Multi-Scale Banking to 45º

Transcription

1 Mult-Scale Bankng to 45º Jeffrey Heer and Maneesh Agrawala Abstract In hs text Vsualzng Data, Wllam Cleveland demonstrates how the aspect rato of a lne chart can affect an analyst s percepton of trends n the data. Cleveland proposes an optmzaton technque for computng the aspect rato such that the average absolute orentaton of lne segments n the chart s equal to 45 degrees. Ths technque, called bankng to 45, s desgned to maxmze the dscrmnablty of the orentatons of the lne segments n the chart. In ths paper, we revst ths classc result and descrbe two new extensons. Frst, we propose alternate optmzaton crtera desgned to further mprove the vsual percepton of lne segment orentatons. Second, we develop mult-scale bankng, a technque that combnes spectral analyss wth bankng to 45. Our technque automatcally dentfes trends at varous frequency scales and then generates a banked chart for each of these scales. We demonstrate the utlty of our technques n a range of vsualzaton tools and analyss examples. Index Terms Informaton vsualzaton, bankng to 45 degrees, lne charts, tme-seres, sparklnes, graphcal percepton 1 INTRODUCTION Few vsualzatons are more common than the lne chart, used to communcate changes n value over a contguous doman, wth the slopes of lne segments encodng rates of change. Mathematcal functons, stock prces, and all varetes of tme-seres data are plotted n ths form. Accordngly, technques whch mprove the graphcal percepton of such dsplays could be of great practcal beneft. Gven the human vsual system s senstvty to lne orentaton [6], t s not surprsng that the relatve orentaton of lne segments n a chart can strongly mpact an analyst s percepton of trends n the data. One way of ntentonally manpulatng these orentatons s through the choce of the aspect rato (wdth/heght) of the chart. Both n hs book Vsualzng Data [2] and elsewhere [1,3,4], Wllam Cleveland demonstrates how the choce of a lne chart s aspect rato can mpact graphcal percepton. Fgure 1 shows plots of average monthly carbon doxde measurements made at the Mauna Loa observatory, an example used n [2]. The frst plot (1a) clearly shows an upward trend n the data. There s an nflecton n the curve, ndcatng that the ncrease n carbon doxde s acceleratng. The second chart (1b) shows the same data plotted at a wder aspect rato. The slow onset and quck decay of the yearly oscllatons s clearly vsble n ths plot. However the nflecton that was vsble n the frst chart becomes dffcult to see. Notng that an average orentaton of 45 maxmzes the dscrmnablty of adjacent lne segments, Cleveland ntroduces a technque called bankng to 45, whch determnes the aspect rato such that the average orentaton of all lne segments n a chart s 45 degrees [2,3,4]. Whle Cleveland s optmzaton procedure makes t easer to see hgher frequency oscllatons n the data, t can obscure lower-frequency trends of nterest. The aspect rato used n Fgure 1b s the result of bankng to 45. As we have noted, the low-frequency nflecton pont s dffcult to dscern n ths chart. Cleveland addresses ths ssue, descrbng a manual process of fttng smooth regresson curves to the data and bankng the resultng low frequency curve. Fndng nterestng trends n the data becomes a tedously teratve tral-and-error process. Users must manually consder each Jeffrey Heer s wth the Computer Scence Dvson of the Unversty of Calforna, Berkeley. E-Mal: jheer@cs.berkeley.edu. Maneesh Agrawala s wth the Computer Scence Dvson of the Unversty of Calforna, Berkeley. E-Mal: maneesh@cs.berkeley.edu. Manuscrpt receved 31 March 2006; accepted 1 August 2006; posted onlne 6 November For nformaton on obtanng reprnts of ths artcle, please send e-mal to: tvcg@computer.org. Fgure 1a. CO 2 measurements, aspect rato = The x-axs shows tme, n monthly ncrements, whle the y-axs shows carbon doxde measurements taken at the Mauna Loa observatory [2]. Note the bend n the trend of ncreasng values, suggestng an acceleratng ncrease. Fgure 1b. CO 2 measurements, aspect rato = The wder aspect rato enables the vewer to see that the ascent of each yearly cycle s more gradual than ts decay. However, the bend n the lower-frequency trend s now dffcult to see. The choce of aspect ratos for Fgures 1a and 1b were automatcally determned usng mult-scale bankng. smoothness level (.e., the frequency scale), bank t to 45º, and then vsually check for an nterestng trend. In ths paper we extend Cleveland s work n two ways. Frst, we explore alternate optmzaton crtera for Cleveland s bankng procedure. These crtera are desgned to fnd an aspect rato that further mproves the vsual percepton of lne segment orentatons. Second, we develop mult-scale bankng, a technque that combnes spectral analyss wth bankng to 45. Our technque automatcally dentfes frequency scales that may be of nterest and then generates a banked chart for each of these scales. The aspect ratos for the charts n Fgures 1a and 1b were automatcally chosen usng our mult-scale bankng approach. Whle our technque fnds the same aspect rato as Cleveland s orgnal approach n Fgure 1b, t also returns the aspect rato whch reveals the low-frequency nflecton pont n Fgure 1a. We have ncorporated mult-scale bankng nto tools for both statc and dynamc vsualzatons, and present the results of applyng the technque on a collecton of real-world data sets.

2 2 BANKING TO 45 Lne charts are typcally used to present the relatonshp between two varables. For a gven value of varable x, vewers can drectly read off the correspondng value of y and vce versa. Vewers can also determne the rate of change of y wth x, by judgng the orentatons of lne segments n the chart. As we have seen, the aspect rato of the chart can sgnfcantly affect the vsual percepton of the orentatons of the lne segments. The challenge n desgnng lne charts s to choose an aspect rato that makes t easy for vewers to dscrmnate lne orentatons n the chart and thereby determne the rates of change. Cleveland [1,2,3,4] was the frst to systematcally address ths challenge. For completeness we revew hs technques and then present several new addtons. 2.1 Maxmzng Orentaton Resoluton Gven a set of n lne segments, the orentaton resoluton s the range of orentatons spanned by the segments and s computed as the maxmum orentaton mnus the mnmum orentaton. Cleveland et al. [1] conducted human-subject experments showng that vewers judge the rato of the slopes of two adjacent lne segments most accurately when the orentaton resoluton between them s maxmzed. They also show that for two lne segments, the resoluton s maxmzed when the absolute value of the orentaton mdangle, the average of the maxmum orentaton and the mnmum orentaton, s 45. Cleveland et al. s argument s as follows. Suppose that s 1 and s2 are the slopes of two lne segments (n the orgnal unts of the data), and that s 1 > s 2. In the physcal dsplay space, for a dsplay rectangle of heght h, wdth w and aspect rato α = w/ h, the physcal slopes become s 1 / α and s 2 / α. The orentaton resoluton, computed wth respect to the physcal slopes of the lne segments, s r( α) = tan ( s / α) tan ( s / α), and the orentaton mdangle s, 1 1 s1 α + s2 tan ( / ) tan ( / α) m( α) = 2 1 α To smplfy the formulaton let f = s2/ s1 and λ = s s. Then, 1 1 r( λ) = tan ( λ/ f ) tan ( λ f ) 1 1 tan ( λ/ f ) + tan ( λ f ) m( λ) = 2 1 At λ = 1, r( λ ) s maxmzed to a value of 2(45 tan ( f )) and m( λ ) = 45. Thus for two lne segments the orentaton resoluton s maxmzed when the average angle between them s 45. Smlarly for two lne segments of negatve slopes the orentaton resoluton s maxmzed at a mdangle of Medan-Absolute-Slope Bankng Wth more than two lne segments t s mpossble to choose a sngle aspect rato such the mdangle between all postvely sloped pars s 45 and all negatvely sloped pars n -45. Instead, Cleveland et al. proposes choosng the aspect rato that sets the medan absolute slope of the lne segments to 1. For a dsplay rectangle of heght h, wdth w and aspect rato α = w/ h, to dsplay n lne segments wth slopes s ( = 1,..., n ) Cleveland et al. fnd α such that the medan s / α = 1. Thus approxmately half the segments wll have an absolute slope of greater than 1 and half wll have an absolute slope of less than 1. The procedure for computng α s straghtforward. Let R y = y max y mn, the range of y values where y range over the set of endponts, smlarly let R x = x max x mn, and let M = medan s. Then settng α = M Rx / Ry satsfes the medan-absolute-slope crteron. Ths closed-form soluton s fast and easy to compute Average-Absolute-Orentaton Bankng More recently, Cleveland [2,3,4] proposes bankng the average orentaton to 45 rather than settng the medan slope to 1. For aspect rato α the orentaton of a lne segment drawn n a chart s 1 gven byθ ( α) = tan ( / α ). We must fnd an aspect rato such that s θ( α ) = 45. (1) n Cleveland ponts out that although there s no closed-form soluton to ths equaton, t s monotone nα. Therefore teratve optmzaton technques (.e., Newton s method) wll converge to a soluton. 2.4 Weghted Average-Absolute-Orentaton Bankng Cleveland further suggests weghtng the average absolute orentaton by the lengths of the lne segments. Denotng the length of the th lne segment by l ( α ), the weghted verson of equaton 1 becomes θ( α) l( α) = 45 (2) l ( α) Agan we must solve ths equaton usng teratve optmzaton, but because the equaton remans monotone n α t can be solved effcently. Cleveland asserts that ths weghtng results n more satsfactory results for the datasets he has tested [4]. It s worth consderng the effect of ths weghtng. In typcal lne charts, the x-values are unformly dstrbuted and therefore the length of a lne segment s determned by the dfference n the y-values of the endponts. In ths case the lnes of longest length are also those of greatest absolute slope, and thus of greatest absolute orentaton. Weghtng by segment length preferentally weghts lne segments wth hgher absolute orentatons (.e., closer to vertcal), more aggressvely optmzng these segments to Bankng by Optmzng Orentaton Resoluton Cleveland s technques are based on the hypothess that maxmzng orentaton resoluton maxmzes the vewer s ablty to perceve the orentaton of lne segments. Yet, both medan-absolute-slope bankng and average-absolute-orentaton bankng are based on centerng ether the slopes or orentatons of lne segments about 1 or 45 respectvely. As descrbed n Secton 2.1 such centerng s optmal for the case of two lne segments. For multple lne segments, however, t s unclear whether centerng generates the optmal soluton. Moreover, the effect of these centerng technques on the overall orentaton resoluton s ndrect. Therefore, we propose drectly optmzng orentaton resoluton. Gven two lne segments and j, the angle between them s r, j ( α) = θ ( α) θ j ( α). Note that r, j s computed as the smallest acute angle between the lne segments, and t therefore always les between 0 and 90. For each par of lne segments, the closer the angle dfference s to 90, the easer t s to vsually perceve the dfference n orentatons. Thus, our approach s to fnd the aspect rato α that maxmzes r, j between all pars of lne segments. r j 2, j In contrast to Cleveland s technques (Sectons 2.2, 2.3 and 2.4), ths approach more drectly optmzes orentaton resoluton snce t consders the angles between lne segments. In some lne charts dscrmnatng the orentatons of successve lnes s more mportant than dscrmnatng orentatons of all pars. In ths case we can optonally choose to apply equaton 3 only to successve pars of lne segments rather than exhaustvely applyng t 2 to all n pars of segments. The result s a local, rather than global, optmzaton of the orentaton resoluton. (3)

3 HEER ET AL.: MULTI-SCALE BANKING TO Average-Absolute-Slope Bankng Bankng Technque co2 prefuse prmtx sun Cleveland et al. [3] have argued that vewers judge orentaton rather medan-slope (ms) than slope when estmatng the rate of change of a curve. They pont medan-slope culled (msc) out that our ablty to dscrmnate two orentatonsθ, θ + δ depends only on δ, but dscrmnatng between two slopes s and average-slope (as) s(1 + r) depends on both s and r. For fxed r and very large or average-slope culled (asc) very small s, the slopes become dffcult to dscrmnate. Ths s why average-orent (ao) Cleveland formulates hs average-absolute-orentaton bankng technques specfcally wth respect to lne segment orentatons. average-orent culled (aoc) As shown prevously, slope and orentaton are related by the average-weghted-orent (awo) arctangent functon. In the case of a sngle lne segment, bankng the average-weghted-orent culled (awoc) orentaton to 45 and bankng the slope to 1 are equvalent 1 operatons, as tan 1 = 45. The medan-absolute-slope procedure global-orent-resoluton (gor) consders only a sngle lne segment (the medan segment), and global-orent-resoluton-culled (gorc) because the arctangent s monotone the medan slope and medan local-orent-resoluton (lor) orentaton concde. In the case of multple lne segments, however, the non-lnearty of the arctangent functon causes the two local-orent-resoluton-culled (lorc) approaches to dverge. Ths non-lnearty necesstates the use of optmzaton technques when computng average-absoluteorentaton bankng. Optmzng the average absolute slope to 1 can be computed quckly and easly, but does not produce the same result as optmzng the average absolute orentaton. For completeness and comparson, we ntroduce average-absolute-slope bankng here. The procedure for bankng the average absolute slope to 1 follows the formulaton of medan slope bankng. As before, α = M Rx / Ry, except we now set M = mean s. As for medan-absolute-slope bankng the soluton for α s closed-form. Table 1. Aspect rato varaton across bankng technques. Each column contans the results of bankng a sngle data set usng twelve dfferent bankers. Each cell contans the computed aspect rato. The bankng technques ntroduced by Cleveland [1,2,3,4] are n bold talcs. Aspect Rato Slopeless Lne Cullng All of the bankng procedures we have presented consder ether the slope or orentaton of every lne segment n the dataset. An addtonal, modfcaton s to cull slopeless lnes those wth ether zero or nfnte slope. Horzontal and vertcal lnes reman unchanged by varatons n aspect rato, yet contrbute to the bankng crtera. For example, ncluson of zero-sloped lnes can affect the calculaton of ether the medan or average orentaton. By removng them from consderaton, the remanng lne segments can be more closely banked to 45. Ths modfcaton can be ncorporated nto any of the prevous bankng routnes. Fgure 2 shows the results of weghted-average-orentaton bankng wth and wthout slopeless lne cullng on an llustratve data set. Only when cullng s appled do the sloped segments reach an average orentaton of 45. Fgure 2. Effects of slopeless lne cullng. Horzontal lnes are typcally ncluded when bankng by average orentaton, resultng n an aspect rato of 1.97 on the left. Cullng these lnes n the bankng optmzaton results n an aspect rato of 4.00 on the rght. In ths case, the sloped segments are set to 45 after the bankng procedure. 2.8 Comparng Bankng Approaches We have appled each bankng technque descrbed n the prevous sectons on four real-world data sets. We compare the results of each of the bankng technques, both wth and wthout the slopeless lne cullng modfcaton, resultng n a total of 12 bankng approaches. The four data sets are: the CO 2 measurements dscussed earler (co2), two months of daly download counts of the prefuse vsualzaton toolkt [5] (prefuse), eght years of share prces for the PRMTX mutual fund (prmtx), and a set of yearly measurements of sunspot actvty (sun). These data sets are dscussed n greater detal n 5 ms msc as asc ao aoc awo awoc gor gorc lor lorc co2 prefuse prmtx sunspot Fgure 3. Aspect rato varaton across bankng technques. Culled and non-culled bankng technques show lttle separaton. The (ao) and (gor) technques produce smlar results, as do (as) and (awo). The (ms) technque produces aspect ratos between those of the other two groups. Secton 3.2 and plots of each data set at varous aspect ratos are presented n Fgures 5-8. The aspect ratos computed by the bankng technques for each data set are shown n Table 1 and Fgure 3. We can see that cullng slopeless lnes has had lttle effect. Ths s to be expected, as the data sets used have few slopeless segments. It s only when the number of slopeless segments accumulates that the cullng modfcaton wll have a pronounced effect. A number of smlartes and dfferences between the technques are also evdent. The non-weghted average-absolute-orentaton (ao) technque produces results smlar to global optmzaton of the orentaton resoluton (gor). Yet, (ao) consders a sngle orentaton per lne segment, whle (gor) consders all pars of lne segments. Therefore (ao) s computatonally more effcent and thus may be preferable to (gor). Smlarly we see that the local resoluton optmzer (lor) produces consstently lower aspect ratos than the global verson. In contrast, the average-weghted-orentaton (awo) and average-absolute-slope (as) bankers consstently produce hgher aspect ratos. As dscussed earler, ths s expected of the weghted-average optmzer, as t preferentally weghts segments wth hgher orentatons and thus flattens those segments more aggressvely. The observed match between (awo) and (as) s partcularly nterestng. Gven the dscusson of Secton 2.6., t may be surprsng that a slope optmzer and an orentaton optmzer provde such smlar output. The average-slope banker has a closed-form soluton and can be computed more effcently than a weghted-average-orentaton bankng. Although Cleveland [2,3,4] recommends usng the weghted optmzer, these results suggest that smlar results may be acheved usng a more computatonally effcent technque. Fnally, the absolute-medan-slope banker (ms) consstently calculates aspect

4 ratos n-between the extremes of the other groups. It can also be computed drectly, wthout need of an optmzer. What remans unclear s whch bankng approach s the most perceptually effectve. A perceptual argument has been made for optmzng the orentaton resoluton (suggestng the (ao) method). However, Cleveland recommends the average weghted orentaton (well matched n our results by the more effcent (as) method). The (ms) method appears to splt the dfference between the two. Exploraton of addtonal data sets and further perceptual experments mght be used to resolve the uncertanty. 3 MULTI-SCALE BANKING TO 45 Each of the bankng approaches dscussed n the prevous secton consder the entrety of the data when determnng the chart aspect rato. As a result, local features are accentuated n the bankng, sometme obscurng larger-scale trends of nterest. For example, the carbon doxde measurements dscussed n the ntroducton ncludes an acceleratng ncrease n values. Ths trend may be dffcult to dscover f one only saw the banked plot of the data (Fgure 1b). As we wll see, ntermedate scales between local and global trends may also contan patterns of great nterest to an analyst. These observatons suggest a need for tools whch ad analysts by dentfyng trends and presentng approprate vsualzatons. Thus, we ntroduce mult-scale bankng, an approach that fnds a set of aspect ratos that optmze the dsplay of trends at varyng scales. A rudmentary approach to mult-scale exploraton s to manually adjust the aspect rato and observe the resultng vews. Many chartng applcatons allow users to change the aspect rato by drectly reszng the data axes and the dsplay wndow. Changng the axs range of a fxed dsplay (for example, by usng range slders) also permts manual bankng, snce the aspect rato changes as the axs scale s vared. Though these approaches enable manual bankng of the data dsplay, exstng tools provde very lttle assstance for choosng the approprate scale. Analysts must manually test a varety of scales and vsually scan the results to fnd the ones that best show trends n the data. Such an teratve search s excessvely tedous, partcularly when analyzng large data sets wth a hgh degree of detal. Cleveland descrbes a manual, teratve process of trend dscovery for mult-scale exploraton [2]. He begns by fttng the data usng loess, a locally weghted regresson technque. Loess consders sequental wndows of a data set; these local regons are weghted and ft to a polynomal regresson curve. The technque requres two parameters: a smoothng parameter α, correspondng to the sze of the wndow, and a parameter λ, specfyng the degree of the fttng polynomal (typcally 1 or 2). The analyst must manually choose values for these parameters, often by tral and error. Once a desred trend curve has been found n ths manner, t can be used as nput to a bankng routne, to compute an aspect rato that optmzes the dsplay of the trend. To then explore addtonal scales, the trend curve s subtracted from the data and the analyss begns anew on the resdual data. Cleveland suggests endng the analyss once the resduals are well ft by normally-dstrbuted random values. The goal of our mult-scale bankng s to accelerate ths analyss process by automatcally dentfyng scales of nterest n the data. 3.1 Method To dentfy the scales of nterest we analyze a frequency doman representaton of the data to look for strong perodc trends. At each scale of nterest we low-pass flter the data to elmnate frequences at hgher scales and then bank the data to compute scale-specfc aspect ratos. To help llustrate our approach, consder a smple data set created by combnng the output of three snusodal oscllators, each operatng at a dfferent frequency and thus creatng a sgnal wth multple scales of nterest. Fgure 4 dsplays the result of applyng mult-scale bankng to ths data set. sn(x) + cos(10x) cos(40x) Aspect Rato = 2.21 Aspect Rato = Aspect Rato = Aspect Ratos Fgure 4. Mult-scale bankng of a mult-frequency sgnal. The data plots at the top of the fgure show the data usng dentfed aspect ratos, wth banked trend lnes shown n red. The power spectrum plot shows the frequency-doman representaton of the data, wth dentfed scales of nterest ndcated by blue bands. The aspect rato plot shows the banked aspect ratos for each possble lowpass flterng of the data, wth the fnal selecton of aspect ratos ndcated by blue bands. We start the analyss by applyng a dscrete Fourer transform (DFT) to the nput data and computng the squared magntude of each of the resultng Fourer coeffcents to form the power spectrum. Ponts of hgh-value n the power spectrum correspond to strong frequency components wthn the nput sgnal. Fgure 4 ncludes the power spectrum for our example data set. Note the promnent spkes for each of the frequency components. Next, we search for trends of nterest n the power spectrum. One approach s to threshold the spectrum and keep all frequency scales for whch the power s greater than the threshold. However, we have observed that spectral energy often occurs n clumps, sometmes contanng local oscllaton. By smoothng the power spectrum over small wndows we can even out these local regons pror to thresholdng. The smoothng s performed by convolvng the power spectrum wth a Gaussan flter, usng a small wndow wdth (typcally 3) and unt standard devaton. We have also expermented wth pre-multplyng the sgnal wth a wndow functon (e.g., Hammng and Hann wndows) but have so far notced lttle change n the results. We then threshold the smoothed spectrum and consder all values that exceed the threshold. By default, we use the mean of the smoothed power spectrum as the threshold value. Ths default has worked well n practce for small to medum sze data sets, though t can be ncreased (or alternatvely, the smoothng wndow can be modfed) to reduce the number of scales chosen by our algorthm. After applyng the threshold we are left wth a set of canddate scales. These canddate scales correspond to strong frequency components and often form a range of contguous ponts n the power spectrum. Proceedng from the lowest to the hghest frequency, we retan only the last (hghest-frequency) value of each range of consecutve values, thereby capturng the total contrbuton of that local regon of energy. We also ensure that the hghest-

5 HEER ET AL.: MULTI-SCALE BANKING TO 45 frequency component of the entre power spectrum s ncluded, representng the trend of the data n ts entrety. These values consttute the scales of nterest, and are ndcated by the blue bands n the power spectrum plot of our example data. The second phase of a mult-scale bankng approach s to create trend curves for each dentfed scale and bank the resultng curve to arrve at a scale-specfc aspect rato. To form the trend curves we low-pass flter the data, removng all frequences hgher than the current scale of nterest. The resultng trend curve s then banked to 45. Fgure 4 shows the resultng banked vews for the sample oscllator data, usng the medan-absolute-slope bankng technque (Secton 2.3). The trend curves are shown n red. Although we compute aspect ratos for each scale of nterest, n some cases the aspect ratos for two or more dstnct scales may be very smlar to one another. Plottng charts at each of these smlar aspect ratos would result n dsplays that are nearly dentcal to one another. Our soluton s to cull aspect ratos that are judged to be hghly smlar. We apply a smple flter that consders each aspect rato n order of ncreasng frequency. An aspect rato s kept only f t dffers from the prevous aspect rato by a threshold scale factor. In practce, a scale factor of 1.25 has worked well for most cases. The fnal result of the mult-scale bankng process s shown n Fgure 4. Note that the low frequency scale n the frst plot becomes more obscured n the subsequent plots. In addton to the data plots and power spectrum, the fgure also shows a graph of aspect ratos for each possble low-pass flterng of the data. The fnal aspect ratos recommended by our algorthm are ndcated as lght blue bands on the aspect rato plot. The aspect rato for the last dentfed scale seen n the power spectrum plot was fltered, as t s close n value to the aspect rato for the prevous scale of nterest. The aspect rato chart also suggests the alternatve approach of analyzng the aspect rato curve drectly. Ths possblty s dscussed further n Secton 4. A pseudocode descrpton of the technque s presented as Algorthm Results We have appled our mult-scale bankng technque to a number of real-world data sets. To compute the bankng, we used Cleveland s medan-absolute-slope technque wth slopeless lne cullng appled. Results are reported usng the same small multples format of Fgure 4. Plots usng each computed aspect rato are shown, ncludng the banked trend curves n red. Each fgure ncludes a plot of the power spectrum, wth dentfed scales ndcated by lght blue bands. The x- axs of the power spectrum shows the frequency ndex, ndcatng the perodcty of the trend (e.g., a value of 8 ndcates that the trend repeats 8 tmes wthn the data set). A plot of the aspect ratos for each possble low-pass flterng s also shown, wth the fltered set of aspect ratos ndcated by lght blue bands Sunspot Cycles The frst data set s a collecton of astronomcal sunspot observatons, a classc example used by Cleveland to llustrate the utlty of the bankng to 45 technque [4]. The data contans values of the Wolfer number, a measure of the sze and number of sunspots observed n a gven year, from 1700 to Results are shown n Fgure 5. Spectral analyss fnds four scales of nterest, at frequency ndces 7, 10, 31, and 36, plus the scale of the data n ts entrety. Flterng the aspect ratos yelds two charts, correspondng to frequency scales 7 and 31. The frst aspect rato of 3.96 shows a low frequency trend ndcatng the oscllaton of hgh-ponts across sunspot cycles. The second aspect rato of shows the oscllaton of the ndvdual sunspot cycles, wth a perod of about 11 years. In ths vew, t becomes apparent that many of the cycles have a steep onset followed by a more gradual decay. MULTIBANK(y) % Returns a set of computed aspect ratos % y : an array of data values to bank w 3 σ 1 α 1.25 Y FFT(y) P Y 2 Z P G(w,σ) T MEAN(Z) % Gaussan flter wndow sze % Gaussan flter standard devaton % Scale threshold for aspect rato % apply dscrete Fourer transform % compute power spectrum % convoluton wth Gaussan flter % mean fltered power, use as threshold % Compute aspect ratos for trends that pass the threshold. % For consecutve runs, only the last value s retaned. for each [1 LENGTH(Z)] f = LENGTH(Z) or (Z > T and Z +1 < T) ar ar BANKTO45(LOWPASS(y, )) AR { ar 1 } % aspect ratos, nclude lowest by default % Keep aspect ratos that are suffcently dfferent from one another for each [2 LENGTH(ar)-1] j LENGTH(AR) f MAX(ar, AR j ) / MIN(ar, AR j ) > α AR AR ar return AR Algorthm 1: Mult-scale bankng. Computes aspect ratos for charts at multple scales by usng spectral analyss to dentfy trends of nterest. Ths pseudocode descrpton assumes the avalablty of an FFT (fast Fourer transform) routne and a routne LOWPASS for low-pass flterng a sgnal subject to a gven perodcty (frequency ndex). The routne BANKTO45 may use any of the bankng routnes dscussed n Secton Carbon Doxde Measurements The next example, also taken from Cleveland, s the prevously ntroduced set of average monthly atmospherc carbon doxde measurements from the Mauna Loa observatory [2]. Results are shown n Fgure 6. Spectral analyss fnds two scales of nterest, at frequency ndces 11 and 33, plus the scale of the data n ts entrety. Flterng the aspect ratos yelds selected vews at scales 11 and 33. The frst aspect rato of 1.17 shows a general trend of ncreasng CO 2 concentratons. A slght bend can be observed, ndcatng the trend s acceleratng. The second aspect rato of 7.87 provdes a clearer vew of yearly oscllatons. Notably, these cycles have a gradual rse and a steeper declne Mutual Fund Performance Fgure 7 shows of the results of our technque appled to fnancal data, n ths case daly closng share values of the T. Rowe Prce Meda and Technology Fund (PRMTX) from 1997 to the present. Spectral analyss fnds one scale of nterest at frequency ndex 38, plus the scale of the data n ts entrety. There s an nterestng lack of domnant frequency components past the frst dentfed scale. Flterng the aspect ratos yelds two charts. The frst aspect rato of 4.23 shows a low frequency oscllaton n values, depctng the dot-com boom and bust followed by a subsequent recovery. The second aspect rato of allows local varatons to be more clearly seen. The second aspect rato s ncluded as a result of our polcy of always ncludng a scale correspondng to the entrety of the data. The correspondng plot s the result of bankng the non-fltered data set to 45.

6 Sunspot Cycles Aspect Rato = 3.96 PRMTX Mutual Fund Aspect Rato = 4.23 Aspect Rato = Aspect Rato = Aspect Ratos Aspect Ratos Fgure 5. Sunspot observatons, The frst plot shows lowfrequency oscllatons n the maxmum values of sunspot cycles. The second plot brngs the ndvdual cycles nto greater relef. Carbon Doxde Measurements Aspect Rato = 1.17 Fgure 7. PRMTX mutual fund performance, The frst plot shows the boom and bust of the dot-com bubble and subsequent recovery. Tthe second plot affords closer consderaton of short-term varatons. Downloads of the prefuse toolkt Aspect Rato = 1.44 Aspect Rato = 2.89 Aspect Rato = 7.87 Aspect Rato = 8.81 Aspect Ratos Fgure 6. Monthly atmospherc CO 2 measurements. The frst plot shows a baselne trend of ncreasng values, wth a slght nflecton. The second plot more clearly communcates the yearly oscllatons. Fgures 5-8 show the results of applyng mult-scale bankng to real-world data sets. Data sets are plotted at each computed aspect rato, wth banked trend lnes shown n red. The power spectrum plot shows a frequencydoman representaton of the data, annotated wth potental scales of nterest. The aspect rato plot shows the banked aspect ratos for each possble lowpass flterng of the data, annotated wth the fnal aspect ratos returned by the algorthm. Aspect Ratos Fgure 8. Daly download counts of the prefuse vsualzaton toolkt. The frst plot shows a general ncrease n downloads. The second plot shows weekly varatons, ncludng reduced downloads on the weekends. The thrd plot enables closer nspecton of day-to-day spkes and decays.

7 HEER ET AL.: MULTI-SCALE BANKING TO 45 Fgure 9. Trend Explorer applcaton, vsualzng a segment of electroencephalogram (EEG) readngs. An nteractve spectrum vew shows trends dentfed by the mult-scale bankng algorthm and enables users to select dfferent flterng ponts, trggerng recalculaton of the aspect rato. By selectng a data range n the lne chart (shown n the mage on the left), users can zoom nto subsets of data (shown n the mage on the rght) Open Source Software Project Trackng The fnal data set conssts of two months of daly project download counts for prefuse, an open source vsualzaton toolkt developed by the authors [5]. A beta release of the toolkt was launched n February 2006, and accordngly we are nterested n how downloadng patterns may have been affected. Results are shown n Fgure 8. Spectral analyss fnds four scales of nterest, at frequency ndces 3, 8, 17, and 26, plus the scale of the data n ts entrety. Flterng the aspect ratos yelds three charts, correspondng to the frst three scales. The frst aspect rato of 1.44 shows an ncrease n the baselne rate of downloads, startng wth the February 9 th release of the beta. The second aspect rato of 2.89 shows an oscllatng pattern that repeats 8 tmes over the two month perod, ndcatng a weekly pattern. Lookng closer, one dscovers a trend that one mght already suspect: fewer downloads occur on the weekends. The thrd aspect rato of 8.81 allows day-to-day oscllatons to be better understood. Interestngly, downloads appear to ncrease mmedately followng a mnor release of the toolkt (occurrng on February 20, March 5, and March 20). The download count then wtnesses a small ntal declne, followed by another ncrease. Then a steeper declne sets n. Ths analyss ndcates that toolkt users may update on dfferent schedules. For example, some users update mmedately whle others wat a few days. 3.3 Applcatons Mult-scale bankng can be frutfully appled n a number of vsualzaton tools. In ths secton we dscuss several applcatons we have bult whch utlze the mult-scale bankng technque. Each of the applcatons was constructed usng the prefuse vsualzaton toolkt [5] Small-Multples Reports One applcaton of mult-scale bankng s for generatng descrptve reports of a data set, usng small multples [7] to dsplay trend data at varous aspect ratos. These summary reports are useful for ganng a statc overvew of a data set and also approprate for use n presentatons or prnted papers. We have wrtten a smple tool for generatng these reports, and have used t to generate the fgures shown n the prevous sectons Sparklnes Bankng can also be appled to ad the generaton of sparklnes, datantense, word-szed graphcs desgned to be ncorporated nto text [8]. For example, a plot of sunspot data mght be ncluded nlne, supportng unnterrupted readng wthout recourse to a separate fgure. The vertcal extent of a sparklne s fxed by the ascent of the typeface, leavng the horzontal extent of the graphc as the sngle degree of freedom. As noted by Tufte [8], bankng can be appled to optmze the sparklne aspect rato, thereby determnng the wdth. We have bult a smple tool that takes a data set and typeface as nput and generates approprately banked sparklne mages. The followng set of sparklnes generated by our tool shows the daly closng share prces of a popular ndex fund and three wellknown technology companes from March 1, 2005 to March 1, Mult-scale bankng was appled separately to each data set. The aspect ratos for the lowest and hghest frequency scales for each data set were collected. We separately averaged the aspect ratos for the hgh frequency scales and the low frequency scales to generate two fnal aspect ratos: ~8.3 for the hgh frequency scale and ~2.9 for the low frequency scale. The hgh frequency plots on the left afford detaled nspecton, whle the low frequency plots on the rght provde a summary of the major trends. More nuanced approaches to bankng multple lnes are dscussed n Secton 4. VFINX GOOG MSFT YHOO Trend Explorer We have also appled mult-scale bankng wthn an nteractve analyss tool, ncorporatng the results of automated analyss nto manual exploraton tasks. Fgure 9 shows our trend explorer applcaton for nteractvely explorng lne charts at varous scales. The lower panel dsplays the data plot and trend curves whle the upper panel provdes a frequency-doman representaton of the data (users can select between vewng ether the power spectrum or the magntude of the Fourer coeffcents). The results of the mult-scale bankng routne are ncluded as lght blue bands wthn the spectrum dsplay, hghlghtng the automatcally dentfed scales of nterest. Users can bank the data dsplay to a partcular trend by movng a red selector wdget wthn the spectrum dsplay, ether by draggng t or by clckng at a desred locaton. In response, the system flters and banks the data, computng a new aspect rato and updatng the data dsplay. As the user moves the selector near an automatcally dentfed trend of nterest, a snap-to effect s used to easly bank the dsplay to that specfc frequency. In the default confguraton, changes n the aspect rato result n a rescalng of the data dsplay, mantanng an overvew of the whole tme seres. Both the horzontal and vertcal dmensons may change n an effort to maxmze the resoluton of the overvew dsplay subject to the aspect rato and avalable dsplay space. The aspect rato s preserved across wndow reszng operatons. The applcaton also supports a fxed data dsplay mode, where changes n the aspect rato nstead result n a change n the dsplayed data range. The current range s dsplayed (and can be manually adjusted) usng a range slder. Ths mode allows users to zoom-n to the data to vew hgh-frequency trends at a hgher dsplay resoluton. Users can perform more nuanced mult-scale analyses by selectng a range of

8 data n the dsplay and then performng a new mult-scale bankng analyss on just that subset, enablng fractal-lke exploraton of the data. Ths s partcularly useful for dvng nto regons wth nterestng trends that do not recur across the data as a whole. 4 CONCLUSION AND FUTURE WORK In ths paper we have nvestgated technques for automatc selecton of aspect ratos for lne chart dsplays, revstng Cleveland s technque of bankng to 45. An emprcal nvestgaton of varous bankng approaches revealed useful correlatons. For example, the data suggest that Cleveland s preferred technque of bankng the weghted average absolute orentaton to 45 may be well approxmated by the more computatonally effcent approach of bankng the average absolute slope to 1. We also explored technques combnng spectral analyss and bankng to automate the selecton of aspect ratos at multple scales. The ntal results are encouragng, dentfyng trends of nterest across multple scales of the data. It s worth dscussng a few ssues that arse when applyng the presented technques n practce. In our provded examples we have presented the banked data on a y-axs range determned by the mnmum and maxmum values n the data. Of course, for quanttatve values on a rato scale t s often more approprate to start the scale at zero. Fortunately, ths does not requre any changes to the bankng routnes, only to the dsplay logc. When the range of values does not nclude zero, the aspect rato ar 0 for a zero-based plot that preserves the banked aspect rato ar b for the data can be computed as ar 0 = ar b (y max - y mn ) / y max. Another ssue to consder s how to properly bank multple curves plotted on the same dsplay. Example scenaros nclude parallel coordnates, mult-lne plots, and algned small multples (as n the sparklne example, 3.3.2). As descrbed by Cleveland, one can bank multple curves plotted over the same scale by smply ncludng the lne segments of each of the curves wthn a sngle bankng optmzaton. For non-algned scales, one can approprately normalze the data and then perform a batch optmzaton usng the normalzed scale. Whle ths approach covers the bankng optmzaton, the spectral analyss component of a mult-scale bankng should be run separately for each curve, creatng a collecton of potental frequency cut-ponts. Each ndvdual curve can then be low-pass fltered subject to a cut-pont, and the fltered curves can then be aggregated for the bankng optmzaton. An nterestng avenue for further research s the task of correlatng the dfferent trends dentfed n each curve. The goal would be to better select cut-ponts for a group of curves beng banked together. Results of mult-scale bankng analyses presented n ths paper have ncluded plots of the aspect ratos for each possble low-pass flterng of the data set. Ths rases an addtonal possblty: forego spectral analyss and nstead compute and then analyze the aspect rato curve drectly. For example, plateaus n the aspect rato curve may provde good vews of the data. Whle feasble, the approach suffers from a few lmtatons. Most notable among these s the computatonal cost, as the technque requres an exhaustve applcaton of flterng and bankng computatons. If the data set sze s large or a bankng routne requrng teratve optmzaton s used ths quckly becomes prohbtvely expensve. Whle we have used low-pass flterng to create trend curves for dentfed scales of nterest, other approaches are possble. Followng the lead of Cleveland, we could subtract lower-frequency trends from the data, and bank the resduals to examne hgher-frequency trends n solaton. Equvalently, we could smply hgh-pass flter the data nstead. Applcatons of band-pass flterng also suggest themselves, for example to examne a local clump of spectral energy, as dscussed prevously. The results of spectral analyss could also be appled to nform alternatve trend lne generaton schemes. Recall that the loess regresson technque used by Cleveland takes two parameters: a smoothng parameter α and a polynomal-degree parameter λ. As ths frst parameter descrbes the sze of a local wndow of consderaton, good values of α may correspond well to the perodcty of frequency components suggested by spectral analyss. We defer detaled exploratons of these possbltes to future work. Fnally, t s worthwhle to consder alternatve means of dentfyng scales of nterest. We have used spectral analyss, whch dentfes perodc patterns across the entre data set. Cleveland has used repeated, manual applcaton of locally weghted regresson to dentfy scales of nterest. Alternatve technques for solatng trends n the data (such as wavelets or automated regresson technques) are possble, and may prove approprate dependent on the context of analyss and the type of data under consderaton. ACKNOWLEDGEMENTS We would lke to thank Dan Goldman for askng the questons that led to our ntal nqures about alternate optmzaton crtera. REFERENCES [1] Cleveland, W. S., M. E. McGll, and R. McGll. The Shape Parameter of a Two-Varable Graph. Journal of the Amercan Statstcal Assocaton, 83: , [2] Cleveland, W. S. Vsualzng Informaton. Hobart Press [3] Cleveland, W. S. A Model for Studyng Dsplay Methods of Statstcal Graphs. Journal of Computatonal and Statstcal Graphcs, 2: , [4] Cleveland, W. S. The Elements of Graphng Data, Revsed Edton. Hobart Press [5] Heer, J., S. K. Card, and J. A. Landay. prefuse: a toolkt for nteractve nformaton vsualzaton. ACM Human Factors n Computng Systems (CHI) [6] Palmer, S. Vson Scence: Photons to Phenomenology. MIT Press [7] Tufte, E. The Vsual Dsplay of Quanttatve Informaton. Graphcs Press, [8] Tufte, E. Beautful Evdence. Graphcs Press, 2006.