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Ingeniería. Revista de la Universidad de Costa Rica
Vol. 34, No. 1: 43-50, Enero-Junio, 2024. ISSN: 2215-2652. San José, Costa Rica
Esta obra está bajo una Licencia de Creative Commons. Reconocimiento - No Comercial - Compartir Igual 4.0 Internacional
Uncertainty in Land Value Modeling of the San José
Metropolitan Region, Costa Rica
La incertidumbre en la modelación de valores del suelo de la Gran Área Metropolitana, Costa Rica
Eduardo Pérez Molina 1 , Darío Vargas Aguilar 2
1 Universidad de Costa Rica, San José, Costa Rica
correo: eduardo.perezmolina@ucr.ac.cr
2 Universidad de Costa Rica, San José, Costa Rica
correo: dario.vargasaguilar@ucr.ac.cr
Recibido: 12/09/2023
Aceptado: 30/11/2023
Abstract
Land value patterns show very distinct spatial associations with accessibility to urban centralities
and physical factors in a territory. However, predictions based on models of this structure can be highly
uncertain, as the underlying data also may show clustering (thus allowing for better predictions in more
densely sampled areas). An assessment of this uncertainty for land value extrapolations in the San José
Metropolitan Region of Costa Rica is presented, via conditional Gaussian simulation, and the determinants
of this uncertainty were explored, to nd spatial strengths and weaknesses in the modeling eorts. The
E-Type prediction from the conditional Gaussian simulation was found to marginally improve on ordinary
kriging methods and it also provided explicit uncertainty patterns, which are the inverse of the land value
prediction. The estimated uncertainty was found to decrease with characteristics that identify suitability
for urban land use (and thus higher land values).
Keywords:
Extrapolation, land values,
sequential Gaussian
simulation, spatial factors,
uncertainty.
Palabras Clave:
Extrapolación,
factores espaciales,
incertidumbre,
simulación gaussiana
secuencial, valor del
suelo.
DOI: DOI:10. 5517/ri.v34i1.56618
Resumen
Los patrones de valor del suelo muestran asociaciones espaciales claras con accesibilidad a
centralidades urbanas y a factores físicos de un territorio. Sin embargo, las predicciones basadas en esta
estructura pueden ser altamente inciertas, dado que los datos mismos también exhiben aglomeración (y,
por tanto, permiten mejores predicciones en las zonas más densamente muestreadas). Se presenta una
evaluación de esta incertidumbre para extrapolaciones de valor del suelo en la Gran Área Metropolitana
de Costa Rica mediante simulaciones gaussianas condicionales y una exploración de los determinantes de
esta incertidumbre, como forma de reconocer fortalezas y debilidades de esta predicción. La predicción
E-Type simulada resultó marginalmente mejor que extrapolaciones mediante kriging ordinario y produjo
una cuanticación espacialmente explícita de la incertidumbre. El patrón de incertidumbre resultó ser un
espejo de los valores del suelo. Se encontró que la incertidumbre se reduce con características asociadas
a mayor aptitud del suelo para usos urbanos y, por tanto, de mayor precio.
PÉREZ, VARGAS: Uncertainty in Land Value Modeling of the San José, Costa Rica. 44
1. INTRODUCTION
The analysis of uncertainty of land value models is a critical
issue for policy formulation [1]. However, while the use of Gaussian
simulation to understand uncertainty has long been applied to
physical land variables (e.g., [2]) and despite kriging having been
applied to land rents for at least 20 years [1], no previous cases of
conditional Gaussian simulation applied to land value modeling
were found .
In general, the analysis of land value in the San José
metropolitan region (GAM) has been fragmentary [1]. Recent
eorts from extension and research projects at the University of
Costa Rica, however, have yielded a data set of real estate listings
that provided the rst synoptic view of real estate prices in the
region [3]. Based on this data, hedonic price models of housing
have been produced [3,4] and the rst eorts at extrapolation of
land values for the entire region (based on kriging and co-kriging)
were developed [1]. To isolate land values, [1] consider in their
analysis only lots —i.e., properties oered in the land market with
no buildings on them and, therefore, with prices only reecting the
attributes of land—; these initial eorts yielded estimates of mean
values and of variance, but they were limited to the kriging and
co-kriging models.
Given the current state of the question, two objectives are
proposed for this paper: rst, to extend previous work on land value
extrapolation (by [1]) to include conditional Gaussian simulation
and, specically, to include uncertainty estimates for the predicted
land value that can be derived with this method; second, to explore
whether the uncertainty of these estimates can be explained by
spatial structure (indeed, by the same spatial structure related to
the point pattern of real estate listings and to the land value pattern
itself).
2. METHODOLOGY
2.1 Land values in the GAM
Data were compiled by [1] from real estate listings published
on the web during 2020-2023. From an original data set of 3670
records with known location and price, extreme values (for the
variables price, lot area, and price per square meter) were ltered
out, resulting in a nal data set of 3196 records. This data set was, in
turn, divided by randomly selecting approximately 10 % of records:
a calibration data set of 2878 records and a validation data set of
318 records result from this data wrangling process.
The nal data sets (of calibration and validation) are shown
in Fig. 1. Locations in panel (a) coincide mainly with the urban
fabric of the GAM. It is worth noting that the calibration data seem
to include a greater proportion of locations in the more central
locations (or, conversely, the calibration data are distributed in a
way that should better represent the peripheral land values). The
land value per square meter was transformed into logarithms (as
in [1], [3], [4] and, more generally, following a standard practice
in the analysis of land values). The logarithmic transformation of
both the calibration and validation data sets (panel (b) of Fig. 1
presents the histogram with a logarithmic scale on the horizontal
axis) show a normal distribution, as should be expected, although
with some degree of skew towards the left (this may be explained
because the ltering process of extreme values is more ecient
in excluding excessively large values of price, area, and price per
unit of area).
Fig. 1. (a) Lot data locations and (b) histogram of price (log. of USD per
square meter) in the GAM (in red, calibration data; in green, validation data).
2.2 Geostatistical analysis and conditional Gaussian simulation
As the nal objective of the modeling eorts is to produce a
spatially explicit prediction of land values per square meter, which
is essentially an extrapolation, ordinary kriging was selected to
generate a linear weighted estimation for land values at unknown
locations from the data set [5], [6], [7].
In kriging, the spatial dependence structure is modeled through
a semivariogram, a function that relates the mean semivariance
(the squared dierences in the Z value for pairs of locations xi, for
locations with known Z values) for all N pairs of locations within
a range of distances h [1],[6],[7]:
The empirical semivariogram is tted by a function with a
specied form; for the GAM, [1] proposed a spheric adjustment;
the gstat package can determine the optimal parameters for this
function, based on the data [8].
( ) =( ) [ ( +) ( )]
( )
(1)
Under the kriging method, the predicted Ẑ values for x
i
locations with
unknown land values result from a weighted average of locations
(with weights wi) with known land value, Z(xi):
PÉREZ, VARGAS: Uncertainty in Land Value Modeling of the San José, Costa Rica. 45
=| ( ) ( )|
(2)
Ordinary kriging chooses the optimal weights by minimizing
kriging variance, which can be determined from the semivariogram
model [5],[6].
The uncertainty estimate for the extrapolation was
determined using sequential Gaussian simulation, which is a
technique to systematically simulate realizations of a random eld.
Given a semivariogram from the data and a random path through
all locations with no known values (such that each location is
only visited once), the sequential Gaussian simulation algorithm
proceeds as follows: (1) it searches for all sampled data and
for all previously simulated locations, (2) it applies kriging to
neighboring points and determines from it the linear estimate
and its variance, (3) sample the value from a normal distribution
with mean and variance from the kriging of the previous step, (4)
assign the sampled value to the location and proceed along the
random path to the next location [5]. The simulation is termed
conditional because it is conditioned on the data, via the kriging.
One hundred instances of land value patterns were simulated
using conditional Gaussian simulation; at each instance, only the
closest 320 points to the location being extrapolated (approximately
10 % of the calibration data) were considered in the kriging. Data
on each simulated instance were back-transformed into their
original units. Based on these simulations, the following metrics
were reported: (1) the E-Type prediction, which is the per location
(i.e., ensemble) mean of the land value [5], (2) the per location
standard deviation and coecient of variation (the coecient of
variation is the ratio of standard deviation to mean), and (3) the
per location 95th and 5th percentiles, as plausible bounds within
which the actual land value should be found. Calculations were
performed using the gstat package [8] of statistical software R [9].
2.3 The determinants of uncertainty: an exploration of social
and physical factors
The pattern of the simulated standard deviations was
analyzed to explore its association with other possible spatial
factors, in line with the objectives proposed. The variance follows
a χ2 distribution. Therefore, to nd the statistical signicance of
the variation in the standard deviation associated to any given
factor, the following approach was employed: for all locations
in the prediction space, (1) histograms were computed for each
spatial factor and the locations were classied into three separate
groups based on limits dened by changes in the histograms of
the spatial factor; (2) a Kolmogorov-Smirnov non-parametric test
was conducted to determine whether the statistical distribution of
the standard deviation of any group was dierent from each of
the other groups (for each factor separately); (3) smooth kernel
densities were estimated for each group (using the geom_density()
function of the package ggplot [10] from R [9]); these densities
were compared. The relative positioning of the dierent kernel
densities (determined by the group) was interpreted to understand
how the factor aected the standard deviation. The general
expectation was that factors associated with greater suitability
for urban land use such as atter terrain and greater accessibility
would present less uncertainty, as they should also be correlated
with greater density of locations with known land values [11].
Following [12], the Kolmogorov-Smirnov test is the most
common instrument to explore the hypothesis of whether two
samples are taken from the same statistical distribution. Taking
two samples, x1, …, xm and y1, …, yn from two distribution
functions, F and G, one may form the empirical distribution
functions Fm: =| {xi: xi ≤ x} |/m and Gn: =| {yi: yi ≤ y} |/n. The test
statistic for the null hypothesis that F = G is given by:
^=( )
(3)
which should be contrasted using probabilities from the cumulative
Kolmogorov distribution [12].
3. RESULTS
As was described, 100 instances of the logarithm of land
values in the GAM were simulated (their back-transformed mean,
95th and 5th percentiles are reported in Fig. 2).
Two important ndings can be seen in Fig. 2, perhaps the
most important is reported in panels.
(d), (e), and (f): these summarize the uncertainty of the
E-Type estimates. The pattern of the coecient of variation is,
approximately, the reverse of the land value predictions (most
clearly seen when compared with the E-Type prediction of panel
(a)). When contrasted with the density of points (Fig. 1, panel
(a)), there is also a clear association. However, the eect of the
algorithm can be seen in that the yellow area of low coecient
of variation extends beyond the more urbanized area predicted
by large land values (the darker blue and purple intervals of Fig.
2, panel (a), which are also the areas with most sale listings in
Fig. 1, panel (a)). The coecient of variation mostly predicts
standard deviations varying between 36 % and 65 % of the mean,
suggesting adequately precise measurements (relatively low
dispersion in the simulated instances); their distribution tends
to be right skewed within this range (shown in the histogram of
Fig. 2, panel (d)).
PÉREZ, VARGAS: Uncertainty in Land Value Modeling of the San José, Costa Rica. 46
Fig. 2. Conditional Gaussian Simulation of land value in the GAM (USD per square meter). (a) E-Type predic-
tion (cell-wise mean of simulated instances), (b) 95th percentile of simulated instances, (c) 5th percentile of
simulated instances, (d) location-wise coecient of variation histogram, (e) descriptive statistics of simulated
predictions and data, (f) coecient of variation map.
The second relevant nding of Fig. 2 corresponds to the
interpretation of panels (a), (b), and (c): in eect, the E-Type
prediction (of panel (a)) is the best estimate of land value; panels
(b) and (c) represent higher and lower bound values for this
prediction: the land value for 90 % of simulated instances was
estimated to lie within the range for each location. An examination
of this more detailed set of maps suggests uncertainties may be
larger than what the overall measures (of validation, discussed
and reported in TABLE I, and of the coecient of variation)
had suggested. By comparing location-wise, for most locations,
the upper (95th percentile) or lower bound (5th percentile) shift
one category. Given the values involved, this represents around
double the back-transformed mean value.
It is also important to understand the quality of the
predictions. TABLE I summarizes the validation exercise results.
Following the methodological approach, 10 % of the lots data
were reserved for validation. For these locations, predictions of
land value were generated using (a) the E-Type prediction of
the conditional Gaussian simulation (the location-wise mean
of all instances) and (b) an ordinary kriging extrapolation, as
benchmark. The error was calculated by subtracting the land
value per square meter (of the data set) from the back-transformed
predicted value. Examining their absolute values, in general, the
error terms showed both models underestimated the actual land
value.
As can be seen in TABLE I, the E-Type land value prediction
is slightly worse than the ordinary kriging: all estimates of
error of the E-Type prediction are somewhat larger than the
corresponding value for ordinary kriging, as is the range is also
smaller. The dierences are very small, in general (at least an
order of magnitude smaller than the error estimate). It is also
worth pointing out that both models have produced very accurate
PÉREZ, VARGAS: Uncertainty in Land Value Modeling of the San José, Costa Rica. 47
predictions: all mean and median error and RMSE estimates are
all less than half of the variable mean (for the land value per
square meter of the validation data set, which is reported in Fig. 2).
The predicted pattern of land values per square meter is
shown in Fig. 2. The pattern coincides with theoretical expectations
and indeed with previous, kriging-based analysis of land values
in the GAM from [1]: land values are larger (shown in dark
blue color) for the centers of San José and Heredia, and the
centers of Alajuela and Cartago are also relatively larger than
their surroundings. Furthermore, lower values are concentrated
on the periphery of the region (rural areas) and the northern zones
of Alajuela and Heredia tend to exhibit larger values than those
of Cartago and San José.
The second objective of this paper, following the estimation
of uncertainties, is to explore if these uncertainties respond
to regularities in space. To do so, ve factors that determine
suitability for urban development (and, in consequence, are related
to land price formation in urban markets) were considered: slope
and elevation, and (Euclidean) distance to the CBD, to the nearest
municipal center and to the nearest main road. For each factor,
three groups of locations were created (except for elevation, for
which only two groups were dened) based on the factor value;
group intervals were generally dened based on the variable
histograms, although for slope, the group limits are related to
statutory building requirements.
TABLE I
VALIDATION OF PREDICTION MODELS OF LAND
PRICE (USD per square meter)
Error measure
Prediction model
Conditional Gaussian
Simulation (E-Type)
Ordinary
Kriging
Root Mean Square Error 149.4 127.7
Mean Absolute Error 110.7 84.3
Median Absolute Error 82.5 56.8
Range of Error -569.2 – 677.0 -437.3 – 765.1
TABLE II
KOLMOGOROV-SMIRNOV STATISTICS FOR
DISTRIBUTION OF STANDARD DEVIATION
FOR DEVELOPED PREDICTIONS GROUPED BY
DETERMINANTS
Comparison D Statistic Prob.
Slope
G1: <30 % vs. G2:>30 % & <50 % 0.110 <0.01
G1: <30 % vs. G3:>50 % 0.137 <0.01
G2:>30 % & <50 % vs. G3:>50 % 0.117 <0.01
Elevation
G1: <1500 masl G2: >1500 masl 0.193 <0.01
Comparison D Statistic Prob.
Distance to CBD
G1: <10 km vs. G2: >10 km & < 25 km 0.412 <0.01
G1: <10 km vs. G3: > 25 km 0.583 <0.01
G2: >10 km & < 25 km vs. G3: > 25 km 0.178 <0.01
Distance to nearest municipal center
G1: <2.5 km vs. G2: >2.5 km & < 7.5 km 0.364 <0.01
G1: <2.5 km vs. G3: > 7.5 km 0.362 <0.01
G2: >2.5 km & < 7.5 km vs. G3: > 7.5 km 0.120 <0.01
Distance to nearest main road
G1: <1 km vs. G2: >1 km & < 7.5 km 0.268 <0.01
G1: <1 km vs. G3: > 7.5 km 0.409 <0.01
G2: >1 km & < 7.5 km vs. G3: > 7.5 km 0.160 <0.01
The Kolmogorov-Smirnov test was used to explore whether
the statistical distribution of standard deviation for each group was
dierent from other groups for the same factor. These results are
shown in TABLE II and, as should have been expected, all test
statistics conrmed the distribution of data of one group is distinct
from other groups. On the one hand, there are sucient simulated
locations (over 28000) for even small dierences to be signicant.
On the other, the larger probability of urbanization associated
with atter zones with greater accessibility to urban centralities
is also associated with both the point pattern of real estate sales
listings [11] –i.e., the sampling density, a key determinant of
uncertainty—and the land value itself.
PÉREZ, VARGAS: Uncertainty in Land Value Modeling of the San José, Costa Rica. 48
Fig. 3. Empirical cumulative distribution functions for standard deviation
of simulated predictions (square of the log. of USD per square meter). Loca-
tions grouped by (a) slope, (b) Euclidean distance to CBD, (c) Euclidean
distance to main roads, (d) elevation, and (e) Euclidean distance to nearest
municipal center.
How each factor aects uncertainty (measured by the
standard deviation of land value of the simulated instances)
suggests urban areas have more diverse land values than zones less
suitable for urban uses. Fig. 3 shows kernel smoothed empirical
distribution densities for the location-wise standard deviation of
simulated instances, grouped by the categories that were used
in constructing TABLE II. The steeper locations (slopes greater
than 50 %) have distinctly larger uncertainty (a sharper peak at
higher value of the distribution) than other groups. This same
pattern is repeated for all variables: greater accessibilities to urban
centralities (the CBD, the nearest municipal center) or the regional
transportation network (main roads), represented by the pink
density function estimate, have all lower peaks at the lower end of
the standard deviation values, suggesting more dispersed values.
In general, the intermediate group of factor values (shown in light
green, Fig. 3) presents intermediate levels of uncertainty and the
group of larger factor values (light blue, Fig. 3), lower levels of
uncertainty (the density functions for intermediate groups are less
right skewed than those for the larger groups of factor values).
Fig. 4. Spatial patterns of determinants of uncertainty. (a) Slope, (b) Euclid-
ean distance to CBD, (c) Euclidean distance to main roads, (d) elevation,
and (e) Euclidean distance to nearest municipal center.
4. SYNTHESIS AND DISCUSSION
The analysis of land value patterns extended previous results
and it has provided further insights, which have contributed
to identify both needs for further study and opportunities for
applications to public policy.
The E-Type prediction from the conditional Gaussian
simulation was found to marginally improve on ordinary kriging
methods. The conditional Gaussian simulation produced, for
validation data, slightly better error measures (RMSE, mean,
median, and range of error) than ordinary kriging (in the analysis
of variations in kriging methods conducted by [1], the dierent
methods tested also resulted in very similar error levels for
validation). This result is indeed not surprising, as the simulations
are conditional on the variogram, and should more iterations had
been simulated, the dierence would have likely been smaller.
On the other hand, in so far as improvements were generated by
the simulations, they were likely related to the improvement of
PÉREZ, VARGAS: Uncertainty in Land Value Modeling of the San José, Costa Rica. 49
over-smoothing limitations in the kriging predictions [5]; but
even this feature could have likely been incorporated into the
kriging by a careful consideration on the number of neighboring
points determining a prediction. Previous exercises of kriging
models did nd limitations due to this over-smoothing problem
that seem to have been improved on by the sequential Gaussian
simulation method (in particular, by better modeling the local
changes at the peri-urban interface of the region); further work
on this issue seems promising.
A distinct advantage of conditional Gaussian simulation is
the spatially explicit measures of uncertainty that can be used
to explore the limitations of the prediction and to more easily
estimate exceedance probabilities [13]; this feature is especially
useful for land value maps in applied scenarios (for example, when
the map predicts land values claimed to be too large by a land
owner, this claim can be easily tested). Further work is required
on this issue (previous comparisons of models estimated from this
data and other data sources suggest systematic underestimation of
land values, particularly for taxation purposes [3]; while outdated
assessments are the simplest explanation, it is also possible that
data sources for the models reported in this paper may be also
partially skewing the results).
It is further worth noting that the literature has detected over-
smoothing problems associated with deterministic methods such as
ordinary kriging that can be overcome with simulation. The current
focus of this study was not the comparison of conditional Gaussian
simulation with other extrapolation predictions; however, this is
regarded as a potential area for further investigation.
The estimated uncertainty patterns are inversely related
to the predicted land value. A very clear and negative spatial
association was identied between the E-Type prediction of
land values per square meter and its standard deviation: in the
urban central area of the GAM, the highest land values (which
coincides both with previous analysis [1] and with theoretical
expectations from urban economics) and lowest uncertainties
were observed. This nding coincides with previous analysis
of the point pattern of real estate listings and its relation to the
determinants of suitability for urban land uses [11].
Indeed, the estimated uncertainty was found to decrease with
characteristics that identify suitability for urban land use (and thus
higher land values). The atter areas of the GAM, which are also
closer to urban centralities (the CBD, main municipal centers),
showed much less uncertainty (smaller location-wise standard
deviation) than zones further away and at higher elevations and
steeper slopes. Therefore, the data set and modeling eorts appear
to demonstrate eciency when predicting urban land values but
also present clear limitations if applied to rural land uses of the
urban periphery.
Despite its importance, hardly any previous case study
reports the use of simulation to understand uncertainty introduced
by interpolation into land or property value predictions (unlike
physical properties of soils, which are derived from similar point
data and for which such analysis seems common). Uncertainty has
been reported as variance of kriging estimates [1] or verication
through out-of-sample prediction [14], in relation to the mean
estimate from this indicator. While theoretical recognition of the
possibility to estimate errors and uncertainty in the context of
land valuation has been acknowledged [15], actual practice has
centered on the accuracy of the mean prediction rather than on
explaining its variance. Uncertainty is important for valuation,
especially when practical applications are performed (such as tax
assessments and potential challenges to these).
In conclusion, the analysis of uncertainties may be critical
for improving urban and regional studies (e.g., the impact of
new infrastructure or of land use regulations) and land value
assessments for tax policy. In this regard, the methods presented
have increased robustness (relative to very local estimates)
because predictions relatively far away from locations with known
values may still benet from their price information via the spatial
dependence encoded in the semivariogram. More importantly, the
estimates of uncertainty permit the assessment of the prediction
for properties that have not been recently sold in the market (and
thus include an inherent check of the prediction which is absent
in isolated tax assessment exercises).
ROLES
Eduardo Pérez Molina: Conceptualization, Methodology,
Software, Formal analysis, Writing - Original Draft
Darío Vargas Aguilar: Data Curation, Writing - Review
& Editing
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