A panel data approach for fashion sales forecasting

Transcription

1 A pael daa approach for fashio sales forecasig Shuyu Tsa-Mig Choi, Na Liu Busiess Divisio, Isiue of Texiles ad Clohig, The Hog Kog Polyechic Uiversiy, Hug Hom, Kowloo, Hog Kog Absrac: Sales forecasig is a impora problem i fashio reail operaios. I his paper, we propose a ovel pael daa based paricle-filer (PDPF) model o coduc fashio sales forecasig. We evaluae he performace of proposed model i erms of sales quaiy ad color red predicio by usig real daa from he fashio idusry. The experimeal resuls provide ovel isighs ad pracical guidace o operaios maagers o he use of pael daa for fashio sales forecasig. Keywords: Fashio sales forecasig, pael daa aalysis, paricle filer Iroducio A accurae sale forecasig is oe of he key success facors of he supply chai maageme i he fashio idusry. For he pas few decades, may forecasig mehods for he fashio produc have bee proposed ad sudied. Tradiioal saisical mehods, such as auo-regressio, expoeial smoohig, ARIMA, SARIMA are probably he mos widely used echiques for fashio sales forecasig. The advaages of hese models are beig fas, simple, maure ad easy o udersad. Bu he facors cosidered by hese models are usually limied (Ni ad Fa 0). Wih he advace of compuig echologies, arificial ielligece (AI) models, which are more versaile ha saisical models, have bee widely employed o impleme fashio sales forecasig i rece years ((Frak, Garg e al. 00) (Au, Choi e al. 008) ad (Yu, Choi e al. 0) ). Despie beig powerful, AI mehods usually require a subsaial amou of ime for coducig forecasig ad he forecasig performace largely depeds o havig sufficie hisorical daa for raiig. The limiaios of boh AI mehods ad saisical mehods hece call for he developme of iovaive ew mehods. This paper also follows his lie of hybrid model forecasig model research bu wih differe focal poi ad mehods.

2 Pael daa models (Aderso ad Hsiao 98), ypically refer o daa coaiig ime series observaios of a umber of idividuals. Therefore, observaios i pael daa ivolve a leas wo dimesios; a cross-secioal dimesio ad a ime series dimesio. Time series ad cross-secioal daa are special cases of pael daa ha are i oe dimesio oly. I usually coais observaios of muliple pheomea obaied over muliple ime periods for he same idividuals. I has bee widely applied i forecasig over he pas decades. I his paper, he pael daa mehod is employed because of is abiliy i capurig he idividual effecs ha exis amog cross-secioal sales daa bu are o capured by he icluded explaaory variables (e.g., he effec from he sales of oher correlaed producs). However, i is difficul for he pael daa mehod aloe o capure he oliear feaures amog he variables (Hsiao 00). To overcome his shorcomig, we propose he use of he paricle filer mehod. I fac, he paricle filer (PF) is a sae-space model ha has bee a powerful ool i modelig ad forecasig dyamic sysems (Do Chug, Li e al. 0). Arulampalam e al. (Arulampalam, Maskell e al. 00) sugges ha, i highly oliear eviromes, a oliear filer such as a paricle filer ca offer a good performace i rackig uexpeced chages. I our work, PF is adoped o predic he ucerai paers sice. Recely, usig he PF mehod, Li e al. (Li, Yu e al. 0) examie a wo-sage model which helps o predic eergy price for differe iercoeced regios. Their aalysis idicaes ha he proposed mehod yields beer ad more sable forecasig resul, compared o some oher commoly adoped AI mehods. I is kow ha fashio producs are quie differe from oher radiioal producs. For isace, here are a lo of sock-keepig-uis (SKUs) eve uder oe sigle produc lie. I geeral, hese SKUs demads are correlaed. Thus, he sales of fashio producs are o oly iflueced by he facors such as size, color, price, ec, bu also he sales of correlaed iems. Moivaed by his, his paper applies he pael daa mehod suppored by he paricle filer o ivesigae he complex relaioship bewee sales amou ad oher ifluece facors i fashio sales forecasig. The pael daa based paricle filer model I he PDPF model, he ime-series red of previous sale, he prices of he iems uder sudy, ad he whole pael impac from he oher correlaed producs are chose o be he decisio variables for coducig sales predicio. The pael daa model allows us o cosruc ad es he more complicaed behavioral models ha he purely cross-secioal or ime-series daa. I also provides he possibiliy of geeraig more accurae predicios for idividual forecasig oucomes ha hose forecasig mehods based solely o ime-series daa. As a remark, he forecasig problem wih very few hisorical daa would be solvable by he pael daa mehod (Hsiao 00).

3 The fashio apparel marke is srogly iflueced by may facors. These facors, commoly called explaaory variables, are ofe ucorollable ad someimes ukow ad also difficul o quaify heir impac (De Toi ad Meeghei 000). I his PDPF model, we icorporae some key facors, such as previous sale, correspodig price, ad he ieracio wih oher correlaed produc iems, io he model. We cosider wo-compoe srucure for he hybrid PDPF fashio sales forecasig model, amely he liear ad o-liear compoes. The pael daa mehod is proposed o ivesigae he relaioship amog hese facors i he liear compoe ad PF is used o hadle he o-liear facors as i is kow o be suiable for such a ask. The fashio sales of iem i durig he ime ierval, deoed by FS i,ca be represeed as follows FSi Si Ni, () where S i ad N i deoe he liear compoe ad he oliear oise, respecively. The commo pael daa model is hece cosruced as: * Sk k Sk P k k, k,...,k;,..., T, () where Si is he sale of iem i durig he ime ierval ; P i is he correspodig price; is coefficie for cross-secio; is coefficie for ime-series; here are K iems for each forecasig caegory. The idepede error erm i disribues over i ad, wih mea zero ad variace u ad is assumed o be ucorrelaed wih price ad previous sale. Assumig ha Si is he forecasig value from he pael daa, he oliear behaviour ha he pael daa ca hardly capure is described as: FS S. () i i i I his paper, PF is adoped o rack he oliear behaviour i he forecasig procedure because i is paricularly useful i dealig wih oliear ad o-gaussia problems (Sarkar 00). I fac, PF is a sequeial Moe Carlo mehodology ad he basic idea is usig paricles o represe he probabiliy desiy fucio (PDF) of sae. Sice he acual sales of fashio produc will be affeced by may o-liear facors, we prese he oliear error model which follows he sae space represeaio i he followig:

4 f (, ) Tasfer Fucio () i i i i y S Measureme Fucio () i i i where i ad i follow Gaussia disribuio i. i. d. N(0, ) ad N(0, i) i,, idepedely. I he measureme equaio, y is he observaio from each forecasig sep. For he PF model, based o (Zhog, Fug e al. 00), we iroduce is aalyical model as follows. Firs, assumig ha he probabiliy desiy fucio (PDF) of he iiial sae p( 0) is kow, he opimized sae esimaio is obaied by calculaig he degree of cofidece of p( ) y: i differe saes, : y represes a se of observaios from period o. The, he codiioal desiy p( y: ) is recursively updaed accordig o Equaios () ad (7). p( y ) p( ) p( y ) d, () : : p( y ) : p( y ) p( y ) p(y y ) :, (7) : where y: is defied as he hisory observaio sequece wih he radom variables. The deomiaor p( ) p( y: ) d is a cosa, which is available from he likelihood fucio ad he saisical characerisic of he observaio oise. PF provides a approximae soluio for he discree-ime recursive updaig of he poserior probabiliy desiy fucio p( ) y:. Uder PF, he poserior disribuio of is approximaed by a collecio of weighed paricles {, } N w. The poserior desiy ca be calculaed by N : w p( y ) ( ), (8) where δ is he dela-dirac fucio ad he weigh w of each paricle is updaed accordig o

5 w p(y ) p( ), (9) w q(, y ) The imporace fucio q(), kow as a proposed codiioal disribuio, is impora i he performace of PF (Gordo, Salmod e al. 99). I geeral, he closer he imporace fucio q() o he disribuio of p( ), he beer he approximaio is. The aim of choosig he opimal imporace fucio is o miimize he variace of he rue weighs so ha degeeracy problem is dimiished i oe way. The deails of choosig he opimal imporace fucio ca be foud i (Djuric, Koecha e al. 00). Case sudy A. Daases I our daase, sales daa of six fashio iems wih seve kids of color, ogeher wih oher relaed properies of he iems are icluded. I our aalysis, i order o check he feasibiliy of our proposed model, we reclassify he weekly daase accordig o iem ad color. I oher words, he relaioship bewee sales ad price will be modeled i wo ypes of daa caegory, amely (i) iem ad (ii) color. The daa se of each caegory coais samples. The firs sample are used as he raiig daa (for esimaig he model parameers) ad he remaiig samples are used o do he forecasig es. Noice ha for sales forecasig, i is commoly explored based o he hisorical daa i a ime-series form (e.g., i mos of he papers reviewed i Secio II). However, i our research, we employ he pael daa based forecasig model by cosiderig he edecy of ime series ad he effec from oher relaed properies of iems. B. Modelig To assess he performace of proposed PDPF model, we cosruc wo differe relaioship models i erms of sales ad color red predicio by pael daa. Modelig sales forecasig As described i Secio III, o esablish a hree-dimesioal relaioship model amog sales, previous sale ad correspodig price, i is ecessary o es wheher he pael daa is saioary or o. Table provides he es resuls which imply ha he probabiliy of havig a commo ui roo is 0. The ull hypohesis ha he commo ui roo of sale ad price series is o-saioary is hece rejeced. I oher words, he pael daases ha we adop o cosruc forecasig model are saioary ad he pael daa esimaio model ca be srucure direcly. Table. Ui roo es resul

6 Mehod Null hypohesis Saisic probabiliy Levi, Li & Chu * commo ui roo process ADF- Fisher Chi-square idividual ui roo process PP- Fisher Chi-square idividual ui roo process *Sigifica a he % level. Table. Hausma es resul Tes Summary Chi-Sq. Saisic Chi-Sq.d.f Probabiliy Cross-secio radom *Esimaed cross-secio radom effecs variace is 0. The Hausma esig resul furher reveals ha a fixed effec model should be cosruced for our forecasig problem. Thus, he liear compoe of fashio produc sale ca be described as follows: * Si m i Si P i i, i,..., N,,..., T, (0) where m represes he effec from he whole pael, * i represes he effecs of hose variables peculiar o he i h iem i more or less he same way over ime., he parameers ad idicae he degree ha he sale of iem i a ime is deermied by he value of he previous sale ad he correspodig price. Noice ha Eq. (0) saes he relaioship amog differe decisio variables, i.e. how he sale is relaed o he previous sale ad is correspodig price. We ow proceed o prese he esimaio of Eq. (0) usig he maximum likelihood esimaor. Table summarizes he oucome of he esimaio procedure usig a pael of daa. coefficie m Table. Esimaio resul of he sale forecasig model esimaio T-Saisic The esimaio resuls show ha he demad of each fashio iem maily depeds o he overall sales red m. Previous sale ad he correspodig price are also impora for explaiig he demad chages of each iem. This resul is cosise wih our expecaios because he price of fashio produc should play he same impora role as i he previous sale. Modelig color forecasig

7 Color forecasig is o esimae he demad of differe color, ad sales amou is used o represe he SKU level sales quaiy. The esig process is similar o he sale forecasig model, ad he resul idicaes ha he color-price pael daa is saioary ad he fixed effec model would be seleced. The fashio color forecasig model could be described as: * CS j m j S j P j j, j,..., N,,..., T. () I his model, we use he oal sale of each colour iem o measure he colour popular edecy, CS j i he above equaio; j represes differe ypes of colour which icludes black, blue, brow, red, whie, gree, grey. The pael daa esimaio resuls are lised i Table. Table. Esimaio resul of he color forecasig model coefficie m esimaio T-Saisic Similar o he esimaio resul of he SKU sale forecasig model, he fashio colour red also maily depeds o he whole marke edecy ad he previous colour red. However, he price effec is much less impora i he color forecasig sceario ha i he SKU sale forecasig sceario. C. Assessme Crieria I he resuls aalysis, he mea squared error (MSE) ad symmeric mea absolue perceage error (SMAPE) are used o measure he forecasig accuracy of our proposed model. They ca be described as: MSE A F SMAPE ( ), () i F A. () ( F A) where F is a vecor of predicios, ad A is he vecor of he rue values. Compuaio aalysis A. Sales Forecasig for Differe Iems To assess he forecasig performaces of he proposed mehod, he Pure Pael Daa mehod i 7

8 which he PF par is o icluded ad PDPF are applied for coducig forecasig wih he same daase. The compariso of he overall forecasig errors for differe iems by usig Pure Pael Daa ad PDPF is show i Table 8. I he aalysis, he sales amou of differe fashio iems iclude T-shir, dress, bag, pa, accessory ad bel are forecased. Noice ha he daa of all iems are colleced i he same ime period so as o eable he pael daa o reveal he ieracios bewee sales of differe iems. MSE ad SMAPE are used o assess he overall forecasig performaces. As show i Table, i is ecouragig o oe ha he proposed PDPF model ouperforms he pure pael daa mehod i mos cases. I is hece clear ha he proposed mehod is suiable for coducig fashio sales forecasig. However, he producs Bag ad Bel are he excepios wih which he proposed PDPF model fails o provide beer forecasig resuls ha he oher mehods. A closer look io he produc feaures reveals ha he sales of Bag ad Bel are likely o be seriously affeced by he sales of oher iems (e.g., T-shir, Dress, Pa), which could be explaied maily by he pael srucure. The pure pael daa mehod is hece very powerful ad appropriae for forecasig he sales of hese iems. As a remark, i he real pracice, here are a lo of fashios iems eed o be forecased i order o make sraegy decisio. However, our proposed PDPF mehod ca forecas a lo of iems simulaeously ad deliver sable ad accurae resuls i geeral. Table. Forecasig compariso for differe iems by usig pure pael daa ad PDPF Iem Pure Pael Daa PDPF MSE SMAPE MSE SMAPE T-shir. 0.0% % Dress.0 7.%.0.8% Bag.00 7.%.7.9% Pa. 0.7% % Accessory.9 0.%. 8.99% Bel.8.%.8.0% Mea..8% % B. Compariso bewee Sales Quaiy Forecasig ad Color Forecasig We compare he forecasig accuracy of he sales (iem) forecasig ad he color (red) forecasig i his subsecio. Firsly, Fig. ad Fig. show he SMAPE of pure pael daa mehod ad he PDPF mehod for boh forecasig schemes. The circle axis deoes he forecasig ime pois while he verical axis refers o he error perceages. From Fig, i ca be observed ha, excep for some special cases ha SMAPE equals 00%, he forecasig accuracy of sales quaiy is more ceralized ad he mea SMAPE is beer ha ha of color forecasig. 8

9 However, i Fig, he siuaio is oally differe. I fac, he PDPF model improves he forecasig accuracy by.97% whe he daa is classified i color while he improveme of iem caegory is oly 0.8% i he same experimeal siuaio. Compared o he pure pael daa mehod, he PDPF model adops PF o capure he oliear feaures ha he pael daa mehod cao describe. Mea SMAPE =.8% Sales Mea SMAPE =.% Color Fig.. SMAPE resuls of Pael Daa for sales ad color. Mea SMAPE = 0.99% Sales Mea SMAPE =. % Color Fig.. SMAPE resuls of hybrid PDPF model for sales quaiy ad color red. Coclusio Nowadays, he fashio idusry is kow o be iformaio drive. The wise use of iformaio for coducig sales forecasig helps a lo i ehacig he operaios maageme of fashio 9

10 compaies. However, fashio sales forecasig is always a challegig problem owig o may ihere feaures such as high volailiy as well as limied daa availabiliy. I order o develop a useful ad iovaive fashio sales forecasig framework, we employ he pael daa mehod ad he paricle filer (PF) approach o form he PDPF model. The core advaage of his hybrid PDPF model is ha he respecive cosruc eables us o cosruc a muli-dimesioal correlaio of he ifluece facors, icludig ime-series red of previous sale, he price of produc iems uder forecas, ad he effecs from oher correlaed produc iem. Refereces Aderso, T. W. ad C. Hsiao (98). "Formulaio ad esimaio of dyamic models usig pael daa." Joural of Ecoomerics 8(): 7-8. Arulampalam, M. S., S. Maskell, e al. (00). "A uorial o paricle filers for olie oliear/o-gaussia Bayesia rackig." IEEE Trasacios o Sigal Processig 0(): Au, K.-F., T.-M. Choi, e al. (008). "Fashio reail forecasig by evoluioary eural eworks." Ieraioal Joural of Producio Ecoomics (): -0. De Toi, A. ad A. Meeghei (000). "The producio plaig process for a ework of firms i he exile-apparel idusry." Ieraioal Joural of Producio Ecoomics (): 7-. Djuric, P. M., J. H. Koecha, e al. (00). "Paricle filerig." Sigal Processig Magazie, IEEE 0(): 9-8. Do Chug, B., J. Li, e al. (0). "Demad learig ad dyamic pricig uder compeiio i a sae-space framework." Egieerig Maageme, IEEE Trasacios o 9(): 0-9. Frak, C., A. Garg, e al. (00). "Forecasig wome's apparel sales usig mahemaical modelig." Ieraioal Joural of Clohig Sciece ad Techology (): 07-. Gordo, N. J., D. J. Salmod, e al. (99). Novel approach o oliear/o-gaussia Bayesia sae esimaio. IEE Proceedigs F (Radar ad Sigal Processig), IET. Hsiao, C. (00). Aalysis of pael daa, Cambridge uiversiy press. Li, X., C. Yu, e al. (0). "Day-ahead elecriciy price forecasig based o pael coiegraio ad paricle filer." Elecric Power Sysems Research 9: -7. Ni, Y. ad F. Fa (0). "A wo-sage dyamic sales forecasig model for he fashio reail." Exper Sysems wih Applicaios 8(): 9-. Sarkar, P. (00). "Sequeial Moe Carlo Mehods i Pracice." Techomerics (): 0-0. Yu, Y., T.-M. Choi, e al. (0). "A Iellige Quick Predicio Algorihm Wih Applicaios i Idusrial Corol ad Loadig Problems." Auomaio Sciece ad Egieerig, IEEE Trasacios o 9(): Zhog, J., Y.-f. Fug, e al. (00). "A biologically ispired improveme sraegy for paricle filer: A coloy opimizaio assised paricle filer." Ieraioal Joural of Corol, Auomaio ad Sysems 8(): 9-. 0