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Add exponential function from Dette & Pepelyshev (2010)
The three-dimensional exponential function from Dette and Pepelyshev for metamodeling exercises has been added to the codebase. The documentation has been updated accordingly. This commit should resolve Issue #331.
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‎CHANGELOG.md

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### Added
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- The tree-dimensional exponential function from Dette and Pepelyshev (2010)
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for metamodeling exercises.
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- The one-dimensional sine function from Higdon (2002) featuring behaviors
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that are different in two scales; for metamodeling (multi-resolution)
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exercises.

‎docs/_toc.yml

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title: Damped Oscillator
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- file: test-functions/damped-oscillator-reliability
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title: Damped Oscillator Reliability
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- file: test-functions/dette-exp
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title: Dette & Pepelyshev (2010) Exponential
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- file: test-functions/flood
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title: Flood
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- file: test-functions/forrester

‎docs/fundamentals/metamodeling.md

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

‎docs/references.bib

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doi = {10.1007/978-1-4471-0657-9_2},
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}
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@Article{Dette2010,
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author = {Dette, Holger and Pepelyshev, Andrey},
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journal = {Technometrics},
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title = {Generalized latin hypercube design for computer experiments},
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year = {2010},
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number = {4},
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pages = {421--429},
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volume = {52},
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doi = {10.1198/tech.2010.09157},
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}
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@Comment{jabref-meta: databaseType:bibtex;}

‎docs/test-functions/available.md

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| {ref}`Damped Cosine <test-functions:damped-cosine>` | 1 | `DampedCosine()` |
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| {ref}`Damped Oscillator <test-functions:damped-oscillator>` | 7 | `DampedOscillator()` |
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| {ref}`Damped Oscillator Reliability <test-functions:damped-oscillator-reliability>` | 8 | `DampedOscillatorReliability()` |
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| {ref}`Dette & Pepelyshev (2010) Exponential <test-functions:dette-exp>` | 3 | `DetteExp()` |
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| {ref}`Flood <test-functions:flood>` | 8 | `Flood()` |
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| {ref}`Forrester et al. (2008) <test-functions:forrester>` | 1 | `Forrester2008()` |
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| {ref}`Four-branch <test-functions:four-branch>` | 2 | `FourBranch()` |

‎docs/test-functions/dette-exp.md

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---
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jupytext:
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formats: ipynb,md:myst
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text_representation:
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extension: .md
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format_name: myst
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format_version: 0.13
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jupytext_version: 1.14.1
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kernelspec:
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display_name: Python 3 (ipykernel)
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language: python
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name: python3
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---
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(test-functions:dette-exp)=
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# Exponential Function from Dette and Pepelyshev (2010)
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```{code-cell} ipython3
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import numpy as np
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import matplotlib.pyplot as plt
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import uqtestfuns as uqtf
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```
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The function is a three-dimensional, scalar-valued function that exhibits
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asymptotic behavior where the function value approaches zero near the origin
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and increases toward a value as the input moves farther away from the origin
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in any direction.
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The function appeared in {cite}`Dette2010` as a test function for comparing
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different experimental designs in the construction of metamodels.
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## Test function instance
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To create a default instance of the test function:
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```{code-cell} ipython3
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my_testfun = uqtf.DetteExp()
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```
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Check if it has been correctly instantiated:
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```{code-cell} ipython3
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print(my_testfun)
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```
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## Description
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The test function is defined as[^location]:
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$$
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\mathcal{M}(\boldsymbol{x}) = 100 \left[ \exp{\left( -\frac{2}{x_1^{1.75}} \right)} + \exp{\left( -\frac{2}{x_2^{1.5}} \right)} + \exp{\left( -\frac{2}{x_3^{1.25}} \right)} \right],
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$$
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where $\boldsymbol{x} = \left( x_1, x_2, x_3 \right)$ is the three-dimensional
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vector of input variables further defined below.
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## Probabilistic input
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The probabilistic input model for the test function is shown below.
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```{code-cell} ipython3
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:tags: [hide-input]
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print(my_testfun.prob_input)
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```
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## Reference results
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This section provides several reference results of typical UQ analyses involving
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the test function.
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### Sample histogram
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Shown below is the histogram of the output based on $100'000$ random points:
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```{code-cell} ipython3
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:tags: [hide-input]
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my_testfun.prob_input.reset_rng(42)
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xx_test = my_testfun.prob_input.get_sample(100000)
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yy_test = my_testfun(xx_test)
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plt.hist(yy_test, bins="auto", color="#8da0cb");
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plt.grid();
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plt.ylabel("Counts [-]");
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plt.xlabel("$\mathcal{M}(X)$");
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plt.gcf().tight_layout(pad=3.0)
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plt.gcf().set_dpi(150);
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```
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## References
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```{bibliography}
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:style: unsrtalpha
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:filter: docname in docnames
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```
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[^location]: see Eq. (4), Section 3.1, p. 424, in {cite}`Dette2010`.

‎src/uqtestfuns/test_functions/__init__.py

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from .currin_sine import CurrinSine
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from .damped_cosine import DampedCosine
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from .damped_oscillator import DampedOscillator, DampedOscillatorReliability
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from .dette import DetteExp
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from .flood import Flood
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from .forrester import Forrester2008
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from .four_branch import FourBranch
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"DampedCosine",
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"DampedOscillator",
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"DampedOscillatorReliability",
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"DetteExp",
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"Flood",
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"Forrester2008",
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"FourBranch",

‎src/uqtestfuns/test_functions/dette.py

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"""
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Module with an implementation of the functions from Dette and Pepelyshev (2010)
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The paper by Dette and Pepelyshev [1] contains several analytical test
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functions used to compare different experimental designs for metamodeling
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applications.
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References
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----------
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1. H. Dette and A. Pepelyshev, “Generalized Latin Hypercube Design for Computer
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Experiments,” Technometrics, vol. 52, no. 4, pp. 421–429, 2010.
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DOI: 10.1198/TECH.2010.09157.
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"""
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import numpy as np
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from uqtestfuns.core.custom_typing import ProbInputSpecs
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from uqtestfuns.core.uqtestfun_abc import UQTestFunFixDimABC
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__all__ = ["DetteExp"]
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def evaluate_exp(xx: np.ndarray) -> np.ndarray:
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"""Evaluate the exponential function from Dette and Pepelyshev (2010).
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Parameters
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----------
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xx : np.ndarray
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M-Dimensional input values given by an N-by-3 array where
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N is the number of input values.
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Returns
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-------
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np.ndarray
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The output of the test function evaluated on the input values.
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The output is a 1-dimensional array of length N.
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"""
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yy = np.sum(np.exp(-2 / xx**([1.75, 1.5, 1.25])), axis=1)
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return 100 * yy
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class DetteExp(UQTestFunFixDimABC):
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"""A concrete implementation of the exponential function."""
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_tags = ["metamodeling"]
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_description = "Exponential function from Dette and Pepelyshev (2010)"
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_available_inputs: ProbInputSpecs = {
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"Dette2010": {
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"function_id": "DetteExp",
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"description": (
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"Input specification for the exponential test function "
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"from Dette and Pepelyshev et al. (2010)"
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),
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"marginals": [
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{
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"name": f"x_{i + 1}",
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"distribution": "uniform",
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"parameters": [0, 1],
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"description": None,
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}
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for i in range(3)
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],
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"copulas": None,
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},
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}
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_available_parameters = None
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evaluate = staticmethod(evaluate_exp) # type: ignore

‎tests/conftest.py

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parameters = np.sort(1 + 2 * np.random.rand(3))
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parameters[[0, 1]] = parameters[[1, 0]]
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# Insert sigma/beta as the second parameter
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parameters = np.insert(parameters, 1, np.random.rand(1))
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parameters = np.insert(parameters, 1, 0.5 + np.random.rand(1))
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elif distribution == "lognormal":
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# Limit the size of the parameters
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parameters = 1 + np.random.rand(2)

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