-s3>` | 2 | `McLainS3()` |
| {ref}`McLain S4 <test-functions:mclain-s4>` | 2 | `McLainS4()` |
| {ref}`McLain S5 <test-functions:mclain-s5>` | 2 | `McLainS5()` |
| {ref}`Oakley & O'Hagan (2002) 1D <test-functions:oakley-1d>` | 1 | `Oakley1D()` |
| {ref}`OTL Circuit <test-functions:otl-circuit>` | 6 / 20 | `OTLCircuit()` |
| {ref}`Piston Simulation <test-functions:piston>` | 7 / 20 | `Piston()` |
| {ref}`Robot Arm <test-functions:robot-arm>` | 8 | `RobotArm()` |
| {ref}`Solar Cell Model <test-functions:solar-cell>` | 5 | `SolarCell()` |
| {ref}`Sulfur <test-functions:sulfur>` | 9 | `Sulfur()` |
| {ref}`Undamped Oscillator <test-functions:undamped-oscillator>` | 6 | `UndampedOscillator()` |
| {ref}`Webster et al. (1996) 2D <test-functions:webster-2d>` | 2 | `Webster2D()` |
| {ref}`Welch et al. (1992) <test-functions:welch1992>` | 20 | `Welch1992()` |
| {ref}`Wing Weight <test-functions:wing-weight>` | 10 | `WingWeight()` |

In a Python terminal, you can list all the available functions relevant
for metamodeling applications using ``list_functions()``
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| {ref}`Damped Cosine <test-functions:damped-cosine>` | 1 | `DampedCosine()` |
| {ref}`Damped Oscillator <test-functions:damped-oscillator>` | 7 | `DampedOscillator()` |
| {ref}`Damped Oscillator Reliability <test-functions:damped-oscillator-reliability>` | 8 | `DampedOscillatorReliability()` |
| {ref}`Dette & Pepelyshev (2010) Curved <test-functions:dette-curved>` | 3 | `DetteCurved()` |
| {ref}`Dette & Pepelyshev (2010) Exponential <test-functions:dette-exp>` | 3 | `DetteExp()` |
| {ref}`Flood <test-functions:flood>` | 8 | `Flood()` |
| {ref}`Forrester et al. (2008) <test-functions:forrester>` | 1 | `Forrester2008()` |
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96 changes: 96 additions & 0 deletions docs/test-functions/dette-curved.md
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---
jupytext:
formats: ipynb,md:myst
text_representation:
extension: .md
format_name: myst
format_version: 0.13
jupytext_version: 1.14.1
kernelspec:
display_name: Python 3 (ipykernel)
language: python
name: python3
---

(test-functions:dette-curved)=
# Highly-Curved Function from Dette and Pepelyshev (2010)

```{code-cell} ipython3
import numpy as np
import matplotlib.pyplot as plt
import uqtestfuns as uqtf
```

The function is a three-dimensional, scalar-valued function.
The function contains no cross-term but curved along each dimension.

The function appeared in {cite}`Dette2010` as a test function for comparing
different experimental designs in the construction of metamodels.

## Test function instance

To create a default instance of the test function:

```{code-cell} ipython3
my_testfun = uqtf.DetteCurved()
```

Check if it has been correctly instantiated:

```{code-cell} ipython3
print(my_testfun)
```

## Description

The test function is defined as[^location]:

$$
\mathcal{M}(\boldsymbol{x}) = 4 \left( x_1 - 2 + 8 x_2 - 8 x_2^2 \right) + \left( 3 - 4 x_2 \right)^2 + 16 \left( x_3 + 1\right)^{0.5} \left( 2 x_3 - 1\right)^2,
$$

where $\boldsymbol{x} = \left( x_1, x_2, x_3 \right)$ is the three-dimensional
vector of input variables further defined below.

## Probabilistic input

The probabilistic input model for the test function is shown below.

```{code-cell} ipython3
:tags: [hide-input]
print(my_testfun.prob_input)
```

## Reference results

This section provides several reference results of typical UQ analyses
involving the test function.

### Sample histogram

Shown below is the histogram of the output based on $100'000$ random points:

```{code-cell} ipython3
:tags: [hide-input]
my_testfun.prob_input.reset_rng(42)
xx_test = my_testfun.prob_input.get_sample(100000)
yy_test = my_testfun(xx_test)
plt.hist(yy_test, bins="auto", color="#8da0cb");
plt.grid();
plt.ylabel("Counts [-]");
plt.xlabel("$\mathcal{M}(X)$");
plt.gcf().tight_layout(pad=3.0)
plt.gcf().set_dpi(150);
```

## References

```{bibliography}
:style: unsrtalpha
:filter: docname in docnames
```

[^location]: see Eq. (5), Section 3.1, p. 424, in {cite}`Dette2010`.
3 changes: 2 additions & 1 deletion src/uqtestfuns/test_functions/__init__.py
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from .currin_sine import CurrinSine
from .damped_cosine import DampedCosine
from .damped_oscillator import DampedOscillator, DampedOscillatorReliability
from .dette import DetteExp
from .dette import DetteCurved, DetteExp
from .flood import Flood
from .forrester import Forrester2008
from .four_branch import FourBranch
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"DampedCosine",
"DampedOscillator",
"DampedOscillatorReliability",
"DetteCurved",
"DetteExp",
"Flood",
"Forrester2008",
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55 changes: 53 additions & 2 deletions src/uqtestfuns/test_functions/dette.py
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from uqtestfuns.core.custom_typing import ProbInputSpecs
from uqtestfuns.core.uqtestfun_abc import UQTestFunFixDimABC

__all__ = ["DetteExp"]
__all__ = ["DetteCurved", "DetteExp"]


def evaluate_exp(xx: np.ndarray) -> np.ndarray:
Expand All @@ -41,6 +41,28 @@ def evaluate_exp(xx: np.ndarray) -> np.ndarray:
return 100 * yy


def evaluate_curved(xx: np.ndarray) -> np.ndarray:
"""Evaluate the highly-curved function from Dette and Pepelyshev (2010).
Parameters
----------
xx : np.ndarray
M-Dimensional input values given by an N-by-3 array where
N is the number of input values.
Returns
-------
np.ndarray
The output of the test function evaluated on the input values.
The output is a 1-dimensional array of length N.
"""
term_1 = 4 * (xx[:, 0] - 2 + 8 * xx[:, 1] - 8 * xx[:, 1] ** 2)
term_2 = (3 - 4 * xx[:, 1]) ** 2
term_3 = (16 * (xx[:, 2] + 1) ** 0.5) * (2 * xx[:, 2] - 1) ** 2

return term_1 + term_2 + term_3


class DetteExp(UQTestFunFixDimABC):
"""A concrete implementation of the exponential function."""

Expand All @@ -51,7 +73,7 @@ class DetteExp(UQTestFunFixDimABC):
"function_id": "DetteExp",
"description": (
"Input specification for the exponential test function "
"from Dette and Pepelyshev et al. (2010)"
"from Dette and Pepelyshev (2010)"
),
"marginals": [
{
Expand All @@ -68,3 +90,32 @@ class DetteExp(UQTestFunFixDimABC):
_available_parameters = None

evaluate = staticmethod(evaluate_exp) # type: ignore


class DetteCurved(UQTestFunFixDimABC):
"""A concrete implementation of the highly-curved function."""

_tags = ["metamodeling"]
_description = "Curved function from Dette and Pepelyshev (2010)"
_available_inputs: ProbInputSpecs = {
"Dette2010": {
"function_id": "DetteCurved",
"description": (
"Input specification for the highly-curved test function "
"from Dette and Pepelyshev (2010)"
),
"marginals": [
{
"name": f"x_{i + 1}",
"distribution": "uniform",
"parameters": [0, 1],
"description": None,
}
for i in range(3)
],
"copulas": None,
},
}
_available_parameters = None

evaluate = staticmethod(evaluate_curved) # type: ignore

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