Proceedigs of the th WSEAS Iteratioal Coferece o APPLIED ad THEORETICAL MECHANICS (MECHANICS '8) The Detectio of Abrupt Chages i Fatigue Data by Usig Cumulative Sum (CUSUM) Method Z. M. NOPIAH, M.N.BAHARIN, S. ABDULLAH, M. I. KHAIRIR AND C. K. E. NIZWAN Departmet of Mechaical ad Materials Egieerig Uiversiti Kebagsaa Malaysia 36 UKM Bagi, Selagor MALAYSIA zm@vlsi.eg.ukm.my Abstract: - The detectio of abrupt chages refers to a time istat at which properties suddely chage, but before ad after which properties are costat i some sese. CUSUM (Cumulative Sum) is a sequetial aalysis techique that is used i the detectio of abrupt chages. The objective i this study is to apply CUSUM techique i aalysig fatigue data for detectio of abrupt chages. For the purpose of this study, a collectio of ostatioary data that exhibits a radom behavior was used. This radom data was measured i the uit of microstrai o the lower suspesio arm of a car. Experimetally, the data was collected for 6 secods at a samplig rate of 5 Hz, which gave 3, discrete data poits. By usig CUSUM method, a CUSUM plot was costructed i moitorig the mea chages for fatigue data. Global sigal statistical value idicated that the data were o Gaussia distributio i ature. The result of the study idicates that CUSUM method is oly applicable for certai type of data with mixed high amplitude i a radom backgroud data. Key-Words: - Abrupt chages, CUSUM, ostatioary data, mea chages, global statistics Itroductio I fatigue data aalysis, data editig plays a importat role i calculatig the damage caused by the stress loadig. The fuctio of fatigue data editig is to remove the small amplitude cycles for reducig the test time ad cost. By usig this approach, large amplitude cycles that cause the majority of the damage are retaied ad thus oly shorteed loadig cosists of large amplitude cycles is produced []. This paper discusses o the use of Cumulative Sum (CUSUM) method for detectio of abrupt chages. A compariso betwee CUSUM ad time series plot for four sets of data were made i order to evaluate whether this method ca be used to extract the importat features that exist i the fatigue data. It is predicted that by usig the CUSUM algorithm, the importat features that uderlies withi the data could be extracted ad suited for fatigue data editig. 2 Literature Backgroud 2. Sigal A sigal is a series of umber that come from a measuremet, typically obtaied usig some recordig method as a fuctio of time [2]. I real applicatios, sigals ca be classified ito two types which are statioary ad o-statioary behavior. A sigal represetig a radom pheomeo ca be characterised as either statioary or ostatioary behaviour. The statioary sigals exhibit the statistical properties remai uchaged with the chages i time. O the other had, statistics of ostatioary sigal is depedet o the time of measuremet [3]. I the case of fatigue research, the sigal cosists of a measuremet of cyclic loads, i.e. force, strai ad stress agaist time. The observatios of a variable were take at equally spaced itervals of time [2]. 2.. Abrupt Chages Abrupt chages are rapid chages that occur with respect to the samplig period of measuremet, if ot istataeously. Whilst, the detectio of abrupt chages refer to tools that help us i decidig whether such a chage occurred i the characteristics of the cosidered object []. The geeric problem of detectig abrupt chages i process parameters has bee widely studied. These chages may be due to shifts that exist i the mea value (edge detectio) or a variatio i sigal dyamics [5]. ISSN: 79-2769 75 ISBN: 978-96-7-6-8
Proceedigs of the th WSEAS Iteratioal Coferece o APPLIED ad THEORETICAL MECHANICS (MECHANICS '8) 2..2 Cumulative Sums CUSUM method is oe of the tools that is used i the detectio of abrupt chages. It was first proposed i 95 ad has bee studied by may authors particularly [6]. The CUSUM method is easy to hadle ad useful for detectig the locatios of chage poits. I particular, it has bee utilized for testig a chage of mea, variace ad distributio fuctio. A advatage of the method lies i the fact that the sample meas, variace ad distributio fuctio are all expressed as the sum of idepedet ad idetically (i.i.d) radom variables [7]. It has the advatage of takig ito accout the history of the forecast series ad is capable of detectig model failure more rapidly whe forecast errors are relatively small [8]. The CUSUM algorithm ca be described i four differet derivatios []. The first is more ituitiobased, ad uses ideas coected to a simple itegratio of sigals with adaptive threshold. The secod derivatio is based o a more formal o-lie statistical approach, similar to the approach used before for itroducig cotrol charts, ad based upo a repeated use of the sequetial probability ratio test. The third derivatio comes from the use of the off-lie poit of view for a multiple hypothesis testig approach. This derivatio is useful for the itroductio of the geometrical iterpretatio of the CUSUM algorithm with the aid of a V-mask. The fourth derivatio is based upo the cocept of opeeded tests []. The CUSUM chart directly icorporates all the iformatio i the sequece of sample by plottig the cumulative sums of the deviatios of the sample values from a target value [6]. I this study, the basic CUSUM equatio is take ito accout. CUSUM is explaied by Equatio () which target µ as the process mea. The cumulative cotrol chart is formed by plottig the quatity i C i = ( x j μ ) () j= agaist the sample umber i. Ci is the cumulative sum up to ad icludig the i th sample. Iteratively CUSUM ca be writte as: C μ, i =,2,3,... (2) ( ) i = xi + Ci where the startig value for the CUSUM, C, is take to be zero. CUSUM may be costructed both for idividual observatios ad for the averages of ratioal subgroups. The case of idividual observatios occurs very ofte i practice, thus that situatio will be treated first. 3 Methodology All data were take from differet road coditios: - campus route, pave ad highway (see Fig. ). Highway Road Coditio Pave Campus Fig. : Mai Factor for Fatigue Data Road Coditio These fatigue sigals have variable amplitude patter i strai format, ad they were measured o the lower suspesio arm of a mid-sized car. Experimetally, the data was collected for 6 secods at a samplig rate of 5 Hz, which gave 3, discrete data poits. The road load coditios were from a stretch of highway road to represet mostly cosistet load features, a stretch of brick-paved road to represet oisy but mostly cosistet load features, ad a i-campus road to represet load features that might iclude turig ad brakig, rough road surfaces ad speed bumps. As the CUSUM method is easy to hadle ad useful i detectig the locatios of chage poits, it was used to detect the chages that exist i the series sigals. The segmetatio of the time series data will be based o how may detectio will be detect by usig the CUSUM method. A flowchart describig the CUSUM method is preseted i Fig. 2 ad it ivolves three stages: the iput sigal ad global statistics parameter; the idetificatio ad extractio of abrupt chages for fatigue damagig evets; ad decisio makig process. The first stage of CUSUM algorithm is to display the three differet types of fatigue data i times series plot ad global statistic parameter. I ormal practice, the global sigal statistical values are frequetly used to classify radom sigals. I this study, mea, root mea square (r.m.s.) ad kurtosis were used [2]. For a sigal with data poits, the mea value of x is give by x = x j j= (3) O the other had, root mea square (r.m.s) value, which is the 2 d statistical momet, is used to ISSN: 79-2769 76 ISBN: 978-96-7-6-8
Proceedigs of the th WSEAS Iteratioal Coferece o APPLIED ad THEORETICAL MECHANICS (MECHANICS '8) quatify the overall eergy cotet of the sigal ad is defied by the followig equatio: / 2 2 r. m. s = x j () i= where x j is the j th data ad is the umber of data i the sigal. The kurtosis, which is the sigal statistical momet, is a global sigal statistic which is highly sesitive to the spikeess of the data. It is defied by the followig equatio: K = ( x ) j x (5) ( r. m. s) j= where r.m.s is the root mea square as calculated i Equatio ad x is the mea value of the sigal data. For a Gaussia distributio, the kurtosis value is approximately 3.. Therefore, kurtosis value which are higher tha 3. idicates the presece of more extreme values tha should be foud i a Gaussia distributio. Kurtosis is used i egieerig for detectio of fault symptoms because of its sesitivity to high amplitude evets [9]. The secod stage of the CUSUM algorithm is to idetify ad extract the abrupt chages for fatigue damagig evets. Areas that exhibit abrupt chages were idetified (circled) ad compared with the origi data. The last stage ivolve with the idetificatio o whether the CUSUM techique ca detect trasiet evets i fatigue data. It is assumed that the CUSUM algorithm ca be used i predictig the abrupt chages that exist i fatigue data. I the algorithm, Equatio (2) was used i calculatig the cumulative sum for all the collected fatigue data. By takig a sum of squares approach for the Equatio (2), it was used as aother method i detectig the fatigue damage evet i the data. Iput time history Display the iput sigal & Global Statistical Parameters th The idetificatio ad extractio of abrupt chages for fatigue damagig evets Fig. 2: The CUSUM algorithm flowchart The CUSUM is represeted as a ordered set of poits C where C is defied as, 2 C = { ci : ci = ( xi μ ) + ci } i =,2,3,...,,, (6) where c is equal to zero x is the i th i elemet of data μ is the global mea statistics The data that was used i this study was the ru uder (CUSUM) algorithm. All the features that have bee idetified by the CUSUM algorithm i the fatigue data were the extracted. The data was the segmeted depedig o the detectio of abrupt chages i the CUSUM plot. CUSUM plot fails, revised algorithm CUSUM applied as abrupt chages tool Implemetatio Results ad Discussio Referrig to Table, the campus route gives the highest value of mea ad RMS which are 72.8 ad 83.3 respectively. Kurtosis results show that all of the data are o-gaussia distributio sice all of kurtosis value exceed 3. ISSN: 79-2769 77 ISBN: 978-96-7-6-8
Proceedigs of the th WSEAS Iteratioal Coferece o APPLIED ad THEORETICAL MECHANICS (MECHANICS '8) Global Statistics Mea RMS Kurtosis Campus Route 72.8 83.3.53 Pave' 58.23 7.53 6.7 Highway 66.32 7.3 3.58 Table : The Global Statistic Usig Equatio (2) i the algorithm, the CUSUM plot did ot show ay sigificat fidigs i detectig the fatigue damage evet/abrupt chages that exist i all selected data. However by usig Equatio (6), it shows that there is a improvemet i detectig the existece of trasiet evet i the fatigue data. Fig. 3, ad 5 show the compariso betwee the CUSUM ad time series plot for fatigue data. The CUSUM plot i Fig. 3 shows that there exist several jumps which related to fatigue damage evet. Amplitude[Microstrai] 5-5 2 3 5 6 Cumulative Amplitude[Microstrai] 6 x 7 2 2 3 5 6 Fig. 3: Compariso betwee CUSUM ad time series plot for Campus Route Data I the Pavé data (Fig. ), the CUSUM method detected oly few sigificat chages i the data. The cumulative value shows oly a smooth lie exist after the iterval of secods although there should be several features that should be extracted by usig the CUSUM plot. This coditio may be due to the existece of oisy data which fluctuated aroud the mea value. From Fig. o the Pave data, the CUSUM method was able to detect oly oe obvious abrupt chages that occurred i the fatigue data (as idicated i the circled area). Fially, Figure 5 shows that the detectio of abrupt chages by usig CUSUM method plot teds to be very small. Referrig to Fig. 5, oly three fatigue damagig evets are detected (by referrig to the circled area). Some fatigue damage evets were ot highlighted by CUSUM method (see the highlight circled area). This may due to the fluctuatio of data aroud the mea value. ISSN: 79-2769 78 ISBN: 978-96-7-6-8
Proceedigs of the th WSEAS Iteratioal Coferece o APPLIED ad THEORETICAL MECHANICS (MECHANICS '8) Amplitude[Microstrai] 5-5 2 3 5 6 Cumulative Amplitude[Microstrai] 8 x 7 6 2 2 3 5 6 Fig. : Compariso betwee CUSUM ad time series plot for Pavé Data Amplitude[Microstrai] Cumulative Amplitude[Microstrai] 5-5 2 3 5 6.5.5 2 x 7 2 3 5 6 Fig.5: Compariso betwee CUSUM ad time series plot for Highway Data ISSN: 79-2769 79 ISBN: 978-96-7-6-8
Proceedigs of the th WSEAS Iteratioal Coferece o APPLIED ad THEORETICAL MECHANICS (MECHANICS '8) 5 Coclusio This study uses CUSUM algorithm i detectig the abrupt chages that exist i time series data for fatigue aalysis. The selected data have may occurreces of trasiet sigals. The fidigs from the study show that the global kurtosis of campus route was the maximum value compared to other data. The study reveals that the CUSUM method is suitable i detectig the existece of fatigue damagig evet for data cotaiig mixed high amplitude i a radom backgroud data. However for data with radom oise ad almost statioary behavior, the CUSUM method is ot that sesitive i detectig the fatigue damagig evet. From this study also, it is suggested that the kurtosis value ca be good idicator for usig the CUSUM method to detect the abrupt chages that exist i fatigue data. 6 Ackowledgemets The authors would like to express their gratitude to Uiversiti Kebagsaa Malaysia ad Miistry of Sciece, Techology ad Iovatio, through the fud of UKM-GUP-BTT-7-25-52, for supportig these research activitie Refereces: [] Abdullah, S., Choi, J.C., Giacomi, J.A., ad Yates, J.R., Bump Extractio Algorithm for Variable Amplitude Loadig, Iteratioal Joural of Fatigue, Vol 28, 25, pp. 675-69. [2] Abdullah, S., Ibrahim, M.D., M.,Nopiah, Z., ad Zahari, A., Aalysis of a variable amplitude fatigue loadig based o the quality statistical approach, Joural of Applied Scieces, Vol 8, 28,59-593 [3] Bedat, J. S. ad Piersol, A. G., Radom Data: Aalysis ad Measuremet Procedures, 2d Editio, Wiley-Itersciece, New York. 986. [] Abdullah, S., Ibrahim, M.D., Zaharim, A., ad M.,Nopiah Z., Statistical Aalysis of a Nostatioary Fatigue Data usig the ARIMA approach, WSEAS Trasactios o Mathematics, Issue 2, 28,59-66 [5] M., Basseville, ad I.V., Nikirov, Detectio of abrupt chages: Theory ad applicatio,3rd editio,993, Pretice-Hall [6] Marko, Baarola, Mika, Ruusue, ad Mika, Pirittima, Some chage detectio ad time series forecastig algorithms for a electroic maufacturig process, Cotrol Egieerig Laboratory Report, A No.26, 25,-27 [7] D.C. Motgomery, Itroductio to statistical quality cotrol, 5 th editio, 25, Wiley Iteratioal Editio [8] S.,Lee, J.,Ha, O.,Na, ad S.,Na, The Cusum test for parameter chage i time series model, Board of the Scadiavia Joural of Statistics, Vol 3, 23,78-796 [9] Qu, L. ad He, Z., Mechaical Diagostics, Shaghai Sciece ad Techology Press, Shaghai, P. R. Chia. 986. ISSN: 79-2769 8 ISBN: 978-96-7-6-8