Pixel Bar Charts: A New Technique for Visualizing Large Multi-Attribute Data Sets without Aggregation

Transcription

1 Pxel Bar Chart: A New Technque or Vualzng Large Mult-Attrbute Data Set wthout Aggregaton Danel Kem*, Mng C. Hao, Julan Lach*, Mechun Hu, Umehwar Dayal Hewlett Packar Reearch Laboratore, Palo Alto, CA Abtract Smple preentaton graphc are ntutve an eay-to-ue, but how only hghly aggregate ata an preent only a very lmte number o ata value (a n the cae o bar chart). In aton, thee graphc may have a hgh egree o overlap whch may occlue a gncant porton o the ata value (a n the cae o the x-y plot). In th paper, we thereore propoe a generalzaton o tratonal bar chart an x-y-plot whch allow the vualzaton o large amount o ata. The bac ea to ue the pxel wthn the bar to preent the etale normaton o the ata recor. Our o-calle pxel bar chart retan the ntutvene o tratonal bar chart whle allowng very large ata et to be vualze n an eectve way. We how that, or an eectve pxel placement, we have to olve complex optmzaton problem, an preent an algorthm whch ecently olve the problem. Our applcaton ung real-worl e-commerce ata how the we applcablty an ueulne o our new ea.. Introucton Becaue o the at technologcal progre, the amount o ata whch tore n computer ncreae raply. Reearcher rom the Unverty o Berkeley etmate that every year about Exabyte o ata generate, wth % avalable only n gtal orm. Toay, computer typcally recor even mple tranacton o everyay le, uch a payng by cret car, ung the telephone an hoppng n e-commerce tore. Th ata collecte becaue bune people beleve that t a potental ource o valuable normaton an coul prove a compettve avantage. Fnng the valuable normaton hen n the ata, however, a cult tak. Vual ata exploraton technque are npenable to olvng th problem. In mot ata mnng ytem, however, only mple graphc, uch a bar chart, pe chart, x-y plot, etc., are ue to upport the ata mnng proce. Whle mple graphc are ntutve an eay-to-ue, they ether: - how hghly aggregate ata an actually preent only a very lmte number o ata value (a n the cae o bar chart or pe chart), or - have a hgh egree o overlap whch may occlue a gncant porton o the ata value (a n the cae o x- y plot). The ueulne o bar chart epecally lmte the uer nterete n relatonhp between erent attrbute uch a prouct type, prce, number o orer, an quantte. The reaon or th lmtaton that multple bar chart or erent attrbute o not upport the covery an correlaton o nteretng ubet, whch one o the man tak n mnng cutomer tranacton ata. For an analy o large volume o e-commerce tranacton [Ec 99], the vualzaton o hghly aggregate ata not ucent. What neee to preent an overvew o the ata but at the ame tme how the etale normaton or each ata tem. In th paper, we ecrbe a new vualzaton technque calle pxel bar chart. The bac ea o pxel bar chart to ue the ntutve an wely ue preentaton paragm o bar chart, but alo ue the avalable creen pace to preent more etale normaton. By colorng the pxel wthn the erent bar accorng to the value o the ata recor, very large amount o ata can be preente to the uer. To make the play more meanngul, two parameter o the ata recor are ue to mpoe an orerng on the pxel n the x- an y-recton. Pxel bar chart can be een a a generalzaton o bar chart. They combne the general ea o x-y plot an bar chart to allow an overlap-ree, nonaggregate play o mult-attrbute ata. Snce pxel bar chart ue each pxel to preent one ata value, they belong to pxel-orente technque. Other pxelorente technque nclue the pral technque [KK 94], the recurve pattern technque [KKA 95], an the crcle egment technque [AKK 96]. Other clae o normaton vualzaton technque nclue geometrc proecton technque (e.g. [In 85, ID 90]), con -bae technque (e.g., [PG 88, Be 90]), herarchcal technque (e.g., [LWW 90, RCM 9, Shn 9]), graph-bae technque (e.g., [EW 93, BEW 95]), whch n general are combne wth ome nteracton technque (e.g., [BMMS 9, AWS 9, ADLP 95]) an ometme alo ome torton technque [SB 94, LRP 95].. From Bar Chart to Pxel Bar Chart A common metho or vualzng large volume o ata to ue bar chart. Bar chart are wely ue an are very ntutve an eay to unertan. Fgure llutrate the ue o a regular bar chart to vualze cutomer trbuton n an e-commerce ale tranacton. The heght o the bar repreent the number o cutomer or erent prouct categore. Bar chart, however, requre a hgh egree o ata aggregaton an actually how only a rather mall number o ata value (only value are hown n Fgure ). Thereore, or ata exploraton o large multmenonal ata, they are o lmte value an are not able to how mportant normaton uch a: *Preently wth the Computer Scence Inttute, Unverty o Contance, Contance, Germany [email protected]; (mng_hao, mhu, ayal)@hpl.hp.com; [email protected]

2 - ata trbuton o multple attrbute - local pattern, correlaton, an tren - etale normaton, e.g., each cutomer prole. Bac Iea o Pxel Bar Chart Pxel bar chart are erve rom regular bar chart (ee Fgure a). The bac ea o a pxel bar chart to preent the ata value rectly ntea o aggregatng them nto a ew ata value. The approach to repreent each ata tem (e.g. a cutomer) by a ngle pxel n the bar chart. The etale normaton o one attrbute o each ata tem encoe nto the pxel color an can be accee an playe a neee. One mportant queton : how are the pxel arrange wthn each bar? Our ea to ue one or two attrbute to eparate the ata nto bar (vng attrbute) an then ue two atonal attrbute to mpoe an orerng wthn the bar (ee Fgure or the general ea). The pxel bar chart can thereore be een a a combnaton o the tratonal bar chart an the x-y agram. y - orerng attrbute x - orerng attrbute vng attrbute Fgure : A Pxel Bar Chart Now, we have a vualzaton n whch one pxel correpon to one cutomer. I the parttonng attrbute reunantly mappe to the color o the pxel, we obtan the regular bar chart hown n Fgure a (Fgure b how the equal-heght-bar-chart" whch we wll explan n the next ecton). Pxel bar chart, however, can be ue to preent large amount o etale normaton. The one-toone correponence between cutomer an pxel allow u to ue the color o the pxel to repreent atonal attrbute o the cutomer or example, ale amount, #o vt, or ale quantty. In Fgure 3a, a pxel bar chart ue to vualze thouan o e-commerce ale tranacton. Each pxel n the vualzaton repreent one cutomer. The number o cutomer can be a large a the creen ze (about.3 mllon). The pxel bar chart hown n Fgure 3a ue prouct type a the vng attrbute an number o vt an ollar amount a the x an y orerng attrbute. The color repreent the ollar amount pent by the correponng cutomer. Hgh ollar amount correpon to brght color, low ollar amount to ark color.. Space-Fllng Pxel Bar Chart One problem o tratonal bar chart that a large porton o the creen pace can not be ue ue to the erng heght o the bar. Wth very large ata et, we woul lke to ue more o the avalable creen pace to vualze the ata. One ea that ncreae the number o playable ata value to ue equal-heght ntea o equal-wth bar chart. In Fgure b, the regular bar chart o Fgure a hown a an equal-heght bar chart. The area (wth) o the bar correpon to the attrbute hown, namely the number o cutomer. I we now apply our pxel bar chart ea to the reultng bar chart, we obtan pace-llng pxel bar chart whch ue vrtually all pxel o the creen to play cutomer ata tem. In Fgure 3b, we how an example o a pace-llng pxel bar chart whch ue the ame vng, orerng, an colorng attrbute a the pxel bar chart n Fgure 3a. In th way, each cutomer repreente by one pxel. Note that pxel bar chart generalze the ea o regular bar chart. I the parttonng an colorng attrbute are entcal, both type o pxel bar chart become cale veron o ther regular bar chart counterpart. The pxel bar chart can thereore be een a a generalzaton o the regular bar chart but they contan gncantly more normaton an allow a etale analy o large orgnal ata et..3 Mult-Pxel Bar Chart In many cae, the ata to be analyze cont o multple attrbute. Wth pxel bar chart we can vualze attrbute value ung mult-pxel bar chart whch ue erent color mappng but the ame parttonng an orerng attrbute. Th mean that the arrangement o ata tem wthn the correponng bar o mult-pxel bar chart the ame,.e., the colore pxel correponng to the erent attrbute value o the ame ata tem have a unque poton n the bar. In Fgure 4, we how an example o three pxel bar chart wth prouct type a the vng attrbute an number o vt an ollar amount a the x an y orerng attrbute. The attrbute whch are mappe to color are ollar amount pent, number o vt, an ale quantty. Note that the pxel n correponng bar n multple bar chart are relate by ther poton,.e., the ame ata recor ha the ame relatve poton wthn each o the correponng bar. It thereore poble to relate the erent bar chart an etect correlaton. 3. Formal Denton o Pxel Bar Chart In th ecton we ormally ecrbe pxel bar chart an the problem that nee to be olve n orer to mplement an eectve pxel placement algorthm. 3. Denton o Pxel Bar Chart For a general enton o pxel bar chart, we nee to pecy the: - vng attrbute (or between-bar parttonng) - orerng attrbute (or wthn-bar orerng) - colorng attrbute (or pxel colorng). In tratonal bar chart there one vng attrbute whch partton the ata nto ont group correponng to the bar. In pace-llng bar chart, the bar correpon to a

3 y - orerng attrbute x - orerng attrbute a) Equal-Wth Bar Chart Fgure : Regular Bar Chart b) Equal-Heght Bar Chart a) Equal-Wth Pxel Bar Chart Fgure 3: Pxel Bar Chart b) Equal-Heght Pxel Bar Chart hgh $ A m o u n t low a) Color=ollar amount b) Color=number o vt c) Color=quantty Fgure 4: Mult-Pxel Bar Chart

4 parttonng o the creen accorng to the horzontal ax (x). 3 where D x, D y, O x, O y, C {A l,, A k, } an D x /D y are the vng attrbute n x-/y-recton, O x /O y are the orerng attrbute n x-/y-recton, an C the colorng attrbute. The mult-pxel bar chart o ale tranacton hown n Fgure 4, or example, are ene by the ve-tuple <prouct type,, no. o vt, ollar amount, C> where C correpon to erent attrbute,.e., number o vt, ollar amount, quantty. Fgure 5: Dvng attrbute on x-ax (e.g., Dx = ) We may generalze the enton o pace-llng pxel bar chart by allowng more than one vng attrbute,.e. one or the horzontal ax (D x ) an one or the vertcal ax (D y ). 3 Fgure 6: Dvng attrbute on x- an y-ax (e.g., Dx =, Dy= Regon) Next, we nee to pecy an attrbute or orerng the pxel n each pxel bar. Agan, we can o the orerng accorng to the x- an the y-ax,.e., along the horzontal (O x ) an vertcal (O y) axe ne each bar. 3 Fgure 7: Orerng attrbute on x- an y-ax (e.g., O x = Dollar Amount, O y=quantty) Fnally, we nee to pecy an attrbute or colorng the pxel. Note that n mult-bar chart we may agn erent attrbute to color n erent bar chart, whch enable the uer to relate the erent colorng attrbute an etect partal relatonhp among them. Note that the vng an orerng attrbute have to tay the ame n orer to o that. Let DB = {,, n } be the ata bae o n ata recor, each = K k, al Al, contng o k attrbute value { a,, a } where A l the attrbute name o value a l. Formally, a pxel bar chart ene by a ve tuple: <D x, D y, O x, O y, C > 3. Formalzaton o the Problem The bac ea o pxel bar chart to prouce ene pxel vualzaton whch are capable o howng large amount o ata on a value by value ba wthout aggregaton. The pecc requrement or pxel play are: - ene play,.e., bar are lle completely - non-overlappng,.e. no overlap o pxel n the play - localty,.e., mlar ata recor are place cloe to each other - orerng,.e., orerng o ata recor accorng to O x, O y. To ormalze thee requrement we rt have to ntrouce the creen potonng uncton : A K Ak Int Int, whch etermne the x-/y-creen poton o each ata recor,.e., ( ) = y) enote the poton o ata recor on the creen, an ( ). x enote the x- coornate an ( ). y the y-coornate. Wthout lo o generalty, we aume that O x = A an O y = A. The requrement can then be ormalze a:. Dene Dplay Contrant The ene play contrant requre that all pxel row (column) except the lat one are completely lle wth pxel. For equal-wth bar chart, the wth w o the bar xe. For a partton p contng o p pxel, we have to enure that p / w : í wth ( í ) = (, ) =.. w, =.. For equal-heght bar chart o heght h the correponng contrant p / h, =.. h : í wth ( í ) = (, ) =... No -Overlap Contrant The no-overlap contrant mean that a unque poton agne to each ata recor. Formally, we have to enure that two erent ata recor are place at erent poton,.e., DB : ( ) ( )., The element ue no attrbute pece.

5 3. Localty Contrant In ene pxel play the localty o pxel play an mportant role. Localty mean that mlar ata recor are place cloe to each other. The parttonng n pxel bar chart enure a bac mlarty o the ata recor wthn a ngle bar. In potonng the pxel wthn the bar, however, the localty property alo ha to be enure. For the ormalzaton, we nee a uncton m(, ) [0 ] whch etermne the mlarty o two ata recor an the nvere uncton o the pxel placement uncton -, whch etermne the ata recor or a gven (x,y)-poton on the creen. The localty contrant can then be expree a w h x= y= w h x= y= m( m( y), y), y + )) + ( x +, y)) mn Note that n general t not poble to place all mlar pxel cloe to each other whle repectng the ene play an no -overlap contrant. Th the reaon why the localty contrant ormalze a a global optmzaton problem. 4. Orerng Contrant The lat contrant whch cloely relate to the localty contrant the orerng contrant. The ea to enorce a one-menonal orerng n x- an y-recton accorng to the pece attrbute O x = A an O y =A. Formally, we have to enure,.. n : a > a ( ). x >,.. n : a > a ( ). y > ( ). x ( ). y Note that orerng the ata recor accorng to the attrbute an placng them n a row-by-row or column-by-column ahon may ealy ulll each one o the two contrant. Enurng both contrant at the ame tme may be mpoble n the general cae. We can ormalze the contrant a an optmzaton problem: w h ( (, ). (, ). + = x y a x y a x y= y). a y). a y + ). a w h ( (, ). (, ). + = x y a x y a x y= ( x +, y). a + ) + + ) mn Note that there may be a trae-o between the x- an the y- orerng contrant. In aton, the optma or the localty an the orerng contrant are n general not entcal. Th ue to the act that the mlarty uncton may nuce a erent optmzaton crteron than the x-/y-orerng contrant. For olvng the pxel placement problem, we thereore have to olve an optmzaton problem wth multple competng optmzaton goal. The problem a typcal complex optmzaton problem whch lkely to be NP-complete an can thereore only be olve ecently by a heurtc algorthm. 3.3 The Pxel Placement Algorthm For the generaton o pxel bar chart, we have to: () partton the ata et accorng to D x an D y ; () etermne the pxel color accorng to C ; an (3) place the pxel o each partton n the correponng regon accorng to O x, O y. The parttonng accorng to D x an D y an the color mappng are mple an traghtorwar to mplement, an thereore o not nee to be ecrbe n etal here. The pxel placement wthn one bar, however, a cult optmzaton problem becaue t requre a two -menonal ort. In the ollowng, we ecrbe our heurtc pxel placement algorthm whch prove an ecent oluton to the problem. The bac ea o the heurtc pxel placement algorthm to partton the ata et nto ubet accorng to O x an O y, an ue thoe ubet to place the bottom- an let-mot pxel. Th prove a goo tartng pont whch the ba or the teratve placement o the remanng pxel. The algorthm work a ollow:. For an ecent pxel placement wthn a ngle bar, we rt etermne the one-menonal htogram or O x an O y, whch are ue to etermne the α-quantle o O x an O y. I the bar uner coneraton ha extenon w x h pxel, we etermne the w, K, ( w) w- quantle or the parttonng o O x, an the h, K, ( h ) h - quantle or the parttonng o O y. The quantle are then ue to etermne the partton X,,X w o O x an Y,,Y h o O y. The partton X,,X w are orte accorng to O y an the partton Y,,Y h accorng to O x.. We can tart now to place the pxel n the lower-let corner,.e., poton (,), o the pxel bar: (,) = mn { } = mn. a { }. a X Y Next we place all pxel n the lower an let pxel row o the bar. Th one a mn (,) = {. a }.. w X = mn, ) =.. Y ( {. a } = h 3. The nal tep the teratve placement o all remanng pxel. Th one tartng rom the lower let to the upper rght. I pxel at poton (-, ) an (, -) are alreay place, the pxel at poton (, ) etermne a We ue a color map, whch map hgh ata value to brght color an low ata value to ark color.

6 (, ) = mn X Y {. a +. a } X Y Becaue we have place the ata n a ata tructure a ntrouce n tep, the pxel to be place at each poton can be etermne n O() tme X Y. I X Y =, we have to teratvely exten the partton X an Y an coner ( X X ) + Y. I th et tll empty, we have to coner ( X X + ) ( Y Y+ ) an o on, untl a ata pont to be place oun. Note that th proceure qute ecent ue to the ata tructure ue. 4. The Pxel Bar Chart Sytem To analyze large volume o tranacton ata wth multple attrbute, pxel bar chart have been ntegrate wth a ata mnng vualzaton ytem [HDHDB 99]. The ytem ue a web brower wth a Java actvator to allow real-tme nteractve vual ata mnng on the web. 4. Sytem Archtecture an Component The pxel bar chart ytem connect to a ata warehoue erver an ue the atabae to query or etale ata a neee. The ata to bul the pxel array kept n memory to upport real-tme manpulaton an correlaton. A llutrate n Fgure 8, the pxel bar chart ytem archtecture contan three bac component:. Pxel array orerng an groupng A pxel array contructe rom the pxel bar chart ve tuple peccaton. One pxel repreent one ata recor,.e., a cutomer. The parttonng algorthm agn each ata recor to the correponng bar accorng to the parttonng attrbute(). The pxel placement mplement a mple veron o the heurtc algorthm preente n ubecton Multple lnke pxel bar chart In mult-bar chart, the poton o the pxel belongng to the ame ata recor reman the ame acro multpxel bar chart or correlaton. The color o the pxel correpon to the value o the electe attrbute. 3. Interactve ata exploraton Th ytem prove multaneou browng an navgaton o multple attrbute. 4. Interactve Data Analy Interactvty an mportant apect o the pxel bar chart ytem. To make large volume o mult-attrbute ataet eay to explore an nterpret, the pxel bar chart ytem prove the ollowng nteracton capablte: () vual queryng; () layere rll-own; (3) multple lnke vualzaton; an (4) zoom. The attrbute ue or parttonng (Dx, Dy), orerng (Ox, Oy), an colorng (C) can be electe an change at executon tme. For entyng correlaton, a ubet o ata tem n a pxel bar chart can be electe to get the pxel correponng to relate attrbute value hghlghte wthn the ame play. A rll-own technque allow the vewng o all relate normaton ater electng a ngle ata tem. When mult-bar chart are preente, pxel ree at the ame locaton acro all the chart wth erent attrbute. In aton to coverng correlaton an pattern, the uer may elect a ngle ata tem to relate all t attrbute value. 5. Applcaton an Evaluaton The pxel bar chart technque ha been prototype n everal e-commerce applcaton at Hewlett Packar Laboratore. It ha been ue to vually mne large volume o ale tranacton an cutomer hoppng actvte at HP hoppng web te. 5. Cutomer Analy The pxel bar chart ytem ha been apple to cutomer buyng pattern an behavor. In Fgure 9, the pxel o the bar chart repreent cutomer makng tranacton on the web. In the reultng pxel bar chart, cutomer wth mlar purchang behavor (.e., prouct type, geographcal locaton, ollar amount, number o vt, an quantty) are place cloe to each other. A tore manager can ue the vualzaton to raply cover cutomer buyng pattern an ue thoe pattern to target marketng campagn. Fgure 9 how the our attrbute o 06,99 cutomer buyng recor. The our pxel bar chart o Fgure 9 are contructe a ollow: () Prouct type the vng attrbute D x; () Dollar amount the x-orerng attrbute O x, Regon y-orerng attrbute O y or 0 Unte State regon; an (3) Regon, ollar amount, number o vt an quantty are the our colorng attrbute C. The uer may oberve the ollowng act: a) Regon attrbute There are 0 erent color to repreent 0 erent regon (labele -0 n Fgure 9a) n the Unte State. The colore wave ncate the number o cutomer n each regon. Regon 9 (larget area) oun to have the larget number o cutomer. Regon 7 (mallet area) ha the leat number o cutomer acro all prouct type. b) Dollar amount attrbute Prouct type 5 ha the mot top ollar amount ale (blue & brown). Type 6 an 7 have a very mall varance acro all regon (ol blue/brown). c) Number o vt attrbute The blue color trbuton n prouct type 4 ncate that cutomer o th prouct type come back more oten than cutomer o other prouct type. ) Quantty attrbute The green color o prouct type 6 ncate that n th category all cutomer bought the ame number o tem acro all regon. It alo obvou that prouct type 4 cutomer have the larget quantte.

7 Clent Pxel Bar Chart Server A) Pxel Array B) Mult-pxel Bar Chart C) Interacton ortng lnkng explorng groupng colorng analyzng Fgure 8: Sytem Archtecture & Component hgh low a) Color: regon b) Color: ollar amount c) Color: no. o vt ) Color: quantty Fgure 9: Mult-Pxel Bar Chart or Mnng 06,99 Cutomer Buyng Tranacton (D x =, D y =, O x =ollar amount, O y =regon, C) hgh cutomer A $345,000 cutomer A 5 vt cutomer A 500 tem low a) Color: ollar amount b) Color: no. o vt c) Color: quantty Fgure 0: Mult-Pxel Bar Chart or Mnng 405,000 Sale Tranacton Recor (D x =, D y =, O x =no. o vt, O y = ollar amount, C)

8 By relatng the multple pxel bar chart o Fgure 9, the uer may oberve that the top ollar amount cutomer come back more requently an purchae larger quantte. 5. Sale Tranacton Analy One o the common queton electronc tore manager ak how to ue the cutomer purchae htory or mprovng prouct ale an promoton. Prouct manager want to unertan whch prouct have the top ale an who are ther top ollar amount cutomer. Whle regular bar chart prove aggregate normaton on the number o cutomer by prouct type (Fgure ), the correponng pxel bar chart nclue mportant atonal normaton uch a the ollar amount trbuton o the ale. Fgure 0 llutrate an example o a mult-pxel bar chart o 405,000 mult-attrbute web ale tranacton. The vng attrbute (D x) agan prouct type; the orerng attrbute are number o vt an ollar amount (O x an O y ). The color (C) n the erent bar chart repreent the attrbute ollar amount, number o vt, an quantty. The ollowng normaton can be obtane: a) Prouct type 7 an 0 have the top ollar amount cutomer (ark color o bar 7, 0 n Fgure 0a). b) The ollar amount pent an #o vt are clearly correlate,.e. or prouct type 4 (lnear ncreae o ark color at the top o bar 4 n Fgure 0b). c) Prouct type 4 an have the hghet quantte ol (ark color o bar 4 an n Fgure 0c). ) By clckng on a pecc pxel (A), we may n out that cutomer A vte 5 tme, bought 500 tem, an pent $345,000 on prouct type 5. It urther nteretng that there are cluter o arker color n bar 4 o Fgure 0c, whch mean that there are certan range o ollar amount ale or whch the quantty ten to be hgher than n other egment. Th obervaton unexpecte an may be ue to enty the cluter o ale tranacton an make ue o the normaton to urther ncreae the ale. Note that the normaton mentone above cannot be etecte by regular bar chart. 6. Concluon In th paper, we preente pxel bar chart, a new metho or vualz ng large amount o mult-attrbute ata. The approach a generalzaton o tratonal bar chart an x-y agram, whch avo the problem o long normaton by aggregaton or overplottng. Intea, pxel bar chart map each ata pont to one pxel o the play. For generatng the pxel bar chart vualzaton, we have to olve a complex optmzaton problem. The pxel placement algorthm an ecent an eectve oluton to the problem. We apply the pxel bar chart ea to real ata et rom an e-commerce applcaton an how that pxel bar chart prove gncantly more normaton than regular bar chart. Acknowlegement Thank to Sharon Beach o HP Laboratore or her encouragement an uggeton, Shu F. W. an Bran O. rom HP Shoppng or provng uggeton an ata. an Graham P. o Aglent Laboratore or h revew an comment. Reerence [ADLP 95] Anupam V., Dar S., Lebre T., Petaan E.: DataSpace: 3-D Vualzaton o Large Databae, Proc. Int. Symp. on Inormaton Vualzaton, Atlanta, GA, 995, pp [AKK 96] Anker M., Kem D. 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