Uniform Sampling Strategies

1
Uniform Sampling Strategies

• Amitay Isaacs, Devendra Ghate

2
Overview

• Sampling Problem
• Uniform sampling
• Random sampling
• Stratified sampling
• Latin Hypercube sampling
• LP?
• How uniform is uniform?

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Sampling Problem

• Given d-dimensional design space
• X X1 X2 Xd
• locate n points X1, X2, , Xn in this
  hypercube
• Normalized space 0 ? Xj ? 1

4
Sampling for Computer Expts.

• For RSM one can use DOE for generating the
  designs
• For computer experiments (e.g. DACE) how does one
  sample?
• Natural Choice - Space Filling
• Uniform contribution from all sub-regions of
  design space

5
Random Sampling

• Coordinate along each axis is sampled from
  uniform distribution
• As N ? ?, random sampling tends to infinity.
• Easy to implement
• Can handle irregular design spaces

(1,1)
x2
O
x1
6
Stratified Sampling

• Partition the design space in N equal regions
• Pick up a point randomly from each region
• More uniform than random sampling for relatively
  less number of points
• Partitioning difficult for irregular design spaces

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Latin Hypercube Sampling

• Partition each coordinate axis in N equal
  regions.
• Pick up a point in each region for each axis,
  Xi,j
• For each Xi,select one of the points Xi,j to
  construct a point in design space
• Same point can be chosen multiple times
• Can handle sampling where input variables have
  specified probability distribution

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Uniform sequences

• All the strategies need to know n before
  sampling
• Some surrogate modeling techniques require
  additional points after first n points have
  been sampled.
• How to add m extra points to the original n
  points, such that mn points have same uniformity
  characteristics?

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LP? Sampling

• Deterministic way of adding points
• Each coordinate axis is partitioned into binary
  intervals
• Each interval is further subdivided at higher
  level

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Measure of Uniformity

• Consider points P1, P2, , PN belonging to unit
  d-dimensional hypercube Kd
• Let G be an arbitrary domain in Kd
• Let SN(G) be number of points in G
• Sequence is Uniform if
• lim N -gt ? SN(G) / N VG where VG is the volume
  of G

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Discrepancy

• P is any arbitrary point in Kd
• GP is the d-dim parallelepiped with diagonal O-P
• Discrepancy
• D sup SN(GP) NVGP

(1,1)
x2
P
O
x1
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Discrepancy Results

• D is plotted for
• dimensions 2, 3, 4, 5
• sampling points from 10 to 400 in steps of 10

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2D
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3D
15
4D
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5D
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References

• Conover, W. J., McKay, M. D., Beckman, R. J.,
  A comparison of three methods for selecting
  values of input variables in the analysis of
  output from a computer code, Technometrics, vol.
  21, no. 2, May 1979.
• Statnikov, Roman B., Matusov, Joseph B.,
  Multicriteria Analysis in Engineering, Kluwer
  Academic Publishers, 2002.