Forecasting the Sales of Music Albums: A Functional Data Analysis of Demand and Supply Side P2P Data

Transcription

1 Forecastng the Sales of Musc Albums: A Functonal Data Analyss of Demand and Supply Sde P2P Data Il-Horn Hann, JooHee Oh, and Gareth James Abstract We predct the sales of musc albums by utlzng demand and supply sde P2P data usng a functonal data analyss (FDA) approach. We fnd that the characterstcs of the functonal form of downloadng behavor explan frst-week sales by more than 6% after controllng for album characterstcs. By updatng our forecasts from 4 weeks to 1 week pror to the album release date, we examne the dynamc changes across dfferent quantles of the sales-dstrbuton for the demand- and supply-sde P2P data. We fnd that the gap between downloadng effect on sales among hgh-quantle vs. low-quantle albums reach the hghest level one week before the release date. Introducton A recent artcle n The Economst (Smmonds 28) reports how frms can beneft from P2P pracy by ganng a real-tme glmpse on the consumers musc tastes and preferences. Our own nteractons wth ndustry experts have shown an ambvalent thnkng regardng P2P networks. Whle many executves almost reflexvely see P2P networks as the enemy, some are seekng ways to use them to ther advantage. For example, move studos have started to seed P2P networks wth move tralers as part of ther vral marketng strategy. In or work, we take advantage of the appearance of musc fles well before the offcal launch date to predct the success of musc albums. Pre-release forecasts of album sales have hstorcally been poor. Good predctors for musc sales are hard to come by, hstorcally, the best predctors were known quanttes such R.H. Smth School of Busness, Unversty of Maryland Marshall School of Busness, Unversty of Southern Calforna 1

2 as the reputaton of an artst and the number of gold and platnum albums (see Lee, Boatwrght, and Kamakura 23). Ths makes the forecast of new releases especally dffcult. In our research, we utlze longtudnal P2P traffc data to forecast album sales. 1 For ths purpose, we employ functonal data analyss (FDA), an emprcal method developed by Ramsay and Slverman (25), to examne the functonal characterstcs of demand- and supply-sde P2P data. By usng functonal data analyss, we capture the smlartes and dfferences of functonal shapes such as trends or curvatures across downloadng hstores. Our approach s smlar to Foutz and Jank (27) who used onlne vrtual stock market prces to predct move box-offce. Data We have three dfferent sources of data. Our data conssts of downloadng data from the Ares peer-to-peer network, sales data for newly released albums n the Bllboard s Top 2 albums, and album specfc characterstcs. We used downloadng data from P2P network and album specfc characterstcs to generate frst-week sales of newly released albums. Our P2P data comes from a leadng P2P ant-pracy and marketng solutons provder. The company actvely montors all major P2P networks and collects data on downloadng and sharng actvtes and provdes servces to all major record labels and move studos. We obtaned downloadng and sharng data from the Ares peer-to-peer network for the tme perod of Aprl 27 to September 27. Ares was chosen prmarly for three reasons: popularty of the P2P network, breadth of coverage of the P2P network, and ablty to montor downloadng and sharng actvtes. The raw data for the network s about 6GB per month; ths ncludes data for downloadng actvtes as well as for sharng. The data ncludes lsts of the name of fle, ttle of album, artst name, genre of musc, unque hash of the fle, and user IP address. We have two types of fle-sharng data, referred as hashes and sources. Each song fle s dentfed by several unque hash codes. Hash-based downloads data s based on daly hashrequests number for each song fles. Whle hash-based downloadng measure represent demand-sde song downloads, source-based downloadng measure represent supply-sde of 1 Bhattacharjee et al. (27) use longtudnal data for a survval analyss. 2

3 downloads. Source-based downloadng measure s based on total unts of global nodes where song-fle s avalable and provded for fle requests. Hash-based and Source-based fle downloads data s processed as daly tme-seres for each album from as early as two-months pror to the release date. Fgure 1-1 and Fgure 1-2 each llustrates average number of daly hash-based and source-based downloads of songs n the albums from two-months pror to the release. For the weekly sales of newly-released albums, we selected new albums from the Bllboard 2 albums chart n the tme perod of May 27 to September 27 provded by SoundScan Inc. 2 Among around 1, albums that appears at least once n the chart from May to July 15, 27, we selected only newly-released albums on that week, ths represents about 2 albums per week. We deleted move soundtracks or re-entered albums due to ther atypcal sales patterns. We ncluded albums that has mnmum three days of downloads pror to the release date from the Ares Peer-to-Peer network. Here, we study total 172 albums of sales predcton usng P2P downloadng pattern and album characterstcs from two months before the release. We ended up from 75 to 152 albums from a month to one week pror release for the pre-release forecastng. We ncluded all the albums that were downloaded more than three days from two-months pror release and updates the album set every week that appears new n every followng week. At one month pror to the release, we have around 75 albums, 12 at three-weeks pror stage, 129 at two-weeks pror and 152 albums a week before release date. Our data ncludes albums whch result frst-week sales range from as low as 3,772 unts to as hgh as 622,827 unts. [Fgure 1-1: P2P Downloads of Albums based on Hash-Request] 2 The data s based on sales from 14, retal outlets, ncludng 4 dfferent chans, 11 mass merchandsers, and over 6 ndependent retal locatons. The average weekly transacton amount to 9-1 mllon albums and the data are sent va model from pont-of-sale regsters n the stores. 3

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5 [Fgure 1-2: P2P Downloads of Albums based on Global-Source] We also collected followng set of album characterstcs varables that have been known as nfluental to forecastng of musc sales. They are genre category of an album, number of daly comments from the YouTube webste before release date, gender of the artst, total number of albums released by an artst, the exstence of sngle albums before release. Whle prevous lterature used weekly number of Rado Arplay as a proxy for marketng varables, we utlzed daly number of comments from the YouTube webste. We also created a dummy for the albums where the sngle album exsts pror to the release. Total number of prevous albums of the artst, genre category of musc, and gender of artsts varables have been known to be nfluental on musc sales. We separated genre varable as Rock, Rap, R&B, Pop, Country and Gospel. Gender varable of the artst s based on male, female and group. Whle we also collected average ratngs from All Musc Gude (AMG) webste, a mxture of professonal and commons ratngs for the album, we excluded the varable because the ratngs were not avalable pror to the release. Table 1 summarzes characterstcs of the data. 5

6 [Table 1: Summary of Album on P2P down2loads, Frst-week Sales and Characterstcs] Varable Obs. Mean Std dev. Mn Max Medan Rank Sales Ave. hash Ave. source YouTube Total albums Sngle AMG ratng Rock Rap/R&B/Pop Country/Gospel Male Group Female Emprcal Pattern of Pracy Curves Ths study attempts to examne functonal characterstcs of pracy curves based on P2P downloadng data for pre-release forecastng purpose. We analyze shapes of pracy curves for the newly launchng albums to understand dfferent types of album nformaton. In partcular, we relate ndvdual album level nformaton grounded on characterstcs of pracy curves to ts mplcaton on frst-week sales. Measurng dynamcs of pracy dsperson n level, speed, and acceleraton along the tme helps understandng shape characterstcs of ndvdual curves. We compare shapes of ndvdual curves to the shape of aggregated market-level pracy curve. Informaton on shape characterstcs of pracy-curves of hgh-sales albums compare to the aggregated market-level s useful for forecastng. Also nformaton on downloadng patterns of low-sales albums wth respect to the market-level realzes not dentcal to that of hgh-sales albums. Based on Functonal Data Analyss (FDA), whch nvolves Functonal Prncpal Component Analyss (FPCA), our analyss heavly reles on the characterstcs of shape and covaraton degree of ndvdual curve wth respect to market level pracy curve. Our approach nvolves three major steps based on the Ramsay and Slverman (25). 6

7 Frst, we derve smooth downloads paths from the observed downloadng hstory data before release. We analyze functonal process of each hash and source based P2P traffc as a measure of early nformaton dsperson for newly launchng albums. Despte ther contnuous nature, lmtatons n measurement capablty allow us to record only dscrete, such as daly, observatons of these curves. Thus, the frst step s to recover, from the observed data, the underlyng contnuous functonal objects by usng smoothng methods. We have x( t ) = µ ( t) + f ( t ) + ε ( t ), where ε s the unobserved error component and the sequence of x are the observed outcomes of the P2P downloads. A major underlyng premse s that the underlyng process, f t ), s dfferentable functon to some order. We appled a flexble and ( computatonally effcent technque called the penalzed smoothng splne (Ruppert et al., 23; Ramsay and Slverman, 25). The penalzed smoothng splne f mnmzes the 2 PNSSE λ ( + λpen 2( f ). The degree of penalzed square error (PNSSE), f x) = [ x f ( t )] departure from a straght lne s measured by defnng a roughness penalty k 2 { D f ( t) } PENk ( f ) = dt, where D k f, k = 1,2, L, denotes the k-th dervatve of the functon f. The smoothng parameter λ controls the trade-off between the data-ft, as measured by the summaton on the left-hand sde of above equaton, and the local varablty of the functon f, measured by the roughness penalty PEN. k 7

8 [Fgure 2: Penalzed smoothed functon of P2P Downloads] Second, we extract the key shape characterstcs that capture the smlartes and dfferences across these downloadng hstores from the functonal Prncpal Components Analyss (FPCA). We summarze characterstcs of level, frst-dervatve, and second-dervatves of hash-based download functon and source-based download functon. Functonal prncpal component analyss s smlar to the prncpal component analyss except for that we now operate on a set of contnuous curves rather than dscrete vectors. For downloadng curves of albums, x ( s),..., xn ( ), we can decompose n the followng form, 1 s x ( t) = ( t) + µ sj PC j ( t), = 1, K, n usng prncpal components curves. We fnd a correspondng = j 1 set of PC curves PC j (s) that maxmze the varance along each component and are orthogonal to one another. We fnd the PC functon PC j (s) whose PCS S = PC ( 1 1 s) x ( s) ds maxmze S subject to PC ds = PC 1. The next step nvolves fndng PC ( ) for whch the PCS 1 1 = S 2 = PC2 ( s) x ( s) ds maxmze S subject to PC 2 = 1 and the constrant PC s) PC ( s) ds. 2 ( 1 = 2 s 8

9 Prncpal component (PC) curves llustrate the drecton of greatest varablty n the curves about ther mean. From the shape of PC curves, we can contrasts the pattern of varaton level n album downloads across the perod. For example, n Fgure 3, frst PC curve from hash-based downloadng curve depcts ncreasng convex curve around 2 weeks before release followng after declnng trend. Frst PC curve from source-based downloadng curve captures ncreasng convex curve around 2 weeks before release followng after ncreasng concave trend. Frst PC curve of demand-sde downloadng curve shows that varaton of daly song fle demand across albums, startng from postve level, slowly decrease untl two weeks pror to the release where t turns to ncrease towards the launchng n average. Whle the varaton of daly demand for the song fles n the early-stage capture the nterest-gap among early-adopters who represent unque preference on yet unknown songs, as more nformaton avalable approachng two weeks pror stage, these preference-orented demands varaton decreases. Varaton among song fle demands start to rapdly ncrease from two-weeks pror to the release date as the early nformaton on songs dffuse wth externalty on ts popularty. In turn, varaton n supply-sde of song-fles ncreases over tme wth dfferent speed. Whle we have ncreasng varety of song fles on supply-sde as tme passes, varaton across song-fle n supply-sde ncrease wth hgh-speed at last two-week perod compare to the early perod. As demand-sde externalty for popular song-fles speed up durng last two week, steeper curve slope of supplyng-sde of song fles n last two week shows the cumulatve degree of expanson for the gap of fle dsperson by ther popularty. [Fgure 3: Frst three PC of Demand- and Supply-sde of Pracy Level curves] st PC 2nd PC 3rd PC st PC 2nd PC 3rd PC The decomposed vectors of downloadng varaton n Fgure 3 allow us to explan general shape of downloadng curves and the drecton of curve varablty across tme. Whle the frst 9

10 prncpal component curve explans almost 7% of varaton, the second and thrd components carry nformaton regardng remaned drecton of dfferent varablty n the downloadng curves. The second PC curves contrast the varaton of downloads for song fles n early and late perod. It depcts decreasng pattern of downloadng curves at early vs. late stage or perod n both demand- and supply-sde. The thrd PC curve llustrates varablty of downloadng patterns for the md-term around three weeks to two weeks pror to the release. Whle demand sde for song fles downloads curve depct convex shape httng the bottom around md-term, supply sde for song downloads curve shows concave shape httng the top around two weeks pror to the release. Demand-sde fle downloadng curves, grounded on hash-requests queres, show that demand-sde nformaton flow on album remans hgh at the early and last stage where the varaton of nterests across albums s also hgh. Supply-sde fle downloadng curves, grounded on unts of source nodes on the network, llustrate the shape of cumulatve amount of nformaton avalable for downloadable song fles at each perod where the varaton of nterests across albums reman hgh or low. Each PC curves n Fgure 3 represent the respectve drecton of varablty of downloadng curves n early, md- and lateterm across tme perods. In Fgure 4, we show an alternatve way to vsualze PC curves accompaned wth mean curve. We llustrate mean curve and frst prncpal component (PC) of downloadng curves wth mean separately on albums that have postve value vs. negatve value of scores, µ ( t) ± s j PC j ( t) where s j s the mean scores of postve value vs. negatve value separately. In each chart, we llustrate mean curve whch depcts market-level downloadng curve, albums wth postve value of PCSs and albums wth negatve value of scores. Real lne n Fgure 4 desgnates a typcal shape of downloadng curves for the albums wth postve value of score. Dashed lne captures the typcal shape of downloadng curves for the albums wth negatve value of scores. The value of mean postve scores (PCS) s represented as the gap between real-lne and dotted-lne (mean-curve) n each chart. Also the average value of negatve scores s the gap between dashed lne and dotted-lne (mean-curve) n each chart. In ths aspect, real lne and dashed lne at Fgure 4 reveal the average level of PCSs for the albums wth postve vs. negatve 1

11 value separately n the drecton of frst PC curve. The value of score (PCS) desgnates the degree of covaraton of downloadng curves wth respect to the drecton of PC curve. Ths means that albums wth hgher scores accompany downloadng curves that follows hghly covaratng patterns wth respect to the frst prncpal component. Hence, downloadng curves of albums wth postve scores follow the pattern of frst prncpal component wth postve scale of covaraton. In contrast, downloadng curves of albums wth negatve scores covarate wth respect to the opposte drecton of the frst PC curve. In the followng part, we relate the value of scores (PCS) to the frst week sales. If the relatonshp between PCS of level functon and the sales s postve, we can analyze the result as that albums wth postve value of score wll result hgh sale after release and vce versa. Hence the value of PCS desgnates frst week sales. The frst raw panel of Fgure 4 llustrates the raw (Level), frst-dervatve (Velocty), and second-dervatve (Accelerator) of demand-sde downloadng curves. The second raw panel curves shows the same based on supply-sde downloadng data. Real lne and dashed lne n each chart can be analyzed n three aspects: respectve level of downloadng amount, the shape of curve change across the perod, and the sze of ncremental gap wth respect to the mean curve of the market. [Fgure 4: Frst PC curve of Level, Velocty and Accelerator of Demand- and Supply-sde Pracy curves] Mean w/ Postve Scores Mean Mean w/ Negatve Scores Mean w/ Postve Scores Mean Mean w/ Negatve Scores Mean w/ Postve Scores Mean Mean w/ Negatve Scores Mean w/ Postve Scores 3 Mean 25 Mean w/ Negatve Scores Mean w/ Postve Scores 12 Mean 1 Mean w/ Negatve Scores Mean w/ Postve Scores 15 Mean 1 Mean w/ Negatve Scores

12 The frst column of charts shows the general shape of downloadng curves n level. Albums wth postve value of score, n general, have hgher demand and supply n downloadng all over the perod whle albums wth negatve value of score, have lower demand and supply n downloadng. Albums wth postve value of scores have ncreasng pattern followng after decreasng towards the release whle albums wth negatve value of scores reman relatvely stagnant. Compare to the mean curve of the market, devaton n fle-demands and supply across albums remans hgh for the albums wth postve value of score. The degree of varaton n fle-demand across albums follow ncreasng pattern before release followng after decreasng pattern as captured by the ncremental gap wth mean curve. Supply-sde of downloads varaton follow ncreasng pattern over the perod as captured by the ncremental gap n the second raw. Ths result mples that f the sales coeffcent of level PCS s postve, then albums wth postve scores wll result hgh sales and the devaton between hgh-sales albums wll be larger than the sales devaton of low-sales albums. The predcted sales gap among albums wth hgh PCS vs. low PCS shows postve and slghtly U-patterned dfference n demand-sde of downloads. In contrast, supply-sde downloads curves predct sales gap based on value of PCS wll show ncreasng pattern towards release date. For ths dfference n nformaton that PC curves capture from demand- and supply-sde data, we combne the PCS from both demand- and supply-sde PC curves n the model. The second column of charts shows the frst-dervatve (velocty) of downloadng curves. Demand-sde downloadng curves show that albums wth postve score grow dramatcally faster than average market level downloads. It also shows that demand for song fles for albums wth negatve score reman lower and sgnfcantly slower down from the market level from ten days pror to the release than that of albums wth postve value of scores. Supply-sde downloads growth remans hgher than market-level untl t slacks down and rapdly back up durng last two-weeks. These velocty curves dstnctvely captures turnng pont of shapes around two weeks pror to the release date. Incremental gap of velocty curves of album wth postve value of score compare to that of market mean curve shows exponentally ncreasng dfference from around 2 weeks pror to the release. These shape characterstcs mples that f the sales coeffcent of velocty PCS s negatve, then albums wth postve value of velocty 12

13 scores wll result lower sales than market average and sales varaton among albums predcted to have low-sales wll be hgher than devaton among albums predcted to result hgh-sales. Whle the demand-sde velocty curve ndcates ths nformaton gap between the group of albums wth postve vs. negatve value of scores from around two-weeks pror to the release, the supply-sde velocty curve captures ths dfference nformaton between groups from very early-stage and late-stage. The thrd column of charts llustrate the second-dervatve (Accelerator) of demand- and supply sde downloadng curves. Whle acceleratng speed of market-level downloads remans stagnant for overall perod, albums wth postve accelerator scores show bg jump n both drecton of curves. Compare to the level- and velocty- curves, accelerator curves convey nformaton from ts bg turnng shape change around the mean curve to the sales. Ths mples that f the sales coeffcent of accelerator PCS s postve, albums wth postve accelerator PCS wll result hgh sales and ther 2 nd dervatve of downloadng curve wll show dramatcally turnng shape around the mean-level downloads. Hence one can predct the level of sales by measurng devaton of accelerator curve n absolute term around. As we examned from Fgure 4, each level-, velocty- and accelerator of downloadng curve conveys dfferent type of nformaton based on ther characterstcs. Descrpton of Pracy curves on Sales: Functonal Regresson We use functonal regresson to predct frst week sales usng PCSs obtaned from prevous steps. Let x (t) be the smooth splne representaton of the th downloadng curve for the album observed from two months pror release. Let yrepresent the frst week sales of an album after release. Functonal regresson establshes a relatonshp between predctor, x (t), and the tem to be predcted, y, as y = f ( x ( t)) + ε, = 1K,, n. Whle ths equaton s dffcult to work drectly because x (t) s nfnte dmensonal, however, for any functon f there exsts a correspondng functon g such that f ( x( t)) = g ( s1, s2, L) where s 1, s 2 Lare the prncpal component scores of x (t). We use ths equvalence to perform functonal regresson wth the functonal prncpal component scores as the ndependent varables. Ths approach s related 13

14 to prncpal components regresson whch s often used for non-functonal, but hgh dmensonal, data. The smplest choce for gwould be a lnear functonal n whch case above equaton becomes y D = β + s j β j + ε some D 1 j=1 for. A somewhat more powerful model s produced by assumng that gs an addtve but nonlnear, functon (Haste and Tbshran, 199). In ths case, above equaton becomes as follows where g j s are non-lnear functons that are estmated as part of the fttng procedure. y D + g j ( sj ) j=1 = β + ε One advantage of usng above equatons to mplement a functonal regresson s that once the scores (PCS) varables, s j s have been computed va the functonal PCA, we can then use standard lnear or addtve regresson to relate yto the prncpal component scores. We can also extend the model by augmentng covarates that contan nformaton about the scores beyond the prncpal components, such as album characterstcs or marketng varables. In the forecastng part, we develop two-types of functonal regresson by augmentng market nformaton on the prncpal component scores. We apply medan and nonlnear functonal regresson on selected varables to dentfy shape characterstcs that have sgnfcant mpact on frst week of musc sales. In ths last step, we selected frst PCS (prncpal components scores) whch explan more than 7% of varaton calculated from the raw and dervatve functons as ndependent varables n the model. We combned PCS varables wth album characterstc varables such as the total number of prevous albums and average daly number of YouTube comments to explan frst-week sales of newly launchng album. We generate pre-release forecasts of sales weekly from as early as a month to a week before release date. To compare our forecastng models, we calculated mean absolute percentage error (MAPE) wth one-album cross-valdaton (CV) for each model. Instead of separatng tranng set as ten-fold methods, we cross-valdated each album n the album set by treatng remanng set as tranng set. Ths ndvdual album-based cross- 14

15 valdaton enabled us to generate unque and nvarable measure of MAPE wth respect to the sample. Estmaton Result We present estmaton result of medan regresson based on albums characterstcs and scores (PCS) obtaned from functonal prncpal component analyss (FPCA). We selected average daly number of the YouTube comments, total number of prevous albums that an artst has, and dummy varable for the exstence of sngle albums as album characterstcs whch turned out to be sgnfcant on sales predcton. From four weeks pror release to a week pror stage, the adjusted R-square and Psuedo R-square of our model ncrease sgnfcantly. Whle adjusted R-square of the model about a month pror release explan frst week sales 56%, t ncrease to 73% at a week pror release. In Table 2, characterstcs varables for the albums show hghly postve relatonshp to the frst week sales at each stage. All varables turn out to be hghly sgnfcant at most stage n the perod except YouTube varable at three weeks pror model and Sngle varable at four weeks pror to the release. We have sx prncpal component score (PCS) varables n the model each representng demand- and supply sde nformaton of level, velocty and accelerator downloadng curves. All PCS varables that capture shape characterstcs grounded on FPCA turned out to be sgnfcant and nfluental on frst-week sales at each stage after controllng the effect of album characterstcs nformaton. Score (PCS) varable based on level curve of both demand- and supply-sde downloadng data shows postve relatonshp on the sales. Degree of covaraton to the same drecton mpled by correspondng PC represents our unt ncrease of ndependent varable based on PCS. A scalar ncrease of covaraton level of demand-sde downloadng curve wth respect to the same drecton of level PC curve ncrease about 38 unts of sales a month pror. In turn, a unt ncrease of covaraton of supply-sde downloadng curve on the drecton of PC-curve ncreases 36 unts of sales a month pror. 15

16 The effects of unt-change on PCS varables based on demand-sde downloadng curves and velocty of supply-sde downloadng curves on sales show ncreasng pattern from 4 weeks pror to 2 weeks pror stage and turn down at a week pror release. In contrast, a unt-change of supply-sde downloads level curve on sales decrease from 4 weeks pror to 2 weeks pror stage and shows upturn to a week pror release. Whle the sales coeffcent of medan regresson n Table 2 measures effects of unt-change of each PCS varables on sales, t cannot capture overall nfluence of each PCS varables on sales. Value of each PCS varables also affect to the sales multpled by sales coeffcent. To compare the effects of PCS varables on sales, Table 3 summarzes the effect of each PCS varable on sales around average value of score (PCS). [Table 2: Estmaton Result of the model at each stage pror to the Release] R- square Album Charac. Hash Source Varables Week-4 Week-3 Week-2 Week-1 Psuedo R R2 (OLS) Adj. R Cons (3.58**) (4.41**) (1.7**) (11.19**) YouTube (3.4**) Total# (2.97**) Sngle (.45) Level (1.57*) Velocty (-1.98*) (.84) (3.95**) (4.31**) (3.53**) (-3.34**) (6.84**) (9.95**) (5.57**) (22.1**) (-27.58**) (4.3**) (5.6**) (2.61**) (9.83**) (-12.42**) Accelerator NA NA NA (14.54**) Level (9.55**) 1.44 (4.72**) 2.74 (2.53*) (18.48**) Velocty (-8.36**) (-5.34**) (-1.93*) (-14.74**) Accelerator NA NA NA (1.3) 16

17 Table 3 summarzes the effect of each PCS varable on sales by multplyng mean value of score (ndependent varables) on the unt-effect of score value ncrease (beta-coeffcent). In the early stage from 4 weeks to 3 weeks pror-release, PCS obtaned from supply-sde downloadng curves are more nfluental on sales compare to the PCS based on demand-sde downloadng nformaton. However, as we approach to the release date from around 2 weeks pror to the release, demand-sde downloadng curves affect frst-weeks sales more than supply-sde nformaton around average level of score value. In a week pror to the release, nstead of downloadng level curves, velocty curve of downloadng data reveals the most nfluental on sales. The sgn of effects of PCS varables on sales s neglgble n Table 3. Average mean value of PCS varables were very small number close to the whch range from a bg negatve value to a bg postve value. For ths reason, we compared absolute term of ts effect on sales to compare relatve nfluence of PCS varables on sales. [Table 3: Mean effects of PCS varables on Sales] Mean Effect (x 1-4 ) Varables Hash Source Week-4 Week-3 Week-2 Week-1 Level Velocty Accelerator NA NA NA 1.24 Level Velocty Accelerator NA NA NA.3 Pre-launch Forecastng wth Cross-Valdaton We present the performance of our proposed model based on augmented functonal regresson usng cross-valdated measure of MAPE. Table 4 presents out-of-sample forecastng results based on MAPE (Mean Absolute Percentage Error) usng medan regresson by updatng the nformaton from one month to a week pror to the release date wth cross-valdaton. We also appled non-lnear regresson based on generalzed addtve model (GAM) and ordnary least squares (OLS), however, medan-regresson outperformed the others n forecastng based on MAPE. Overshootng was severe n two other methods. We present and compare all our forecastng result based on cross-valdated measure of MAPE for hold-out sample. Crossvaldaton method based on an album rotates samplng each and every album n the data set as 17

18 hold-out and treat remanng set as tranng sample. Ths unque album-based cross-valdaton enables us to obtan hold-out set nvarant, unque measure of errors to calculate MAPE. Fgure 5 llustrates our forecastng model. At each stage from four weeks to a week pror to the release date, we use avalable set of downloadng data to estmate coeffcents of download functons. Instead of usng estmated curves of past data, we used downloadng curves for last two-weeks perod predcted from the downloadng functons from avalable data. [Fgure 5: Descrpton on Forecastng Model] We found the last two-week perod of downloadng data more nformatve to generate sales forecasts. At a month pror to the release, however, downloadng data for last two-week perod are not avalable. Hence, we predcted downloadng data for the last two week perod usng the estmated functons based on avalable data from 8weeks pror to a month pror stage. Table 4-1 summarzes our pre-release forecastng results at each week startng from a month pror to the release. It shows decreasng pattern of MAPE based on average measure as we update nformaton from downloadng data every week. Whle average measure of MAPE s as hgh as from 1% to 2% of sales, medan-measure of MAPE ranges as low as from 48% to 73%. Hgh value of average-measure of MAPE reveals the varety of albums wth sales that ranges from very low to very hgh n our data sample. In our data sample, we have albums that realze MAPE from as low as.2% to as hgh as 19% of sales. Ths range of MAPE mples that any MAPE forecastng result that s not based on our cross-valdaton method for each album generate only an nstance of MAPE n ths range dependng on the holdout set. Our unque and 18

19 holdout-nvarable measure of MAPE result shows as low as 48. 4% based on medan from two weeks pror to the release. [Table 4-1: Proposed Model - Augmented Functonal Regresson] Cross Valdated MAPE Hold-Out sample Mn Max Medan Average Week Week Week Week Table 4-2 supports our forecastng model llustrated n Fgure 5. Benchmark model n Table 4-2 results better predcton for the sales whch used real downloadng data for last twoweek perod although the data s not really avalable at 4 weeks pror stage. Our forecastng model s bult on ths benchmark model usng predcted data for the perod where the downloadng data seems the most nformatve n forecastng sales. A week pror release, our model usng predcted data, performs even slghtly better than usng real data at the last stage. Ths result mght mply as we approach to the release date, at the last week, nose n downloadng data dramatcally ncrease whch create dfference n forecastng result wth our model based on predcton usng the real data untl two weeks pror stage. [Table 4-2: Benchmark for Proposed model based on Real data] Cross Valdated MAPE Benchmark: Hold-Out sample Mn Max Medan Average Week Week Week Week Comparng Alternatve Models To fully understand the advantages of FDA, we compare two models of FDA wth 3 non- Functonal models. Two functonal models are named as Functonal Regresson and Augmented Functonal Regresson and the three non functonal models are Album Characterstc, Meta Bass 19

20 and Augmented Bass. Table 4 classfes all the models based on ther use of nformaton across curves and the types of varables. The functonal regresson approach has three man strengths. Frst t s able to ncorporate nformaton from other types of albums to mprove predcton accuracy. Second, t mplements a non-parametrc fttng procedure so t s not restrcted by parametrc assumptons. Thrd, t utlzes the functonal nature of the downloadng curves. We chose the three comparson models to gan an understandng of the gans from each of these strengths. For example, Album Characterstcs are parametrc and non-functonal and doesn t ncorporate nformaton from characterstcs of other albums. In contrast, Meta bass and Augmented Meta Bass models extend Class Bass can ncorporate nformaton from characterstcs of other albums but are parametrc and non-functonal so they llustrate the mprovement from borrowng strength across curves. Functonal Regresson and Augmented Functonal Regresson ncorporate nformaton from other albums, and are flexble to be non-parametrc and functonal. [Table 4: Comparson of Alternatve Models] Album Characterstcs Meta Bass Model Augmented Bass Model Functonal Regresson Augmented Functonal Regresson Model Medan Reg. GAM YouTube comments, Total number of albums, Sngle Bass Parameters: market potental (m), nnovator coeff. (p), mtators coeff. (q) Bass Parameters (m,p,q) and album characterstcs Prncpal Component Scores (PCS) from level, velocty and accelerator curves Prncpal Component Scores (PCS) from level, velocty and accelerator curves and album characterstcs varables Non-parametrc lnear regresson around the medan (5% quantle) value of sales by mnmzng sum of absolute value of errors Non-lnear regresson based on Generalzed Addtve Model (GAM) Album Characterstcs Model s just based on the three albums characterstcs varable, descrbed n the Table 4. Ths s base-lne model that doesn t utlze any shape nformaton compare to the Bass-related model or functonal-type model. Meta-Bass model s an extended model of Class Bass Model. The Classc Bass Model (Bass, 1969) fts each curve n the sample separately by estmatng the followng models: 2

21 ( p+ q) t = [ F( t) F( t 1) ] + ε, F ( t) = ( p+ q) t s( t) m t 1 e 1+ ( q / p) e where t= tme perod, s (t) = sales at tme t, p= coeffcent of nnovaton, q= coeffcent of mtaton, m= cumulatve market potental. We estmate the model va the genetc algorthm because Venkatesan et al. (24) provde convncng evdence that the genetc algorthm provdes the bests method for fttng the Bass model relatve to all pror estmaton methods. For each curve, we use the downloadng curves usng smooth splne to estmate the three Bass parameters, m, p, and q. We then extend the Classc Bass model to use nformaton across curves. To do so, we frst estmate m, p, and q for each curve usng the genetc algorthm, as outlned above. Then, for each tem to be predcted, we ft the non-lnear addtve model, y = β + g m ) + g ( p ) + g ( q ) + ε, 1( 2 3 to the estmaton sample where, g 1, g 2, and g3are smoothng splnes as defned prevously. We used the estmated parameters from ths addtve model and the estmates of m, p, and qfor each curve n the holdout sample to compute the correspondng tem to be predcted for each of the holdout curves. Not that the estmaton of m, p, and qcan also be done usng a Bayesan formulaton wth a pror on { m p, q},. The Augmented Meta-Bass s the same non-lnear addtve model used for the Meta-Bass except that we add on album characterstcs for each the albums, thus: y = β + g1( m ) + g 2 ( p ) + g 3( q ) + g k ( zk ) + ε, k k = 1 where zkdenotes th k varable of album characterstcs for each album. Note that the Meta-Bass and Augmented Meta-Bass are extensons of the Classc Bass that make use of all of the nformaton across curves, rather than just utlzng each curve ndvdually. Snce, usng nformaton across curves s an essental feature of functonal regresson, dong so puts the Meta Bass and the Augmented Bass on the same platform as the FDA models. 21

22 For the Functonal Regresson model, we compute three prncpal component scores, the frst each on the downloadng curves, x (t), on the velocty curves, '( t), and on the accelerator curves, "( t). The prncpal component scores on the velocty and accelerator x curves are computed n an dentcal fashon to that for the pracy curves except that we utlze the dervatve of x (t). We then use these three scores as the ndependent varables n an addtve regresson model, as shown n the equaton at prevous parts. We then used the estmated parameters of ths equaton and the data from curves n the holdout sample, to compute the tems to be predcted n the holdout sample. Our second functonal approach enhances the power of Functonal Regresson by addng album characterstcs varable for each the albums, as wth the Augmented Meta-Bass model. Hence, the Augmented Functonal Regresson model nvolves estmatng a non-lnear addtve x model on the estmaton sample as follows y g j ( sj ) + g k ( zk ) ε where the j j= 1 k = 1 = β + g s are smoothng splnes. We then compute the albums to be predcted for each curve n the holdout sample from the estmated values of the above parameters and the data n each curve n the holdout sample. Ths model s drectly comparable to Augmented Meta Bass as both models use nformaton across curves and from products. Model Comparson We present our forecastng results wth comparson n fve dfferent models on each stage for whch Functonal Regresson and Augmented Functonal Regresson outperforms each of the other methods. We compare the Functonal Regresson model to the 3 other nonfunctonal models to assess the ablty of functonal-based model n predcton. In Table 5, Functonal regresson and Augmented Functonal Regresson are superor to the Album characterstcs model, Meta Bass model, and Augmented Bass model based on averagemeasure of MAPE at all perod. From a month to a week pror to the release, Functonal Regresson outperforms for at least 4% based on MAPE at any gven week comparsons wth three other non-functonal models. [Table 5 : Comparson of Cross-valdated MAPE wth Benchmarkng models ] 22

23 MAPE Comparson : Holdout sample Only Album characterstcs Average Medan Meta Bass model Augmented Bass model Functonal Regresson Week Augmented Functonal Regresson Week Week Week Week Week Week Week The performance of Functonal Regresson s mxed when compared wth a model based on album characterstcs n terms of medan-based MAPE. From four weeks to three weeks pror release models, Album characterstcs model and Augmented Bass model outperforms Functonal Models. However, from 2 weeks pror to the release, Augmented Functonal Model, our proposed model, outperforms for at least 3% based on medan-based MAPE for any gven benchmarkng models. It appears that downloadng data n early perod around a month pror release ncludes bg outlers resultng nformaton dfference n downloadng curves. Two functonal-based models capture the pattern of downloadng curves of these outlers that have sgnfcantly hgh level of downloads better than non-functonal based model, such as Album Characterstcs model, n sales forecastng. That s shown n the forecastng performance gap of functonal-related models between average-based MAPE vs. medan-based MAPE. Functonal-based models clearly outperform all other models measured by average-mape. Downloadng data convey more nformaton as we approach to the release date around two-week pror stage and Augmented Functonal model whch combnes both albums characterstcs nformaton and downloads nformaton across the curves performs the best. It also appears that the Augmented Functonal Model s slghtly superor to the Functonal Model from a week before the release based on average-mape. We can nterpret that the nformaton acqured across shapes of functonal curves embed more nose around a week pror to the release perod. Augmented Functonal model that ncorporate albums characterstcs nformaton reduces the 23

24 effects of nose obtaned form downloadng data. From an early stage of predcton around a month pror to the release, these shape nformaton reduces the lower boundary of forecastng error, at the same tme, produces overshootng n upper boundary due to the uncertanty. Whle models relyng album characterstcs appears to perform well based on medan-mape n the very early stage of forecastng, from two weeks pror to the release, as more nformaton penetrate n the market, functonal-based model outperforms n both mean-based and medanbased measure of MAPE. 24

25 Appendx [Table 2: Par-wse Correlaton] Varable Sales Ave. hash Sales 1 Ave..686** 1 hash (.) Ave..6921**.9151** source (.) (.) YouTube.5468**.7846** (.) (.) Total.1796* albums (.184) Sngle.3655**.2142** (.) (.48) Ave. source 1.75** (.).2241** (.31) YouTube Total albums 1.242** (.14) * (.596) Sngle [Fgure : Functonal shape of Hash- and Source- based P2P downloads] 1 25

26 [Fgure 2-2: Penalzed smoothed functon based on Global-Source] Album level analyss We analyze characterstcs of downloadng curves wth examples of ndvdual albums n hgh-, medum-, and low- range of sales. Table 2 summarzes each three albums from three categores of sales volume where the hghest range starts from more than 4K and the lowest volume range lmts less than 1K of sales. For the albums summarzed n Table 2, Fgure 6 contrasts tme-changng pattern PCSs of each albums n the sales category from 4 weeks to a week pror to the release. [Table 1: Hash- and Source-based PCSs by Sales] Hash Source Sales Range Level Velocty Accelerator Level Velocty Accelerator Mean Sales > 4K (55) K < Sales < 4K (53) Sales < 1K (44)

27 Table 1 summarzes average prncpal component scores of albums wth respect to the frst prncpal components of level, frst-dervatve, and second dervatve of downloadng curves by sales. Average prncpal component score (PCS) of albums wth sales more than 4, unts s for hash-based level curves where that of source-based downloads s Average PCS value s strctly hgher among hgh-sales albums compare to that of medum or low sales albums where unts sold range around from 1, to 4, or below 1,. Ths mples that demand-sde downloadng curves of hgh-sales albums covary about tmes wth a unt varaton of frst PC curve. For the supply-sde downloadng curves, hgh-sales albums covary wth the shape of frst PC curve around tmes of unt change. Negatve value of average PCS mples opposte drecton of covaraton wth respect to the drecton of PC curve. For example, average value of PCS of albums range between 1K and 4K of sales s and that of albums wth sales lower than 1K unts s Albums wth mddle range of sales shows hgher degree of covaraton,. 64 tmes, wth frst PC of velocty curve n the opposte drecton compare to the albums wth that of (. 32 tmes) low range of sales under 1K. From Table 1, we can confrm that downloadng curves of albums predct the level of sales from the degree of covaraton wth respect to the drecton of PC. Albums wth hgher PCS that covary more wth the drecton of PC result n hgher sales. Albums wth sales more than 4K shows the hghest covaraton level n the absolute term based on average value of PCS. It also shows that the level curves ndcates the most dfferences n terms of absolute dfference of covaraton level (PCS) among downloadng curves of albums by sales. In turn, velocty curves ndcate the most dfference n terms of percentage change of covaraton level (PCS) accordng to the unt change of PC. For example, albums wth more than 4K unt sales covary wth the drecton of PC curve ( ) tmes wth respect to the unt varaton based on source-based level curves. In turn, level and accelerator curves show around 16% dfference of covaraton n terms of percentage change between hgh-sales and medum-level sales albums. Velocty curve reveal around 19% dfference of covaraton level between hgh-sales vs. medum sales group of albums. Hence, each type of curves-level, velocty, and accelerator- reveal unque characterstcs as ndcator functon to predct sales based on the downloadng curves. 27

28 [Table 2 : Illustraton of Albums by Sales level] Hgh Sales (Sales > 4K ) Medum Sales (1K < Sales < 4K) Low Sales (Sales < 1K) Artst Album Sales T. I. T. I. VS TIP 467,737 MAROON 5 IT WON T BE SOON BEFORE LONG R. Kelly DOUBLE UP 385,93 CARTEL CARTEL 28,79 LIL WYTE ONE & ONLY 15,556 ROCKET SUMMER DO YOU FEEL 15,88 WITHIN HEART OF EVERYTHING 7,1 TEMPTATION PAUL VAN DYK IN BETWEEN 6,84 TOMAHAWK ANONYMOUS 4,96 [Fgure 5: Illustraton of Downloadng curvs of Albums ] 28

29 Prncpal component scores of album ttled T. I. VS TIP, wth more than 4K unts of sales, shows bgger n absolute value, compare to that of albums wth less than 4K sales from 4 weeks to a week pror to the release. In fgure 6, whle PCSs of album ttled DO YOU FEEL and ANONYMOUS slghtly change around -2 to 2 range for hash-based downloadng curves, PCS of T. I. VS TIP realzes n about 4 tmes bgger range. Ths shows that our PCS value based on shape of downloadng curves for an ndvdual album ndcates the sales from four weeks pror to the release by level of covaraton, denoted by PCS, wth respect to the prncpal component. [Fgure 6: Hash- and Source-based PCS of Level functon] (H)T.I. VS TIP (M)DO YOU FEEL (L)ANONYMOUS (H)T.I. VS TIP (M)DO YOU FEEL (L)ANONYMOUS Hash- and Source-based PCS of Velocty functon -6 29

30 2 1 (H)T.I. VS TIP (M)DO YOU FEEL (L)ANONYMOUS (H)T.I. VS TIP (M)DO YOU FEEL (L)ANONYMOUS [Fgure 4 : Hash- and Source-based PC Scores by Sales ] Sales > 4K (55) 1K < Sales < 4K (53) Sales < 1K (44) Sales > 4K (55) 1K < Sales < 4K (53) Sales < 1K (44) Level Velocty Accelerator. -5. Level Velocty Accelerator References Bhattacharjee,S., Gopal, R.D., Lertwachara, K., Marsden, J.R. and Telang,R (27), The effect of Dgtal Sharng Technologes on Musc Markets: A Survval Analyss of Albums on Rankng Charts, Management Scence. Dellarocas, C. (23), The Dgtzaton of Word of Mouth: Promse and Challenges of Onlne Feedback Mechansms, Management Scence, 49(1) Ramsay, J.O. (2), Functonal components of varaton n handwrtng, Journal of the Amercan Statstcal Assocaton, 95(449): 9-15 Ramsay, J.O. and Slverman, B.W. (1996, 25), Functonal Data Analyss (Frst, Second ed.), Sprnger-Verlag, New York Koenker, R., and Bassett, G. J. (1978), Regresson Quantles, Econometrca 46(2), 33-5 Ruppert, D., Wand, M.P., and Carroll, R.J. (23), Semparametrc Regresson, Cambrdge Unversty Press, Cambrdge G19 Bass, F. (1995), Emprcal generalzatons and Marketng Scence: A Personal Vew, Marketng Scence, 14(2), G6-3

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