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WORKING PAPER
Latent segmentation of urban space through residential
location choice
Tomás Cox Oettinger (1)(2)(4)
Ricardo Hurtubia González (2)(3)(4)(5)
1. Department of Urbanism, Universidad de Chile.
2. Department of Transport Engineering and Logistics, Pontificia Universidad Católica de Chile.
3. School of Architecture, Pontificia Universidad Católica de Chile.
4. Centre for Sustainable Urban Development (CEDEUS)
5. Complex Engineering Systems Institute (ISCI)
Contact information: tcox2@uc.cl / rhg@ing.puc.cl
Vicuña Mackenna 4860, Macul, Santiago, Chile / (56 2) 2 354 4270
Abstract
Understanding the preferences of households in their location decisions is key for residential
demand forecast and urban policy making. Accounting for preference heterogeneity across agents
is key for the modelling process but, we argue, not enough to completely describe location choice
behavior. Due to place-specific conditions, the same agent may have different preferences
depending on the sector of the city considered as potential location, a phenomena known as spatial
heterogeneity. Segmenting the city by defining zones where agents are supposed to behave
similarly has been a common modelling solution, assigning different zonal preference parameters
in the estimation process. This has been usually done with two-step methods, where spatial
segmentation is done independently of the location choice process, something that could bias
estimation results. We propose and test a one-step model for simultaneous estimation of location
preference parameters and spatial segmentation, therefore accounting for heterogeneity across
agents and space. The model is based on Ellickson´s bid-auction approach for location choice and
latent class models. We test our model with a case study in Santiago, Chile and compare it with
other models for spatial segmentation. In terms of predictive power, our approach outperforms a
model with no zones, a model with zones defined exogenously, and a clustering-based two-step
model. This novel approach allows for a better conceptual ground for urban predictive models with
spatial segmentation.
Keywords: Spatial Heterogeneity, Location Choice Models, Latent Classes.
1. Introduction
Modelling households’ location decisions is key to understand past and future patterns of urban growth and
change, which helps to plan transport and services infrastructure, and to design policies that guide towards
a better and more sustainable urban development.
The work by Alonso (1964) was the first to model the spatial distribution of different types of households
according to their specific willingness to pay (WP) for a location as a function of its attributes, such as
accessibility and built surface. In this model, each location is assumed to be auctioned, and the household
with the highest bid (correlated with its WP) “wins” the location. This defines both the spatial distribution
of households and the prices of real estate goods. Most present models of location choice are based on later
formulations by McFadden (1978) and by Ellickson (1981). In both approaches, households evaluate each
location in terms of its attributes and dwelling characteristics, and their probability of choosing the location
depends on this evaluation (a utility function in McFadden’s and a WP function in Ellickson’s).
In location choice models, WP functions are generally based on the interaction between a vector of location
attributes and agent characteristics, and a vector of unobserved parameters that represent the marginal
contribution of each attribute and characteristic to the WP. These parameters are specific to each type of
agent and are modelled as to represent their preferences in the choice process. Besides demographic
heterogeneity we can also find heterogeneity across (types of) places, known as spatial heterogeneity. As
demographic or agent heterogeneity can be included by segmenting agents according to their
characteristics, also spatial heterogeneity can be included by segmenting locations according to their spatial
attributes and assigning segment-specific preferences.
In this paper we argue that spatial segmentation and how it affects valuation of attributes (preferences) is a
complex issue, as it is part of the spatial cognition of city dwellers. If we want to achieve a spatial
segmentation that maximizes the likelihood that parameters are representative of preferences in each zone,
we cannot predefine zones only from differences in built or urban attributes as it is done, for example, in
cluster analysis (Jain, 2010; MacQueen, 1967). Location preferences should play an active role in the spatial
segmentation process, something that can’t be achieved when segmentation is defined before the estimation
of preference parameters, but can be done with a joint estimation of both spatial segmentation and
preferences.
In order to do so, we propose a model based on the bid-auction approach, where agents not only have a WP
function, but also have an heterogeneous perception of urban space which can be described by a spatial
segmentation function, with parameters that are estimated jointly with the WP parameters, following a
latent class modelling approach (Kamakura & Russell, 1989).
This model presents a novel simultaneous approach to spatial segmentation and location choice, which we
affirm is ahead of previous (two-step) zone-based segmentation methodologies, thanks to zones defined by
a classification function based on parameters that are estimated in order to maximize the likelihood of
reproducing the phenomenon. This method is also behavior-based, in a model formulation that follows
agent segmentation process and is consistent with microeconomic theory. We apply the model to household
location data for Santiago de Chile and compare results with those obtained when using a model with no
segmentation, and other two models where segmentation is done in a first step (exogenous zones and
cluster-based zones). Model comparison is done using a validation subsample.
The paper is structured in five sections. After this introduction, section two discusses the issue of spatial
heterogeneity in general terms; section three details the proposed modelling framework; section four
explains the mathematical formulation of the bid-auction localization model. Section five presents the
proposed model, conceptually and mathematically. Section six presents the data and implementation of the
proposed model. Part seven presents the results and the comparison with other approaches. Part eight
concludes the paper.
2. Sources and methods for spatial heterogeneity
In location choice models, heterogeneity is dealt with when the modeler accounts for different behavior for
different types of agents, understanding that people have a complex nature and that a general rule or set of
drivers is not enough. The differences in behavior under the same conditions, for different types of people,
is also observed across space. Spatial heterogeneity means that model parameters are not stationary across
space (Anselin, 1988) which means that, for different reasons, the same individual facing the same
conditions will behave different in different parts of the territory.
2.1. Sources of Spatial Heterogeneity
Fotheringham et al. (2002) identify three reasons for non-stationary parameters in space. Two of them
belong to modelling shortcomings: one related to the possibility of data samples being different across
space and the other to the existence of non-observed variables that are correlated with spatial variations.
The third one is proper to the spatial phenomena itself, related to contextual effects that affect the valuation
of location attributes by individuals. The fact that the same person could react in different ways to the same
stimuli, depending on the location of the city where she or he stands, provides evidence for the existence
of some underlying qualitative aspects of places that interact with observed attributes. Neighborhood effects
(Becker & Murphy, 2009; Durlauf, 2003; Sampson, Morenoff, & Gannon-Rowley, 2002) may explain the
synergies among certain urban attributes that can act as a multiplier of their effects on behavior. As
traditional model parameters represent the marginal effect of one additional unit of an attribute in the level
of the explained variable, it is natural to think that this effect is not constant, as it can be sensitive (or
relative) to the levels of other urban attributes in the same location.
Different levels of urban attributes can represent states of saturation or scarceness of that attribute. In places
where the attribute is abundant, it is possible that the valuation of that attribute is lower than in places where
the attribute is scarce, meaning the preference parameter for that attribute should vary across space. For
example, the same additional square meter of green area can have a greater effect (larger parameter value)
if the location in question has a high built density (and therefore scarcer green areas) than if it’s a low
density location. In a complex system as a city, attributes interact in complex ways, so it is reasonable to
assume that the effect of urban attributes on behavior is not isolated from the magnitude of other attributes
(Abbott, 1997).
One of the first systematic works exploring how neighborhoods are recognized and affect behavior was the
research by Lynch (1960), which surveyed inhabitants of three cities in the US to obtain maps of how they
perceived their neighborhoods. Lynch identified five elements (zones, barriers, paths, milestones, nodes)
that people can recognize as characteristic of a city and that are related to the reading that people make to
orient themselves and be able to "navigate" through the city; a concept that was later investigated as mental
maps applied to space (Gould & White, 1974). Other research (Nasar, 1990; Salesses, Schechtner, &
Hidalgo, 2013) has identified how people not only identify sectors, but also apply different valuations based
on their urban attributes. This is consistent with general theories from psychology (for example Gestalt
(Wertheimer & Riezler, 1944), fragmentation or chunking (Gobet et al., 2001; Miller, 1956), mental models
(Johnson-Laird, 2010)) and indicates that people tend to group or add information in elements to simplify
the abundant information of the context, and be able to handle it efficiently.
2.2. Methods for identifying spatial heterogeneity
Spatial heterogeneity is a special case of heterogeneity in general, for which we can find an early example
in Quandt (1958), who used different functions in a linear regression for different subsets of observations.
There are different technics to structure the variation of parameters across space. The simplest one, that can
be usually seen in hedonic price (Rosen, 1974) and location choice models, is to use an exogenous
zonification (administrative or functional mainly), where each zone has a different set of parameters
corresponding to the observations in the zone. Exogenous zonification, however, presents the shortcoming
of the Modifiable Areal Unit Problem (MAUP) (Openshaw, 1984), which recognizes that zone-based
spatial analysis can have different outcomes depending on the zonification used. This means that, since
exogenous segmentation can be arbitrary, so can be the results.
Some techniques address this issue by estimating location-specific parameters, running a regression for
each zone or area only using observations within a distance (Moving windows regression, MWR) (Chica-
Olmo, 1995; Dubin, 1992) or using decreasing weights for observations depending on distance to location
(Geographically weighted regression, GWR) (Fotheringham et al., 2002; Páez, Long, & Farber, 2008).
These methods have the advantage of not having to rely on arbitrary zones (but the size of the moving
window and the decreasing weights function can be arbitrary).
2.3. Housing submarkets as a form of spatial heterogeneity
One approach to spatial heterogeneity is the study of submarkets in the literature of hedonic price models.
These models characterize the price of a real state as a function of its attributes but, since the studies of
Schnare and Struyk (1976) and Palm (1978), it has been recognized that one single price function is not
representative of the heterogeneity of different zones of the city (see also Adair, Berry, & McGreal, 1996;
Galster, 1996; Goodman & Thibodeau, 1998; Watkins, 2001). Several studies (for a recent reference, see
for example Jang & Kang, 2015), have used functional zones such as center and peripheries, each one with
a different price function which is estimated independently. Zonification is used to estimate different
preference parameters or marginal prices specific to each zone, accounting for heterogeneity in the spatial
structure of the city. But, as discussed before, exogenous or ex ante defined zones does not necessarily
resemble the real change in preference behavior of agents in the city.
A more sophisticated definition of zones has been achieved with two-step techniques, where the first step
uses attribute-based aggregation methods, such as Principal Component Analysis or Clustering to identify
homogenous areas in terms of spatial attributes or homogenous sets of housing units in terms of their built
characteristics (Bourassa, Hamelink, Hoesli, & Macgregor, 1999; Rosmera & Lizam, 2016). Other two-
step methods are based, for example, in Space Syntax to define submarkets (Xiao, 2017), but this last one
is basically analogous to clustering methods, using road network attributes instead of built or land use
attributes. In a second step, modelers run a regression with different parameters for each of the homogenous
zones defined in the first step. This approximation generates better results than exogenous zones, but we
argue that homogeneity in attributes is not the same as homogeneity in preferences. A two-step method
cannot be used to identify preference-based homogeneous zones, as preferences can only be identified in
the second step.
Besides from this problem (or maybe related), it has been shown that estimation-based, two-step definitions
of submarkets are unable to outperform models using zones based on submarkets defined by real estate
agents (Bourassa, Hoesli, & Peng, 2003; Keskin & Watkins, 2017).
3. Problem and proposed model: simultaneous estimation
We argue that in order to define zones, or submarkets, that really group similar preferences, a simultaneous
estimation method (one step) has to be used. The proposed simultaneous estimation is based on defining
two sets of parameters: submarket-specific preference parameters, and submarket classification function
parameters. Both sets of parameters are estimated jointly to better capture the phenomena and reduce bias
(Ben-Akiva et al., 2002).
We propose that this joint estimation can be achieved for location choice models by using Latent Class
Models (LCM) (Kamakura & Russell, 1989). These models estimate the probability of individuals
belonging to a certain class of decision maker as a function of her characteristics, while simultaneously
estimating the preference parameters for each of the classes considered in the model.
LCM have been widely used to model heterogeneity in preferences for location choice across decision
makers (Cox & Hurtubia, 2019; Ettema, 2010; Liao, Farber, & Ewing, 2014; Lu, Southworth, Crittenden,
& Dunhum-Jones, 2014; Olaru, Smith, & Taplin, 2011; Walker & Li, 2007). The cited literature uses latent
classes to identify classes of households and real estate developers or, in general terms, agents that search
for locations, based in a traditional choice framework as described by McFadden (1978). LCM allows
estimating a different set of preference parameters for each class of agent. The probability of choosing a
location is a total probability that considers the probabilities of being part of each class, and the probabilities
of choosing that location conditional on belonging to a particular class.
In the authors’ knowledge, LCM have only been applied in location choice models under a “traditional”
choice approach, in which the classes segment households and give a different set of parameters to each
class. However, the framework has not been applied in the context of a bid auction approach which,
although mathematically analogous to the choice approach in its formulation, has a totally different
interpretation and allows for the introduction of endogenous heterogeneity in preferences across space. In
this matter, the authors have found only one reference of spatial segmentation using LCM (Sarrias, 2019),
who evaluated changes in subjective evaluations of well-being, but not directly applied to location choice
and with far lesser detail in spatial attributes used for segmentation.
4. Bid-auction approach in Location choice models
Location choice models are based on the assumption of agents facing a set of location alternatives. Each
alternative reports a utility, which depends on the alternative attributes, agent characteristics, and a set of
preference parameters. Building on McFadden’s (1978) choice model, and Alonso’s (1964) work, Ellickson
(1981) proposed the bid auction approach which is appropriated for location decisions in a households and
real estate market interaction.
The bid auction model can be derived from a utility maximization problem, in which an agent chooses a
location from a set of different locations in the city, trading off with consumption of other goods ().
Besides consumed goods, the agent’s utility depends on a vector of preference parameters and a vector
of attributes of each location . A budget constraint is added, in which rent for location plus expenditure
in other goods (priced at ) have to be equal or less than the agents´ available income .
(1)
Assuming equality to clear from the budget constraint and replacing it in , we obtain an indirect utility
function () and the utility maximization problem simplifies to choosing the location that maximizes :
(2)
Considering a fixed referential maximum utility level
(expected by the agent), we can clear the rent
which, if assumed endogenous, represents the willingness to pay (WP) of the agent for that location, in
order to reach the reference utility (Jara-Díaz & Martínez, 1999). Endogenous rent then becomes the
willingness to pay agent h for location ():
(3)
If the utility function of (1) has a (quasi) linear form, the willingness to pay function can be simplified and
written in terms of two components: one specific to the agent, related to the income level and expected
maximum utility, and one related to the preferences the agent has for attribute location (Martinez, 2000):
(4)
Assuming an i.i.d Gumbell distributed error term associated to the (accounting for unobserved
attributes) the probability that agent is the highest bidder for location , and therefore locates there, is
defined by a logit function where is non-identifiable a scale parameter (McFadden, 1973):
(5)
From a sample of located agents (segmented by type) and the attributes of their location, and using
maximum likelihood estimation, this model can identify, for each type of agent, the marginal WP for each
attribute considered in the WP function.
The bid auction approach has been used in several Transport and Land Use Interaction (LUTI) models such
as MUSSA (Martínez, 1996), ILUTE (Salvini & Miller, 2005) and IRPUD (Wegener, 2011). Several
research papers on the literature about location choice also use this approach (Chattopadhyay, 1998; Gross,
Sirmans, & Benjamin, 1990; Hurtubia & Bierlaire, 2014; Hurtubia, Martinez, & Bierlaire, 2019; Muto,
2006)).
5. Proposed latent spatial-segmentation model
The proposed model applies a latent class model framework (LCM) to a bid-auction location choice model.
Mathematically, applying LCM to bid-auction framework is relatively similar to doing so for a choice
framework, but the interpretation and application differs in substantial aspects.
Figure 1: Latent Classes applied in a bid-auction framework (source: the authors).
In a bid-auction model with latent classes, the class membership function applies to the location, understood
as the agent that receives the bids (e.g the owner of the property or the land). Therefore, the class-specific
preference parameters can be interpreted as the location-seeking agents (households and firms) having a
different valuation of urban attributes conditional to the class of the location.
In simple terms, an agent will valuate differently an amenity depending on the class of the location. If the
distribution of the magnitudes of the attributes is somehow continuous in the city (which indeed happens,
due to spatial dependence; see Anselin, 1988) classes of locations can be related to neighborhoods.
Departing from the bid-auction model presented above, we modify equation (5) so the probability
is conditional to each class of locations or submarkets:
(6)
Each agent will have a different depending on the submarket or class of the location where they are
bidding to, because the is function of a set of preferences parameters which are conditional to the
class of location.
Simultaneously, each location will have a probability of belonging to a class which, according to the
standard formulation of LCMs, is a logit probability based on a class membership function for which
we assume an error term i.i.d Gumbell and a non-identifiable scale parameter :
(7)
As we are segmenting space into zones or neighborhoods, the class membership function depends on
location attributes , instead of agents characteristics, unlike previous applications of LCM to location
choice (see for example Walker and Li (2007) and Hoshino (2011)). A vector of parameters is estimated,
which represent the marginal contribution of each location attribute to the probability of belonging to a
class.
Given the probability that agent gets the location , conditional to the class of the location, for all agents
and locations (eq 6), and also the probability that location belongs to class , for all locations and classes
(eq 7), the probability that agent h gets location , unconditional to class membership is:
(8)
Using equation (8), maximum likelihood estimation can be used to identify parameters and from
observations of agents’ location decisions, without requiring any information regarding submarket
structure. This approach avoids an ex-ante definition of the membership of locations to submarkets and,
instead, infers how agents’ perceive locations as part of a submarket, and accordingly variate their
preferences.
In this specific model implementation, we use an estimation method proposed by Lerman & Kern (1983),
where the maximum likelihood is not only targeting to reproduce the actual localizations of agents, but also
minimizing the difference between winning WP and observed price (in this case monthly rent) paid by the
located agent. We use this method because it allows to identify the scale parameter for each class () and,
therefore, enables the identification of parameters for all type of agents (otherwise parameters for one type
of agent have to be fixed and the other parameters estimated relative to them). This scaling of parameters
renders estimated values of WP with the same magnitude as observed prices.
As we are including latent classes, first we have to specify a Lerman & Kern likelihood function specific
to each class, and then specify a final likelihood function considering the probabilities of each class.
Equation (9) shows the likelihood function conditional to each class , where is the observed rent in
the location ,
is the highest bid modelled for location conditional to that location being part of class
, and is the scale parameter of the class-specific logit function. This function is calculated for each
location .
(9)
The final likelihood function to be maximized, not conditional to class, is:
(10)
The likelihood function is basically maximizing the joint probability that, for each location, the winning
agent has the highest bid, and that the highest bid is equal to the observed rent.
6. Application to Santiago case study
The proposed model was tested with a database of households from the 2012 Origin Destination Survey for
Santiago (SECTRA, 2015), each with socio economic variables and exact georeference. Location attributes
were calculated for each location using a Geographical Information System (GIS). With this information,
besides the proposed model, a base model (with no spatial heterogeneity) and two other alternative
approaches (cluster-based zones and administrative zones) were estimated for comparison purposes. Direct
likelihood was measured with a validation sample, for the proposed model and the alternative approaches,
in order to compare predictive power.
6.1. Urban structure of Santiago
The spatial structure of Santiago depends on its particular history and national hierarchy. With 6,123,000
inhabitants (INE, 2018), it is the main city of Chile in terms of population, economic activity and
administrative power. Santiago has evolved from a traditional compact city to a fragmented and globalized
city (Borsdorf, 2003), where both densification and expansion development patterns have been observed in
recent decades (Cox & Hurtubia, 2016; Vergara Vidal, 2017)
In terms of urban structure, the city of Santiago answers to the latest stage of the model described by
Borsdorf (2003), where there is a main Central Business District (CBD) based on the historical center, from
which departs a wedge of high income residential areas (towards the north-east in the case of Santiago),
with an spine in its central axis of more modern commercial and office areas (Providencia and Avenida Las
Condes). In the case of Santiago, this commercial spine is attracting more protagonism in later years (Suazo,
2017). Borsdorf’s model describes the location of high income households as very differentiated from low
income households, which locate in broad areas beside industrial corridors. In the case of Santiago, this
areas correspond to north-west, west and south peripheral areas of the city. Also described in this model,
and observed by other authors (de Mattos, 1999; Sabatini & Salcedo, 2010) is the later fragmentation of
this sectorial model towards a more network-based urbanization, with growing system of highways that
connect so-called “globalization artifacts”, such as malls, airport, gated communities and industrial parks,
sprawling on the fringes of the city, many times inserted in but not actually communicated with low income
areas. Although this relatively new “leap-frog” urbanization hasn´t followed the sectorial residential
segregation seen in past decades, the segregation is still being reproduced but in a lower scale (Sabatini,
2015).
If the proposed model adequately reproduces how people perceive city areas when choosing location, this
depiction of the city coming from urban geography should be somehow observed when mapping the
resulting spatial segmentation of the model.
6.2. Data
Household data was extracted from the Santiago 2012 Origin-Destination Survey (SECTRA, 2015),
accounting for 18,624 observations, from which 14,172 were used for estimation (exclusion was based on
lack of some key attributes for some observations). The survey considers 790 zones as its basic spatial
analysis unit, we use these zones to compute some of the attributes describing each location (such as average
income and accessibility measures). Households were segmented into three categories according to the
educational level (EL) of the head of household. Low EL correspond to 0 to 11 years of education
1
, middle
EL to 12 to 15 years of education and high EL to 16 or more years. The map in Figure 2 presents the spatial
distribution of households, as well as the general structure of the city while Table 1 characterizes these
types of households. Table 2 describes the attributes used to describe each location and their sources.
COD
Level
Years of
Education
Number of
Households
%
Lo-EL
Low Educational Level
0 to 11
6620
37.1%
Mid-EL
Middle Educational Level
12 to 15
7774
43.6%
Hi-EL
High Educational Level
16 +
3436
19.3%
TOTAL
17830
1
In Chile, having 12 years of education implies finishing the compulsory high school degree. However, a large part
the population does not achieve this educational level.
Table 1: Segmentation of households according to educational level (EL).
Variable
Unit
Description
Source
Mean
Min
Max
Monthly Rent
Million
CLP
Monthly rent paid by household in
million chilean pesos (CLP)
Origin Destination
Survey (SECTRA, 2012)
0.19
0.01
5
Accessibility to
Industry (transit)
-
Gravitational with negative
exponential function weighted by
industry surface in destination zone.
Own calculation based
on Internal Revenue
Service (2014) and
SECTRA (2015)
1807
33
4536
Accessibility to
commerce
(transit)
-
Gravitational with negative
exponential function weighted by
commerce surface in destination
zone.
Own calculation based
on Internal Revenue
Service (2014) and
SECTRA (2015)
2262
46
6096
Accessibility to
Industry (car)
-
Gravitational with negative
exponential function weighted by
industry surface in destination zone.
Own calculation based
on Internal Revenue
Service (2014) and
SECTRA (2015)
5082
1031
6934
Accessibility to
commerce (car)
-
Gravitational with negative
exponential function weighted by
commerce surface in destination
zone.
Own calculation based
on Internal Revenue
Service (2014) and
SECTRA (2015)
5894
1048
8583
Distance to
closest Subway
Station
km
Euclidian distance from household
to closest subway station as of 2012
Own calculation in QGIS
4.74
0.03
49.84
Distance to
closest highway
exit
km
Euclidian distance from household
to closest highway exit as of 2012
Own calculation in QGIS
2.04
0.03
13.27
Zonal average
income
Million
CLP
Average income of the households
in the OD Zone.
Origin Destination
Survey (SECTRA, 2012)
0.66
0.14
4.95
Built surface
m2
Average built surface of residential
units in the block of the household
Internal Revenue Service
(2014)
31
49.59
207.3
Built density in
zone
coef
Total built surface divided by zone
area.
Internal Revenue Service
(2014)
0.38
0
4.59
Table 2: Variables evaluated for the model. *CLP: Chilean Peso.
Figure 2: Location of Survey Households according to their Educational Level.
One limitation of the database is the lack of information on built surface for the dwelling of each observed
household´s location. As a proxy, we use the average built surface in the zone of the dwelling.
Accessibility for each zone in mode was calculated with a gravitational measure (Hansen, 1959),
following:
(11)
where is the amount of opportunities (e.g. built surface of commerce, or industry) in each of the possible
destination zones,
is the travel time in mode (car or transit) between pair of zones and , and is
an impedance parameter (we used , which gave the higher significance and likelihood in the
estimation stage, while also reproducing observed distributions of trip-lengths). For accessibility to
commerce and industry, built surface of each land use in all of the 790 OD Survey zones was extracted
from the Internal Revenue Service registry for 2014. Travel times by car and transit between each OD
Survey zone was obtained from the strategic transport model for Santiago (ESTRAUS, SECTRA (2016) )
which is calibrated with the same OD Survey travel data.
Figure 3: Maps showing attributes used for the spatial segmentation function (latent class model) and the clusterization. The
attributes are Built density in zone (left) and Average income in zone (right).
6.3. Estimation results
Several specifications for the Willingnes to Pay () and Class Membership () functions were
explored. Attributes that were not significant were left out, unless they were significant for more than one
household type and class, in which case they were kept. A benchmark model was estimated, parallel to the
estimation of the proposed model. The benchmark (base) model uses the same variables, household types
and approach (bid-auction and Lerman & Kern estimation) as the proposed model, but has no segmentation
of locations. Table 3 shows estimation results for both models.
For this implementation of the proposed model, the zonal average income and zonal built density (see
Figure 3) were used to classify the city locations in two different types (classes) of locations. In the base
model these attributes were included directly in the WP function. From the sign of the class membership
parameters, the probability of membership to Class One improves in zones with higher income and density.
Therefore Class One can be labelled as wealthy and dense locations. By opposition, class two corresponds
to sparse and low income locations.
BASE MODEL
(NO CLASSES)
PROPOSED MODEL
(2 CLASSES)
Observations
17830
17830
Null model log-likelihood
-258759
-258759
Final log-
likelihood
-77213
-72709
Attribute
Household
Type
Coefficient (t-test)
Class 1
Class 2
(Dense/Wealthy)
(Sparse/Low Income)
Accessibility to
Commerce by
transit
Low-EL
-0.0000965 (-0.36)*
-0.00438 (-1.51)*
0.000613 (3.65)
Mid-EL
0.00351 (14.57)
0.00329 (3.16)
0.00255 (13.54)
Hi-EL
0.00741 (18.74)
0.0019 (2.05)
0.0054 (8.45)
Low-EL
0.000398 (1.06)*
0.00801 (1.94)*
-0.000453 (-1.89)*
Accessibility to
Industry by
transit
Mid-EL
-0.00429 (-12.49)
-0.0064 (-4.33)
-0.00263 (-9.79)
Hi-EL
-0.0112 (-19.67)
-0.00542 (-3.87)
-0.0054 (-5.81)
Accessibility to
Commerce by
car
Low-EL
0.00109 (3.41)
0.00855 (2.08)
-0.000142 (-0.68)*
Mid-EL
-0.00203 (-7.05)
-0.000289 (-0.22)*
-0.00257 (-11.18)
Hi-EL
0.005 (10.37)
0.00983 (7.68)
-0.00607 (-6.19)
Accessibility to
Industry by car
Low-EL
-0.000958 (-2.42)
-0.0144 (-2.89)
0.000561 (2.18)
Mid-EL
0.00333 (9.02)
0.00158 (1.02)*
0.00355 (12.57)
Hi-EL
-0.00186 (-2.86)
-0.00929 (-5.77)
0.00766 (6.79)
Distance to
nearest subway
station
Low-EL
0.0166 (0.91)*
0.0215 (0.07)*
0.0374 (3.37)
Mid-EL
0.0994 (5.61)
0.167 (1.33)*
0.0691 (5.8)
Hi-EL
0.299 (8.23)
0.868 (8.51)
0.137 (3.92)
Distance to
nearest highway
exit
Low-EL
-0.0385 (-0.74)*
-1.6 (-2.13)
0.0372 (1.19)*
Mid-EL
0.0000959 (0)*
-0.188 (-0.82)*
0.0528 (1.56)*
Hi-EL
-0.00546 (-0.06)*
-0.459 (-2.32)
0.033 (0.29)*
Average Built
surface in zone
Low-EL
0.00696 (1.26)*
0.127 (3.2)
0.0144 (4.48)
Mid-EL
0.0597 (13.23)
0.128 (9.09)
0.0458 (16.02)
Hi-EL
0.146 (27.74)
0.238 (22.88)
0.0627 (8.18)
Average Income
in Zone
Low-EL
-0.0482 (-0.09)*
Mid-EL
7.68 (30.99)
Hi-EL
7.61 (33.93)
Built density in
zone
Low-EL
-2.77 (-6.62)
Mid-EL
0.657 (3.07)
Hi-EL
0.554 (2.4)
Household type
constant
Low-EL
5.92 (7.06)
5.36 (10.43)
Mid-EL
-4.65 (-6.04)
2.13 (3.97)
Hi-EL
-28.7 (-22.04)
-8.66 (-5.92)
Class Membership Variables
Class 1
Class 2
Intercept
-9.22 (-31.85)
Average Income in Zone
10.8 (26.64)
Built density in zone
1.86 (5.87)
0.164 (169.84)
0.0907 (65.96)
0.28 (116.92)
Table 3: Estimation parameters.
Parameters and are the scale parameters for the choice model of each class, identifiable thanks to the
Lerman & Kern (1983) estimation method we used.
Built surface (as said before, we use average built surface in the zone of the dwelling as a proxy for this
variable) is, as expected, a relevant attribute. From the base model, we observe that higher EL households
are willing to pay more than other households for additional built space. But from the latent class model
we can also see that these households are willing pay as much as almost four times more for an additional
square meter, if the location is in a wealthier and denser zone.
Accessibility to commerce, by transit and car, is always positive in the base model but, when including
latent classes, we can some differences. Accessibility by car increases willingness to pay in wealthy areas,
but decreases it in low income/density areas for al types of households (except mid-EL households which
are indifferent to this attribute in dense and wealthy areas). This can be interpreted as households assigning
a positive value to the type of commerce usually observed in wealthy areas, with the opposite occurring in
low income areas. This result is consistent with our expectations, considering the fact that high income
municipalities are capable to minimize or mitigate the negative externalities of commercial activities
(congestion, garbage production, landscape impact of buildings), while lower income municipalities
usually don’t have enough resources to control this.
Being distant to subway stations is always positive, which is expected considering we are including other
variables that account for accessibility. In the latent class model we see that being distant is more important
in wealthier zones than in low income zones, and this difference is more critical for High EL households.
These may be due to reasons similar to those exposed for the commerce case, considering the negative
externalities metro stations can produce in their immediate surroundings
This differences in parameters between classes is consistent with our hypothesis that spatial heterogeneity
in preferences can be captured by using latent classes as different zones in the city. The latent class model
has a significantly better likelihood the benchmark model, which indicates that this new dimension of
heterogeneity introduced helps to better reproduce the location choice phenomenon.
6.4. Spatial distribution of class membership of locations
Once the class membership parameters of the model are estimated, they can be used to evaluate the
probability of each location of belonging to each spatial class all over the city. The map in Figure 4 shows
the spatial distribution of these probabilities, indicating a clear segmentation of the city, consistent with the
well-known socioeconomic segregation patterns of Santiago (Sabatini, 2006). Class 1 locations (blue),
related to dense and high-income zones, are clearly correlated with what is known as the “high income
wedge” of Santiago. However, there are several places with a high probability of belonging to class 1
outside of this wedge, where a combination of density and relatively high income is observed. Most of the
city has a high probability of belonging to class 2 (yellow), which are mostly the extended peripheries, with
low density and medium and low income.
Some isolated zones with high probability of belonging to class 1 are newer private developments, where
medium high income households have located. These projects answer to the typology of gated
communities, locating outside the high income area of the city and being mainly disconnected to their
immediate lower income context (Borsdorf, Hidalgo, & Sánchez, 2007) and many times surrounded by
hills to reinforce their “enclave” condition.
Figure 4: OD Survey Households´ locations and their probability of membership to a wealthy high density zone (class 1 in the
model) according to the proposed model..
It can be seen that few locations are neither yellow nor blue. As it is shown in histogram in figure 5, most
of the locations (55%) fall below the 1% probability of belonging to class 1 while 16% do so for class 2.
This is evidence that perception or valuation of urban attributes in Santiago not only variates for different
zones, but also that this differences have clear cuts, building strong perceptual urban limits.
As the location classification in this model is estimated simultaneously with the model for the location
decision process of each household, this can be interpreted as households perceiving the city as a two clearly
different sets of zones, where they will apply different valuations of urban attributes.
Figure 5: Histogram of the probability of membership to class 1 for every location (households’ location in ODS 2012).
6.5. Comparison to alternative models: Exogenous zones and attribute-based clusters
There are different approaches to include spatial heterogeneity in a location choice model. In order to
compare the effectiveness of the proposed method, we compare its results with those of two alternative
approaches: a model with exogenous zones, and two models with zones based on clustering of location
attributes. We used the same attributes as in the base and proposed model. The same two attributes that
were used for the segmentation function in the latent class model were used for the clustering process in
the cluster models.
For the model of exogenous zones, we use the seven macro-zones of the Origin Destination Survey that we
have used in this work, and we estimated the location model with preference parameters specific to each
zone. Map in figure 6 shows the households colored according to the zone where they belong.
Table 5 (in Annex) shows the estimated preference parameters of each household for each attribute in each
zone, for the exogenous zone model. The principal result to notice is that the likelihood (-76.511) is lower
than the one obtained with the proposed model while, as it can be expected, it´s better than the likelihood
for the base model (with no zones). Parameters are mostly significant at 95%, and with significantly
different values for each zone, indicating that these zones, although defined with no explicit market
considerations, are still capable to define submarkets where preferences are different. Accessibilities by car
where excluded as the model was not able to be estimated with all the attributes, which is probably due to
the high amount of parameters involved.
As for the cluster-based models, they are expected to have better performance than the exogenous zones
model, as locations are grouped following differences in some of their attributes. We used a k-means
method (MacQueen, 1967) for the cluster-based models, where each possible location was assigned to one
cluster (we estimated two models: one with seven clusters, in order to compare with exogenous zones
model, and one with two clusters, similar to the number of classes in the proposed model). The assignation
criteria was to group locations with similar level of two attributes: Zonal Average Income and Built Density
(the same attributes used in the proposed model to generate the latent classes). ¡Error! No se encuentra el
origen de la referencia. shows the spatial distribution of the clusters.
Table 6 and 7 (in Annex) shows the estimation results for these model. The log-likelihood of these two
models is higher than the one for the exogenous model and the base model (no zones), but still is not higher
than the one for the proposed model. Most of the parameters are significant at a 95% level and show
differences between zones, indicating they do represent different zones.
39%
16%
9% 3% 3% 2% 2% 2% 1% 1% 0% 1% 1% 0% 1% 1% 1% 1% 3%
13%
0
2000
4000
6000
8000
0 - 0.05
0.05 - 0.1
0.1 - 0.15
0.15 - 0.2
0.2 - 0.25
0.25 - 0.3
0.3 - 0.35
0.35 - 0.4
0.4 - 0.45
0.45 - 0.5
0.5 - 0.55
0.55 - 0.6
0.6 - 0.65
0.65 - 0.7
0.7 - 0.75
0.75 - 0.8
0.8 - 0.85
0.85 - 0.9
0.9 - 0.95
0.95 - 1
Frecuency (# locations)
Probability of membership to Class 1 (Wealthy Dense)
Distribution of Probability of merbership to Class 1
(Wealthy Dense)
Figure 6: Survey Households colored according to the Survey zone they belong (zones used to estimate the exogenous zones
model).
Figure 7: Survey households colored according to the cluster they belong (clusters used to estimate the seven clusters model).
Figure 8: Survey households colored according to the cluster they belong (clusters used to estimate the two clusters model).
In order to validate the proposed model against the other models, we reestimated all the models with a
random sample of 90% of the locations, and then with the remaining 10% we calculated the probability that
the observed winning household also wins the bid in the model, for each location in the validation sample.
Table 4 shows a summary of model-fit and information statistics for all the estimated models using the
validation sample.
Model
Log-Likelihood
Number of
parameters
% significant
parameters
(95%)
AIC
BIC
No spatial heterogeneity
-7,608
31
74%
15,278
15,448
7 Exogenous zones
-7,534
109
55%
15,287
15,885
2 Cluster-based zones
-7,494
56
57%
15,100
15,408
7 Cluster-based zones
-7,450
193
46%
15,285
16,344
Endogenous zones (proposed model)
-7,216
50
76%
14,532
14,806
Table 4: Model Likelihood comparison.
The proposed model shows better log-likelihood than the other models, and also better percentage of
significant parameters. A loglikelihood ratio test is performed for the proposed model against each of the
other models, showing it is significantly better than the others by at least 95%. To account for the relation
between the number of parameters and the likelihood, we also calculate the Akaike Information Criterion
(AIC)(Akaike, 1998) and Bayesian Information Criteriorn (BIC) (Schwarz, 1978) for each model, where
the proposed model also has a better performance.
7. Conclusions and discussion
We propose a discrete choice model that allows to include spatial heterogeneity with probabilistic zones
(fuzzy limits among zones) that are defined endogenously, following a one-step estimation of location
preferences and zone segmentation parameters.
The main conclusion is that the proposed location model, with endogenous spatial heterogeneity (LCM
applied to the bid auction approach), outperforms other common approaches in terms of direct likelihood
when applied to out-of-sample data, indicating better forecasting abilities. Also, the proposed model is
parsimonious, as it has better likelihood with less parameters than the other models. This is confirmed by
the Aikaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) as is shown in table 4.
These results are based on own our formulations of the alternative models, but we tried to keep them as
standard as we could and with similar and comparable criteria for all models.
When we apply the segmentation function to the map of Santiago, two clear zones emerge, which are
closely related to the income stratification of this city. The proposed method helps not only to identify
zones, but also to observe how fuzzy or clear-cut are the boundaries among zones, which can relate to
perceived barriers or to smooth transitions between zones.
We think this modelling approach provides a significant contribution in approaching spatial segmentation
from a behavioral perspective. As the segmentation method is integrated in the microeconomic modelling
of the decision process, segmentation parameters can be interpreted as the decision maker criteria of
segmentation of the city that assist him in the location choice process. It can be interpreted as individuals
dealing with qualitative (classes) and quantitative (WPs) aspects in their decision process.
The proposed approach can be easily extended to hedonic price models, usually found in the real estate
submarkets literature (Bourassa et al., 1999; Rosmera & Lizam, 2016). The approach can also be applied
to other behavior taking place in the urban context that could be influenced by spatial attributes, such as
mobility patterns and their relation to the built environment (see Oliva et al. (2018) for an early example of
this)
Acknowledgments
The authors would like to acknowledge the support provided by the Center for Sustainable Urban
Development (CEDEUS, CONICYT/FONDAP 15110020), the Complex Engineering Systems
Institute, ISCI (CONICYT PIA/BASAL AFB180003), FONDECYT (project N°1180605), and the
CONICYT PhD Scholarship for the corresponding author (2015-2019).
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8. ANNEX
EXOGENOUS ZONES MODEL
Observations
17830
Null model log-likelihood
-258759
Final log-likelihood
-76511
Attribute
Household
Type
Zone 1
Zone 2
Zone 3
Zone 4
Zone 5
Zone 6
Zone 7
Accessibility
to Commerce
by transit
Low-EL
-0.000337 (-
1.46)*
0.000264
(1.1)*
-0.000766 (-
1.63)*
-0.000837 (-
1.49)*
0.000475
(2.32)
0.000804
(3.09)
0.000885
(0.62)*
Mid-EL
-0.000476 (-
2.15)
-0.00105 (-
4.51)
-0.0000457 (-
0.21)*
0.00165
(4.93)
0.000754
(3.53)
0.000798
(3.85)
0.0026
(1.64)*
Hi-EL
Accessibility
to Industry by
transit
Low-EL
Mid-EL
Hi-EL
Accessibility
to Commerce
by car
Low-EL
Mid-EL
Hi-EL
-0.00143 (-
4.51)
-0.00129 (-
4.68)
0.00119
(6.63)
0.00203
(5.08)
-0.00234
(-7.5)
-0.000322 (-
1.21)*
-0.00222 (-
2.79)
Accessibility
to Industry by
car
Low-EL
Mid-EL
Hi-EL
Distance to
nearest
subway
estation
Low-EL
-0.151 (-
4.04)
0.18 (1.09)*
-0.154 (-
0.71)*
2.03 (1.51)*
-0.148 (-
2.04)
-0.165 (-
1.18)*
0.00224
(0.11)*
Mid-EL
0.0322
(1.07)*
-0.879 (-6)
-0.848 (-
5.28)
1.1 (1.23)*
-0.116 (-
1.54)*
-0.765 (-
5.84)
0.0438
(1.85)*
Hi-EL
-0.355 (-
5.26)
-0.719 (-
3.05)
-1.52 (-
11.31)
-3.11 (-2.48)
-0.304 (-
2.15)
-1.62 (-8.3)
0.0745
(1.63)*
Average Built
surface in
zone
Low-EL
0.00394
(0.4)*
-0.000768 (-
0.04)*
0.0858 (2.44)
0.014
(0.41)*
-0.0258 (-
1.68)*
-0.0258 (-
1.68)*
0.0267
(1.5)*
Mid-EL
0.0447
(6.15)
0.0655 (3.6)
0.225 (18.53)
0.0272
(1.4)*
0.0679
(4.28)
0.0402
(2.71)
-0.0203 (-
1)*
Hi-EL
0.0468
(2.69)
0.0867
(2.69)
0.244 (28.63)
-0.0675 (-
2.1)
0.163
(5.35)
0.125 (6.17)
0.11 (3.54)
Average
Zonal Income
Low-EL
1.03 (0.95)*
-3.05 (-2.44)
-0.524 (-
0.37)*
1.67 (0.58)*
-0.153 (-
0.1)*
-4.21 (-3.26)
-3.5 (-
1.86)*
Mid-EL
9.2 (10.83)
17.8 (16.66)
1.36 (2.52)
0.398
(0.22)*
7.71
(5.18)
9.8 (12.86)
5.66 (2.8)
Hi-EL
24.9 (14.45)
25.1 (11)
5.83 (15.94)
10.4 (4.65)
28.8
(14.81)
16.3 (18.78)
10.3 (3.45)
Built Density
in Zone
Low-EL
0.203
(0.13)*
-0.207 (-
0.17)*
-2.79 (-
1.64)*
-1.42 (-
1.42)*
-1.92 (-
0.96)*
-5.06 (-2.79)
-0.716 (-
0.12)*
Mid-EL
4.88 (3.63)
-0.737 (-
0.63)*
0.539 (0.91)*
0.496
(1.21)*
-8.71 (-
4.19)
0.435
(0.32)*
14.3 (1.99)
Hi-EL
5.3 (1.82)*
0.971
(0.38)*
0.298 (0.82)*
0.441
(0.95)*
-6.23 (-
1.5)*
-0.547 (-
0.22)*
-34.1 (-
2.26)
Household
constant
Low-EL
8.4 (16.32)
8.4 (16.32)
8.4 (16.32)
8.4 (16.32)
8.4
(16.32)
8.4 (16.32)
8.4 (16.32)
Mid-EL
0.64 (1.54)*
0.64 (1.54)*
0.64 (1.54)*
0.64 (1.54)*
0.64
(1.54)*
0.64 (1.54)*
0.64
(1.54)*
Hi-EL
-9.34 (-6.99)
-9.34 (-6.99)
-9.34 (-6.99)
-9.34 (-6.99)
-9.34 (-
6.99)
-9.34 (-6.99)
-9.34 (-
6.99)
0.168
(169.54)
Table 5: estimation parameter of the exogenous zones model
ATTRIBUTE-BASED 7 CLUSTER
MODEL
Observations
17830
Null model log-
likelihood
-258759
Final log-likelihood
-75556
Attribute
Household Type
Cluster 1
Cluster 2
Cluster 3
Cluster 4
Cluster 5
Cluster 6
Cluster 7
Accessibility to
Commerce by
transit
Low-
EL
0.00198
(1.02)*
-0.000495
(-0.82)*
0.000771
(0.78)*
0.00291
(0.45)*
0.00028
(0.84)*
0.0021
(0.38)*
-0.00882 (-
1.15)*
Mid-
EL
-0.00276 (-
2.54)
0.00268
(6.13)
0.0017
(1.72)*
-0.000372
(-0.13)*
0.00111
(3.16)
-0.00653 (-
3.04)
0.00839
(1.74)*
Hi-EL
0.000356
(0.25)*
0.00436
(7.18)
0.00456
(1.84)*
-0.00277 (-
1.21)*
0.00212
(2.37)
-0.00466 (-
3.19)
0.00975
(4.11)
Accessibility to
Industry by
transit
Low-
EL
-0.00401 (-
1.76)*
0.00092
(1.22)*
0.000858
(0.82)*
-0.00639 (-
0.98)*
-0.000119
(-0.26)*
-0.00168 (-
0.21)*
0.0238
(1.64)*
Mid-
EL
0.00253
(1.72)*
-0.00425 (-
7.66)
-0.00209 (-
1.94)*
-0.0012 (-
0.45)*
-0.000528
(-1.1)*
0.00985
(3.34)
-0.0207 (-
2.32)
Hi-EL
-0.000514 (-
0.24)*
-0.00622 (-
7.76)
-0.00497 (-
1.81)*
0.000268
(0.11)*
-0.000639
(-0.53)*
0.00656
(3.26)
-0.0194 (-
4.4)
Accessibility to
Commerce by
car
Low-
EL
-0.00225 (-
1.05)*
0.0018
(2.6)
0.00109
(1.17)*
-0.0104 (-
0.93)*
0.000284
(0.71)*
0.000177
(0.02)*
0.029
(2.51)
Mid-
EL
0.00684 (5)
-0.00285 (-
5.85)
-0.00199 (-
2.18)
0.00413
(0.87)*
-0.00315 (-
7.65)
0.0224
(6.56)
0.00411
(0.64)*
Hi-EL
0.0116
(5.56)
-0.000168
(-0.24)*
-0.00533 (-
2.24)
0.0191
(4.91)
-0.00579 (-
5.48)
0.0297
(12.83)
0.012 (4)
Accessibility to
Industry by car
Low-
EL
0.00359
(1.45)*
-0.00214 (-
2.8)
-0.00157 (-
1.54)*
0.0116
(0.95)*
0.000148
(0.3)*
-0.000692
(-0.06)*
-0.037 (-
2.46)
Mid-
EL
-0.00844 (-
4.97)
0.00457
(8.63)
0.0029
(2.87)
-0.00285 (-
0.54)*
0.00278
(5.57)
-0.0257 (-
6.27)
0.00183
(0.22)*
Hi-EL
-0.0144 (-
5.62)
0.00217
(2.63)
0.00653
(2.5)
-0.0179 (-
3.93)
0.00435
(3.48)
-0.0323 (-
11.67)
-0.00475 (-
1.22)*
Distance to
nearest subway
estation
Low-
EL
0.21 (0.16)*
0.0308
(0.3)*
0.00572
(0.28)*
6.41
(1.38)*
0.0484
(0.77)*
3.04
(1.44)*
1.56
(1.92)*
Mid-
EL
1.5 (1.55)*
-0.12 (-
1.76)*
0.0593
(2.66)
-0.429 (-
0.2)*
-0.0551 (-
0.84)*
1.3 (1.77)*
0.338
(0.73)*
Hi-EL
4.68 (3.54)
0.0818
(0.88)*
0.0545
(1.09)*
-2.26 (-
1.4)*
-0.444 (-
2.35)
3.05 (6.74)
1.46 (6.91)
Distance to
nearest highway
exit
Low-
EL
-0.285 (-
0.33)*
-0.361 (-
2.27)
0.0214
(0.33)*
7.79 (2.23)
-0.0726 (-
0.71)*
-2.33 (-
1.94)*
-0.116 (-
0.08)*
Mid-
EL
1.18 (2.2)
0.292
(2.75)
-0.0681 (-
0.94)*
1.74
(1.16)*
-0.0841 (-
0.77)*
-1.37 (-
2.89)
-2.26 (-
4.21)
Hi-EL
-1.53 (-2.31)
0.113
(0.79)*
-0.265 (-
1.61)*
0.163
(0.16)*
-0.244 (-
0.83)*
-2.51 (-
7.55)
-2.31 (-
9.51)
Average Built
surface in zone
Low-
EL
0.00786
(0.19)*
0.00036
(0.02)*
0.0035
(0.51)*
0.301
(2.64)
-0.0027 (-
0.22)*
-0.0027 (-
0.22)*
0.0472
(0.67)*
Mid-
EL
0.141 (6.41)
0.0581
(5.4)
0.00533
(0.7)*
0.186 (4.1)
0.131
(10.39)
0.0957
(3.78)
0.0458
(1.31)*
Hi-EL
0.161 (6.18)
0.109
(9.16)
0.0163
(0.98)*
0.138
(3.86)
0.242
(7.58)
0.0578
(3.68)
0.0265
(1.99)
Average Zonal
Income
Low-
EL
-0.834 (-
0.18)*
1.93
(1.09)*
3.04
(1.88)*
-13.1 (-
1.24)*
0.132
(0.11)*
2.64
(0.43)*
-1.85 (-
0.46)*
Mid-
EL
2.92 (1.21)*
9.95 (9.3)
14.7 (7.96)
-1.06 (-
0.26)*
19.5 (15.8)
2.51
(1.22)*
1.35
(1.01)*
Hi-EL
14.1 (5.25)
18.9 (16.2)
29.8 (6.68)
7.6 (2.35)
33.4
(10.56)
9.39 (8.32)
3.17 (6.38)
Built Density in
Zone
Low-
EL
0.0226
(0.01)*
2.84
(1.13)*
-5.51 (-
1.76)*
0.699
(0.48)*
-3.34 (-
2.23)
-6.47 (-
1.51)*
-21 (-
1.09)*
Mid-
EL
10.5 (5.51)
2.85
(1.74)*
11.4 (3.74)
0.318
(0.55)*
-2.04 (-
1.29)*
-4.63 (-
3.2)
-8.89 (-
1.26)*
Hi-EL
12.1 (6.44)
-4.7 (-
2.24)
-16.8 (-
2.37)
-0.829 (-
1.38)*
-3.53 (-
0.86)*
-6.3 (-7.4)
-26.2 (-
8.58)
Household
constant
Low-
EL
5.72 (6.24)
5.72 (6.24)
5.72 (6.24)
5.72 (6.24)
5.72 (6.24)
5.72 (6.24)
5.72 (6.24)
Mid-
EL
-4.91 (-5.59)
-4.91 (-
5.59)
-4.91 (-
5.59)
-4.91 (-
5.59)
-4.91 (-
5.59)
-4.91 (-
5.59)
-4.91 (-
5.59)
Hi-EL
-21.3 (-
13.44)
-21.3 (-
13.44)
-21.3 (-
13.44)
-21.3 (-
13.44)
-21.3 (-
13.44)
-21.3 (-
13.44)
-21.3 (-
13.44)
0.175
(167.84)
Table 6: Estimation parameters of the seven clusters model.
ATTRIBUTE-BASED 2 CLUSTER MODEL
Observations
17830
Null model log-likelihood
-258759
Final log-likelihood
-76170
Attribute
Household Type
Cluster 1
Cluster 2
Accessibility to Commerce
by transit
Low-EL
0.000447 (1.22)*
-0.0022 (-0.89)*
Mid-EL
-0.00263 (-7.59)
-0.000142 (-0.11)*
Hi-EL
-0.0065 (-10.01)
-0.000636 (-0.51)*
Accessibility to Industry by
transit
Low-EL
-0.0000768 (-0.3)*
0.00187 (1.02)*
Mid-EL
0.00252 (10.4)
-0.00115 (-1.2)*
Hi-EL
0.00518 (11.44)
-0.00171 (-1.94)*
Accessibility to Commerce
by car
Low-EL
0.00103 (3.36)
-0.00137 (-0.48)*
Mid-EL
-0.00279 (-10)
0.0104 (6.56)
Hi-EL
-0.000588 (-1.14)*
0.0251 (17)
Accessibility to Industry by
car
Low-EL
-0.000896 (-2.35)
0.000997 (0.28)*
Mid-EL
0.00354 (9.97)
-0.0101 (-5.15)
Hi-EL
0.00232 (3.4)
-0.0239 (-13.26)
Distance to nearest subway
estation
Low-EL
0.0194 (1.1)*
0.178 (0.26)*
Mid-EL
0.0594 (3.34)
0.193 (0.63)*
Hi-EL
0.155 (4.2)
0.798 (3.86)
Distance to nearest highway
exit
Low-EL
Mid-EL
-0.028 (-0.59)*
-1.26 (-3.91)
Hi-EL
0.106 (1.17)*
-2.34 (-10.87)
Average Built surface in
zone
Low-EL
-2.31 (-3.14)
-0.87 (-0.93)*
Mid-EL
2.72 (4.25)
0.175 (0.45)*
Hi-EL
1.9 (1.52)*
-1.3 (-3.79)
Average Zonal Income
Low-EL
0.0279 (0.1)*
-0.00536 (-0.01)*
Mid-EL
14.5 (34.11)
0.786 (0.97)*
Hi-EL
24.7 (43.52)
5.66 (12.93)
Built Density in Zone
Low-EL
-2.31 (-3.14)
-0.87 (-0.93)*
Mid-EL
2.72 (4.25)
0.175 (0.45)*
Hi-EL
1.9 (1.52)*
-1.3 (-3.79)
Household constant
Low-EL
5.75 (7.53)
Mid-EL
-5.21 (-6.65)
Hi-EL
-28.2 (-19.99)
0.171 (168.87)
Table 7: Estimation parameters of the two cluster model.
