# RECORD ecb8bacfa12ff5fad71503a1b459329f9f420bc4
# TITLE: A House Price Modeling Based on Clustering and Kriging: The Medellín Case
# YEAR: 2022.0  VENUE: The Review of Regional Studies  DOI: 10.52324/001c.66198
# AUTHORS: H. Villada-Medina; J. F. Rendón-García; C. Ramírez-Dolores; G. Alcalá
# PAGES: 23  CANDIDATE TABLE PAGES (1-based): [11, 12, 15]
# 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.


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(2022) 52, 321–343
A House Price Modeling Based on Clustering and
Kriging: The Medell´ın Case ∗
Hern´an D. Villada-Medinaa, Juan F. Rend´on-Garc´ıaa,
C´esar A. Ram´ırez-Doloresb, and Gerardo Alcal´ac
aFacultad de Ciencias Econ´omicas y Administrativas, Instituto Tecnol´ogico Metropolitano, Colombia
bInstituto de Energ´ıas Renovables, Universidad Nacional Aut´onoma de M´exico, M´exico
cCentro de Investigaci´on en Recursos Energ´eticos y Sustentables, Universidad Veracruzana, M´exico
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 Medell´ın, 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 Mat´ern 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.
Keywords: clustering; geostatistics; k-means; kriging; mass appraisal; value map.
JEL Codes: C21, R12, R32
1.
INTRODUCTION
House prices are estimated individually or collectively. According to Ling and Archer (2018),
traditional approaches for individual appraisal of real estate can be classified into three main
categories: the cost approach, the income approach, and the sales comparison approach. The
cost approach is used mainly to appraise newly built properties, and the income approach
is for commercial and investment real estate. The sales comparison approach consists of
comparing the prices of a sample of residential properties sold in the area of the subject
∗We wish to thank two anonymous referees for providing helpful comments. All remaining errors are our
own. Villada-Medina and Rend´on-Garc´ıa are Full-time Professors at Instituto Tecnol´ogico Metropolitano,
Colombia. Ram´ırez-Dolores is an Assistant Professor at Universidad Nacional Aut´onoma de M´exico, M´exico,
and Alcal´a is a Full-time Professor at Universidad Veracruzana, M´exico. Corresponding Author: Gerardo
Alcal´a, E-mail: galcala@uv.mx
(c) Southern Regional Science Association 2022
ISSN 1553-0892, 0048-49X (online)
www.srsa.org/rrs


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VILLADA-MEDINA et al: MODELING BASED ON CLUSTERING AND KRIGING
331
Table 2: Descriptive statistics of the different clusters.
Case
Clustering
Submarket
Submarket
Average
Average
Average
Average
Average
Type of
criterion
ID
price [COP]
area [m2]
age [years]
number
number of
property
of bedrooms
bathrooms
apartment-house
Case A
1
1
1
519,400,000
164.50
21.77
3.19
2.73
74-17
2
2
166,086,677
64.77
12.76
2.54
1.63
175-27
2 submarkets
2
1
3
519,400,000
140.00
26.43
3.12
2.46
92-27
2
4
166,086,677
65.44
8.12
2.48
1.63
157-17
3
1
5
394,000,000
138.13
24.92
3.23
2.46
98-31
2
6
195,000,000
62.40
8.19
2.35
1.59
151-13
4
1
7
391,000,000
136.60
23.41
3.19
2.51
110-29
2
8
185,262,248
58.82
8.47
2.33
1.48
139-15
5
1
9
400,800,000
135.90
23.61
3.20
2.49
108-31
2
10
162,980,576
58.99
8.19
2.31
1.50
141-12
Case B
1
1
11
394,800,000
202.40
21.20
3.29
3.00
27–8
2
12
167,000,000
131.90
21.37
3.09
2.41
63-12
3 submarkets
3
13
656,500,000
60.51
12.09
2.49
1.59
159-24
2
1
14
405,446,667
182.80
19.77
3.24
2.86
43-15
2
15
149,893,490
96.54
30.86
2.90
2.03
59-12
3
16
593,900,000
64.59
7.45
2.49
1.63
147-17
3
1
17
207,000,000
174.96
17.89
3.26
2.82
49-16
2
18
193,146,395
100.16
30.18
3.14
2.08
55-17
3
19
579,400,000
60.69
7.83
2.33
1.56
145-11
4
1
20
214,277,778
162.23
25.30
3.20
2.77
71-16
2
21
177,743,646
53.35
8.39
1.74
1.35
78-5
3
22
493,300,000
77.31
13.50
3.09
1.82
100-23
5
1
23
158,168,081
148.00
26.17
3.25
2.57
73-31
2
24
201,390,715
48.45
9.90
2.03
1.01
66-13
3
25
421,100,000
80.25
9.58
2.76
2.09
110-0
Case C
1
1
26
109,749,367
211.30
21.37
3.30
3.00
19–8
2
27
257,780,079
51.43
10.22
2.29
1.43
115-14
4 submarkets
3
28
679,300,000
88.35
17.23
2.97
1.99
60-13
4
29
118,561,633
144.80
21.94
3.14
2.61
55-9
2
1
30
250,069,288
168.40
15.14
3.16
2.72
45-12
2
31
451,976,562
74.76
24.43
2.82
1.74
42-19
3
32
587,900,000
60.17
6.24
2.41
1.59
130-8
4
33
154,327,869
151.10
36.32
3.19
2.60
32-5
3
1
34
177,529,049
185.00
20.18
3.31
2.95
40-15
2
35
362,000,000
75.26
8.55
3.09
1.83
79-13
3
36
597,400,000
53.83
7.92
1.73
1.38
78-4
4
37
222,174,543
102.19
31.44
3.03
2.09
52-12
4
1
38
163,084,765
185.00
20.34
3.28
3.06
40-13
2
39
221,046,875
47.99
9.71
2.00
1.01
66-11
3
40
607,100,000
78.25
6.94
2.70
2.07
90-2
4
41
110,457,143
103.60
29.49
3.18
2.07
53-18
5
1
42
258,226,182
147.48
26.24
3.10
2.65
83-0
2
43
230,704,225
46.04
8.82
1.57
1.02
50-1
3
44
442,638,554
73.11
8.92
2.78
1.91
116-0
4
45
121,425,490
115.90
20.81
3.33
1.93
0-43
circular and Mat´ern ones (80% and 10%, respectively). In addition, according to Table 3,
the best variogram was obtained for clustering criterion 2. For both submarkets, the best fit
model was the circular one. Nevertheless, submarket 2 was adjusted to an R-Squared value
of 0.5832; and submarket 1, to 0.12. The cutoff distance and lag width of submarket 2 were
7741 m and 968 m, respectively.
In Case B (three submarkets), Table 4 indicates that the circular model was the most
common option (40%), followed by the Gaussian, exponential, Mat´ern, and spherical mod-
els (26.7%, 13.3%, 13.3%, and 6.7%, respectively). Additionally, as shown in Table 3, the
variogram best described a submarket was obtained for clustering criterion 2. Submarket
3 was best fitted using the circular model (with an R-Squared value of 0.5811), while sub-
markets 1 and 2 were best fitted using the Mat´ern and spherical models (with an R-Squared
value of -0.1258 and 0.2201, respectively). This negative value suggests that the null model’s
prediction is more accurate than that of kriging.
©Southern Regional Science Association 2022.


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332
The Review of Regional Studies 52(3)
Table 3: Models Selected via Leave-One-Out Cross-Validation.
Case
Clustering
Submarket
Submarket
Model
Lag
Cutoff
R-Squared
RMSE
MAPE
Observation
Criterion
ID
Width
Distance
[%]
Points (Np)
Case A
1
1
1
Mat´ern
701
5608
0.0037
200821692
33.23
86
2
2
Circular
645
3871
0.4478
80970818
36.94
177
2 submarkets
2
1
3
Circular
561
5608
0.1200
225096888
53.41
109
2
2
4
Circular
968
7741
0.5832
101771961
36.02
154
3
1
5
Circular
623
5608
0.1816
215563505
49.76
119
2
6
Circular
774
3871
0.5121
103455828
37.44
144
4
1
7
Circular
1051
8412
0.1628
210399681
47.76
131
Case A
2
8
Circular
1003
9031
0.4336
95228844
38.16
132
5
1
9
Circular
1051
8412
0.1813
210224706
47.94
130
2
10
Mat´ern
1290
6452
0.4748
93068789
37.46
133
Case B
1
1
11
Gaussian
1202
8412
-0.1893
209728785
28.05
32
2
12
Gaussian
565
4522
0.2326
127167193
29.02
72
3 submarkets
3
13
Mat´ern
1290
9033
0.3944
73197888
34.47
159
2
1
14
Mat´ern
1930
9650
-0.1258
208438795
26.82
54
2
15
Spherical
1594
7972
0.2201
86942896
42.34
65
2
3
16
Circular
1106
7743
0.5811
98650579
32.87
144
3
1
17
Exponential
1206
9650
-0.1006
198375039
26.48
60
2
18
Gaussian
759
6834
0.2918
86607028
34.97
66
3
19
Gaussian
1807
9033
0.5287
95304322
39.36
137
4
1
20
Circular
2103
8412
0.1359
205963996
35.01
82
2
21
Circular
954
7630
0.5008
93897000
40.65
71
3
22
Exponential
748
5984
0.4838
100414468
33.73
110
5
1
23
Circular
623
5608
0.1711
222583902
48.66
97
2
24
Circular
1908
7630
0.4721
65323203
32.32
66
3
25
Circular
486
3400
0.5527
105530392
30.75
100
Case C
1
1
26
Spherical
1103
5515
-0.0403
178616456
22.51
26
2
27
Exponential
430
3871
0.2737
55315971
30.44
110
3
28
Circular
971
6795
0.3968
91120137
32.20
67
4 submarkets
4
29
Circular
1574
7872
0.0995
159473948
29.25
60
2
1
30
Spherical
526
3158
0.0605
188054416
26.46
56
2
31
Exponential
499
3492
0.1456
72819134
41.23
55
3
32
Circular
717
6452
0.5286
92037204
33.71
119
4
33
Circular
1402
5608
0.0166
147003301
32.82
33
3
1
34
Gaussian
981
9814
-0.1942
220260213
29.10
51
2
35
Gaussian
532
4787
0.6244
99754816
30.22
83
3
36
Circular
2225
8902
0.5565
90656313
38.97
71
3
4
37
Spherical
340
3395
0.2171
89545708
34.30
58
4
1
38
Mat´ern
1052
4206
-0.0618
192524506
25.89
50
2
39
Circular
1908
7630
0.4761
65452585
34.23
65
3
40
Circular
876
4381
0.5250
106857235
32.79
84
4
41
Circular
629
5659
0.2603
103575352
34.67
64
Case C
5
1
42
Spherical
342
3417
0.1736
199870272
43.01
78
2
43
Circular
1090
7630
0.5219
73716461
35.25
44
4 submarkets
3
44
Circular
798
4788
0.5221
103316140
32.89
104
4
45
Circular
885
5313
0.1645
250953969
50.10
37
Note: The clustering criteria include five different variables (area, age, bedrooms, bathrooms, and type of
property). Clustering criterion 1 considers the first variable (area); clustering criterion 2, the first two
variables (area and age); and so on. The best-modeled submarket is shown in bold, and its corresponding
clustering criterion rows use grey background.
Table 4: Summarized Results of the Models Selected From Table 3.
Number of
Lag width
Cutoff distance
Exponential
Gaussian
Spherical
Circular
Mat´ern
Clusters
[m]
[m]
[%]
[%]
[%]
[%]
[%]
All cases
45
1018
6418
6.7
15.6
11.1
55.6
11.1
Case A
10
867
6461
0
0
0
80
20
Case B
15
1219
7434
13.3
26.7
6.7
40
13.3
Case C
20
944
5634
5
15
20
55
5
In Case C (four submarkets), Table 4 shows that the circular model was the most common
one (55%), followed by the spherical, Gaussian, exponential, and Mat´ern (20%, 15%, 5%, and
©Southern Regional Science Association 2022.


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VILLADA-MEDINA et al: MODELING BASED ON CLUSTERING AND KRIGING
333
5%, respectively). Also, according to Table 3, the variogram that best described a submarket
was obtained by applying clustering criterion 3. The best fit model for submarket 2 was the
Gaussian one (with an R-Squared value of 0.6244, a cutoff distance of 4787 m, and a lag
width of 532 m); for submarket 1, the Gaussian model (with an R-Squared value of -0.1942);
for submarket 3, the circular model (with R-Squared values of 0.5565); and for submarket
4, the spherical model (with R-Squared values of 0.2171).
According to Table 3 for the grey highlighted clustering criteria, Case A showed only one
of the two submarkets properly modeled (a relatively high R-Squared value). For Case B
there was a relatively high R-Squared value, an intermediate value, and a low value for each
of the three submarkets. For Case C, clustering criteria 3, yielded two submarkets properly
modeled with the highest R-Squared values (0.62 and 0.56, respectively).
Submarkets and Clustering Criterion Effect
The effect of the number of submarkets is observed by considering the weighted average
values for the R-squared and MAPE values for each of the three cases (Table 5). Since
the weighted average includes poorly modeled submarkets, high MAPE and low R-squared
values are obtained.
The best behavior is found for Case B, with mean R-Squared and
MAPE values of 0.3446 and 35.05, respectively.
The clustering criterion effect is observed by taking the weighted average of the five
different clustering criteria (Table 6). When comparing clustering criterion 1 (area) and
clustering criterion 2 (area and age), it can be noticed an improvement in the modeling as
the R-Squared increases from 0.2720 to 0.3409. It means that the variable of age provides
additional and valuable information for the model.
On the other hand, clustering criteria 3 and 4, present no substantial improvement com-
pared to clustering criterion 2. A similar explanation could be that the number of bedrooms
and bathrooms depends on the area variable. Finally, clustering criterion 5 presents the
highest R-Squared value, which means that prediction results are improved if the model is
provided with information about the type of housing unit (i.e., apartment or house).
Table 5: Weighted Average Values for Cases Obtained from Table 3
Case
Mean R-Squared
Mean MAPE[%]
Case A
0.3370
41.51
Case B
0.3446
35.05
Case C
0.3164
33.54
From this point on, we will present a complete analysis (as shown in Figure 3) of Case B
(three submarkets) using clustering criterion 2 (area and age), as these two factors produce
the best models according to the criteria discussed previously. Figure 4 shows the spatial
distribution of the three different submarkets in Medell´ın, Colombia, in Case B.
Moreover, Table 7 provides a spatial description of the three submarkets defined ap-
plying clustering criterion 2 and includes the following information: submarket, number of
data (ND), data density, number of paired comparisons among data, percentage of paired
©Southern Regional Science Association 2022.


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VILLADA-MEDINA et al: MODELING BASED ON CLUSTERING AND KRIGING
335
Table 7: Spatial Data Statistics of Case B (Three Submarkets) Using
Clustering Criterion 2 (Area and Age)
Submarket
Number of
Data
Paired
Percentage of Paired
Distance Within Which
Mean
SD of
Mean
SD of
Data
Density
Comparisons
Comparisons Within
50 % of Paired
Distance
Distance
Log(Price)
Log(Price)
(NP)
(NP/m2)
1 km
Comparisons Can Be Found
(m)
(m)
(COP)
(COP)
1
54
0.5376
1431
4.40
8225
4041
2755
20.1481
0.36760714
2
65
0.6471
2080
3.65
8202
4515
2407
19.0574
0.54374478
3
144
1.4335
10296
3.65
11510
4979
2510
18.8644
0.70816505
Figure 5 shows selected semi-variograms of the three models analyzed in this study (i.e.,
Mat´ern, spherical, and circular) and the modeling parameters (nugget, range, and partial
sill) for the three submarkets. Additionally, the validation indexes obtained via LOOCV are
presented in Table 8, which is actually a subset of Table 3. Figure 5 indicates that the three
submarkets are best modeled using large cutoff distance values and lag widths. Nevertheless,
only submarket 3 yields proper results according to the R-Squared index.
Figure 5: Selected Semi-variogram Models for the Set with Three
Submarkets Using Clustering Criterion 2 (Area and Age).
Table 8: Leave-One-Out Cross-Validation of the Model Selected for Case B
(Set with Three Submarkets) Using Clustering Criterion 2 (Area and Age).
Submarket
Cluster ID
Model
Lag
Cutoff
R-squared
RMSE
MAPE
Width
Distance
1
14
Mat´ern
1930
9650
-0.1258
208438795
26.82
2
15
spherical
1594
7972
0.2201
86942896
42.34
3
16
circular
1106
7743
0.5811
98650579
32.87
The selected variograms were used to obtain the mean values on the residential property
map, as shown in Figure 6 shows the interpolated kriging map (which is masked by the
transport network) of each cluster in Medell´ın. This is an example of the isoline maps that
can be obtained from kriging interpolation; in this case, it shows the results of 3 submarkets
applying criterion 2 (area and age). In Table 8, the mean absolute percentage error (MAPE)
of submarkets 1, 2, and 3 reached 26.82%, 42.34%, and 32.87%, respectively.
©Southern Regional Science Association 2022.


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336
The Review of Regional Studies 52(3)
Figure 6: Map of Predicted Property Prices (Set With 3 Clusters)
Obtained Applying Criterion 2 (Area and Age)
Figure 7 shows the Multidimensional Index of Living Conditions (MILC) according to the
Encuesta de Calidad de Vida (Quality of Life Survey) 2 in the year 2018 of the 16 comunas
(districts) of Medell´ın. This index serves as a global indicator of living conditions (on a scale
from 1 to 100) for each district (Alcald´ıa de Medell´ın, 2019). According to the isoline maps
of Figure 7, the southeast area of the city, which is district 14 (El Poblado), concentrates
the most expensive housing. In fact, the MILC of district 14 is the highest.
The two-stage methodology approach offers interesting insights into the behavior of each
submarket in relation to the living standards in the area of the dwellings. According to
Figure 6, a dwelling of the submarkets 1 or 2 can cost up to three times more for being
located in an area with a high MILC (such as in the southeast of the city), rather than
being located in the northern side of the city (districts 1 to 7), where the MILC is under the
average of the city (49.3). These differences in prices are much more noticeable for properties
of submarket 3 (typically recently built and small apartments, according to Table 2). In this
case, a dwelling can cost up to six times more for being located in the district with the
highest MILC.
The predictions plotted in Figure 8 show the discrepancies in prices (log was applied for
the sake of visualization) between Medell´ın Property Price Register and the values predicted
for the third cluster, which is the best fit by a line with intercept 5.86 and slope 0.69. The fact
that the dashed line (adjusted to the points) is above the solid line in the early phases of the
chart indicates that, when property values are estimated, the kriging method overestimates
low values and underestimates high ones.
2The Encuesta de Calidad de Vida (Quality of Life Survey) is an instrument whose purpose is to monitor and
measure the socioeconomic conditions of the inhabitants of the 16 districts and 5 townships that make up
the city of Medell´ın. It is a primary source of information that allows knowing indices on issues of marked
importance such as population, housing, households, education, workforce, health, and social security.
©Southern Regional Science Association 2022.