Fuzzy c-Means Clustering, Entropy Maximization, and Deterministic and Simulated Annealing

20 Pages·2012·1.35 MB·English

by  Makoto Yasuda

ChCahpatpetre r130
Fuzzy c-Means Clustering, Entropy Maximization,
and Deterministic and Simulated Annealing
MakotoYasuda
Additionalinformationisavailableattheendofthechapter
http://dx.doi.org/10.5772/48659
1.Introduction
Many engineering problems can be formulated as optimization problems, and the
deterministicannealing(DA)method[20]isknownasaneffectiveoptimizationmethodfor
such problems. DA is a deterministicvariant of simulated annealing (SA) [1, 10]. The DA
characterizes the minimizationproblemof costfunctions as the minimization ofHelmholtz
freeenergywhichdependsona(pseudo)temperature,andtrackstheminimumoffreeenergy
while decreasing temperature and thus it can deterministically optimize the function at a
giventemperature[20]. Hence,theDAismoreefficientthantheSA,butdoesnotguarantee
aglobaloptimalsolution. ThestudyontheDAin[20]addressedavoidanceofthepoorlocal
minimaofcostfunctionofdataclustering.Thenitwasextensivelyappliedtovarioussubjects
such as combinational optimization problems[21], vector quantization [4], classifierdesign
[13],pairwisedataclustering[9]andsoon.
On the other hand, clustering is a method which partitions a given set of data points into
subgroups,andisoneofmajortoolsfordataanalysis. Itissupposedthat,intherealworld,
clusterboundariesarenotsoclearthatfuzzyclusteringismoresuitablethancrispclustering.
Bezdek[2] proposed the fuzzy c-means (FCM) which is now well known as the standard
techniqueforfuzzyclustering.
Then, after the work of Li et al.[11] which formulated the regularization of the FCM with
Shannon entropy, Miyamoto et al.[14] discussed the FCM within the framework of the
Shannon entropybasedclustering. Fromthehistoricalpointofview, however, itshouldbe
notedthatRoseetal.[20]firststudiedthestatisticalmechanicalanalogyoftheFCMwiththe
maximumentropymethod,whichwasbasicallyprobabilisticclustering.
Tomeasurethe“indefiniteness”offuzzyset,DeLucaandTermini[6]definedfuzzyentropy
after Shannon. Afterwards, some similar measures from the wider viewpoints of the
indefinitenesswereproposed[15,16]. Fuzzyentropyhasbeenusedforknowledgeretrieval
fromfuzzydatabase[3]andimageprocessing[31],andprovedtobeuseful.
Tsallis [24] achieved nonextensive extension of the Boltzmann-Gibbs statistics. Tsallis
postulated a generalization form of entropy with a generalization parameter q, which, in a
©2012Yasuda,licenseeInTech.Thisisanopenaccesschapterdistributedunderthetermsofthe
CreativeCommonsAttributionLicense(http://creativecommons.org/licenses/by/3.0),whichpermits
unrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperly
cited.
272 2Simulated Annealing – Advances, Applications and Hybridizations Will-be-set-by-IN-TECH
limitofq→1,reachestheShannonentropy.Lateron,Menardet.al.[12]derivedamembership
functionbyregularizingFCMwiththeTsallisentropy.
In this chapter, by maximizing the various entropies within the framework of FCM, the
membershipfunctionswhichtakethefamiliarformsofthestatiticalmechanicaldistribution
functionsarederived. Theadvantagetousethestatisticalmechanicalmembershipfunctions
is that the fuzzy c-means clustering can be interpreted and analyzed from a statistical
mechanicalpointofview[27,28]
Afterthat,wefocusontheFermi-Diraclikemembershipfunction,because,ascomparedtothe
Maxwell-Boltzmann-like membership function, the Fermi-Dirac-like membership function
hasextraparametersα s(α correspondstoachemicalpotentialinstatisticalmechanics[19],
k k
and kdenotesadatapoint), whichmakeitpossibletorepresentvariousclustershapeslike
former clustering methods based on such as the Gaussian mixture[7], and the degree of
fuzzy entropy[23]. α s stronglyaffect clustering resultsand they mustbe optimized under
k
anormalizationconstraintofFCM.Ontheotherhand,theDAmethod,thoughitisefficient,
does not give appropriate values of α s by itself and the DA clustering sometimes fails if
k
α sareimproperlygiven. Accordingly,weintroduceSAtooptimizeα sbecause,aspointed
k k
above, both ofDAand SAcontain the parameter correspondingto the systemtemperature
andcanbenaturallycombinedasDASA.
Nevertheless,thisapproachcausesafewproblems.(1)Howtoestimatetheinitialvaluesofα s
k
underthe normalizationconstraint? (2)Howtoestimate theinitial annealing temperature?
(3)SAmustoptimizearealnumberα [5,26].(4)SAmustoptimizemanyα s[22].
k k
LinearapproximationsoftheFermi-Dirac-likemembershipfunctionisusefulinguessingthe
initialα sandtheinitialannealingtemperatureofDA.
k
InordertoperformSAinamanyvariablesdomain,α stobeoptimizedareselectedaccording
k
toaselectionrule. Inanearlyannealingstages,mostα sareoptimized. Inafinalannealing
k
stage, however, only α s of data which locate sufficiently away from all cluster centers are
k
optimized because their memberships might be fuzzy. Distances between the data and
the cluster centers are measured by using linear approximations of the Fermi-Dirac-like
membershipfunction.
However, DASAsuffersafewdisadvantages. Oneofthem isthat itisnotnecessarilyeasy
tointerpolatemembershipfunctionsobtainedbyDASA,sincetheirvaluesarequitedifferent
eachother.Thefractalinterpolationmethod[17]issuitablefortheseroughfunctions[30].
Numerical experiments show that DASA clusters data which distribute in various shapes
moreproperlyandstablythansingleDA.Also,theeffectivenessofthefractalinterpolationis
examined.
2.Fuzzyc-means
wLehticXh=sho{ux1ld,..b.e,xdni}v(idxked=in(txo1k,c..c.l,uxstkpe)rs∈CR=p) b{eCa1,d..a.t,aCsce}t.inLaetpV-dim=en{svio1,n.a.l.,revacl}(svpiac=e,
(v1,...,v p)) be the centers of clusters and u ∈ [0,1](i = 1,...,c;k = 1,...,n) be the
i i ik
membershipfunction.Alsolet
n c
J = ∑ ∑u (d )m (m>1) (1)
ik ik
k=1i=1
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be the objective function of the FCM where d = (cid:4)x −v(cid:4)2. In the FCM, under the
ik k i
normalizationconstraintof
c
∑u =1 ∀k, (2)
ik
i=1
theLagrangefunctionL isgivenby
FCM
(cid:2) (cid:3)
n c
L = J− ∑ η ∑u −1 , (3)
FCM k ik
k=1 i=1
where η is the Lagrange multiplier. Bezdek[2] showed that the FCM approaches crisp
k
clusteringasmdecreasesto+1.
3.Entropymaximization ofFCM
3.1.Shannonentropymaximization
First, we introduce the Shannon entropy into the FCM clustering. The Shannon entropy is
givenby
n c
S=− ∑ ∑u logu . (4)
ik ik
k=1i=1
Underthenormalizationconstraintandsettingmto1,thefuzzyentropyfunctionalisgiven
by (cid:2) (cid:3)
n c n c
δS− ∑ α δ ∑u −1 −β ∑ ∑δ(u d ), (5)
k ik ik ik
k=1 i=1 k=1i=1
whereα and βaretheLagrangemultipliers,andα mustbedeterminedsoastosatisfyEq.
k k
(2).ThestationaryconditionforEq.(5)leadstothefollowingmembershipfunction
u = e−βdik . (6)
ik ∑cj=1e−βdjk
andtheclustercenters
v = ∑nk=1uikxk. (7)
i ∑nk=1uik
3.2.Fuzzyentropymaximization
WethenintroducethefuzzyentropyintotheFCMclustering.
Thefuzzyentropyisgivenby
n c
Sˆ=− ∑ ∑{uˆ loguˆ +(1−uˆ )log(1−ˆu )}. (8)
ik ik ik ik
k=1i=1
Thefuzzyentropyfunctionalisgivenby
(cid:2) (cid:3)
n c n c
δSˆ− ∑ α δ ∑uˆ −1 −β ∑ ∑δ(uˆ d ), (9)
k ik ik ik
k=1 i=1 k=1i=1
274 4Simulated Annealing – Advances, Applications and Hybridizations Will-be-set-by-IN-TECH
whereα andβaretheLagrangemultipliers[28].ThestationaryconditionforEq.(9)leadsto
k
thefollowingmembershipfunction
1
uˆ = , (10)
ik eαk+βdik+1
andtheclustercenters
v = ∑nk=1uˆikxk. (11)
i ∑nk=1uˆik
InEq. (10),βdefinestheextentofthedistribution[27]. Equation(10)isformallynormalized
as
uˆ = 1 /∑c 1 . (12)
ik eαk+βdik+1 j=1eαk+βdjk+1
3.3.Tsallisentropymaximization
Letv˜ andu˜ bethecentersofclustersandthemembershipfunctions,respectively.
i ik
TheTsallisentropyisdefinedas
(cid:2) (cid:3)
S˜=− 1 ∑n ∑c u˜q −1 , (13)
q−1 k=1i=1 ik
whereq∈Risanyrealnumber.Theobjectivefunctionisrewrittenas
n c
U˜ = ∑ ∑u˜qd˜ , (14)
ik ik
k=1i=1
whered˜ =(cid:4)x −v˜ (cid:4)2.
ik k i
Accordingly,theTsallisentropyfunctionalisgivenby
(cid:2) (cid:3)
n c n c
δS˜− ∑ α δ ∑u˜ −1 −β ∑ ∑δ(u˜qd˜ ). (15)
k ik ik ik
k=1 i=1 k=1i=1
ThestationaryconditionforEq.(15)yieldstothefollowingmembershipfunction
u˜ = {1−β(1−q)d˜ik}1−1q, (16)
ik Z˜
where
c
Z˜ = ∑{1−β(1−q)d˜jk}1−1q. (17)
j=1
Inthiscase,theclustercentersaregivenby
v˜ = ∑nk=1u˜qikxk. (18)
i ∑nk=1u˜qik
Inthelimitofq→1,theTsallisentropyrecoverstheShannonentropy[24]andu˜ approaches
ik
u inEq.(6).
ik
Fuzzyc-MeansClustering,EFnutrzopzyyM cax-iMmizeaationn,sa nCdlDuestetremrininistgic,a EndnStimroulpatyed MAnanexailimngization, and Deterministic and Simulated Annealing5 275
4.Entropymaximization andstatistical physics
4.1.ShannonentropybasedFCMstatistics
IntheShannonentropybasedFCM,thesumofthestates(thepartitionfunction)forthegrand
canonicalensembleoffuzzyclusteringcanbewrittenas
n c
Z= ∏ ∑e−βdik. (19)
k=1i=1
BysubstitutingEq.(19)forF=−(1/β)(logZ)[19],thefreeenergybecomes
(cid:4) (cid:5)
F=−1 ∑n log ∑c e−βdik . (20)
β
k=1 i=1
Stable thermal equilibrium requires a minimization of the free energy. By formulating
deterministicannealingasaminimizationofthefreeenergy,∂F/∂v =0yields
i
v = ∑nk=1uikxk. (21)
i ∑nk=1uik
ThisclustercenteristhesameasthatinEq.(7).
4.2.FuzzyentropybasedFCMstatistics
In a group of independent particles, the total energy and the total number of particles are
givenby E = ∑ (cid:7)n and N = ∑ n, respectively,where(cid:7) representstheenergyleveland
l l l l l l
n representsthe numberofparticlesthat occupy (cid:7). Wecanwritethesumofstates, orthe
l l
partitionfunction,intheform:
ZN = ∑ e−β∑l(cid:7)lnl (22)
∑l(cid:7)lnl=E,∑lnl=N
whereβistheproductoftheinverseoftemperatureTandk (Boltzmannconstant).However,
B
itisdifficulttotakethesumsin(22)countingupallpossibledivisions.Accordingly,wemake
thenumberofparticlesn avariable,andadjustthenewparameterα(chemicalpotential)so
l
astomake∑ (cid:7) n = Eand∑ n = Naresatisfied. Hence,thisbecomesthegrandcanonical
l l l l l
distribution,andthesumofstates(thegrandpartitionfunction)Ξisgivenby[8,19]
∞ ∞
Ξ= ∑(e−α)NZN =∏ ∑ (e−α−β(cid:7)l)nl. (23)
N=0 l nl=0
ForparticlesgovernedbytheFermi-Diracdistribution,Ξcanberewrittenas
Ξ=∏(1+e−α−β(cid:7)l). (24)
l
Also,n isaveragedas
l
1
(cid:6)n (cid:7)= (25)
l eα+β(cid:7)l+1
whereαisdefinedbytheconditionthatN=∑ (cid:6)n (cid:7)[19].HelmholtzfreeenergyFis,fromthe
l l
relationshipF=−k TlogZ ,
B N
276 6Simulated Annealing – Advances, Applications and Hybridizations Will-be-set-by-IN-TECH
(cid:6) (cid:7) (cid:4) (cid:5)
F = −kBT logΞ−α∂∂αlogΞ =−1β ∑log(1+e−α−β(cid:7)l)+αN . (26)
l
Takingthat
(cid:7)
E=∑ l (27)
eα+β(cid:7)l+1
l
intoaccount,theentropyS=(E−F)/Thastheform
S=−k ∑{(cid:6)n (cid:7)log(cid:6)n (cid:7)+(1−(cid:6)n (cid:7))log(1−(cid:6)n (cid:7))}. (28)
B l l l l
l
Ifstatesaredegeneratedtothedegreeofν,thenumberofparticleswhichoccupy(cid:7) is
l l
(cid:6)N(cid:7)=ν(cid:6)n (cid:7), (29)
l l l
andwecanrewritetheentropySas
(cid:6) (cid:7) (cid:6) (cid:7)
S=−k ∑{(cid:6)Nl(cid:7)log(cid:6)Nl(cid:7) + 1− (cid:6)Nl(cid:7) log 1− (cid:6)Nl(cid:7) }, (30)
B ν ν ν ν
l l l l l
whichissimilartofuzzyentropyin(8). Asaresult,u correspondstoagraindensity(cid:6)n (cid:7)
ik l
andtheinverseofβin(10)representsthesystemorcomputationaltemperatureT.
In the FCM clustering, note that any data can belong to any cluster, the grand partition
functioncanbewrittenas
n c
Ξ= ∏∏(1+e−αk−βdik), (31)
k=1i=1
which, fromthe relationship F = −(1/β)(logΞ−α ∂logΞ/∂α ), gives the Helmholtz free
k k
energy (cid:4) (cid:5)
F=−1β ∑n ∑c log(1+e−αk−βdik)+αk . (32)
k=1 i=1
TheinverseofβrepresentsthesystemorcomputationaltemperatureT.
4.3.CorrespondencebetweenFermi-Diracstatisticsandfuzzyclustering
In the previoussubsection, we have formulated the fuzzy entropy regularized FCM as the
DA clustering and showed that its mechanics was no other than the statistics of a particle
system(theFermi-Diracstatistics). Thecorrespondencesbetweenfuzzyclustering(FC)and
the Fermi-Diracstatistics (FD) are summarized in TABLE 1. The major differencebetween
fuzzy clustering and statistical mechanics is the fact that data are distinguishable and can
belong to multiple clusters, though particles which occupy a same energy state are not
distinguishable.Thiscausesasummationoramultiplicationnotonlyonibutonkaswellin
fuzzyclustering. Thus,fuzzyclusteringandstatisticalmechanicsdescribedinthispaperare
notmathematicallyequivalent.
• Constraints: (a)Constraintthatthesumofallparticles NisfixedinFDiscorrespondent
withthenormalizationconstraintinFC.Energylevellisequivalenttotheclusternumber
Fuzzyc-MeansClustering,EFnutrzopzyyM cax-iMmizeaationn,sa nCdlDuestetremrininistgic,a EndnStimroulpatyed MAnanexailimngization, and Deterministic and Simulated Annealing7 277
Fermi-DiracStatistics FuzzyClustering
Constraints (a)∑nl=N(b)∑(cid:7)lnl=E (a)∑c uik=1
l l i=1
DistributionFunction (cid:6)nl(cid:7)=(cid:8)eα+β1(cid:7)l+1 uik= eαk+β1dik+1
Entropy S=−kB∑ (cid:6)Nνl(cid:7)log(cid:6)Nνl(cid:7) SFE=−∑n ∑c {uikloguik
(cid:8)+(cid:6)1−(cid:6)Nνll(cid:7)(cid:7)logl(cid:6)1−(cid:6)Nνll(cid:7)(cid:7)(cid:9) +(1−ukik=)1lio=g1(1−uik)}
l l
Temperature(T) (given) (given)
(cid:10) (cid:11) (cid:10) (cid:11)
PartitionFunction(Ξ) ∏ 1+e−α−β(cid:7)l ∏n ∏c 1+e−αk−βdik
(cid:4) l (cid:5) k(cid:4)=1i=1 (cid:5)
FreeEnergy(F) −β1 ∑log(1+e−α−β(cid:7)l)+αN −β1 ∑n ∑c log(1+e−αk−βdik)+αk
l k=1 i=1
Energy(E) ∑l eα+β(cid:7)(cid:7)ll+1 k∑=n1i∑=c1eαk+βddiikk+1
Table1.CorrespondenceofFermi-DiracStatisticsandFuzzyClustering.
i. Inaddition,thefactthatdatacanbelongtomultipleclustersleadstothesummationon
k.(b)ThereisnoconstraintinFCwhichcorrespondstotheconstraintthatthetotalenergy
equalsEinFD.Wehavetominimize∑nk=1∑ci=1dikuikinFC.
• Distribution Function: In FD, (cid:6)n (cid:7) gives an average particle number which occupies
l
energylevell,becauseparticlescannotbedistinguishedfromeachother.InFC,however,
dataaredistinguishable,andforthatreason, u givesaprobabilityofdatabelongingto
ik
multipleclusters.
• Entropy:(cid:6)N(cid:7)issupposedtocorrespondtoaclustercapacity. ThemeaningsofSandS
l FE
willbediscussedindetailinthenextsubsection.
• Temperature:Temperatureisgiveninbothcases1.
• Partition Function: The fact that data can belong to multiple clusters simultaneously
causestheproductoverkforFC.
• FreeEnergy: HelmholtzfreeenergyFisgivenby−T(logΞ−α ∂logΞ/∂α )inFC.Both
k k
SandS equal−∂F/∂Tasexpectedfromstatisticalphysics.
FE
• Energy:TherelationshipE=F+TSorE= F+TS holdsbetweenE,F,TandSorS .
FE FE
4.4.MeaningsofFermi-Diracdistributionfunctionandfuzzyentropy
In the entropy function (28) or (30) for the particle system, we can consider the first term
to be the entropy of electrons and the second to be that of holes. In this case, the physical
limitationthatonlyoneparticlecanoccupyanenergylevelatatimeresultsintheentropythat
formulatesthestateinwhichanelectronandaholeexistsimultaneouslyandexchangingthem
makes no difference. Meanwhile, what correspondto electronand hole infuzzy clustering
are the probability of fuzzy event that a data belongs to a cluster and the probability of its
complementaryevent,respectively.
Fig.2showsatwo-dimensionalvirtualclusterdensitydistributionmodel. Alatticecanhave
atmostonedata.LetM bethetotalnumberoflatticesandm bethenumberoflatticeswhich
l l
1IntheFCM,however,temperatureisdeterminedasaresultofclustering.
278 8Simulated Annealing – Advances, Applications and Hybridizations Will-be-set-by-IN-TECH
haveadatainit(markedbyablackbox). Then,thenumberofavailabledivisionsofdatato
latticesisdenotedby
W =∏ Ml! (33)
m !(M −m )!
l l l l
which, from S = k logW (the Gibbs entropy), gives the form similar to (30)[8]. By
B
extremizing S, we have the mostprobable distributionlike(25). In this case, as thereis no
distinctionbetweendata,onlythenumbersofblackandwhitelatticesconstitutetheentropy.
Fuzzyentropyin(8),ontheotherhand, givestheamountofinformationofweatheradata
belongstoafuzzyset(orcluster)ornot,averagedoverindependentdatax .
k
Changing a viewpoint, the stationary entropy values for the particle system seems to be a
requestforgivingthestabilityagainsttheperturbationwithcollisionsbetweenparticles. In
fuzzyclustering,datareconfigurationbetweenclusterswiththemoveofclustercentersorthe
change of clustershapes is correspondentto this stability. Letus representdata densityby
(cid:6)(cid:7). If data transfer from clusters Ca and Cb to Cc and Cd as a magnitude of membership
function, the transition probability from {...,Ca,...,Cb,...} to {...,Cc,...,Cd,...} will
be proportional to (cid:6)Ca(cid:7)(cid:6)Cb(cid:7)(1−(cid:6)Cc(cid:7))(1−(cid:6)Cd(cid:7)) because a data enters a vacant lattice.
Similarly, the transition probability from {...,Cc,...,Cd,...} to {...,Ca,...,Cb,...} will be
proportionalto(cid:6)Cc(cid:7)(cid:6)Cd(cid:7)(1−(cid:6)Ca(cid:7))(1−(cid:6)Cb(cid:7)). Intheequilibriumstate,thetransitionsexhibit
balance(thisisknownastheprincipleofdetailedbalance[19]).Thisrequires
(cid:6)Ca(cid:7)(cid:6)Cb(cid:7) = (cid:6)Cc(cid:7)(cid:6)Cd(cid:7) . (34)
(1−(cid:6)Ca(cid:7))(1−(cid:6)Cb(cid:7)) (1−(cid:6)Cc(cid:7))(1−(cid:6)Cd(cid:7))
Asaresult,ifenergyd isconservedbeforeandafterthetransition,(cid:6)C(cid:7)musthavetheform
i i
1−(cid:6)C(cid:6)iC(cid:7) (cid:7) =e−α−βdi (35)
i
orFermi-Diracdistribution
1
(cid:6)C(cid:7)= , (36)
i eα+βdi+1
whereαandβareconstants.
Consequently,theentropylikefuzzyentropyisstatisticallycausedbythesystemthatallows
complementary states. Fuzzy clustering handles a data itself, while statistical mechanics
handlesalargenumberofparticlesandexaminesthechangeofmacroscopicphysicalquantity.
ThenitisconcludedthatfuzzyclusteringexistsintheextremeofFermi-Diracstatistics,orthe
Fermi-Diracstatisticsincludesfuzzyclusteringconceptually.
4.5.TsallisentropybasedFCMstatistics
Ontheotherhand,U˜ andS˜satisfy
S˜−βU˜ = ∑n Z˜1−q−1, (37)
1−q
k=1
whichleadsto
∂S˜
=β. (38)
∂U˜
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M1
M3
M2
Figure1.Simplelatticemodelofclusters. M1,M2,...representclusters.Blackandwhiteboxrepresent
whetheradataexistsornot.
Equation(38)makesitpossibletoregardβ−1asanartificialsystemtemperatureT[19].Then,
thefreeenergycanbedefinedas
F˜ =U˜ −TS˜=−1 ∑n Z˜1−q−1. (39)
β 1−q
k=1
U˜ canbederivedfromF˜as
U˜ =− ∂ ∑n Z˜1−q−1. (40)
∂β 1−q
k=1
∂F˜/∂v˜ =0alsogives
i
v˜ = ∑nk=1u˜qikxk. (41)
i ∑nk=1u˜qik
5.Deterministic annealing
TheDAmethodisadeterministicvariantofSA.DAcharacterizestheminimizationproblem
of the cost function as the minimization of the Helmholtz free energy which depends on
the temperature, and tracks its minimum while decreasingthe temperatureand thus itcan
deterministicallyoptimizethecostfunctionateachtemperature.
Accordingto the principleof minimal freeenergyin statistical mechanics, the minimum of
the Helmholtz free energy determines the distribution at thermal equilibrium [19]. Thus,
formulatingthe DAclusteringas aminimization of(32) leadsto ∂F/∂v = 0at the current
i
temperature, and gives (10) and (11) again. Desirable cluster centers are obtained by
calculating(10)and(11)repeatedly.
In this chapter, we focusonapplication of DAto the Fermi-Dirac-likedistributionfunction
describedintheSection4.2.
5.1.LinearapproximationofFermi-Diracdistributionfunction
The Fermi-Diracdistributionfunction can be approximatedby linear functions. That is, as
showninFig.1,theFermi-Diracdistributionfunctionoftheform:
280 1S0imulated Annealing – Advances, Applications and Hybridizations Will-be-set-by-IN-TECH
1.0
f(x)
g(x)
f(x), g(x)[]0.5
0 -α-1 √(-α/β) -α+1
√(-αβ ) √(-αβ )
[x]
Figure2.TheFermi-Diracdistributionfunction f(x)anditslinearapproximationfunctionsg(x).
1
T
Tnew
x)] 0.5
f([ Δx
0
xnew x
[x]
Figure3.DecreasingofextentoftheFermi-Diracdistributionfunctionfromxtoxnewwithdecreasing
thetemperaturefromTtoTnew.
1
f(x)= (42)
eα+βx2+1
isapproximatedbythelinearfunctions
⎧ (cid:6) (cid:7)
⎪⎪⎪⎪⎪⎪⎨1.0 (cid:6)x≤ −ακ−1 (cid:7)
κ α 1 −α−1 −α+1
g(x)= − x− + ≤x≤ , (43)
⎪⎪⎪⎪⎪⎪⎩0.02 2 2 (cid:6)−ακ+1 ≤x(cid:7) κ
κ