Design of Experiments for Extreme Values Patrick J Laycock - PowerPoint PPT Presentation

Design of Experiments for Extreme Values Patrick J Laycock University of Manchester Jesús López Fidalgo Universidad de Castilla-La Mancha

EXTREME VALUES Can arise in statistical analysis for various reasons, in particular – when only max or min can be recorded – breaking strain of a chain with several links – when only max or min is normally recorded – winning times; pit-depths in corroded metal => max-stable extreme value distributions

The generalized extreme value (GEV) family of distributions   1                    F y ( ) exp{ [1 ( y ) ] } 0 y , Types I, II or III when  =0,  <0 or  >0 The distribution has mean :            (1 ) 1             (1 ) provided 1  and standard deviation  1 1       2 2              1 2 1 provided    2

Constant variance model GEV         T x y  ( ) with location parameter i i i i     T E y [ | x ] x i i 0 i          where ( )[1 (1 )] and 0       2            . 2 Var y [ | x ] ( ) 1 2 1   i i . So, all the usual experimental designs can be justified for the estimation of  0 and  

Constant variance model ….. Using ordinary least-squares, the intercept and residual variance can subsequently be used to provide moment estimators for  and  .     MLE regular for 1 1 2 2 . Full maximum likelihood produces a parameter- dependent information matrix, with the usual design requirements for some form of prior information concerning the parameters.

Constant coefficient of variation models GEV       y  ( ) with a common link function :- i i i            T T ( x ) and ( x ) i 0 i i 0 i           T T giving ( x ) and ( x ) . , i 0 i i 0 i .      and hence a constant coefficient of variation 0 0

Constant coefficient of variation model … with log link         T T ( x ) exp( x ) j z with z exp( x ) j i i ji ji ji   y log( y ) so that, setting i i          T E y [ | x ] x we have i i i 0 i   where   log( ) . 0 0 And y i * has (approximately) constant variance, independent of  and x . So the usual experimental designs can be approximately justified for the estimation of  0 *, 

Optimality Criteria  D-optimality:- maximize det{I     -optimality:- minimize trace {I     G-optimality :- (linear models version)   2 ˆ d=d(x, )=Var {y(x)}/ (x) minimize over x  

 For non-linear/non-normal models designs are generally only locally optimal (depend on  ), but robustness can be examined  G-optimality :- (nonlinear version)        1 d x ( , , ) tr I x { ( , ) M ( , ) } where       m ( , ) I ( , x ) ( dx )  ij ij

•Fisher’s Information matrix:       f f   I E           t i j     2     f ( y , , x )  t  ( ) E          t i j  

General Equivalence Theorem Kiefer & Wolfowitz(1960), Lynda V White (1973) nonlinear • The following conditions on a design measure are equivalent   ( ) i is D ( )-optimum   (ii) is G ( )-optimum    (iii) sup d x ( , , ) k  x

Max pit depths of immersed coupons 90 80 70 60 pit depth (mils) 50 40 30 20 10 0 0 100 200 300 400 500 600 immersion time (hours)

Laycock et al (1990) modelled this Pierpoline pitting corrosion data using maximum likelihood  ,  i =  0 t i  and found estimation with  i =  0 t i  0  + 0.317)  0 = 6.322  + 1.721)  = 0.401  + 0.162)  = 0.376  + 0.047) The original design and ten others were compared at these parameter values

relative efficiencies for competing designs Design A B C D E #support 9 pt 9 pt 9 pt 9 pt 9 pt  - %eff 100 152 172 180 218 D - %eff 100 111 115 116 121 Equiv n 19 17.1 16.6 16.4 15.6 Max x d(x,  6.20 3.45 2.74 2.34 2.10 Max x d(x,  4.67 4.02 3.72 3.52 3.49

relative efficiencies for competing designs, continued F G H I J K Design #support 3 pt 3 pt 3 pt 4 pt 3 pt 2 pt  - %eff 403 255 403 234 213 404 142 126 126 121 121 142 D - %eff 13.4 15.1 15.1 15.7 15.7 13.4 Equiv n Max x d(x,  1.59 1.85 1.59 1.96 2.03 1.54 Max x 3.28 3.53 3.36 3.60 3.61 3.35 d(x, 

Conclusions Overall, it seemed worthwhile to base designs on the log(t) scale (designs d,g,h,i,j) and to choose Bailey spacing, based on     sin 2 i p / , i 0,1  , p - suitably scaled - with a reasonable number of distinct design points (designs h, j), since this combination gives both good efficiency on all criteria examined here and soundly based or improved analysis techniques. Robustness of these conclusions was verified by varying the parameters about their assumed values.

Strength of materials Leadbetter et al(1980, p272, Example 14.2.1) paper-making data strips 5cm wide, x cms long 3.0000 2.9000 2.8000 KNewtons per metre 2.7000 2.6000 2.5000 2.4000 2.3000 2.2000 2.1000 2.0000 0 200 400 600 800 1000 1200 length, x in cms mean = (a0x^-k/k)G(1+k)  0 = 2,  = 0.1 x data lay in [8, 1000]

• Strength of materials I Leadbetter et al(1980, p272) model the breaking strength of materials as a function of tested length, x, using GEV dns for minima. They set  i =  0 x i -  and Type III as Weibull dn (x-->-x) lower bound  = 0 , implying    x 1       2   2                  ( x ) (1 ) and 0 i 1 2 1   i 0 i i  so constant coefficient of variation. But taking logs here does not produce a (approx) linear model, since the ‘regression’ parameter is the highly non-linear parameter,  y i = ln(  i ) =   (  0 ,  ) –  ln(x i )

TRIAL DESIGNS FOR Leadbetter et al(1980, p272, Example 14.2.1) paper-making data strips 5cm wide, x cms long 9 h 8 g 7 f f f 6 DESIGN e e e 5 d d 4 c c c c c c c c c c c 3 b b b bb b b b b b b 2 a a a a a a a a a a a 1 0 0 200 400 600 800 1000 1200 length, x in cms

Design a b c d e f g h D- %efficiency 100 108 116 147 125 125 69 6 equivalent n 11 10 9.5 7.5 8.8 8.8 16 174

Conclusions The one point designs (g and h) are technically feasible but very inefficient. The classic two point design at the interval extremes is as usual D-optimal, but more points are desirable for model-testing. The spacing suggested by Bailey(1982), based on     sin 2 i p / , i 0,1  , p - suitably scaled - with three or more distinct design points (designs c or f), give both good efficiency and improved analysis techniques. Robustness of these conclusions was verified by varying the parameters about their assumed values.

Bailey R.A. (1982) The decomposition of treatment degrees of freedom in quantitative factorial experiments. J. Roy. Stat. Soc., B , 44 , 63--70. Leadbetter M.R., Lindgren G. & Rootzen H. (1983) Extremes and Related Properties of Random Sequences and Processes . Springer, New York. White, L.V. (1973) An extension of the General Equivalence Theorem to nonlinear models. Biometrika , 60 , 345-348. Laycock, P.J., Cottis R.A. & Scarf P. A. (1990) Extrapolation of extreme pit depths in space and time. J. Electrochem. Soc., 137, 64—69.