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 Ladch, Mechun Hu, Umehwar Dayal Sotware Technology Laboratory HP Laboratore Palo Alto HPL-00-9 Aprl th, 00* E-mal: (mng_hao, mhu, dayal)@hpl.hp.com, kem@normatk.un-kontanz.de, ulan@ladch.de pxel, bar chart, web tranacton, vualzaton Smple preentaton graphc are ntutve and eay-to-ue, but how only hghly aggregated data and preent only a very lmted number o data value (a n the cae o bar chart), and may have a hgh degree o overlap whch may occlude a gncant porton o the data value (a n the cae o the x-y plot). In th paper, we thereore propoe a generalzaton o tradtonal bar chart and x-y-plot whch allow the vualzaton o large amount o data. The bac dea to ue the pxel wthn the bar to preent the detaled normaton o the data record. Our o-called pxel bar chart retan the ntutvene o tradtonal bar chart whle allowng very large data et to be vualzed n an eectve way. We how that, or an eectve pxel placement, we have to olve complex optmzaton problem, and preent an algorthm whch ecently olve the problem. Our applcaton ung real-world e-commerce data how the wde applcablty and ueulne o our new dea. * Internal Acceon Date Only Approved or External Publcaton Preently wth the Computer Scence Inttute, Unverty o Contance, Contance, Germany Copyrght Hewlett-Packard Company 00

2 Pxel Bar Chart: A New Technque or Vualzng Large Mult-Attrbute Data Set wthout Aggregaton Danel Kem*, Mng C. Hao, Julan Ladch*, Mechun Hu, Umehwar Dayal Hewlett Packard Reearch Laboratore, Palo Alto, CA (mng_hao, mhu, dayal)@hpl.hp.com Abtract Smple preentaton graphc are ntutve and eayto-ue, but how only hghly aggregated data and preent only a very lmted number o data value (a n the cae o bar chart), and may have a hgh degree o overlap whch may occlude a gncant porton o the data value (a n the cae o the x-y plot). In th paper, we thereore propoe a generalzaton o tradtonal bar chart and x-y-plot whch allow the vualzaton o large amount o data. The bac dea to ue the pxel wthn the bar to preent the detaled normaton o the data record. Our o-called pxel bar chart retan the ntutvene o tradtonal bar chart whle allowng very large data et to be vualzed n an eectve way. We how that, or an eectve pxel placement, we have to olve complex optmzaton problem, and preent an algorthm whch ecently olve the problem. Our applcaton ung real-world e-commerce data how the wde applcablty and ueulne o our new dea.. Introducton Becaue o the at technologcal progre, the amount o data whch tored n computer ncreae very quckly. Reearcher rom the Unverty o Berkeley etmate that every year about Exabyte o data generated, wth % avalable only n dgtal orm. Today, computer typcally record even mple tranacton o everyday le, uch a payng by credt card, ung the telephone and hoppng n e-commerce tore. Th data collected becaue bune people beleve that t a potental ource o valuable normaton and could provde a compettve advantage. Fndng the valuable normaton hdden n the data, however, a dcult tak. Vual data exploraton technque are ndpenable to olvng th problem. In mot data mnng ytem, however, only mple graphc, uch a bar chart, pe chart, x-y plot, etc., are ued to upport the data mnng proce. Whle mple graphc are ntutve and eay-to-ue, they ether: - how hghly aggregated data and actually preent only a very lmted number o data value (a n the cae o bar chart or pe chart), or - have a hgh degree o overlap whch may occlude a gncant porton o the data value (a n the cae o x-y plot). The ueulne o bar chart epecally lmted the uer ntereted n relatonhp between derent attrbute uch a product type, prce, number o order, and quantte. The reaon or th lmtaton that multple bar chart or derent attrbute do not upport the dcovery and correlaton o nteretng ubet, whch one o the man tak n mnng cutomer tranacton data. For an analy o large volume o e-commerce tranacton [Ec 99], the vualzaton o hghly aggregated data not ucent. What needed to preent an overvew o the data but at the ame tme how the detaled normaton or each data tem. In th paper, we decrbe a new vualzaton technque called pxel bar chart. The bac dea o pxel bar chart to ue the ntutve and wdely ued preentaton paradgm o bar chart, but alo ue the avalable creen pace to preent more detaled normaton. By colorng the pxel wthn the derent bar accordng to the value o the data record, very large amount o data can be preented to the uer. To make the dplay more meanngul, two parameter o the data record are ued to mpoe an orderng on the pxel n the x- and y-drecton. Pxel bar chart can be een a a generalzaton o bar chart. They combne the general dea o x-y plot and bar chart to allow an overlap-ree, non-aggregated dplay o mult-attrbute data. Snce pxel bar chart ue each pxel to preent one data value, they belong to the cla o pxelorented technque. Other pxel-orented technque nclude the pral technque [KK 94], the recurve pattern technque [KKA 95], and the crcle egment technque [AKK 96]. Other clae o normaton vualzaton technque nclude geometrc proecton technque (e.g. [In 85, ID 90]), con-baed technque (e.g., *Preently wth the Computer Scence Inttute, Unverty o Contance, Contance, Germany kem@normatk.un-kontanz.de ; ulan@ladch.de

3 a). Equal-Wdth Bar Chart Fgure : Regular Bar Chart b). Equal-Heght Bar Chart [PG 88, Bed 90]), herarchcal technque (e.g., [LWW 90, RCM 9, Shn 9]), graph-baed technque (e.g., [EW 93, BEW 95]), whch n general are combned wth ome nteracton technque (e.g., [BMMS 9, AWS 9, ADLP 95]) and ometme alo ome dtorton technque [SB 94, LRP 95].. From Bar Chart to Pxel Bar Chart A common method or vualzng large volume o data to ue bar chart. Bar chart are wdely ued and are very ntutve and eay to undertand. Fgure llutrate the ue o a regular bar chart to vualze cutomer dtrbuton n an e-commerce ale tranacton. The heght o the bar repreent the number o cutomer or derent product categore. Bar chart, however, requre a hgh degree o data aggregaton and actually how only a rather mall number o data value (only value are hown n Fgure ). Thereore, or data exploraton o large multdmenonal data, they are o lmted value and are not able to how mportant normaton uch a: - data dtrbuton o multple attrbute - local pattern, correlaton, and trend - detaled normaton, e.g., each cutomer prole (age, ncome, locaton, etc.). Bac Idea o Pxel Bar Chart Pxel bar chart are derved rom regular bar chart (ee Fgure a). The bac dea o a pxel bar chart to preent the data value drectly ntead o aggregatng them nto a ew data value. The approach to repreent each data tem (e.g. a cutomer) by a ngle pxel n the bar chart. The detaled normaton o one attrbute o each data tem encoded nto the pxel color and can be acceed and dplayed a needed. One mportant queton : how are the pxel arranged wthn each bar? Our dea to ue one or two attrbute to eparate the data nto bar and then ue two addtonal attrbute to mpoe an orderng wthn the bar (ee Fgure or the general dea). The pxel bar chart can thereore be een a a combnaton o the tradtonal bar chart and the x-y dagram. Now, we have a vualzaton n whch one pxel correpond to one cutomer. I the parttonng attrbute redundantly mapped to the color o the pxel, we obtan the regular bar chart hown n Fgure a (Fgure b how the equal-heghtbar-chart" whch we wll explan n the next ecton). Pxel bar chart, however, can be ued to preent large amount o detaled normaton. The one-to-one correpondence between cutomer and pxel allow u to ue the color o the pxel to repreent an addtonal attrbute o the cutomer - or example, ale amount, number o vt, or ale quantty. In Fgure 3a, a pxel bar chart ued to vualze thouand 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 product y - orderng attrbute x - orderng attrbute parttnng attrbute Fgure : Pxel Bar Chart

4 a). Pxel Bar Chart Fgure 3: Pxel Bar Chart b). Space Fllng Pxel Bar Chart type a the parttonng attrbute and number o vt and dollar amount a the x and y orderng attrbute. The color repreent the dollar amount pent by the correpondng cutomer. Hgh dollar amount correpond to brght color, low dollar amount to dark color.. Space-Fllng Pxel Bar Chart One problem o tradtonal bar chart that a large porton o the creen pace can not be ued due to the derng heght o the bar. Wth very large data et, we would lke to ue more o the avalable creen pace to vualze the data. One dea that ncreae the number o dplayable data value to ue equal-heght ntead o equal-wdth bar chart. In Fgure b, the regular bar chart o Fgure a hown a an equal-heght bar chart. The area (wdth) o the bar correpond to the attrbute hown, namely the number o cutomer. I we now apply our pxel bar chart dea to the reultng bar chart, we obtan pace-llng pxel bar chart whch ue vrtually all pxel o the creen to dplay cutomer data tem. In Fgure 3b, we how an example o a pace-llng pxel bar chart whch ue the ame parttonng, orderng, and colorng attrbute a the pxel bar chart n Fgure 3a. In th way, each cutomer repreented by one pxel. Note that pxel bar chart generalze the dea o regular bar chart. I the parttonng and colorng attrbute are dentcal, both type o pxel bar chart become caled 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 and allow a detaled analy o large orgnal data et..3 Mult-Pxel Bar Chart In many cae, the data to be analyzed cont o multple attrbute. Wth pxel bar chart we can vualze attrbute value ung mult- pxel bar chart whch ue derent color mappng but the ame parttonng and orderng attrbute. Th mean that the arrangement o data tem wthn the correpondng bar o mult-pxel bar chart the ame,.e., the colored pxel correpondng to the derent attrbute value o the ame data tem have a unque poton n the bar. In Fgure 4, we how an example o three pxel bar chart wth product type a the parttonng attrbute and number o vt and dollar amount a the x and y orderng attrbute. The attrbute whch are mapped to color are dollar amount pent, number o vt, and ale quantty. Note that the pxel n correpondng bar n multple bar chart are related by ther poton,.e., the ame data record ha the ame relatve poton wth each o the correpondng bar. It thereore poble to relate the derent bar chart and detect correlaton. 3. Formal Denton o Pxel Bar Chart In th ecton we ormally decrbe pxel bar chart and the problem that need to be olved n order to mplement an eectve pxel placement algorthm. 3. Denton o Pxel Bar Chart For a general denton o pxel bar chart, we need to pecy the: - dvdng attrbute (or between-bar parttonng) - orderng attrbute (or wthn-bar orderng) - colorng attrbute (or pxel colorng). 3

5 $ A m o u n t a) C=dollar amount b) C=number o vt c) C=quantty Fgure 4: Mult Pxel Bar Chart Chart (D x =, D y =, O x =number o vt, O y =dollar amount, C) In tradtonal bar chart there one dvdng attrbute whch partton the data nto dont group correpondng to the bar. In pace-llng bar chart, the bar correpond to a parttonng o the creen accordng to the horzontal ax (x). 3 Next, we need to pecy an attrbute or orderng the pxel n each pxel bar. Agan, we can do the orderng accordng to the x- and the y-ax,.e., along the horzontal (O x ) and vertcal (O y ) axe nde each bar. 3 Fgure 5: Dvdng attrbute on x-ax (e.g., D x = ) We may generalze the denton o pace-llng pxel bar chart by allowng more than one dvdng attrbute,.e. one or the horzontal ax (D x ) and the one or the vertcal ax (D y ). 3 Fgure 6: Dvdng attrbute on x- and y-ax (e.g., D x =, D y = Regon) Fgure 7: Orderng attrbute on x- and y-ax (e.g., O x = Dollar Amount, O y = Quantty) Fnally, we need to pecy an attrbute or colorng the pxel. Note that n mult-bar chart we may agn derent attrbute to color n derent bar chart, whch enable the uer to relate the derent colorng attrbute and detect partal relatonhp among them. Note that the dvdng and orderng attrbute have to tay the ame n order to do that. Let DB = {d,, d n } be the data bae o n data record, each contng o k attrbute value d = { a, K, ak }, al Al, where A l the attrbute name o value a l. Formally, a pxel bar chart dened by a ve tuple: 4

6 <D x, D y, O x, O y, C > where D x, D y, O x, O y, C {A l,, A k, } and D x /D y are the dvdng attrbute n x-/ydrecton, O x /O y are the orderng attrbute n x-/y-drecton, and C the colorng attrbute. The mult-pxel bar chart o ale tranacton hown n Fgure 4, or example, are dened by the ve-tuple <product type,, no. o vt, dollar amount, C> where C correpond to derent attrbute,.e., number o vt, dollar amount, quantty. 3. Formalzaton o the Problem The bac dea o pxel bar chart to produce dene pxel vualzaton whch are capable o howng large amount o data on a value by value ba wthout aggregaton. The pecc requrement or pxel dplay are: - dene dplay,.e., bar are lled completely - non-overlappng,.e., no overlap o pxel n the dplay - localty,.e., mlar data record are placed cloe to each other - orderng,.e., orderng o data record accordng to O x, O y. To ormalze thee requrement we rt have to ntroduce the creen potonng uncton : A K Ak Int Int, whch determne the x-/y-creen poton o each data record d,.e., ( d ) = y) denote the poton o data record d on the creen, and ( d ). x denote the x-coordnate and ( d ). y the y-coordnate. Wthout lo o generalty, we aume that O x = A and O y = A. The requrement can then be ormalzed a:. Dene Dplay Contrant The dene dplay contrant requre that all pxel row (column) except the lat one are completely lled wth pxel. For equal-wdth bar chart, the wdth w o the bar xed. For a partton p contng o p pxel, we have to enure that p / w : d í wth ( d í ) = (, ) =.. w, =.. For equal-heght bar chart o heght h the correpondng contrant p / h, =.. h : d í wth ( d í ) = (, ) =... No -Overlap Contrant The no-overlap contrant mean that a unque poton agned to each data record. Formally, we have to enure that two derent data record are placed at derent poton,.e., d d DB : ( d ) ( d )., 3. Localty Contrant In dene pxel dplay the localty o pxel play an mportant role. Localty mean that mlar data record are placed cloe to each other. The parttonng n pxel bar chart enure a bac mlarty o the data record wthn a ngle bar. In potonng the pxel wthn the bar, however, the localty property alo ha to be enured. For the ormalzaton, we need a uncton m(d, d ) [0 ] whch determne the mlarty o two data record and the nvere uncton o the pxel placement uncton -, whch determne the data record or a gven (x,y)-poton on the creen. The localty contrant can then be expreed 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 dene dplay and no-overlap contrant. Th the reaon why the localty contrant ormalzed a a global optmzaton problem. 4. Orderng Contrant The lat contrant whch cloely related to the localty contrant the orderng contrant. The dea to enorce a one-dmenonal orderng n x- and y-drecton accordng to the peced attrbute O x = A and O y =A. Formally, we have to enure,.. n : a > a ( d ). x >,.. n : a > a ( d ). y > ( d ). x ( d ). y Note that orderng the data record accordng to the attrbute and 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: The element ued no attrbute peced. 5

7 w x= h ( y). a (, ). x y + a 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 trade-o between the x- and the y-orderng contrant. In addton, the optma or the localty and the orderng contrant are n general not dentcal. Th due to the act that the mlarty uncton may nduce a derent optmzaton crteron than the x-/y-orderng 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 and can thereore only be olved ecently by a heurtc algorthm. 3.3 Pxel Placement Algorthm For the generaton o pxel bar chart, we have to - partton the data et accordng to D x and D y - determne the pxel color accordng to C - place the pxel o each partton n the correpondng regon accordng to O x, O y. The parttonng accordng to D x and D y and the color mappng are mple and traghtorward to mplement, and thereore do not need to be decrbed n detal here. The pxel placement wthn one bar, however, a dcult optmzaton problem becaue t requre a twodmenonal ort. In the ollowng, we decrbe our heurtc pxel placement algorthm whch provde an ecent oluton to the problem. The bac dea o the heurtc pxel placement algorthm to partton the data et nto ubet accordng to O x and O y, and ue thoe ubet to place the bottom- and let-mot pxel. Th provde a good 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 determne the onedmenonal htogram or O x and O y, whch are ued to determne the α-quantle o O x and O y. I the bar under conderaton ha extenon w x h pxel, we determne the w, K, ( w ) w-quantle or the We ue a colormap whch map hgh data value to brght color and low data value to dark color. parttonng o O x, and the h, K, ( h ) h - quantle or the parttonng o O y. The quantle are then ued to determne the partton X,,X w o O x and Y,,Y h o O y. The partton X,,X w are orted accordng to O y and the partton Y,,Y h accordng to O x.. We can tart now to place the pxel n the lower-let corner,.e., poton (,), o the pxel bar: (,) = d mn { } = mn d. a { } d. a d X d Y Next we place all pxel n the lower and let pxel row o the bar. Th done a d mn (,) = { d. a }.. w d X = d mn, ) = d.. Y ( { d. a } = h 3. The nal tep the teratve placement o all remanng pxel. Th done tartng rom the lower let to the upper rght. I pxel at poton (-, ) and (, -) are already placed, the pxel at poton (, ) determned a (, ) = d mn { d a + d a }. d X. Y X Y Becaue we have placed the data tructure a ntroduced n tep, the pxel to be placed at poton can be determned n O() tme X Y. I X Y =, we have to teratvely extend the partton X and Y and conder d ( X X ) + Y. I th et tll empty, we have to conder d X X ) ( Y Y ) ( + + and o on, untl a data pont to be placed ound. Note that th procedure qute ecent due to the data tructure ued. 4. The Pxel Bar Chart Sytem To analyze large volume o tranacton data wth multple attrbute, pxel bar chart have been ntegrated wth a data mnng vualzaton ytem [Hao 99]. The ytem ue a web brower wth a Java actvator to allow real-tme nteractve vual data mnng on the web. The web nterace are baed on tandard HTML and 6

8 Clent Server Pxel Bar ortng groupng Pxel Bar Chart Mult- Pxel Bar lnked colorng Interacton explorng data Fgure 8: Sytem Archtecture & Component Java applet, whch are ued to explore relatonhp and to retreve data wthn a regon o nteret. The erver ntegrated wth the data warehoue and the mnng engne. The uer at the clent de vually explore the data by dynamcally acceng the large mult-attrbute tranacton wth complex relatonhp through HTML page n a web brower. 4. Sytem Archtecture and Component The pxel bar chart ytem connect to a data warehoue erver and ue the databae to query or detaled data a needed. The data to buld the pxel array kept n memory to upport realtme manpulaton and correlaton. A llutrated n Fgure 8, the pxel bar chart ytem archtecture contan three bac component:. Pxel array orderng and groupng A pxel array contructed rom the pxel bar chart ve tuple peccaton. One pxel repreent one data record,.e., a cutomer. The parttonng algorthm agn each data record to the correpondng bar accordng to the parttonng attrbute(). The pxel placement mplement a mpled veron o the heurtc algorthm preented n ubecton Multple lnked pxel bar In mult-bar chart, the poton o the pxel belongng to the ame data record reman the ame acro mult-pxel bar chart or correlaton. The color o the pxel correpond to the value o the elected attrbute (uch a prce, number o order, etc.). 3. Interactve data exploraton Th ytem provde multaneou browng and 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 data et eay to explore and nterpret, the pxel bar chart ytem provde the ollowng nteracton capablte: vual queryng layered drll-down / detal-on-demand multple lnked vualzaton zoom n and out o the pxel bar chart The attrbute ued or parttonng (Dx, Dy), orderng (Ox, Oy), and colorng (C) can be elected and changed at executon tme. For dentyng correlaton, a ubet o data tem n a pxel bar chart can be elected to get the pxel correpondng to related attrbute value hghlghted wthn the ame dplay. A drlldown technque allow the vewng o all related normaton ater electng a ngle data tem. When mult-bar chart are preented, pxel rede at the ame locaton acro all the chart wth derent attrbute. In addton to dcoverng correlaton and pattern, the uer can elect a ngle data tem to relate all t attrbute value. 5. Applcaton and Evaluaton The pxel bar chart technque ha been prototyped n everal e-commerce applcaton at Hewlett Packard Laboratore. It ha been ued to vually mne large volume o ale tranacton and cutomer hoppng actvte at HP hoppng web te. 5. Cutomer Analy The pxel bar chart ytem ha been appled to cutomer buyng pattern and 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., product type, geographcal locaton, dollar amount, number o vt, and quantty) are placed cloe to each other. A tore manager can ue the vualzaton to rapdly dcover cutomer buyng pattern and ue thoe pattern to target marketng campagn. Fgure 9 how the our attrbute o 06,99 cutomer buyng record. The our pxel bar chart o Fgure 9 are contructed a ollow: 7

9 hgh low a) Color: Regon b) Color: Dollar amount c) Color: No. o Vt d) Color: Quantty Fgure 9: Mult-Pxel Bar Chart or Mnng 06,99 Cutomer Buyng Tranacton (D x =, D y =, O x =dollar amount, O y =regon, C) - Product type the dvdng attrbute D x product type - Dollar amount the x-orderng attrbute O x Regon the y-orderng attrbute O y or 0 Unted State regon - Regon, dollar amount, number o vt, and quantty are the our colorng attrbute C Many mportant act may be dcovered n Fgure 9 (a, b, c, d). In the bar or the derent attrbute, the uer may oberve the ollowng act: a) Regon attrbute There are 0 derent color to repreent 0 derent regon (labeled -0 n Fgure 9a) n the Unted State. The colored wave ndcate the number o cutomer n each regon. Ater analyzng cutomer dtrbuton, regon 9 (larget area) ound to have the larget number o cutomer. Regon 7 (mallet area) ha the leat number o cutomer acro all product type. b) Dollar amount attrbute Product type 5 ha the mot top dollar amount ale (blue & brown). The dollar amount ale o product type 6 and 7 have a very mall varance acro all regon (old blue/brown). c) Number o vt attrbute The blue color dtrbuton n product type 4 ndcate that cutomer o th product type (conumable) come back more oten than cutomer o other product type. d) Quantty attrbute The green color o product type 6 ndcate that n th category all cutomer bought the ame number o tem acro all regon. It alo obvou that product type 4 cutomer have the larget quantte. By relatng the derent bar chart o the mult bar chart o Fgure 9, the uer may oberve or example the ollowng cluter and trend: - Regon 4 ha the mot cutomer but regon 9 the mot protable wth the mot requent vt and the larget quantte. - The top dollar amount cutomer come back more requently and 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 product ale and promoton. Product manager want to undertand whch product have the top ale and who are ther top dollar amount cutomer. An e-commerce manager, or example, need to anwer queton a to whch product type have the hghet dollar amount cutomer, how oten the cutomer come back and or whch product. Thee analye may alo be ued to determne whch product may be mpacted when the tore ue coupon. 8

10 cutomer A $345,000 cutomer A 5 vt cutomer A 500 tem hgh low a) Color: dollar amount b) Color: no. o vt c) Color: quantty Fgure 0: Mult-Pxel Bar Chart or Mnng 405,000 Sale Tranacton Record (D x =, D y =, O x =no. o vt, O y = dollar amount, C) Whle regular bar chart provde only aggregated normaton on the number o cutomer by product type (Fgure ), the correpondng pxel bar chart nclude mportant addtonal normaton uch a the dollar amount dtrbuton o the ale. More peccally, the pxel bar chart provde the ollowng addtonal normaton: - Dollar amount veru product dtrbuton - Each cutomer detal normaton can be drlled down a needed. Fgure 0 llutrate an example o a mult-pxel bar chart o 405,000 mult-attrbute web ale tranacton. The dvdng attrbute (D x ) agan product type; the orderng attrbute are number o vt and dollar amount (O x and O y ). The color (C) n the derent bar chart repreent the attrbute dollar amount, number o vt, and quantty. From Fgure 0, the ollowng normaton about the web ale can be obtaned: a) Product type 0 and product type 7 have the top dollar amount cutomer (dark color o bar 7 and 0 n Fgure 0a). b) The dollar amount pent and the number o vt are clearly correlated, epecally or product type 4 (lnear ncreae o dark color at the top o bar 4 n Fgure 0b). c) Product type 4 and have the hghet quantte old (dark color o bar 4 and n Fgure 0c). d) By clckng on a pecc pxel (A), we may nd out that cutomer A vted 5 tme, bought 500 tem, and pent $345,000 on product type 5. It urther nteretng that there are cluter o darker color n bar 4 o Fgure 0c, whch mean that there are certan range o dollar amount ale or whch the quantty tend to be hgher than n other egment. Th obervaton unexpected and may be ued to denty the cluter o ale tranacton and make ue o the normaton to urther ncreae the ale. Note that the normaton mentoned above cannot be detected by regular bar chart. 6. Concluon In th paper, we preented pxel bar chart, a new method or vualzng large amount o mult-attrbute data. The approach a generalzaton o tradtonal bar chart and x-y dagram, whch avod the problem o long normaton by aggregaton or overplottng. Intead, pxel bar chart map each data pont to one pxel o the dplay. For generatng the pxel bar chart vualzaton, we have to olve a complex optmzaton problem. The pxel 9

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