Feature-Based Recognition of Control Chart Patterns: A Generalized Approach

Transcription

1 Qualty Technology & Quanttatve Management Vol. 5, o. 3, pp. 3-, 8 QTQM ICAQM 8 Feature-Based Recognton of Control Chart Patterns: A Generalzed Approach Susanta Kumar Gaur 1 and Shankar Chakraborty 1 SQC & OR Unt, Indan Statstcal Insttute, Inda Department of Producton Engneerng, Jadavpur Unversty, Inda (Receved August 5, accepted February 7) Abstract: Control chart patterns (CCPs) can be assocated wth certan assgnable causes creatng problems n the manufacturng processes and thus, the recognton of CCPs can accelerate the dagnostc search, for those causes. Researches n developng CCP recognton systems have tradtonally been carred out usng standardzed or scaled process data to ensure the generalzed applcablty of such systems. Whereas standardzaton of data requres addtonal efforts for estmaton of the mean and standard devaton of the underlyng process, scalng of process data leads to loss of dstncton between the normal and stratfcaton patterns because a stratfcaton pattern s essentally a normal pattern wth unexpected low varablty. In ths paper, a new approach for generalzaton of feature-based CCP recognton system s proposed, n whch the values of extracted shape features from the control chart plot of actual data become ndependent of the process mean and standard devaton. Based on a set of sx shape features, eght most commonly observed CCPs ncludng stratfcaton pattern are recognzed usng heurstc and artfcal neural network technques and the relatve performance of those approaches s extensvely studed usng synthetc pattern data. Keywords: Artfcal neural network, control chart patterns, generalzaton of features, heurstcs, pattern recognton. 1. Introducton tatstcal process control (SPC) s one of the most effectve approaches n total qualty S management (TQM) and control charts, prmarly n the form of X chart, reman among the most mportant tools of SPC. A process s consdered to be out-of-control when any pont falls beyond the control lmts or the control charts dsplay unnatural (nonrandom) patterns [13]. Whle the former condton can easly be dentfed, the latter s dffcult to recognze precsely and effectvely because the control chart patterns (CCPs) are usually dstorted by random nose that occurs naturally n a manufacturng process. The Statstcal Qualty Control Handbook [] ncludes varous types of control chart patterns (CCPs). Among these, there are eght basc CCPs, e.g. normal (OR), stratfcaton (STA), systematc (SYS), cyclc (CYC), ncreasng trend (UT), decreasng trend (DT), upward shft (US) and downward shft (DS), as shown n Fgure 1. All other patterns are ether specal forms of basc CCPs or mxed forms of two or more basc CCPs. Only the OR pattern s ndcatve of a process contnung to operate under controlled condton. All other CCPs are unnatural and assocated wth mpendng problems requrng pre-emptve actons. The Shewhart control charts consder only the current sample data pont to determne the status of a process. Hence, they do not provde any pattern-related nformaton when the process s out-of-control. Many supplementary rules lke zone tests or run rules have been developed to assst the users n detectng the CCPs [15, 16]. However, the applcaton of all

2 4 Gaur and Chakraborty these rules can result n excessve number of false alarms. Detecton of an unnatural pattern allows the users to have more nsght about the process n addton to the n-control/out-of-control decson. The classfed patterns can help the users to dentfy the potental problems n the process and provde clues or gudelnes for effectve trouble-shootng. Hence, CCP recognton s the most mportant supplement and enhancement to the conventonal control chartng technque. Recently, a bulk of research ntatves have been drected towards developng computer-based algorthms for automated recognton of varous control chart patterns amng to montor the manufacturng processes n real tme envronment. The motvaton for those research works s to take advantage of the wdespread use of automated data acquston systems for computer chartng and analyss of the manufacturng process data. The basc approach remans as defnng a movng observaton or SPC montorng wndow consstng of the most recent data ponts and then detectng the control chart pattern that appears n the wndow. The data ponts are the sample average values when the process mean s montored usng X chart. The approaches adopted by the researchers for developng automated CCP recognton systems range from the applcaton of expert systems [4, 18, 1] to artfcal neural networks (A) [, 6, 8, 1, 17, 19, ]. The advantage of an expert system or rule-based system s that t contans the nformaton explctly. Ths explct nature of the stored nformaton facltates the provson of explanatons to the operators as and when requred. However, the use of rules based on statstcal propertes has the dffculty that smlar statstcal propertes may be derved for some patterns of dfferent classes, whch may create problems of ncorrect recognton. The use of A technques has the advantage that t does not requre the provson of explct rules or templates. Rather, t learns to recognze patterns drectly through typcal example/sample patterns durng a tranng phase. One demert of A s that the nformaton t contans, are mplct and vrtually naccessble to the users. Ths creates dffcultes n understandng how a partcular classfcaton decson has been reached and also n determnng the detals of how a gven pattern resembles wth a partcular class. In addton, there s no systematc way to select the topology and archtecture of a neural network. In general, ths has to be found emprcally, whch can be tme consumng. Most of the reported works used raw process data as the nput vector for control chart pattern recognton. Only a few researchers [5, 8, ] have attempted to recognze varous CCPs usng extracted features from raw process data as the nput vector. Whereas Pham and Wan [] and Gaur and Chakraborty [5] have used extracted shape features from the control chart plot, Hassan et al. [8] have utlzed extracted statstcal features from the nput data set. The use of raw process data s computatonally neffcent. It requres large neural network sze due to whch tranng of the A becomes dffcult and tme consumng. Use of extracted shape features from the control chart plot or statstcal features from the nput process data can reduce the network sze and learnng tme. However, extracton of statstcal features requres consderably large number of observatons and ther nterpretaton s also not easly comprehendble to the practtoners. Moreover, the statstcal features lose nformaton on the order of the data. The shape features can be extracted from smaller number of observatons wthout losng order of the data. The feature-based approach also provdes a greater choce of recognton technques. Pham and Wan [] and Gaur and Chakraborty [5] have demonstrated that properly developed heurstcs based on the extracted shape features from control chart plot can effcently dfferentate varous CCPs. The feature-based heurstc approach has a dstnct advantage

3 Feature-Based Recognton of Control Chart Patterns 5 that the practtoners can clearly understand how a partcular pattern has been dentfed by the use of relevant shape features. Snce extracted shape features represent the man characterstcs of the orgnal data n a condensed form, both the feature-based heurstc and A approaches can facltate effcent pattern recognton. From mplementaton vewpont, a developed feature-based CCP recognzer should be applcable to any general process. Ths objectve s usually fulflled by extractng features from the plot of standardzed or scaled process data, nstead of extractng features from the plot of actual process data. The process data are standardzed usng the transformaton th zt ( y t )/, where y t and z t are the observed and standardzed values at t tme pont respectvely, and and are the process mean and standard devaton respectvely. On the other hand, the process data are scaled to (, 1) or (1, +1) nterval usng the lnear transformaton z ( y y )/( y y ) or z ( y y )/( y y ) 1 t t mn max mn t t mn max mn respectvely. The standardzaton of process data assumes that the mean and standard devaton values are known a pror. In practcal applcatons, these values are estmated wth addtonal efforts. A large estmaton error for the process mean and standard devaton can also affect the recognton performance adversely. The advantage of scalng of process data s that t requres no addtonal effort. However, the dstncton between normal and stratfcaton patterns s lost when the process data are scaled to (, 1) or (1, 1) nterval, snce a stratfcaton pattern s essentally a normal pattern wth unexpected low varablty. However, many real lfe stuatons demand for a CCP recognton system that wll be capable of detectng stratfcaton pattern too and therefore, scalng of the nput process data s not often acceptable. In ths paper, a new generalzed approach for control chart pattern recognton s proposed, n whch the values of the extracted shape features from the plot of actual data are made ndependent of the process mean and standard devaton. Based on a set of sx shape features, eght most commonly observed CCPs ncludng stratfcaton pattern are recognzed usng tree-structured classfcaton and A technques and ther relatve performance s extensvely studed. Ths paper s organzed n dfferent sectons. The selected set of useful features s presented n secton. The proposed approach for generalzaton of the features s dscussed n secton 3. Secton 4 dscusses the desgn of the pattern recognzers followed by the expermental procedure n secton 5. Secton 6 provdes the results and dscussons and secton 7 concludes the paper (a) ormal (b) Stratfcaton (c) Systematc (d) Cyclc (e) Increasng trend (f) Decreasng trend (g) Upward shft Fgure 1. Varous control chart patterns. (h) Downward shft

4 6 Gaur and Chakraborty. Selected Features Shape features of control chart patterns can be extracted from dfferent consderatons [5] and many of them can be hghly correlated. However, a good CCP recognzer should be capable to dfferentate dfferent patterns wth hgh accuracy usng a mnmum number of features and the correlaton among those features should be as small as possble. Lower s the assocaton among the features, hgher wll be the predcton stablty [14]. In ths paper, therefore, a set of sx features, whch are havng farly low correlaton among themselves, are chosen. These features along wth the mathematcal expressons for ther extractons are descrbed as below: (). Rato between varance of the data ponts ( SD ) and mean sum of squares of errors of the least square (LS) lne representng the overall pattern (RVE): RVE ( y ) 1 ( ) ( ) y y y y t t ( t t ). (1) where t c ( 1,,.., ) s the dstance of th tme pont of observaton from the orgn n the control chart plot, c s the constant lnear dstance used to represent a samplng nterval n the control chart plot, y s the observed value of a qualty th characterstc at tme pont, s the sze of the observaton wndow, t 1 t / and y 1 y /. The magntude of RVE for normal, stratfcaton, systematc and cyclc patterns s approxmately one, whle for trend and shft patterns, t s greater than one. (). Sgn of slope of the LS lne representng the overall pattern (SB): The slope (B) of the LS lne ftted to the data ponts n an observaton wndow s gven by the followng equaton: 1 1 B y t t t t. () It may be noted that the feature B has two characterstcs,.e. (a) ts absolute value (AB), and (b) ts sgn (SB). Lkewse RVE, the magntude of AB can dfferentate the four patterns that hang around the centerlne,.e. normal, stratfcaton, systematc and cyclc patterns from trend and shft patterns. It has been observed that RVE s more powerful than AB n dscrmnatng these two groups of patterns. Here, therefore, only the sgn characterstc of the slope of the LS lne representng the overall pattern has been taken nto consderaton as a useful feature. The sgn of the slope can be negatve or postve. Thus, the feature SB can be vewed as a categorcal varable, whch s f the sgn s negatve, and 1 otherwse. It can dscrmnate decreasng versus ncreasng trend and downward versus upward shft patterns. (). Area between the overall pattern and the LS lne per nterval n terms of SD (ALSPI): ALSPI ALS /( 1) SD (3) where ALS s the area between the pattern and the ftted LS lne representng the overall pattern. The value of ALS can be easly computed by summng the areas of the trangles and trapezums that are formed by the LS lne and overall pattern. The computaton,

5 Feature-Based Recognton of Control Chart Patterns 7 method s descrbed n detals n the next secton. The magntude of ALSPI s the hghest for STA pattern, lowest for the SYS pattern and ntermedate for all other patterns. (v). Average absolute slope of the straght lnes passng through the consecutve ponts (AASBP): AASBP y y t t. (4) The magntude of AASBP s the hghest for SYS pattern, lowest for STA pattern and ntermedate for all other patterns. The basc dfference between trend and shft patterns s that n case of trend patterns, the departure of observatons from the target value occurs gradually and contnuously, whereas n case of shft patterns, the departure occurs suddenly and then the observatons hang around the departed value. The followng two features are extracted after segmentaton of the observaton wndow so that these two types of devatons can be dscrmnated, and for ther extractons, two types of segmentatons approaches,.e. (a) pre-defned segmentaton, where the segment szes reman fxed, and (b) crteron-based segmentaton, where the segment szes may vary n order to satsfy a desred crteron, have been used. (v). Range of slopes of straght lnes passng through sx par-wse combnatons of mdponts of four equal segments (SRAGE): The feature SRAGE s extracted based on pre-defned segmentaton of the observaton wndow. In ths segmentaton approach, the total length of the data plot s dvded nto four equal segments consstng of ( 4) data ponts, as shown n Fgure. It s assumed that the total number of observatons () s so chosen that t can be dvsble by 4. The behavor of the process wthn a segment can be represented by the mdpont of the segment, whch s gven as k( 4) 1 k( 4) 1 t ( 4), y ( 4), k k where k = 1, ( /41), ( /4 1), (3 /4 1) for the frst, second, thrd and 4 fourth segment respectvely. A combnaton of two mdponts can be obtaned n C 6 ways mplyng that sx straght lnes can be drawn passng through the mdponts of these four segments. So the feature SRAGE can be extracted usng the followng expresson: SRAGE = maxmum ( s jk ) mnmum ( s jk ), (5) where j 1,,3; k,3,4; j k. s jk s the slope of the straght lne passng through th the mdponts of j and k th segments. The magntude of SRAGE wll be hgher for shft patterns than trend patterns. The magntude of SRAGE wll also be hgher for CYC pattern than OR, STA and SYS patterns, unless each segment of the cyclc pattern conssts of a complete cycle.

6 8 Gaur and Chakraborty Observed value Tme pont of samplng Fgure. Four segments of equal sze n a pattern. (v). Sum of absolute slopes of the LS lnes ftted to the two segments (SASPE): If the tme pont of occurrence of a shft can be known and two LS lnes are ftted to the observatons before and after the shft, each lne wll be approxmately horzontal to the X-axs. However, the tme pont of occurrence of the shft cannot be known exactly. Therefore, a crteron for segmentaton of the observaton wndow nto two segments s consdered. Here, the defned crteron s mnmzaton of the pooled mean sum of squares of errors (PMSE) of the two LS lnes ftted to the two segments. Assumng that at least 8 data ponts are requred for fttng a LS lne, least square lnes are ftted to all possble two segments and the segmentaton whch leads to the mnmum PMSE s chosen. Then, the feature SASPE s extracted usng the followng equaton: j 1 SASPE B, (6) th where B j s the slope of the LS lne ftted to j segment. The magntude of SASPE wll be hgher for trend patterns than shft patterns. Table 1 shows the values of par-wse correlaton coeffcents among the selected features computed from a set of tranng samples (see secton 5.1). The table reveals that the degrees of assocaton among the selected features are consderably low. j

7 Feature-Based Recognton of Control Chart Patterns 9 Table 1. Par-wse correlaton coeffcents among the selected features. Selected features RVE SB ALSPI AASBP SRAGE SASPE RVE SB ALSPI AASBP SRAGE SASPE Generalzaton of Features The necessary condtons for generalzaton of the extracted features from the data plot of a normal process are determned frst. The th observaton from a normal process can be modeled as y r, where, and r are the values of process mean, standard devaton and standard normal varate respectvely. Therefore, replacng y by r and t by c n equatons (1), () and (4), the values of the correspondng features are obtaned as follows: RVE ( r ) ( 1) ( ) ( ) r r r r ( ), (7) B c r ( ) 1 c 1, (8) AASBP r r 1. (9) c It s noted from equaton (7) that the value of RVE s gven by a functon of r only for a gven. Ths mples that the feature RVE s always ndependent of and. The equatons (8) and (9) ndcate that for gven, the values of B and AASBP wll also be expressed as a functon of r only, f c 1. It can be shown that the value of slope of the LS lne ftted to any number of observatons wll be ndependent of and, f c 1. Ths s ndcatve that the feature SASPE wll also be ndependent of the two process parameters under the condton that each samplng nterval s represented by a lnear dstance equal to one standard devaton. On the other hand, the condton for negatve or postve sgn of the slope (B) wll depend on the value of the numerator n equaton (8). The sgn wll be negatve,.e. SB = f 1 c r ( ) or, r ( ). So, the condton for negatve (or smlarly postve) sgn of the slope does not depend on the values of, and c. Ths mples that the value of SB wll be ndependent of the 1

8 1 Gaur and Chakraborty process mean and standard devaton wthout requrng any condton on representaton of the samplng ntervals n the control chart plot. As t s noted from equaton (3), t s necessary to derve the values of ALS and SD for computaton of the value of ALSPI. For smplcty, let us consder a possble appearance of the frst three data ponts on a control chart plot, as shown n Fgure 3 and calculate the respectve ALS and SD values. A B I C E J K D Fgure 3. Computaton of ALS. F G In Fgure 3, BF represents the LS lne, yˆ b ˆ + bt 1, where, yˆ s the predcted observaton at th tme pont, and ˆb and ˆb 1 are the two constant terms. The constants ˆb and ˆb 1 represent the ntercept and slope of the LS lne respectvely. The values of ˆb and ˆb 1 can be estmated as below: ˆb y bˆ t 1 where, r r ( ) 1 1 rp Q 3, (1) P 1 P, a functon of only for gven, and Q r ( ), a functon of r and only for gven. 3 1 ˆ 3 y t t bˆ 1 r ( ) 1 t t c Q c P. (11) The ponts A, D and G, n Fgure 3, are the plots of the frst three observatons on the control chart. Two straght lnes IE and KF are drawn parallel to the mean lne and the lengths of these lnes represent the samplng nterval n the control chart plot. Therefore, t can be shown that IE = KF = c. AB = rp Q Q y1 y ˆ1 = r1. (1) P P

9 Feature-Based Recognton of Control Chart Patterns 11 DJ = rp Q Q y y ˆ = r. (13) P P GF = rp Q 3Q y3 y ˆ3 = r3. (14) P P It may be noted that n Fgure 3, ABC and CDJ are smlar trangles. The trangles BCI and CEJ are also smlar. Thus, usng the propertes of smlar trangles, t can be found that IC BC AB CE CJ DJ or, rp Q Q r1 c CE Q P CE rp Q Q r P P or, CE rp Q Q r c P P rp Q Q rp Q Q r1 r P P P P (15) IC c CE rp Q Q r1 c P P rp Q Q rp Q Q r1 r P P P P. (16) Here, the straght lnes IC and CE represent the heghts of ABC and CDJ respectvely. Therefore, the value of ALS s obtaned as ALS area of ABC + area of CDJ + area of trapezum DGFJ c [ rp Q Q r1 P P rp Q Q rp Q Q r1 r P P P P + rp Q Q r P P rp Q Q rp Q Q r1 r P P P P

10 1 Gaur and Chakraborty rp + Q Q rp Q 3Q r r3 ]. P P P P (17) and the value of SD s gven by the followng expresson: 3 3 ( y y) ( r r) 1 1 SD. (18) 31 Thus, the value of ALSPI, whch can be computed usng equaton (3), wll become a functon of r, and c. However, f the constant lnear dstance, c s made equal to one process standard devaton, the value of ALSPI wll be gven by a functon of r only mplyng that the magntude of ALSPI wll become ndependent of the values of process mean and standard devaton. The feature SRAGE s defned based on the segmentaton of the observaton wndow nto four equal segments, as shown n Fgure. The value of the slope of the straght lne drawn through the mdponts of frst and second segments.e. s 1 wll be gven by the followng expresson: 1 1 y ( 4) ( 4) ( 4) ( 4) 1 ( 4) 1 1 ( 4) ( 4) s 1 1 c ( 4) ( 4) ( 4) 1 1 y c ( 4) ( 4) r r ( 4) 1 1 ( 4) ( 4) c ( 4) 1 1. The above expresson reveals that the value of s 1 wll be gven by a functon of r only f c 1. Smlarly, the values of s jk (for all j and k ) wll also be ndependent of the two process parameters, when c 1. Thus, the value of SRAGE wll be ndependent of the values of the two process parameters under the condton that each samplng nterval n the control chart plot s represented usng a lnear dstance equal to one standard devaton of the underlyng process. All unnatural patterns are generated due to addton of some extraneous systematc varatons nto a normal process. All these extraneous varatons can be expressed n terms of process standard devaton and consequently, the equaton for modelng the observatons from an unnatural pattern can be expressed n the form of the equaton as used for modelng a normal pattern,.e. y a coffcent. For example, the equaton for modelng an UT pattern s y r g, where, g w s the gradent and w s a constant. The same equaton can be expressed as y ( r w ). The dependence propertes of a feature wth respect to process mean and standard devaton for all the unnatural patterns wll, therefore, be smlar to normal pattern. It can, therefore, be concluded that the magntudes of all the extracted features from the control chart plot n an observaton wndow wll become ndependent of process mean and standard devaton under the condton that each nterval n the plot s represented by a lnear dstance equal to one standard devaton. All the features, therefore, are extracted here assumng that a samplng nterval n the control chart plot s represented by a lnear dstance, c 1.

11 Feature-Based Recognton of Control Chart Patterns Control Chart Pattern Recognton 4.1. Feature-Based Heurstcs Ths technque uses smple IF (condton) THE (acton) heurstc rules. The condtons for the rules set the threshold values of the features and the actons are the classfcaton decsons. The set of heurstcs, arranged as a decson tree, can provde easly understood and nterpreted nformaton regardng the predctve structure of the data for varous features. Gven a set of extracted features from the learnng samples, classfcaton tree programs lke CART (Classfcaton and regresson trees) [1] and QUEST (Quck, unbased, effcent statstcal tree) [1] can generate a set of heurstcs based on these features arranged as bnary decson tree, whch can be used for recognton of control chart patterns (CCPs). Both these tree-structured classfcaton algorthms allow automatc selecton of the rght-szed tree that has the optmal predctve accuracy. The procedures for the rght-szed tree selecton are not foolproof, but at least, they take the subjectve judgment out of the process of choosng the rght-szed tree and thus avod over fttng and under fttng of the data. The CART performs an exhaustve grd search of all the possble unvarate splts whle developng a classfcaton tree. CART searches can be lengthy when there are a large number of predctor varables wth many levels and t s based towards choosng the predctor varables wth more levels for splts [1]. On the other hand, QUEST employs modfcaton of the recursve quadratc dscrmnant analyss and ncludes a number of nnovatve features for mprovng the relablty and effcency of the classfcaton tree that t computes. QUEST s fast and unbased. QUEST s lack of bas n varable selecton for splts s a dstnct advantage when some predctor varables have few levels and other have many. Moreover, QUEST does not sacrfce the predctve accuracy for speed [11]. It s planned, therefore, to adopt QUEST algorthm for determnng the tree-structured classfcaton rules for CCP recognton. 4.. Feature-Based eural etwork Many researchers [6, 8, 1, 17, 19, ] have successfully used multlayer perceptron (MLP) wth back propagaton learnng rule to solve the pattern classfcaton problems. The advantage of ths type of neural network s that t s smple and deally suted for pattern recognton tasks [9]. One man dsadvantage of the back propagaton learnng rule s reported to be ts longer convergence tme. However, the problem may not be sgnfcant for applcaton to pattern classfcaton, snce tranng s envsaged as an nfrequent and off-lne actvty. The MLP archtecture s, therefore, used here for developng the CCP recognton system. The basc structure of an MLP archtecture comprses an nput layer, one or more hdden layer(s) and an output layer. The nput layer receves numercal values from the outsde world and the output layer sends nformaton to the users or external devces. The number of nodes n the nput layer s set accordng to the actual number of features used,.e. sx. The number of output nodes s set correspondng to the number of pattern classes,.e. eght. The processng elements n the hdden layer are used to create nternal representatons. Each processng element n a partcular layer s fully connected to every processng element n the succeedng layer. There s no feed back to any of the processng elements. Prelmnary nvestgatons are carred out to choose the number of hdden layers, number of nodes n these layers and select an effcent back propagaton tranng algorthm by conductng experments coded n MATLAB usng ts A toolbox [3]. Based on the

12 14 Gaur and Chakraborty results of these ntal nvestgatons, t s decded to use one hdden layer for the neural network and the number of nodes n the hdden layer s set to 15. Thus, the A structure s Snce the supervsed tranng approach s used here, each pattern presentaton s tagged wth ts respectve label. The target values for the recognzer s output nodes are represented by a vector of 8 elements, e.g. the desred output vector for OR and STA patterns are [.9,.1,.1,.1,.1,.1,.1,.1] and [.1,.9,.1,.1,.1,.1,.1,.1] respectvely. The maxmum value (.9) dentfes the correspondng node that s expected to secure the hghest output for a pattern consdered to be correctly classfed. Gradent descent wth momentum and adaptve learnng rate (trangdx) (trangdx s the code avalable n MATLAB for the tranng algorthm) s found to provde reasonably good performance and more consstent results. It s also more memory-effcent. The trangdx s, therefore, chosen here for tranng of the neural network. The actvaton functons used are hyperbolc tangent (tansg) for the hdden layer and sgmod (logsg) for the output layer. The hyperbolc tangent functon transforms the layer nputs to output range from 1 to +1 and the sgmod functon transforms the layer nputs to output range from to Expermental Procedure 5.1. Sample Patterns Sample patterns are requred for assessng the usefulness of varous possble shape features n dscrmnatng dfferent patterns as well as developng/valdatng a CCP recognzer. Ideally, sample patterns should be collected from a real manufacturng process. Snce, a large number of patterns are requred for developng and valdatng a CCP recognzer, and as those are not economcally avalable, smulated data are often used. Ths s a common approach adopted by the other researchers too. Table. Equatons and parameters for control chart pattern smulaton. Control chart Pattern parameters Parameter values Pattern equatons patterns Mean ( ) 8 ormal y r Standard devaton ( ) 5 Stratfcaton Random nose ( ). to.4 y r Systematc Systematc departure (d) 1 to 3 y r d( 1) Cyclc Ampltude (a) Perod (T) 1.5 to.5 8 and 16 y r asn( / T) Increasng Gradent (g).5 to.1 y r g trend Decreasng Gradent (g) -.1 to -.5 y r g trend Shft magntude (s) 1.5 to.5 y r ks ; Upward shft Shft poston (P) 9, 17, 5 k = 1 f P, else k = Downward Shft magntude (s) -.5 to -1.5 y r ks ; shft Shft poston (P) 9, 17, 5 k = 1 f P, else k = ote: = dscrete tme pont at whch the pattern s sampled ( = 1,.., 3), th r = random value of a standard normal varate at tme pont, and th y = sample value at tme pont. Snce a large wndow sze can decrease the recognton effcency by ncreasng the tme requred to detect the patterns, an observaton wndow wth 3 data ponts s

13 Feature-Based Recognton of Control Chart Patterns 15 consdered here. The equatons along wth the correspondng parameters used for smulatng the eght basc CCPs are gven n Table. The values of dfferent parameters for the unnatural patterns are randomly vared n a unform manner between the lmts shown. A set of 4 (58) sample patterns are generated from 5 seres of standard normal varates. It may be noted that each set contans equal number of samples for each pattern class. Ths s so done because f a partcular pattern class s traned more number of tmes, the neural network wll become based towards that partcular pattern. 5.. Experments Multple sets of learnng samples as well as test samples are needed to rgorously evaluate the recognton and generalzaton performances of the heurstc and A-based CCP recognzers that may be developed based on the selected set of shape features. Here, ten sets of tranng and ten sets of test samples of sze 4 each are generated for the purpose of the expermentaton. Only dfference between these twenty sets of sample patterns s n the random generaton of standard normal varate and values of dfferent pattern parameters wthn ther respectve lmts. Each set of tranng samples s subjected to the classfcaton tree analyss usng QUEST algorthm wth the followng parameters: Pror probabltes for dfferent patterns: proportonal to class sze Msclassfcaton cost of a pattern: equal for all the patterns Stoppng rule: Prune on msclassfcaton error Stoppng parameters: (a) value of n for Mnmum n rule = 5, and (b) value of for Standard Error rule = 1.. Ths results n ten dfferent classfcaton trees gvng ten heurstc-based CCP recognzers. These recognzers are labeled n Table 3. The recognton performance of each heurstc-based recognzer s then evaluated usng all the ten sets of test samples. On the other hand, a traned A can only accept a certan range of nput data. The extracted shape features are, therefore, scaled (pre-processed) such that ther values fall wthn (1, +1) before ther presentaton to A for the learnng process. The neural network s traned ten tmes by exposng t separately to the ten sets of tranng samples wth the followng tranng parameters: Maxmum number of epochs = 5 Error goal =.1 Learnng rate =.1 Momentum constant =.5 Rato to ncrease learnng rate = 1.5 Rato to decrease learnng rate =.7 The tranng s stopped whenever ether the error goal has been acheved or the maxmum allowable number of tranng epochs has been met. In ths process, ten dfferent A-based recognzers are developed. All these A-based recognzers have the same archtecture and dffer only n the tranng data sets used. These recognzers are labeled.1.1 n Table 4. The recognton performance of each A-based recognzer s then evaluated usng all the ten sets of test samples. As one of the ams of ths expermentaton s to compare the relatve performance of the two types of recognzers, the correspondng heurstc and A-based recognzers are developed usng the same set of tranng samples. 6. Results and Dscussons The results obtaned durng the tranng and verfcaton phases of the developed heurstc and A-based CCP recognzers are gven n Tables 3 and 4 respectvely. It s observed that the recognton performance s qute good for both these two types of

14 16 Gaur and Chakraborty recognzers. The overall mean percentage of correct recognton acheved by the heurstc-based recognzer at the tranng and verfcaton (recall) phases are 96.79% and 95.4% respectvely, and the overall mean percentage of correct recognton acheved by the A-based recognzer at the tranng and recall phases are 96.38% and 95.9% respectvely. It may be noted that the recognton performance of both these two types of recognzers at the recall phase s nferor to the recognton accuracy that s acheved durng the tranng phase. Ths decrease n recognton accuracy durng the recall phase s more for the heurstc-based recognzer than the A-based recognzer. Ths s ndcatve that the predctve performance for A-based recognzer s better than heurstc-based recognzer. The percentage of correct recognton for heurstc and A-based recognzers at recall phase ranges from 94.55% to 95.78% and from 95.5% to 96.6% respectvely. Pared t-tests ( =.1) are conducted for 1 pars of heurstc and A-based recognzers for ther performance n terms of percentage of correct classfcaton. The results of statstcal sgnfcance tests are summarzed n Table 5. These results suggest that the dfference n recognton accuracy between these two types of recognzers s sgnfcant. Ths confrms that A-based recognzers gve better recognton performance compared to heurstc-based recognzers. Table 3. Tranng and verfcaton (recall) performance of heurstc-based recognzers. Tranng phase Verfcaton phase Recognzer umber of Correct Correct classfcaton (%) number splts n the tree classfcaton (%) Mean Standard devaton Overall mean Table 4. Tranng and verfcaton performance of A-based recognzers. Tranng phase Verfcaton phase Recognzer umber of Correct Correct classfcaton (%) number epochs classfcaton (%) Mean Standard devaton Overall mean

15 Feature-Based Recognton of Control Chart Patterns 17 Table 5. Statstcal sgnfcance tests for dfference n recall performance. H : H : 1 Hypothess recall (A-Heurstc) recall (A-Heurstc) tstatstcs t crtcal Decson Reject H On the other hand, the standard devaton (sd) of correct recognton percentage durng the recall phase s observed to be consstently hgher for heurstc-based recognzers than A-based recognzers. The statstcal sgnfcance for ths dfference s carred out usng F-test. The crtcal value of F 9,9 ( =.1) s 5.35, whereas the computed F-statstcs Heurstc A ( sd / sd ) for all the ten pars of heurstc and A-based recognzers s found to be greater than 5.5. Ths mples that A-based recognzer has better predctve consstency than heurstc-based recognzer. However, heurstc-based recognzer has the dstnct advantage that the practtoners can clearly understand how a partcular pattern class s dentfed by the use of relevant shape features. Ths advantage can compensate the poorer recognton and predctve consstency of heurstc-based recognzer. The best heurstc-based recognzer n terms of consstency of recognton performance s recognzer no. 1.5, and ts heurstc rules n the form of classfcaton tree are shown n Fgure 4. ALSPI <=.98 STA AASBP <=1.831 SYS SASPE <=.435 SRAGE <=.16 RVE<=1.333 CYC SB=1 SB= SRAGE <=.417 US CYC ALSPI <=.571 ALSPI <=.579 SRAGE <=.48 SB= DT UT RVE<=1.85 RVE<=1.89 RVE<=1.13 RVE<=1.17 ALSPI <=.618 SASPE <=.316 OR UT OR UT DT OR DT DS OR DS CYC RVE<=1.47 DT ALSPI <=.487 CYC OR Fgure 4. Classfcaton tree for recognton of control chart patterns.

16 18 Gaur and Chakraborty 6.1. Senstvty Studes for Pattern Recognton In ths paper, all the shape features are extracted consderng that the samplng nterval n the control chart plot s represented by a constant dstance, c 1, where = 5 (known). In realty, the value of s to be estmated. Therefore, t s planned to study the senstvtes of both the heurstc and A-based recognzers wth respect to the estmaton error of. To study the senstvtes of pattern recognzers, t s necessary to generate addtonal test patterns, consderng the values of dfferent from ts known value and then subjectng the patterns to classfcaton by the developed recognzers. All these test data sets should be smulated usng the same seres of standard normal data to ensure that the sequences of randomness are smlar n all these test sets; otherwse the effect of changes n the sequences of randomness wll be confounded wth the effect of departure of from ts known value. A set of 5 tme seres wth 3 standard normal data, from whch the sample patterns leadng to the heurstcs shown n Fgure 4 are developed s taken for generaton of the addtonal test patterns. On the other hand, care s also taken so that the effect of random changes n the values of pattern parameters s not confounded wth the effect of devaton of from ts known value. For the purpose of generaton of addtonal test patterns, therefore, the values of varous pattern parameters are fxed as: =8, =.3, d =, a =, T =16, g =.75, s = and P =16. These values are mostly the mdponts of the ranges of respectve pattern parameters, as shown n Table. Usng the above mentoned tme seres of standard normal data and pattern parameters, fve addtonal test pattern sets wth sze 4 each are generated consderng the values of as 4.5, 4.75, 5., 5.5 and 5.5 (mplyng 1% devaton of from ts known value) and they are labeled 1 to 5. These addtonal test patterns are then subjected to classfcaton usng the heurstc and A-based recognzers (recognzer no. 1.5 and.5 respectvely). The acheved recognton performance s shown n Table 6. It s observed that for both the recognzers, the recognton performance s the maxmum for the addtonal pattern set number 3, whch s generated assumng = 5. More and more s the devaton n the value of, poorer becomes the recognton performance. However, the recognton performance remans reasonably well wthn the 1% devaton of for both the recognzers. The ranges of varaton n recognton performance for the heurstc and A-based recognzers are.45 and 1.45 respectvely. Ths mples that the performance of A-based recognzer s relatvely less senstve to estmaton error of the process standard devaton than heurstc-based recognzer. Table 6. Recognton accuracy (%) for addtonal test patterns. Type of recognzer Addtonal sets of test patterns Range Heurstc recognzer A recognzer Conclusons A new approach for generalzaton of feature-based CCP recognton system s presented. In ths approach, values of the extracted shape features from the control chart plot of actual process data become ndependent of mean and standard devaton of the underlyng process. The advantage of the proposed approach s that one has to know or

17 Feature-Based Recognton of Control Chart Patterns 19 estmate the process standard devaton only, whereas n other tradtonal approaches lke standardzaton of process data, both the process mean and standard devaton values need to be known or estmated. The proposed approach s also advantageous over scalng of the process data nto (, 1) or (1, +1) nterval snce n ths approach, the dstncton between normal and stratfcaton patterns s preserved. Two CCP recognzers are developed based on a set of sx shape features usng heurstc and A technques, whch can recognze all the eght basc CCPs ncludng stratfcaton pattern. The relatve performance of the two feature-based approaches for CCP recognton s extensvely examned and ther senstvtes wth respect to estmaton error of process standard devaton are also studed. It s observed that the A-based recognzer acheves better recognton performance than the heurstc-based recognzer. The results also ndcate that the A-based recognzer gves more consstent performance and s less senstve to estmaton error of process standard devaton than the heurstc-based recognzer. However, the heurstc-based recognzer has the dstnct advantage that the practtoners can clearly understand how a partcular pattern s dentfed wth the help of relevant shape features. Ths advantage can compensate the poorer recognton performance and consstency of the heurstc-based recognzers. References 1. Breman, L., Fredman, J. H., Olshen, R. A. and Stone, C. J. (1984). Classfcaton and Regresson Trees. Wadsworth and Brooks, Monterey, CA.. Cheng, C. S. (1997). A neural network approach for the analyss of control chart patterns. Internatonal Journal of Producton Research, 35(3), Demuth, H. and Beale, M. (1998). eural etwork Toolbox User s Gude. Math Works, atck, MA. 4. Evans, J. R. and Lndsay, W. M. (1988). A framework for expert system development n statstcal qualty control. Computers and Industral Engneerng, 14(3), Gaur, S. K. and Chakraborty, S. (7). A study on the varous features for effectve control chart pattern recognton, Internatonal Journal of Advanced Manufacturng Technology, 34(3-4), Guh, R. S., Zorrassatne, F., Tannock, J. D. T. and O Bren, C. (1999). On-lne control chart pattern detecton and dscrmnaton a neural network approach. Artfcal Intellgence n Engneerng, 13(4), Guh, R. S. and Shue, Y. R. (5). On-lne dentfcaton of control chart patterns usng self-organzng approaches. Internatonal Journal of Producton Research, 43(6), Hassan, A., ab Baksh, M. S., Shaharoun, A. M. and Jamaluddn, H. (3). Improved SPC chart pattern recognton usng statstcal features. Internatonal Journal of Producton Research, 41(7), Haykn, S. (1999). eural etwork: A Comprehensve Foundaton. Prentce-Hall, Englewood Clffs, J. 1. Hwarng, H. B. and Hubele,. F. (1993). Back-propagaton pattern recognsers for X control charts: methodology and performance. Computers and Industral Engneerng, 4(), Lm, T.-S., Loh, W.-Y. and Shh, Y.-S. (1997). An emprcal comparson of decson trees and other classfcaton methods. Techncal Report 979, Department of Statstcs, Unversty of Wsconsn, Madson.

18 Gaur and Chakraborty 1. Loh, W.-Y., and Shh, Y.-S. (1997). Splt selecton methods for classfcaton trees. Statstca Snca, 7, Montgomery, D. C. (1). Introducton to Statstcal Qualty Control. John Wley and Sons, ew York. 14. Montgomery, D. C. and Peck, E. A. (198). Introducton to Lnear Regresson Analyss. John Wley and Sons, ew York. 15. elson, L. S. (1984). The Shewhart control chart test for specal causes. Journal of Qualty Technology, 16(4), elson, L. S. (1985). Interpretng Shewhart X control chart. Journal of Qualty Technology, 17(), Perry, M. B., Spoerre, J. K. and Velasco, T. (1). Control chart pattern recognton usng back propagaton artfcal neural networks. Internatonal Journal of Producton Research. 39(15), Pham, D. T. and Oztemel, E. (199 a ). XPC: an on-lne expert system for statstcal process control. Internatonal Journal of Producton Research, 3(1), Pham, D. T. and Oztemel, E. (199 b ). Control chart pattern recognton usng neural networks. Journal of System Engneerng, (4), Pham, D. T. and Wan, M. A. (1997). Feature-based control chart pattern recognton. Internatonal Journal of Producton Research, 35(7), Swft, J. A. and Mze, J. H. (1995). Out-of-control pattern recognton and analyss for qualty control charts usng LISP-based systems. Computers and Industral Engneerng, 8(1), Western Electrc Company. (1958). Statstcal Qualty Control Handbook, Western Electrc, Indanapols. Authors Bographes: Susanta Kumar Gaur s a faculty member n the Statstcal Qualty Control & Operatons Research Dvson of Indan Statstcal Insttute (ISI) at Kolkata, Inda. Besdes teachng, he s hghly nvolved n consultancy works n varous ndustres and appled research n the feld of qualty control, qualty assurance and qualty management. He has publshed several papers n varous natonal and nternatonal journals durng past 8 years. Shankar Chakraborty s Reader n the Producton Engneerng Department of Jadavpur Unversty at Kolkata, Inda. He s hghly engaged n teachng and research. Hs research nterests nclude the feld of ntellgent manufacturng, qualty functon deployment, applcaton of analytcal herarchy process and artfcal neural networks. He has publshed many papers n varous natonal and nternatonal journals durng last 1 years.