1# RECORD ecb8bacfa12ff5fad71503a1b459329f9f420bc42# TITLE: A House Price Modeling Based on Clustering and Kriging: The Medellín Case3# YEAR: 2022.0 VENUE: The Review of Regional Studies DOI: 10.52324/001c.661984# AUTHORS: H. Villada-Medina; J. F. Rendón-García; C. Ramírez-Dolores; G. Alcalá5# PAGES: 23 CANDIDATE TABLE PAGES (1-based): [11, 12, 15]6# ABSTRACT: In this study, house prices are modeled using a mixed two-stage model for mass appraisal employing valuations of second-hand housing units conducted in Medellín, Colombia. In the first stage, submarkets of houses that share non-spatial attributes are created using clustering; in the second stage, the spatial dependency is incorporated into the house price estimation using kriging. The best results were obtained when the sample was divided into three submarkets using property area and age as the classification criterion and later applying a Matérn kriging model to submarket 1, a spherical kriging model to submarket 2, and a circular kriging model to submarket 3. These results may provide further guidance to enhance mass appraisal practice in other Latin American cities as well as potentially other cities in developing countries.789===== PAGE 1 (score 6, numeric-paren 0, table-words 0, var-words 6) =====1011(2022) 52, 321–34312A House Price Modeling Based on Clustering and13Kriging: The Medell´ın Case ∗14Hern´an D. Villada-Medinaa, Juan F. Rend´on-Garc´ıaa,15C´esar A. Ram´ırez-Doloresb, and Gerardo Alcal´ac16aFacultad de Ciencias Econ´omicas y Administrativas, Instituto Tecnol´ogico Metropolitano, Colombia17bInstituto de Energ´ıas Renovables, Universidad Nacional Aut´onoma de M´exico, M´exico18cCentro de Investigaci´on en Recursos Energ´eticos y Sustentables, Universidad Veracruzana, M´exico19Abstract: In this study, house prices are modeled using a mixed two-stage model for mass appraisal20employing valuations of second-hand housing units conducted in Medell´ın, Colombia. In the first stage,21submarkets of houses that share non-spatial attributes are created using clustering; in the second stage,22the spatial dependency is incorporated into the house price estimation using kriging. The best results were23obtained when the sample was divided into three submarkets using property area and age as the classification24criterion and later applying a Mat´ern kriging model to submarket 1, a spherical kriging model to submarket252, and a circular kriging model to submarket 3. These results may provide further guidance to enhance mass26appraisal practice in other Latin American cities as well as potentially other cities in developing countries.27Keywords: clustering; geostatistics; k-means; kriging; mass appraisal; value map.28JEL Codes: C21, R12, R32291.30INTRODUCTION31House prices are estimated individually or collectively. According to Ling and Archer (2018),32traditional approaches for individual appraisal of real estate can be classified into three main33categories: the cost approach, the income approach, and the sales comparison approach. The34cost approach is used mainly to appraise newly built properties, and the income approach35is for commercial and investment real estate. The sales comparison approach consists of36comparing the prices of a sample of residential properties sold in the area of the subject37∗We wish to thank two anonymous referees for providing helpful comments. All remaining errors are our38own. Villada-Medina and Rend´on-Garc´ıa are Full-time Professors at Instituto Tecnol´ogico Metropolitano,39Colombia. Ram´ırez-Dolores is an Assistant Professor at Universidad Nacional Aut´onoma de M´exico, M´exico,40and Alcal´a is a Full-time Professor at Universidad Veracruzana, M´exico. Corresponding Author: Gerardo41Alcal´a, E-mail: galcala@uv.mx42(c) Southern Regional Science Association 202243ISSN 1553-0892, 0048-49X (online)44www.srsa.org/rrs454647===== PAGE 11 (score 39, numeric-paren 0, table-words 7, var-words 7) =====4849VILLADA-MEDINA et al: MODELING BASED ON CLUSTERING AND KRIGING5033151Table 2: Descriptive statistics of the different clusters.52Case53Clustering54Submarket55Submarket56Average57Average58Average59Average60Average61Type of62criterion63ID64price [COP]65area [m2]66age [years]67number68number of69property70of bedrooms71bathrooms72apartment-house73Case A74175176177519,400,00078164.507921.77803.19812.738274-1783284285166,086,6778664.778712.76882.54891.6390175-27912 submarkets92293194395519,400,00096140.009726.43983.12992.4610092-2710121024103166,086,67710465.441058.121062.481071.63108157-17109311011115112394,000,000113138.1311424.921153.231162.4611798-3111821196120195,000,00012162.401228.191232.351241.59125151-13126412711287129391,000,000130136.6013123.411323.191332.51134110-2913521368137185,262,24813858.821398.471402.331411.48142139-15143514411459146400,800,000147135.9014823.611493.201502.49151108-31152215310154162,980,57615558.991568.191572.311581.50159141-12160Case B1611162116311164394,800,000165202.4016621.201673.291683.0016927–8170217112172167,000,000173131.9017421.371753.091762.4117763-121783 submarkets179318013181656,500,00018260.5118312.091842.491851.59186159-241872188118914190405,446,667191182.8019219.771933.241942.8619543-15196219715198149,893,49019996.5420030.862012.902022.0320359-12204320516206593,900,00020764.592087.452092.492101.63211147-172123213121417215207,000,000216174.9621717.892183.262192.8222049-16221222218223193,146,395224100.1622530.182263.142272.0822855-17229323019231579,400,00023260.692337.832342.332351.56236145-112374238123920240214,277,778241162.2324225.302433.202442.7724571-16246224721248177,743,64624953.352508.392511.742521.3525378-5254325522256493,300,00025777.3125813.502593.092601.82261100-232625263126423265158,168,081266148.0026726.172683.252692.5727073-31271227224273201,390,71527448.452759.902762.032771.0127866-13279328025281421,100,00028280.252839.582842.762852.09286110-0287Case C2881289129026291109,749,367292211.3029321.372943.302953.0029619–8297229827299257,780,07930051.4330110.223022.293031.43304115-143054 submarkets306330728308679,300,00030988.3531017.233112.973121.9931360-13314431529316118,561,633317144.8031821.943193.143202.6132155-93222323132430325250,069,288326168.4032715.143283.163292.7233045-12331233231333451,976,56233474.7633524.433362.823371.7433842-19339334032341587,900,00034260.173436.243442.413451.59346130-8347434833349154,327,869350151.1035136.323523.193532.6035432-53553356135734358177,529,049359185.0036020.183613.313622.9536340-15364236535366362,000,00036775.263688.553693.093701.8337179-13372337336374597,400,00037553.833767.923771.733781.3837978-4380438137382222,174,543383102.1938431.443853.033862.0938752-123884389139038391163,084,765392185.0039320.343943.283953.0639640-13397239839399221,046,87540047.994019.714022.004031.0140466-11405340640407607,100,00040878.254096.944102.704112.0741290-2413441441415110,457,143416103.6041729.494183.184192.0742053-184215422142342424258,226,182425147.4842626.244273.104282.6542983-0430243143432230,704,22543346.044348.824351.574361.0243750-1438343944440442,638,55444173.114428.924432.784441.91445116-0446444745448121,425,490449115.9045020.814513.334521.934530-43454circular and Mat´ern ones (80% and 10%, respectively). In addition, according to Table 3,455the best variogram was obtained for clustering criterion 2. For both submarkets, the best fit456model was the circular one. Nevertheless, submarket 2 was adjusted to an R-Squared value457of 0.5832; and submarket 1, to 0.12. The cutoff distance and lag width of submarket 2 were4587741 m and 968 m, respectively.459In Case B (three submarkets), Table 4 indicates that the circular model was the most460common option (40%), followed by the Gaussian, exponential, Mat´ern, and spherical mod-461els (26.7%, 13.3%, 13.3%, and 6.7%, respectively). Additionally, as shown in Table 3, the462variogram best described a submarket was obtained for clustering criterion 2. Submarket4633 was best fitted using the circular model (with an R-Squared value of 0.5811), while sub-464markets 1 and 2 were best fitted using the Mat´ern and spherical models (with an R-Squared465value of -0.1258 and 0.2201, respectively). This negative value suggests that the null model’s466prediction is more accurate than that of kriging.467©Southern Regional Science Association 2022.468469470===== PAGE 12 (score 26, numeric-paren 0, table-words 5, var-words 7) =====471472332473The Review of Regional Studies 52(3)474Table 3: Models Selected via Leave-One-Out Cross-Validation.475Case476Clustering477Submarket478Submarket479Model480Lag481Cutoff482R-Squared483RMSE484MAPE485Observation486Criterion487ID488Width489Distance490[%]491Points (Np)492Case A493149414951496Mat´ern49770149856084990.003750020082169250133.235028650325042505Circular50664550738715080.44785098097081851036.945111775122 submarkets513251415153516Circular51756151856085190.120052022509688852153.41522109523252425254526Circular52796852877415290.583253010177196153136.02532154533353415355536Circular53762353856085390.181654021556350554149.7654211954325446545Circular54677454738715480.512154910345582855037.44551144552455315547555Circular556105155784125580.162855921039968156047.76561131562Case A56325648565Circular566100356790315680.43365699522884457038.16571132572557315749575Circular576105157784125780.181357921022470658047.94581130582258310584Mat´ern585129058664525870.47485889306878958937.46590133591Case B5921593159411595Gaussian59612025978412598-0.189359920972878560028.0560132602260312604Gaussian60556560645226070.232660812716719360929.02610726113 submarkets612361313614Mat´ern615129061690336170.39446187319788861934.476201596212622162314624Mat´ern62519306269650627-0.125862820843879562926.8263054631263215633Spherical634159463579726360.22016378694289663842.34639656402641364216643Circular644110664577436460.58116479865057964832.876491446503651165217653Exponential65412066559650656-0.100665719837503965826.4865960660266118662Gaussian66375966468346650.29186668660702866734.9766866669367019671Gaussian672180767390336740.52876759530432267639.366771376784679168020681Circular682210368384126840.135968520596399668635.0168782688268921690Circular69195469276306930.50086949389700069540.6569671697369822699Exponential70074870159847020.483870310041446870433.737051107065707170823709Circular71062371156087120.171171322258390271448.6671597716271724718Circular719190872076307210.47217226532320372332.3272466725372625727Circular72848672934007300.552773110553039273230.75733100734Case C7351736173726738Spherical73911037405515741-0.040374217861645674322.5174426745274627747Exponential74843074938717500.27377515531597175230.44753110754375528756Circular75797175867957590.39687609112013776132.20762677634 submarkets764476529766Circular767157476878727690.099577015947394877129.25772607732774177530776Spherical77752677831587790.060578018805441678126.4678256783278431785Exponential78649978734927880.14567897281913479041.2379155792379332794Circular79571779664527970.52867989203720479933.71800119801480233803Circular804140280556088060.016680714700330180832.82809338103811181234813Gaussian8149818159814816-0.194281722026021381829.1081951820282135822Gaussian82353282447878250.62448269975481682730.2282883829383036831Circular832222583389028340.55658359065631383638.97837718383839484037841Spherical84234084333958440.21718458954570884634.30847588484849185038851Mat´ern85210528534206854-0.061885519252450685625.8985750858285939860Circular861190886276308630.47618646545258586534.2386665867386840869Circular87087687143818720.525087310685723587432.7987584876487741878Circular87962988056598810.260388210357535288334.6788464885Case C8865887188842889Spherical89034289134178920.173689319987027289443.0189578896289743898Circular899109090076309010.52199027371646190335.25904449054 submarkets906390744908Circular90979891047889110.522191210331614091332.89914104915491645917Circular91888591953139200.164592125095396992250.1092337924Note: The clustering criteria include five different variables (area, age, bedrooms, bathrooms, and type of925property). Clustering criterion 1 considers the first variable (area); clustering criterion 2, the first two926variables (area and age); and so on. The best-modeled submarket is shown in bold, and its corresponding927clustering criterion rows use grey background.928Table 4: Summarized Results of the Models Selected From Table 3.929Number of930Lag width931Cutoff distance932Exponential933Gaussian934Spherical935Circular936Mat´ern937Clusters938[m]939[m]940[%]941[%]942[%]943[%]944[%]945All cases94645947101894864189496.795015.695111.195255.695311.1954Case A9551095686795764619580959096009618096220963Case B964159651219966743496713.396826.79696.79704097113.3972Case C97320974944975563497659771597820979559805981In Case C (four submarkets), Table 4 shows that the circular model was the most common982one (55%), followed by the spherical, Gaussian, exponential, and Mat´ern (20%, 15%, 5%, and983©Southern Regional Science Association 2022.984985986===== PAGE 13 (score 50, numeric-paren 0, table-words 20, var-words 9) =====987988VILLADA-MEDINA et al: MODELING BASED ON CLUSTERING AND KRIGING9893339905%, respectively). Also, according to Table 3, the variogram that best described a submarket991was obtained by applying clustering criterion 3. The best fit model for submarket 2 was the992Gaussian one (with an R-Squared value of 0.6244, a cutoff distance of 4787 m, and a lag993width of 532 m); for submarket 1, the Gaussian model (with an R-Squared value of -0.1942);994for submarket 3, the circular model (with R-Squared values of 0.5565); and for submarket9954, the spherical model (with R-Squared values of 0.2171).996According to Table 3 for the grey highlighted clustering criteria, Case A showed only one997of the two submarkets properly modeled (a relatively high R-Squared value). For Case B998there was a relatively high R-Squared value, an intermediate value, and a low value for each999of the three submarkets. For Case C, clustering criteria 3, yielded two submarkets properly1000modeled with the highest R-Squared values (0.62 and 0.56, respectively).1001Submarkets and Clustering Criterion Effect1002The effect of the number of submarkets is observed by considering the weighted average1003values for the R-squared and MAPE values for each of the three cases (Table 5). Since1004the weighted average includes poorly modeled submarkets, high MAPE and low R-squared1005values are obtained.1006The best behavior is found for Case B, with mean R-Squared and1007MAPE values of 0.3446 and 35.05, respectively.1008The clustering criterion effect is observed by taking the weighted average of the five1009different clustering criteria (Table 6). When comparing clustering criterion 1 (area) and1010clustering criterion 2 (area and age), it can be noticed an improvement in the modeling as1011the R-Squared increases from 0.2720 to 0.3409. It means that the variable of age provides1012additional and valuable information for the model.1013On the other hand, clustering criteria 3 and 4, present no substantial improvement com-1014pared to clustering criterion 2. A similar explanation could be that the number of bedrooms1015and bathrooms depends on the area variable. Finally, clustering criterion 5 presents the1016highest R-Squared value, which means that prediction results are improved if the model is1017provided with information about the type of housing unit (i.e., apartment or house).1018Table 5: Weighted Average Values for Cases Obtained from Table 31019Case1020Mean R-Squared1021Mean MAPE[%]1022Case A10230.3370102441.511025Case B10260.3446102735.051028Case C10290.3164103033.541031From this point on, we will present a complete analysis (as shown in Figure 3) of Case B1032(three submarkets) using clustering criterion 2 (area and age), as these two factors produce1033the best models according to the criteria discussed previously. Figure 4 shows the spatial1034distribution of the three different submarkets in Medell´ın, Colombia, in Case B.1035Moreover, Table 7 provides a spatial description of the three submarkets defined ap-1036plying clustering criterion 2 and includes the following information: submarket, number of1037data (ND), data density, number of paired comparisons among data, percentage of paired1038©Southern Regional Science Association 2022.103910401041===== PAGE 15 (score 24, numeric-paren 0, table-words 9, var-words 4) =====10421043VILLADA-MEDINA et al: MODELING BASED ON CLUSTERING AND KRIGING10443351045Table 7: Spatial Data Statistics of Case B (Three Submarkets) Using1046Clustering Criterion 2 (Area and Age)1047Submarket1048Number of1049Data1050Paired1051Percentage of Paired1052Distance Within Which1053Mean1054SD of1055Mean1056SD of1057Data1058Density1059Comparisons1060Comparisons Within106150 % of Paired1062Distance1063Distance1064Log(Price)1065Log(Price)1066(NP)1067(NP/m2)10681 km1069Comparisons Can Be Found1070(m)1071(m)1072(COP)1073(COP)1074110755410760.53761077143110784.40107982251080404110812755108220.148110830.367607141084210856510860.64711087208010883.65108982021090451510912407109219.057410930.5437447810943109514410961.433510971029610983.651099115101100497911012510110218.864411030.708165051104Figure 5 shows selected semi-variograms of the three models analyzed in this study (i.e.,1105Mat´ern, spherical, and circular) and the modeling parameters (nugget, range, and partial1106sill) for the three submarkets. Additionally, the validation indexes obtained via LOOCV are1107presented in Table 8, which is actually a subset of Table 3. Figure 5 indicates that the three1108submarkets are best modeled using large cutoff distance values and lag widths. Nevertheless,1109only submarket 3 yields proper results according to the R-Squared index.1110Figure 5: Selected Semi-variogram Models for the Set with Three1111Submarkets Using Clustering Criterion 2 (Area and Age).1112Table 8: Leave-One-Out Cross-Validation of the Model Selected for Case B1113(Set with Three Submarkets) Using Clustering Criterion 2 (Area and Age).1114Submarket1115Cluster ID1116Model1117Lag1118Cutoff1119R-squared1120RMSE1121MAPE1122Width1123Distance112411125141126Mat´ern11271930112896501129-0.12581130208438795113126.82113221133151134spherical113515941136797211370.2201113886942896113942.34114031141161142circular114311061144774311450.5811114698650579114732.871148The selected variograms were used to obtain the mean values on the residential property1149map, as shown in Figure 6 shows the interpolated kriging map (which is masked by the1150transport network) of each cluster in Medell´ın. This is an example of the isoline maps that1151can be obtained from kriging interpolation; in this case, it shows the results of 3 submarkets1152applying criterion 2 (area and age). In Table 8, the mean absolute percentage error (MAPE)1153of submarkets 1, 2, and 3 reached 26.82%, 42.34%, and 32.87%, respectively.1154©Southern Regional Science Association 2022.115511561157===== PAGE 16 (score 7, numeric-paren 1, table-words 1, var-words 4) =====115811593361160The Review of Regional Studies 52(3)1161Figure 6: Map of Predicted Property Prices (Set With 3 Clusters)1162Obtained Applying Criterion 2 (Area and Age)1163Figure 7 shows the Multidimensional Index of Living Conditions (MILC) according to the1164Encuesta de Calidad de Vida (Quality of Life Survey) 2 in the year 2018 of the 16 comunas1165(districts) of Medell´ın. This index serves as a global indicator of living conditions (on a scale1166from 1 to 100) for each district (Alcald´ıa de Medell´ın, 2019). According to the isoline maps1167of Figure 7, the southeast area of the city, which is district 14 (El Poblado), concentrates1168the most expensive housing. In fact, the MILC of district 14 is the highest.1169The two-stage methodology approach offers interesting insights into the behavior of each1170submarket in relation to the living standards in the area of the dwellings. According to1171Figure 6, a dwelling of the submarkets 1 or 2 can cost up to three times more for being1172located in an area with a high MILC (such as in the southeast of the city), rather than1173being located in the northern side of the city (districts 1 to 7), where the MILC is under the1174average of the city (49.3). These differences in prices are much more noticeable for properties1175of submarket 3 (typically recently built and small apartments, according to Table 2). In this1176case, a dwelling can cost up to six times more for being located in the district with the1177highest MILC.1178The predictions plotted in Figure 8 show the discrepancies in prices (log was applied for1179the sake of visualization) between Medell´ın Property Price Register and the values predicted1180for the third cluster, which is the best fit by a line with intercept 5.86 and slope 0.69. The fact1181that the dashed line (adjusted to the points) is above the solid line in the early phases of the1182chart indicates that, when property values are estimated, the kriging method overestimates1183low values and underestimates high ones.11842The Encuesta de Calidad de Vida (Quality of Life Survey) is an instrument whose purpose is to monitor and1185measure the socioeconomic conditions of the inhabitants of the 16 districts and 5 townships that make up1186the city of Medell´ın. It is a primary source of information that allows knowing indices on issues of marked1187importance such as population, housing, households, education, workforce, health, and social security.1188©Southern Regional Science Association 2022.1189