1.4. Sampling

Sampling Strategy, Experimental Design

MOEA selection strategy:
1.Random sampling 2.Best sampling 3.Tournament sampling 4.Tournament+Best sampling
Links:

https://www.google.nl/search?sclient=psy-ab&client=safari&rls=en&q=github+sampling+python&oq=github+sampling+python&gs_l=serp.3..0i22i30k1l2.2824.3846.0.4517.7.7.0.0.0.0.91.551.7.7.0….0…1c.1.64.psy-ab..0.7.538…33i160k1.G42G3jxX1XY&pbx=1&biw=1680&bih=961&dpr=2&cad=cbv&bvch=u&sei=-e5HWNLGHsrNgAbDjruoAg#q=github+sampling+strategy+python

https://en.wikipedia.org/wiki/Sampling_(statistics)

https://en.wikipedia.org/wiki/Latin_hypercube_sampling

https://docs.scipy.org/doc/numpy/reference/routines.random.html

Factorial Designs:
samFullFact, samFracFact, samFF2n, samPlackettBurman
Response-Surface Designs:
samBoxBehnken, samCentralComposite
Randomized Designs:
samLatinHypercube

1.4.1. samBoxBehnken

surrogate.sampling.samBoxBehnken()[source]

Create a Box-Behnken design

Parameters:
  • n – The number of factors in the design
  • center – The number of center points to include (default = 1).
Returns:

The design matrix

This code was originally published by the following individuals for use with Scilab:

  • Copyright (C) 2012 - 2013 - Michael Baudin
  • Copyright (C) 2012 - Maria Christopoulou
  • Copyright (C) 2010 - 2011 - INRIA - Michael Baudin
  • Copyright (C) 2009 - Yann Collette
  • Copyright (C) 2009 - CEA - Jean-Marc Martinez

website: forge.scilab.org/index.php/p/scidoe/sourcetree/master/macros

Much thanks goes to these individuals. It has been converted to Python by Abraham Lee.

Example: 
>>> samBoxBehnken(3)
array([[-1., -1.,  0.],
       [ 1., -1.,  0.],
       [-1.,  1.,  0.],
       [ 1.,  1.,  0.],
       [-1.,  0., -1.],
       [ 1.,  0., -1.],
       [-1.,  0.,  1.],
       [ 1.,  0.,  1.],
       [ 0., -1., -1.],
       [ 0.,  1., -1.],
       [ 0., -1.,  1.],
       [ 0.,  1.,  1.],
       [ 0.,  0.,  0.],
       [ 0.,  0.,  0.],
       [ 0.,  0.,  0.]])

1.4.2. samCentralComposite

surrogate.sampling.samCentralComposite()[source]

Central composite design

Parameters:
  • n – The number of factors in the design.
  • center – A 1-by-2 array of integers, the number of center points in each block of the design. (Default: (4, 4)).
  • alpha –

    A string describing the effect of alpha has on the variance.

    alpha can take on the following values:

    1. ’orthogonal’ or ‘o’ (Default)
    2. ’rotatable’ or ‘r’
  • face –

    The relation between the start points and the corner (factorial) points.

    There are three options for this input:

    1. ’circumscribed’ or ‘ccc’: This is the original form of the central composite design. The star points are at some distance alpha from the center, based on the properties desired for the design. The start points establish new extremes for the low and high settings for all factors. These designs have circular, spherical, or hyperspherical symmetry and require 5 levels for each factor. Augmenting an existing factorial or resolution V fractional factorial design with star points can produce this design.
    2. ’inscribed’ or ‘cci’: For those situations in which the limits specified for factor settings are truly limits, the CCI design uses the factors settings as the star points and creates a factorial or fractional factorial design within those limits (in other words, a CCI design is a scaled down CCC design with each factor level of the CCC design divided by alpha to generate the CCI design). This design also requires 5 levels of each factor.
    3. ’faced’ or ‘ccf’: In this design, the star points are at the center of each face of the factorial space, so alpha = 1. This variety requires 3 levels of each factor. Augmenting an existing factorial or resolution V design with appropriate star points can also produce this design.
Returns:

The design matrix with coded levels -1 and 1

This code was originally published by the following individuals for use with Scilab:

  • Copyright (C) 2012 - 2013 - Michael Baudin
  • Copyright (C) 2012 - Maria Christopoulou
  • Copyright (C) 2010 - 2011 - INRIA - Michael Baudin
  • Copyright (C) 2009 - Yann Collette
  • Copyright (C) 2009 - CEA - Jean-Marc Martinez

website: forge.scilab.org/index.php/p/scidoe/sourcetree/master/macros

Much thanks goes to these individuals. It has been converted to Python by Abraham Lee.

Note

  • Fractional factorial designs are not (yet) available here.
  • ‘ccc’ and ‘cci’ can be rotatable design, but ‘ccf’ cannot.
  • If face is specified, while alpha is not, then the default value of alpha is ‘orthogonal’.
Example: 
>>> samCentralComposite(3)
array([[-1.        , -1.        , -1.        ],
       [ 1.        , -1.        , -1.        ],
       [-1.        ,  1.        , -1.        ],
       [ 1.        ,  1.        , -1.        ],
       [-1.        , -1.        ,  1.        ],
       [ 1.        , -1.        ,  1.        ],
       [-1.        ,  1.        ,  1.        ],
       [ 1.        ,  1.        ,  1.        ],
       [ 0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ],
       [-1.82574186,  0.        ,  0.        ],
       [ 1.82574186,  0.        ,  0.        ],
       [ 0.        , -1.82574186,  0.        ],
       [ 0.        ,  1.82574186,  0.        ],
       [ 0.        ,  0.        , -1.82574186],
       [ 0.        ,  0.        ,  1.82574186],
       [ 0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.        ]])

1.4.3. samFullFact

surrogate.sampling.samFullFact()[source]

Create a general full-factorial design

Parameters:levels – An array of integers that indicate the number of levels of each input design factor.
Returns:The design matrix with coded levels 0 to k-1 for a k-level factor

This code was originally published by the following individuals for use with Scilab:

  • Copyright (C) 2012 - 2013 - Michael Baudin
  • Copyright (C) 2012 - Maria Christopoulou
  • Copyright (C) 2010 - 2011 - INRIA - Michael Baudin
  • Copyright (C) 2009 - Yann Collette
  • Copyright (C) 2009 - CEA - Jean-Marc Martinez

website: forge.scilab.org/index.php/p/scidoe/sourcetree/master/macros

Much thanks goes to these individuals. It has been converted to Python by Abraham Lee.

Example: 
>>> samFullFact([2, 4, 3])
array([[ 0.,  0.,  0.],
       [ 1.,  0.,  0.],
       [ 0.,  1.,  0.],
       [ 1.,  1.,  0.],
       [ 0.,  2.,  0.],
       [ 1.,  2.,  0.],
       [ 0.,  3.,  0.],
       [ 1.,  3.,  0.],
       [ 0.,  0.,  1.],
       [ 1.,  0.,  1.],
       [ 0.,  1.,  1.],
       [ 1.,  1.,  1.],
       [ 0.,  2.,  1.],
       [ 1.,  2.,  1.],
       [ 0.,  3.,  1.],
       [ 1.,  3.,  1.],
       [ 0.,  0.,  2.],
       [ 1.,  0.,  2.],
       [ 0.,  1.,  2.],
       [ 1.,  1.,  2.],
       [ 0.,  2.,  2.],
       [ 1.,  2.,  2.],
       [ 0.,  3.,  2.],
       [ 1.,  3.,  2.]])

1.4.4. samLatinHypercube

surrogate.sampling.samLatinHypercube()[source]

Generate a latin-hypercube design

Parameters:
  • n – The number of factors to generate samples for
  • samples – The number of samples to generate for each factor (Default: n)
  • criterion – Allowable values are “center” or “c”, “maximin” or “m”, “centermaximin” or “cm”, and “correlation” or “corr”. If no value given, the design is simply randomized.
  • iterations – The number of iterations in the maximin and correlations algorithms (Default: 5).
Returns:

An n-by-samples design matrix that has been normalized so factor values are uniformly spaced between zero and one.

This code was originally published by the following individuals for use with Scilab:

  • Copyright (C) 2012 - 2013 - Michael Baudin
  • Copyright (C) 2012 - Maria Christopoulou
  • Copyright (C) 2010 - 2011 - INRIA - Michael Baudin
  • Copyright (C) 2009 - Yann Collette
  • Copyright (C) 2009 - CEA - Jean-Marc Martinez

website: forge.scilab.org/index.php/p/scidoe/sourcetree/master/macros

Much thanks goes to these individuals. It has been converted to Python by Abraham Lee.

Example:

A 3-factor design (defaults to 3 samples):

>>> samLatinHypercube(3)
array([[ 0.40069325,  0.08118402,  0.69763298],
       [ 0.19524568,  0.41383587,  0.29947106],
       [ 0.85341601,  0.75460699,  0.360024  ]])

A 4-factor design with 6 samples:

>>> samLatinHypercube(4, samples=6)
array([[ 0.27226812,  0.02811327,  0.62792445,  0.91988196],
       [ 0.76945538,  0.43501682,  0.01107457,  0.09583358],
       [ 0.45702981,  0.76073773,  0.90245401,  0.18773015],
       [ 0.99342115,  0.85814198,  0.16996665,  0.65069309],
       [ 0.63092013,  0.22148567,  0.33616859,  0.36332478],
       [ 0.05276917,  0.5819198 ,  0.67194243,  0.78703262]])

A 2-factor design with 5 centered samples:

>>> samLatinHypercube(2, samples=5, criterion='center')
array([[ 0.3,  0.5],
       [ 0.7,  0.9],
       [ 0.1,  0.3],
       [ 0.9,  0.1],
       [ 0.5,  0.7]])

A 3-factor design with 4 samples where the minimum distance between all samples has been maximized:

>>> samLatinHypercube(3, samples=4, criterion='maximin')
array([[ 0.02642564,  0.55576963,  0.50261649],
       [ 0.51606589,  0.88933259,  0.34040838],
       [ 0.98431735,  0.0380364 ,  0.01621717],
       [ 0.40414671,  0.33339132,  0.84845707]])

A 4-factor design with 5 samples where the samples are as uncorrelated as possible (within 10 iterations):

>>> samLatinHypercube(4, samples=5, criterion='correlate', iterations=10)

1.4.5. samOptimalLHC

surrogate.sampling.samOptimalLHC()[source]

Generates an optimized Latin hypercube by optimizing the Morris-Mitchell criterion for a range of exponents and plots the first two dimensions of the current hypercube throughout the optimization process.

Parameters:
  • n – number of points required
  • Population – number of individuals in the evolutionary operation optimizer
  • Iterations – number of generations the evolutionary operation optimizer is run for
Returns:

X optimized Latin hypercube

Note

high values for the two inputs above will ensure high quality hypercubes, but the search will take longer. generation - if set to True, the LHC will be generated. If ‘False,’ the algorithm will check for an existing plan before generating.

1.4.6. samPlackettBurman

surrogate.sampling.samPlackettBurman()[source]

Generate a Plackett-Burman design

Parameters:n – The number of factors to create a matrix for.
Returns:An orthogonal design matrix with n columns, one for each factor, and the number of rows being the next multiple of 4 higher than n (e.g., for 1-3 factors there are 4 rows, for 4-7 factors there are 8 rows, etc.)

This code was originally published by the following individuals for use with Scilab:

  • Copyright (C) 2012 - 2013 - Michael Baudin
  • Copyright (C) 2012 - Maria Christopoulou
  • Copyright (C) 2010 - 2011 - INRIA - Michael Baudin
  • Copyright (C) 2009 - Yann Collette
  • Copyright (C) 2009 - CEA - Jean-Marc Martinez

website: forge.scilab.org/index.php/p/scidoe/sourcetree/master/macros

Much thanks goes to these individuals. It has been converted to Python by Abraham Lee.

Example:

A 3-factor design:

>>> samPlackettBurman(3)
array([[-1., -1.,  1.],
       [ 1., -1., -1.],
       [-1.,  1., -1.],
       [ 1.,  1.,  1.]])

A 5-factor design:

>>> samPlackettBurman(5)
array([[-1., -1.,  1., -1.,  1.],
       [ 1., -1., -1., -1., -1.],
       [-1.,  1., -1., -1.,  1.],
       [ 1.,  1.,  1., -1., -1.],
       [-1., -1.,  1.,  1., -1.],
       [ 1., -1., -1.,  1.,  1.],
       [-1.,  1., -1.,  1., -1.],
       [ 1.,  1.,  1.,  1.,  1.]])

1.4.7. samRandom

surrogate.sampling.samRandom()[source]

samRandom

Parameters:n – default 2
Returns:

Note

Not sphinx doc!! 20170214 Encoding: utf-8

module numpy.random.mtrand

from /System/Library/Frameworks/Python.framework/Versions/2.7/Extras/lib/python/numpy/random/mtrand.so by generator 1.138

no doc

Links:
https://docs.scipy.org/doc/numpy/reference/routines.random.html

1.4.8. samRandomLHC

surrogate.sampling.samRandomLHC()[source]

Generates a random latin hypercube within the [0,1]^k hypercube

Parameters:
  • n – desired number of points
  • k – number of design variables (dimensions)
  • Edges – if Edges=1 the extreme bins will have their centers on the edges of the domain
Returns:

Latin hypercube sampling plan of n points in k dimensions