Efficient Mining of Partial Periodic Patterns in Time Series Database

10 Pages·0.11 MB·English

by  Han J., Dong G., Yin Y.

Efficient Mining of Partial Periodic Patterns
in Time Series Database
Jiawei Han (cid:0) Guozhu Dong (cid:1) Yiwen Yin
SchoolofComputingScience DepartmentofComputerScienceandEngineering SchoolofComputingScience
SimonFraserUniversity WrightStateUniversity SimonFraserUniversity
[email protected] [email protected] [email protected]
Abstract whereeverypointintimecontributes(preciselyorapproxi-
mately)tothecyclicbehaviorofthetimeseries. Forexam-
Partial periodicity search, i.e., search for partial peri- ple,allthedaysintheyearapproximatelycontributetothe
odicpatternsintime-seriesdatabases,isaninterestingdata seasoncycle of theyear. A usefulrelated typeofperiodic
mining problem. Previous studies on periodicity search patterns,calledpartialperiodicpatterns,whichspecifythe
mainlyconsiderfindingfullperiodicpatterns,where every behaviorofthetimeseriesatsomebutnotallpointsintime,
pointintimecontributes(preciselyorapproximately)tothe havenotreceivedenoughattention. Anexamplepartialpe-
periodicity.However, partialperiodicityisverycommonin riodic pattern may state that Jim reads the Vancouver Sun
practice since it is more likely that only some of the time newspaper from 7:00 to 7:30 every weekday morning but
episodesmayexhibitperiodicpatterns. his activities at other times do not have much regularity.
Wepresentseveralalgorithmsforefficientminingofpar- Thus,partialperiodicityisalooserkindofperiodicitythan
tialperiodicpatterns,byexploringsomeinterestingproper- fullperiodicity,anditexistsubiquitouslyintherealworld.
tiesrelatedtopartialperiodicity,suchastheAprioriprop- Thepurposeofthecurrentpaperistofillthegapbyconsid-
ertyandthemax-subpatternhitsetproperty,andbyshared eringtheefficientminingofpartialperiodicpatterns.
mining of multiple periods. The max-subpattern hit set Most methods for finding full periodic patterns are ei-
property is a vital new property which allowsus to derive therinapplicableto orprohibitivelyexpensiveforthemin-
the counts of all frequent patterns from a relatively small ing of partial periodicpatterns, becauseof themixture of
subsetofpatternsexistinginthetimeseries. Weshowthat periodiceventsandnon-periodiceventsinthesameperiod.
mining partial periodicity needs only two scans over the Forexample, FFT(FastFourierTransformation)cannotbe
timeseriesdatabase,evenforminingmultipleperiods. The applied to mining partial periodicity because it treats the
performance study shows our proposed methods are very time-series as an inseparable flow of values. Some peri-
efficientinmininglongperiodicpatterns. odicitydetectionmethods candetectsomepartial periodic
patterns, but only if the period, and the length and timing
Keywords. Periodicitysearch, partialperiodicity,time-
ofthesegmentinthepartialpatternswithspecificbehavior
seriesanalysis,dataminingalgorithms.
areexplicitlyspecified.Forthenewspaperreadingexample,
weneed to explicitly specify details such as“find the reg-
ularactivitiesofJimduringthehalf-hourafter7:00forthe
1.Introduction periodof hours.” Anaiveadaptationofsuchmethodsto
(cid:4)(cid:6)(cid:5)
ourpartial periodicpattern miningproblem wouldbe pro-
Finding periodic patterns in time series databases is an hibitively expensive, requiring their application to a huge
importantdata miningtask with many applications. Many numberofpossiblecombinationsofthethreeparametersof
methodshavebeendevelopedforsearchingperiodicitypat- length,timing,andperiod.
ternsinlargedatasets[8].However,mostpreviousmethods Besides full periodicity search, there are many recent
on periodicitysearch are on mining full periodic patterns, studies on time series data mining: Most concentrate on
symbolicpatterns,althoughsomeconsidernumericalcurve
(cid:2) ResearchwassupportedinpartbyresearchgrantsfromtheNatural patterns in time series. Agrawal and Srikant [3] devel-
SciencesandEngineeringResearchCouncilofCanadaandtheNetworks
oped an Apriori-like technique [2] for mining sequential
ofCentresofExcellenceProgramofCanada
(cid:3) patterns. Mannilaet al.[10] consider frequentepisodesin
Partofthis workwas donewhile visiting SimonFraser University
duringhissabbaticalfromUniversityofMelbourne,Australia. sequences, where episodes are essentially acyclic graphs
1
of events whose edges specify the temporal before-and- oripropertyandthemax-subpatternhitsetproperty,andby
afterrelationalshipbutwithouttiming-intervalrestrictions. sharedminingofmultipleperiods. Themax-subpatternhit
Inter-transactionassociationrulesproposedbyLuetal.[9] set propertyisavitalnew propertywhichallowsto derive
are implication rules whose two sides are totally-ordered the counts of all frequent patterns from a relatively small
episodeswith timing-intervalrestrictions (onthe events in subset of patterns mined from the time series. We show
the episodes and on the two sides). Bettini et al. [5] con- thatminingpartialperiodicityneedsonlytwoscansoverthe
siderageneralizationofinter-transactionassociationrules: timeseriesdatabase,evenforminingmultipleperiods. The
these are essentially rules whose left-hand and right-hand performance study shows our proposed methods are very
sidesareepisodeswithtime-intervalrestrictions. However, efficient. The proposed methods are also robust that can
unlikeours,periodicityisnotconsideredinthesestudies. beappliedinavarietyofcasesincludingminingmultiple-
Similartoourproblem,theminingofcyclicassociation levelpartialperiodicityandminingpartialperiodicitywith
rules by O¨zden, et al. [12] also considers the mining of perturbationandevolution.
(cid:7)
somepatternsofarangeofpossibleperiods. Observethat The remaining of the paper is organizedas follows. In
cyclic association rules are partial periodic patterns with Section 2, conceptsrelated to partial periodicity are intro-
perfectperiodicityinthesensethateachpatternreoccursin duced. InSection3,methodsforminingpartialperiodicity
everycycle,with confidence.Theperfectnessinperi- in regard to both single and multiple periods are studied.
(cid:8)(cid:10)(cid:9)(cid:11)(cid:9)(cid:13)(cid:12)
odicityleadstoakeyideausedindesigningefficientcyclic InSection4, theimplementationofanovel data structure,
associationruleminingalgorithms: Assoonasitisknown namely the max-subpattern tree, for facilitatingthe count-
thatan associationrule does not hold at a particularin- ingofthehitmaximalpatterns,andthederivationoftheset
(cid:14)
stantoftime,wecaninferthat cannothaveperiodswhich offrequentpatternsfromthehitmaximalpatterns,arepre-
(cid:14)
includethistimeinstant. Forexample,ifthemaximumpe- sented. In Section 5, a comparison of the performance of
riodofinterestis anditisdiscoveredthat doesnot theproposedalgorithmsisreported.Weconcludeourstudy
(cid:15)(cid:17)(cid:16)(cid:19)(cid:18)(cid:21)(cid:20) (cid:14)
holdinthefirst timeinstants,then cannothaveany inSection6.
(cid:15)(cid:17)(cid:16)(cid:19)(cid:18)(cid:22)(cid:20) (cid:14)
periods. This idea leads to the useful “cycle-elimination”
strategyexplored in thatpaper. Sincereal lifepatternsare
2 ProblemDefinition
usuallyimperfect,ourgoalisnottomineperfectperiodicity
andthus“cycle-elimination”basedoptimizationwillnotbe
consideredhere. (cid:23) Assumethatasequenceof (cid:26) timestampeddatasetshave
AnApriori-likealgorithmhasbeenproposedformining beencollectedinadatabase. Foreachtimeinstant(cid:27),let(cid:28)(cid:30)(cid:29)
imperfectpartialperiodicpatternswithagiven(single)pe- beasetoffeaturesderivedfromthedatasetcollectedatthe
riod in arecent studyby twoofthe currentauthors[7]. It instant.Thus,thetimeseriesoffeaturesisrepesentedas,
isaninterestingalgorithmforminingimperfectpartialpe-
(cid:31)!
riodicity. However, withadetailedexaminationofthedata (cid:28) (cid:28) (cid:28)(cid:30)’
(cid:7)#" (cid:23)(cid:13)"%$(cid:10)$%$&" $
characteristicsofpartial periodicity,wefoundthatApriori
pruninginminingpartialperiodicitymaynotbeaseffective Let( betheunderlyingsetoffeatures. Wewillalsouse
asinminingassociationrules. the“don’tcare”character) ,whichcanmatchanysingleset
of features. We define a pattern as a non-
Ourstudyhasrevealedthefollowingnewcharacteristics * * *&-
empty sequence over (cid:7),. +(cid:10)W+%+e will use
ofpartialperiodicpatternsintimeseries: TheApriori-like . /0(cid:4)(cid:6)13254(cid:6)687(cid:6)9;:<4#)=7 >*8>
todenotethelengthof ,andwillsaythat istheperiod
propertyamongpartialperiodicpatternsstillholdsforany * >*?>
of the pattern . Let the -length of be the
fixed period, but it does not hold forpatternsbetweendif- * ( * * * -
numberof whichcontainslettersfrom . A(cid:7) +(cid:10)p+%a+tternwith
ferent periods. Furthermore, there is a strong correlation *@(cid:29) (
-length isalsocalledan -pattern. Moreover, asubpat-
amongfrequenciesofpartialpatterns. ( (cid:27) (cid:27)
tern of a pattern is a pattern
Themaincontributionsofthispaperareasfollows. We * * * - *(cid:17)A *(cid:17)A *@-A
such that and have(cid:7)th+(cid:10)e+%+same length, and (cid:7) +(cid:10)+%+for
considertheefficientminingofpartialperiodicpatterns,for * *@A *@A(cid:29)CB *@(cid:29)
every position where . For example, the pattern
asingleperiodaswellasforasetof periods. Wepropose (cid:27) *(cid:17)A(cid:29)ED )
F F isoflength anditisof -length (i.e.,itisa
several mining algorithms, by exploring some interesting )G4 7(cid:6)J8K L ( (cid:5)
4-patte"IrHn);andF F and aretwoofthe
properties related to partial periodicity such as the Apri- )M4 7N)O) )O) J8K (cid:4)(cid:13)P
subpatternsofF F "IH . H
)Q4 7#J8K
(cid:24) Itisimportanttopointoutthat[12]concentratesontheeliminationof Thefrequency co"(cid:22)uHntandconfidenceofapattern ina
*
candidateitemsetsfortheassociationruleminingalgorithm,althoughthe timeseries aredefinedas
(cid:28) (cid:28)R’
cycle-eliminationstrategydoesleadtoasmallreductiononthenumberof (cid:7)(cid:6)"(cid:10)$%$(cid:10)$(cid:22)"
patternswhenweprocessthetimeseriesfromlefttoright.
(cid:25) SUT F
Notethatamodifiedstrategy,wherewestopconsideringcertainpat- K%V(cid:6)WXK%(cid:26) WU(cid:26)\[(cid:17)/0*%9 >]4(cid:6)(cid:27)N>(cid:6)(cid:9)(cid:30)^_(cid:27),‘ba (cid:26)cJ
H(cid:17)Y H(cid:17)Z "
ternsassoonasthelengthofthetimeseriestobeprocessedisnotenough
d
tomaketheconfidencehigherthanthethreshold,canbeused. If isasingletonwewillomitthebrackets,e.g.,wewrite as .
e&f g(cid:17)h(cid:10)i h
2
thestring s istruein , 3 Methods for mining partial periodicity in
(cid:28) (cid:29)kjl(cid:22)jm (cid:28) (cid:29)kjl(cid:22)jmnjl(cid:21)j7=>
and (cid:7) +%+(cid:10)+ timeseries
SUT
S K(cid:10)V(cid:6)WUK%(cid:26) WX(cid:26)\[(cid:17)/o*(cid:10)9
H(cid:17)Z (cid:26) /0*%9 H(cid:17)aY H(cid:17)Z " In this section, we explore methods for mining partial
periodicity in a time series, proceeding from mining par-
where is the maximum number of periodsof length
a >*8> tial periodicity for a single given period to mining partial
contained in the time series (i.e., is the positiveinteger
a periodicity for a specified range of periods (i.e., multiple
such that ). Each segment of the
ap>*?>q^r(cid:26)s‘t/0atuv(cid:8)(cid:10)9%>*?> periods).
form , where , is called a
(cid:28) (cid:29)kjl(cid:22)jm (cid:28) (cid:29)kjl(cid:21)jmnjl(cid:22)j (cid:9)w^x(cid:27)y‘xa
periodsegm(cid:7) e+(cid:10)n+%t+. Wesayapattern istrue in
* * * - 3.1 Miningpartialperiodicityforsingleperiod
theperiodsegmentortheperiodsegmentm(cid:7) +(cid:10)a+%tc+ hes ,if,for
*
each position , either is or all the letters in occur
(cid:27) * (cid:29) ) * (cid:29) 3.1.1 Single-periodapriorimethod
in the set of features in the segment. Thus, if is a
(cid:27){z}| *@A
subpatternof , thenthesetofsequencesthatcanmatch Apopularkeyidea usedintheefficientminingofassocia-
* *
isasubsetofsequencesthatcanmatch . tion rules is the Apriori property discovered in [2]: If one
*@A
subset of an itemset is not frequent, then the itemset itself
cannotbefrequent. Thisallowsustousefrequentitemsets
Example2.1 Forexample,F isapatternofperiod ;its
)(cid:127)~ (cid:128) ofsize asfiltersforcandidateitemsetsofsize .
frequencycountin thefeatureseries F F F is2; (cid:27) (cid:27)\u5(cid:8)
and its confidence is , where 3 is the4#~m"IHa7(cid:6)x~imK(cid:10)u~mH nK(cid:10)uJ mber Interestingly,foreachperiod(cid:134) , thepropertysupporting
(cid:23) theApriori“trick”stillholds:
ofperiodsoflength 3.. Thefrequencycountof F in
4#~ 7(cid:6))
F 4(cid:6)~ J\7(cid:6)K F 4#~ 7 F(cid:13)F ~ isalso (cid:23) . "(cid:22)H Property3.1 [Apriorionperiodicity]Eachsubpatternof
"(cid:22)H(cid:6)" "IH H .
afrequentpatternofperiod isitselfafrequentpatternof
(cid:134)
Similartominingassociationrules[2],wesaythatapat- period(cid:134) .
tern is a frequent partial periodic pattern in a time se-
Theproofisbasedonthefactthatpatternsaremorerestric-
ries if its confidenceis larger thanor equal to athreshold,
tivethantheir subpatterns. Suppose isasubpatternofa
S . Theminingoffrequentpartialperiodicpatterns *@A
aC(cid:27)k(cid:26) (cid:26) frequentpattern . Then isobtainedfrom bychanging
in a tiH(cid:17)mZ e series is to discover, possibly with some restric- * *(cid:17)A *
somesetofletterstoasubsetor . Hence ismorerestric-
tions,allthefrequentpatternsoftheseriesforoneperiodor ) *
tivethan andthusthefrequencycountof isgreaterthan
arangeofspecifiedperiods. Morespecifically,theinputto *@A *@A
orequaltothatof . Thus isfrequentaswell.
miningincludes: * *(cid:17)A
An algorithm for mining partial periodic patterns for a
given fixed period based on this Apriori “trick” was pre-
(cid:129) Atimeseries(cid:31) .
sented in [7]. We include a simplied version here for the
sakeofcompleteness.
(cid:129) Aspecifiedperiod;orarangeofperiodsspecifiedbytwo
integers and . Algorithm3.1 [Single-periodApriori]Findallpartialpe-
(cid:130) (cid:132)U(cid:27){(cid:133)?(cid:132)
Z(cid:6)(cid:131) riodic patternsforagivenperiod satisfyingagiven con-
(cid:134)
(cid:129) Aninteger indicatingthattheratioofthelengthsof(cid:31) fidence threshold min conf in time-series (cid:31) , based on the
a
Aprioriproperty3.1.
andthepatternsmustbeatleast . Thiswillensurethat
a
thepatternsminedwouldbeofvaluetotheapplicationat Method.
hand.
1. Find ,thesetoffrequent1-patternsofperiod ,byac-
(cid:135) (cid:134)
cumula(cid:7)tingthefrequencycountforeach1-patternineach
Remark: Sometimes the derivation of the feature series
whole periodsegment and selectingamong them whose
fromtheoriginaldataseriesisquiteinvolved,andtheinter-
frequencycountisnolessthan S(cid:138)(cid:137) ,where
actionoftheperiodicpatternswiththederivationoffeatures a(cid:136)(cid:27)}(cid:26) (cid:26) a a
isthemaximumnumberofperiods. H@Z
mayleadtoimprovedperformance. Henceitisworthwhile
tocombinetheminingofthefeaturesfromthedatasetswith
2. Find all frequent -patterns of period , for from 2 up
(cid:27) (cid:134) (cid:27)
themining ofthepatterns, as isthe caseforthemining of
to ,basedontheideaofApriori,andterminateimmedi-
(cid:134)
cyclicassociationrules[12]. Forourworkontheminingof
atelywhenthecandidatefrequent -patternsetisempty.
(cid:27)
frequentpartialperiodicpatternsthough,thisinteractionis
notusefulforachieving computationaladvantageandthus Analysis.
we will assume that we are dealing with the feature time Number of scans over the time series. Step 1 of the
seriesinourstudy. algorithmneedstoscanthetimeseries(cid:31) once.Step2needs
3
toscan(cid:31) upto timesintheworstcase. Thusthetotal Obviously, the derivation of frequent -patterns is still
(cid:134)(cid:139)2(cid:140)(cid:8) (cid:8)
numberofscansisnomorethantheperiod . an effective way to dramatically reduce the candidate set
(cid:134)
Spaceneeded.(1)AtStep1,supposethereexistatotalof tobeexaminedlaterbecausethereareusuallyonlyasmall
S distinctfeaturesatpositions in(cid:31) , number of features being frequent at a particular position
(cid:29) (cid:27) (cid:134)cuR(cid:27) /oa(cid:141)2(cid:142)(cid:8)(cid:10)9(cid:143)(cid:134)cuR(cid:27)
where isthenumbersuchthat" "(cid:10)$%$(cid:10)(cid:31)$@" . We but there couldbe alarge number offeaturesappearing in
a aR(cid:134)(cid:144)^(cid:145)> >(cid:11)‘s/0a(cid:141)u(cid:136)(cid:8)(cid:10)9{(cid:134)
need - S unitsofspace toholdthecounts.Intheworst theposition. Thisisespeciallytruewhentheaveragenum-
(cid:146) (cid:29)(cid:143)(cid:147) (cid:29) (cid:148)
casewhen(cid:7) every featureisdistinctintheentiretimeseries beroffeaturesperpositionislargerthan . Thus
¥ (cid:16)N(cid:29)’ (cid:7)ƒo§o’(cid:11)¤U'
(cid:31) , we need j(cid:149)\j units of space . After Step 1, we ourdiscussionwillbefocusedonhowtoreducethesearch
(cid:146) (cid:29)(cid:143)(cid:147) >(cid:28)R(cid:29)(cid:17)> P
onlyneed >(cid:135) > un(cid:7)itsofspacetokeep(cid:135) ,thesetoffrequent effortafterthesetoffrequent (cid:8) -patterns,(cid:135) ,isfound.
(cid:8) -patterns in(cid:7) (cid:31) . (2) At Step 2, the m(cid:7)aximum(cid:150) number of Ourkeyideaisbasedonthenotionsofm(cid:7) ax-patternsand
candidatesubpatternsthatwemaygenerateis >(cid:135) > hitpatterns,definednext.
(cid:7) u
(cid:150) (cid:150) (cid:4) (cid:151) A candidate (frequent) max-pattern, “(cid:159)(cid:16)(cid:19)(cid:18)(cid:22)(cid:20) , is the
>(cid:135) > >(cid:135) > j(cid:152)\(cid:153)&j .Considering maximalpatternwhichcanbegeneratedfrom ,thesetof
(cid:7) u u (cid:7) (cid:4) 2C>(cid:135) >(cid:154)2(cid:138)(cid:8) (cid:135)
(cid:128) (cid:151) +%+%+ >(cid:135) > (cid:151) (cid:7) frequent -patterns. For example, if thefrequen(cid:7)t1-pattern
thatwe stillneed (cid:7)space to keepthe set of frequent 1- (cid:8)
>(cid:135) > set is F , thecandidate
patterns,thetotalamo(cid:7) untofspaceneededis j(cid:152) (cid:153) j inthe 4 )O)(cid:139))(cid:139)) )(cid:13)~,)(cid:139))(cid:13)) )«) )(cid:139)) )O)O)(cid:11)J8)87
(cid:4) 2p(cid:8) max-patternisF " .No"ticetHhata"positioninthecandidate
worsecaseinthiscomputation. However, theaveragecase ~ J8)
max-patternmayHbeallowedtohave adisjunctionofmore
should be much smallerthan the worst casesince if every
thanonenon- letter.Forexample,ifthefrequent1-pattern
featureisdistinctinthetimeseries,thenthereisnoneedto )
setis F ,the
find periodic patterns. The existence of any periodicity in 4 )(cid:139))O)«) )(cid:13)~ )O)(cid:13)) )(cid:11)~ )«)(cid:13)) )O) )«) )G)(cid:139))(cid:13)J8)87
candidatemax-p"atte(cid:7)rnisF" (cid:23) " . H "
thetimeserieswillreducethememoryneeded. 4#~ ~ 7 J8)
(cid:7)(cid:6)" (cid:23) H
Letthe -lengthofthecandidatemax-pattern, ,be
( “(cid:159)(cid:16)(cid:19)(cid:18)(cid:22)(cid:20)
. A subpattern of is hit in a period segment
3.1.2 Single-periodmax-subpatternhitsetmethod >“(cid:159)(cid:16)(cid:19)(cid:18)(cid:22)(cid:20)X> “(cid:159)(cid:16)G(cid:18)(cid:22)(cid:20)
(cid:31) of (cid:31) if itisthemaximalsubpatternof in (cid:31) . For
(cid:29) “(cid:159)(cid:16)G(cid:18)(cid:22)(cid:20) (cid:29)
Althoughthe Aprioritrick mayreduce thesearch spacein example, for F , the hit subpatternfor
“ (cid:16)G(cid:18)(cid:22)(cid:20) 4#~ ~ 7 J8)
partial periodicity mining in a similar way as association aperiodsegment (cid:31) F (cid:7)(cid:10)" (cid:23) H is F
(cid:29) 4#~ ~ 7 4#J J 7#K 4(cid:6)~ ~ 7(cid:19))
rulemining,itisimportanttonotethatthedatacharacteris- , because it is true in (cid:31) (cid:7)#"an(cid:23)dH%n(cid:7)one(cid:7)(cid:10)"of(cid:23) its superp(cid:7)(cid:10)a"tte(cid:23) rns
)(cid:13)) (cid:29)
ticsin thetwocases arevery different. Inminingassocia- F ,F ,andF ,isin(cid:31) . The
4#~ ~ 7 )‹) 4(cid:6)~ ~ 7?)\J?) 4(cid:6)~ ~ 7 J8) (cid:29)
tionrules,thenumberoffrequent -itemsetsshrinksquickly hits(cid:7)(cid:10)e"t,(cid:23) H,ofatim(cid:7)(cid:10)e" s(cid:23)eries(cid:31) isthese(cid:7)(cid:10)t" o(cid:23)faHllhitsubpatterns
(cid:27) ›
as increasesbecauseofthesparsityoffrequent -itemsets of in(cid:31) .
(cid:27) (cid:27) “(cid:159)(cid:16)(cid:19)(cid:18)(cid:21)(cid:20)
in a large transaction database. However, in mining par-
Theusefulnessofhitmax-patternsis: Wecanderivethe
tialperiodicity,veryoftenthenumberoffrequent -patterns
(cid:27) completesetofpartialperiodicpatterns,fromthefrequency
shrinksslowly(when )as increases. Theslowspeed
(cid:27),(cid:155)v(cid:8) (cid:27) countsofallthehitmaximalsubpatternsof .Thiswill
ofdecreaseinthenumberoffrequent -patternsisduetoa “ (cid:16)(cid:19)(cid:18)(cid:22)(cid:20)
(cid:27) bedetailedbelow.
strongcorrelationbetweenfrequenciesofpatternsandtheir
We would like to give an estimate of the buffer size
subpatterns.Wenowillustratethispoint.
needed in computation based on the idea of hit patterns.
Example3.1 Supposewehavetwofrequent1-patterns,F ) One upper bound of the buffer size is estimated in terms
and )(cid:13)~ , such that (cid:26) S / F )(cid:11)9 (cid:9) and (cid:26) S /0)(cid:11)~(cid:17)9 (cid:9) , of , the total number of periods in (cid:31) . , the size of
in a time-series (cid:31) .H(cid:17)Z Then it must$(cid:157)(cid:156)be theH(cid:17)Zcase that (cid:9) $(cid:157)^(cid:156) theahit set in a time series (cid:31) , should be no>›(cid:140)big> ger than ,
mH(cid:17)Z e(cid:26)nSts/ Fth~(cid:17)a9(cid:139)t^vma(cid:9)tc$(cid:157)(cid:156)h,Fas~ mexaptlcahinbeodthbeF l)owan.dS)(cid:11)in~ c,ea(cid:26)llS p/ eF r~(cid:17)i9(cid:159)od^_$(cid:157)(cid:158)se(cid:9) g- ic.aen.,g>›(cid:140)en>fier^flatea a.tTmhoisstiosnoebvhiiotussubspinactteerena,chanpderaiohditsseugbmpeaant-t
holds.Toderivetheotherinequality,letF(cid:160) dH(cid:17)eZnotethepred$(cid:157)i(cid:156)- ternmaybehitinmorethanoneperiodsegment.Theother
catethataletterisnotF ,similarly~(cid:160) .TheconfidenceofF(cid:160) ) in upperbound ofthebuffer sizeisestimatedin termsof the
(cid:31) isatmost (cid:9) (cid:8) ,because (cid:26) S /(cid:22)F(cid:160) )(cid:13)9 (cid:8)Q2 (cid:26) S / F )(cid:13)9 . Sim- maximalnumberofpatternsthatcanbegeneratedfrom ,
ilarly, (cid:26) S /o) ~%(cid:160)$ 9q^¡(cid:9) (cid:8) .SiH(cid:17)nZce (cid:26) S / F ~(cid:17)9(cid:159)¢v(cid:8),H(cid:17)Z2 (cid:26) S /(cid:22)F(cid:160) )(cid:11)9c2 the set of frequent 1-patterns. Since each hit pattern of(cid:135) (cid:31)(cid:7)
H(cid:17)Z (cid:26)TSh/oe)H(cid:17)Z~(cid:17)(cid:160)s9lo,witfroeldlouwctsio$thnaotfH(cid:17)Zt(cid:26)heS H(cid:17)/sZFe~(cid:17)t9Qo¢bf c(cid:9)a$n(cid:158) d.idateH(cid:17)Zfrequent - iilsaartsoubthpeatatenranlyosfis“(cid:159)p(cid:16)(cid:19)er(cid:18)(cid:22)f(cid:20)o,rmwhedichinisAglgeonreirtahtmed3f.r1o,mthe(cid:135) (cid:7)s,izseimo-f
(cid:27)
patternsas(cid:27) growsmakestheAprioripruningofAlgorithm t(cid:150)he set of subp(cid:150) atterns which can b(cid:150) e generated from (cid:135) (cid:7) is
3.1lessattractive.Isthereabetterway? >(cid:135) > >(cid:135) > >(cid:135) > j(cid:152) (cid:153) j .
(cid:7) u (cid:7) u u (cid:7) (cid:4) 2(cid:176)(cid:8)
£ (cid:8) (cid:151) (cid:4) (cid:151) +(cid:10)+%+ >(cid:135) > (cid:151)
Theunitofspaceisthespaceneededtoholdthefeatureidentifierand Therefore, , the size of the hit set (cid:7)in a time series (cid:31) ,
itsassociatedcount, anditssizeis usually2-8bytes, dependingon the >›(cid:140)>
shouldbenobiggerthan j(cid:152) (cid:153) j . Combiningbothupper
imp⁄lementation. (cid:4) 25(cid:8)
Thisisequaltothetotalspacethatthetimeseriesoccupies. bounds,wehave
4
Property3.2 [The bound of hit set] The size of the hit 2. Scan (cid:31) once. During the scan, for each period seg-
setisboundedbytheformula, j(cid:152) (cid:153) j , ment,ifitshitsetisnonempty,dothefollowing:addthe
>›(cid:140)>–^†aC(cid:27)k(cid:26)n4#a (cid:4) 25(cid:8)‡7
where isthetotalnumberofperiodsin (cid:31) , an" d isthe max-subpatternintothehitsetbuffer(withtheassociated
a (cid:135)
setoffrequent1-patterns. (cid:7) countinitializedto1)ifitisnotalreadythere;otherwise,
increasethecountofthemax-subpatternbyone. Thehit
Using this formula, we can calculate the bound of the setbufferisimplementedintheformofamax-subpattern
maximalbuffersizeneededintheprocessing:Giventheset tree,anoveldatastructure,tobediscussedinSection4.
offrequent1-patterns, , themaximal(additional) buffer
(cid:135)
size needed for registerin(cid:7) g the counts of all the maximal 3. After the scan, derive the frequent patterns from the hit
set. Wewilldiscusshowtoimplementthefindingofthe
subpatternsof is j(cid:152) (cid:153) j .
“ (cid:16)G(cid:18)(cid:22)(cid:20) aC(cid:27)k(cid:26)n4#a (cid:4) 25>(cid:135) >#2¡(cid:8)‡7
" (cid:7) countsofthehitpatternsandhowto usethesecountsto
Thispropertyisveryusefulinpractice. Forexample, if
derivethefrequentpatternsinSection4. Itturnsoutthat
we found 500 frequent 1-patterns when calculating yearly
bothcanbedoneefficiently.
periodic patterns for 100 years, the buffer size needed is
at most 100; on the other hand, if we found 8 frequent
Analysis.
1-patterns for calculating weekly periodicpatterns for 100
Numberofscansover the timeseries. Thefirststepof
years,thebuffersizeneededisatmost (cid:4)‡·O2 2s(cid:8) (cid:4)‡(cid:5)?(cid:181) . thealgorithmneedstoscan (cid:31) once. Thesecondstepneeds
Wecanalwaysselectthesmalleroneinestima(cid:158)tingthemax- to scan (cid:31) one more time. Thus the total number of time-
imalbuffersizeneededincomputation.
seriesscansis2,independentoftheperiod .
(cid:134)
Beforeturningtoourhit-setbasedalgorithm,weexam-
Space needed. (1) The space needed for Step 1 is the
inethe probabilitydistributionsofmaximal subpatternsof
sameasAlgorithm3.1. AfterStep1,weneed unitsof
>(cid:135) >
“(cid:159)(cid:16)(cid:19)(cid:18)(cid:22)(cid:20) . spacetokeep ,thesetoffrequent -patternsin(cid:7) (cid:31) . (2)At
(cid:135) (cid:8)
thesecondstep,(cid:7) suppose thereare frequent -patterns
>(cid:135) > (cid:8)
Heuristic3.1 [Popularity of longer subpatterns] The in (cid:31) . AccordingtoProperty3.2,thet(cid:7)otalspaceneededfor
probabilitydistributionofthemaximalsubpatternsof“ (cid:16)G(cid:18)(cid:22)(cid:20) the hit set is at most aC(cid:27)k(cid:26)n4#a (cid:4) j(cid:152) (cid:153) j 2(cid:176)(cid:8)‡7 , where a is the
is usually denser for longer subpatterns (i.e., with the ( - totalnumberofperiodsin(cid:31) . "
lengthcloserto )thantheshorterones.
>“(cid:159)(cid:16)(cid:19)(cid:18)(cid:21)(cid:20)–>
In comparison with Algorithm 3.1, Algorithm 3.2 re-
ThisheuristiccanbeobservedinExample3.1.Fromtheex- duces the total number of scans of the time series from
(cid:134)
ample,wehave T F8¶ F T F F (the length of the period) to 2, and it also uses much less
(cid:9) ^E(cid:134) ~%4(cid:6)a *(cid:10)WU~o(cid:134) [o[IK (cid:26)n/ ~(cid:17)9 ~#7•^
,but T F8$(cid:157)¶(cid:158) FZ T F F . Inmost buffer space in the computation in most cases. This can
(cid:9) (cid:134) ~%4(cid:6)a *%WU~o(cid:134) [o[IK (cid:26)n/ ~(cid:17)9 )=7(cid:138)^‚(cid:9) (cid:8)
ca$(cid:157)(cid:156)ses,theeZxistenceofashortmax-subpatternind$ icatesthat also be seen from the followingobservation: Suppose the
the nonexistence of some non- -letter, which reduces the hit subpattern for a period segment is F , which is not
) ~ J
chanceforthecorrespondingnon- letterpatternstoreach in the hit set yet. We need only one unHit space to reg-
)
highconfidence.Thuswehavetheheuristic. ister the string and its count 1. However, for the Apriori
This heuristics will imply that the number of nodes in technique, thecandidate2-patternsto begeneratedwillbe
the tree data structureof the next section is usually small. F F F , 3-patterns to
4 ~G)R) ) ) )(cid:142))(cid:13)J )(cid:13)~ ) )(cid:13)~,)RJ )(cid:30)) J\7
It is also useful for efficient buffer management: In order be gene"ratedH w"ill be "F H "F "F H , and the
4 ~ ) ~N)(cid:30)J ) J )(cid:11)~ J\7
to reducetheoverallcost ofaccess,the longersubpatterns 4-patternswillbe F ,H pl"usweh"avetHou"pdaH tethecount
4 ~ J\7
shouldbearrangedtobemoreeasilyaccessible(suchasput associatedwitheach oH fthem. Thus, itisexpected thatthe
inmainmemory)thantheshorterones. max-subpatternhitsetmethodmayhavebetterperformance
Wenowpresentamainalgorithmforminingpartialpe- inmostcases. Wewillcomparetheperformanceofthetwo
riodic patterns for a given period, which is based on the algorithmsinSection5.
discussionsabove.
3.2 Miningpartialperiodicitywithmultipleperi-
Algorithm3.2 [Max-subpatternhit-set]Findallthepar- ods
tialperiodicpatternsforagivenperiod inatime-series(cid:31) ,
(cid:134)
based on the max-subpattern hit-set, for a given min conf Mining partial periodicity for a given period covers a
threshold. goodsetofapplicationssincepeopleoftenliketomineperi-
Method. odicpatternsfornaturalperiods,suchasannually,quarterly,
monthly,weekly,daily,orhourly.However,certainpatterns
1. Scan (cid:31) once to find , the set of frequent 1-patterns may appear at some unexpected periods, such as every 11
(cid:135)
of period , using Step(cid:7) 1 of Algorithm 3.1. Form the years,orevery14hours.Itisinterestingtoprovidefacilities
(cid:134)
candidatemax-pattern, ,from . tomineperiodicityforarangeofperiods.
“ (cid:16)G(cid:18)(cid:22)(cid:20) (cid:135)
(cid:7)
5
Toextendpartial periodicityminingfromoneperiod to Algorithm3.4 [Shared mining of multiple periods]
multipleperiods,onemightwishtoextendtheideaofApri- Shared mining of all the partial periodicpatterns for a set
ori to computing partial periodicity among different peri- ofperiodsin agivenrangeofinterest, , in time-
(cid:134) (cid:134)X…
ods, that is, to use the patterns of small periods as fil- series(cid:31) ,withthegivenmin conf thresho(cid:7)(cid:10)l"(cid:10)d$%.$%$I"
(cid:134)
ters for candidate patterns of periods of the form for
„(cid:6)(cid:134) Method.
an integer . This will work if all frequent patterns
„s(cid:155)”(cid:8)
of period are frequent patterns of period . Unfortu- 1. Scan(cid:31) once,forallperiods intherangeofinterest,do
„(cid:6)(cid:134) (cid:134) (cid:134) ‰
nately,thisisnotthecase. Forexample,forthetimeseries thesameasStep1inAlgorithm3.2.
fF(cid:8)(cid:10)r»(cid:6)~oH(cid:5)mJ.FpS~aHurKptiFpa~olHspJeeFrti~hoHedK i,ccH(cid:17)opZna(cid:26)fitStde/0er)(cid:139)nncs)eoHftJ?hp9ree rsihoo(cid:8)(cid:10)dl»(cid:13)d(cid:4)(cid:4) ,aisasn(cid:9)fid$l(cid:128)[email protected](cid:26)fIofS rw/ cH eaJ8n9uds ie- (cid:134)Tpe(cid:7)hr"%ai$(cid:10)to$%di$(cid:22)s",(cid:134)(cid:134)‰f…o,)r,ufisailnnldgp(cid:135)etrh(cid:7)ie/`o(cid:134)ds‰sa9 m,(cid:134) t‰eheiSnsteettphoe1frfaarnsegqieuneonAftli1gn-otperairttehtsemtrn(3is..eo1.f,.
date partial periodicpatternsof period (cid:5) , we willmiss the Foreachsetoffrequent1-patternsofperiod(cid:134) ‰ ,formthe
parGtiaivlepnertihoadticwpeacttaenrnno)Gte) xHteJ n.dtheApriori“trick”tomul- candidatemax-pattern,“(cid:159)(cid:16)(cid:19)(cid:18)(cid:22)(cid:20)\/`(cid:134) ‰ 9 ,from(cid:135) (cid:7) /`(cid:134) ‰ 9 .
2. Scan(cid:31) once,forallperiods intherangeofinterest,do
tipleperiods,oneobviouswaytominepartialperiodicpat- (cid:134)(cid:11)‰
thesameasStep2inAlgorithm3.2.
ternsforarangeofperiodsistorepeatedlyapplythesingle-
periodalgorithmforeachperiodintherange. Asimilarprocesswhichwillnotbeexplainedindetail.
Analysis.
Algorithm3.3 [Looping over single period computa-
Numberofscansover the timeseries. Thefirststepof
tion]Findallthepartialperiodicpatternsforasetofperiods thealgorithmneedstoscan (cid:31) once. Thesecondstepneeds
inagivenrangeofinterest,(cid:134) (cid:134)X… ,inthetime-series(cid:31) , to scan (cid:31) one more time. Thus the total number of time-
withthegivenmin conf thres(cid:7)(cid:10)h"(cid:10)o$%l$%d$I."
seriesscansis2,independentoftheperiod .
(cid:134)
Method. Spaceneeded. Thetotalspacerequiredintheworstcase
issameasinAlgorithm3.3.
1. for each period in the range of interest (i.e.,
(cid:134)(cid:13)‰
), apply Algorithm 3.2 (“max-subpattern hit- Algorithm3.4exploressharedprocessingatminingpar-
(cid:134) (cid:134)U…
se(cid:7)(cid:6)t"%”$(cid:10))$%o$(cid:22)n" period . tial periodicityfor multiple periods. The advantage of the
(cid:134)(cid:13)‰
method is that we only need two scans of time series for
Analysis. mining partial periodicity for multiple periods. The over-
Numberofscansoverthetimeseries.Sinceeachperiod head of the method is thatalthough it reduces the number
willtake2scansofthetimeseries,thetotalnumberofscans of scans to 2, it will require more space in the process-
ofthetimeseriesis (cid:137) . ing of each scan than the multiplescan methodbecause it
(cid:4) „
Spaceneeded.Forcomputingpartialperiodicityforperi- needs to register the correspondingcounts for each period
odsfrom to ,thespacerequiredisbasicallythesumof (for ). However,sincethesharedfeatureswill
(cid:134) (cid:134) … (cid:134)(cid:13)‰ (cid:8)(cid:139)‘w´ˆ‘_„
spacefore(cid:7)ach . Noticethatthespacerequiredforinitial sharethespaceaswell(withcountsincremented),andthere
(cid:134) ‰
Step1computationisstill j(cid:149)\j intheworstcasesince should be many sharedfeaturesin periodicitysearch (oth-
the space once used in com(cid:146) p(cid:29)(cid:143)(cid:147)u(cid:7)ta>(cid:28)(cid:30)tio(cid:29)In> for period , can be erwise, why mining periodicity?), the space required will
(cid:134)(cid:13)‰
hardlyapproachtheworstcase.Therefore,itshouldstillbe
reinitialized and reused for computing other periods. But
weneedintotal … unitsofspacetokeepdiffer- anefficientmethodinmanycasesforminingpartialperiod-
ent sets of frequ(cid:146)en‰ t(cid:147) 1(cid:7) ->(cid:135)pa(cid:7) t/(cid:154)t(cid:134)(cid:13)e‰(cid:13)rn9%s> , where is the set of icitywithmultipleperiods.
(cid:135) /(cid:154)(cid:134)(cid:13)‰(cid:6)9
frequent -patternsin (cid:31) derivedforperio(cid:7)d . Similarly,it
(cid:8) (cid:134)(cid:11)‰
takesatmost … j(cid:152) (cid:153)}(cid:190)-%¿I(cid:192) j unitsofspaceto 4 Derivationofallpartialpatterns
(cid:146) ‰ (cid:147) aC(cid:27)k(cid:26)n4#a(cid:30)‰ (cid:4) 2!(cid:8)(cid:6)7
computeall,wher(cid:7)e isth"etotalnumberofperiods in
a ‰ (cid:134) ‰
(cid:31) . Inthis section, weexamine the implementationconsid-
erations of our proposed algorithms. Algorithm 3.1 is an
Algorithm 3.3 provides an iterative method for mining Apriori-likealgorithmwhichcanbeimplementedsimilarly
partialperiodicityformultipleperiods. However, whenthe asotherApriori-likealgorithmsforminingassociationrules
number of periods is large, we still need a good number (e.g. [2]). Algorithm 3.2 forms the basis for all the three
of scans to mine periodicity for multiple periods. An im- remainingalgorithmsandrequiresnewtrickstoachieveef-
provementtotheabovemethodistomaximallyexplorethe ficiency, and thus ourdiscussionisfocusedonits efficient
miningofperiodicityformultipleperiodsinthesamescan, implementation.
whichleadstothesharedminingofperiodicityformultiple Algorithm 3.2 consists of two steps: Step 1, scan the
periods,asillustratedbelow. time series once and find frequent 1-pattern set ; and
(cid:135)
(cid:7)
6
10
Step 2, scan the time series one moretime, collect the set
a{b1, b2}*d*
of the max-subpatterns hit in (cid:31) , and derive the set of fre-
quent patterns. The implementation of Step 1 is straight- ~a ~d
~b1 ~b2
forward and has been discussed in the presentationof Al- 50 40 32
0
gorithm3.1. However, Step2isnontrivialandneedssome *{b1,b2}*d* ab2*d* ab1*d* a{b1,b2}***
gooddatastructuretofacilitatethestorageofthesetofmax- ~a ~a ~a
~b1 ~b1 ~b2
subpatternshitin(cid:31) andthederivationofthesetoffrequent ~d ~b2
~b1
patterns. 8 ~b2 18 5 19~d 0 ~d 2
*b2*d* *b1*d* *{b1,b2}*** a**d* ab2*** ab1***
Anewdatastructure,calledmax-subpatterntree, isde-
signedto facilitatetheregistration of thehit countofeach
max-subpattern and derivation of the set of frequent pat- Figure 1. A max-subpattern tree to store the set of
terns,asillustratedinFigure1. Itsdesignisnowoutlined. max-subpatternshitinthetime-series.
Themax-subpatterntreetakesthecandidatemax-pattern
astherootnode,whereeachsubpatternof with
“(cid:159)(cid:16)(cid:19)(cid:18)(cid:22)(cid:20) “(cid:159)(cid:16)G(cid:18)(cid:22)(cid:20)
one non- letter missingis a direct child node of the root.
) 2. If the node is found, increase its count by 1. Other-
The tree expands recursively, according to the following wise,create(cid:131)anewnode (withcount1)anditsmissing
rules. A node , if containing more than 2 non-) letters, ancestor nodes(only tho(cid:131)seon thepath to , with count
mayhaveaseto(cid:131) fchildren,eachofwhichisasubpatternof 0), if any, and insert it (orthem) into the c(cid:131) orresponding
with one more non- lettermissing. Notice that anode
) place(s)ofthetree.
c(cid:131) ontaining only 2 non- letters will not have any children
)
Forexample,iftheveryfirstmax-subpatternnodefound
sinceeveryfrequent-1patternisalreadyin .Importantly,
wedonotcreate anodeif neitherthe node(cid:135) n(cid:7) oritsdescen- in (cid:31) is )(cid:13)~ )(cid:142)J8) for “(cid:159)(cid:16)(cid:19)(cid:18)(cid:22)(cid:20) F 4(cid:6)~ ~ 7R)(cid:138)J8) , we will
dant(s) containing more than 1 non- letter is hit in (cid:31)¯˜ . createthen(cid:7)ode)(cid:13)~ ),J8) (withcount1(cid:7) )",a(cid:23) ftercreatingtwo
) ancestornodes(wi(cid:7)thcount0): =F (which
Eachnodehasa“count”field(whichregistersthenumber 4#~ ~ 7?)‹J8)
istherootofthetree),and (cid:131)=(cid:7) (cid:7) " (cid:23) (whichis
of hits of the current node), aparent link (which is nil for )84(cid:6)~ ~ 7N)GJ8)
’s child, following the F (cid:131) l(cid:23)ink). T(cid:7)h"e(cid:23)node is
the root), and a set of child links; each child link points a (cid:160) )(cid:11)~ )(cid:139)J8)
childandisassociatedwithacorrespondingmissingletter. (cid:131)«(cid:7) ’schild,followingthe~(cid:160) link. (cid:7)
(cid:131)G(cid:23) (cid:23)
Alinkcanbenilwhenthecorrespondingchildhasnotbeen
Analysis.
hit.
Letthetotalnumberofnon- lettersin be . For
) “(cid:159)(cid:16)G(cid:18)(cid:22)(cid:20) (cid:26) ƒ
Notice that a non-) letter position of a max-subpattern amax-subpattern containing (cid:26)U¸ ((cid:26)U¸¯(cid:155)†(cid:8) )non-) letters,
inamax-subpatterntreemaycontainasetofletters,which weneedtofollow(cid:131) linkstofindthenodeandcreate
(cid:26) ƒ 2p(cid:26)U¸
matchesthesetoflettersatthepositioninaperiodsegment. atmost new nodesin theworstcase. There-
(cid:26) ƒ 2b(cid:26)U¸pu(cid:145)(cid:8)
Forexample,for“ (cid:16)(cid:19)(cid:18)(cid:21)(cid:20) =F 4(cid:6)~ ~ 7X)cJ8) ,themax-subpattern fore,thetimecomplexityofnodesearchandnodecreation
o)(cid:13)f) ,thaendpethrieosdegsemgemnetnwtiFll4#c~o(cid:7)(cid:10)n" t~(cid:7)(cid:10)r(cid:23)i"b7(cid:13)u4(cid:23) tH%e(cid:7)(cid:10)o"(cid:22)H(cid:17)n(cid:23)e7#cJ o. uK n(cid:148) titsoFth4#i~s(cid:7)(cid:10)n" ~o(cid:23)d7Ge.) swuibllpabtetelrensswtihllancre(cid:26)aƒt.e eAitlhseor, osinnlcye0enacohdein(swehrteinonitohfitms)axo-r
The update of the max-subpattern tree is performed as lessthan nodes,thetotalnumberofthenodesinthetree
(cid:26) ƒ
follows. is less than (cid:137) , where is the size of the hit set.
(cid:26) ƒ >›(cid:140)> >›(cid:140)>
Algorithm4.1 [Insertion in the max-subpattern tree]
Insertamax-subpattern foundduringthe scanof (cid:31) into Ingeneral,toinsertasubpatternweneedtobothlocate
themax-subpatterntree (cid:131) . thepositionandupdatethecountofthenodeifthenodeis
˘
found,orotherwiseinsertoneorseveralnewnodes.
Method.
Example4.1 Let Figure 1 be the current max-subpattern
1. Startingfromtherootofthetree,findthecorresponding
tree . Toinserta(max)subpatternF intothetree,we
nodebycheckingthemissingnon-) letterinorder. searc˘ hthetreestartingwiththeroot, ~ (cid:7) )c)(cid:13)) F .
“(cid:159)(cid:16)(cid:19)(cid:18)(cid:22)(cid:20) 4#~ ~ 7(cid:6))(cid:13)J8)
Forexample,foramax-patternnode inatreewith The first non- letter missing is and the seco(cid:7)n" d(cid:23) non-
)(cid:13)~ )(cid:11)J8) ) ~ )
theroot, “(cid:159)(cid:16)G(cid:18)(cid:22)(cid:20) F 4#~ ~ 7(cid:19))«J?) ,there(cid:7) aretwoletters,F letter missing is J . Thus we first(cid:23)follow the (cid:160)~ branch to
and~ ,missing. Then(cid:7)o"de(cid:23) canbefoundby(1)following nodeF ~ )(cid:127)J8) ,andthenfollowtheJ(cid:160) branch.Sin(cid:23)cethenode
theF (cid:23)link(markedas“ F ”inFigure1)to , F (cid:7) islocated,itscountisincrementedby1.
(cid:160) ˙ )84#~ ~ 7X)cJ8) ~ )G)(cid:13))
and then(2)followingthe ~(cid:160) linkto )(cid:13)~ )OJ8) ,(cid:7) a" s(cid:23)shown (cid:7)
inFigure1. (cid:23) (cid:7) Before discussing the derivation of the set of frequent
patterns,weneedtointroducetheconceptofreachablean-
¨ weshowsuchanodeh%(cid:201) (cid:25) (cid:0)˚(cid:0)(cid:17)(cid:0) usingadottedboxinFigure1. cestors.Sincethetraversalandcreationofthechildrenofa
7
nodeinthemax-subpatterntreefollowthenon- letterpo- (cid:129) scantreeTtofindfrequencycountsofthesecandi-
)
sitionorder,someoftheancestornodesofanodemaynot date patterns and eliminate the non-frequent ones.
bedirectlylinkedto anode. For example, in Figure1, the Noticethatthefrequencycountofanodeisthesum
node F is linkedto onlyone parent F but not of the count of itself and those of all of its reach-
)(cid:139))(cid:13)J8) ~ )OJ8)
the other F (note: this missing link is(cid:23) marked by a ableancestors. Ifthederivedfrequent -patternset
~ )(cid:139)J?) (cid:27)
dashedlinei(cid:7)ntheFigure). isempty,return.
Ingeneral, the set ofreachable ancestors ofa node
in amax-subpatterntree is thesetofallthe nodesin (cid:131) , 7
˘ ˘
whicharepropersuperpatternsof . Itcanbecomputedas Analysis.
ofonll“(cid:159)ow(cid:16)Gs(cid:18)(cid:22):(cid:20) (,1w)hdiecrhiviesaroluisgth(cid:131)ly(cid:16) thoefmpoiss(cid:131)istiinogn-lewttiesersdfirfofemre(cid:131)ncbea,s(e2d) shoLwenttihnethteotaanlanluymsisbeorfoAflgnoorni-th) mle4tt.e1r,sthine“tim(cid:16)Ge(cid:18)(cid:22)(cid:20)cobmep(cid:26)clƒe.xAitys
thesetoflinkedancestorsconsistsofthosepatternswhose forsearchinganodeislessthan . Sincethereareatmost
(cid:26)cƒ
omfinssoitn-glinlektetderasnfcoersmtorasparroeptheorsperepfiaxtteorfns(cid:131) w(cid:16) h,oasnedm(i3s)sitnhgelseet-t (cid:4) ’8((cid:209) in2¡cl(cid:26)cudƒ innogdaelslttohebmeigsesninegradteedscfernodmanthtse),maanxd-pthaettreernartereaet
˘
tersformapropersublist(butnotprefix)of(cid:131) (cid:16) . most >›(cid:140)> reachableancestorsin˘ ,where >›(cid:140)> isthesizeof
thehitset,theworstcasetimecomplexityforderivationof
Example4.2 We compute the set of reachable ancestors allthefrequentpatternsisO( (cid:137) ’ (cid:209) (cid:137) ),i.e.,propor-
(cid:26) ƒ (cid:4) >›w>
for a node )(cid:204))3)(cid:13)J8) in a max-subpattern tree with root tionalto ’ andthesizeofthehitset,butexponentialto(cid:26) ƒ
“(cid:159)(cid:16)(cid:19)(cid:18)(cid:22)(cid:20) F 4(cid:6)~ ~ 7O)RJ8) . The list of missingnon-) letters (i.e.,propHortionaltothesizeofthetreethatcanbegenerated
is ˝F ~ ~ . T(cid:7)h" us(cid:23) ,thesetoflinkedancestorsis(1)6(cid:160) (miss- by“ (cid:16)G(cid:18)(cid:22)(cid:20) ).Sinceaninfrequentnodewillreducethenumber
ingn"ot(cid:7)h"in(cid:23)Ig˛,whichistheroot);(2)F(cid:160) (i.e.,missingF ,which ofcandidatesto begeneratedin thefuturerounds,thereal
isthenode)84(cid:6)~ ~ 7;)(cid:127)J8K );and(3)F(cid:160) ~(cid:160) (i.e.,missingF ,then processingcostisusuallymuchsmallerthanthecostinthe
missing ~ , wh(cid:7)i"ch(cid:23) is the node )(cid:13)~ )R(cid:7) J8K ). The set of not- worstcase.
linkedanc(cid:7)estorsis: (corre(cid:23) spondingtothemissing
)(cid:13)~ )«J8)
letterpattern F(cid:160) (cid:160)~ ), F ~ (cid:7) )(cid:139)J8) (correspondingto (cid:160)~ ), F )O)(cid:13)J8) We illustrate how to derive the frequent „ -patterns for
(correspondingt(cid:23)o~(cid:160) (cid:160)~ (cid:23) ),andF ~ ),J?) (correspond(cid:7) ingto(cid:160)~ ). „(cid:138)(cid:155)v(cid:8) fromthemax-subpatterntree˘ .
Inotherwords,one(cid:7) ca(cid:23) nfollow(cid:7)thelinkswhosemarkisn(cid:23)ot
Example4.3 Let Figure 1 be the derived max-subpattern
J(cid:160) inorderedway(toavoidvisitingthesamenodemorethan tree ,and S(cid:210)(cid:137) . Wecantraversethemax-
once)andcollectallthenon- nodesreachedin . ˘ a(cid:136)(cid:27)}(cid:26) (cid:26) a (cid:5)8L
(cid:131) ˘ subpatterntreetoH(cid:17)Zfindallthefrequent -patternsfor
„ „—(cid:155)†(cid:8)
as follows. Starting at level 2, we have the followingfre-
Essentiallythereisatreetraversalforeachfixedpattern,
quentpatterns: (68), (68),
exceptthatwedonotvisitanodeanditsdescendantsifthe 4#)(cid:11)~ )(cid:159)J8) )(cid:13)~ )(cid:159)J8) )84#~ ~ 7(cid:211))(cid:159))(cid:13))
(47),F (119),(cid:23) F (92),(cid:7) F (84) .(cid:7) W" (cid:23)eshow
nodeisnotanancestorpatternofourcurrentpattern. )G)(cid:13)J8) ~ )(cid:159))(cid:13)) ~ )(cid:159))(cid:13)) 7
thederivationof (cid:23) (68) here:(cid:7)sincethelistofmiss-
Thederivationofthefrequent -patternsisperformedas )(cid:13)~ )(cid:139)J8)
„ inglettersinthisnod(cid:23) eis F ,itssetofreachableancestors
follows. ˝ ~
is 4 6(cid:160) , F(cid:160) , (cid:160)~ 7 , and thus its fr(cid:7)(cid:212)e˛quent count = 10 + 0 + 50 +
8 (itself) =(cid:7) 68. Since level-2 has no infrequent nodes, we
Algorithm4.2 [Derivation of frequent patterns from
search all the nodes at level-1 and have the followingfre-
max-subpattern tree] The derivation of the frequent -
„ quentpatterns: F (60),F (50) ;Sincethereis
patterns for all , given a max-subpattern tree , by an 4 ~ );J8) ~ ),J8) 7
„ ˘ onenodeinfrequent(cid:23),level-0(root(cid:7))hasnofrequentpatterns.
Apriori-liketechnique.
Noticealthoughweonlysavedonenodecomputationinthis
Method. case,itwillsavemuchmorewhenthetreeislargeandthere
aremoremissingnodes.
1. The set of frequent -patterns is derived in the first
(cid:8) (cid:135)
scanofAlgorithm3.2. (cid:7) Fromtheaboveexample,onecanseethattherearemany
frequent -patternswithsmall thatcanbegeneratedfrom
„ „
2. Themax-subpatterntree isderivedinthesecondscan
˘ amax-subpatterntree.Inpracticalapplications,peoplemay
ofAlgorithm3.2. Thesetoffrequent -patterns( )
„ „(cid:144)(cid:155)(cid:176)(cid:8) onlybeinterestedinthesetofmaximalfrequentpatterns
isderivedasfollows.
insteadofallfrequentpatterns,whereasetofmaximalfre-
for to do quent patterns is a subset of the frequent pattern set and
(cid:27)(cid:127)ˇ (cid:4) >(cid:135) > 4
(cid:7) every otherpatternin the setis asubpatternofan element
(cid:129) derivecandidatepatternswith -length fromfre- intheset. Forexample,ifthesetoffrequent pattern(for
( (cid:27) „
quentpatternswith -length by“ -way ) is F F F , the set of maximal
( /0(cid:27)@2C(cid:8)%9 /0(cid:27)(cid:22)u—(cid:8)(cid:10)9 „w(cid:155)(cid:213)(cid:8) 4 ~(cid:159))•) )(cid:11)~ ) ) ) ~ )=7
join”. frequentpatternsis" F H " . H " H
4 ~ )87
H
8
Ifauserisinterestedinderivingthesetofmaximalfre- of frequent1-patterns)are fora fixed , and they are con-
(cid:134)
quentpatterns, the MaxMineralgorithmdeveloped byBa- trolledbythechoiceofsomeappropriateconfidencethresh-
yardo[4]isagoodcandidate.Thesuccessofthisalgorithm old.Wefoundthatotherparameters,suchasthenumberof
stems from generatingnew candidatesby joining frequent featuresoccurringatafixedpositionandthenumberoffea-
itemsetsandlookinghead. However,itstillrequirestoscan tures in the time series, do not have much impact on the
(cid:31) up to period times in the worst case. The mixture of performanceresultandthus theyare not consideredin the
(cid:134)
max-subpattern hit set method and the MaxMiner can get tests.
rid of this problem and will be more efficient than pure
MaxMiner. The details of the new method will be exam- 5.2 Performancecomparisonofthealgorithms
inedinfutureresearch.
Figure 2 shows there is a significant efficiency gain by
5 Performancestudy max-subpatternhit-set over Apriori. In this figure, the
maximalpattern length(the maximal -length of frequent
(
In this section we report a performance study which partial periodic patterns) grows from to . The other
(cid:4) (cid:8)%(cid:9)
compares the performance of the periodicitymining algo- parameters are kept constant: and .
(cid:134) L(cid:13)(cid:9) >(cid:135) > (cid:8)(cid:10)(cid:4)
rithmsproposedin thispaper. Inparticular,wegiveaper- We run two sets of tests, one with the length of(cid:7) the time
formance comparison between the single-period Apriori series being and the other being . As
(cid:8)%(cid:9)(cid:13)(cid:9) (cid:9)(cid:11)(cid:9)(cid:13)(cid:9) L(cid:13)(cid:9)(cid:11)(cid:9) (cid:9)(cid:13)(cid:9)(cid:11)(cid:9)
algorithm (Algorithm 3.1) (or simply called Apriori), and we can see, the r"unning time of max-subpatte"rnhit-set
themax-subpatternhit-setalgorithm(Algorithm3.2)(or is almost constant for both cases, while Apriori is almost
simplyhit-set)appliedtoasingleperiod. linear. When MAX-PAT-LENGTH is , the gain by
¢
This comparison indicates that there is a significant max-subpatternhit-setover Aprioriisabo(cid:158) utdouble. We
gaininefficiencybymax-subpatternhit-setoverApriori. expectthisgainwillincreaseforlargerMAX-PAT-LENGTH.
Since there is more gain when applied to multiple pe-
riods by using max-subpatternhit-set, it is clear that Time (seconds) Apriori 500k
7000
max-subpatternhit-setisthewinner.
The performance study is conducted on a Pentium 166 6000
machinewith64megabytesmainmemory,runninginWin- 5000
dows/NT.TheprogramiswritteninMicrosoft/VisualC++.
4000
HitSet500k
5.1 TestingDatabases 3000
2000
Apriori 100k
Eachtesttimeseriesisasynthetictime-seriesdatabases
1000 HitSet100k
generated using a randomized periodicity data generation
Max-Pat-Length
algorithm. From a set of features, potentially frequent 1- 2 4 6 8 10
patternsarecomposed. Thesizeofthepotentiallyfrequent
1-patterns is determined based on a Poisson distribution. Figure 2. Performance gain when
These patterns are generated and put into the time-series MAX-PAT-LENGTHincreases:(cid:134) L(cid:13)(cid:9) , >(cid:135) > (cid:8)(cid:10)(cid:4) .
(cid:7)
accordingtoanexponentialdistribution.
Itisimportanttonotethat,thegainshowninFigure2is
LENGTH thelengthoftimeseries donebykeepingeverythinginmemory,andbyconsidering
(cid:149)
aperiod onlyone period. Ingeneral, thiswill be unlikelythecase,
(cid:134)
MAX-PAT-LENGTH themaximal -lengthof andmax-subpatternhit-setwillperformeven betterthan
(
frequentpatterns Aprioriforthefollowingreasons:
thenumberoffrequent1-patterns
>(cid:135) (cid:7) > (cid:129) In general, the time series of features may need to be
storedondisk,dueto factorssuchaseach maycon-
Table1.Parametersofsynthetictimeseries (cid:28)(cid:30)(cid:29)
tainthousandsoffeaturesandthelengthofthetimeseries
can be longer. When the time series is stored on disk,
The basic parameters used to generate the synthetic there would be a large amount of extra disk-IO associ-
databasesarelistedinTable1. TheparametersofLENGTH ated withApriori, but notwithmax-subpatternhit-set
(cid:149)
(thelengthoftimeseries)and (aperiod)areindependently since it only requires two scans. Even when the
(cid:134)
chosen. The parameters of MAX-PAT-LENGTH (the max- time series is not stored on disk, Apriori will need
imal -length of frequent patterns) and (the number to go over this huge sequence many more times than
( >(cid:135) >
(cid:7)
9
max-subpatternhit-set. Thusmax-subpatternhit-set periods, mining periodicassociation rulesbased onpartial
willbefarbetterthanApriori. periodicity,andquery-andconstraint-basedminingofpar-
tial periodicity [11]. We are studying these problems and
(cid:129) When there are a range of periods to consider, implementingouralgorithmsforminingpartialperiodicity
max-subpatternhit-set can find all frequent patterns inadataminingsystemandwillreportourprogressinthe
in two scans but Apriori will require many more future.
scans, depending on the number of periods and the
-length of the maximal frequent patterns. Hence
( References
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10