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On Sequential Experimental Design for Empirical Model-Building under Interval Error Sergei Zhilin, Altai State University, Barnaul, Russia

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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

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3 Regression under Interval Error Model structure xTxT + … x1x1 x2x2 xpxp y Input variables x = (x 1,…,x p ) T measured without error Output variable y measured with error Linear-parameterized modeling function Model parameters to be estimated Measurement error “Interval” error means “unknown but bounded”:

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4 Regression under Interval Error Each row (x j, y j, j ) of the measurements table constrains possible values of the parameter with the set Values of the parameter consistent with all constraints form an uncertainty set

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5 Set of feasible models Regression under Interval Error Fitting data with the model y = 1 + 2 x 11 22 x y In (x, y) domain In ( 1, 2 ) domain Uncertainty set A is unbounded = not enough data to build the model Uncertainty set A Set of feasible models

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6 Regression under Interval Error Problems that may be stated with respect to uncertainty set A Interval estimates of Point estimates of 22 11 11 11 22 22 Model parameters estimation 11 ^ 22 ^

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7 Regression under Interval Error Problems that may be stated with respect to uncertainty set A Point estimate of y Interval estimate of y Prediction of the output variable value for fixed values of input variables x y y(x)y(x) y(x)y(x) x y(x)y(x) ^

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8 –Sequential experimental design –Simultaneous experimental design Experimental Design: Refining Context Product or process optimization Model quality optimization Experiment Analysis (Is the model quality satisfactory?) Design for ~1 observation End Begin Experiment Analysis Design for N observations End Begin

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9 Experimental Design for Regression under Interval Error –model –design space –design matrix –measurements –error bounds –information matrix – covariance matrix –standardized variance function of y(x, ) Notations

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10 Experimental Design for Regression under Interval Error Design optimality criteria –Classical NameMinimizes D -optimality (volume of joint confidence interval) G -optimality (maximal variance of prediction) –Interval ( by M.P. Dyvak ) NameMinimizes I D -optimalitysquared volume of A I E -optimalitysquared maximal diagonal of A I G -optimalitymaximal prediction error D = (X T X) –1 d(x) = x T Dx Depend only on X, hence are applicable for interval error as well I E - and I G - optimality are equivalent for spherical design space and n > p

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11 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 ( p=n ) and use 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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12 x next = I E Design(, X, Y, E) Experimental Design for Regression under Interval Error How to use the information which Y brings? 22 11 1.Find out the direction a of maximal spread of A : 2.Next experimental point x next is selected in such a way that it induces the constraint orthogonal to a has maximal norm (width of constraint ) w Uncertainty set A(X,Y,E)

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13 i = 0; repeat x = I E Design(, X i, Y i, E i ); Experimental Design for Regression under Interval Error I E -optimal sequential design ( X 0, Y 0, E 0 ) – initial dataset

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14 Experimental Design for Regression under Interval Error I E -optimal sequential design ( X 0, Y 0, E 0 ) – initial dataset y = measurement in x with error ; i = i + 1; until i > N or IA(X i, Y i, E i ) is small; i = 0; repeat x = I E Design(, X i, Y i, E i );

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15 Experimental Design for Regression under Interval Error Simulation study 1. Comparison of I E - and D -optimal sequential designs under zero errors repeat until i > 9 I E -optimal sequential design D -optimal sequential design repeat until i > 9

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16 Experimental Design for Regression under Interval Error Simulation study 1. D -optimal sequential design results Variables domainParameters domain Volume(A) = 4 2 IA = [0.45, 1.55] [1.45, 2.55] Volume(IA) = ,7 1,5,9 2,6,10 4,8 22

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17 Experimental Design for Regression under Interval Error Simulation study 1. I E -optimal sequential design results Variables domainParameters domain Volume(A) = 2 IA = [0.59, 1.41] [1.60, 2.40] Volume(IA) = 0.66 22

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18 Experimental Design for Regression under Interval Error Simulation study 2. Comparison of I E - and D -optimal sequential designs under error which follows truncated normal distribution 3 Errors are simulated by – truncated normal distribution { 3 uniformly distributed points from }

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19 Experimental Design for Regression under Interval Error for r = 1 to 1500 do until i > N repeat end for { 3 uniformly distributed points from }; { 3 random values from }; random value from ; if then Simulation study 2

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20 Experimental Design for Regression under Interval Error Simulation study 2. Results for Number of selected points N Number of winnings k, (1500 – k) 0%0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% I E -Design D

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21 Experimental Design for Regression under Interval Error Simulation study 2. Results for Number of selected points N Number of winnings k, (1500 – k) 0%0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% I E -Design D

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22 Experimental Design for Regression under Interval Error The “cost” of I E -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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23 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 I E -optimal sequential design procedures I E -optimal sequential design for high-dimensional models demands for special computational techniques

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