Analytical Study of Fractal Dimension Types in the Context of SPC Technical Paper. Noa Ruschin Rimini, Irad Ben-Gal and Oded Maimon

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1 Aalytcal Study of Fractal Deso Types the Cotext of SPC Techcal Paper oa Rusch R, Irad Be-Gal ad Oded Mao Departet of Idustral Egeerg, Tel-Avv Uversty, Tel-Avv, Israel Ths paper provdes a aalytcal study of the fractal deso types' propertes whe pleeted as otorg statstcs of the fractal SPC, as suggested by the Fractal SPC ethod. The purpose of ths study s to aalyze the characterstcs of the proposed otorg statstcs the cotext of the suggested fractal SPC, ad to explore the relatos betwee the fractal deso types' easures ad the otored process' characterstcs. The overall purpose of ths study s to provde gudeles o the establshet of a ufed otorg schee that tegrates all fractal deso fors to a sgle aoaly-detecto ad decso-ag tool. The deso of forato s proposed as the a otorg statstc of the fractal SPC ethod suggested ths wor. Cosequetly, a aalytcal study of ts propertes was already gve Rusch-R et al. 0. The followg Sectos, therefore, focus o a aalytcal study of the box coutg ad the correlato desos.. Aalytcal Study of the Box Coutg Deso I order to explore the propertes of box coutg deso the cotext of the suggested fractal SPC, we refer to the studes of the Asyptotc Equpartto Property AEP, as dscussed Cover ad Thoas 99. Accordg to Cover ad Thoas 99, The EAP property s foralzed the followg theore: f,,..., are..d. ~ p x, the log H probablty, p,,..., for large values of. Ths eables to dvde the set of all sequeces to two sets: the typcal set, where the saple etropy s close to the true etropy, ad the o-typcal set, whch cotas the other sequeces. Cover ad Thoas 99 provde the followg defto for the typcal set: The typcal set A wth respect to px s the

2 set of sequeces x, x,..., x wth the followg property: H H p x, x,..., x. Cover ad Thoas 99 state that ay property that s proved for the typcal sequeces wll the be true wth hgh probablty ad wll detere the average behavor of a large saple. We cla, ad later o show, that the propertes of the Box Coutg deso, the cotext of the fractal SPC proposed ths wor, ca be draw fro the propertes of the typcal set. We cotue, therefore, to descrbe the propertes of the typcal set, as explaed Cover ad Thoas 99: If x, x,...,, the H log p x, x,..., x H x A Pr{ A } for suffcetly large H A, where A deotes the uber of eleets the set A. A H for suffcetly large. I order to deostrate the correlatos betwee the typcal set ad the Box Coutg Deso propertes the cotext of the fractal SPC pleetato, we refer to equato 3. As explaed Rusch-R et al. 0, the coputato of the Box Coutg deso the cotext of suggested Fractal SPC, s accoplshed by coutg the uber of crcles a certa resoluto, occuped by data pots. The uber of occuped crcles s deoted. The Box coutg deso s the calculated accordg to equato 3, whe r. As descrbed Rusch-R et al. 0, accordg to the proposed appg ethod, every crcle of radus represets a specfc -legth subsequece of the orgal process. Cosequetly, deotes the uber of -legth subsequece types that appeared a process saple. Usg the ters related to the typcal set as defed above, we cla that for large values of, there exst the followg correlatos betwee the typcal set ad the Box Coutg Deso propertes: The typcal set A ca correspod to a process saple.

3 The sets of sequeces x, x,..., x defed for the typcal set ca correspod to the sets of -legth subsequeces exstg a process saple. A, whch deotes the uber of eleets set A, ca correspod to, whch deotes the uber of -legth subsequece types that appeared a process saple. The above correlatos allow us to coclude that the propertes of the Box Coutg deso could be derved fro the propertes o the typcal set. Specfcally, ths leads us to draw the followg sgfcat cocluso: The predcted value, whch deteres the predcted value of the Box Coutg deso, ca be straghtly derved fro the dstrbuto of data pots the process p x. As show, H A ~, where A deotes the uber of eleets the set A. cosequetly, oe ca coclude that H ~ 3 Ad that the cotrolled lts are defed by: H x H x We refer to the cases whch s ot large eough. These cases do ot fulfll the basc AEP assupto, ad therefor eed to be addressed separately. These cases are ore coplcated sce the followg propertes of the asyptotc case do ot apply:. I the asyptotc case, all -legth subsequeces have a equal occurrece probablty. Oppostely, cases s ot large eough, each -legth subsequece has ts ow occurrece probablty, whch ca be dfferet fro the other -legth subsequeces occurrece probabltes.. I the asyptotc case, the occurrece probablty of all -legth subsequeces equal to H. Ths does ot apply cases s ot large eough. I order to detere the relatos betwee the otored process characterstcs ad the expected value of the Box Coutg deso these cases, we refer to the ultoal dstrbuto. We deote the probablty of the occurrece of each type of -legth sub-sequeces p, as deoted Rusch-R et al. 0. The ultoal dstrbuto s defed as the jot dstrbuto of a set of rado

4 varables, whch are the uber of occurreces of the possble outcoes a sequece of ultoal trals. For a sequece of legth ad for types of outcoes, recall that the probablty ass fucto of the ultoal dstrbuto s: f! x x x,..., x ;, p,..., p p x, x,..., x p,..., p, whe x x!... x! We cla that the ultoal dstrbuto best fts to odel the uber of occurreces of each type of -legth sub-sequeces, each havg a probablty of occurrece p, a saple of sze, cases whe the -legth sub-sequeces are..d. ote that o-asyptotc cases, the subsequece legth s sgfcatly saller tha the saple sze,.e.,. The exact uber of -legth subsequeces a saple sze s, sce the subsequeces are detered by a 'sldg wdow'. Wthout loss of geeralty, ad for the sae of splcty, sce, we assue, heceforth, that ~. Cosequetly, we assue that the overall uber of -legth subsequeces a saple sze equals. Recall that we deoted the uber of category types a process by, hece, the uber of types of -legth sub-sequeces s deoted by. Accordgly, the probablty ass fucto the case of odelg the uber of occurreces of each type of -legth sub-sequece s coputed by: f! x x x,..., x p p p x x x p p ;,,...,,,...,,...,, whe x x!... x! ote that the expected value of the Box Coutg deso s detered by the value, whch deotes the expected uber of -legth subsequece types a process saple of sze. ote that the uber of occurreces of each type of - legth subsequece s ot drectly related to the coputato. Moreover, the specfc types of -legth subsequece that occur are ot reflected by the coputato. The relevat property s the su of dfferet -legth subsequece types that occurred a process saple. We cla, that order to predct the value of, as well as to pre-detere ts cotrol lts, oe eeds to base coputatos

5 o the ultoal dstrbuto propertes. We start by provdg gudeles for such coputato. The proble of pre-deterg ca be rephrased as follows: assug -legth sub-sequece types, each havg a probablty of occurrece p, a saple of sze. Oe eeds to copute the followg probabltes: p l l,,...,. The values of wth the hghest probabltes wll represet the predcted values. The cotrol lts are establshed based o the values wth probabltes that are saller tha the specfed type Ι error rate. I order to calculate p l, oe eeds to copute, based o the probablty ass fucto of the ultoal dstrbuto, probablty of all cobatos of -legth subsequeces resultg wth l types of dfferet -legth subsequeces, the su all probabltes to obta p l. the Ths coputato should be repeated for l,,...,. Ths procedure ca be descrbed by the followg algorth:. Italze: l=. For the coputato of p l, calculate, based o the probablty ass fucto of the ultoal dstrbuto, the probablty of all cobatos of - legth subsequeces a saple of sze, resultg wth l types of dfferet - legth subsequeces, the su all probabltes to obta p l. 3. If l < creet l l, ad go to step. If l= go to step Select the predcted values l by the ode of p l aog all l,,..., 5. Select cotrol lts of by the dstrbuto percetle of p l detered by a specfed pre-requred type Ι error rate. To llustrate stage of the algorth, we show a exaple of the procedure requred order to copute p,.e., the probablty of saples cosstg of exactly two types of -legth subsequeces. The frst step s to calculate, based o the probablty ass fucto of the ultoal dstrbuto, the probablty of the occurrece of a specfc par of -legth subsequece types, a saple of sze. The

6 secod step, s to repeat ths coputato for all pars of -legth subsequece types, wth j,.e. for a total of j! coputed probabltes. Ths procedure ca be descrbed as follows:! pars. The thrd step, s to su all! p, j j p, j, j j!!! p p j Accordgly, a slar procedure s requred order to detere p,..., l l,. We have show, that the value, whch defes the value of the Box Coutg deso, as well as ts cotrol lts, ca be pre-detered based o the ultoal dstrbuto propertes. We ephasze that ths procedure s coputatoally prohbted. The sgfcace of ths study les the fact that t defes the relatos betwee the Box Coutg deso's expected values ad the otored process' characterstcs. Moreover, t provdes the ablty to aalytcally detere the expected value of the Box Coutg deso, assug oe ows the dstrbuto of data pots the orgal process p x. ote, that the coclusos of ths study are restrcted to the cases whch the -legth sub-sequeces are..d. Further research should be doe order to provde gudeles to detere the expected value of the Box Coutg deso whe -legth sub-sequeces are ot..d.. Aalytcal Study of the Correlato Deso I order to explore the propertes of the correlato deso the cotext of the suggested fractal SPC, we refer to the studes of Grassberger ad Procacca 983. Grassberger ad Procacca 983 suggested ths fractal deso easure sce the coputato of exstg fractal deso easures, such as the box coutg deso, were very dffcult the cases D >. Accordg to Grassberger ad Procacca 983, the correlato deso s obtaed by cosderg correlatos betwee pots of a log te-seres. They deote the pots of the log te-seres

7 by } { t },...,,...,, where s a arbtrary but fxed te creet. They state that although ost pars, wth j wll be dyacally j ucorrelated pars of essetally rado pots, the pots le o the attractor. Therefore, they wll be spatally correlated. They suggest to easure the spatal correlato wth the correlato tegral C C l, defed accordg to: {uber of pars, j whose dstace s less tha }. They j state that the correlato tegral s related to the stadard correlato fucto F c r l by C d rc r. They suggest, that for F r j, j j 0 sall 's, C grows le a power D cor C ~, ad that D cor ca be tae as a ost useful easure of a fractal deso. I other words, Grassberger ad Procacca's defto of the correlato fractal deso D ca be descrbed by equato 5. cor I order to aalyze the propertes of the correlato deso the cotext of the suggested fractal SPC, we refer to Rusch-R et al. 0. As explaed, the coputato of the Correlato deso the cotext of suggested Fractal SPC s accoplshed by selectg as the radus of a crcle at a certa resoluto, the calculatg the correlato deso accordg to equato 5, whe. It s also explaed Rusch-R et al. 0, that the correlato deso, deoted by D cor, easures the probablty that two pots chose at rado wll be wth a radus dstace of each other,.e., easures the probablty for the occurrece of slar -legth sub-sequece types. I order to detere the relatos betwee the otored process characterstcs ad the expected value of the Correlato deso these cases, we refer, oce ore, to the ultoal dstrbuto. We deote the probablty of the occurrece of each type of -legth sub-sequeces p, as deoted Rusch-R et al. 0. The ultoal dstrbuto s defed Secto. As stated Secto, we cla that the ultoal dstrbuto best fts to odel the uber of occurreces of each type of -legth sub-sequeces, each havg a probablty of occurrece p, a saple of sze. The probablty ass fucto the case of odelg the uber of occurreces of each type of -legth sub-sequece s also gve Secto. ote

8 that the expected value of the Correlato deso s detered by the value C whe,.e., C, whch s derved by the uber of pars of slar -legth subsequece types that appeared a process saple of legth. We cla, that order to obta the predcted value C, as well as to pre-detere the cotrol lts, oe eeds to base coputatos o the ultoal dstrbuto propertes. We start by provdg gudeles for such coputato. The proble of pre-deterg C ca be rephrased as follows: assug havg a probablty of occurrece -legth sub-sequece types, each p a saple of sze, oe eeds to base coputatos o the followg ultoal probablty ass fucto:! x x p x, x,..., x p p,...,, whe x x!... x! The values of x, wth the hghest probabltes wll derve the p,..., predcted C value. Cosequetly, we suggest to base the coputato of the pre- detered value C o the coputato of the ea of a ultoal dstrbuto. The ea of a ultoal dstrbuto s gve by: E p,,.... Hece, the ea the case of odelg the uber of occurreces of each type of -legth sub-sequece s coputed by: E p,,.. C. Moreover, we suggest to base the coputato of the cotrol lts o the coputato of the varace of a ultoal dstrbuto. The varace of a ultoal dstrbuto s gve by: p - p,,.... Accordgly, the varace the case of odelg the uber of occurreces of each type of -legth sub-sequece s coputed by: p - p,,... ote that the coputato of C cosders the uber of pars of slar -legth subsequece types. We deote the uber of slar -legth subsequeces of type by s. Hece, C s coputed by: s! s s C.! s! We suggest that the value of s,,.. ca be predcted by E p,,... The descrbed above stateets lead us to suggest the

9 followg algorth order to copute C, for the cases whe the -legth subsequeces are..d.:. Italze: =. Copute E p 3. If < creet, ad go to step. If = go to step Copute C E C by: C E C E! E!! E E E E Sce: E b E E a be a... E E E... E cases s..d I order to copute Var C, recall: C! s s!! s s s s. Accordgly, we suggest the followg algorth order to copute the varace Var C for the cases whe the -legth sub-sequeces are..d.:. Italze: =. Copute Var p - p 3. If < creet, ad go to step. If = go to step Copute predcted Var C by: Var C 4 [ E Var ] Var Var cases Sce: s..d

10 Var a Var Var b b Var Var... Var... Var cases s..d Var Var Var cases s..d Var E Var E Var Var Var cases s..d Var [ E Var ] Var cases s..d The above algorths eable us to select the predcted value C as the coputed value of E C. Cotrol lts are coputed by C Var C, whe s detered by a specfed requred type Ι error rate. We have show, that the value C, whch defes the value of the Correlato deso, as well as ts cotrol lts, ca be pre-detered based o the ultoal dstrbuto propertes. The sgfcace of ths study les the fact that t defes the relatos betwee the Correlato deso's expected values ad the otored process' characterstcs. Moreover, t provdes the ablty to aalytcally detere the expected value of the Correlato deso, assug oe ows the dstrbuto of data pots the orgal process p x. ote, that the coclusos of ths study are restrcted to the cases whch the -legth sub-sequeces are..d. Further research should be doe order to provde gudeles to detere deso whe -legth sub-sequeces are ot..d. the expected value of the Correlato It s also portat to ote that the correlato deso dffers fro the other fractal deso types the fact that t does ot dvde the space to two desoal grd cells of sde sze r. I fact, t oly cosders whether the dstace betwee two pots s saller tha. Ths dfferece should be cosdered whe explorg the propertes of the correlato deso the cotext of the suggested fractal SPC. Whe dealg wth both forato ad Box Coutg deso types, ad selectg r as the radus of a crcle, havg the a-pror owledge of the exact locatos ad radus szes of crcles created by the crcle trasforato algorth, we esured all pots sde the sae grd fall to the sae crcle,.e. represet the sae -legth subsequece. However, ths s ot the case whe dealg wth the correlato deso. The Correlato deso cosders the dstace betwee two pots. Hece, specfc

11 requreets should be defed, order to esure that the coclusos regardg the propertes of the correlato deso the cotext of the suggested fractal SPC are vald. I order to esure that the suggested gudeles regardg the correlato deso are vald, t should be esured that all spatally correlated pots,.e., all pots wth a dstace whch s saller tha, fall to the sae crcle,.e. represet the sae -legth sub-sequece. Ths requres that the dstace betwee crcles wll be greater tha. Otherwse, spatally correlated pots ca be located two dfferet eghborg crcles. I such a case, our suggested coclusos are ot vald. I order to fulfll ths requreet, ad based o the deftos ad proof gve Appedx A, oe should choose as a fucto of the uber of category types so that t fulfls: 3 R D 3 s. 3. Aalytcal Study of Fractal Deso Types Propertes Coclusos I ths techcal paper, we have explored three types of fractal deso: box coutg deso, deso of forato, ad correlato deso, the cotext of the suggested fractal SPC. The three fractal deso types ad the relatos aog the were studed through a set of experets sulatg a set of real-world process scearos. These experets exaed the fractal deso types' sestvty to dfferet fors of aoales. We the provded a aalyss regardg the propertes of each fractal deso type, ad preseted gudeles for pre-deterg each fractal deso type's ea value ad cotrol lts based o the otored process characterstcs. Accordg to the results descrbed the prevous Sectos, ths Secto presets our coclusos regardg the relatos betwee all three fractal deso types ad provdes gudeles o how to tegrate all three deso types to a sgle schee, oe that eables both process otorg ad root cause aalyss. Table suarzes the coclusos regardg the fractal deso types' sestvty to dfferet fors of aoales, by presetg all real-world scearos exaed Rusch-R et al., ad scorg each fractal deso's sestvty to each process devato type. The scorg schee s preseted as follows: for each real-world process devato scearo, the fractal deso type's relatve sestvty

12 s gve scores -3, whe score 3 dcates that the fractal deso was the ost sestve deso type for the specfc scearo, etc. If the fractal deso s ot sestve at all to a certa type of process devato, ts score ths rubrc wll be 0. Followg are the deso types' sestvty results: Table : Fractal deso types' sestvty to dfferet process scearos Process Scearo Box Coutg Deso Iforato Deso Correlato Deso Marov Process 3 Patter-reoccurrg process 0 3 Marov Patter-reoccurrg 3 process Dstrbuto-based process 3 Marov process wth outlers Relyg o the uercal experets, as well as o the aalytcal study of fractal deso types, we coclude that geeral the forato deso outperfors both Box Coutg ad Correlato deso ters of zg both type ad type errors. Therefore, as descrbed the suggested fractal SPC fraewor preseted Rusch-R et al. 0, we suggest to utlze the forato deso as the a otorg statstc, yet to explot both Box Coutg ad Correlato desos for root cause aalyss whe a chagg pot s detected. evertheless, t should be oted, that ths decso should be based o the specfc real-world scearo ad otorg requreets. For exaple, f every process outler should dcate a chagg pot ad the plcatos of avodg a type error are uch ore severe tha that of type error, as ght occur, for exaple, the fraud detecto area, the Box Coutg deso should be cosdered as a a otorg statstc. For exaple whe ltg the dscusso to the above-etoed processes, oe ca use the type of out-of-cotrol sgals, to dcate whch type of devato the process occurred. Wheever the forato deso detects a chagg pot, oe ay use the followg gudeles for the purpose of root cause aalyss:

13 If the Box Coutg deso detects a chagg pot ad the Correlato deso does ot dcate a chagg pot oe ca coclude the devato s caused by the occurreces of a few ew sybols or ew -legth subsequeces. If the Correlato deso detects a chagg pot ad the Box Coutg deso does ot dcate a chagg pot oe ca coclude the devato s caused by chages patters of -legth subsequeces, but o ew sybols or ew -legth subsequeces have occurred. If both Correlato deso ad Box Coutg deso detect a chagg pot, but the chage Correlato deso s ore evdet, oe ca coclude the devato s caused by chages correlatos of -legth subsequeces, cludg the occurrece of a few ew -legth subsequeces, such as ay occur a dstrbuto-based process scearo. If both Correlato deso ad Box Coutg deso detect a chagg pot, but the chage Box Coutg deso s ore drastc, oe ca coclude the devato s caused by the occurrece of a few ew sybols or ew -legth subsequeces, wthout drastc chages correlatos of -legth subsequeces, such as ay occur certa Marov processes. I order to coplete the root-cause aalyss procedure, as well as to provde a coprehesve vew of the process aoales, the suggested ethod offers the teractve Fractal SPC otorg cotrol chart, as llustrated Rusch-R et al. 0. The Fractal SPC otorg cotrol chart eables oe to vsually lear ad aalyze the otored process. Oe ca reveal uderlyg correlatos, patters ad ew -legth subsequeces both ' cotrol' ad 'out of cotrol' saples, order to obta a coprehesve schee for root cause aalyss ad assgable causes detfcato. Acowledgeet Ths research was supported by THE ISRAEL SCIECE FOUDATIO grat o. 36/0.