A practical approach to combine data mining and prognostics for improved predictive maintenance

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1 A practcal approach to combne data mnng and prognostcs for mproved predctve mantenance Abdellatf Bey- Temsaman +32 (0) fmtc.be Marc Engels +32 (0) fmtc.be Andy Motten +32 (0) fmtc.be Steve Vandenplas +32 (0) fmtc.be Agusman P. Ompusunggu +32 (0) fmtc.be 1 Flanders MECHATRONICS Technology Centre. Celestjnenlaan 300D, B-3001, Leuven, Belgum ABSTRACT Orgnal Equpment Manufacturer companes (OEMs) are facng more and more the challenge to ncrease the effcency and reduce the cost for the servce of ther equpment over ther lfecycle. A predctve mantenance strategy, where the optmal tme to schedule a servce vst s forecasted based on the condton of the equpment, s often proposed as an answer to ths challenge. However, predctve mantenance approaches are frequently hampered. Frst, by the lack of knowledge of the features those gve a good ndcaton of the condton of the equpment. Second, by the processng power needed for predcton algorthms to forecast the future evoluton of the selected features, especally, when large measurements are collected. In most cases, ths processng power s not avalable from the machne s processor. To overcome these problems, we propose n ths paper to combne two approaches that are currently used separately: data mnng and prognostcs. The proposed method conssts of two steps. Frst, data mnng and relablty estmaton technques are appled to hstorcal data from the feld n order to optmally dentfy the relevant features for the condton of the equpment and the assocated thresholds. Secondly, a predcton model s ftted to the lve data of the equpment, collected from customer s premses, for predctng the future evoluton of these features and forecastng the tme nterval to the next mantenance acton. To overcome the lmted processng power of the machne s processor, ths predcton part s computed on a local server whch s remotely connected to the machne. The proposed method proved always to retreve, from the datasets, the relevant feature to be forecasted. Valdaton has been done for two dfferent ndustral cases. A frst prototype of the predctve module s mplemented n some copers and s runnng n lve condtons, snce November 2008, n order to check the forecast robustness. Frst results showed that ths module offers a very good ndcaton on when part replacement would be requred, wth some level of uncertanty whch decreases over tme. Calculated busness cases showed that ths module wll be hghly benefcal for the company. Savngs of approxmately 4,8 mllon/year worldwde are estmated. Ths estmate was manly calculated by reducng labour n reactve servce vsts and stock costs. Categores and Subject Descrptors D.2.7 [Dstrbuton, Mantenance and Enhancement] D.4.8 [Performance] C.4 [Performance of systems] General Terms Performance, Relablty, Algorthms, Management 1. INTRODUCTION Condton based mantenance (CBM), also called Predctve mantenance (PdM), has evdent benefts to OEMs, ncludng reducng mantenance cost, ncreasng machne relablty and operaton safety, and mprovng tme and resources management [1]. A sde to sde comparson between PdM and tradtonal correctve or preventve mantenance programs s made n [2]. From ths survey, t was concluded that major mprovements can be acheved n mantenance cost, unscheduled machne falures, repar downtme, spare parts nventory, and both drect and ndrect overtme premums, by usng PdM. Although these benefts are well llustrated, two major problems hamper the mplementaton of predctve mantenance n ndustral applcatons. Frst, the lack of knowledge about the rght features to be montored and second, the requred processng power for predctng the future evoluton of features, whch s often not avalable on the machne s processor. For the last problem, the present work fts n an archtecture where the machnes at the customer sde are connected to a central offce. The lve measurement data are processed by a server at ths central offce. Ths allows usng standard computers wth off the shelf software tool, wth vrtually unlmted processng power and storage capacty. Next to condton montorng of the machne and predctve mantenance schedulng, the central server can also be used for provdng other servces to the customer. For the former problem, data mnng technques proved to be useful for relevant features extracton. It has been proved n [3][4] that applcaton of data mnng technques to data, such as acoustc emsson for the montorng of corroson processes, s very useful for extractng relevant features whch can be used as parameters for machne dagnoss and/or prognostcs. However, n many other ndustral applcatons, no clear physcal

2 understandng of the process s avalable and therefore to retreve these relevant features, a clear methodology s requred. The man contrbuton of ths paper s such a practcal methodology whch combnes data mnng and predcton technques used consecutvely n order to perform an accurate predctve mantenance schedulng. Ths methodology s called the IRIS-PdM approach (see Secton 6). Unlke standards such as Cross Industry Standard Process for Data Mnng (CRISP-DM) [5], whch ndcated, n a hgh level, the dfferent steps to be followed n order to apply data mnng to ndustral data, the IRIS-PdM approach descrbed n ths paper proposes specfc algorthms whch can be appled drectly n dfferent ndustral applcatons. Furthermore, up-tll-now prognostc has been tackled as an ndependent step from data mnng, assumng the predcton of a completely known parameter [6]. On the contrary, n the IRIS-PdM approach, prognostcs s an ntegral part of the flowchart and makes use of the relevant features extracted n data mnng step. In ths way, the IRIS-PdM approach enables the possblty to compare dfferent features evoluton and combne them to mprove the accuracy of remanng lfetme forecast. The IRIS-PdM approach conssts manly of: () A data mnng step on hstorcal data where data preparaton, data reducton and relevant features extracton are performed. () A prognostcs step, where optmal thresholds are retreved usng relablty estmaton methods and predcton algorthms are appled on lve data to estmate the remanng tme for the relevant features to reach the thresholds. Ths remanng lfe tme can be used to establsh an optmal predctve mantenance. Ths paper s organzed as follows. In Secton 2, general descrptons of the remote connectvty platform and the IRIS- PdM approach are gven. In Secton 3, the applcaton of the IRIS-PdM approach to an ndustral dataset s llustrated, wth a revew of data preparaton steps, data reducton technques and the most mportant data mnng modelng technques used to optmally dentfy relevant features. In Secton 4, a descrpton of the prognostcs step s gven, wth a revew of relablty estmaton technques to determne optmal thresholds and predcton algorthms to forecast the optmal tme to schedule a mantenance acton. Fnally, conclusons are gven n Secton GENERAL DESCRIPTIONS 2.1 Remote Connectvty Platform Remote connectvty to customer premses offers the OEMs a contnuous montorng of ther products, whch results n a long servce lfe of the products. Ths s thanks to the recommendatons that OEMs can gve to ther customers for an optmal usage of the machnes as well as the optmal mantenance schedulng before the machne or a sub-part of the machne approaches the end of lfe. Furthermore, mprovement of mantenance logstcs plannng may be acheved. The work presented n ths paper fts wthn the framework of such a platform, schematcally shown n fgure 1. A human expert n a remote assstance center, at the machne bulder sde connects remotely to the customer premses, through a secure nternet connecton, and collects the data contnuously or perodcally from the machne. Usng dfferent local software tools, ncludng data mnng software, dfferent ntellgent servces can be provded such as Predctve Mantenance (PdM). Remote assstance center provdng ntellgent servces: CBM, PdM, Secure Internet Connecton Fgure 1. Schematc of the remote connectvty platform. 2.2 IRIS-PdM Approach In order to optmally forecast the predctve mantenance actons, a practcal approach has been developed where dfferent steps are followed startng from the avalable hstorcal database of machnes runnng n the feld to the PdM schedulng. The dfferent steps of the IRIS-PdM approach are summarzed n fgure 2. Data mnng block Hstorcal and lve data Prognostcs block Unfed data format Data preparaton Optmal thresholds (relablty estmaton) Data reducton Advanced predcton Relevant features Features selecton Fgure 2. IRIS-PdM approach. steps. Customer premses Case Study Predctve Mantenance Estmaton The IRIS-PdM approach proved to be easy to transfer from one ndustral applcaton to another. It was manly tested for predctve mantenance of copy machnes, but t also proved to be feasble for predctve mantenance of hgh-end mcroscopes. The startng pont s the hstorcal database of the machnes. In general, such a database exsts for almost all machne manufacturers, but only a lmted amount of nformaton s currently used. The next step conssts of data preparaton, ncludng data transformaton from the orgnal to unfed data formats and data cleanng such as removng outlers or calculatng mssng values. Note that the data preparaton step may be tme consumng snce the unfed data format should be compatble wth the data mnng software tool and stll retan a physcal nterpretaton of the data. The data modelng step conssts of two sub-steps. Frstly, the data reducton sub-step where the Independence Sgnfcance Feature

3 method s used to reduce sgnfcantly the number of attrbutes. Ths method s descrbed n Secton Secondly, the features selecton sub-step where the relevant features are extracted out of the selected attrbutes usng a decson tree (DT) method. Ths sub-step s descrbed n Secton The choce of these methods was based on requrements of processng speed, accuracy, and nterpretablty of the results. The fnal step conssts of prognostcs, tself dvded nto two substeps. Frst, relablty estmaton methods are appled to hstorcal database n order to dentfy the optmal thresholds of the selected features. These technques are descrbed n Secton 4.1. Secondly, a predcton algorthm s appled to lve data n order to forecast the tme to schedule the optmal PdM. The predcton model used n ths paper s based on slope calculaton and called Weghted Mean Slope (WMS) model. Ths model s descrbed n Secton 4.2. In next sectons the dfferent steps of IRIS-PdM approach are descrbed n detals and llustrated for an ndustral dataset. 3. APPLICATION OF IRIS-PdM APPROACH TO A DATASET EXAMPLE In ths secton, we llustrate the dfferent steps of the IRIS-PdM approach, descrbed n the prevous secton, on an ndustral mantenance database. The dataset used n ths secton, s a large hstorcal mantenance database of more than 2000 copers. Every coper has one or more data fles correspondng to the hstory of servce vsts. One data fle contans measurements of the sensors/counters that the copers are equpped wth and nformaton added by the servce techncan, that llustrates the mantenance acton type performed durng hs vsts. Ths mantenance type can be correctve, n case a replacement of a broken part s done, or preventve, n case the part s changed before t s actually broken. As mentoned prevously, the preventve mantenance s decded by the servce techncan and not always based on a physcal understandng of the degradaton process. 3.1 Data Preparaton Step The data preparaton conssts of transformng the nformaton contaned n the orgnal data fles of dfferent machnes to a unfed format. The orgnal data s structured n dfferent fles of dfferent formats. The unfed format s a matrx contanng columns correspondng to the dfferent attrbutes whch represent sensors/counters and lnes correspondng to observatons at a servce vst. Two extra columns wth the component replacement nformaton and mantenance acton type are added. The dfferent data fles are concatenated nto one sngle fle. Snce enough data was avalable and mssng data can be consdered as mssng completely at random n the studed applcaton, the mssng values were smply dscarded from the dataset. A schematc of such a data set format s gven n table 1. Based on the dataset, a matrx of more than 1000 attrbutes (features) and 1 mllon objects was generated. 3.2 Data Modelng Step Data mnng algorthms are generally evaluated accordng to three dfferent crtera [7] 1. Interpretablty: how well the model helps to understand the data 2. Predctve accuracy: how well the model can predct unseen stuatons 3. Computatonal effcency: how fast the algorthm s and how well t scales to very large databases The mportance of these crtera dffers from one applcaton to another. For our use, the nterpretablty and scalng to large databases were essental. A summary of the technques mplemented n popular data mnng software, s gven n table 2. Attrbute 1 Object F11 Object F12 Object F13... Object F1M Table 1. Schematc of a unfed data format Attrbute 2 Object F21 Object F22 Object F23... Object F2M Attrbute N Object FN1 Object FN2 Object FN3 Object F Output 1 Part not Part Part not Part Table 2. Important data mnng methods and assocated algorthms Technque Clusterng Classfcaton Conceptual clusterng Dependency modelng Regresson Rules based modelng Descrpton The data modelng step s dvded nto two sub-steps () data reducton, and () features selecton Data reducton The method chosen for data reducton s called Independence Sgnfcance Feature (ISF) technque. Ths method, ntally descrbed n [8], s meant to quckly and nexpensvely dscard features whch seem obvously useless for the dvson of the data n multple classes. Unlke the Prncpal Components Analyss (PCA) method, the ISF method does not generate new features and therefore retanng the physcal meanng of the orgnal features. The ISF method proved also to be much qucker than other methods lke correlaton or entropy reducton. For example, the processng tme made by ISF method to calculate sgnfcance of the top 100 attrbutes out of the database descrbed n Secton 3.1 s approxmately 2.5s whle Spearman correlaton method made 20s and Entropy reducton method made around 130s. The ISF method conssts on measurng the mean of a feature for all classes wthout worryng about the relatonshp to other features. The larger the dfference between means, the better separaton between classes (better sgnfcance).... Unsupervsed machne learnng to group objects n dfferent classes Assgn unknown objects to well establshed classes Qualtatve language to descrbe the knowledge used for clusterng Descrbes the dependency between varables Summarzaton Provdes a compact descrpton of a subset of data Determnes functons that lnks a gven contnuous varable to other K-Means, AutoClass Output 2 No mantenance Correctve mantenance No mantenance Preventve mantenance Method(s) Neural Networks, K-NN, SVM Decson Tree PCA, Dendrogram, ISF Statstcal reportng Vsualzaton ANN, Regresson Tree, ML Regresson Generate rules that descrbe tendency Assocaton rules of data

4 In our case, ths dramatcally reduces the number of canddate predctors to be consdered n the fnal selecton process. The ISF method reduces the data from more than 1000 features to ~100 features, usng as an output of the two classes correspondng to the replacement nformaton (Output 1 n table 1). The mathematcal formula to calculate the sgnfcance, for every attrbute, s gven as: X X Sg = (1) Wth S 1 2 S X 1 X 2 s s = + X 1 X 2 n n 1 2 Where Sg :Sgnfcance X : mean valueof the nput objects n class s 2 : the varance of the nput objects n class n : number of samples n class Only features wth hgh sgnfcance measure are retaned for further processng. When the attrbutes are ordered accordng to decreasng sgnfcance, ths results n the graph of fgure 3. As can be seen on ths graph, only the frst 100 attrbutes have hgher sgnfcance than 0.2. Ths threshold was chosen, snce the most related attrbutes to the classfcaton output have sgnfcance hgher than ths value. These attrbutes are selected to be further processed n the next step. (2) algorthm mplementatons are qute robust and can deal wth multvarate and mult-classes analyses. In [9] a descrpton of a decson tree method s gven. It s a topdown tree structure consstng of nternal nodes, leaf nodes, and branches. Each nternal node represents a decson on a data attrbute, and each outgong branch corresponds to a possble outcome. Each leaf node represents a class. Inducng a decson tree s done by lookng to the optmal attrbute to be used at every node of the tree. In order to do that, the followng steps are followed [10]: (1) a quantty known as Entropy ( H ) s calculated for every attrbute m H ( s1, s2,..., sm) = p log2( p ) (3) = 1 wth s the number of samples belongng to the class C, ( m possble classes) for the calculated attrbute. p s the rato of number of samples n each class dvded by the total number of samples (can be understood as a probablty). (2) the expected nformaton E (A) s computed for each attrbute v s1 j + s2 j smj E( A) = H ( s1 j, s2 j,..., smj ) (4) j = 1 s where v s the number of dstnct values n the attrbute A and sj s the number of samples of class C n a subset by parttonng on the attrbute A. S j obtaned Fgure 3. Sgnfcance measure versus attrbutes Fgure 4. Decson tree model for relevant feature extracton Features selecton Once the data reducton s done as explaned n the prevous secton, the relevant features extracton method s appled. In ths paper the Decson Trees (DT) method has been chosen. The man advantage of ths method s the ease to nterpret the results and transform them to f-then rules, whch can be easly understood by an ndustral user. Furthermore, decson trees (3) Compute the nformaton gan G (A) usng G A) = H s, s,..., s m E( (5) ( ) ) ( 1 2 A (4) select attrbute havng hghest nformaton gan to be test attrbute (5) Splt the node and repeat steps (1) to (4) tll all values belong to the same class C or all attrbutes were used.

5 In ths work, the CART decson tree mplementaton n Matlab statstcs toolbox was used for a relevant feature selecton. Fgure 4 shows a decson tree used to retreve the relevant features from the lst of the selected features obtaned n Secton The purpose of ths classfcaton problem, s to dentfy from the lst of features, whch features are the most relevant to predct a mantenance acton. The classfcaton output used n ths step s the mantenance type nformaton (Output 2 n table 1). Note that n order to reduce the sze of the three a prunng method s used. Ths prunng method calculates the statstcal sgnfcance of a further splt of a data and stops the process when ths sgnfcance becomes too low. By keepng the sze of the three under control, t remans feasble to physcally nterpret the classfcaton. In the resultng decson tree for our dataset, more than 95% of preventve mantenances were performed when the feature wth number x72 s hgher than the value ~2800 (rght hand branch). The pattern proposed by decson tree can be vsualzed also by the scatter plot n fgure 5. Ths fgure shows a clear separaton between values of feature x72 for the preventve mantenance class and the rest of the classes, at the threshold of ~2800. The preventve mantenance observatons around zero value of y-axs are performed by servce techncans for testng purposes. By lookng back to the physcal meanng of attrbute x72, a meanngful relatonshp to preventve mantenance can be carred out. In order to check the accuracy of decson trees (DT) method towards other well-known data mnng methods, such as k-nn (Nearest Neghbor) [11], a 5 fold cross valdaton check s carred out wth both methods and vsualzed usng confuson matrces. The correspondng results are shown n table 3 and table 4, respectvely for the k-nn and DT method. An overall accuracy of 88% ± 0.5% was acheved for decson trees algorthm versus only 77% ± 0.9% for k-nn algorthm. Fgure 5. Classfcaton usng the relevant feature extracted by DT Table 3. Confuson matrx for k-nn method Table 4. Confuson matrx for DT method Predcton Total 92.9% 4.6% 2.4% Actual 21.3% 5.3% 73.3% 20.1% 1.9% 77.8% Where, and stand, respectvely, for no mantenance, correctve and preventve mantenances. It s clearly shown from the tables that msclassfcaton usng the DT method s much lower than for the k-nn method. As an example, 77.8% of preventve mantenances are correctly classfed usng the DT method, versus only 39.3% usng the k- NN method. Note that the low percentage of the correct classfcaton usng both methods for correctve mantenance s manly due to the correspondng low number of samples n the complete data set. The value 2800 shown n fgure 5 can be used as a threshold for the montorng of the selected feature. Note that, n order to fne tune ths threshold n an optmal way, statstcal methods, such as relablty estmaton could be used. Ths latter together wth predcton algorthms are dscussed n the next Secton. 4. PROGNOSTICS Prognostcs n the IRIS-PdM approach conssts on two steps. () the relablty estmaton step where optmal thresholds are calculated form the hstorcal database for the selected features and () the predcton step where a predcton algorthm s appled to lve data to forecast the tme for schedulng a mantenance acton. The former step s descrbed n Secton 4.1, whle the latter step s descrbed n Secton Relablty Estmaton In the IRIS-PdM approach, relablty estmaton s appled to the hstorcal database n order to estmate the optmal thresholds for the selected features. These thresholds are gong to be used for forecastng the remanng tme untl the next mantenance acton. Ths estmaton conssts of fttng a lfe tme dstrbuton model on the data and dentfyng the falure rate at a gven feature s value [12]. For our dataset, a Webull dstrbuton model fts qute well the data. Fgure 6 shows the Webull dstrbuton model compared to the lfe measurements of the machnes. Ths model can be used to determne the optmal threshold by lookng to an acceptable falure rate of the studed components. In our case we retan the threshold of ~2800 for the feature x72, whch corresponds to ~20% falure rate (rght sde graph). Predcton Total 67.6% 9.9% 21.7% Actual 57.3% 15.9% 26.7% 49% 11.6% 39.3%

6 Fgure 6. Webull dstrbuton model ftted to the data 4.2 Predcton Algorthm In ths step, tme seres of lve data are analyzed. The man goal s to obtan a model of the evoluton of the selected feature based on past observatons n order to predct the future evoluton. Combnng ths predcton wth the optmal threshold, as defned n the prevous secton, allows estmatng the remanng tme before replacement of the part. The accuracy of ths remanng tme depends strongly on the choce of the model. Two classes of models can be dentfed n lterature: the models whch allow a non-lnear trend predcton and the ones whch allow a lnear trend predcton of tme seres. For the former case, neural network s a good example, whch s extensvely used n stock market predcton [13]. It uses at least two parameters: the number of hdden nodes and the weght decay. These two settngs are dynamcally changng to adapt to the observatons. For lnear trend predcton methods, exponental smoothng technques have been used snce long tme [14][15]. In these technques some model parameters need to be optmzed n order to mnmze the error between model and data. Ths can be problematc n the presence of steep changes n the data evoluton versus tme. Therefore, a model based on weghted mean slopes (WMS) estmaton has been developed. Ths model works robustly, even when data changes abruptly. In ths model, the predcton s performed by lookng to the recent observatons and ft an optmal model to them. Y = y y,..., s the lve tme Suppose the prmary data {, 1 2 y n } sequence of the selected feature at tme T { t t,..., } = 1, 2 t n. The weghted mean slope (WMS) s calculated as: yk yk 1 Sk = ; tk tk 1 k. Sk k WMS = k k { k = 2,..., n} The predcton value at tme t + m, s gven by yt + m = yt + WMS. m (7) (6) Fgure 7. Forecast of remanng tme to mantenance acton An example of WMS predcton model appled to lve data of one coper s shown n fgure 7. The uncertanty bounds are based on varances of dfferent machnes usages, whch have been calculated off-lne. The WMS model has the ablty to follow flat zones, where machnes are not operatonal, whch s not possble to acheve wth a smple regresson or exponental smoothng models. In ths fgure three zones correspondng to low, medum and hgh rsk zones are used. For each zone a dfferent threshold can be set dependng crtcalty of an unscheduled repar for the customer. The remanng days to perform a mantenance acton are dsplayed wth a confdence nterval for each zone. 5. CONCLUSIONS In ths paper, a practcal methodology called the IRIS-PdM approach has been presented and dscussed. Ths approach conssts of dfferent steps makng use of data mnng, relablty estmaton technques and predcton algorthms n order to extract the relevant feature and use t n prognostcs to predct the remanng tme untl a preventve mantenance acton s requred. Independent Sgnfcance Feature (ISF) was successfully appled on an ndustral data set for data reducton. Next, the most relevant features were extracted by means of the Decson Tree classfcaton method. Comparson between the k-nn and the DT methods proves that DT s an accurate classfcaton method. A Weghted Mean Slopes (WMS) model was appled for predcton of the remanng tme to schedule a mantenance acton. Ths model works robustly, even for abruptly changng data. The methods descrbed n ths approach can be broadly and robustly appled to dfferent ndustral data sets and mantenance databases. The results can also be easly nterpreted and transformed to f-then rules, allowng nsght and easy nteracton wth the results. 6. ACKNOWLEDGMENT Ths research fts n the framework of IRIS (Intellgent Remote Industral Servces) project whch s fnancally supported by the Dutch governmental agences: Mnstry of Economy Affars, Provnce of Noord-Brabant, Provnce Lmbrug and Samenwerkngsverband Rego Endhoven, n the framework of

7 Peken n de Delta (PID) program (Project N o. EZ 1/5262). IRIS s a cooperaton of the project partners Soux Remote Solutons, Assembléon, Océ Technologes, FEI Company, FMTC (Belgum) and the Techncal Unversty of Endhoven. The IRIS consortum was created n cooperaton wth the BOM (Brabantse Ontwkkelngs Maatschappj) and BO4. The authors wsh to thank all the IRIS project partners for ther valuable advces and revew of ths research. 7. REFERENCES [1]. J. Blar & A. Shrkhodae, Dagnoss and prognoss of bearngs usng data mnng and numercal vsualzaton technques, p , Proceedngs of the 33 rd Southeastern Symposum on System theory, [2]. R. K. Mobley, An ntroducton to predctve mantenance, Van Nostrand Renhold, [3]. G. Van Djck, Informaton theoretc approach to feature selecton and redundancy assessment, PhD thess, Katholeke Unverstet Leuven, Aprl [4]. R. Isermann, Fault-dagnoss systems, Sprnger [5]. C. Shearer, The CRISP-DM model: the new blue prnt for data mnng, Journal of data warehousng, Vol. 5, Nr. 4, p , [6]. K. M. Goh, T. Tjahjono & S. Subramanam, A revew of research n manufacturng prognostcs, p , IEEE Internatonal Conference on Industral Informatcs, [7]. R. Duda, P. Hart & D. Stork, Pattern classfcaton, 2 nd edton, 2001 by John Wley & Sons, Inc. [8]. W. Sholom, I. Ntn, Predctve data mnng: a practcal gude, Morgan Kaufmann, [9]. P. Geurts, Contrbutons to decson tree nducton, PhD thess, Unversty of Lège, Belgum, [10]. J. You, L. & S. Olafsson, Mult-attrbute decson trees and decson rules, Chap. 10, Sprnger, Hedelberg, Germany, pp , 2006 [11]. Y. Zhan, H. Chen, G. Zhang, An optmzaton algorthm of K-NN classfcaton, Proceedng of the ffth nternatonal conference on machne learnng and cybernetcs, Dalan, August p [12]. P. Yadav, N. Choudhary, C. Blen, Complex system relablty estmaton methodology n the absence of falure data, Qualty and relablty engneerng nternatonal, 2008; 24: [13]. D. Komo, C. J. Cheng & H. Ko, Neural network technology for stock market ndex predcton, p , Internatonal symposum on speech, mage processng and neural networks, Aprl 1994, Hong Kong. [14]. R. G. Brown, F. R. Meyer, The fundamental theorem of exponental smoothng, A. D. Lttle, Cambrdge, , 1961 [15]. E. S. Gardner, Exponental smoothng: the state of the arts Part II, Internatonal journal of forecastng 22 (2006)