Pattern Recognition Techniques applied to Evaluation Engineering Problem

Transcription

1 90 IJCSNS Internatonal Journal of Computer Scence and Network Securty, VOL.9 No., November 009 Pattern Recognton Technques appled to Evaluaton Engneerng Problem Mara Teresnha Arns Stener*; Anselmo Chaves Neto**; Sílva Nede Bráulo; Valdr Alves UFPR *Engneerng Producton Department; **Statstcal Department CP: 908; CEP: , Curtba, Paraná, Brazl Summary The purpose of ths paper s to present a Pattern Recognton methodology composed by Multvarate Statstcal Analyss technques, n order to buld a Multple Lnear Regresson statstcal model to evaluate real estates accordng to ther characterstcs (varables, attrbutes). Frst, a Clusterng Analyss was appled to the data of each urban estate class (apartments, houses or plots) to obtan homogeneous clusters wthn each class. Next, the Prncpal Components Analyss (P.C.A.) was appled to solve the multcollnearty problem that may exst among the varables n the model. The scores of the prncpal components are then the new ndependent varables and wth them, the Multple Lnear Regresson model was adjusted for each cluster of smlar estates, wthn each class. Ths methodology was appled to estates n the cty of Campo Mourão, Paraná, Brazl. The model for each smlar cluster wthn each class of evaluated estates presented an adequate adjustment to the data and a satsfactory predctve capacty. Keywords:Evaluaton Engneerng, Clusterng; Prncpal Components Analyss, Multple Lnear Regresson.. Introducton The real estate market s one of the most dynamc areas of the tertary economc sector and ts man dffcultes to evaluate goods come from the estates characterstcs (attrbutes, varables), whch are qute heterogeneous and can keep a relaton between them. Estate evaluaton, whether for tax collecton, for sale, securty for fnancng or others, n general s subjectvely made, based upon the personal experence of estate managers, and of other professonals, who compare the data of the estate that s beng negotated wth those other estate transactons. In most cases, no scentfc procedure s systematcally used for ths purpose. The purpose wth ths paper s to propose a Pattern Recognton methodology based on statstcal technques, able to forecast an estate s value by consderng the hstorcal records of smlar estates. These value records are those defned n deals that were closed n the past. For such, we consdered as a case study the estate market n the cty of Campo Mourão, Paraná, Brazl, and n the apartments, houses and plots classes. Ths way, once a statstcal model s obtaned for better representng the analyzed market, durng a certan perod, one wll be able to forecast the market value (prce) of any estate wth the maxmum possble precson. Ths paper s organzed as follows: n Secton, the problem n the Evaluaton Engneerng area s delmted by presentng the man norms and concepts related to the theme and some related papers are dscussed; n Secton 3, we descrbe the data for the practcal problem under consderaton. In Secton 4, we succnctly descrbe the statstcal technques used n ths work and also present the proposed Pattern Recognton methodology; n Secton 5, s descrbed the results of applyng the proposed methodology to the problem s data. Fnally, n Secton 6, the conclusons for the work are presented.. Evaluaton Engneerng Accordng to [6], Evaluaton Engneerng s as a part of engneerng that ncludes knowledge from ths area, from archtecture and others (socal, exact and of nature) wth the purpose of techncally determnng the value of a certan good, ts rghts, fruts and reproducton costs, thus subsdzng decsons wth respect to values and nvolvng goods of all natures. Its practtoners may be: engneers, archtects, agronomsts, each one wthn ther professonal qualfcatons and accordng to the laws of the Federal Engneerng and Archtecture Councl (or Conselho Federal de Engenhara e Arqutetura - CONFEA). The frst works n the Evaluaton Engneerng area publshed n Brazl, of whch there are records, are dated of the begnnng of the 0 th century. Methods to evaluate plots were ntroduced n 93 and from 99 on they started to have a systematzed use [7]. The Brazlan Assocaton of Techncal Norms (or Assocação Braslera de Normas Técncas - ABNT) s the Natonal Forum for Norms. The frst norms for estate evaluaton appeared n the md 950s and were organzed by publc enttes and nsttutes. The frst pre-project of ABNT norms n Evaluaton Engneerng s dated 957 and the frst Brazlan Norm for Evaluaton of Urban Estates s dated 977, NB-50/77 [6]. Ths norm was revsed n 989 and orgnated NBR 5676 (or NB-50/89), regstered at INMETRO. Accordng to NB-50/89, real estates may be classfed accordng to: use (resdental, commercal, ndustral, nsttutonal or mxed); class (plot, apartment, house, offce, store, shed, garage vacancy, mxed, hotel, hosptal, theater, club or recreaton areas); and clusterng (allotted Manuscrpt receved November 5, 009 Manuscrpt revsed November 0, 009

2 IJCSNS Internatonal Journal of Computer Scence and Network Securty, VOL.9 No., November area, house condomnum, apartment buldng, housng development, store group, offce buldng, group of offce buldngs, group of store unts, shoppng center or ndustral complex). However, we must pont out that ths work only used data that correspond to estate from the apartment, house and plot classes. It s nterestng to notce that a part of an estate s value can be consdered random because there are countless nfluences n defnng ts value, ths s, one may thnk of the estate s fnal value based on a most probable value, ncreased or decreased of and unpredctable part and accordng to certan punctual nfluences. Ths way, an estate s value follows ths statstcal model: Y = μ + ε, where Y s the negotated value (prce); μ s the probable value and ε s the stochastc dsturbance term; thus, the expectaton for Y s E(Y) = μ.. For further detals consult [3], among others. Accordng to [8], the real estate market has a behavor that s dfferent from other goods markets due to the specal characterstcs estates show, especally the countless sources of dvergence and dssmlarty they present, thus makng mpossble to compare them drectly. Among the factors that dstngush estates from one another, one can menton: long lfe, fxed spatal poston, sngularty, hgh maturng term and hgh cost of unts. ABNT (NBR5676/90) [] dvdes the evaluaton methods nto two great groups: drect and ndrect methods. A method s consdered as beng drect when the value resulted from the evaluaton does not depend on others [6]. Drect methods are dvded nto market data comparatve method (defnes values by comparng smlar market data) and mprovements reproducton costs comparatve method (approprates the mprovements value). Accordng to [6], the use of drect methods has been preferred and when there s enough market data for ther use, they are the choce. A method s consdered ndrect when t needs the results from some drect method. Indrect methods are dvded nto ncome method (defnes the value n functon of an exstng revenue or forecasted by the good n the market, ths s, by the good s economc value); unevolutonal method (value s estmated by techncal-economcal feasblty studes for ts use) and the resdual method (t calculates the dfference between the estate s total value and the mprovements value, consderng the marketablty factor). Regardng precson levels, evaluaton tasks may be classfed as follows: expedtous strctness level (the value s obtaned wthout usng any mathematcal nstrument), normal (uses statstc methods and there are requrements wth respect to data collecton and treatment), strct (the value, whch s a result of the method employed, shall have a maxmum confdence level of 80%, wth null hypothess tested to the maxmum sgnfcance level of 5%) and the strct specal level, whch s characterzed by fndng a statstcal model, the most comprsng as possble, ths s, one that ncorporates the greatest number of characterstcs that may contrbute to form the value. The functon estmated to form the value must be effcent but not based. The null hypothess over the regresson model must be rejected only to the maxmum sgnfcance level of % (ANOVA). Null hypothess for the regresson model parameters should be tested to the sgnfcance level of 0% for the unlateral test (test t ) or 5% on each branch of the blateral test. The followng basc condtons should be analyzed wth respect to resdues of the model adjusted to the data: have a Gaussan dstrbuton, varance homogenety and ndependence. Thus, resdues must be Gaussan, ndependent and dentcally dstrbuted, ths s, ε ~N(0,σ ). There are some papers, n the lterature, that deal wth Evaluaton Engneerng. One can menton, for nstance, the work of [], whch compares the predctve performance of Artfcal Neural Networks wth the Multple Regresson Analyss, for sellng resdental houses. Several comparsons were made between the two models varyng: data sample sze, functonal specfcaton and tme forecast. In the work [], the authors examne the effect a vew to a lake (Lake Ere, E.U.A.) has on the value of a house. In ths study were consdered those prces based on the transacton of houses (market prce). Results show that besdes the varable vew, whch s sgnfcantly more mportant than the others, bult area and plot sze are also mportant. In [5], the authors compared the Lnear Regresson and the Artfcal Neural Networks technques to carry out an estmate of costs for sellng or rentng estates n the cty of Porto Alegre, RS, Brazl. Two databases were evaluated: ),600 estates offered for sale, 0 attrbutes each and ) 500 estates offered for rental, 85 attrbutes each. From the total number of attrbutes, only sx were selected to tran the models. In [9], t s presented two tools for evaluaton engneerng: generalzed lnear models and Neural Networks appled to 50 urban plots from three dstrcts n the cty of Recfe, PE, Brazl. In [], t s also made a comparatve study between the use of Neural Networks and Multple Regresson Analyss to estmate the sales value of real estates, regardng the offer of 7 mddle and low ncome apartments n the real estate market n the cty of Belo Horzonte, MG, Brazl. In [4], the author presented a work that uses Neural Networks to determne the nfluence varable accessblty has upon the value of urban plots, comparng them wth the Multple Regresson model, n two ctes n São Paulo s countrysde (São Carlos and Araçarguama), Brazl, The mentoned varable presented a weght over the fnal estate s prce greater than 34%.

3 9 IJCSNS Internatonal Journal of Computer Scence and Network Securty, VOL.9 No., November Problem Descrpton The use of the Pattern Recognton methodology proposed here was appled to urban estates from the classes, apartments, houses and plot, n the cty of Campo Mourão, Paraná, Brazl. The sample was bult wth 9 estates (classes), beng 44 from the apartment class, 5 from the house class and 4 from the plot class. They are all located n the cty s urban area and 80 of these are located nn the cty s central area. Attrbutes are of the qualtatve and quanttatve types; the apartments are lsted n Attachment and, as can be notced, they total, already dvded nto clusters (further sectons), 7 n cluster A, 9 n cluster B and 9 n cluster C. 4. Pattern Recognton Methodology The Pattern Recognton methodology to reach the goal conssts of the followng statstcal technques from the Multvarate Analyss area:. Clusterng Analyss: through ths technque we try to determne the clusters of homogeneous tems for each class of estate. In ths analyss we used the Eucldan Dstance and Ward s method was used as connecton method.. After formng the homogeneous clusters, dscrmnants were bult wth two purposes: evaluate the consstency of the clusters that were obtaned and also allocate future tems n each one of the clusters that form each class. 3. Followng, the Prncpal Components Analyss was appled to each one of the clusters, from each class, to substtute the values of the orgnal varables by the prncpal components scores and crcumvent the eventual multcollnearty problem.. Fnally, a Multple Lnear Regresson model was adjusted for each one of the clusters of each estate class. The cash prce, called value, was consdered the answer varable to the model. 4. Descrpton of the Statstcal Technques Multvarate Analyss s a set of technques used, among others, to solve problems related to: ) Covarance structure of random vector (summarzed n the covarance or correlaton matrx) through Prncpal Components Analyss; Factor Analyss and Canonc Correlaton Analyss; ) Items Clusterng (Cluster Analyss); 3) Pattern Recognton and Classfcaton [0]. In ths secton we wll succnctly descrbe the multvarate statstcal technques that were used. a) Clusterng Analyss Clusterng Analyss conssts n a technque that has the purpose of formng homogeneous clusters of objects (estates). Clusters are formed based on ther dstances (Eucldan, Mahalanobs, among others) or smlartes and on a connecton method between the partal clusters. The dstance that s usually used s the Eucldan Dstance: p d( x, y) = ( x y ). The mostly used connecton = method s Ward s, whch mnmzes the loss of nformaton when jonng two clusters, by usng the crteron of mnmzng the sum quadratc n error, SQE = (x j x)'(x j x). ] j= b) Quadratc Dscrmnaton Score for Recognton and Classfcaton In ths study we used the recognton and classfcaton rule based on the mnmum total probablty of error defned by the quadratc score for populaton (cluster), gven by: Q d ln ( ) ( ) x x ( ) = Σ μ Σ x μ + ln( p ) =,,, g, where p s the probablty that ths tem belongs to populaton Π ; μ and Σ are respectvely the average vector and the covarance matrx of populaton. These parameters are generally unknown and, therefore, we work wth ther estmates x and S. Wth respect to the p, one can take them as the proportons of the clusters groups szes; x 0 s recognzed as belongng to Π k f: d Q Q k (x 0 ) > d (x 0 ) =,,..., g, wth k. c) Prncpal Components Analyss Be the random vector x wth p correlated components. Ths relatonshp s structure can be summarzed n covarance matrx Σ or n correlaton matrx ρ. It s known from the Spectral Decomposton Theorem that Σ = PΛP or ρ = PΛP, where P s the orthogonal egenvectors matrx and Λ s the egenvalues dagonal matrx. Thus, there are p non-correlated Prncpal Components represented by lnear combnatons Y = e, whch recompose ths covarance structure, where e and λ, =,,,p are, respectvely, the egenvectors and egenvalues of Σ or ρ. Besdes, t s well known that V(Y ) = λ expresses the mportance of each prncpal component. A number m < p of Prncpal Components can represent a sgnfcant part of the total varaton and t s possble to use them nstead of the p orgnal varables. A crteron to determne the number of Prncpal Components to be consdered was suggested by Kaser, n 960 [0]. It conssts n takng a

4 IJCSNS Internatonal Journal of Computer Scence and Network Securty, VOL.9 No., November number m of Prncpal Components that s equal to the number of egenvalues λ. Moreover, t s nterestng to consder the part of the varaton explaned by the m Prncpal Components above (around) 90%, ths s, by extendng the mentoned crteron, egenvalues smaller than may be consdered, provded they are close to. When applyng the Prncpal Components Analyss, the scores of ts m prncpal components are obtaned. Ths way, matrx of the model of order n x p s transformed nto matrx E of order n x m, m < p, correspondng to the scores of the m Prncpal Components. d) Multple Lnear Regresson for Forecastng In order to obtan the value of a varable Y n functon of other varables, ndependent from one another, we use a Multple Lnear Regresson model, gven by: Y = β + ε; where Y s the observed answers vector of the n observatons (estates), s the model s matrx of order n x p; ε s the errors vector of dmenson n and β (to be estmated) s the parameters vector of dmenson p. Once defned the tem s (estate) cluster k of class l, based on the Clusterng and on the Recognton and Classfcaton, the adjusted Multple Lnear Regresson model s used to estmate the estate s j value ' by: yˆ = esc βˆ, where esc j s the components scores j j vector βˆ s the parameters estmated vector. 5. Results The cluster of the apartment class were formed by the Clusters Analyss descrbed n tem a of Secton 4., above. The result ndcated that three clusters make up the apartments class, as shown n Fgure, below. Cluster contans 38.64% of the analyzed apartments, cluster has.73% and cluster 3 has 38.64%, ( 000) Dstance 0,8 0,6 0,4 0, 0 9 Dendrogram Ward's Method - Eucldean Dstance 8 partments , , , ( 000) Fgure. Dendrogram of the three classes formed wth the 44 apartments After the Cluster Analyss, a Dscrmnant Analyss was made, usng the Quadratc Scores, as descrbed n tem b of Secton 4., showng that the classfcaton of the 44 apartments nto three classes (clusters) were corrected. We have that from the 7 apartments that belong to class, from the 0 that belong to class and from the 7 that belong to class 3, all were also classfed correctly. Ths way, we have a precson of 00%. Thus, results were consstent: from the 44 observatons that adjust to the model, 00% were correctly classfed. The nterpretaton of the clusters that were obtaned was made accordng to the attrbutes n each class, beng 7 from cluster A, 9 from cluster B and 9 from cluster C, as we have already mentoned. Next, we present the most determnng aspects n each one of the three clusters: Cluster : all apartments are located downtown; they are located n buldngs wth at least seven floors; wth at least a garage vacancy; buldngs have glazed coverng; have a mnmum area of 60 m ; at least one elevator; more than two bedrooms; all have a sute; all of them have complete mad lodgngs; and prces are hgher than R$5, Cluster : all apartments are located downtown; they are located n buldngs wth at least 3 floors; buldngs have glazed facng, or of marble or grante; more than one garage vacancy; have a mnmum area of 0 m ; buldngs newer than 5 years; more than two bedrooms; all have a sute; all of them have more than one elevator; all have complete mad lodgngs; and prces are hgher than R$75, Cluster 3: all apartments have only one garage vacancy; exclusve area s smaller than 3 m ; low buldngs and prces range from R$30, to R$0, Next, the Prncpal Components Analyss was appled to the data of the orgnal explcatve data and obtanng m = 6 components and ther scores, as shows Table, below. Comp. Number Egenvalue Percent of Varance Cum. Percent Table. Prncpal Components Analyss of the apartments that belong to cluster (m = 6) Through the results n Table, at the end, we can notce that the frst component has hgher weghts n the orgnal varables (n boldface): dstance from schools (dschool); dstance from supermarkets (dsmarket); dstance from hosptals (dhosptal); preservaton condtons (conservaton) and number of bathrooms (bath). The second component has hgher weghts n varables: number of elevators (elevator); number of bedrooms (nbedr) and the estate s apparent age (ageapparent). The thrd component has hgher weghts n varables: number of elevators (elevator); number of rooms (nrooms); how the buldng s covered (pbuldng); number of bathrooms

5 94 IJCSNS Internatonal Journal of Computer Scence and Network Securty, VOL.9 No., November 009 (bath); number of sttng rooms (nstrooms); fnshng qualty (fnqualty). The fourth component has hgher weghts n varables: how the buldng s covered (pbuldng); number of floors the buldng has (nfloor); area bult (area); actual age (agereal); vacancy n garage (vacancy); number of sttng rooms (nstrooms); fnshng qualty (fnqualty). The ffth component has the hghest weghts: number of vacances n the garage (vacancy); how the buldng s covered (pbuldng); number of floors the buldng has (nfloor) and number of elevators (elevator). Fnally, the sxth component has hgher weghts n varables: how the buldng s covered (pbuldng); preservaton level (conservaton) and number of vacances n the garage (vacancy). The scores the sx components suppled for the 7 apartments are n Table 3, at the end. These are the explcatve varables values that were consdered to adjust the lnear regresson model. Whle adjustng the Multple Lnear Regresson model Y = β + ε, t was notced that the ffth and sxth components were not sgnfcantly mportant, because ther p-values were greater than 0.05 and, therefore, they were dscarded and only the frst four were consdered, as shown n Table 4, at the end. The R statstcs that measures the adjustment s qualty s gven by: R = n = n = ( yˆ y),com 0 < R ( y y) < Suppled the value of R = , ths s, the adjusted model explans around 96% of the market s prce varablty. Therefore, the Multple Lnear Regresson equaton for the apartments belongng to class, whch descrbes the relaton between prce and the four ndependent components s gven by the followng equaton: () Prce = Y Y Y Y 4 The Analyss of Varance, contaned n Table 5, at the end, shows that the hypothess of no regresson s rejected, ths s, the model above s truly sgnfcant. The necessary premses to use the lnear model and the appled tests were all checked and satsfed by the resdues, ths s, ε ~ N(0, σ ). The values forecasted by equaton (), adjusted, and the observed values and the error percentages n the forecast a presented n Table 6, at the end. In the same way, the analyss carred out for the 0 apartments that belong to class resulted n sx Prncpal Components that explan 9.44% of the orgnal data s varablty and to the scores that compose the model s matrx of order (0 x 6) the Multple Lnear Regresson model was adjusted. The adjustment s determnaton coeffcent was of R = , ths s, the adjusted model explans almost 00% of the market prce s varablty. As for the 7 apartments that belong to cluster 3, the Prncpal Components Analyss showed that the frst seven components explan % of the orgnal data s varablty. Adjustng the model to the matrx of scores of order (7 x 7) suppled a determnaton coeffcent of R = , ths s, the adjusted model explans close to 90% of the market prce s varablty. 6. Conclusons In ths paper we propose a Pattern Recognton methodology based on multvarate statstcal technques to forecast prces of urban real estate. Ths methodology s composed by the followng technques: Clusterng Analyss, n whch smlar estates are clustered n terms of ther attrbutes; Quadratc Determnant Scores, n whch the consstency of those clusters s checked and one has a crteron for allocatng a new tem. Next, the Prncpal Components Analyss s appled n order to obtan m < p components, as well as ther ndependent scores to substtute the orgnal p varables, thus crcumventng the multcollnearty problem. Fnally, the Multple Lnear Regresson model of values vector Y s adjusted aganst the explcatve varables summarzed n matrx E wth order (n x m), ths s, Y = Eβ + ε, whch supples an estmate of estate s x 0 value through equaton ŷ 0 = ' ˆβ e 0, where e 0 s the vector of correspondent scores. Ths methodology was appled to the other two estate classes (5 houses and 4 plots) wth a result that was consdered qute satsfactory. The qualty of the adjustment to the varables, now truly ndependent, generated the determnaton coeffcents shown n Table 7. Class Cluster Cluster Cluster 3 Cluster 4 Apts Houses Plots Table 7. Values of R for the clusters of the three classes

6 IJCSNS Internatonal Journal of Computer Scence and Network Securty, VOL.9 No., November Therefore, gven a new estate n the cty of Campo Mourão (apartment, house or plot), of whch one wshes to have a value estmate, ntally the cluster to whch ths estate belongs, must be checked and the quadratc scores must be appled. Once the cluster s dentfed, one can use the Multple Lnear Regresson model that corresponds to such cluster. For the apartment class, cluster, the defned model s presented n equaton (), secton 5. Ths same way, we have the models for the other stuatons. Ths methodology s generc and can be used for any cty by obtanng the models defntons for the several dfferent stuatons n each cty. The multvarate Pattern Recognton methodology that was presented for forecastng real estates prces s relable, hghly approprate and reaches results wth qute satsfactory precson levels. Ths way, t may serve as support for estate managers when defnng estates prces, as well as form people and companes who want to realstcally evaluate ther assets. One must be aware that the defned Multple Lnear Regresson models must be perodcally readjusted due to the country s hghly dynamc economy and growth. References [] ABNT (Assocação Braslera de Normas Técncas), Avalação de Imóves Urbanos (NBR 5676 e NBR 50), ABNT, Ro de Janero, 004. [] BOND, M. T.; SEILER, V. L.; SEILER, M. J. Resdental Real Estate Prces: a Room wth a Vew, The Journal of Real Estate Research, v. 3, n., p. 9-37, 00. [3] BRAÚLIO, S. N. Proposta de uma Metodologa para a Avalação de Imóves Urbanos baseada em Métodos Estatístcos Multvarados, Dssertação de Mestrado em Métodos Numércos em Engenhara (Programação Matemátca), UFPR, Curtba, PR, 005. [4] BRONDINO, N. C. M. Estudo da Influênca da Acessbldade no Valor de Lotes Urbanos atrabés do uso de Redes Neuras, Tese de Doutorado em Engenhara Cvl (Transportes), USP-São Carlos, SP, 999. [5] CECHIN, A. L.; SOUTO, A. & GONZÁLEZ, M. A. Análse de Imóves através de Redes Neuras Artfcas na Cdade de Porto Alegre, Scenta, v.0, n., p. 5-3, 999. [6] DANTAS, R.A. Engenhara de Avalações: uma Introdução à Metodologa Centífca, São Paulo: Pn, 003. [7] FIKER, J. Avalação de Imóves Urbanos, São Paulo: Pn, 997. [8] GONZÁLEZ. M. A. S. & FORMOSO, C. T. Análse concetual das dfculdades na determnação de modelos de formação de preços através da análse de regressão, Engenhara Cvl UM, 8, p , 000. [9] GUEDES, J. C. Duas Ferramentas Poderosas a Dsposção do Engenhero de Avalações: Modelos Lneares Generalzados e Redes Neuras, Anas do I COBREAP, Guarapar, ES, 00. [0] JOHNSON, R. A. & WICHERN, D. W. Appled multvarate statstcal analyss, New Jersey: Prentce Hall, 998. [] NGUYEN, N. & CRIPPS, A. Predctng Housng Value: A Comparson of Multple Regresson Analyss and Artfcal Neural Networks, The Journal of Real Estate Research, vol., no. 3, p , 00. [] PELLI NETO, A. & ZÁRATE, L. E. Avalação de Imóves Urbanos com a utlzação de Redes Neuras Artfcas, Anas do IBAPE II COBREAP, Belo Horzonte, MG, 003. Mara Teresnha Arns Stener got her Master's and Ph.D.'s degrees n Producton Engneerng, on Operatons Research area, at Federal Unversty of Santa Catarna, Brazl, and her Pos-Doc, at the Technologcal Insttute of Aeronautcs, São José dos Campos, SP, Brazl. She s an Assocate Professor at Federal Unversty of Paraná, Curtba, Paraná, Brazl. She lectures on Engneerng Undergraduate Programs and on Numercal Methods n Engneerng Graduate Program. e-mal: tere@mat.ufpr.br Anselmo Chaves Neto got hs Master's degree n Statstcal at UNICAMP, Campnas, SP, Brazl and hs Ph.D.'s degree n Electrcal Engneerng at Catholc Pontfíca Unversty of Ro de Janero, RJ, Brazl. He s an Assocate Professor at Federal Unversty of Paraná, Curtba, Paraná, Brazl. He lectures on Engneerng and Statstcal Undergraduate Programs and on Numercal Methods n Engneerng Graduate Program. e-mal: anselmo@ufpr.br Sílva Nede Bráulo got her Master s degree n Numercal Methods n Engneerng at Federal Unversty of Paraná, Curtba, Paraná, Brazl. Valdr Alves got hs Master s degree n Numercal Methods n Engneerng at Federal Unversty of Paraná, Curtba, Paraná, Brazl.

7 96 IJCSNS Internatonal Journal of Computer Scence and Network Securty, VOL.9 No., November 009 Table. Weghts of the orgnal varables n each one of the sx Prncpal Components of the apartments that belong to class varable component component component component component component pbuldng elevator vacancy area nfloor level nrooms nstrooms nbedr bath dschool dhosptal dsmarket fnqualty conservaton agereal ageapparent Table 3. Scores of the sx prncpal components of the apartments that belong to cluster component component component component component component Real States

8 IJCSNS Internatonal Journal of Computer Scence and Network Securty, VOL.9 No., November Table 4. Adjustment of the Multple Lnear Regresson Model for the apartments that belong to class and t Test Parameter Estmate Standard Error t Statstc p-value CONSTANT PCOMP_ PCOMP_ PCOMP_ PCOMP_ Table 5. Analyss of Varance of the Regresson Model s Adjustment for apartments belongng to class Source Sum of Squares Df Mean Square F-Rato p-value Model E0 4.45E Resdual.049E9.6868E8 Total E0 6 Observed Value y (R$) Table 6. Results for the 7 apartments n Cluster Forecasted Value Absolute Error ŷ (R$) y - ŷ (R$) Percentage Error (y - ŷ ) (%) 30, ,75.00, , , , ,73.00, , , , , , , , , ,54.00, , , , , , , , , , , , , , , , , , , ,864.00, , , , , ,8.00 7,

9 98 IJCSNS Internatonal Journal of Computer Scence and Network Securty, VOL.9 No., November 009 ATTACHMENT Lst of attrbutes for apartments and ther clusters Attrbutes Descrpton Categores Apts. Cluster A pbuldng Identfes how the buldng s = pantng covered = glazed coverng 3 = ceramc 4 = marble / grante level conservaton agereal Score related to the floor n whch the apartment s located. Identfes the estate s preservaton condtons. Score related to the buldng s chronologcal age (mrrors the technologcal state). ageapparent Score related to the apparent buldng s age. dschool Identfes dstance from schools = up to 500 meters = 500 to 800 meters 3 = more than 800 meters Apts. Cluster B Apts. Cluster C = st to 3 rd floors = 4 th to 6 th floors 3 = 7 th to 9 th floors 4 = 0 th or hgher = bad = regular 3 = good 4 = excellent = more than 0 years = 5 to 0 years 3 = 0 to 5 years 4 = 5 to 0 years 5 = to 5 years 6 = up to year (dem) dhosptas Identfes dstance from hosptals (dem) dsmarket Identfes dstance from (dem) supermarkets. local posapartam fnqualty Classfes the dstrct and other characterstcs of where the resdence s. Identfes the apartment s poston n relaton to the buldng (front, sde or back). Identfes the several fnshng levels. = valung 0 = ndfferent - = devalung = front = sde 3 = back = low = normal 3 = hgh nfloor Number of floors the buldng has. Quantty elevator Indcates the number of elevators Quantty n the buldng. area Indcates the apartment s area Area expressed n square meters. vacancy Indcates the number of vacances Quantty for cars, avalable for the apartment. nbedr Indcates the number of bedrooms Quantty n the apartment. madlod Indcates the exstence (or not) of 0 = nexstent mad lodgngs. = exstent sute Indcate the presence (or not) of 0 = nexstent sutes. = exstent nstrooms Indcates the number of sttng Quantty rooms n the apartment. nrooms Indcates the total number of Quantty rooms the apartment has. bath Indcates the number of bathrooms Quantty n the apartment. Total Attrbutes (source: Imoblára Tapowk, Guarapuava, Paraná, Brazl)