Wuhan Key

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Decison Making With Uncertainty And Data Mining

David L. OlsonDepartment of Management

University of Nebraska

Lincoln, NE 6888!"#$%"'( #)'!#'%

*+- "'( #)'!8

Dolson/0nl.ed0

Des1eng 203 &corresponding a0t1or(

4c1ool of 50siness

University of 4cience and ec1nology of 71ina

efei +n10i '""'6 9.:. 71inadas1/0stc.ed0

Keywords- M0ltiple attrib0te decision making &M+DM(; data mining; 0ncertainty ; *0

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Abstract.

Data mining is a ne=ly developed and emerging area of comp0tational intelligence t1at offers

ne= t1eories, tec1ni>0es, and tools for analysis of large data sets. ?t is e@pected to offer more

and more s0pport to modern organi

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b0siness processes. 1e field of data mining aims to improve decision making by foc0sing on

discovering valid, compre1ensible, and potentially 0sef0l kno=ledge from large data sets.

1is paper presents a brief demonstration of t1e 0se of Monte 7arlo sim0lation in grey

related analysis. 4im0lation provides a means to more completely describe e@pected res0lts, to

incl0de identification of t1e probability of a partic0lar option being best in a m0ltiattrib0te

setting. 1e ne@t section describes a Monte 7arlo sim0lation of res0lts of decision tree analysis

of real credit card data. Monte 7arlo sim0lation provides a means to more completely assess

relative performance of alternative decision tree models. :elative performance of crisp and

f0

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reflect 0ncertainty as e@pressed by f0

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+ntFnio B".6 ".8C B".) ".$C B".' ".#C B".# ".8C B"." ".#C B".# ".)C B".) %.""C

*Gbio B".' ".#C B"." ".'C B".6 ".8C B"." ".6C B"." ".)C B"." ".'C B"." ".#C

+lberto B".# ".6C B".'" ".8"C B".6 ".8C B"." ".8"C B". ".$"C B".'" ".#C B".) %.""C

*ernando B".8 %.""C B". ".)C B".6 ".8C B".% ".6C B"." ".)"C B".# ".8"C B". ".)"C

?sabel B"." ".$C B".6 ".$C B".# ".6C B".6 ".$C B"." "."C B".# ".8"C B"." ".$"C

:afaela B".6 ".8C B".% ".C B".# ".6C B".' ".)C B"." ".#C B".# ".8"C B".%" ".C

+ll of t1ese inde@ val0es are positive. 1e ne@t step of t1e grey related met1od is to standardi0ences based

on t1e optimal =eig1ted interval n0mber val0e for every alternative. 1is is defined as t1e

interval n0mber for eac1 attrib0te defined as t1e ma@im0m left interval val0e over all

alternatives, and t1e ma@im0m rig1t interval val0e over all alternatives. *or "1, t1is =o0ld yield

t1e interval n0mber B".%), ".C. 1is reflects t1e ma@im0m =eig1ted val0e obtained in t1e data

set for attrib0te "1. able gives t1is vector, =1ic1 reflects t1e range of val0e possibilities

&entries are not ro0nded(-

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able - :eference N0mber Iector

"1 "2 "# "$ "% "& "'

Ma@&Min( ".%)"" ".''" "."' ".%6' "."' "."'' ".%8)

Ma@&Ma@( "."" ".'' ".'" ".#)" ".#"" ".'#"" ".""

Distances are defined as t1e ma@im0m bet=een eac1 interval val0e and t1e e@tremes generated.

able # s1o=s t1e calc0lated distances by alternative.

able #- Distances *rom +lternatives to :eference N0mber Iector

Distances "1 "2 "# "$ "% "& "'+ntFnio B."#, ."'C B", "C B."', .%'C B.", ."C B."#, .'"'C B", ."%C B", "C

*Gbio B.%', .%$'C B.'%, .8C B", "C B."8), .%C B.""), .

"6)C

B."', .%6C B.%), .

"'C

+lberto B."8, .%''C B.%6, ."8'C B", "C B."), .")C B", "C B."%', .

%"C

B", "C

*ernando B", "C B.%', .%%C B", "C B.%', .%C B.""), ."$C B", "C B.%, .%6C

?sabel B."), ."%)C B.", "C B."%, ."6C B", "C B."#, .%8C B", "C B."6', .

"C

:afaela B."#, ."'C B.%8, .C B."%, ."6C B.%, .%C B."#, .'"'C B", "C B.%6', .

'#)C

1e ma@im0m distance for eac1 alternative to t1e ideal is identified as t1e largest distance

calc0lation in eac1 cell of able #. 1ese ma@ima are s1o=n in able .

able - Ma@im0m Distances

Distances "1 "2 "# "$ "% "& "'

+ntFnio "."' " ".%' "." ".'"' "."% "

*Gbio ".%$' ".8 " ".% "."6) ".%6 "."'

+lberto ".%'' ".%6 " ".") " ".%" "*ernando " ".%' " ".% "."$ " ".%6

?sabel ".") "." "."6 " ".%8 " "."6'

:afaela "."' ". "."6 ".% ".'"' " ".'#)

6

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+ reference point

(C(&,(&B,...,(C'&,('&B,(C%&,(%&&B """"""" nunuuuuuU +++

= is establis1ed as

t1e ma@im0m of entries in eac1 col0mn of able ). 1is point 1as a minim0m of " and a

ma@im0m of ".8". 10s t1e reference point is B", ".8C. Ne@t t1e met1od calc0lates t1e

ma@im0m distance bet=een t1e reference point and eac1 of t1e 2eig1ted Matri@ 7 val0es.

5ased 0pon =eig1t interval n0mber standardi0ence

(C(&,(&B,...,(C'&,('&B,(C%&,(%&&B """"""" nunuuuuuU +++

= , t1e form0la

for t1is calc0lation is given as follo=s.

JC,B(C&(,&BJma@ma@JC,B(C&(,&BJ

JC,B(C&(,&BJma@ma@JC,B(C&(,&BJminmin(&

""""

""""

++++

++++

+

+

=

ikikki

ikik

ikikki

ikikki

i

cckukucckuku

cckukucckuku

k

21ere & (,"& + ( is called resolving coefficient. 1e smaller is, t1e greater its

resolving po=er. ?n general, B"%C .1e val0e of may c1ange according to t1e practical

sit0ation.

:es0lts by alternative are given in able 6-

able 6- 2eig1ted Distances to :eference 9oint

Distances "1 "2 "# "$ "% "& "' A(erages

+ntFnio ".)8)%# % ".6%6""" ".)$8%# ".#8)#' ".$'))%% % ).*)1%12

*Gbio ".""""" ". % ".6'"## ".)#"8 ".8#6' ".8888$ ).%*)$$%

+lberto ".6%%%%% ".8#6' % ".)%$6'6 % ".6#)"$ % ).'**)#'

*ernando % ".6%6""" % ".6'"## ".68%#%6 % ".8#6' ).''11#2

?sabel ".) ".86%6$ ".)6')6 % ".%6))$ % ".)#$"' ).*)$&%1

:afaela ".)8)%# ".68#'% ".)6')6 ".68%'" ".#8)#' % ".#)"" ).&$2'*2

)

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1e average =

=n

i

ii kn

r%

(&%

& mi ,...,',%= ) of t1ese =eig1ted distances is 0sed as t1e

reference n0mber to order alternatives. 1ese averages reflect 1o= far a=ay eac1 alternative is

from t1e nadir, along =it1 1o= close t1ey are to t1e ideal, m0c1 as in O94?4. 1is set of

n0mbers indicates t1at ?sabel is t1e preferred alternative, alt1o0g1 +ntFnio is e@tremely close,

=it1 +lberto and *ernando close be1ind. 1is closeness demonstrates t1at t1e f0

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Xis t1e random n0mber dra=n &=1ic1 is t1e area(

?fX!-

( ) ( )

"!

aaaaaaXaX

+

++=

%'#%'% &8(

?f! X!$K-

( )'' aaK

!XaX

+= &$(

?f!$KX-

( ) ( ) ( )"!

aaaaaaXaX

+

+=

%'##%# &%"(

O0r calc0lation is based 0pon dra=ing a random n0mber reflecting t1e area &starting on

t1e left &a%( as ", ending on t1e rig1t &a#( as %(, and calc0lating t1e distance on t1e !a@is. 1e

sim0lation soft=are 7rystal 5all =as 0sed to replicate eac1 model %,""" times for eac1 random

n0mber seed. 1e soft=are enabled co0nting t1e n0mber of times eac1 alternative =on.

9robabilities given in able ) are t10s simply t1e n0mber of times eac1 alternative 1ad t1e

1ig1est val0e score divided by %,""". 1is =as done ten times, 0sing different seeds. 1erefore,

mean probabilities and standard deviations &std( are based on %",""" sim0lations. 1e Min and

Ma@ entries are t1e minim0m and ma@im0m probabilities in t1e ten replications s1o=n in t1e

table.

able )- 4im0lated 9robabilities of 2inning for Uniform *0

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seed6789 0.381 0.000 0.179 0.046 0.394 0.000

seed7890 0.343 0.000 0.199 0.02 0.406 0.000

seed8901 0.328 0.000 0.201 0.04 0.426 0.000

seed9012 0.33 0.000 0.189 0.048 0.410 0.000

seed0123 0.360 0.000 0.183 0.03 0.404 0.000

!in 0.328 0.000 0.168 0.040 0.384 0.000!ean 0.34 0.000 0.189 0.047 0.410 0.000

!a" 0.381 0.000 0.210 0.03 0.429 0.000

std 0.017 0.000 0.012 0.004 0.01 0.000

3.%. +nalysis of :es0lts

1e res0lts for eac1 system =ere very similar. Differences =ere tested by t!test of

differences in means by alternative. None of t1ese difference tests =ere significant at t1e ".$

level &t=o!tailed tests(. 1is establis1es t1at no significant difference in interval or trape

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21ile t1is e@ample is on a small set of data, t1e intent =as to demonstrate =1at co0ld be

done in t1at conte@t co0ld be applied on large!scale data sets as =ell. O0r proposal is 0ni>0e to

o0r kno=ledge, proposing t1e 0se of sim0lation to more f0lly 0se grey!related data t1at more

acc0rately reflects t1e real problem. ?f t1is co0ld be done =it1 small!scale data sets, o0r

contention is t1at it can also be done =it1 large!scale data sets in a data mining conte@t.

$. Grey elated Decision +ree Model

Arey related analysis is e@pected to provide improvement over crisp models by better reflecting

t1e 0ncertainty in1erent in many 10man analystsH minds. Data mining models based 0pon s0c1

data are e@pected to be less acc0rate, b0t 1opef0lly not by very m0c1. o=ever, grey related

model inp0t =o0ld be e@pected to be stabler 0nder conditions of 0ncertainty =1ere t1e degree of

c1ange in inp0t data increased.

2e applied decision tree analysis to a small set &%,""" observations total( of credit card

data. Originally, t1ere =as one o0tp0t variable &=1et1er or not t1e acco0nt defa0lted, a binary

variable =it1 % representing defa0lt, " representing no defa0lt( and 6 available e@planatory

variables. 1ese variables =ere analy

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1e e@planatory variables incl0ded five binary variables and one categorical variable,

=it1 t1e remaining '" being contin0o0s. o reflect f00e decision

trees =ere obtained, =it1 form0las again given belo=. + total of seven e@planatory variables

=ere 0sed in t1ese fo0r categorical decision trees.

1ese models =ere t1en entered into a Monte 7arlo sim0lation &s0pported by 7rystal

5all soft=are(. + pert0rbation of eac1 inp0t variable =as generated, set at five different levels of

pert0rbation. 1e intent =as to meas0re t1e loss of acc0racy for crisp and grey related models.

1e model res0lts are given in t1e seven model reports in t1e appendi@. 4ince different

variables =ere incl0ded in different models, it is not possible to directly compare relative

acc0racy as meas0red by fitting test data. o=ever, t1e means for t1e acc0racy on test data for

eac1 model given in able $ s1o= t1at t1e crisp models declined in acc0racy more t1an t1e

categorical models. 1e col0mn 1eadings in able $ reflect t1e degree of pert0rbation sim0lated.

%'

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able $- Mean Model +cc0racy

Model "ris, ).2% ).%) 1.)) 2.)) #.)) $.)) ).2%

7ontin0o0s % ".)" ".)" ".)" ."68 ".6) ".66 ".6 ".)"

7ontin0o0s ' ".6) ".6) ".6) ".6) ".6) ".66 ".66 ".6)

7ontin0o0s ".)% ".)% ".)" ".6$ ".6) ".6) ".66 ".)%"ontinuous ).&-# ).&-# ).&-) ).&*) .&') ).&&' ).&%' ).&-#

7ategorical % ".)" ".)" ".68 ".6) ".66 ".66 ".6 ".)"

7ategorical ' ".)" ".)" ".)" ".6$ ".68 ".6) ".6) ".)"

7ategorical ".)" ".)" ".)" ".6$ ".6$ ".68 ".6) ".)"

7ategorical # ".)" ".)" ".)" ".6$ ".68 ".6) ".6) ".)"

"ategorical ).')) ).')) ).&-% ).&** ).&'* ).&') ).&&% ).'))

1e f0

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tec1ni>0e offers more insig1ts to assist o0r decision making in f0

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B8C 9earl, ., 9robabilistic reasoning in intelligent systems, Net=orks of 9la0sible

inference, Morgan a0fmann, 4an Mateo,7+ %$88.

B$C Aa0 2.L., 50e1rer D... Iag0e sets. ?EEE rans, 4yst. Man, 7ybern, '&%$$( 6%"!6%#

%

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+99END?- Models and t1eir res0lts

7ontin0o0s Model %-

I!&$3&.$$454I!%*31.%$464I!&3#.-14546777

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proportion

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0.68 0.69 0.70 0.72 0.73

1000 Trials 994 /ispla$edFore+ast ,ont 1 a++)ra+$

est matri@-

Model " Model % +cc0racy

+ct0al " # %6

+ct0al % %# ') ".)"

4im0lation acc0racy of %"" observations, %""" sim0lation r0ns

pert0rbation B!".',".'C ".6)!".)

pert0rbation B!".","."C ".6!".)#

pert0rbation B!%,%C ".6'!".)pert0rbation B!','C ".8!".)#

pert0rbation B!,C ".)!".)#pert0rbation B!#,#C ".6!".)

0. 0.60 0.6 0.70 0.7

%6

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7ontin0o0s Model '-

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1000 Trials 991 /ispla$ed

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Model " Model % +cc0racy+ct0al " #" %$

+ct0al % %# ') ".6)


pert0rbation B!".',".'C ".6!".)%pert0rbation B!".","."C ".6!".)%

pert0rbation B!%,%C ".6"!".)#

pert0rbation B!','C ".8!".)

pert0rbation B!,C ".!".)8pert0rbation B!#,#C ".!".)6

0. 0.60 0.6 0.70 0.7

%)

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7ontin0o0s Model -

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+ct0al % %# ') ".)%


pert0rbation B!".',".'C ".6!".)6pert0rbation B!".","."C ".6!".)6

pert0rbation B!%,%C ".$!".))

pert0rbation B!','C ".#!".)$

pert0rbation B!,C ".!".)8pert0rbation B!#,#C ".!".)6

0. 0.60 0.6 0.70 0.7

%8

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7ategorical Model '-

I!&$89high94I!%$89lo94

I!"D:89mid94I!#'89lo9464574if!"D:89lo9454677

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+ct0al " #' %)

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pert0rbation B!".',".'C ".6!".)

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pert0rbation B!%,%C ".6%!".)6pert0rbation B!','C ".8!".)6

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0.60 0.6 0.70 0.7

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7ategorical Model -

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0.60 0.6 0.70 0.7

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7ategorical Model #-

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