Response Modeling Methodology: Empirical Modeling for Engineering and Science (Series on Quality, Reliability and Engineering Statistics) (Series on Quality, Reliability and Engineering Statistics)

Response Modeling Methodology

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Series on Quality, Reliability and Engineering Statistics

Response Modeling Methodology Empirical Modeling for Engineering and Science

Haim Shore Ben-Gurian University of the Negev, Israel

We World Scientific NEW JERSEY · LONDON · SINGAPORE · BEIJING · SHANGHAI · HONG KONG · TAIPEI · CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Series on Quality, Reliability and Engineering Statistics — Vol. 8 RESPONSE MODELING METHODOLOGY (RMM) Empirical Modeling for Engineering and Science Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-102-1

Printed in Singapore.



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(7.4)

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136

RESPONSE MODELING METHODOLOGY

individual observations may then be derived numerically either from the quantile function [(8.1) or (8.2)] or from the density function of W. Both methods will now be developed for (8.1). Corresponding expressions may similarly be developed for (8.2). Estimating the Variance of W (Based on Eq. 8.1) Method I Re-write (8.1) as W = Q1(Z1; ω1) + Q2(Z2; ω2),

(8.13)

where Q1(Z1; ω1) = (α/λ)[(η+σε1Z1)λ – 1] + µ2 + σε2ρZ1, Q2(Z2; ω2) = σε2(1-ρ2)(1/2) Z2,

(8.14)

with, respectively, parameters' vectors ω1 = {α, λ, µ2, σε1, σε2ρ}, ω2 = σε2(1-ρ2)(1/2).

(8.15)

Note, that ω1 contains five parameters while ω2 has only one parameter. Since Z1 and Z2 are independent random variables so are Q1 and Q2, and for the r-th non-central moment of W we obtain E(Wr) = µr'(W) =

r

r

j =0

 

∑  j M

j

(Q1 ) M r − j (Q2 ) ,

(8.16)

where Mj(Q1) and Mr-j(Q2) are the j-th and (r-j)-th non-central moments of Q1 and Q2, respectively, namely: Mk(Q1) =

∫

∞

−∞

{(α/λ)[(η+σε1z)λ – 1] + µ2 + σε2ρz}k φ(z)dz (8.17)

Mk(Q2) =

∫

∞

−∞

[σε2(1-ρ2)(1/2) z]k φ(z)dz

= [σε2(1-ρ2)(1/2)]k µk(Z) where µk(Z) is the k-th standard normal moment.

(8.18)

CHAPTER 8 ESTIMATING THE RELATIONAL MODEL

137

The variance of W for individual observations may be estimated from (8.16)-(8.18), and inserted into the minimization routine (8.10). Method II This method uses an approximate expression for the density function of W, fW, given in (8.20) below. The variance of W may then be estimated from µr'(W) =

∫

∞

−∞

w r f W ( w)dw ,

Var(W) = µ2'(W) - [µ1'(W)]2 ,

(8.19)

where µr'(W) is the r-th non-central moment of W. Note, that applying Method I may encounter difficulties if the expression (η+σε1Z1) becomes negative during numerical integration (imaginary values may be encountered, especially if η has values close to zero). Therefore, we may either select a translation constant for η that ensures non-negative values for (η+σε1Z1) (we have discussed earlier why such a translation is legitimate), or we may opt for using Method II in calculating weights for the W-NL-LS procedure. 8.4.3. Stage II - Estimating the RMM "Error Parameters" {ρ, σε1, σε2} At this stage, the error parameters, {ρ, σε1, σε2}, are estimated via a ML routine, based on the approximate log-likelihood function of the RMM error distribution. In Section 9.1, an expression for the density function (d.f.) of W is developed, which requires numerical integration. This may render application of an ML estimation procedure a numerically prohibitive undertaking. We therefore opt for a Taylor expansion which would eliminate the need for integration. The result is a relatively simple expression for the d.f. of W, which is based on the expected values of the first four terms in the Taylor expansion of the conditional density fWε1(wε1,θ ). The resulting expression appears in (9.10), where η=1. With η as an additional argument, we have:



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CHAPTER 10 FITTING PROCEDURES

169

and Z2, respectively, and ω1= {α, λ, µ2, σε1, σε2ρ}, ω2= {σε2(1-ρ2)(1/2)}, are the parameters that need to be determined (six parameters altogether). Note that QS1 and QS2 differ from Q1 and Q2, respectively, given in (9.16). Since Z1 and Z2 are independent standard normal variables, the r-th moment of WS is Mr(WS) = E[(WS)r] =

r

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j =0

 

∑  j M

j

(QS1 ) M r − j (QS 2 ) , (10.10)

where Mj(QS1) and Mr-j(QS2) are the non-central j-th and (r-j)-th moments of QS1 and of QS2, respectively. For the latter we have Mk(QS1) = E[(QS1)k] =

∫

∞

−∞

{{(α/λ)[(η+σε1z)λ – 1] + µ2 + σε2ρz - µ}/σ}k φ(z)dz , (10.11) Mk(QS2) =

∫

∞

−∞

[(σε2/σ)(1-ρ2)(1/2) z]k φ(z)dz

= [(σε2/σ)(1-ρ2)(1/2)]k µk(Z),

(10.12)

where µk(Z) is the k-th moment of the standard normal distribution. For k odd, we have identically µk(Z)=0. For k even, we have for the first few moments µ2(Z) = E(Z2) =1, µ4(Z) = 3, µ6(Z) = 15, µ8(Z) = 105. This implies that to calculate a moment of any desirable degree via (10.10) only numerical integration of (10.11) is needed. To identify the parameter set, {ω1, ω2}, the following routine needs to be applied: Find {ω1, ω2} that minimize the objective function 6

OF = [M1 (WS)]2 +

∑

[Mr (WS)/µr(WS) - 1]2,

(10.13)

2

where µr(WS)= E[(WS)r] is the given exact r-th moment of WS (the standardized W), namely, the r-th moment associated with the distribution that we wish to represent by the RMM model (these moments may be derived from available explicit expressions for the moments, or numerically calculated with the exact density function).





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Figure 10.2. Plots of the quantile function (approximate and exact, left) and of the deviation (approximate minus exact, right) for the Gamma (3,2). Horizontal axis is the standard normal quantile (z). Vertical axis (left plot) is the standardized gamma quantile.





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CHAPTER 19 RMM DISTRIBUTIONAL APPROXIMATIONS

329

λ = -0.61228883, S1 = -0.11105481, S2 = 0.44334159, α = -6.37309208.

(19.22)

[the value of α was derived from (19.21) with the above values of S1 and S2; Also the coefficients of z in (19.20), S1 and S2, have different signs, a possibility that is anticipated from (19.7)]. Introducing from (19.22) into (19.20) and thence into (19.17), the error plot in Figure 19.4 is obtained. The maximum absolute error is reduced from 2(10)-6 (Figure 19.3) to 6(10)-7. Although (19.20) has only three parameters that need to be determined from the fitting procedure (α is not), it has accuracy better than the four- parameter approximation (19.18). As a final comment, we note two interesting features of the new approximations for the normal CDF. First, they are non-polynomial. This implies that their high accuracy emanates not from the large number of parameters that characterize polynomial approximations (refer to Johnson et al., 1994), but possibly because the RMM original model is in some way related to the normal distribution. This shows in that the accuracy obtained from

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366

RESPONSE MODELING METHODOLOGY

Similarly, from (21.24), the parameters of (21.22) are A1 = 2.023, B1 = 4.976, A2 = 4.864, B2 = 4.756.

(21.28)

Introducing the parameters in (21.27) into (21.5), a piece-wise linear approximation is obtained for the quantile of Y in terms of the standard normal quantile. From this, the r-th non-central moment of Y is µr' = E(Yr) =

∫

0

−4

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4

+

∫ [σ ( A + C ) z − 0.7978Cσ + µ ] φ ( z)dz , r

0

(21.29)

where φ(z) is the density function of the standard normal variable, and the limits of integration were set arbitrarily as z=-4 and z=4. Similarly, introducing the parameters in (21.28) into (21.22), we obtain for the r-th non-central moment of Y: µr' = E(Yr) =

∫

0

−4

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4

+

∫ [ A z + B ] φ ( z )dz 0

r

2

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(21.30)

From (21.29) and (21.30), the first four moments are µ = µ1', σ2 = µ2' − (µ1')2, Sk = [µ3' - 3µ2' µ1' + 2(µ1')3] / σ3, Ku = [µ4' - 4µ3' µ1' + 6µ2' (µ1')2 - 3(µ1')4] / σ4 - 3

(21.31)

Table 21.2 displays the resulting first four moments. In this table, (21.22) [the un-standardized (21.4)] is fitted by the two-moment fitting procedure, with parameters given by (21.24). Also, (21.5) is fitted by the three-moment procedure, with parameters calculated from (21.20) (namely, Sk is approximately preserved). For both models, the first three moments are well preserved. Note, in particular, that for (21.22) only moments of second degree at most took part in the fitting procedure. However, the skewness measure is preserved.

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