INTERNETTRAFFICDATAFLOWFORECASTBYRBF
NEURALNETWORKBASEDONPHASESPACE
RECONSTRUCTION
LuJinjun,WangZhiquan
1,2
1
(1.AutomationDepartment,NanjingUniversityofScience&Technology,Nanjing,210094,P.R.China;2.ModernEducationalTechnologyCenter,NantongVocationalCollege,Nantong,226007,P.R.China)Abstract:CharacteristicsoftheInternettrafficdataflowarestudiedbasedonthechaostheory.Aphasespacethatisisometricwiththenetworkdynamicsystemisreconstructedbyusingthesinglevariabletimeseriesofanetworkflow.Someparameters,suchasthecorrelativedimensionandtheLyapunovexponentarecalculated,andthechaoscharacteristicisprovedtoexistinInternettrafficdataflows.Aneuralnetworkmodelisconstruct-edbasedonradialbasisfunction(RBF)toforecastactualInternettrafficdataflow.Simulationresultsshowthat,comparedwithotherforecastsoftheforward-feedbackneuralnetwork,theforecastoftheRBFneuralnet-workbasedonthechaostheoryhasfasterlearningcapacityandhigherforecastingaccuracy.
Keywords:chaostheory;phasespacereconstruction;Lyapunovexponent;Internetdataflow;radialbasisfunc-tionneuralnetwork
CLCnumber:TP273 Documentcode:A ArticleID:1005-1120(2006)04-0316-07
INTRODUCTION
OnforecastingcommunicationInternettraf-ficdataflows,thetraditionalmethodadoptsmathematicalstatistics.Andthemodernmethodsareasfollows:themodelpredictionmethodbasedonFARIMA(p,d,q)
[2]
[1]
showinghowchaostheorycanpreditInternetda-taflows.Ref.[7]provedthatanInternetopera-tionflowischaracterizedbyselfsimilarity.Ref.[8]provedthattherewasaconsanguineousrela-tionbetweenselfsimilarityandchaos.Moreover,someeigenvalueshavethesamevalues,andanewmodelbytheseresultsisusedforstudyingInternetdataflows.BasedonTakenstheoremofareconstructionphasespaceintegratingwithcon-cretedataofInternetdataflows,thecorrelativedimensionandtheLyapunovexponentarecalcu-lated,andchaoscharacteristicsareprovedtoex-istinInternetdataflowsinthispaper.Then,aradialbasisfunction(RBF)neuralnetworkmodelofanInternetdataflowisconstructedandapre-dictionsampleoftheInternetdataflowisgiven.SimulationresultsshowthattheRBFneuralnet-workmodelofanInternetdataflowbasedonthechaostheory,comparedwithotherforward-feed-
,thefuzzyadaptivepre-
dictionmethodbasedonthefractionalconformi-tyslidingaveragemodel[3,4]andthemethodforpredictingInternetdataflowsusingfuzzyjudg-mentrules.Atthepresent,tacklingthetimeseriesproblemusingchaosdynamicsisahottopicanditisappliedtomanydomains,suchastheturbulentflow,thebiology,andtheeconomics,etc.
[6]
[5]
.ButthepapersforstudyingInternetdata
flowsbasedonthechaostheoryarefewandonlyremainonthestudyofcharacteristicsofInternetdataflows.Itinspiresustostudynewways
Foundationitems:SupportedbytheNationalNaturalScienceFoundationofChina(6037406);theNaturalScienceFoundationofJiangsuProvince(BK2004132);theSpecializedResearchFoundationfortheDoctoralProgramofHigherEducationofChina(2002088025).
Receiveddate:2006-04-26;revisionreceiveddate:2006-10-01 E-mail:ljj@mail.ntvc.edu.cnNo.4LuJinjun,etal.InternetTrafficDataFlowForecastbyRBFNeuralNetwork……317
backneuralnetworkmodels,hasafasterlearningspeedandhigherforecastingaccuracy.1.1 Determinationofthebestembeddeddimen-sionmandthebesttimedelay Inthepaper,astandardG-Palgorithmischosentodefinethebestembeddeddimensionm
[10]
andthebesttimedelay.Thestepsforcalcu-latingthecorrelativedimensionareasfollows:
Step1 Reconstructtheseriestomakethembey(t1),…,y(tN)withhighdimensionsbyusingthephasereconstructionmethod;
Step2 Calculatethecorrelativedimension;
1C(r,m)=Nlim2H(r-→∞N∑i,j=1
y(ti)-y(tj))
whereH(s)istheHeavisidefunction.
Step3 Inaproperrangeofr,C(r,m)isthepowerfunctionofr,i.e.C(r,m)=r
d(m)
N
1 PHASESPACERECONSTRUC-TIONOFINTERNETDATA
FLOW
Inthephasespacereconstruction,theevolvementofthearbitraryweightisfoundbyotherweightscorrelatingwitheachother.Thustheinformationofcorrelatingweightsishiddeninthedevelopmentprocessofoneofthearbitraryweights.Thehighdimensionphasespaceisre-constructedbyusingthephasespacereconstruc-tiontheory,andthenthedynamiccharacteristicsoftheformersystemsarereconstructedbythetimeserieswithasinglevariable.Phasespacere-constructionandthetechnologyofthephasespacewereproposedinRef.[9].Therearetwoimportantparametersinthephasespacerecon-struction,i.e.thebestembeddeddimensionm
.Thehomeomorphismbe-andthetimedelay
tweenthereconstructedphasespaceandthefor-mersystemisgiveninthefollowing.
IntheoriginaltimeseriesX1,X2,…,XN,ac-cordingtoTakensTheorem[9],thereisaproperembeddeddimensionm≥2d+1,anddisthecor-relativedimensionofthechaosattractor.Recon-structingthephasespaceRusingthetimedelayisasfollows
{Y(i)}={x(i),x(i+),x(i+2),…,
x(i+(m-1))} i=1,2,…,M
(1)
wheremistheembeddeddimension,thetimedelay,andY(i)theithphasedotinaphasespace.ThenumberofallphasedotsisM=N-(m-1),andthelastphasedotis
Y(M)=(XM,XM+,…,XN)
(2)
WiththeInternetdataflowevolvinginthem-Dphasespace,thederivedphasespaceisdif-ferentialhomeomorphismwiththechaosseriesinthesenseoftopologyequivalence,i.e.thetrackofthephasedotY(i)keepsthecharacteristicsofthechaosseriessystem,thecorrelativedimensiondandtheLyapunovexponentofthechaossystemthesame.m
(3)
;
Step4 Taketheembeddeddimensionmasavariable,andcontinuouslyincreasem.RepeatSteps(2,3)untildimensiond(m)doesnotvarywithm.Thenthederivedd(m)canbetakenasthecorrelativedimensionofthistimeseries.1.2 Calculationofcorrelativedimensioninac-tualInternetdataflow
TherepresentativeInternetdatapAugTL1smeasuredintheBelllabareusedinthepaper,whichcontain3142dataandthetimedelayde-[8]
finedbyaself-correlativefunction.
Atfirst,aself-correlativefunctionofchaostimeseriesisgiven.Thenthegraphofthecorrel-
ativefunctionwithrespecttotimeisdrawn.Accordingtothenumericalresults,whenthecor-relativefunctiondecreasestotheoriginalvalue1-1/e,theobtainedtimeisthetimedelayofthereconstructionphasespace.
ProgrammingandcalculatingusingS-plussoftware,theself-correlativemapoftheInternetdataflowpAugTL1sisshowninFig.1.Choosethetimedelayas“1”accordingtothecorrela-tivemethod,chooseproperrtocalculatethecor-relativeintegrationC(r),andmakethemapln(r)～ln(c(r,m)).ProgrammingandcalculatingusingMATLABsoftware,Fig.2isderived.InFig.2,differentcurvesrepresentthattheembed-deddimensionmtakesdifferentvalues,andtheir318TransactionsofNanjingUniversityofAeronautics&AstronauticsVol.23
slopesarethecorrelativedimensions.Itcanbeseenwhenthevalueoftheembeddeddimensionissmall,thedistancebetweenthecurvesbecomessmallandgoesparallel.Itindicatesthatthecor-relativedimensionstendtobeastablevalue.Thatis,thereexistsasaturationcorrelativedi-mension.Whenmisequalto9,theslopeofthecurvedoesnotincrease.Thenthelinearregres-sioncalculatingonln(r)～ln(c(r,m))istaken.Theslopeis4.1265,sothecorrelativedimensionofpAugTLlswouldbed=4.1265.Thisshowsthatatleastfivevariablesareneededtoexpressthesystem.
acteristicsofasystematictrackevolvementpro-cess,measuringsensitivitytoadependenceoninitialconditions.Exponentsymboliccompositioncandefinewhetherthesystematicevolvementcancausechaosphenomenaanddistinguishtypesofattractors.
AmongLyapunovexponents,thelargestex-ponentisthemostimportant.Oneisthatthenumberwhoseproductmultipliedbyitistherangeofpredictabletimelengthsinalongperiodevolvement.Theotheristhatthesystemischaotic,sothereisatleastonepositiveLyapunovexponent,
mappingtrackexponentdiverging
speedfromnearbyinitialconditions.
ThemainmethodsofcalculatingLyapunovexponentsarethedefinitionmethod,theJocobian
[5][11]
method,theWolfmethod,thesmalldata
[12]
methodandsoon.ThesmalldatamethodisadoptedtodefinethelargestLyapunovexponent.
Concretestepsofcalculatingareasfollows:
Step1 LetYbethetrackmatrixofthe
Fig.1 pAugTL1sself-correlativemap
phasespacereconstruction
Y=(Y1,Y2,…,YM)T
(4)
Itisreconstructedbyatimeseries{x1,x2,…,xn}withNlengthunits.Here,Yicanbede-rivedthroughEq.(1).
Theoriginaltimeseries{x1,x2,…,xn}ismadeafastFouriertransformation(FFT).ThenthemeanperiodPiscalculated.
Step2 Findthenearestneighborofeverypoint.MarkthenearestneighborasYjforpointYj.Itcanbeobtainedbytravellingthroughthe
Fig.2 ln(r)～ln(c,(r,m))map
minimumdistancebetweenallthepointsandthereferencepointsYj.Thiscanbeexpressedas
dj(0)=min‖Yj-Yj‖
Yj
ReconstructingthephasespaceaccordingtoTakensTheorem,mshouldsatisfym≥2d+1.Basedontheabovediscussion,thecorrelativedi-mensionoftheInternetdataflow
sequence
pAugTLlsdoesnotincreaseastheincreasingoftheembeddeddimensionafterm=9.Theembed-deddimensionhereistherequiredembeddedspacedimension.
1.3 CalculatingmethodofLyapunovexponent TheLyapunovexponentexpressesthechar-(5)
wheredj(0)istheminimumdistancefromthejthpointtothenearestneighborpointofthejthpoint,and‖Yj-Yj‖theEuclideannorm.
Step3 ForeverypointYjinthephasespace,calculatethedistancebetweenthetwonearestneighboringpointsaftertheithtimestepdj(i)
dj(i)=Yj+i-Yj+i i=1,2,…,min(M-j,M-j)(6)No.4LuJinjun,etal.InternetTrafficDataFlowForecastbyRBFNeuralNetwork……319
Step4 Foreveryi,y(i)istheaveragecal-culationoflndj(i)totheallj,thatis
1y(i)=lndj(i)(7)
qt∑j=1
whereqisnumberofallnon-zerodj(i).
Accordingtotheabovemethods,theInter-netdataflowtimeseriespAugTL1siscalculatedbyusingMATLABsoftwaretoobtaintheLya-punovexponentmap,asshowninFig.3.Thelineslopeisobtainedas0.035byapplyingthemethodofleastsquares.SotheLyapunovexpo-nentoftheInternetdataflowtimeseriesis0.035,whichshowsthatchaoscharacteristicsex-istintheInternetdataflowsequence.
signalinthelocalregions.ThemostcommonlyusedbasefunctionistheGaussfunction
‖x-ci‖
Ri(x)=exp- i=1,2,…m(8)
2i2wherexisann-Dinputvector;cithecenteroftheithbasefunction,havingthesamedimensionasx;theithperspectivevariable,decidingthewidthofthecenterofthisbasefunction;mthenumberofaperceptionunit;and‖x-ci‖thedistancebetweenxandci.
TheinputlayerimplementsthenonlinearmappingfromxtoRi(x)andtheoutputlayerim-plementsthelinearmappingfromRi(x)toyk,
Fig.3 Lyapunovexponentcalculationmap
i
2n
Fig.4 RBFneuralnetwork
i.e.
m
yk=
∑w
i=1
ik
Ri(x) k=1,2,…,r(9)
2 FORECASTOFINTERNET
TRAFFICDATAFLOWBASEDONRBFNEURALNETWORK
2.1 StructureofRBFneuralnetwork
AccordingtotheembeddeddimensionmofthechaostimeseriesoftheInternetdataflowde-rivedabove,andtakingm=9asthenumberofinputlayerknots,theRBFneuralnetworkismadeupofthreelayers,andthestructureisshowninFig.4.Theknotsoftheinputlayeronlydelivertheinputsignalstothehiddenlayer.Theknotsofthehiddenlayercanbeexpressedasara-diationfunctionlikeaGaussfunction,butthefunctionoftheoutputknotsisthesimplelinearfunction.
Thefunctionaryfunction(basefunction)ofthehiddenlayerknotswillrespondtotheinputwhereristhenumberoftheoutputknotsandwiktheweight.
2.2 LearningalgorithmofRBFneuralnetwork ThecenterofthebasefunctionofthehiddenunitnumberandtheweightsforRBFneuralnet-worksarealldecidedbylearning.Theorthogonalleastsquare(OLS)algorithmisavalidalgorithmofmanyRBFnetworklearningalgorithms
[13]
.In
theOLSalgorithm,onesampleisselectedeachtime,andtheincreasinggainoftheoutputvari-anceismaximized.Thenthebestunitnumberofthehiddenlayerisobtainedandoutputweightsarecalculated.BecausetheMISOstructureisusedinthecommonpredictionmodel,soassum-ingthat
T
y=[y(1),…,y(N)]isanexpectationout-
putseries;320TransactionsofNanjingUniversityofAeronautics&Astronautics
n
s
Vol.23
p=[p1,p2,…,pn]isahiddenlayeroutput
T
matrix,wherepi=[pi(1),pi(2),…,pi(N)],
Untilthensthstep,when1-
∑[err]
j=1
j
<,
1≤i≤n;w=[w1,w2,…,wn]Tisaweightoutputmatrix;E=[(1),…,(N)]Tanerrorseriesofalearnedsample.
pissingulardecomposed,p=C・A,Aisann×nordersingularmatrix
10
A=
0
Ciisorthogonal,so
CC=H
ementis
nT
theprocessisfinished,where0<<1isanal-lowableerrorbeingselected.Thusthecenterdot
iscanbedefined,andthentheoutputmatrixwcalculatedbyEq.(15).
2.3 RBFneuralnetworkpredictingmodelofIn-ternetdataflowtimeseries
121…
1323…
……10
1n2na(n-1)n
1
(10)
Topredicty(i)ofthetimeseries,theinput
vectorisdefinedasx(t)=[y(t-1),y(t-2),…,
T
y(t-n)],whichisasamplespaceofnsamples,wherenisahistoricaldatalength.SothewholeInternetstructureisequivalentasy(t)=f(y(t-1),y(t-2),…,y(t-n))。
Becausethecorrelativeextentofformerandlatterdataaredifferentinthedifferenttimese-ries,especiallyinthecomplicatedtimeseries,thepredictionresultsofdifferenthistoricaldatalengthsaredifferentandthehistoricaldatalengthsindifferentstagesaredifferent,too.Thispaperdoesnotadoptmethodsaccordingtoexperi-encesandtrial.Incontrast,itcutstheformerda-tabeforeprediction.ThenmakethepredictionwiththedifferenthistoricaldatabyusingtheRBFnetwork.Finally,choosethelengthoftheleastmeansquareerrortooptimizethehistoricaldatalengthtoenhancethepredictiveperformanceofthenetwork.
Cisann×nordermatrixandthecolumnvector
(11)
whereHisadiagonalmatrix,andthediagonalel-
hi=cc=
Tii
∑c(t)c(t) 1≤i≤n
i
i
i=1
(12)
ThesolutionbasedonOLSis
=H-1CTygorand
cTiyg=T 1≤i≤n
cici=gAw
(13)(14)(15)
Therecursionprocesstodefinethecenterdotofthehiddenlayerisasfollows:
Step1 Forany1≤i≤n,calculate
ci1=pi
g1=(c1)y/(c1)c1[err]=(g)(c)c/yy
search[err]i11=max[err]i1, 1≤i≤nandchoose c1=ci11=pi1ik-1,calculate
=cpi/(cc) 1≤j≤k
k-1
i
jk
Tj
Tjj
i1
i1
2
iTi11
T
i
iT
i
Ti
(16)(17)(18)(19)(20)
3 FORECASTSIMULATION
Accordingtotheabovementionedtheoryandmethod,theRBFneuralnetworkbasedontheOLSalgorithmisappliedtothetimeseriesfore-casting.A3142historicalInternetdataflowisusedtoforecastthe200futureInternetdataflow,asshowninFig.5.
Thelayernumberofaneuralnetworkmodel
isselectedtobethree.Theinputnumberis9andtheoutputnumberis1.Toavoidoverfitting,theallowableerrorcannotbetoosmallandtrainingsampledatacannotbetoolarge.Here,thebestembeddeddimensionm=9isselectedastheopti-malhistoricaldatalength,andtheallowableerrorissettobe0.01.2800networkdataareselected Stepkk≥2,forany1≤i≤n,i≠i1,…,i≠
(21)(22)(23)(24)(25)(26)ik
cik=pi-
∑c
j=1
ijkj
gik=(cik)Ty/(cik)Tcik[err]i1=(gik)2(cik)Tcik/yTy
search [err]=max{[err], 1≤i≤n,i≠i1,…,i≠ik-1}
k-1
i1k
andchoose ck=c=pik-ikk
∑cj=1ikjkj
No.4LuJinjun,etal.InternetTrafficDataFlowForecastbyRBFNeuralNetwork……321
precisioncannotbeenhancedbyincreasingtheunitnumberoftheneuralnetworkonthehiddenlayer.BecauseinitializationoftheBPnetworkisrandom,theoutputsaredifferent,whiletheout-putsoftheRBFnetworkarethesame.Here,thebestresultoftheBPnetworkisshowninFig.7.
Fig.5 HistoricalInternetdataflow
froma3142Internetdataflowasthetrainingsample.After1000timestraining,a200Inter-netdataflowisselectedasaforecastingsample,andthesimulationcurveisshowninFig.6(sam-pledataandforecastingdataarenormalized.).
Fig.7 BPforecastcurvesofInternetdataflow
Inaddition,althoughtheadaptivelearningratemomentumgradientBPalgorithmisadoptedtotrainthenetwork,thelearningspeedisstillslow.Takingtheexperimentasanexample,theRBFnetworkusesonly2swhiletheBPnetworkuses3min.Table1showsforecasterrorsof10dotsinRBFandBP.Itcanbeseenthatthemean
Fig.6 RBFforecastcurvesofInternetdataflow
squareerror(MSE)ofBPisabout1.39whilethatofRBFisabout0.73.
FromFigs5,6andTable1,itisconcludedthat,comparedwiththeBPneuralnetworkmethod,theRBFneuralnetworkmethodbasedonthephasespaceismoreaccurateandhasafasterspeed.
TheBPnetworkisusedtoforecastandselectdoublehiddenlayers.Samplesanddatacondi-tionsarethesameasthatinRBF,andtheallow-ableerroris0.001.Thefirstandthesecondlay-ersadopttwoneuralunitsbecausetheforecasting
Table1 ForecasterrorofRBFandBP
No.12345678910Realvalue91.888.0187.4986.284.7986.6593.79101.4108.84113.06ForecastvalueofBP92.9289.5986.2887.2886.7287.2395.12102.51107.31114.95BPrelativeerror/%1.221.80-1.381.252.280.671.421.09-1.411.67ForecastvalueofRBF91.988.6187.0186.4586.2687.1394.54102.01107.9113.72RBFrelativeerror/%
0.110.68-0.550.291.730.550.800.60-0.860.581.39
0.73
Root-mean-squareerrorofBP
Root-mean-squareerrorofRBF
322TransactionsofNanjingUniversityofAeronautics&AstronauticsVol.23
4 CONCLUSION
ThereconstructionphasespacetheoryisadoptedtostudythechaoscharacteristicsoftheInternetdataflow.ChaosphenomenaareprovedtoexistintheInternetdataflowbycalculatingthecorrelativedimensionandtheLyapunovexpo-nent.ThechaosInternetdataflowtimeseriesispredictedusingtheRBFneuralnetworkbasedontheOLSalgorithm.Forthemethod,thestruc-tureissimple,thelearningspeedisfast,theforecastingprecisionishigh,andtheachievedre-sultsaresatisfactory.BecausethereisnolocalminimumproblemintheRBFnetwork,theOLSalgorithmconstructsapproximatelyoptimalnet-worksautomatically,andsolvestheover-fitting
problemtoagreatextentandenhancesthegener-alizedabilityofthenetwork.However,forecast-ingofInternetdataflowsbasedonthechaosthe-oryisonlyaninitialattempt.WiththedeepstudyofchaosphenomenaofInternetdataflows,thecharacteristicvariablesm,andnumericalcalcu-lationwillbesimplified.Meanwhile,thefore-castingprecisionwillbeenhanced,thereliabilitywillbehigherandthespeedwillbefaster.References:
[1] GarrettMW,WilhingerW.Analysis,modelingand
generationofself-similarVBRvideotraffic[C]//ProceedingsSIGCOMM94.NewYork:ACMPress,1994:269-280.
[2] ChenBor-Sen,YangYu-Suarg,LeeBote-Kuen,et
al.FuzzyadaptivepredictiveflowcontrolofATM
networktraffic[J].CJJIEEETransactionsonFuzzySystems,2003,M11(4):568-581.
[3] LiuJiakun,JinZhigang,XueFei,etal.Trafficpre-dictionanditsapplicationusingFARLMAmodels[J].JournalofElectronicsandInformationTechnol-ogy,2001,23(4):403-407.(inChinese)
[4] ChenLiang,WangXiaofan,HanZhengzhi.Control-lingbifurcationandchaosinInternetcongestioncon-trolmodel[J].InternationalJournalofBifurcation&Chaos,2004,14(5):1863-1876.(inChinese)
[5] JoachimH,WernerL.Lyapunovexponentsfroma
timeseriesofacausticchaos[J].PhysicalReviewA,1989,39(4):2146-2152.
[6] LanWen.Abriefsurveyondynamicalsystem[J].
294.(inAdvancesinMathematics,2002,31(4):293-Chinese)
[7] LelandW.Ontheself-similarnatureofEthernet
traffic[J].IEEE/ACMTransactionsonCommunica-tionsonNetworking,1994,2(1):1-15.
[8] LuJinhu,LuJunan,ZhenShihua.Analysisandits
applicationofchaostimeseries[M].Wuhan:WuhanUniversityPress,2002.(inChinese)
[9] TakensF.Detectingstrangeattractorsinturbulence
[J].LectureNotesinMath,1981,898:366-381.
[10]GrassbergerP,ProcacciaI.Dimensionandentropy
ofstrangeattractorsfromafluctuatingdynamicsap-proach[J].Physica,1984,13D:34-35.
[11]WolfA,SwiftJB,SwnneyHL,etal.Determining
Lyapunovexponentsfromatimeseries[J].Physics
317.D,1985,16(2):285-[12]RosensteinMT,CollinsJJ,DelucaCJ.Apractical
methodforcalcultinglargestLyapunovexponentsindynamicalsystems[J].PhysicaD,1993,65(1):117-134.
[13]ChenS,CowanCFN,GrantPM.Orthogonal
leastsquareslearningalgorithmforradialbasisfunc-tionnetworks[J].IEEETransactionsonNeuralNet-works,1991,2(2):302-309.
基于相空间重构的网络流量RBF神经网络预测
陆锦军1,2 王执铨1
(1.南京理工大学自动化学院,南京,210094,中国;2.南通职业大学现代教育技术中心,南通,226007,中国)
摘要:应用混沌理论,分析了网络流量,用单变量的网络流量时间序列重构与网络动力系统等距同构的相空间,进而计算了实际网络的关维数和Lyapunov指数,并证实了网络流量存在混沌特性;据此建立了基于径向基函数(RBF)神经网络的模型,并对实际网络数据流进行了预测。仿真
结果表明,相对于其他前馈神经网络预测,基于混沌理论的RBF神经网络预测方法学习速度快,预测精度高。关键词:混沌理论;重构相空间;Lyapunov指数;网络流量;
RBF神经网络
中图分类号:TP273
基金项目:
国家自然科学基金(6037406)资助项目;江苏省自然科学基金(BK2004132)资助项目;高等学校博士学科点
专项科研基金(202088025)资助项目。
因篇幅问题不能全部显示，请点此查看更多更全内容