On Sequential Experimental Design for Empirical Model-Building under Interval Error

1
On Sequential Experimental Design for Empirical
Model-Buildingunder Interval Error

• Sergei Zhilin,
• sergei_at_asu.ru
• Altai State University,Barnaul, Russia

2
Outline

• Regression under interval error
• Experimental design refining context
• Classical and interval design optimality
  criteria
• Sequential experimental design for regression
  models under interval error
• Comparative simulation study of classical and
  interval sequential design procedures
• Conclusions

3
Regression under Interval Error

• Model structure

Input variables x (x1,,xp)T measured without
error
Output variable ymeasured with error
Linear-parameterized modeling function
Model parametersto be estimated
Measurement error

• Interval error means unknown but bounded

4
Regression under Interval Error
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Regression under Interval Error

• Fitting data with the model y ?1 ?2x

In (x, y) domain
In (?1, ?2) domain
Uncertainty set A is unbounded not enough data
to build the model
?2
y
Set of feasible models
Uncertainty set A
Set of feasible models
Uncertainty set A
?1
x
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Regression under Interval Error

• Problems that may be stated with respect to
  uncertainty set A

• Model parameters estimation

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Regression under Interval Error

• Problems that may be stated with respect to
  uncertainty set A

• Prediction of the output variable value for
  fixed values of input variables

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Experimental Design Refining Context

• Sequential experimental design

• Simultaneous experimental design

• Product or process optimization
• Model quality optimization

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Experimental Design for Regression under Interval
Error

• Notations

• model

• design space

• information matrix

• design matrix

• measurements

• error bounds

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Experimental Design for Regression under Interval
Error

• Design optimality criteria
• Classical
• Name Minimizes
• D -optimality (volume of joint confidence
  interval)
• G -optimality (maximal variance of prediction)
• Interval (by M.P. Dyvak)
• Name Minimizes
• ID -optimality squared volume of A
• IE -optimality squared maximal diagonal of A
• IG -optimality maximal prediction error

Depend only on X,hence are applicable for
interval error as well
D (XTX)1
IE- and IG-optimality are equivalent for
spherical design space and n gt p
d(x) xTDx
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Experimental Design for Regression under Interval
Error

• Motivation
• Classical methods of experimental design use
  only an information which X brings, nor Y, nor E
• Interval methods of experimental design developed
  by Dyvak work for saturated designs (pn)
  anduse X and E, nor Y.
• Does using of information, which Y contains,
  allow to improve the quality of constructed
  model or to increase the speed of sequential
  experimental design procedure?

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Experimental Design for Regression under Interval
Error

• How to use the information which Y brings?

Uncertainty set A(X,Y,E)

• Find out the direction a of maximal spread of A

• Next experimental point xnext?is selected in
  such a way that it
• induces the constraint orthogonal to a
• has maximal norm (width of constraint
  )

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Experimental Design for Regression under Interval
Error

• IE-optimal sequential design

(X0, Y0, E0) initial dataset
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Experimental Design for Regression under Interval
Error

• IE-optimal sequential design

(X0, Y0, E0) initial dataset
y measurement in x with error ?
i i 1
until i gt N or IA(Xi, Yi, Ei) is small
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Experimental Design for Regression under Interval
Error

• Simulation study 1. Comparison of IE- and
  D-optimal sequential designs under zero errors

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Experimental Design for Regression under Interval
Error

• Simulation study 1. D-optimal sequential design
  results

Variables domain
Parameters domain
1,5,9
3,7
2,6,10
4,8
Volume(A) 0.6400 ? 4?2
IA 0.45, 1.55?1.45, 2.55
Volume(IA) 1.21
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Experimental Design for Regression under Interval
Error

• Simulation study 1. IE-optimal sequential design
  results

Variables domain
Parameters domain
Volume(A) 0.5077 ? ??2
IA 0.59, 1.41?1.60, 2.40
Volume(IA) 0.66
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Experimental Design for Regression under Interval
Error

• Simulation study 2. Comparison of IE- and
  D-optimal sequential designs under error which
  follows truncated normal distribution

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Experimental Design for Regression under Interval
Error
Simulation study 2
for r 1 to 1500 do
repeat
random value from
until i gt N
if
then
end for
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Experimental Design for Regression under Interval
Error

• Simulation study 2. Results for

1500
1250
1000
Number of winnings k, (1500 k)
750
500
250
I
-Design
E
D
-Design
0
0
5
10
15
20
25
Number of selected points N
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Experimental Design for Regression under Interval
Error

• Simulation study 2. Results for

1500
1250
1000
Number of winnings k, (1500 k)
750
500
250
I
-Design
E
D
-Design
0
0
5
10
15
20
25
Number of selected points N
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Experimental Design for Regression under Interval
Error

• The cost of IE-optimal design
• The problem of finding maximal spread direction
  of A
• is a concave quadratic programming problem
  (CQPP)
• It is proved that CQPP is NP-hard, i.e. solving
  time of the problem exponentially depends on its
  dimension (the number of input variables p)
• To overcome the difficulties we need to use
  special computational means (such as parallel
  computers) or we can limit ourself with
  near-optimal solutions

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Conclusions

• Interval model of error allows to use the
  information about measured values of output
  variable for effective sequential experimental
  design
• The results of the performed simulation study
  give a cause for careful analytical investigation
  of properties of IE-optimal sequential design
  procedures
• IE-optimal sequential design for high-dimensional
  models demands for special computational
  techniques