APPLICATION OF CLUSTERING METHODS IN BANK S PROPENSITY MODEL

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1 APPLICATION OF CLUSTERING METHODS IN BANK S PROPENSITY MODEL Sergej Srota Haa Řezaková Abstract Bak s propesty models are beg developed for busess support. They should help to choose clets wth a hgher potetal (probablty) to buy a product. Propesty models are mostly created by usg logstc regresso. It s a classfcato task for clet s segmetato based o soco-demographcs, products ad trasacto characterstcs. A smlar classfcato task ca be solved by cluster aalyss. Therefore, the am of ths cotrbuto s to mprove a propesty model for cosumer loa by addg a ew varable, whch dcates the clet s belogg to a certa cluster based o the applcato of some methods of cluster aalyss. We apply two clusterg methods. The frst oe s the k-meas algorthm, whch belogs to cetrod-based clusterg. The secod oe s TwoStep cluster aalyss proposed for large data sets ad herarchcal clusterg use. The effect of addg a ew varable s evaluated by comparg the total respose rate the curret propesty model agast the respose rate ew propesty models wth the cluster varable. Key words: logstc regresso, cluster aalyss, propesty model JEL Code: C25, C38, D12 Itroducto Bak s propesty models are created for clet s segmetato, where logstc regresso s mostly used for modelg. Cluster aalyss could be also used for classfcato tasks. As Arpo ad Caas (2016) metoed, usg cluster aalyss ca mprove the success of propesty models, where each model ca be bult for each cluster. Other authors, Rudolph et al. (2016), also used the combato of cluster aalyss ad propesty models, where objects a data set could be clustered by propesty scores. Gawrysak et al. (2001) metoed, that usg cluster aalyss ca lead to a more accurate estmato of the depedet varable by the regresso fucto. It s assumed that the frst step, the objects the data set are dvded to clusters 1431

2 usg cluster aalyss. I the secod step, after ths dvso, values of the depedet varable are estmated each created cluster. Estmated regresso parameters ca be dfferet for each cluster. There are also other uses of cluster aalyss, see e.g. (Km et al., 2016, Qa ad Wu, 2011, Svarc, 2016, Yag et al., 2016). Based o these examples, the dea of ths cotrbuto s about applcato of cluster aalyss before creatg a bak s propesty model by logstc regresso. The am of ths paper s to mprove a propesty model for cosumer loa by addg a ew varable, whch dcates that the clet belogs to a certa cluster based o the applcato of some method of cluster aalyss. We suppose a curret propesty model for cosumer loa order to predct the probablty of buyg a product wth a certa umber of explaatory varables. The success of propesty models s measured by the respose rate. I the text below, the prcples of logstc regresso ad selected cluster methods, whch are k- meas ad TwoStep, are descrbed. Ths theoretcal part s followed by the results of the aalyses. I these aalyses, cluster varables are created by usg the above metoed methods for clusterg to the dfferet umber of clusters. After these, sx ew propesty models are bult by logstc regresso, where the puts are the same explaatory varables as the curret propesty model ad moreover, a ew explaatory cluster varable (oe for each model) s created by cluster aalyss. The descrbed process ca be cosdered as combg cluster aalyss wth logstc regresso. For the aalyses performace, statstcal program IBM SPSS Modeler s used. 1 Bak s propesty models Classfcato ad predctve statstc models have become popular the bakg world wth the developmet of computg techologes. I bakg, propesty models are used for clet s classfcato accordg to ther tedecy to buy a bakg product. Oe specfc propesty model s developed for each bak product. Clet s tedecy s estmated by the value of the score, whch dcates the probablty of buyg a bakg product. The value of the score les the terval 0, 1 ad t s calculated by logstc regresso. After the calculato, clets are dvded to 10 groups (there s oe dvso for oe product) by values of the score. I the teth group, there are clets wth the hghest values of the score the terval 0.9, 1. The success of clet s classfcato accordg to ther tedecy to buy bak s product ca be measured by the respose rate. It expresses the proporto of clets who really bought the product over the 1432

3 total umber of selected clets based o the value of the score. More about modelg ca be foud (Loff ad Berry, 2011). 1.1 Logstc regresso Logstc regresso for predcto of probablty π of buyg a product s mostly used for developg bak s propesty models. As metoed e.g. (Agrest, 2002), logstc regresso the bary target varable Y s cosdered, where Y = 1 meas, that clet bought a product ad Y = 0 meas, that clet dd t buy ay products. Probablty that Y = 1 ca be deoted as P(Y = 1) = π. I the classc lear regresso, the fte rage of possble values for the target varable s cosdered. The problem s, that we eed a fte rage of possble values the terval 0, 1. It ca be passed, f we cosder o-lear relatoshp betwee a set of explaatory varables X ad the target varable Y for each object ( our case for each clet) the well-kow model of logstc regresso: βx e P ( Y 1X), (1) βx 1 e where β s a vector of parameters of the regresso fucto. Estmated parameters ca be obtaed by the maxmum lkelhood method. 1.2 Methods of cluster aalyss Relatoshps betwee objects a data set ca be aalyzed by a cluster aalyss. A geeral process of cluster aalyss s about dvdg objects to clusters a way that objects the same cluster are more smlar to each other tha objects a dfferet cluster. There are two geeral groups of clusterg methods accordg to dfferet ways of clusterg. The frst oe s herarchcal clusterg, where each stage we obta a dfferet umber of clusters. The secod oe s cetrod-based clusterg (a cetrod s a vector of statstcs, e.g. averages or medas, calculated for dvdual varables from values correspodg to the objects assged to a certa cluster). For each of these methods, t s cosdered that oe object ca be oly oe cluster. For aalyses proposed ths paper, the k-meas ad TwoStep methods were chose. For comparg the success of these clusterg methods, the slhouette coeffcet wll be used. It measures clusters qualty as metoed e.g. (Löster, 2016) K-meas method The well-kow method belogg to partto clusterg s the k-meas algorthm. It s determed for objects characterzed by quattatve varables. The algorthm eeds to defe 1433

4 the umber of clusters to whch objects wll be assged. Fally, each object belogs oly to oe cluster. The algorthm s teratve; at every step the cetrod s calculated for each cluster. The cetrod represets the average values of varables for each gve cluster. Object s parttog to clusters s optmal, whe the followg fucto s mmzed: f KM k u h h1 1 2 x x, (2) h where u h = 1, f the -th object ( = 1,,, where s the umber of objects) belogs to the h-th cluster (h = 1,..., k, where k s the umber of clusters), otherwse u h = 0; x h s a vector of average values of varables for each gve h-th cluster, ad. s the Eucldea dstace. The metoed fucto (2) s mmzed uder followg codtos: k h1 u h 1 for 1, 2,...,, (3) 1 u h 0 for h 1, 2,..., k. (4) TwoStep cluster aalyss For object clusterg a large data set TwoStep cluster aalyss s a sutable method, whch s a modfcato of herarchcal clusterg methods. Ths method uses the BIRCH algorthm (Balaced Iteratve Reducg ad Clusterg usg Herarches). The dea s to create auxlary clusters, whch are clustered a herarchcal way the ext step. Ths method uses ether the Eucldea dstace or the log-lkelhood dstace. At the begg of the TwoStep cluster aalyss, the umber of clusters ca be defed, as the k-meas algorthm. The rage of clusters ca be defed IBM SPSS Modeler. More about the BIRCH algorthm ca be foud (Zhag et al., 1996) Slhouette coeffcet The slhouette coeffcet combes the cocepts of cluster coheso ad cluster separato. Values of the slhouette coeffcet are the terval 1, , where 1 s the worst ad 1 s the best assgmet of objects clusters. The bass of ths coeffcet s the evaluato of assgmet of dvdual objects to ts cluster. Ths evaluato for the -th object ca be expressed as the value:, (5) max,

5 where jc h d j h, 1 m gh jc g d g j, dj s the Eucldea dstace betwee the -th ad j-th objects, ad h (g) s the umber of objects the h-th (g-th) cluster. The slhouette coeffcet s the average value calculated over all objects a data set,.e. 1. (6) A example of graphcal presetato of the value s show Fg. 1. Fg. 1: Cluster qualty measurg IBM SPSS Modeler Source: ow costructo wth IBM SPSS Modeler 2 Applcato of clusterg methods a propesty model I geeral, a modelg base preparato for a propesty model s very demadg of ts sze, because t cotas thousads of varables represetg products ad trasactoal characterstcs for hudreds of thousads of clets. I our case, the data set of about clets s aalyzed. The target varable Y represets formato, whether a clet bought (Y = 1) or dd t buy (Y = 0) a cosumer loa. Due to the cluster aalyss purposes, the best 100 quattatve explaatory varables were selected for the modelg base. These 100 varables were chose based o the correlato comparso betwee the target ad explaatory varables. Ths type of varable selecto s avalable IBM SPSS Modeler Feature Selecto ode. The referece model for the evaluato of ew propesty models wll be a model based oly 1435

6 o applcato of logstc regresso wth the same 100 explaatory varables. It wll be amed the curret propesty model the followg text. For the creato of ew propesty models, the frst step, the ew cluster varables were created wth usg the k-meas ad TwoStep clusterg methods (wth the Eucldea dstace) wth a cosderato of 5, 10 ad 15 exstg clusters (oe ew varable for each method ad type). Cluster aalyss was appled o the whole modelg base ( objects ad 100 varables). After the frst roud of calculatos, sx ew propesty models were bult, where the score was calculated based o the applcato of logstc regresso. Explaatory varables for each of the ew propesty models were: oe cluster varable from the frst step ad varables, whch were used the curret propesty model. Ths modelg base was dvded to trag ad testg part the 70:30 rato for modelg purposes, where the success of the curret ad each ew propesty model was compared based o the respose rate the testg part. As metoed above, the respose rate s also flueced by the umber of selected clets, so for calculatos ad comparsos, clets wth the score value of at least 0.7 wll be selected. The curret propesty model has the total respose rate of % o the selected testg modelg part. Tab. 1 shows the respose rate of ew propesty models cludg cluster varable obtaed by the k-meas algorthm wth a cosderato of 5, 10 ad 15 clusters. It also presets values of the slhouette coeffcet, whch measures the cluster qualty. The hghest respose rate s % the case wth 10 clusters cosderato, although the value of the slhouette coeffcet s less tha other cases (0.4 vs. 0.5). Tab. 1: The success of ew propesty models combed wth the k-meas algorthm umber of clusters slhouette coeffcet respose rate ( %) 5 clusters clusters clusters Source: ow costructo I Tab. 2, the respose rates of ew propesty models are show, whch are combed wth TwoStep cluster aalyss. There s also 5, 10 ad 15 clusters cosderato the data set. TwoStep cluster aalyss was developed for large data set, but the respose rate s worse tha 1436

7 usg the k-meas algorthm these three cases. Eve though the respose rate s lower, comparso wth the curret propesty model, t s slghtly hgher. Tab. 2: The success of ew propesty models combed wth TwoStep cluster aalyss umber of clusters slhouette coeffcet respose rate ( %) 5 clusters clusters clusters Source: ow costructo Cocluso The respose rate the ew propesty models s slghtly hgher after cludg the cluster explaatory varable to the logstc regresso. Ths ew varable dcates the clet belogg to a certa cluster based o the applcato of the k-meas algorthm or TwoStep cluster aalyss. Three varats were cosdered for each cluster method, whch vared oly the umber of clusters the data set (5, 10 ad 15 clusters). Values of the slhouette coeffcet, whch measures the qualty of the created clusters, were betwee 0.2 ad 0.5 for the troduced types of clusterg methods, so t ca be cosdered as a good result. I the used data set, geeral, the propesty models combed wth the k-meas algorthm had a hgher respose rate comparso wth TwoStep cluster aalyss. The curret propesty model had the respose rate of %. Based o the hghest respose rate, the best ew propesty model was the model, where k-meas clusterg method was used wth cosderato of 10 clusters the data set. The respose rate was % ad the value of the slhouette coeffcet was 0.4, whch dcates good clusters qualty. O the opposte ed, the ew propesty model wth the lowest respose rate was the model, where the cluster varable was created by TwoStep cluster aalyss wth the cosderato of 5 clusters the data set. The respose rate was % ad the value of the slhouette coeffcet was 0.3. Based o the acheved results, all sx ewly created propesty models could be put to real deploymet, but due to oly a slght mprovemet, there s o reaso to replace the curret propesty model. Due to the provded aalyss of the use of clusterg methods propesty models created by logstc regresso, t s possble to explore further for ehacemets. 1437

8 Ackowledgmet Ths paper was processed wth the cotrbuto of log term sttutoal support of research actvtes by Faculty of Iformatcs ad Statstcs, Uversty of Ecoomcs, Prague. Refereces Agrest, A. (2002). Categorcal data aalyss (2d ed.). NYC: Joh Wley & Sos. Arpo, B. & Caas, M. (2016). Propesty score matchg wth clustered data. A applcato to the estmato of the mpact of caesarea secto o the Apgar score. Statstcs Medce, 35(12), Gawrysak, P., Okoewsk, M., & Rybsk, H. (2001). Regresso yet aother clusterg method. I Itellget Iformato Systems Physca-Verlag HD, pp Km, J. K., Kwo, Y., & Pak, M. C. (2016). Calbrated propesty score method for survey orespose cluster samplg. Bometrka, 103(2), Loff, G. S. & Berry, M. J. (2011). Data mg techques: for marketg, sales, ad customer relatoshp maagemet. NYC: Joh Wley & Sos. Löster, T. (2016). Determg the optmal umber of clusters cluster aalyss. I Löster T., Pavelka T. (Eds.), 10th Iteratoal Days of Statstcs ad Ecoomcs. Praha: Meladrum, pp Qa, G. & Wu, Y. (2011). Estmato ad selecto regresso clusterg. Europea Joural of Pure ad Appled Mathematcs, 4(4), Rudolph, K. E., Colso, K., Stuart, E. A., & Aher, J. (2016). Optmally combg propesty score subclasses. Statstcs Medce, 35(27), Svarc, M. (2016). Propesty score matchg ad alteratve approach to dvduals matchg for couterfactual evaluato purposes. I Reff, M., Gezk, P. (Eds.), Proceedgs of the 2016 Iteratoal Scetfc Coferece o Quattatve Methods Ecoomcs Multple Crtera Decso Makg XVIII, pp Yag, S., Imbes, G. W., Cu, Z., Fares, D. E., & Kadzola, Z. (2016). Propesty score matchg ad subclassfcato observatoal studes wth mult level treatmets. Bometrcs, 72(4), Zhag, T., Ramakrsha, R., & Lvy, M. (1996). BIRCH: A effcet data clusterg method for very large databases. ACM SIGMOD Record, 25(2),

9 Cotact Ig. Sergej Srota Uversty of Ecoomcs, Prague, Departmet of Statstcs ad Probablty Sq. W. Churchll 1938/4, Prague 3, Czech Republc prof. Ig. Haa Řezaková, CSc. Uversty of Ecoomcs, Prague, Departmet of Statstcs ad Probablty Sq. W. Churchll 1938/4, Prague 3, Czech Republc 1439