Criteria of Database Quality Appraisement and Choice Stochastic Models in Prediction of Real Estate Market Value

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1 Criteria of Database Quality Appraisemet ad Choice Stochastic Models i Predictio of Real Aa BARAŃSKA, Polad Key words: cadastre, real estate, valuatio, property taxes, market value SUMMARY Databases of real estates, aalysed i cosideratio of the coherece, are the groud for estimatio of evaluatio model i liear form (or o-liear, reducible to the liear model) of multiple regressio. Covariace matrix for predicted values of real estates determiig the regressio model is the basis for calculatio of defied parameters that are matrix trasformatio ivariats. O the basis of these parameters, choice rules of the optimum evaluatio model for real estate market value predictio ca be formulated. Three followig parameters are proposed for market iformatio aalysis: coefficiet of determiatio R, i.e. stadard measure of reliace i estimated evaluatio det K model: R = 1, det K0 parameter determied from trace of covariace matrix: 1 tr{ Cov( W )} σtr = W śr parameter determied from determiat of covariace matrix: o σ det = det{ Cov( W )} Wśr K correlatio matrix, icludig complete correlatio coefficiets betwee all pairs of variables (r ij ), det K determiat of correlatio matrix, det K0 determiat of sub-matrix, created by elimiatig the first verse ad the first colum i matrix K, i.e. correlatio coefficiets cocerig depedig variable (price) tr{cov(w) trace of covariace matrix for predictios market values of real estates settig evaluatio model, det{cov(w)} determiat of covariace matrix for predictios market values of real estates settig evaluatio model, W śr average value of predictios market values of real estates settig evaluatio model, umber of real estates used for models estimatio. u umber of idepedet variables appearig i model. 1 u 1/13

2 Criteria of Database Quality Appraisemet ad Choice Stochastic Models i Predictio of Real Aa BARAŃSKA, Polad 1. INTRODUCTION Evaluatio of real estates became last time a very importat ad idispesable elemet of real estates maagemet. Cosequetly, ew, improved methods of estimatio are eeded. O the basis of real estate assessmet regulatios, there are may ways of applyig statistical methods ad of creatig multidimesioal mathematical models, which describe as well as possible a give market of real estates. Multidimesioal character of the problem results from the multitude of factors ifluecig directly the real estate value. Creatio of mathematical model for a chose market is a problem extremely complex, because it ivolves a appropriate preparatio of modellig database as regards its completeess ad reliability, takig ito cosideratio a optimum selectio of variables. To fid the "best" model, it is ecessary to examie a great amout of iformatio i database. Therefore, it is ecessary to formulate some criteria of choosig a optimal model, as well as the criteria allowig the estimatio of the quality of modellig database form. It cocers mostly relatio betwee the quatity of variables ad the base size. The methodology i this field develops dyamically. There are may suggestios, i literature ad i practical assessmets, of applyig differet formulas or eve total mathematical models. This is the reaso to formulate criteria of choosig a model optimum o a give market for evaluatio of real estates. The author proposes the selectio of such a model accordig to the previously determied ivariat parameters. O the groud of these parameters values, for a database icludig real estates of a give local market, selectio criteria allowig to choose a optimal model of pricig ad estimatig database quality ca be formulated. The reaso to udertake such a task is the ecessity of doig a geeral valuatio of real estates i Polad. I accordace with Art. 16 of the law cocerig maagemet of real estates, the cadastral value must take ito cosideratio the differeces betwee attributes of particular real estates ad it should approximate their market value. The cadastral value is used to determie the basis of taxatio of a real estate ad i case of official activities whe realisatio is based o the market value of a real estate. /13

3 . DEFINITIONS OF INVARIANT PARAMETERS Three followig parameters are suggested for a aalysis of market iformatio: coefficiet of determiatio R, i.e. stadard cofidece measure of estimated model of pricig: det K R = 1, det K0 (1) 1 r01 r0... r0 k r 10 1 r1... r1 k K = r0 r r k () rk 0 rk 1 rk... 1 K correlatio matrix, icludig coefficiets of total correlatio withi each pair of variables (r ij ), det K determiat of correlatio matrix, det K 0 determiat of submatrix, created by cacellig of the first row vector ad the first colum i matrix K, i.e. correlatio coefficiets cocerig depedet variable (price). R coefficiet of liear multiple correlatio, determiig the degree of matchig the hyperplae to the poit patter represetig prices ad attributes of idividual real estates. 1-R is a coefficiet of o-coformace of the model ad the real estates market values used for estimatio of a pricig model. parameter settled o the basis of covariace matrix trace, determied by formula: 1 tr{ Cov{ W}} σ tr =, (3) W śr tr{cov(w)} trace of covariace matrix for predicted market values of real estates establishig a pricig model, W śr mea value of predicted real estates market values establishig umber of real estates used to estimate a model. parameter calculated from the covariace matrix determiat, determied by formula: det{cov(w) 1 u σ = det{ Cov( W )}, (4) det W śr determiat of covariace matrix for predicted market values of real estates establishig a pricig model, 3/13

4 W śr u mea value of predicted real estates market values establishig a pricig model, umber of idepedet variables occurrig i a model. The aalysis of formulas (3) ad (4) idicates that these quatities costitute some kid of measure of dispersio roud the mea model value, so they ca be objective parameters applied to formulate criteria of reliability of databases used i modellig of market value. 3. METHODOLOGY OF INVESTIGATION Iformatio o lad properties for housig as the object of commercial traffic was used as startig material for ivestigatio. It comes from 10 differet local markets of properties. Great differeces of prices ad variable dyamics of trasactios are characteristic features of aalysed markets. Gathered databases cotai 0 to 130 properties. Iformatio icludes totally 530 lad properties. The market of estates is created for every tow (commue) separately. For big tows, separate estate markets are created for particular quarters. It is caused by difficulty to fid attributes (variables) easily idetifiable ad measurable, which could be used to trasform prices to oe commo market of properties. I majority of cases, properties prices are positively differet, eve i commues of similar umber of ihabitats. Factors like: labour market, attractiveess of a tow, eviromet purity, ecoomic prospects, situatio, ladscape, are decisive i pricig. We ca otice also differet reactios to local factors, visible i characteristics of verified models. Achieved data have bee subject of a pre-treatmet aimig to prepare pricig models to the tests. Prelimiary aalyses permitted to defie methods of groupig achieved data, isolated features ifluecig estate value ad imagied ifluece rate of respective features. By this treatmet, databases are brought to coectivity allowig to acquire more reliable results of ivestigatio. I modellig process of real estates market, first, i every database, a multidimesioal liear model as liear multiple regressio has bee tested: u F( X, a) = w = a a (5) i i 0 + X ik * k = 1 w i model value of i-th estate i a give database, X i attributes value vector for i-th estate (1xu), X ik value of attribute k for i-th estate, a 0 free term i the model, a vector of multiple regressio coefficiet ((u+1)x1), a k coefficiet of regressio stadig by attribute k. If the aalysed market of real estates was steady, variability of prices i relatio to respective attributes would be liear. However, the market of real estates i Polad beig ot stable, k 4/13

5 variability of prices ofte is ot liear ad sometimes chages are eve abrupt. Cosequetly, i cases where liear model proved ureliable, estimatio of liear model parameters has bee doe, proceedig as follows: For each attribute ad each uit price, a fuctio i form geeral is defied: Y = g(x k ), (6) X k variable represetig the value of the attribute k, Y prices of real estates i database. The fuctio g may have very differet forms. The followig elemetary fuctios have bee cosidered: liear, polyomial, power, logarithmic, expoetial, irratioal. I mathematical statistics, there is o aalytical method eablig a optimal selectio of a suitable form of fuctio. Reliability measure for regressio model (6), describig variability of market prices i relatio to the established attribute, is the square of the curviliear correlatio coefficiet q: q [ y g( X k )] 1 (7) [ y yˆ] i i= 1 = i= 1 y i uit price of i-th estate i database, ) y mea value (arithmetic mea) of estate price from a database, g(x k ) predicted uit price for attribute k of a real estate, resultig from admitted o-liear pricig model. This coefficiet value may be a criterio for selectig suitable form of fuctio g. i Selected forms of fuctio g were used to create multidimesioal models, i.e. global o-liear fuctios F for respective databases: W = F(X, a) (8) W set of predicted prices geeratig a pricig model, X multidimesioal variable represetig real estate attributes, a vector of model parameters. Respective fuctios g for differet attributes were withi fuctio F iterrelated by a depedece additive or multiplicative. Parameters of all aalysed regressio models were estimated with least squares method. Each of estimated o-liear models has bee reduced to a liear model (5) with the assistace of Taylor series expasio. 5/13

6 Reliability of estimated models has bee verified by testig the hypothesis o variace equality of the part explaied by regressio model ad of the part uexplaied. Fisher- Sedecor test has bee used here, at the sigificace level p = 0,05, for which test statistics has the followig form: R m F = * (9) 1 R m 1 R coefficiet of multiple liear correlatio, size of trial (quatity of real estates i a database), u quatity of idepedet variables i a model, m = u + 1 quatity of estimated parameters of a pricig model. The above test ot oly examies the absolute variace ratio of explaied ad uexplaied parts, but also takes ito accout the ecessity to retai i a model right proportios betwee the quatity of cases ad the quatity of ukows. I cosequece, some (9) models have bee elimiated, eve from amog these havig a very high (above 0,90) absolute value of R. 4. DETERMINATION OF INVARIANTS VALUES The last step was the determiatio of values of previously defied ivariat parameters (1,3,4). Covariace matrix for model values of estates has bee determied as follows: T T 1 Cov( W ) = ˆ σ 0 X ( X X ) X (10) ˆ σ 0 uloaded estimator of remaider variace (determiig iaccuracy of model parameters estimatio): T T T Y Y aˆ X Y ˆ σ 0 = (11) u 1 1 x11 L x1 u 1 x x = 1 L u X matrix cotaiig umbers 1 ad idepedet variables M M O M 1 x L x a = 1 a 0 a 1 M a u u (attributes), vector of multiple liear regressio coefficiets, 6/13

7 Y = y 1 y M y vector of depedet radom variable (prices of estates). I sum, o all estates databases, may differet models determiig estate market value have bee tested. First, all models with multiple correlatio coefficiet less tha 0,60 have bee elimiated. For each of remaiig 97 models, after the correlatio matrix for variables occurrig i model ad covariace matrix for estate model values establishig a pricig model have bee determied, the values of three defied ivariats were calculated. Thereby, 97 sets of these three umbers, were achieved. The results are preseted i Table 1. Besides the values of ivariats, the table cotais: the ame of the tow where the data used to modellig were acquired, the type of model ad costat values describig the database ad the model. They are: umber of real estates i a database, umber of idepedet variables i a model u, umber of estimated parameters i a pricig model m, umber of degrees of freedom k = - m, remaider variace - σ. Table 1; Values of Ivariats 0 Nr MIEJSCOWOŚĆ ZMIENNOŚĆ MODEL u m k σ 0 R σ tr σ det 1 Bolesław Liiowa ,807,936,166,05308 Bolesław Liiowa ,06,70,17, Bolesław Liiowa ,364,73,183, Bolesław Liiowa ,930,78,03, Bolesław Liiowa ,643,814,189, Bolesław Liiowa ,76,836,01,037 7 Bolesław Liiowa ,750,864,1, Bolesław Liiowa K-P-D-M ,86,681,10, Bolesław ieliiowa ,63,894,148, Bolesław ieliiowa ,35,994,057, Bolesław ieliiowa ,35,994,057, Bolesław ieliiowa ,355,990,05, Bolesław ieliiowa ,6,99,047, Bolesław ieliiowa ,66,994,050, Bolesław ieliiowa ,708,976,076, Bolesław ieliiowa ,834,966,080, Bolesław ieliiowa ,365,768,114, Bolesław ieliiowa ,365,768,114, Bolesław ieliiowa ,178,800,13, /13

8 Nr MIEJSCOWOŚĆ ZMIENNOŚĆ MODEL u m k σ 0 R σ tr σ det 0 Bolesław ieliiowa ,064,84,138, Bolesław ieliiowa ,654,934,111,00000 Bolesław ieliiowa ,986,949,077, Bolesław ieliiowa ,784,964,07, Bolesław ieliiowa ,715,963,065, Busko Zdrój liiowa ,875,810,090, Busko Zdrój liiowa ,515,781,084, Busko Zdrój ieliiowa a ,7,94,057, Busko Zdrój ieliiowa b ,151,95,05,005 9 Busko Zdrój ieliiowa 3a ,436,951,049, Busko Zdrój ieliiowa 4a ,954,960,051, Busko Zdrój ieliiowa 4b ,193,955,054, Busko Zdrój ieliiowa 5a ,116,957,046, Busko Zdrój ieliiowa 5b ,108,953,048, Busko Zdrój ieliiowa ,1,949,047, Krowodrza ieliiowa 1a ,053,695,34, Krowodrza II liiowa ,711,919,040, Krowodrza II liiowa ,446,949,03, Krowodrza II ieliiowa ,668,90,047, Krowodrza II ieliiowa 1a ,673,951,037, Krowodrza II ieliiowa 1b ,56,96,033, Nowy Sącz liiowa ,934,811,147,009 4 Nowy Sącz liiowa ,438,803,140, Nowy Sącz liiowa ,78,879,10, Nowy Sącz liiowa ,086,877,11, Nowy Sącz liiowa ,458,898,105, Nowy Sącz liiowa ,983,895,098, Nowy Sącz ieliiowa ,975,859,150, Nowy Sącz ieliiowa ,94,859,151, Proszowice liiowa ,468,894,191, Proszowice liiowa ,913,93,197, Proszowice ieliiowa ,016,964,11, Proszowice ieliiowa ,016,964,11, Proszowice ieliiowa ,610,963,114, Proszowice ieliiowa ,648,963,115, Proszowice ieliiowa 5' ,84,975,116, Proszowice ieliiowa 5a ,690,96,115, Proszowice ieliiowa 5'a ,96,975,115, Proszowice ieliiowa 5b ,463,998,046, Proszowice ieliiowa ,560,963,11, Przeworsk liiowa ,006,605,033, Przeworsk ieliiowa ,158,643,038, /13

9 Nr MIEJSCOWOŚĆ ZMIENNOŚĆ MODEL u m k σ 0 R σ tr σ det 6 Rzeszów liiowa ,689,84,060, Rzeszów liiowa ,81,856,05, Rzeszów liiowa ,845,875,048, Rzeszów ieliiowa ,447,80,061, Rzeszów ieliiowa 1a ,610,85,053, Rzeszów ieliiowa 1b ,785,870,049, Rzeszów ieliiowa ,637,84,060, Rzeszów ieliiowa a ,13,854,05, Rzeszów ieliiowa b ,500,871,048, Rzeszów ieliiowa ,5,85,060, Rzeszów ieliiowa 3a ,904,856,05, Rzeszów ieliiowa 3b ,196,873,048, Rzeszów ieliiowa ,640,84,059, Rzeszów ieliiowa 4a ,669,857,051, Rzeszów ieliiowa 4b ,648,876,047, Świdik liiowa ,96,615,094, Świdik liiowa ,487,676,088, Świdik liiowa ,6,600,073, Świdik liiowa ,308,65,069, Świdik liiowa ,6,711,079, Świdik liiowa ,57,711,074, Świdik ieliiowa ,09,640,093, Świdik ieliiowa ,775,736,089, Świdik ieliiowa ,974,704,093, Świdik ieliiowa ,615,760,085, Świdik ieliiowa ,437,816,080, Świdik ieliiowa I ,671,839,083, Świdik ieliiowa II ,515,833,079, Świdik ieliiowa III ,671,834,081, Świdik ieliiowa IV ,515,89,078, Trzyciąż liiowa ,864,743,063, Trzyciąż ieliiowa ,19,859,058, Trzyciąż ieliiowa ,853,898,049, Trzyciąż ieliiowa ,580,98,041, Trzyciąż ieliiowa ,89,88,053, Trzyciąż ieliiowa ,53,934,035, STUDY ON DEPENDENCE OF INVARIANTS ON QUANTITIES DESCRIBING THE DATABASE OF REAL ESTATES AND THE PRICING MODEL I order to formulate criteria for real estates database ad model reliability estimatio, several scatter diagrams of ivariats depedece o costat quatities describig a database ad applied pricig model. O each of these diagrams, a tred lie, estimated with least squares 9/13

10 method, has bee put o. It eables to cofirm the occurrece or o-occurrece of ay relatios betwee these quatities ad, i cosequece, to draw coclusios cocerig the model or the database. Diagrams of ivariats depedece o followig parameters have bee made: database size, umber of idepedet variables i a model, umber of degrees of freedom, remaider variace. Selected diagrams of the whole set are preseted below: Scatter diagram (third ivariat - database size) 0,055 0,045 0,035 sigma det 0,05 0,015 0,005-0, Scatter diagram (third ivariat - umber of idepedet variables) 0,055 0,045 0,035 sigma det 0,05 0,015 0,005-0, u <8 >=8 10/13

11 Superficial diagram 3D sigma det (, u) 0,001 0,008 0,014 0,0 0,07 0,033 0,04 0,046 0,05 0,059 poad 6. CONCLUSIONS The mai purpose of the study was the presetatio of possibilities for valuatio of a database ad a model applied to estimate the real estate market value, usig defied covariace matrix trasformatio parameters ivariats for model values of estates. After the parameters of about a hudred models, tested o 10 databases of differet size (18 to 13 estates) ad of differet amout of features take ito accout (4 to 18), have bee estimated, ad after the values of three ivariats: R, σ tr, σ det have bee determied, o the basis of these ivariats scatter diagrams i cojuctio with characteristics of databases ad models, the followig coclusios ca be formulated: Optimal quatity of real estates i a database for modellig the estates values should correspod to threefold umber of determied parameters of a pricig model. It must be poited that this umber should be, at least, twice as large as the umber of determied model parameters, ad that its elargig (more tha quadruple umber of parameters) i most cases does ot brig improvemet of the model. Maximum umber of idepedet variables should ot exceed 14 parameters. Optimum umber of parameters (attributes) should be determied by the prelimiary aalysis of real estates market. The umber of degrees of freedom i a pricig model should be cotaied betwee 8 ad 4. The selectio of a database for estimatio of a pricig model ca be ackowledged to be optimal if the value of ivariat σ det is situated i the iterval: σ det (0,0008; 0,0055). 11/13

12 The selectio of model for estimatio of real estates values ca be ackowledged as satisfyig whe the value of determiatio coefficiet R fulfils the iequality: R 0,837. Usig the parameter σ tr, criteria of selectig a database for estimatio of a pricig model ca ot be formulated, because the value of this parameter shows a strog fluctuatio. From the compariso of coclusios 4 ad 6 it may be observed that for selectio of a correct pricig model, cosideratio of othig but variaces of model values is ot sufficiet. It is ecessary to take ito accout the whole covariace matrix for model values Cov(W). REFERENCES Barańska A. 003 Kryteria stosowaia modeli stochastyczych w predykcji rykowej wartości ieruchomości. Rozprawa doktorska, Akademia Góriczo Huticza, Wydział Geodezji Góriczej i Iżyierii Środowiska. Kraków Barańska A. 004 Wybór cech ieruchomości do modelowaia matematyczego wartości rykowej a przykładzie kilku baz ieruchomości grutowych. UWND AGH, Geodezja (t. 1) Barańska A., Mitka B. 00 Statystycze przygotowaie baz daych do dalszych aaliz. IX Krajowa Koferecja Komputerowe Wspomagaie Badań Naukowych, Polaica Zdrój, 4-6 paździerika 00 r. Czaja J. 001 Metody szacowaia wartości rykowej i katastralej ieruchomości. KOMP- SYSTEM, Kraków Czaja J., Preweda E. 000 Aaliza ilościowa różych współczyików korelacji a przykładzie sześciowymiarowej zmieej losowej. UWND AGH, Geodezja (tom 6). Czaja J., Preweda E. 000 Aaliza statystycza zmieej losowej wielowymiarowej w aspekcie korelacji i predykcji. UWND AGH, Geodezja (tom) Statistica PL, 1997, podręczik użytkowika. StatSoft, Kraków BIOGRAPHICAL NOTES Name: Employmet: AGH Uiversity of Sciece ad Techology i Krakow Polad (from ) Degree: Philosophy Doctor of Geodetic Scieces (AGH Krakow, ) Study: Doctor Study at the Faculty of Miig Surveyig ad Eviromet Egieerig (AGH Krakow, ) Degree: Master of Geodetic Sciece, Specializatio: Real Estate Admiistratio (AGH Krakow, ) Study: Master Study at the Faculty of Miig Surveyig ad Eviromet Egieerig (AGH Krakow, ) Degree: Master of Mathematics, Specializatio: Applied Mathematics (Jagielloia Uiversity Krakow, ) Study: Master Study at the Faculty of Mathematics ad Physics Faculty (Jagielloia Uiversity Krakow, ) 1/13

13 Publicatios: Zastosowaie uogólioych modeli liiowych w wyceie ieruchomości A. Barańska, Geodezja, tom 5, zeszyt, 1999 r. Zasady przeprowadzaia powszechej taksacji ieruchomości w Polsce A. Barańska, J. Czaja, B. Kryś, Prace Naukowe Istytutu Górictwa Politechiki Wrocławskiej Nr 90, Seria: Koferecje Nr 7, Wrocław 000 r. Kryteria stosowaia modeli stochastyczych w predykcji rykowej wartości ieruchomości A. Barańska, Geodezja, tom 8, zeszyt 1, 00 r. Wstępa ocea wiarygodości baz ieruchomości A. Barańska, XVIII Jesiea Szkoła Geodezji, zeszyty aukowe Akademii Roliczej we Wrocławskiej r 45, sekcja geodezji urządzeń rolych XX, 00 r. Statystycze przygotowaie baz daych do dalszych aaliz A. Barańska, B. Mitka, materiały koferecyje IX Krajowej Koferecji Komputerowe Wspomagaie Badań Naukowych, Wrocławskie Towarzystwo Naukowe, Wrocław Polaica Zdrój 4-6 paździerik 00 r. Modele Statystycze w zagadieiu testowaia baz daych o ieruchomościach A. Barańska, P. Zając, materiały koferecyje IX Krajowej Koferecji Komputerowe Wspomagaie Badań Naukowych, Wrocławskie Towarzystwo Naukowe, Wrocław Polaica Zdrój 4-6 paździerik 00 r. Aaliza w czasie stau ryku ieruchomości grutowych a tereie Polski połudiowo wschodiej A. Barańska, Geodezja, tom 1, 004 r. Wybór cech ieruchomości do modelowaia matematyczego wartości rykowej a przykładzie kilku baz ieruchomości grutowych A. Barańska, Geodezja, tom, 004 r. Participatio: 4 cofereces i years with 4 papers CONTACTS Uiversity of Sciece ad Techology Faculty of Miig Surveyig ad Evirometal Egieerig Terrai Iformatio Departmet al. A. Mickiewicza Krakow POLAND Tel Fax abara@agh.edu.pl 13/13