skandInsurance
`
`
`
`September 1995 - Volume 62, N0 3
`PUBLISHED at THE AMERICAN RISK AND
`INSURANCE Assocumon, INC.
`
`
`
`
`
`THEJOURNAL‘or
`
`INVITED ARTICLE
`
`Insurance Regulation in Transition
`
`by Robert W Klein
`
`SYMPOSIUM ON INSURANCE
`
`SOLVENCY AND FINANCE
`
`An Equilibrimn Model of Insurance
`Pricing and Capitalization
`
`5}! Greg Taylor
`
`Fuzzy Techniques ofPattern Recognition
`in Risk and Claim Classification
`
`by Ricfmrdfl. Derrzg andK1zy:zt0_'f]l/I.
`Osraszewslei
`
`Insurer Capital Structure Decisions and tire
`Viability of Insurance Derivatives
`
`by Scatt.E. Harrington, Steven l/. Marin, and
`Greg [Vie/mm
`
`The Valuation of Option Features in 7
`Retirement Benefits ‘
`
`by Michael "S/Jerri:
`
`Examining Changes in Rmerves Using
`Soocliastic Interest Models
`
`5} Siu- Wei Lai and Edward W7. Frees
`
`Solvency Risk and the Tax Sheltering '
`Behavior of Property-liability Insurers
`6} Richard PanAm[end
`l/. Vswcznatfi
`
`NOTES AND COMMUNICATIONS
`
`l’iI:Ealls of the‘ Current Experience Rating
`Plan: Comment
`
`by ll?/illzlzrn R. Gillam
`
`Pitfalls of the Current Experience Rating
`Plan: Reply
`by Steven 15. Mar}; andAn‘/mr E. Parry
`
`Liberty Mutual Exhibit 1037
`
`Liberty Mutual v. Progressive
`CBM2013-00009
`
`4 Page 00001
`
`Liberty Mutual Exhibit 1037
`Liberty Mutual v. Progressive
`CBM2013-00009
`Page 00001
`
`

`

`© The Journal afflisk and Insurance, 1995 Vol. 62-, No. 3, 447-482
`
`
`
`Fuzzy vTe¢hniquresp not Pattern ,R‘?¢°l9fiiliO.n't
`RisK_-.ClI.‘|d Claim Clidssificslfion. Q
`i
`-
`' Richard A. De'n-ig*--r-.*---
`’
`Krzyszt-of M.*=0st‘aszew'skie
`
`T
`
`'
`
`t
`
`”cABS11bkCTYvi*“
`
`-xv
`
`,
`
`, Applications of fuzzy set theory to property-liability and lifezinsurance have emerged
`. pin the last few years through the work of Lernaire (1990),
`and I_)en__-i_g__ (1-993,
`'
`r_1994),_and Ostaszewski (_l9_93f)‘. This article Vcontinuesithat linetof research
`providing an
`‘
`overview of
`pattern recognition techniques arid"usin'g"the-n_1 ii1‘clu'ster'-i'r1'g for risk_ a.r1i:l'
`-clairns classification. The clas§ic'c'lustei"ing problem if groujiifigftowns i'nto'rating.’_territo+ "
`-
`' Vries (DuMou'chel,'1983;- Conger, __'1987),-is revisited using: ‘these fu'zzy_rnethods_and'—1987'
`through 199.0 Massachusetts automobile_.i_ns_ur_an'ce data.-The new-problem of classifying —.
`claims in terms of suspected fraud is a1so_addressed_using. these same fuzzy methods and
`data drawnfrom a study of 1989 bodily injury liability {claims in Massachusetts.,
`_
`
`_
`
`,
`
`-'
`
`'
`
`“
`
`' Ifitfétiusiifiiiitn
`
`experiment -was
`In '-1'96-1', -Eilsberg presented the following-paradox.
`designed with two urns,-each containing 100-balls,‘ of which the first one was
`known to contain'50-red-balls and 50 black ba1l’s§=whi1e no further'infor1natio'1_1
`was giveniaboul: the contents‘ of the otl1er'i1rn. If asked to bet on the‘c'olor of
`a ball drawn?-from one": of the urns, most people ‘were found ’-_indiffer¢ent"a's- to
`which color they would choose no matter whether: the 5ball was drawn: from the
`first or the second
`-On-the -other hand; Ellsberg found that if people‘ 5were
`asked which -‘uni they would prefer to usepfotf betting on either-color; they.
`
`Richard A. Derrig is Senior Vice President of the Autdmobile Insurers Bureau of Massachué
`setts ‘and Vice President—Reseamh for the Insurance Fraud Bureau of Massachusetts. Krz-ysztof
`Ostaszewski is Associate.Profcssor-of- Mathematics and Actuarial Program Director at the Univer~
`sity_of_Louisvil1e.
`.
`i
`_-
`I
`_
`—
`t
`;_
`_
`_,
`-
`-
`-,
`Kizysztof Ostaszewski has worked on this project at the University of Louisville with financial
`support from the Actuarial Education and Research Fund, and
`support from AERF is grateful-
`1y acknowledged. The authors thank Jeff Strong and Robert Roesch of the Automobile Insurers '
`Bureau for invaluable help in prograrnming and performing "calculations involved in this project,
`Herbert I. Wcisberg for suggesting the fuzzy clustering of fraud assessment data, Ruy Cardoso for
`helpful comments on an -early draft, Julie Jannuzzi for production of the document, and one
`anonymous reviewer. _
`'
`'
`'
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`448
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`‘The Journal of Risk and Insurance
`
`.
`
`consistently favored the first um (no matter what color they were asked to bet
`on).
`What seems to be present in this experiment is the participants’ perception i
`of uncertainty. When we say “uncertainty,” the usualassociation is with “prob-
`ability.” The Ellsberg paradox illustrates that some other formof uncertainty F
`can indeed exist. Probability theory provides no basis for the outcome of the
`Ellsberg experiznent.
`0
`-
`e
`Klir and Fo1gl,r:(l_9-‘88).._ analyze_ the semantic acontexti oftlie term “uncertain"
`_ and arrive at theeconclusions"thtat-I fl1'ere’are' two'mairf typesof uncertainty, cap-
`tured by the terms “vagueness? and"‘arnbiguity.” Vagueness is associated with
`the difficulty of"making sharp orprecise distinctions among objects. “Ambigu-
`ity” is caused by situations where the choice between two or more alternatives
`is unspecified. The basic set of axioms ofprobability theory originating from
`Kolmogorov, rests on the assumption that the outcome of a random event can_
`be observed and identified with -precision. Any vagueness of observation is
`considered negligible, or not significant topthe construction of the theoretical
`model. Yet one caI_111'0_t escape the [conclusion that forms_ of uncertainty repre-
`sented
`vagueness of observationss, humanperceptions, and interpretations,
`are missing from probabilisticmodels, which: equate uncertainty with random’-
`ness i(i.te.,- a sophisticated version of «ambiguity).'
`'_
`7
`-
`0
`V
`a
`Several reasons ‘may exist for wanting to search‘ for models of a__form of
`uncertainty other than randomness. One is that vagueness is unavoidable.’ Giv:—
`en imprecision of natural language, or perception of the phenomena
`observed, vagueness becomes a major factor in any attempt to model or predict-
`the coursetof events.,But-there is more. When the .. phenomena observed be-
`come so complex that exact-iineasurement involving all‘ features considered
`significant would be impossible,‘ or_.longer than economically: feasible for
`study, mathematical precision is often abandonediinc favor -of more workable
`simple,- but vague, “common sen_se’_’ models. Thus, complexity of the problem
`may be another cause_-of vagueness.
`:
`-
`,
`_
`-
`These reasons.were thezdriving force behind the develop‘men,t.of. the fuzzy
`'set,.theory.'(‘FSTI')-. This field: of applied mathematics has become a-_d_yna'mic_
`research and applications field, with success stories ranging from a fuzzy logic
`rice cooker to an artificial intelligence in control of Japan’s Sendai subway
`system. The main idea of fuzzy set theory is to -propose a model of uncertainty
`different from that given by probability, precisely because a different form-of
`uncertaintyisbeing modeledf
`-
`_'
`.
`M
`j-
`-
`,
`-
`Fuzzy set theory was created in -Zadeh’s (1965) historic article. To present
`this basic idea, recall that a characteristic fimction of a subset E of a universe
`of discou‘rse'U is defined as
`‘
`'
`l
`'
`E
`'
`.---
`E Iif
`I-E
`.
`A7580‘) = {0 if E,
`
`E
`
`' In other words, the characteristic function describes the membership ofan _
`element x in a set E. It equals one if x is a member of E, and zero otherwise.
`
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`Fuzzy Techniques of Pattern'Recognition in Risk ‘and Claim Classification
`
`449'
`
`Zadeh challenged the idea that membership in all sets behaves -in the man4
`ner described above. -One example would be the set of “tall people.” We con-‘
`sistently talk about the set of “tall people,” yet understand that the concept
`used. is not precise. A person who is 5"11" is tall only to a certain degree, and
`yet such a person is not “not tall.”' Zadeh writes,
`'
`‘
`'
`
`The notion of fuzzy set provides a convenient point of departure for the_ construction
`of a conceptual framework which parallels in many respects the framework used in -
`' the case of ordinary sets, but is more general than the latter and, potentially, may '
`prove to have a much-wider scope of applicability, particularly in the fields of pat-
`tern classification and information processing, Essentially, such a framework pro-= _
`vides a natural way of dealing with problems in which the source of imprecision is
`the absence of sharply defined criteria of class membership rather than the presence__
`of "random variables._
`'
`"
`'
`'
`h
`
`'
`
`_
`
`,
`
`In the fuzzy set theory, membership of an element in a set is described by
`the membership function 'of“the set.'I‘f U"is tlieiuiiverseof discourse, and E" is
`_a, fuzzy subset of U, the membership function ].1-,3:lLl-—>[_(_),.1] assigns to every‘
`element x in the set Iii its degree of membership pE(ir)_. We write either (E,}1Ee)f
`or E~ for that fuzzy set, to distinguish from the standard set notation E. The.
`membership function is a .generalization of the characteristic _function,of arr
`ordinary set. Ordinary sets are termed crisp sets in fuzzysets theory. They are
`considered a special case—a, fuzzy setaiscgrlsp if, and only if, its membership
`function does not have fractional values,
`'
`On the basis ofthis definition, one_then.de.velops such concepts asset
`retic operations on fuzzy sets (union, intersection, etc.),_a_s well as the notions
`of fuzzy numbers, fuzzy relations, fuzzy ari_th_me_tic,i and approztimate reasoning '
`(known popularly as“fuzzy l0,gic”).:Pa_ttem recognitio,n,_or_the search for
`structure in data, provided the early impetus for deve1o_pinggFS'.l‘ becauseof
`fundamental involvement of human perception (Dubois and Prade,
`l9_.8Q)_ and
`the inadequacy of standard mathematics" to deal ‘with compleqg. and ill-defined
`systems (Bezdek and Pal,‘ 19.92). The formal development began with. Zadeh
`(196_-_5_)i'int1ioducing the principal concepts of FS_T. Acomplete presentation of .
`FST is "provided in Zimmennan (1991).
`,
`.
`,_
`_;
`The first recognitionof _FST applicability to the problem of insurance._under¥
`is due to DeWit (1982). Lernaire (1990) sets out a more extensive
`agenda for FST in insurance theory, most.,not_ably in the financial aspects
`the business._ Under the auspices of the Society of Actuaries,_Osta'sze_wski
`(1993) assembled a large number of possible applications of fuzzy set theory
`in actuarial science. His presentation-includes such areas as'.econorni_cs of rislc,
`time value of money, individual and collective models or risk, classification,
`assumptions, conservatism, and adjustment.
`and Deirig ('_19,93, 1994)
`complement that work by exploringrrapplications of fuzzy sets to property-_
`liability insurance, forecasting: and pricing p_r_ob1em_s.,
`H
`',
`_
`_,
`.
`Here, we present a method of flizzyipattern rccovgnitionrfor risk,pa._nvd claims
`classification. We applygfuzzy patt_e1jn.re_cognition to two problems in Massa} _
`chusetts private passenger automobile insurance: defining rating territories and
`classifying claims with regard to their suspected fraud content. Dubois and
`
`i
`
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`

`

`450
`
`_
`
`M
`
`._
`
`M
`
`-
`
`,_-T.{ze;Jourfnpal of Risk and -Insurance
`
`Prade (1930), Bezdek (1981), and Kandel(1982)__p_1:ovide-oyerviews of fuzzy
`techniques p'in.pa_tten1 recognition. Zimmerman -(1991) and Bezdelg and Pal
`(1992) provide other valuable references on the subject,
`it
`p
`.
`.;_
`.
`M Tl1e._concept_of a fuzzy set_-and_ the_mat_her_natical a1gorithrns_ ineededtto-:-irn;
`plement classification using fuzzy techniques._is-describedin the next section.
`Grouping towns in Massachusetts into rating territories for risk [classification
`purposes is viewed asa'_'fuzzy clustering ‘problein becauseniany "town's" can be
`strongly're_Ia'ted to tp\_ivo,.orfmorei_territo1ies, thereby creatingjavhordcr problem:
`to which of several related territories should a town be assigned.~We also
`explore the--influenceof geographical--proximityr on the‘ resulting..fuzz-by territo-
`ries and classification -of-claims by their suspected fraudulent-'content.“A final
`section summarizes’and"-providessome alternatiiie
`‘directions for
`FST in risk and claims classification problems.
`i
`'
`
`1 “
`7.] 7A1”g.ofrit!i!sS.£9?eF!iiériCié§éifi¢étisiILiL g
`" i L¢maae'c1990)
`Ostaszeitrski (1993) point out that :insu‘rance*r'isk=-ciassi= .
`fication often resorts ‘either to vague‘ rnethods_—”asi in the case of using niii1tiple- t
`ill-’c_lefi'netl»pe"rsonal criteria-_'to identify "good‘ ti:
`methods
`that are_ezces§iye1y precise'%4—as'i11‘the"case ofa person ‘who
`to-classify as"
`apreferred iriis’l'c*for life‘ insurance application beeausethis‘ or her body ‘weight
`eirceeds the stated Iiniit by'ha]f ‘a pound. Kandel (-‘li98'2),'w‘17'i'ti'r‘1g f1‘orr1'a' differ-
`ent perspective, says: A “In a very fimdamentaI’w"ay; the"intirna'te - relation be—'-
`tween t_lIe';th'eory”of fuzzy sets‘ and the theory of pattern recognition and ciassi- _
`gfication re‘sts‘on the "fact'th'at most real-*v'vor1d*cIasses are ifiizzy in-natars.'* 'I_‘h'is
`isjexactly t11e'reason" that we propose to utilize"-the n1ethodoio'gy'of 'fiizzy;clu'sf—
`te_r'in'g- in‘ ‘tenitoiial-"cl’as'sification
`to e_XteI'1d= that method ‘to“-_‘-classifyinitgj
`c1‘a'ir'I1s_fo'r susPi‘:§3tecl”fraud.
`'
`"i
`.
`'
`0“ *
`4-
`j Kaindel (1"9's-2) classifies" irariousi te'ch‘niques”of_ fuzzy pattern ireéiogifitioii.
`Syntactic: ‘techniques’ apply "When the pattern 3sojtight' "is" related . toithe” ‘foirnal
`structure of tl1e'languag'e.-Seimémtic techniques ’Aapply‘to’tiiose "producing fuzzy
`partitions‘ of data sets.‘-According: to‘ Eezdek and ‘Pal-"('ip992);' are first choice
`faced by a pattern recognition system designer‘ is 'that”of process description;
`‘The designer 'may"choose"fror_ri among-syntactic, numeric'a'1, contextual, rule-
`based,’ hybrid," and‘ fi1zz'y- prdeess descriptions." Feature e'analysis_’-‘is theinext
`design"step,j in Which‘ (generally given iniithe -form of -a data" vector-*co'r'1"—
`inforrn‘ati'o'n- "about the analyzed objects) r'nay'”be ‘s1'ibjected"to p‘repi‘o‘4
`cessing, displays, and extraction. Next, semantic clustering a1g‘orithms,‘genei'ati
`' g actuaI‘sti1ictu£res
`data, are”3idei1'tified. Finally, the designer 4-addresses
`ciiiste*r'irL'3ilidity and__opti'rna1ity.'
`_
`7
`s
`T‘
`"
`T
`- We use a fuzzy pa-ttern‘ recogriition 'tec‘hni'que‘ given" by Bezdek _(19’s1)-: In
`the classification of -B-ezdek and Pal (1:9_9_2)',i it can be described ‘as ‘ta numerical
`process description, fuzzy c-means iterative semantic algoritiin-1.‘_'Becaiise the
`data We analyze 'are’in'the' form of numerical vectors (i:e:, vectors- in a-e’1iclide—
`an space), with'a11um'ber of clusters sou_ght'predeterr'nii1ed, we "consider ‘the
`I-,
`
`'
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`_
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`Fuzzy Techniques of Pattern Recognition in Risk and Claim Classification
`
`451
`
`fuzzy c-means technique most appropriate. Bezdek et al. (1-987) discuss the
`convergence properties of the algorithm.
`a
`'
`The task is to divide :1 objects, where n is a natural number, each represent-
`ed by a vector in a p-dimensional euclidean space
`
`x1, x2,..., x,','
`
`(co_ordi.nates of the vectors are known as features), into_-c, 2 Sc <_ n, categori-
`cally lroInogeneous_ subsets called clusters. The _,objects belonging to the same"
`cluster should be similar, and the objects in. diffe_r_ent_ clusters shouldpbe as
`dissimilar aslpossible. The number of clusters, c,_ is ‘specified in advance‘. _If the
`membership, function __of objects in clusters“ talceslon fractional vjal1ies,the_n' we
`have__fLizzy _ch_1sters.
`pro_ces_s is calleddustering.
`"
`'
`'
`'
`u
`'
`A Any'c1ustefing_ method must answer two fundamental qnlestionsfl
`properties of the dataset should be used,"
`_way_ should they be
`‘used to‘identify’:clustersl.'.Once the algorithrnf those two conditions is
`specified; _there“a_1'e_,' of course, more
`=c[1'1estio'ns,'such, as Whether
`algorithin is effective for
`possible jsetsof ‘data; as well as the 'questi'o'i1.of.
`validity of c1usters_'(s'ee Kandel, 1982, and'7Bezdek_,'and“P.al,_ ‘I992, fora discus; ..
`sionof this"problem);e '
`‘
`'
`;
`fl
`-
`a
`-‘:Ris_k"cIassification
`to distinguish risks for the purposes" of rating*'an'd
`underwriting. In claims processing; the purpose is" to identify -claims -suspected
`of fraud for special processingnnd route norisuspicibus clainfs through normal -
`adjusting channels. Insurance risks and claims are both described here by c_er_-
`- tain data patterns. The pattern recognition algorithm does the “detective work”
`of finding..clusters_of;sirn-ilar risks and-claims.~ 0
`A
`thedata...-set be ,
`_
`0
`*
`
`-
`
`:
`_
`T
`‘
`‘X =9 txg.-i<a';:... 3<..}"-
`X is assumed to be a finite subset of a p—-dimensional euclideanlspace RP. Each
`'
`x, =' (x,,,i;-»x,,,;..x,,',,), 1: = 1,- 2,
`n
`
`'
`
`is called a feature vector, while each xkj, where j = 1, 2,..., p, is the jtl1feature
`of the vector xk.
`0
`.
`A partition of the data set X‘into'fuzzy clusters is described by the set of
`membership fnnctions of the clusters (note that such a description could also“-
`apply to crisp clusters, with the membership "function meaning sirnplyjthe
`characteristic function). The clusters are denoted by S1, S-2,..., Sc with the corre-"
`sponding membership functions 3181, 1132,
`115 .In other words, we will con—
`struct c clusters that are fuzzy sets.
`-' A- c X in-"matrix conta-iningthe values. of the membership functions -of the
`fuzzy clusters
`'
`‘
`"
`
`l
`= In-S -
`is afuzzy c—partition if it.-satisfies the following conditions;
`
`K
`
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`

`

`
`
`452
`
`The»Joumat'ofRisk and insurance"
`
`i
`
`i=1
`
`pS_(Xk) = i re: each 1; = 1,
`'
`
`u
`
`11,
`
`0 S E ]lS.(Xk) S n for each i =_ 1, 2,..., c.
`=1
`
`(1)
`
`[(2)
`
`Condition (1) says that each feature vector xi: has its total membership value
`of one divided‘ among all clusters, and condition (2) states that the sum of
`membership degrees of featureve'ctors'in' a given cluster does not exceed the
`total ‘number of feature vectors.
`5
`'
`'
`'
`Given the above definition, let us now present the fuzzy c-means algoritlftn
`of Bezdek (1981), also used in Ostaszewski (1993). :l‘he iterative algorithm
`consists of four steps; we‘_add a fifth step to make the result operational. The-
`first step sets out a working definition of distance between feature vectors (the
`vector. norm) and _an initial starting partition. The seco'nd'stepi_identifies' the-
`center ofeach cluster in the partition: The third step recalculates the men_1be_r-
`ship functions of the partition as nonnalized distances from the step 2 centers. T
`The fourth step "checks the distance between successive partiti_ons_. to determine
`if,.the_iteration procedure should be stopped.
`'I'he__fifth step d'is_car_ds_’ small
`membership values (below some predetermined ot, 0 <__0t < 1)‘ to make the.-
`partition operational. The five formal steps follow.
`
`'
`7
`
`Stép 1,
`
`Choose c, an integer betweentwo a'nd.n,-has the number-uof clusters into»
`which the data is partitioned. Choose a positive parameter m,-and a symmetric,
`positive-definite p x p matrix G. Define the vector norm I!
`{la , using the
`transpose operator T, by
`
`“K1; H vi
`
`<3)
`
`i
`
`Set"the§iteration ‘counting’ pararrieter ll equal to zero, and choose the
`fuzzy partition
`
`lSiSc.lSk_<._n
`
`‘[ps(i0)(xk)]
`fl(Oi'
`Choose a parameters >0 (this number will indicate when to’ stop the iteration
`process).
`Note that the columns of the fuzzy partition matrix, numbered one through
`11, correspond to data vectors, and each column gives degrees of membership
`of the data point in clusters. one through c. The maI1ix3.norn1'|| NC, is suitably
`chosen in such a way that two data vectors with great similarities are relatively
`
`Page 00007
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`

`Fuzzy Techniques of Pattern Recognition in Risk and Claim Classification
`
`453
`
`close to each other, while dissimilar data are set apart. Although no perfect
`measure of such relationship exists, we can adjust the scale of xk coordinates
`by introducing appropriate diagonal entries, and any known correlations of
`coordinates can be represented in the nondiagonal entries. The size ofjthe
`matrix G corresponds to the number of coordinates in data vectors.
`The main idea of ‘the algorithm is to produce reasonable centers for clusters
`of data, and then. group data vectors aroundcluster centers which. are reason-'-
`ably close to them; Unlike in standard crisp-"algorithms; fractional cluster mem-'
`bership is allowed, which gives us flexibility to adjust for any otherwise desir-
`able phenorriena.
`0
`.
`.
`r
`-
`-
`
`Step 2
`
`Calculate the fuzzy cluster centers {vi‘“)}.l=1_2,___,cgiven by the following formu-
`
`la:
`
`_H_ 212:1 (]'lS(iE)(Xk))m -Xk
`___j.___..___.._
`Ego (’.*§(f'r(’-_‘I<)).~.In .
`_
`.
`_
`.
`for i[= 1; ‘2,._.., dc.
`The cluster centers arenierely weighted averages of data vectors; Weights
`are given by the mth powers of tl1eiinembershi_p. degree._iBezdek et a1. (1987)
`discuss the influerice of the scaling factor In," as well as convergence of'th'e
`resulting algorithm.
`
`I
`
`E [(4)
`
`V (0i
`
`T
`A
`p
`_
`Step3
`Calculate the new partition (i.e., membership matrix) .. ..
`
`I"J""”’ =
`
`'
`
`T
`
`_
`km) "
`ipsi
`(x")'J1§sc,1sIcsn
`
`,
`
`'
`
`where
`
`—
`.._(0.2_..,.
`‘J.
`_
`1
`i'ti5i“”(Xk) = ~Z“““*-i0 V’ Ti H” " it
`
`
`
`2‘; — 0
`3.1.
`0
`Ilxk-vj‘“’_l2-,
`.
`_
`where i = 1; '2;.-..,' c, and 16': l, 2,...,>n;
`If xk =_l vi“), however, fonnu_laA(5) cannot be us‘ed.,In tl1at'casle,iwe -set
`
`r
`
`
`
`(5)
`
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`454
`
`—
`
`,
`
`‘The Journal ofARi._slc and Insurance
`
`'*‘siM("k) ”{01—irIfi<'kss 1,1} = 1,
`
`c.
`
`-
`
`a
`
`0 (6)
`
`-' This step of the «algorithm carries us from the previous membership matrix
`(numbered 12).tothe nextzone (numbered. fl.+ 1). One can interpret formula‘.(5)
`as follows: if the vector norm measures the similarity of two data .vectors,.the
`(m-1)st root of its reciprocal is a form of measure of dissimilarity, and formula
`(5) assigns a new membership: degree by relating the dissimilarity with a given
`cluster centerfto the .“totale=-dissirrrilarityt present.” Formula (5)-5-is, however, -"a
`result of a longer optimization procedure discussed further by Bezdek et-all
`(1987).
`
`Step 4
`By using the natural matrix norm, or the extension. of H [JG to the
`norm, or by choosing a different matrix norm more suitable to the problem,
`calculate
`'
`-
`
`.;A = IIUM 9 "W116-
`If A > 3, repeat steps 2, 3, and 4. ‘Otherwise, stop at some iteration count 0*.
`This “stopping procedure” is a standard numerical analysis technique——if yet
`another iteration does not change much, the result is the best possible. Clearly,
`the i_ procedure "rests on the _assumption
`the a_lgorithrn’s convergence, but
`luclcily, the proof of that convergence exists, Bezdek et
`(l987).'
`'_
`"
`
`Step 5
`
`The final fuzzy matrix U’ 2* is structured for operational use by means of the-
`normalized occur, for some 0 < a < '1. Quite simply, all membership function
`values less than or are replaced with zero and the function is renonnalizedj
`(sums to_ one) to preserve partition condition (1). For small 0:, the resulting
`partition is still fuzzy; for large on (or mar-cuts, where the largest membership
`value is set equal to one all others are zero), the resulting partitions are likely _
`to be crisp.
`i
`'
`0
`'
`
`Automobile) Rating Territories in_.Massachusetts '
`As Conger (1987) points out,
`
`In Massachusetts, -thepast ten‘ years have witnessed the evolution of an increasingly
`sophisticated system of methodologies for determining the definitions of rating
`territories for private passenger automobile insurance. ‘In ._contrast
`to territory,
`schemes in other states, which tend to group geographically contiguous towns, these .
`Massachusetts mefliodoiogies have had asitlreir goalthe g'roupii1g'of towns witlr”
`similar expected losses per exposure, regardless of the geographic contiguity or non-
`contiguity of the grouped towns.
`
`Page 00009
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`
`
`Fuzzy Techniques of Pattern Recognition in Risk and Claim Classification
`
`455
`
`.
`
`-
`
`Note the ambiguous nature of “similar expected losses,” adecidedly fuzzy
`concept.
`g
`_
`_
`g
`'
`g
`g
`f
`_
`The methodology used for territorialirating results in a final combined five-
`coverage pure pren1ium'index for each of the 360 towns (or,'rnore“ p'recisely,
`350 towns and ten areas into which Bos_ton iscdivided for automobile rating
`purposes). A complete description of the empirical Bayes procedure for deter—'
`mining the biennial individual and combined coverage town indices from four .
`. years of data is given in DuMouchel (1983).. The indices, which are numbers"
`relatively close to 1' representing expected losses’ in relation to those of the
`entire state e_xpectecl.losses, are then ordered and-.territories.. are .cr.eate_d_.,by
`partitioning that linear ordering.-' Because frequent switches from one ;h3I'1‘i_tory
`to another are undesirable but inevitable, numerous restrictions on moving
`towns from one territory to another exist in actualregulatory practice. Once
`territory clusters are set for" a rating year, five individual coverage rates are
`determined using that single clustering,’ one which may or may not be appro—,
`priate for each coverage, but which is assumed to be equitable overall.
`_
`Such difficulties and irnprecisions in groupings warrant an investigation of
`fuzzy clustering. Resulting fuzzy clusterswouldbe much. more flexible, be-_ .
`cause a town belonging partially to two orrhore territories could be"assigned
`to one of them if regulatory limitations dictate unique assignments of towns to
`territories. Although stability of territory assignment is desirable and conve-
`nient, the system of clustering towns into territories should meet the standard -
`responsiveness criterion for risk classification. Towns have an incentive‘ to
`reduce their relative loss costs by. maintaining their roads, safety engineering,
`and law enforcement, if those actionszbring about lower premiums. When the
`system is not responsive, or slow to respond, ‘the incentives can be diminished
`or lost.
`'
`..
`The pure premium indices are calculated for the following coverages -for all:
`350 towns: bodily "injury liability (A-1 ar1dcB), personal injury-protection (A-
`2), property damagefliability (PDL), collision, comprehensive, and a sixth
`category comprising the five individualcoverages combined. We use those
`values as the coordinates of vectors xk, k = 1, 2, '3,..., 350, representingthe’
`towns in the data space. This implies that we treat-the data. space as six-dimenfi
`sional, as six parameters are used to describe towns. In our calculations, we‘ .
`use either the five coverage indices (five—d_-i_rnen_sionai vectors) or the combined '
`index (one-dimensional vectors) but not both. The data for the 1993 indices 0
`(based on the 1987 through 1990 data) for towns in Bristol County are given
`in Appendix A. Data for all .350 tow-ns and Boston are available in Automobile
`Insurers Bureau (1992) or from the authors. We begin by illustrating the algo-
`rithm for a manageable "set of towns: the twenty towns of Bristol County,
`Massachusetts.
`-
`0
`
`'
`
`_
`
`‘In general, the partitioning is accomplished’ by grouping towns within five to six -percent
`intervals on either side of the statewide averageindex ‘of one.
`-
`-
`-
`
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`

`
`
`
`
`"456
`
`-
`
`.
`
`-
`
`The Journal of Risk "and Insurance
`
`The Bristol County Algorithm
`The
`clustering for Bristol County is the indicated 1993 territory as- I
`signment groupings relabeled one to five? The initial five-coverage partition
`matrix is '
`A
`_
`V
`i
`'0
`
`. fi(0) 2 [pS(‘0)(X-Q]
`
`-
`iSi§.1SIrS20
`
`’
`
`where; pS§°’(xk)- represents the membership of towmxk incluster Si; and it
`I
`equals-one if the town" is in the territory, or zero if it is not;
`.
`-
`We also set the stopping parameter 3 = 0.05, and III =' 2.-The initial ‘cluster
`centers are icalculated as
`'
`'
`t
`
`.
`(0)
`
`v.
`
`0
`20‘
`:
`a.E1:'=1t(’.“s(a°)("'=))2 xk
`
`= :_..__j‘- - __-.
`
`
`
`
`
`0
`
`'
`
`I
`
`.
`
`0
`
`I
`
`j(=1 (pS(;O,),(xk))%
`
`0
`
`2,‘..., 5. We proceedto: ‘evaluate thenew partition Inatrixi T
`u“’p=p[pSg>(xk)]1gg_wfl?
`_
`=
`
`'
`
`I"
`
`_
`-_r (8) -
`
`for i it
`
`where
`
`A
`
`1
`
`.~
`"
`
`0
`
`1
`
`_
`
`7
`
`0
`
`E;
`
`(X)
`
`— (V500
`
`
`
`.
`
`'
`
`pS(i1)(Xk) =
`
`5
`. _
`
`,.
`-
`_
`P)
`P ( Kr
`5 Zp—l
`(23%
`)
` >311; gm (<xr>.;== j<v,-‘sip
`where the subscript p refers touone of theofiive pure premium--coordinates of:"a
`town, and i = 1, 2,..., 5, k = 1, .2,..., 20, and g1, are weights representing the a
`distribution’ of losses across -coverage?
`>-
`0
`'
`= vim’; however, formula (6)'must be used." In that case, we set
`
`2 For illustrative purposes, the town of Fairhaven, which was assigned to 1993 Territory 9, is
`included with those towns in Territory 8. Fall River is included with New :-Bedford. Actual 1993’
`rating territories are subject to judgmental adjustments and capping and are not always those
`shown here.
`
`3'I'he coverage weight distribution, using 1990 exposures times 'four—year pure premiums, is
`[(gii) = (02229, 0.1109, 0.2048, 0.3210, 0.1404); (gfi) = 0 ifi aej, 1‘ S i, j S 5)].
`
`Page 00011
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`Fuzzy Techniques of Pattern Recognition in Risk and-Claim rCla.s's§flcation
`
`457-
`
`'
`lifkzi,
`__
`(1)
`%i@Q"0fiksLk=L2W2Q#LLwi
`
`Now we calculate the distance between the initial partition matrix. U (0) and the
`new partition matrix Um, by taking the simple matrix norm
`
` (10)
`
` J’
`
`.
`
`If A < 8 = 0.05, the process is stopped. Otherwise, theiterative algorithrn
`continues. The results of the calculation, with an on-cut of 0.2, are presented in
`Table 1.
`.
`
`Table 1
`
`Fuzzy Town Cluster Membership Values
`_
`[for Bristol County, Massachusetts
`‘
`i
`_ Membership Values
`,_
`..
`7
`pl.’
`
`9
`
`Initial
`Cluster
`
`.
`
`‘us!
`
`0
`
`an
`
`E,’
`
`1
`1
`2
`2
`2
`2
`2
`2
`3
`3
`
`3
`3
`4
`4
`4
`
`_4
`4
`4
`
`1
`1
`- 0.32
`10.40
`-0.58" 9
`0 '
`00
`0
`0.25‘
`0
`
`0
`0
`0.22
`3' 0.2-3
`0
`1
`1
`0
`0
`O
`
`-
`
`0
`0
`0.46
`0.38 f
`= 0.42 2-
`0
`0
`1
`0.43
`1
`
`0
`'0
`0
`0
`0
`
`0
`0
`0 '
`
`0
`0
`0
`0
`0
`
`0.37
`0
`0
`
`1
`1
`0
`0
`0.37
`
`0.30
`0
`0
`
`0
`0
`0
`-‘ 0.
`0
`' ‘0-
`0
`0
`0.32
`0
`
`0
`0
`1-
`1
`0.63
`
`0.33
`1
`' 1
`
`Town Name
`
`Mansfield
`Emnhrmmnmnmgh
`Dighton
`Rehoboth
`Norton
`_ Freetown
`Berkley
`Raynham
`Seelconlc
`Easter:
`
`Attleboro
`Dartmouth
`Somerset
`Swansea
`Taunton
`
`Westport
`Acushnet
`Fairhaven
`
`II,‘
`
`'
`
`Sum
`
`O
`0
`0
`0
`0_
`0
`0
`0
`0
`0
`
`0
`0
`O
`0
`0
`
`0‘
`0
`0'
`
`14
`1
`1
`1
`1
`l
`1
`1 -
`1
`1
`
`_l;
`
`'-
`
`1’
`1
`1 .
`
`1r
`1
`1
`
`1
`1
`0
`0
`0
`t 0
`5
`Fall--River
`1
`.
`I
`0
`0
`'0
`0” 0
`_5
`New Bedford
`
`Sum
`3.54 -
`2.82
`6.35
`5.29
`22
`20
`'
`
`' Note: C—means fuzzy clustering algorithm, with five-coverage dam pattern, ninth iterationstopping
`
`pmuneter 0.0499 (0.05, ot-cut_= 0.2, no geographical variables.-
`
`Figurestl-1 and 02-display the. resunltsrolf-tlle -transiiutionlufrorn
`clusters to final fuzzy clusters‘. Figure
`displays the,-20 Bristol County tovignsi
`
`Page00012
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`

`

`458
`
`'
`
`The Journal of Risk and Insurance
`
`C
`_
`_
`Figure 1 _
`M
`In-itia1.Tem'tor;ial_ Town Clustering by Combined Index Territory
`for Bristol County, Massachusetts
`
`
`
`
`-
`_Figure 2.
`.
`.
`Fuzzy Town C_lustering by Five Coverage Indices
`for»:Bristol County, Massachusetts
`
`9-mlumm-I
`
`‘——n¢m_uanb«
`
`'
`
`_0'.-0.2.
`
`grouped into their initial C1ilSteI?S- in: increasing combined?index=-order:‘rFor=
`example, Town 1 (Mansfield) has the lowest combined_index value (0.80l8)
`and is-in-'-the lowest ranked territory, while-Tow‘n=2@_:(New Bedford}-has :the
`highest index (102977)-'and‘sis in the‘ highest r'aink_ed territory.
`T‘
`'
`-
`-0'
`'0
`=
`
`Page 00013
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`

`
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`
`Fuzzy Techniques of Pattern Recognition in Risk and Claim Classification
`
`439
`
`Figure 2 shows the fuzzy clusteringiresults that provide for the incorpora-
`tion of five-dimensional data (individual coverages), as well-as. thefractional
`assignments (fuzziness) to-the clusters. With. fuzzy ‘clustering, towns tend to
`become associated with nearby clusters as well as with their “home” cluster.
`Town 8 (Raynham) becomes associated with fuzzy cluster 3 and has little
`association- (less than 0.2) with its original home cluster 2. Town 5 (Norton)
`with home cluster_2 splitspinto fuzzy clusters 1 and 3. These movenientsgare
`typical of fuzzy clustering results.
`i
`'
`'
`
`,
`
`T
`
`'
`
`Geographical Proximity
`We also perform a calculation adding two more features for each tow_1i1,—fL_—lits”
`geographical coordinates divided by the coordinates of the town with_.the
`est Massachusetts coordinates, Nantucket (the division is performedto. adjust
`thescale and to match the other fe_a'tures, which are all close to one)..f.i]}y _
`performing the a_lgorithm_on these vectors, including geographical coordinates,
`awe increase the chance of arriving at clusters that are not only actuarially
`similar, but relatively close geographically.
`=
`=
`.-
`This calculation is performed in the same manner as before, but with seven
`feature variables. We show results for pure premium data weighted 50 percent
`and geographical variables 50 percent, but ally relative weighting scheme,can'
`be used to reflect the modeler ’s preference fongeographic dependence of ‘
`toties. The 50/50 results areflpresented in Table 2. Note that _a full 50_ percent
`weight onthe two geograpllical coordinates producedonly slight differenqese_:__ ;
`fromlthe five variable centers shown in Table 1. Recall that other states,.,I_1se
`geographical proximity as an impo