Latin Hypercube and Orthogonal sampling Emine Şule Yazıcı Koç University Joint work with Kevin Pamela Burrage Diane Donovan Thomas A McCourt Bevan Thompson Population ID: 615776
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Slide1
Estimates of the coverage of parameter space by Latin Hypercube and Orthogonal sampling
Emine Şule Yazıcı
Koç University
Joint
work
with
:Kevin, Pamela Burrage, Diane Donovan,
Thomas A McCourt
,
Bevan ThompsonSlide2
Population of ModelsPopulation of Models (POM) offer
a methodology:
for introducing inherent variability to capture the
underlying
dynamical
processes,
while simultaneously varying multiple parameters
.
UNCERTAINTY
QUANTIFICATIONSlide3
Erosion and the
Great Barrier Reef
Regular/systematic sweeps sampling
erosionSampling at distinct times varying by longitude and latitudeSlide4
Latin Hypercube SamplingFirst discretize the parameter space
10-20
21-30
31-40
41-50
51-60
61-7071-801-2X2-3X3-4X
4-5X5-6X6-7X7-8X
Parameter 1: Age: 10-80
Parameter 2: Income: 1-8Slide5
Latin Hypercube Sampling
Slide6
3-D Latin Hypercube
SamplingSlide7
LHT is an
OS if
n = p
d
points are distributed evenly across sub-blocks.Some research shows:Uniformity of small dimensional margins.Improved representation of the underlying variability.A form of variance reduction.Better screening for effective parameters.Equally fast implementation.
Orthogonal Sampling (OS)Slide8
Overlapping Latin Trials
A LHT
for
d=2
and
n=6
k=2
trials
k=3
trials
k=4
trials
k=1
trials
k=5
trialsSlide9
We want to calculate the
expected
coverage
of the parameter spaceSlide10
Expected Coverage of the Parameter Space
by
k
trials
X
i
represents the expected intersection size of m arbitrary trials Slide11
How to calculate xi(n)Slide12
How many LHT’s are there of dimension d?There are n!d-1
different trials of dimension d
There are 8!
2
trials of dimension 3Slide13
The number of ways to choose with repetitionRemember that the number of ways to choose k elements from a set of size v
allowing repetition is Slide14
How many different selections of m
d-
trials
are
there?There aredifferent collections of d-trials of size m Slide15
Fix any row of a d-trialGiven any cell of a Latin Hypercube Trial of dimension d
Intersection of how many d-trial collections of size m contains this cell? Slide16
Calculating xi(n) for LHSTheorem
T
ake
a
set of m LHTs of dimension d on
[n]. The expected number of ordered d-tuples common to all m LHTs isSlide17
Calculating xi(n) for OSs Slide18
How many OT’s are there of dimension d?Let n=pd
There are (p
d-1
)!
dp
different trials of dimension d
XXXXXXXXXSlide19
How many OS’s are
there
of
dimension
d?
1 1 11 1 21 2 11 2 2 2 1 1 2 1 22 2 12 2 2(1,.) (1,.) (1,.)
(1,.) (1,.) (2,.)(1,.) (2,.) (1,.)(1,.) (2,.) (2,.)(2,.) (1,.) (1,.)(2,.) (1,.) (2,.)(2,.) (2,.) (1,.)(2,.) (2,.) (2,.)Assume dimension 3 so n=23p=2 and q= 43214There are p.d functions each having q! different choices. So a total of q!p.d=(pd-1)!pd different OT of dimension d.Slide20
How many OTs contain a fixed d-tuble?
1 1 1
1 1 2
1 2 1
1 2 2
2 1 1 2 1 22 2 12 2 2(1,.) (1,.) (1,.)(1,.) (1,.) (2,.)(1,.) (2,.) (1,.)(1,.) (2,.) (2,.)
(2,.) (1,.) (1,.)(2,.) (1,.) (2,.)(2,.) (2,.) (1,.)(2,.) (2,.) (2,.)1 1 1There are (pd-1)!pd OTs each having n=pd d-tuples. There are a total of nd d-tuples, so each d-tuple occurs at n(pd-1)!pd / nd OTs. Slide21
Calculating xi(n) for OSTheorem
T
ake
a
set of m OTs of dimension d
on [n]. The expected number of ordered d-tuples common to all m OTs isSlide22
Expected Coverage of the Parameter Space
by
k
trials
X
i
represents the expected intersection size of m arbitrary trials Slide23
Projection onto subspacesSlide24
Expected Coverage of the 2- Dimensional
Subspaces
X
i
represents
the expected intersection size of m arbitrary trials Slide25
Edge coverage of d-trialsAn (i,j)-
edge
of a d-
tuple
a=(a
1,a2,…,ad) is an ordered pair (ai, aj)There are edges in a d-tuple and total different
edges in a d-trial.There are a total of many possible edges.Slide26
How many d-trials contain a fixed edge?
di
fferent
d-trials containing a
fixed edge
There are (a1,a2,a3…,ad) Defines a 2-dimensional subspaceSlide27
The number of common edges to
all
m d-
trials
in LHSlide28
Situation so far Slide29
How to analyze:Binomial expansion
gives
:
AlsoSlide30
How to analyzeSlide31
How to analyzeSlide32
How to analyze
usingSlide33
Theoretical bounds on percentage coverageTheorem In the case of Latin Hypercube sampling and Orthogonal sampling (with
n=
p
d
)
the expected percentage coverage of parameter space is given bySlide34
Simulation Results (t=2)
d
=5Slide35
Simulation Results (t=3)
d
=5Slide36
Simulation Results (t=4)
d
=5