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

1 ( ) exp{ [1 ( ) ] } , F y y y                      The generalized extreme value (GEV) family of distributions Types I, II or III when =0, <0 or >0 The distribution has mean :

 

1

(1 ) (1 ) 1 provided          



          

and standard deviation

   

1 2

2

1 1 2 1 2 provided                   

Constant variance model

( ) with location parameter

i i i i

y GEV        

Tx



[ | ]

i i i

E y    

T

x x

   

2 2

[ | ] ( ) 1 2 1

i i

Var y                x where ( )[1 (1 )] and         

.

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 .

.

1 1 2 2

MLE regular for     Full maximum likelihood produces a parameter- dependent information matrix, with the usual design requirements for some form

• f prior information concerning the parameters.

Constant coefficient of variation models

( ) with a common link function :-

i i i

y GEV       

. .

( ) and ( ) giving ( ) and ( )

i i i i i i i i

                    

T T T T

x x x x     

,

and hence a constant coefficient of variation

Constant coefficient of variation model … with log link

.

So the usual experimental designs can be approximately justified for the estimation of 0*, 

( ) exp( ) exp( )

j

i i ji ji ji j z with z

x



     



T T

x x so that, setting

log( )

i i

y y

 

we have log( )  

 

[ | ]

i i i i

E y   

  

  

T

x x where And yi

* has (approximately) constant variance,

independent of  and x.

Optimality Criteria

• D-optimality:- maximize det{I
• -optimality:- minimize trace {I



• G-optimality :- (linear models version)

minimize

• ver x  

2

ˆ d=d(x, )=Var {y(x)}/ (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

( , , ) { ( , ) ( , ) } ( , ) ( , ) ( )

ij ij

d x tr I x M where m I x dx         

 

  

• Fisher’s Information matrix:

2 ( , , ) ( )

t t t

f f I E i j f y x E i j



                                   

 

General Equivalence Theorem

Kiefer & Wolfowitz(1960), Lynda V White (1973) nonlinear

• The following conditions on a design measure

are equivalent

( ) is ( )-optimum (ii) is G ( )-optimum (iii) sup ( , , )

x

i D d x k      





Max pit depths of immersed coupons

immersion time (hours) pit depth (mils)

Laycock et al (1990) modelled this Pierpoline pitting corrosion data using maximum likelihood estimation with i = 0ti

 , i = 0ti  and found

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

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

relative efficiencies for competing designs, continued

1.54 2.03 1.96 1.59 1.85 1.59

Maxx d(x,

3.35 3.61 3.60 3.36 3.53 3.28

Maxx d(x, Equiv n D - %eff  - %eff Design #support

404 213 234 403 255 403 13.4 15.7 15.7 15.1 15.1 13.4 142 121 121 126 126 142 K 2 pt J 3 pt I 4 pt H 3 pt G 3 pt F 3 pt

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

• 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.

 

sin 2 / , 0,1 , i p i p   

Strength of materials

Leadbetter et al(1980, p272, Example 14.2.1) paper-making data strips 5cm wide, x cms long

2.0000 2.1000 2.2000 2.3000 2.4000 2.5000 2.6000 2.7000 2.8000 2.9000 3.0000 200 400 600 800 1000 1200 length, x in cms KNewtons per metre mean = (a0x^-k/k)G(1+k)

x data lay in [8, 1000]

0 = 2,  = 0.1

• 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 = 0xi
• 

and Type III as Weibull dn (x-->-x) lower bound  = 0, implying 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, 

yi = ln(i) = (0,) – ln(xi)

   

1 2

2

( ) (1 ) and 1 2 1

i i i i

x x

 

        

 

             

TRIAL DESIGNS FOR Leadbetter et al(1980, p272, Example 14.2.1) paper-making data strips 5cm wide, x cms long

a a a a a a a a a a b b b bb b b b b b b c c c c c c c c c c c d e e e f f f h a d g

1 2 3 4 5 6 7 8 9 200 400 600 800 1000 1200

length, x in cms DESIGN

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

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

• 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.

 

sin 2 / , 0,1 , i p i p   

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.