![CHAPTER TWO
Mathematical preparation
Hajime Seya1
, Yoshiki Yamagata2
1
Departments of Civil Engineering, Kobe University, Kobe, Hyogo, Japan
2
Center for Global Environmental Research, National Institute for Environmental Studies,
Tsukuba, Ibaraki, Japan
Contents
2.1 Definitions of notations 9
2.2 The classical linear regression model 10
2.2.1 The classical linear regression model and violation of typical assumptions 10
2.2.2 Endogeneity 12
2.2.3 Spatial autocorrelation of error term and heteroskedastic variance 16
2.3 The generalized linear model 17
2.4 The additive model 19
2.5 The basics of Bayesian statistics 23
2.5.1 Bayes’ theorem 23
2.5.2 The Markov chain Monte Carlo method 24
2.5.3 Bayesian estimation of the classical linear regression model 28
References 30
2.1 Definitions of notations
In this book, scalars are shown in fine italics a, and vectors and matrices
are shown in bold a (there are instances of lowercase characters and upper-
case characters). Moreover, when a is a column vector and A is a matrix, ai is
the ith component of a, ai is a vector from a with the ith component
removed, Ai is the ith row of A, Ai,j is the component on row i column j
of A, and Ai is a matrix from A with the ith row removed. Furthermore,
I expresses an identity matrix, O a square matrix from 0, 1 a column vector
of 1, 0 a column vector of 0, and A1
and A0 are the inverse matrix of A and
the transposed matrix of A, respectively. Dimensions of matrices and vectors
are omitted, where obvious, from the context. However, depending on
their necessity, these are written as A[n] for an nth order square matrix and
A[n m] for a matrix with n rows and m columns. In addition, as much as
Spatial Analysis using Big Data
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![consisting of bk, and ε is an N 1 error term vector consisting of εi. If we
rearrange such that Xh[1;x], bh[b1;b0
2]’, we have
y ¼ Xb þ ε: (2.2.3)
In the CLR model, the following four assumptions are usually made:
(1) X is exogenous.
(2) Given X, the conditional expected value of y is Xb, and the
conditional expected value of ε is 0. That is, E[εjX] ¼ 0.
(3) Given X, the error term ε satisfies:
Var½εjX ¼ s2
εI½N ¼
0
B
B
B
@
s2
ε 0 / 0
0 s2
ε 1 «
« 1 1 0
0 / 0 s2
ε
1
C
C
C
A
: (2.2.4)
Eq. (2.2.4) implies independent and identically distributed error terms.
(4) The rank of X is K. That is, an inverse matrix exists in (X’X).
In addition to these four assumptions, the following assumption is typi-
cally made:
(5) εwN(0s2
εI½N); that is, normal distributed error terms in population.
This assumption implies
y w N
Xb; s2
εI½N
: (2.2.5)
With assumption (5), the ordinary least squares (OLS) estimator of b
follows the normal distribution, making it possible to perform a hypothesis
test on its significance when the number of observations in a sample is rather
small (when large, we can apply the central limit theorem).
The OLS estimator b
bols of the CLR model is obtained by minimizing
the sum of squares b
ε'olsb
εols of residual vector:
b
εols ¼ y Xb
bols (2.2.6)
The first-order condition of optimization yields
X'
y ¼ X'Xb
bols (2.2.7)
With assumption (4), there is an inverse matrix in (X’X), and therefore
the OLS estimator of b can be obtained from
b
bols ¼ ðX'XÞ1
X'
y (2.2.8)
Mathematical preparation 11](https://image.slidesharecdn.com/33827-250403131549-a66ed5b4/85/Spatial-analysis-using-big-data-methods-and-urban-applications-Yamagata-23-320.jpg)
![Also, the variance of b
bols is given by:
Var
b
bols
¼ s2
εðX'XÞ
1
(2.2.9)
Since s2
ε is usually unknown, we substitute s2
ε with the estimator
b
s2
ε;ols ¼ b
ε'olsb
εols=ðN KÞ (2.2.10)
Note that the fitted value of y is given by b
y ¼ Xb
bols, and thus if this is
substituted into Eq. (2.2.8), we obtain b
y ¼ XðX'XÞ1
X'y. Here,
PX ¼ X
X'X
1
X' is a projection matrix that produces b
y from y, and
for this reason, it is termed a hat matrix. Similarly, MX ¼ I[N]PX is an
operator that creates a residual from y. These operators are also important
in the restricted maximum likelihood (REML) method described elsewhere.
Where assumptions (1)e(4) hold, the OLS estimator becomes the best
linear unbiased estimator (BLUE), and this property is known as the
Gauss-Markov theorem. Unfortunately, however, it is rare in empirical
analyses that all these assumptions are satisfied, and in many cases, violations
of assumptions (1)e(3) occur in particular. Therefore, later we explain the
consequences of violating assumptions (1)e(3), and the countermeasures
to them. Note that with respect to assumption (4), it is possible to satisfy
this assumption by removing the explanatory variable(s) that causes perfect
multicollinearity.
2.2.2 Endogeneity
When deviating from assumptions (1) and (2) (i.e., when a correlation
between the explanatory variable and error term occurs), the OLS estimator
lacks both consistency and unbiasedness. Cases where x has a measurement
error, where an important variable is missing from the model (omitted
variable bias), and where x and y are jointly determined (simultaneity/
reverse causality) are regarded as examples of this kind of situation. As a
countermeasure to this problem, the instrumental variable (IV) method is
applied to obtain consistent estimators for the regression coefficients.
The IV(s) must have correlation to the endogenous explanatory variables
conditionally on the other covariates, although they cannot have correlation
to the error term conditionally on the other covariates. The latter condition
rule out any direct effect of the instruments on the dependent variable or
any effect running through omitted variables. This is called the exclusion
restriction. As its generalization, the two-stage least squares (2SLS) method
as well as the generalized method of moments (GMM) can also be used.
12 Hajime Seya and Yoshiki Yamagata](https://image.slidesharecdn.com/33827-250403131549-a66ed5b4/85/Spatial-analysis-using-big-data-methods-and-urban-applications-Yamagata-24-320.jpg)
![Since these methods are used for estimating parameters of the spatial eco-
nometric models, let us briefly describe their basics here.
We will now explicitly introduce endogenous variables into the CLR
model, which we can express as follows:
y ¼ Xb þ _
X _
b þ ε; (2.2.11)
where we take X to be an N K explanatory variable matrix consisting of
constant terms and exogenous variables, and _
X as an N L explanatory
variable matrix consisting of endogenous variables. In addition, b denotes a
K 1 regression coefficient vector corresponding to exogenous variables,
and _
b denotes an L 1 regression coefficient vector corresponding to
endogenous variables. Here, because _
X is an endogenous variable, it has a
correlation to the error term (Cov[εi, _
xl;i]s0, c l ¼ 1, ., L). Hence we
may consider introducing IVs, say Z[NP], which correlates with _
X, but
does not correlate with ε. For the sake of identification, the degree P of Z
must be greater than or equal to the number of endogenous variables L.
Under such a scenario, the IV estimator can be obtained by applying the
2SLS method. That is, when the explanatory variable matrix is rearranged to
form Rh
X; _
X
, a two-stage estimation is performed, such that R is
projected onto the plane spanned by S h [X;Z], uncorrelated to the error
term and the estimated b
R value obtained, the next y is regressed on
b
R; not R. This is a simple idea of using b
R, which does not have correlation
to the error term. If a valid IV is used, the 2SLS estimator may be consistent.
The 2SLS estimator of parameter €
bh
h
b0
; _
b0
i0
is finally obtained from
b
€
b2sls ¼
b
R
0
b
R
1
b
R
0
y; (2.2.12)
Var
h
b
€
b2sls
i
¼ s2
ε
R0 b
R
1
; (2.2.13)
where b
R ¼ ðS0SÞ1
S0R, and therefore €
b2sls ¼
h
R0SðS0SÞ1
S0R
i1
R0SðS0SÞ1
S0y. Since s2
ε is unknown, s2
ε can be substituted with an estimate
using
b
s2
ε;2sls ¼ b
ε'2slsb
ε2sls=ðN KÞ; (2.2.14)
Mathematical preparation 13](https://image.slidesharecdn.com/33827-250403131549-a66ed5b4/85/Spatial-analysis-using-big-data-methods-and-urban-applications-Yamagata-25-320.jpg)
![where hs(y,X,b) is also an R 1 vector-valued function. If the number of
the estimand, K, of b matches the number of moment conditions R, as in the
case of the CLR model, it is possible to obtain those parameters using the
MM. However, where R K the parameters that exactly satisfy
Eq. (2.2.21) are generally not present. Therefore, we consider determining
parameters that minimize a quadratic form such as
hsðy; X; bÞ0
Vhsðy; X; bÞ (2.2.22)
The estimator obtained in this way is a GMM estimator. The GMM esti-
mator is known to have consistency and asymptotic normality under very
general conditions (Hayashi, 2000). Here, V is the weight assigned to
each condition and Hansen (1982) showed that under some regularity con-
ditions, the minimum variance of the GMM estimator can be achieved by
using the following equation:
V ¼
1
N
X
N
i¼1
hðyi; Xi; bÞhðyi; Xi; bÞ0
#1
(2.2.23)
However, in order to obtain the estimator V in Eq. (2.2.23), it is neces-
sary to substitute the corresponding estimates for b. Therefore, a two-step
estimation is performed: calculating b
b
ð0Þ
gmm using suitable initial value of
weight (e.g., identity matrix value), substituting it into Eq. (2.2.23) to obtain
b
V
1
, and finally obtaining the GMM estimator by minimizing Eq. (2.2.22).
Here, we can easily show that the 2SLS estimator is a special case of the
GMM estimator. Now, let’s return to Eq. (2.2.11). When we set the IVs to
Z[NP], we can define S h [X;Z][N(KþP)]. Now, by adding the moment
condition relating to Z to that of the CLR model, the following equation is
obtained:
E½S0
ε ¼ 0½ðKþPÞ1 (2.2.24)
Replacing the left side with a sample analogous, we obtain
S0
y R€
b
N
; (2.2.25)
where Rh
X; _
X
, €
bh
h
b'
; _
b'
i'
. The GMM estimator for €
b is obtained by
minimizing the quadratic form
hs
y; X; _
X; Z; €
b
0
Vhs
y; X; _
X; Z; €
b
; (2.2.26)
Mathematical preparation 15](https://image.slidesharecdn.com/33827-250403131549-a66ed5b4/85/Spatial-analysis-using-big-data-methods-and-urban-applications-Yamagata-27-320.jpg)
![Fortunately, however, if the structure of S is known, using the generalized
least squares method, b0s BLUE can be obtained:
b
bgls ¼
X0
X1
X
1
X0
X1
y. (2.2.32)
Of course, S is usually not known, and it is necessary to set N N el-
ements of S with any assumption.
Simply speaking, in the modeling of geostatistical data (geostatistical
model), the varianceecovariance matrix is directly constructed as a function
of distance. Meanwhile, in lattice data modeling (spatial econometric model),
it is indirectly constructed through structuring the dependency between the
data or error terms (e.g., autoregression type and moving average type). These
differences are explained in more detail in Chapters 4 and 5.
2.3 The generalized linear model
In the CLR model introduced in Eq. (2.2.1), if we assume that the
error term follows a normal distribution, it can be expressed as:1
E½yi ¼ mi ¼ Xib; yiwN
mi; s2
ε
; (2.3.1)
where Xi denotes the vector consisting of the ith row of X. Here, needless to
say, it is not mandatory to assume normal distribution. The GLM can handle
a wide class of distributions called an exponential distribution family, where
we have
f ðE½yiÞ ¼ f ðmiÞ ¼ Xib (2.3.2)
Note that the relationship between mi and Xib are modeled by nonlinear
function f(,). The function f(,) is called a link function, and this kind of
model is called a generalized linear model. The GLM has the following
characteristics:
[1] Dependent variable yi follows a distribution belonging to the exponen-
tial distribution group.
[2] f(mi) and Xib have a linear relationship.
Poisson distribution and binomial distribution, in addition to the normal
distribution, are well-known distributions belonging to exponential distribu-
tion family. Commonly used link functions differ, depending on the
1
For the sake of simplicity, we do not distinguish between the stochastic variable Yi and its observation
yi.
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![Z1 ¼
2
6
4
x1;1 k1;1
þ
/
x1;1 k1;Q1
þ
« 1 «
x1;N k1;1
þ
/
x1;N k1;Q1
þ
3
7
5;
Z2 ¼
2
6
4
x2;1 k2;1
þ
/
x2;1 k2;Q2
þ
« 1 «
x2;N k2;1
þ
/
x2;N k2;Q2
þ
3
7
5:
Define Rh[X;Z]; €
bh
b0
; b0
0, and a matrix D as
D ¼
0
B
B
@
O½22 O½2Q1 O½2Q2
O½Q12
e
l
2
1 I½Q1Q1 O½Q1Q2
O½Q22 O½Q2Q1
e
l
2
2 I½Q2Q2
1
C
C
A; (2.4.6)
where e
l1;e
l2 0 denotes the Lagrangian multipliers. The penalized least
squares estimate for €
b can be obtained by minimizing
y R€
b
2
þ €
b
0
D€
b; (2.4.7)
where jj . jj denotes a vector norm. The optimization yields
b
€
be
l
¼ ðR0
R þ DÞ
1
R0
y: (2.4.8)
When e
l1
or e
l2
takes a value close to 0, overfitting the data tends to
occur. When the value takes a relatively large value, on the other hand,
we cannot fully capture the nonlinearity between x1(or x2) and y. Hence
the calibration of these parameters are very important.
Here, let’s consider substituting fixed effects b1,h and b2,h with random
effects u1;hwi:i:d:N
0; s2
1u
and u2;hwi:i:d:N
0; s2
2u
, respectively, and
formalizing the model as a mixed model. This will smoothen the linear
spline function, and avoid data overfitting (Ruppert et al., 2003, p.109).
Moreover, a further advantage is that there are many statistical packages
for mixed models. Substituting b1,h, b2,h by u1,h, u2, h in Eq. (2.4.7) and
dividing by s2
ε, we obtain
1
s2
ε
y Xb Zu
2
þ
e
l
2
1
s2
ε
u1
2
þ
e
l
2
2
s2
ε
u2
2
; (2.4.9)
22 Hajime Seya and Yoshiki Yamagata](https://image.slidesharecdn.com/33827-250403131549-a66ed5b4/85/Spatial-analysis-using-big-data-methods-and-urban-applications-Yamagata-34-320.jpg)
![3.1.2 Specification of the spatial weight matrix
The choice of weight matrix W affects both estimations and inferences (e.g.,
Florax and Folmer, 1992; Griffith, 1996; Stakhovych and Bijmolt, 2009).
However, currently, there are few guidelines for selecting the correct W
(Anselin, 2002). Following Stakhovych and Bijmolt (2009), we classified
the means of providing W into the following three types:
[A] Completely exogenous
[B] Determining from the data
[C] Estimating
[A] is a typical method provided by whether area boundaries are in
contact (contiguity-based W) as previously mentioned, as well as the inverse
distance, Delaunay triangulation, and so forth (LeSage, 1999).
For [B], Aldstadt and Getis (2006) proposed an algorithm termed
AMOEBA (a multidirectional optimum ecotope-based algorithm),2
in which
the geometric form of spatial clusters are identified, similar to a region
growing algorithm of image segmentation. The distance need not be limited
to the geographic one. We can consider other types of distances, including
social economic distances (e.g., difference of gross domestic product [GDP]),
migration flow (Conway and Rork, 2004), technological similarity (Lychagin
et al., 2016), among others. These methods may also be categorized to [B].
However, we need to note that methods for parameter estimations and
inferences of spatial econometric models have been developed for exogenous
W. If W is endogenous, we need to rely on recently proposed parameter
estimation methods for endogenous W (e.g., Qu and Lee, 2015; Zhou
et al., 2016). Kostov (2010) noted that the reason why W based on geograph-
ical distance is often used is that the exogeneity is automatically satisfied.
There are very few studies classified as [C]. Fern
andez-V
azquez et al.
(2009) tried to estimate W by generalized maximum entropy and generalized
cross-entropy techniques. In a panel setting when the number of time series
variables can grow faster than the number of time points for data, studies
employed (graphical) LASSO to estimate W (Ahrens and Bhattacharjee,
2015; Moscone et al., 2017; Lam and Souza, 2019).
Because research regarding the estimation of the elements of W in [C] is
still under development, it is common to select the most adequate W from
prepared candidates. We could mention Kostov (2010) based on boosting
and LeSage and Pace (2009), who attempted a model selection using
2
GIS software that implements AMOEBA is available at the website of the first author (http://www.acsu.
buffalo.edu/wgeojared/tools.htm). It can also be implemented using the AMOEBA package of R.
Global and local indicators of spatial associations 37](https://image.slidesharecdn.com/33827-250403131549-a66ed5b4/85/Spatial-analysis-using-big-data-methods-and-urban-applications-Yamagata-48-320.jpg)
![posterior model probabilities based on a Bayesian approach.3
The latter
Bayesian approach is useful for the selection of both X and W (LeSage and
Pace, 2009). Kelejian (2008) and Kelejian and Piras (2011) extended the
J-test, which is a representative method of nonnested model selection, to
the spatial econometric model. Meanwhile, Mur and Angulo (2009) noted
the usefulness of information criterion (comprehensibility and clarity of results
in the sense of selecting the unique best model). Similarly, Stakhovych and
Bijmolt (2009) stated that using information criterion (e.g., Akaike’s informa-
tion criterion [AIC]) increases the possibility of correct model specification.
Seya et al. (2013) attempted to simultaneously select W and X to minimize
the AIC by applying a technique termed transdimensional simulated
annealing that combines reversible jump Markov chain Monte Carlo and
simulated annealing. Other attempts include not model selections but model
ensembles. We can mention those based on Bayesian model averaging
(LeSage and Parent, 2007) and weighted average least squares (WALS)
(Seya et al., 2014).4
3.1.3 Standardization of the spatial weight matrix
The spatial lag model (SLM), described in Chapter 6, is a representative
spatial econometric model. For the maximum likelihood estimation of the
parameters of this model, calculation of the term ðI rWÞ1
is necessary
during the process of estimation (here r is a parameter that indicates the
degree of spatial autocorrelation). In many previous studies, jrj 1 is
assumed in the analogy of time series models, but the singular point of
ðI rWÞ is not necessarily outside the range of ð 1; 1Þ. Therefore,
some standardization is typically performed on W to guarantee the existence
of the inverse matrix. The most widely used standardization is row-
standardization (also termed row-normalization), in which the elements of
each row of W sum to unity. With the contiguity-based W (Eq. 3.1.1),
row-standardization may be performed as
W ¼
0
B
B
B
B
B
B
@
0 1=2 1=2 0 0
1=3 0 0 1=3 1=3
1=2 0 0 1=2 0
0 1=3 1=3 0 1=3
0 1=2 0 1=2 0
1
C
C
C
C
C
C
A
(3.1.5)
3
The Spatial Econometrics Toolbox of Matlab is available at the website of the first author.
4
See Magnus and De Luca (2016) for details about the WALS.
38 Hajime Seya](https://image.slidesharecdn.com/33827-250403131549-a66ed5b4/85/Spatial-analysis-using-big-data-methods-and-urban-applications-Yamagata-49-320.jpg)
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- 6. Spatial Econometrics and Spatial Statistics SPATIAL ANALYSIS USING BIG DATA Methods and Urban Applications Edited by YOSHIKI YAMAGATA Center for Global Environmental Research National Institute for Environmental Studies Tsukuba, Ibaraki, Japan HAJIME SEYA Departments of Civil Engineering Graduate School of Engineering Faculty of Engineering Kobe University, Kobe, Hyogo, Japan
- 7. Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-813127-5 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Candice Janco Acquisition Editor: Scott J Bentley Editorial Project Manager: Redding Morse Production Project Manager: Debasish Ghosh Cover Designer: Matthew Limbert Typeset by TNQ Technologies
- 8. Contributors Toshihiro Hirano Kanto Gakuin University, Yokohama, Kanagawa, Japan Daisuke Murakami The Institute of Statistical Mathematics, Tachikawa, Tokyo, Japan Hajime Seya Departments of Civil Engineering, Kobe University, Kobe, Hyogo, Japan Yoshiki Yamagata Center for Global Environmental Research, National Institute for Environmental Studies, Tsukuba, Ibaraki, Japan Takahiro Yoshida Center for Global Environmental Research, National Institute for Environmental Studies, Tsukuba, Ibaraki, Japan ix j
- 9. Preface The world today is experiencing a technological revolution caused by the internet of things (IoT), big data, and artificial intelligence (AI). Consequently, interest in using spatial big data for practical purposes has boomed in recent years. Hundreds of new applications are emerging over a wide variety of areas such as weather forecasting, car navigation, restaurant/ hotel recommendation, and so forth. New types of data are also becoming available through IoT applications and the scope of spatial analysis models being drastically enhanced by AI and machine-learning techniques. In the urban context, these technologies are often associated with another international interestdsmart cities. The implementation of a combination of these technologies in urban environments could exceed our original expectations. Managing these systems requires spatial statistical researchers to take on the responsibility of data scientists analyzing complex urban spatial big data, addressing new societal challenges, and contending with real-world issues like climate change by developing new applications. The purpose of this book is to provide graduate-level students and urban researchers alike with the basic theories and methods of spatial statistics and spatial econometrics necessary for developing new urban analytic appli- cations utilizing spatial big data. Therefore, in the beginning emphasis is centered on mathematical formulation of spatial statistical and econometric methods, complemented with new developments for analyzing spatial big data. Initially more attention is given to describing the fundamental theories in an illustrative manner, to be useful for applied researchers (Methods). Utilizing programming illustrations, using R code, we describe how to empirically implement the methodologies presented in the previous chapters (Implementations). Following this, varieties of empirical application examples relating to the spatial big data analytical methods are explained with special emphasis placed on climate change mitigation issues in the urban spatial planning context (Applications). We believe by combining foundational methods of spatial statistics and spatial econometrics with practical empirical applications, researchers and practitioners may have a better understanding of how spatial big data analysis can be applied to facilitate evidence-based urban spatial planning. In the empirical part of this book, the focus is on models exemplifying these concepts, shown being utilized for climate change mitigation solutions in xi j
- 10. cities in Japan. The research methods presented in this book are general enough to be applicable for addressing other different type of issues in different cities as well as other multispatial scales such as regional or national levels. The authors also believe that this book’s contents can contribute to supporting researchers, practitioners, and students in conducting practical spatial analyses using big data for their studies or others. Spatial statistics (econometrics) refers to statistical (econometric) analysis conducted on data with position coordinates, that is, spatial data. The utilization of spatial aspects of data in statistical analyses can enhance and improve the reliability of our models and analysis. One of the key concepts in spatial statistics and spatial econometrics is spatial dependency or spatial autocorrelation, defined in Tobler’s first law of geography:“Everything is related to everything else, but near things are more related than distant things.” Although historically the most popular academic areas for spatial autocorrelation analysis were, and are, ecology and genetics; the scope of application for spatial statistics has expanded substantially to include geography and regional science, medicine and epidemiology (Waller and Gotway, 2004), criminology (Townsley, 2009), image analysis (Curran and various studies in Atkinson, 1998), remote sensing (Cressie, 2018), minology (Journel and Huijbregts, 1978), soil science (Goovaerts, 1999), climate science (Elsner et al., 2011), and the water field (Ver Hoef et al., 2006), among others, reinforcing the fact that knowledge is cumulative. First, let us briefly trace the lineage of spatial autocorrelation analysis. The first use of spatial autocorrelation analysis is often attributed to John Snow’s cholera map in the mid 19th century. Snow created a disease map for the Soho area of London, in which the distribution of cholera patients and the distribution of water pumps were superimposed onto the city map. In doing so he found that the data was concentrated and distributed, with cholera patients clustered around water pumps in Broadstreet. This is 30 years before the discovery of Vibrio cholerae by Robert Koch. Snow’s analysis, which acquired useful information by mapping spatial accumulation and autocorrelation of cholera, is considered to be the first real case of spatial data analysis. It would take 100 years before Moran’s development of the I statistic (1948; 1950), and Geary’s C statistic (1954) made it possible to quantitatively evaluate the presence or absence of spatial autocorrelation. In the 1960 and 1970s, in the field of quantitative geography, spatial autocorrelation was regarded as one of the most basic and important prob- lems, leading researchers to develop stronger analytical modeling methods for spatial data (e.g., Curry, 1966; Cliff and Ord, 1973). Building on these xii Quasi real-time energy use estimation using Google’s Popular Times dataPreface
- 11. techniques, the scientific field of spatial econometrics, which examines the relationship between data in discrete spaces (zones of cities, towns, etc.) and flows of econometric geography, emerges in regional science. However, the field of econometrics has now become an independent field of study, with many articles published in mainstream econometrics journals (Anselin, 2010; Arbia, 2011). Separately, another form of spatial statistics arose in the field of natural sciences. This academic field was established from mining science and treated spatial data as a continuous quantity in space (Matheron, 1963). In geostatistics, the dependency between data is described as a direct function of distance. Once a function is identified, the dependency between data at any location can be expressed using it, which enables spatial prediction of data at any point. This is a major feature of modeling in geostatistics. Regardless of field, according to Cressie (1993), spatial data may be categorized into: (1) geostatistical data (2) lattice data (3) point patterns The term spatial statistics is sometimes used to represent an academic discipline dealing with geostatistical data or point patterns. However, spatial statistics, as a comprehensive system of analysis, is better defined in relation to all three data types noted by Cressie (1993). Among these, our book deals only with the spatial data categorized in geostatistical and lattice data. , We have little experience with point patterns; therefore, for more detail we refer to comprehensive texts such as Diggle (2013). Further, several key topics fall outside our scope including spatial data acquiring, sampling, handling, processing, and mining. Our focus is on spatial data analysis (or modeling). Methods for processing geostatistical data were developed from geo- statistics, whereas lattice data analysis arose from spatial econometrics (Anselin, 1988). Although there are many similarities in modeling tech- niques of geostatistics and spatial econometrics, Anselin (1986) describes it as “each approach tends to be self-contained, with little cross-reference shown in published articles.” Different development histories make it relatively rare for either field to reference the other’s papers. Compounding this is a sort of mutually exclusive entry barrier resulting from different designations being used for the same model, depending on the field, causing confusion. In fact, texts that cover methods of spatial econometrics Preface xiii
- 12. and geostatistics are limited to Haining (1990, 2003) and Chun and Griffith (2013), for example. Brunsdon and Comber (2018) cover the implementa- tion of R code in both fields but have few theoretical descriptions. Thus, in this book, using our latest research, we explain the methods employed by both fields as much as possible while supporting their implementation using R coding. Moreover, facilitating the need for better modeling techniques for spatial big data, beyond just reviewing recent efforts, we introduce implementation methods utilizing R. This book consists of three parts: Methods (Chapters 2e6), Implementations (Chapter 7), and Applications (Chapters 8e11). The content of each chapter is briefly explained as follows. Chapter 1: Defines spatial data and its two major features, “spatial autocorrelation” and “spatial heterogeneity.” It provides a solid foundation upon which all subsequent chapters are built. Chapter 2: Provides the basic mathematical preparation necessary for spatial statistical and econometric analysis. We describe the classical regression model that is the basis of this book followed by explanations about the applied regression models, generalized linear model and additive model. Of course, readers who are familiar with regression models may skip this chapter. In addition, since spatial statistical models in recent years are often subjected to theoretical development based on Bayesian statistics, it is also explained here. Chapter 3: Introduces measures (test statistics) related to the existence of spatial autocorrelation in data called global indicators of spatial association, followed by reviews on the measures related to where the spatial auto- correlation occurs, called local indicators of spatial association (LISA). Also, the spatial weight matrix, which is an important tool for spatial eco- nometrics, is explained. Chapters 4 and 5: Explain the modeling techniques of the geostatistical data and lattice data, respectively. For the latter, it especially focuses on spatial econometrics. Applications to spatial big data is also discussed. Chapter 6: Introduces geographically weighted regression (GWR) and eigenvector spatial filtering approaches that have been developed in quantitative geography. Their recent advances, especially in terms of computation, are also explained. Chapter 7: Provides implementations with R. We chose R because of its barrier-free nature (available for free and easy to learn), which is important for students, as well as the existence of a lot of excellent packages that include many specialized functions for analyzing spatial big data. xiv Quasi real-time energy use estimation using Google’s Popular Times dataPreface
- 13. As examples of spatial data, we use well-known housing price data in Lucas county (Ohio, USA), available through spData package of R. The data size is 25,357, thus medium-sized. Chapters 8e11 illustrate the application of spatial statistical/econo- metrics techniques for urban planning issues, especially focusing on climate change mitigation. Each chapter uses various original data, and in this sense our application chapters do not offer current standard applications. However, we believe such applications would become more and more important in the future as aforementioned. The details of each chapter is explained as follows: Chapter 8: Illustrates spatial modeling by combining multiple spatial data. This kind of topic is becoming important today as an increasing number of spatial data in different forms are available. We first introduce a geostatistical approach to estimate temperatures in an intraurban scale by combining weather monitoring data and geo-tagged tweets relating to heat. Then, this approach is employed in an empirical study in Tokyo. Chapter 9: Illustrates two GPS data analyses to quantify goodness of walking environment. The first study applies a quantitative geographic approach (GWR model; see Chapter 6) to investigate local determinants of a walking environment, focusing on the impact of a pedestrian network structure on the number of pedestrians. The second analysis applies LISA (see Chapter 3) to quantify the heat-wave risk for pedestrians in an intraurban scale. Chapter 10: Applies a spatial econometric approach to a spatially explicit downscaling of socioeconomic scenarios; the resulting socioeconomic scenarios by 0.5-degree grids are useful to evaluate regional climate risks in the future. We first apply the spatial econometric model to project city population growth in several future scenarios. Then, the result is used in the downscaling. Chapter 11: Illustrates an estimation of quasi real-time energy consump- tion in each building using Google’s popular time data, which records quasi real-time human locations/activities collected from users of Google Map on smartphones. We apply the geostatistical compositional kriging model to the popular time data for a ward in central Tokyo. Although this book is as self-comprehensive as possible, a basic (under- graduate level) knowledge of statistics and econometrics is useful. Therefore, readers who are not familiar with statistics and econometrics are encouraged to familiarize themselves with these topics before reading this book. After reading our book, readers who are interested in spatial statistics and Preface xv
- 14. spatial econometrics may also refer to more advanced textbooks such as Cressie and Wikle (2011) for spatial statistics, and Kelejian and Piras (2017) for spatial econometrics. Yoshiki YAMAGATA and Hajime SEYA May 1, 2019 References Anselin, L., 1986. Some further notes on spatial models and regional science. Journal of Regional Science 26 (4), 799e802. Anselin, L., 1988. Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, Dordrecht. Anselin, L., 2010. Thirty years of spatial econometrics. Papers in Regional Science 89 (1), 3e25. Arbia, G., 2011. A lustrum of SEA: recent research trends following the creation of the Spatial Econometrics Association (2007e2011). Spatial Economic Analysis 6 (4), 377e395. Brunsdon, C., Comber, L., 2018. An Introduction to R for Spatial Analysis and Mapping, second ed. SAGE Publications Ltd, London. Chun, Y., Griffith, D.A., 2013. Spatial Statistics and Geostatistics: Theory and Applications for Geographic Information Science and Technology. SAGE Publications Ltd, Thousand Oaks. Cliff, A.D., Ord, J.K., 1973. Spatial Autocorrelation. Pion, London. Cressie, N.A.C., 1993. Statistics for Spatial Data, Revised Edition. Wiley, New York. Cressie, N.A.C., Wikle, C.K., 2011. Statistics for Spatio-Temporal Data. Wiley, New York. Cressie, N.A.C., 2018. Mission CO2ntrol: a statistical scientist’s role in remote sensing of atmospheric carbon dioxide. Journal of the American Statistical Association 113 (521), 152e168. Curran, P.J., Atkinson, P.M., 1998. Geostatistics and remote sensing. Progress in Physical Geography 22 (1), 61e78. Curry, L., 1966. A note on spatial association. The Professional Geographer 18 (2), 97e99. Diggle, P.J., 2013. Statistical Analysis of Spatial and Spatio-Temporal Point Patterns. third ed. CRC Press, Boca Raton, FL. Elsner, J.B., Hodges, R.E., Jagger, T.H., 2011. Spatial grids for hurricane climate research. Climate Dynamics 39 (1e2), 21e36. Geary, R.C., 1954. The contiguity ratio and statistical mapping. The Incorporated Statis- tician 5 (3), 115e145. Goovaerts, P., 1999. Geostatistics in soil science: state-of-the-art and perspectives. Geoderma 89 (1e2), 1e45. Haining, R., 1990. Spatial Data Analysis in the Social and Environmental Sciences. Cambridge University Press, Cambridge. Haining, R., 2003. Spatial Data Analysis: Theory and Practice. Cambridge University Press, Cambridge. Journel, A.G., Huijbregts, C.J., 1978. Mining Geostatistics. Academic Press, London. Kelejian, H.H., Piras, G., 2017. Spatial Econometrics. Academic Press, Cambridge. Matheron, G., 1963. Principles of geostatistics. Economic Geology 58 (8), 1246e1266. Moran, P.A.P., 1948. The interpretation of statistical maps. Journal of the Royal Statistical Society B 10 (2), 243e251. xvi Quasi real-time energy use estimation using Google’s Popular Times dataPreface
- 15. Moran, P.A.P., 1950. A test for the serial dependence of residuals. Biometrika 37 (1e2), 178e181. Townsley, M., 2009. Spatial autocorrelation and impacts on criminology. Geographical Analysis 41 (4), 452e461. Ver Hoef, J.M., Peterson, E., Theobald, D., 2006. Spatial statistical models that use flow and stream distance. Environmental and Ecological Statistics 13 (4), 449e464. Waller, L.A., Gotway, C.A., 2004. Applied Spatial Statistics for Public Health Data. Wiley, New York. Preface xvii
- 16. CHAPTER ONE Introduction Yoshiki Yamagata1 , Hajime Seya2 1 Center for Global Environmental Research, National Institute for Environmental Studies, Tsukuba, Ibaraki, Japan 2 Departments of Civil Engineering, Kobe University, Kobe, Hyogo, Japan Contents 1.1 The definition of spatial data 1 1.2 Characteristics of spatial data: spatial autocorrelation and spatial heterogeneity 3 1.2.1 Spatial autocorrelation 3 1.2.2 Spatial heterogenity 4 References 5 1.1 The definition of spatial data Data relating to geospatial information is used in our everyday lives. In this book, based on the Japanese Basic Act on the Advancement of Utilizing Geospatial Information promulgated in 2007, we define the term geospatial information as (1) information that represents the position of a specific point or area in geospace (including temporal information pertaining to said informa- tion, hereinafter referred to as positional information); and/or (2) any information associated with this information. We term the data, which relates to geospatial information, as spatial data. In addition, the aforementioned geospatial and geographic information are taken to have the same meaning. Naturally, there are various other methods of defining spatial data (e.g., Waller and Gotway, 2004, pp. 38e39). Currently, the single most important book concerning spatial statistics is, undoubtedly, Cressie (1993). This great work spans 900 pages, and has served as a “dictionary” in this field for many years. The first chapter classified spatial Spatial Analysis Using Big Data ISBN: 978-0-12-813127-5 https://doi.org/10.1016/B978-0-12-813127-5.00001-1 © 2020 Elsevier Inc. All rights reserved. 1 j
- 17. data into geostatistical data, lattice data, and point patterns.1 We begin this section with an outline of these data. Let < be the whole set of real numbers, and let s ˛ <d be a spatial po- sition in Euclidean space of dimension d (usually d ¼ 2 or 3)2 and let Y(s) be a random (possibly multivariate) quantity at position s. The spatial process3 is defined as fYðsÞ: s ˛ Dg (D 3 <d shows the domain). d ¼ 2 corresponds to two-dimensional spatial coordinates (e.g., x and y planar coordinates), and d ¼ 3 is where the height dimension is added to this (e.g., elevation). As previously described, spatial data is data with location and attributes. Here, the term data is frequently used to correspond with observed values. In this book, the realization of the spatial process Y(s), namely the observed value, is expressed as y(s).4 Cressie and Wikle (2011) have developed a clear discussion by clearly separating a data model (DM) relating to y(s) and a process model (PM) relating to Y(s). In this book, DM and PM are used separately when needed, and the three types of spatial data are defined as follows. • Geostatistical data y(s) are the realized values obtained from the geostat- istical process Y(s), where s varies continuously within a fixed domain D. For example, elevation corresponds to this data type. • Lattice data y(s) are the realized values from the lattice process Y(s), where s varies on a countable subdomain of a fixed domain D. For example, socioeconomic data gathered from areas such as municipalities and satellite remote sensing image data at the pixel level correspond to this data type. • Spatial point patterns data y(s) are the realized values from the spatial point process, which is a spatial process relating to the position of a randomly occurring event where D itself is random. Event data, such as crime, correspond to this type. In addition, following Cressie and Wikle (2011, p.18), we assume a spatio-temporal process that adopts a time axis, expressing where t moves continuously within T3< as fYðs; tÞ: s ˛D; t ˛Tg and where it moves discretely as fYtðsÞ: s ˛D; t ˛Tg. 1 Note that among the standard texts with some reputation, Banerjee et al. (2014) refer to (a) and (b) as point-referenced and areal data, respectively, while Schabenberger and Gotway (2005) present (b) as lattice/regional data (other classification names are identical). 2 d > 3 is often used in the field of computer experiments. 3 Often also called a random field. 4 Arbia (2006) discusses the importance of distinguishing between Y(s) and y(s). However, there seems to be little awareness of this point in applied research. 2 Yoshiki Yamagata and Hajime Seya
- 18. However, our book basically focuses on spatial process/data, not spatio- temporal process/data. Now, observation points are taken to be si (i ¼ 1, ., N). In this book, the univariate random variable is expressed as either Y(si) or Yi, and its actual value as y(si) or yi. 1.2 Characteristics of spatial data: spatial autocorrelation and spatial heterogeneity According to Anselin (1988), the characteristics of spatial data are spatial autocorrelation and spatial heterogeneity. The term spatial dependence is also often used to mean the former of the two. Naturally, autocorrelation and dependency are not identical terms. However, in practice, both are often used to mean autocorrelation, and in this book, we advance the argument that they are mutually interchangeable, as in Anselin and Bera (1998, p. 240). In addition, although the term spatial correlation is often used, it is more appropriate to use the term autocorrelation rather than correlation for the correlation that occurs due to spatial positioning in the same variable (Getis, 2008). 1.2.1 Spatial autocorrelation Spatial autocorrelation, as shown in Fig. 1.2.1, is generally classified into positive spatial autocorrelation, in which neighboring data show similar trends, and negative spatial autocorrelation, in which neighboring data show notably different values. These are known as Tobler’s (1970) First Law of Geographydeverything is related to everything else, but near things are more related than distant things. The latter, shown in a checkerboard pattern in Fig. 1.2.1, is not always easy to explain intuitively; however, when considering, for example, the spatial distribution of forests or crops, negative spatial autocorrelation may occur if appropriate thinning is not performed Posive Spaal autocorrelaon Random distribuon Negave Spaal autocorrelaon Figure 1.2.1 A representation of spatial autocorrelation and are binary variables taking 1 and 0, respectively. Introduction 3
- 19. due to competition for necessary nutrients. See Griffith and Arbia (2010) for details concerning negative spatial autocorrelation. Mathematically, spatial autocorrelation is expressed by the following moment condition (Anselin and Bera, 1998): Cov yi; yj ¼ E yiyj EðyiÞ$E yj s0; cisj: (1.1) Here, yi and yj show data at points si ˛D; sj ˛D. Various phenomena showing spatial autocorrelation exist around us. Let us take an example of the official land prices (Land Market Price Publication) in Japan, published by the Ministry of Land, Infrastructure, Transport and Tourism. One method used by real estate surveyors is transaction comparison, where land is evaluated by comparing the surrounding transaction prices. As a result, there is the possibility of spatial autocorrelation occurring in appraisal value (Tsutsumi and Seya, 2009). In another example, in the field of spatial epidemiology, the visualization and detection of disease clustering is of particular interest. For example, while it is known that the risk of each dis- ease occurring depends on the region (because of aspects like region-specific foods and culture), a positive spatial autocorrelation is evident where this kind of risk is spatially concentrated. Since illness is a type of event (number) data, a specific method is required to test for spatial concentration (e.g., Waller and Gotway, 2004). Here, we wish to briefly describe the difference between spatial autocorrelation and temporal autocorrelation. The dependency relationship of time series is modeled on the idea that the causal chain between the prior phenomenon and the phenomenon of interest follows the direction of progress, and that the phenomenon at a given point in time exerts no influ- ence on those prior to that point in time. Conversely, spatial autocorrelation is characterized by simultaneous occurrence in multiple directions with accom- panying feedback (Anselin, 2009). Although the details are described later, this relationship complicates the estimations and inferences. 1.2.2 Spatial heterogenity Spatial heterogeneity refers to the uneven distribution of a trait, event, or relationship across a region (Anselin, 2010). From a statistical point of view, it results in unstable model structure in space (function form and/or regression coefficient). One example is the segmented market of real estate (e.g., Islam and Asami, 2009). In a cross section, however, care needs to be taken since there are many cases in which spatial heterogeneity is indistinguishable from spatial 4 Yoshiki Yamagata and Hajime Seya
- 20. autocorrelation. For example, when the residuals from a regression analysis form positive spatial clusters, this can be interpreted as both spatial heteroge- neity (group level variance heterogeneity) and spatial autocorrelation (con- centration of similar residuals) (Anselin, 2001). Therefore, in spatial econometrics, it is common to impose structure on the problem through the specification of a model, coupled with extensive specification testing for potential departures from the null model (Anselin and Bera, 1998). However, approaches nonparametrically specifying unknown patterns of spatial heterogeneity have also been developed (Kelejian and Piras, 2017), which will also be explained in this book. References Anselin, L., 1988. Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, Dordrecht. Anselin, L., 2001. Spatial econometrics. In: Baltagi, B. (Ed.), A Companion to Theoretical Econometrics. Blackwell, Oxford, pp. 310e330. Anselin, L., 2009. Spatial regression. In: Fotheringham, S., Rogerson, P. (Eds.), The SAGE Handbook of Spatial Analysis. Sage Publications Inc, Los Angeles, pp. 255e276. Anselin, L., 2010. Thirty years of spatial econometrics. Papers in Regional Science 89 (1), 3e25. Anselin, L., Bera, A.K., 1998. Spatial dependence in linear regression models with an intro- duction to spatial econometrics. In: Ullah, A., Giles, D.E. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York, pp. 237e289. Arbia, G., 2006. Spatial Econometrics: Statistical Foundations and Applications to Regional Growth Convergence. Springer, New York. Banerjee, S., Carlin, B.P., Gelfand, A.E., 2014. Hierarchical Modeling and Analysis for Spatial Data, second ed. Chapman Hall/CRC, Boca Raton. Cressie, N.A.C., 1993. Statistics for Spatial Data, Revised Edition. Wiley, New York. Cressie, N.A.C., Wikle, C.K., 2011. Statistics for Spatio-Temporal Data. Wiley, New York. Getis, A., 2008. A history of the concept of spatial autocorrelation: a geographer’s perspective. Geographical Analysis 40 (3), 297e309. Griffith, D.A., Arbia, G., 2010. Detecting negative spatial autocorrelation in georeferenced random variables. International Journal of Geographical Information Science 24 (3), 417e437. Islam, K.S., Asami, Y., 2009. Housing market segmentation: a review. Review of Urban Regional Development Studies 21 (2e3), 93e109. Kelejian, H.H., Prucha, I.R., 2007. HAC estimation in a spatial framework. Journal of Econometrics 140 (1), 131e154. Schabenberger, O., Gotway, C.A., 2005. Statistical Methods for Spatial Data Analysis. Chapman Hall/CRC, Boca Raton. Tobler, W., 1970. A computer movie simulating urban growth in the Detroit region. Economic Geography 46 (2), 234e240. Tsutsumi, M., Seya, H., 2009. Hedonic approaches based on spatial econometrics and spatial statistics: application to evaluation of project benefits. Journal of Geographical Systems 11 (4), 357e380. Waller, L.A., Gotway, C.A., 2004. Applied Spatial Statistics for Public Health Data. Wiley, New York. Introduction 5
- 21. CHAPTER TWO Mathematical preparation Hajime Seya1 , Yoshiki Yamagata2 1 Departments of Civil Engineering, Kobe University, Kobe, Hyogo, Japan 2 Center for Global Environmental Research, National Institute for Environmental Studies, Tsukuba, Ibaraki, Japan Contents 2.1 Definitions of notations 9 2.2 The classical linear regression model 10 2.2.1 The classical linear regression model and violation of typical assumptions 10 2.2.2 Endogeneity 12 2.2.3 Spatial autocorrelation of error term and heteroskedastic variance 16 2.3 The generalized linear model 17 2.4 The additive model 19 2.5 The basics of Bayesian statistics 23 2.5.1 Bayes’ theorem 23 2.5.2 The Markov chain Monte Carlo method 24 2.5.3 Bayesian estimation of the classical linear regression model 28 References 30 2.1 Definitions of notations In this book, scalars are shown in fine italics a, and vectors and matrices are shown in bold a (there are instances of lowercase characters and upper- case characters). Moreover, when a is a column vector and A is a matrix, ai is the ith component of a, ai is a vector from a with the ith component removed, Ai is the ith row of A, Ai,j is the component on row i column j of A, and Ai is a matrix from A with the ith row removed. Furthermore, I expresses an identity matrix, O a square matrix from 0, 1 a column vector of 1, 0 a column vector of 0, and A1 and A0 are the inverse matrix of A and the transposed matrix of A, respectively. Dimensions of matrices and vectors are omitted, where obvious, from the context. However, depending on their necessity, these are written as A[n] for an nth order square matrix and A[n m] for a matrix with n rows and m columns. In addition, as much as Spatial Analysis using Big Data ISBN: 978-0-12-813127-5 https://doi.org/10.1016/B978-0-12-813127-5.00002-3 © 2020 Elsevier Inc. All rights reserved. 9 j
- 22. possible, mathematical notations will follow the standard form given by Abadir and Magnus (2002). 2.2 The classical linear regression model Before explaining spatial statistics and spatial econometrics, we explain the basis of the classical linear regression (CLR) model. Also, the generalized linear model (GLM), the additive model, and Bayesian statistics are introduced here. Particularly, in recent years, many statistical estima- tions, inferences, and predictions with regression models have been based on the Bayesian statistical theory, and therefore it is desirable to understand its basic principles. In fact, Banerjee et al. (2014) (for geostatistics or spatial statistics) and LeSage and Pace (2009) (for spatial econometrics), two of the standard texts, have developed theories that are reliant mostly upon Bayesian statistics. 2.2.1 The classical linear regression model and violation of typical assumptions In the CLR model, the following relationship is established for all observed values yi at position si (i ¼ 1, ., N): yi ¼ b1 þ X K k¼2 xk;ibk þ εi: (2.2.1) where, yi denotes a dependent variable, xk,i (k ¼ 2, ., K) denotes an exogenous explanatory variable, b1 is a constant, bk is the regression coeffi- cient corresponding to xk,i, and εi is the error term. Expressing Eq. (2.2.1) as a matrix yields the following equation: 0 B B B @ y1 y2 « yN 1 C C C A ¼ b1 0 B B B @ 1 1 « 1 1 C C C A þ 0 B @ 2 6 4 x2;1 / xK;1 « 1 « x2;N / xK;N 3 7 5 1 C A 0 B B B @ b2 b3 « bK 1 C C C A þ 0 B B B @ ε1 ε2 « εN 1 C C C A ; (2.2.2) or y ¼ b11 þ xb2 þ ε: where y is an N 1 dependent variable vector containing yi, 1 is an N 1 vector containing 1, x is an N (Ke1) exogenous explanatory variables matrix consisting of xk,i, b2 is a (Ke1) 1 regression coefficient vector 10 Hajime Seya and Yoshiki Yamagata
- 23. consisting of bk, and ε is an N 1 error term vector consisting of εi. If we rearrange such that Xh[1;x], bh[b1;b0 2]’, we have y ¼ Xb þ ε: (2.2.3) In the CLR model, the following four assumptions are usually made: (1) X is exogenous. (2) Given X, the conditional expected value of y is Xb, and the conditional expected value of ε is 0. That is, E[εjX] ¼ 0. (3) Given X, the error term ε satisfies: Var½εjX ¼ s2 εI½N ¼ 0 B B B @ s2 ε 0 / 0 0 s2 ε 1 « « 1 1 0 0 / 0 s2 ε 1 C C C A : (2.2.4) Eq. (2.2.4) implies independent and identically distributed error terms. (4) The rank of X is K. That is, an inverse matrix exists in (X’X). In addition to these four assumptions, the following assumption is typi- cally made: (5) εwN(0s2 εI½N); that is, normal distributed error terms in population. This assumption implies y w N Xb; s2 εI½N : (2.2.5) With assumption (5), the ordinary least squares (OLS) estimator of b follows the normal distribution, making it possible to perform a hypothesis test on its significance when the number of observations in a sample is rather small (when large, we can apply the central limit theorem). The OLS estimator b bols of the CLR model is obtained by minimizing the sum of squares b ε'olsb εols of residual vector: b εols ¼ y Xb bols (2.2.6) The first-order condition of optimization yields X' y ¼ X'Xb bols (2.2.7) With assumption (4), there is an inverse matrix in (X’X), and therefore the OLS estimator of b can be obtained from b bols ¼ ðX'XÞ1 X' y (2.2.8) Mathematical preparation 11
- 24. Also, the variance of b bols is given by: Var b bols ¼ s2 εðX'XÞ 1 (2.2.9) Since s2 ε is usually unknown, we substitute s2 ε with the estimator b s2 ε;ols ¼ b ε'olsb εols=ðN KÞ (2.2.10) Note that the fitted value of y is given by b y ¼ Xb bols, and thus if this is substituted into Eq. (2.2.8), we obtain b y ¼ XðX'XÞ1 X'y. Here, PX ¼ X X'X 1 X' is a projection matrix that produces b y from y, and for this reason, it is termed a hat matrix. Similarly, MX ¼ I[N]PX is an operator that creates a residual from y. These operators are also important in the restricted maximum likelihood (REML) method described elsewhere. Where assumptions (1)e(4) hold, the OLS estimator becomes the best linear unbiased estimator (BLUE), and this property is known as the Gauss-Markov theorem. Unfortunately, however, it is rare in empirical analyses that all these assumptions are satisfied, and in many cases, violations of assumptions (1)e(3) occur in particular. Therefore, later we explain the consequences of violating assumptions (1)e(3), and the countermeasures to them. Note that with respect to assumption (4), it is possible to satisfy this assumption by removing the explanatory variable(s) that causes perfect multicollinearity. 2.2.2 Endogeneity When deviating from assumptions (1) and (2) (i.e., when a correlation between the explanatory variable and error term occurs), the OLS estimator lacks both consistency and unbiasedness. Cases where x has a measurement error, where an important variable is missing from the model (omitted variable bias), and where x and y are jointly determined (simultaneity/ reverse causality) are regarded as examples of this kind of situation. As a countermeasure to this problem, the instrumental variable (IV) method is applied to obtain consistent estimators for the regression coefficients. The IV(s) must have correlation to the endogenous explanatory variables conditionally on the other covariates, although they cannot have correlation to the error term conditionally on the other covariates. The latter condition rule out any direct effect of the instruments on the dependent variable or any effect running through omitted variables. This is called the exclusion restriction. As its generalization, the two-stage least squares (2SLS) method as well as the generalized method of moments (GMM) can also be used. 12 Hajime Seya and Yoshiki Yamagata
- 25. Since these methods are used for estimating parameters of the spatial eco- nometric models, let us briefly describe their basics here. We will now explicitly introduce endogenous variables into the CLR model, which we can express as follows: y ¼ Xb þ _ X _ b þ ε; (2.2.11) where we take X to be an N K explanatory variable matrix consisting of constant terms and exogenous variables, and _ X as an N L explanatory variable matrix consisting of endogenous variables. In addition, b denotes a K 1 regression coefficient vector corresponding to exogenous variables, and _ b denotes an L 1 regression coefficient vector corresponding to endogenous variables. Here, because _ X is an endogenous variable, it has a correlation to the error term (Cov[εi, _ xl;i]s0, c l ¼ 1, ., L). Hence we may consider introducing IVs, say Z[NP], which correlates with _ X, but does not correlate with ε. For the sake of identification, the degree P of Z must be greater than or equal to the number of endogenous variables L. Under such a scenario, the IV estimator can be obtained by applying the 2SLS method. That is, when the explanatory variable matrix is rearranged to form Rh X; _ X , a two-stage estimation is performed, such that R is projected onto the plane spanned by S h [X;Z], uncorrelated to the error term and the estimated b R value obtained, the next y is regressed on b R; not R. This is a simple idea of using b R, which does not have correlation to the error term. If a valid IV is used, the 2SLS estimator may be consistent. The 2SLS estimator of parameter € bh h b0 ; _ b0 i0 is finally obtained from b € b2sls ¼ b R 0 b R 1 b R 0 y; (2.2.12) Var h b € b2sls i ¼ s2 ε R0 b R 1 ; (2.2.13) where b R ¼ ðS0SÞ1 S0R, and therefore € b2sls ¼ h R0SðS0SÞ1 S0R i1 R0SðS0SÞ1 S0y. Since s2 ε is unknown, s2 ε can be substituted with an estimate using b s2 ε;2sls ¼ b ε'2slsb ε2sls=ðN KÞ; (2.2.14) Mathematical preparation 13
- 26. where b ε2sls ¼ y R € b b2sls (2.2.15) Note the use of R instead of b R. Since the coefficient estimator in the 2SLS method can be obtained asymptotically, it can be divided either by (NK) or N. The 2SLS method is also used as the parameter estimation method of the spatial lag model (SLM), which is one of the representative models of the spatial econometrics that will be introduced in Chapter 5. GMM, instead, can be applied for the parameter estimation of the spatial error model (SEM), which is also introduced in Chapter 5. Now, our explanation turns to the GMM. First let us explain the method of moments (MM), which also can be used as the parameter estimation of the CLR model. The MM is a method of estimating parameters by using moment conditions that a model should satisfy. The moment condition in the CLR model is the absence of correlation between the explanatory variables and the error term; that is, E½X0 iεi ¼ 0½K1; ci ¼ 1; .; N (2.2.16) Here, since Xi is a 1 K vector expressing the ith row component of X, the conditional expression becomes set K. In matrix form, we can write this condition as E½X0 ε ¼ 0½K1 (2.2.17) The corresponding sample moment condition can be obtained as X0ε N ¼ X0ðy XbÞ N ¼ 0½K1: (2.2.18) Hence with assumption (4), the MM estimator may be given as b bmm ¼ ðX0 XÞ 1 X0 y; (2.2.19) and thus it is understood to be identical to the OLS estimator. Here, let us write the moment condition of the CLR model shown in Eq. (2.2.16) in a more general way: E½hðyi; Xi; bÞ ¼ 0½R1; (2.2.20) Here, h(yi,Xi,b) is an R 1 vector-valued function. If we substitute with the sample moment condition, this becomes: hsðy; X; bÞ ¼ 1 N X N i¼1 hsðyi; Xi; bÞ; (2.2.21) 14 Hajime Seya and Yoshiki Yamagata
- 27. where hs(y,X,b) is also an R 1 vector-valued function. If the number of the estimand, K, of b matches the number of moment conditions R, as in the case of the CLR model, it is possible to obtain those parameters using the MM. However, where R K the parameters that exactly satisfy Eq. (2.2.21) are generally not present. Therefore, we consider determining parameters that minimize a quadratic form such as hsðy; X; bÞ0 Vhsðy; X; bÞ (2.2.22) The estimator obtained in this way is a GMM estimator. The GMM esti- mator is known to have consistency and asymptotic normality under very general conditions (Hayashi, 2000). Here, V is the weight assigned to each condition and Hansen (1982) showed that under some regularity con- ditions, the minimum variance of the GMM estimator can be achieved by using the following equation: V ¼ 1 N X N i¼1 hðyi; Xi; bÞhðyi; Xi; bÞ0 #1 (2.2.23) However, in order to obtain the estimator V in Eq. (2.2.23), it is neces- sary to substitute the corresponding estimates for b. Therefore, a two-step estimation is performed: calculating b b ð0Þ gmm using suitable initial value of weight (e.g., identity matrix value), substituting it into Eq. (2.2.23) to obtain b V 1 , and finally obtaining the GMM estimator by minimizing Eq. (2.2.22). Here, we can easily show that the 2SLS estimator is a special case of the GMM estimator. Now, let’s return to Eq. (2.2.11). When we set the IVs to Z[NP], we can define S h [X;Z][N(KþP)]. Now, by adding the moment condition relating to Z to that of the CLR model, the following equation is obtained: E½S0 ε ¼ 0½ðKþPÞ1 (2.2.24) Replacing the left side with a sample analogous, we obtain S0 y R€ b N ; (2.2.25) where Rh X; _ X , € bh h b' ; _ b' i' . The GMM estimator for € b is obtained by minimizing the quadratic form hs y; X; _ X; Z; € b 0 Vhs y; X; _ X; Z; € b ; (2.2.26) Mathematical preparation 15
- 28. and its sample analogous yields: S0 y R€ b N !0 V S0 y R€ b N ! ; (2.2.27) where V is obtained by: V ¼ s2 εS0S N 1 (2.2.28) From the first-order condition of optimization, the GMM estimator of € b is given by: b € bgmm ¼ h R0 SðS0 SÞ 1 S0 R i1 R0 SðS0 SÞ 1 S0 y: (2.2.29) It is apparent that this equation is identical to the 2SLS estimator (Eq. 2.2.12). In addition, the asymptotic distribution is also consistent with that obtained from the 2SLS method. For further details, please refer to Hayashi et al. (2000). 2.2.3 Spatial autocorrelation of error term and heteroskedastic variance Next, we examine violations of assumption (3), where the error term does not satisfy homoskedasticity and/or no-autocorrelation. In both cases the OLS estimator is unbiased, but not efficient. Particularly in the case of spatial data, there are many instances where no-autocorrelation does not hold due to spatial autocorrelation stems from unobserved factors. Now, let us expand the CLR model in the following manner: y ¼ Xb þ u; (2.2.30) Var½u ¼ E½uu0 ¼ S; (2.2.31) with S ¼ 0 B B B @ Var½u1 Cov½u1; u2 / Cov½u1; uN Cov½u2; u1 Var½u2 / Cov½u2; uN « « 1 « Cov½uN ; u1 / / Var½uN 1 C C C A ; where, S is termed a varianceecovariance matrix, which is a matrix with variance in the diagonal terms and covariance in nondiagonal terms. If S ¼ s2 εI does not hold, the OLS estimator is not BLUE, and the standard error estimator has bias, which may result in erroneous inference. 16 Hajime Seya and Yoshiki Yamagata
- 29. Fortunately, however, if the structure of S is known, using the generalized least squares method, b0s BLUE can be obtained: b bgls ¼ X0 X1 X 1 X0 X1 y. (2.2.32) Of course, S is usually not known, and it is necessary to set N N el- ements of S with any assumption. Simply speaking, in the modeling of geostatistical data (geostatistical model), the varianceecovariance matrix is directly constructed as a function of distance. Meanwhile, in lattice data modeling (spatial econometric model), it is indirectly constructed through structuring the dependency between the data or error terms (e.g., autoregression type and moving average type). These differences are explained in more detail in Chapters 4 and 5. 2.3 The generalized linear model In the CLR model introduced in Eq. (2.2.1), if we assume that the error term follows a normal distribution, it can be expressed as:1 E½yi ¼ mi ¼ Xib; yiwN mi; s2 ε ; (2.3.1) where Xi denotes the vector consisting of the ith row of X. Here, needless to say, it is not mandatory to assume normal distribution. The GLM can handle a wide class of distributions called an exponential distribution family, where we have f ðE½yiÞ ¼ f ðmiÞ ¼ Xib (2.3.2) Note that the relationship between mi and Xib are modeled by nonlinear function f(,). The function f(,) is called a link function, and this kind of model is called a generalized linear model. The GLM has the following characteristics: [1] Dependent variable yi follows a distribution belonging to the exponen- tial distribution group. [2] f(mi) and Xib have a linear relationship. Poisson distribution and binomial distribution, in addition to the normal distribution, are well-known distributions belonging to exponential distribu- tion family. Commonly used link functions differ, depending on the 1 For the sake of simplicity, we do not distinguish between the stochastic variable Yi and its observation yi. Mathematical preparation 17
- 30. distribution that we assume. For example, a logarithmic link function is gener- ally used for Poisson distribution and its generalization, negative binominal dis- tribution. These commonly used link functions are called canonical link functions, and are convenient for practical use because the maximum likeli- hood estimates of parameters can easily be calculated by applying the iterative reweighted least squares (IRLS) method. In the following, as an example of the GLM, we explain the Poisson regression model, which is often used in spatial statistics. Now, let yi be the number of occurrences of an event. Events {y1, ., yN} are assumed to be mutually independent, and it is assumed that the frequency of an event’s occurrence is very small. Given such a situation, the probability distribution of yi is yiwPoissonðmiÞ; (2.3.3) and a Poisson distribution can approximate this situation well. Famous ex- amples of phenomena that can be described by a Poisson distribution include the number of spelling mistakes when writing a page of text, and the number of traffic fatalities in a year in a given region. The probability mass function of the Poisson distribution is given by pðyijmiÞ ¼ m yi i expðmiÞ yi! ; (2.3.4) where yi! denotes the factorial of yi. In the Poisson distribution, yi takes an infinite value with yi˛{0,1,2,.,N}. An important property of the Poisson distribution is that expected value ¼ variance ¼ mi, and since the shape of the distribution is defined by one parameter, mi, it has the advantage of being extremely easy to use. In actual data, however, variance expected value, termed overdispersion, often occurs. In this case it is possible to use a negative binomial distribution that extends the Poisson distribution and assumes that the variations of mi follow the gamma distribution. The probability mass function of the negative binomial distribution is given by pðyijmiÞ ¼ G yi þ y1 yi!Gðy1Þ y1 y1 þ mi y1 mi y1 þ mi yi ; (2.3.5) and the expected value and variance are mi and miþymi 2 , respectively. It is therefore possible to adjust overdispersion using the parameter y. In the Poisson (or negative binomial) distribution, the logarithmic function 18 Hajime Seya and Yoshiki Yamagata
- 31. lnðmiÞ ¼ Xib (2.3.6) is often used as a link function. It follows the mean component given as mi ¼ expðXibÞ ¼ exp b1 þ x2;ib2 þ . þ xK;ibK (2.3.7) Here, we introduce offset term to Eq. (2.3.7). Let ni be the arbitral total number (e.g., population) of geographical unit i (i ¼ 1, ., N), and let yi be the number of occurrences of events. Then it is natural to think that the larger ni becomes, the number of event occurrences increases. However, if we set a dependent variable as a dimensionless quantity (i.e., ratio), it is no longer possible to distinguish 1/2 from 2/4 (i.e., information is lost), and it becomes unclear what kind of distribution we should assume for this ratio variable. As an alternative, therefore, we can assume mi ¼ ni exp b1 þ x2;ib2 þ . þ xK;ibK ; (2.3.8) where the expected value of yi, mi, is modeled such that it is proportional to ni. Taking the natural logarithm of both sides and in vector form, we obtain lnmi ¼ lnni þ Xib; (2.3.9) where ln ni is a constant called an offset term, which is a known constant included in the model, and can be easily incorporated in parameter estimation. Parameter b of the Poisson regression model can be estimated using the maximum likelihood method via the IRLS method. For more details about GLM, see, for example, Wood (2017). 2.4 The additive model As we have discussed, the GLM is a generalized linear model, in the sense that E(yi) and linear component Xib are related using a nonlinear function f(,). However, it is possible that f(mi) and the explanatory variables have nonlinear relationships. Hence natural extension is f ðmiÞ ¼ gðXibÞ; (2.4.1) where g(,) is a nonlinear function. Fig. 2.4.1 shows a case in which Seya et al. (2011) constructed a hedonic model in which rent data for Tokyo’s 23 wards in FY 2006 were regressed on several explanatory variables, and for one of the explanatory variables, the logarithm of occupied area, a partial residual plot (one index of a nonlinearity judgment by plotting xk;i b bkþ b εi with respect to explanatory variable xk,i) was created. From this figure, it can Mathematical preparation 19
- 32. be seen that the relationship with rent changes significantly around 1.9 (60e80 m2 occupied area). Where this kind of nonlinearity exists, in the CLR model, the relationship between the explanatory variable and the data cannot be expressed well. Now, among the nonlinear functions g(,), a model that expresses the sum of several smoothing functions like f ðmiÞ ¼ Xib þ g1ðx1iÞ þ g2ðx2iÞ þ .; (2.4.2) is comparatively easy to handle, and is termed the generalized additive model (Hastie and Tibshirani, 1990; Wood, 2017). Since this model contains both the parametric term Xib, and the nonparametric term g1(x1,i)þg2(x2,i)þ., it can be understood to be a semiparametric regression model (Ruppert et al., 2003). Of these, the nonparametric terms are often specified to use penalized spline functions. Next, for the sake of simplicity, we will explain the additive model (AM), assuming that f(,) itself is linear ( f(mi) ¼ mi), and also we assume the case of two explanatory variables. When data (yi; x1,i; x2,i) is obtained, the AM can be formulated as follows: yi ¼ b0 þ x1;ib1 þ x2;ib2 þ g1 x1;i þ g2 x2;i þ εi; (2.4.3) 1.2 1.4 1.6 1.8 2.0 2.2 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 ) t n e r ( l a u d i s e R + t n e n o p m o C Figure 2.4.1 Partial residual plot (y axis:xk;i b bk;i þ b εi, x axis:xk,i). The R function crPlots (car package) was used for the plot. Dotted line: linear estimator; solid line: LOESS esti- mator (smoothing). 20 Hajime Seya and Yoshiki Yamagata
- 33. where g1 and g2 are smoothing functions, respectively, and εi is the inde- pendent and identically distributed error term. When a smoothing function is specified by a linear spline function, the following equation is obtained: yi ¼ b0 þ x1;ib1 þ x2;ib2 þ X Q1 h¼1 b1;h x1;i k1;h þ þ X Q2 h¼1 b2;h x2;i k2;h þ þ εi; (2.4.4) where (xk)þ is an operator that is 0 for xk, and xk for xk, and k expresses a “not.” b1,h and b2,h are the corresponding parameters. For example, let us suppose that the relationship between x1andy changes greatly in the vicinity of x1 ¼ 0.6. In this case, with k1 ¼ 0.6, for values less or greater than 0.6, we should attach a difference in the relationship between x1and y (see Ruppert et al., 2003, for an explanation of the figure on this point). Since multiple such points can usually be seen, multiple nots are placed corresponding to each explanatory variable, respectively. In the example of Eq. (2.4.4), Q1 nots are allocated to the variable x1, and Q2 nots are allocated to variable x2. If there are too many nots Q1 and Q2, issues of overfitting the data may occur. Hence we can estimate parameters using the penalized least squares method with restrictions on possible values of b1,h and b2,h. There are various methods for this restriction; for example, max b1;h m, P b1;h m, P b2 1;h m(m is an appropriate constant). The third restriction may lead to a type of ridge estimator. Expressing Eq. (2.4.4) in a matrix form, we obtain: y ¼ Xb þ Zb þ ε; (2.4.5) where X ¼ 2 6 6 6 4 1 « 1 x1;1 « x1;N x2;1 « x2;N 3 7 7 7 5 ; b ¼ h b0; b1; b2 i0 ; b ¼ b1;1; .; b1;Q1 ; b2;1; .; b2;Q2 0 ; Z ¼ ½Z1; Z2; Mathematical preparation 21
- 34. Z1 ¼ 2 6 4 x1;1 k1;1 þ / x1;1 k1;Q1 þ « 1 « x1;N k1;1 þ / x1;N k1;Q1 þ 3 7 5; Z2 ¼ 2 6 4 x2;1 k2;1 þ / x2;1 k2;Q2 þ « 1 « x2;N k2;1 þ / x2;N k2;Q2 þ 3 7 5: Define Rh[X;Z]; € bh b0 ; b0 0, and a matrix D as D ¼ 0 B B @ O½22 O½2Q1 O½2Q2 O½Q12 e l 2 1 I½Q1Q1 O½Q1Q2 O½Q22 O½Q2Q1 e l 2 2 I½Q2Q2 1 C C A; (2.4.6) where e l1;e l2 0 denotes the Lagrangian multipliers. The penalized least squares estimate for € b can be obtained by minimizing y R€ b 2 þ € b 0 D€ b; (2.4.7) where jj . jj denotes a vector norm. The optimization yields b € be l ¼ ðR0 R þ DÞ 1 R0 y: (2.4.8) When e l1 or e l2 takes a value close to 0, overfitting the data tends to occur. When the value takes a relatively large value, on the other hand, we cannot fully capture the nonlinearity between x1(or x2) and y. Hence the calibration of these parameters are very important. Here, let’s consider substituting fixed effects b1,h and b2,h with random effects u1;hwi:i:d:N 0; s2 1u and u2;hwi:i:d:N 0; s2 2u , respectively, and formalizing the model as a mixed model. This will smoothen the linear spline function, and avoid data overfitting (Ruppert et al., 2003, p.109). Moreover, a further advantage is that there are many statistical packages for mixed models. Substituting b1,h, b2,h by u1,h, u2, h in Eq. (2.4.7) and dividing by s2 ε, we obtain 1 s2 ε y Xb Zu 2 þ e l 2 1 s2 ε u1 2 þ e l 2 2 s2 ε u2 2 ; (2.4.9) 22 Hajime Seya and Yoshiki Yamagata
- 35. where u ¼ ½u0 1; u0 20 , u1 ¼ u1;1; .; u1;Q1 0 , andu2 ¼ u2;1; .; u2;Q2 0 . Here, if we look at s2 1u ¼ s2 ε . e l 2 1, s2 2u ¼ s2 ε . e l 2 2, Eq. (2.4.9) can be no other than the general criterion for obtaining the best linear unbiased predictor (BLUP) of b and u (Ruppert et al., 2003, p.100). In other words, the penalized least squares method of Eq. (2.4.9) is equivalent to finding the BLUP in the mixed model. Hence let us reconstruct the model to be the standard form of the mixed model as y ¼ Xb þ Zu þ ε ¼ R€ b þ ε; (2.4.10) Cov u ε ¼ 2 6 6 4 s2 1;uI½Q1 O O O s2 2;uI½Q2 O O O s2 εI½n 3 7 7 5: (2.4.11) We can estimate € b with, for instance, the REML (Ruppert et al., 2003, pp.100e101) method. The estimator of € b can be obtained as b € be l ¼ ðR0 R þ DÞ 1 R0 y: (2.4.12) Here, as in the hat matrix in the linear regression model, the matrix RðR0R þ DÞ1 R0, converts an observed value into a fitted value, and its trace gives a degree of freedom to the model. This value can be interpreted as a measure of smoothness of the smoothing function, and it can also be obtained for each explanatory variable (Ruppert et al., 2003, pp.175e176). In this way, the degree of nonlinearity can be quantified. Moreover, whether nonlinearity ought to be considered can be judged statistically using a likelihood ratio test, which is possible to implement using the standard mixed model software. The AM is a model that forms the foundation of the geo-additive model, which is sometimes used for modeling large spatial data (see Chapter 4). Note that geo-additive model is an extension of the GLM framework using AM; please refer to Wood (2017) in regard to this. 2.5 The basics of Bayesian statistics 2.5.1 Bayes’ theorem In standard statistics (frequentism), we try to estimate parameters by assuming that they have a specific fixed value. However, in the Bayesian sta- tistical framework, we believe parameters to have a (probability) distribution. Mathematical preparation 23
- 36. This distribution, called a prior distribution, needs to be given to the analyst in advance. In Bayesian estimation, we obtain the posterior distributions by updating the prior distributions using observed data based on the Bayes’ the- orem, and performing a statistical inference (Bayesian inference) based on that posterior distribution. Since the choice of prior distributions affect the results of parameter estimation there is a position that we should, as much as possible, opt to give noninformative prior distributions. Since now, because of the development of computation techniques including the Markov chain Monte Carlo method (MCMC) and the Hamiltonian Monte Carlo, even when given noninformative prior distributions, it is possible to derive and evaluate the posterior distributions through simulation. When taking the prior probability density function of parameter q (q3J ) to be p(q), and the likelihood function to be p(yjq), the posterior probability density function of q is obtained as pðqjyÞ ¼ pðq; yÞ R pðq; yÞdq ¼ pðyjqÞpðqÞ mðyÞ fpðyjqÞpðqÞ; (2.5.1) where p(qjy) incorporates all prior information and data in the form of the prior distribution and likelihood function, respectively. The m(y) ¼ !p(q,y) dq term is a scaling constant, representing the marginal probability density function of y. Since this term is often difficult to evaluate analytically, and does not depend on q; it is often abbreviated as in Eq. (2.5.1). In Bayesian estimation, it is possible to perform flexible modeling by incorporating prior beliefs and knowledge about parameters into the model through the prior distribution in this way. In fact, there are also approaches to modeling spatial autocorrelation through prior distribution (see Congdon, 2010). 2.5.2 The Markov chain Monte Carlo method After obtaining the posterior distribution for q, statistical inferences may be made about the parameters. To that end, when taking the parameter of interest to be qj, it is necessary to use integral calculus to remove nuisance parameters qj, in which we have no interest. In this way we obtain a marginal probability density function, such that pðqjjyÞ ¼ Z pðqjyÞdqj: (2.5.2) By using this marginal probability density, point estimation/interval estimation for qjcan be performed. Unlike standard statistics, although the obtained parameters have a distribution, there are many cases where the 24 Hajime Seya and Yoshiki Yamagata
- 37. point estimate b qjis of our interest. For instance, the average, median mode of the marginal posterior distribution is used as the point estimate value. For interval estimation, we can use the Bayesian credible interval 100(1a)% for example. When a ¼ 0.05, the credible interval indicates that parameter b qj is included in the interval (l, m) with 95% probability. This concept is more intuitive and easier to understand than confidence intervals in frequentism. In the formula, we define Pr l b qj m ¼ Z m l p b qj y dqj ¼ 1 a; 0 a 1: (2.5.3) Parameter inference in the regression model (significance testing) is performed by asking whether or not 0 is included in the credible interval. For example, looking at the 95% credible interval, if (0:5 b qj 1:0), since 0 is not included within the credible interval, we can judge the parameter as being positively significant. However, since multiple integrations of Eq. (2.5.2) are difficult to perform where q has large dimensions, an approx- imation calculation is often necessary. Various Monte Carlo integration methods have been proposed to date,2 and in particular, the MCMC method Gelfand and Smith (1990) brought to the statistical sciences has been an epoch-defining method that can efficiently evaluate high- dimensional integration, contributing greatly to the development and practical implementation of Bayesian statistics3 . The MCMC method, as the name suggests, is a Monte Carlo method using a Markov chain. Prior to the MCMC method, many methods were based on independent sampling from the distribution; however, the MCMC method uses the Monte Carlo method for sample sequences with serial correlation. Here, a Markov chain is used to generate a sample series with a serial correlation. In Markov chains, there is a property that, when repeated a sufficient number of times from an appropriate initial value, the distribution of stochastic samples converges on an invariant distribution under regular conditions. Therefore, by constructing a Markov chain such that this invariant distribution becomes a posterior distribution, stochastic samples of the Markov chain can be used as probability samples from the posterior distribution. Representative algorithms include the Gibbs sampler and the MetropoliseHastings (MH) algorithm. We explain these next. 2 Rue and Martino’s (2007) integrated nested Laplace approximation method for function approximation without simulation has also seen major development in recent years. 3 In fact, the first author, Professor Alan Gelfand, is a prominent spatial statistician. Mathematical preparation 25
- 38. The Gibbs sampler is an algorithm for when a full conditional distribu- tion is available. When the probability density function of the posterior dis- tribution is taken to be p(qjy), the density function of the conditional posterior distribution can be expressed as follows: pðq1jy; q2; .; qJ Þ pðq2jy; q1; q3; .; qJ Þ « pðqJ jy; q1; .; qJ1Þ (2.5.4) When it is possible to generate random samples from each conditional distribution, we can use the Gibbs sampler as follows: (0) Set the initial value: qð0Þ ¼ q ð0Þ 1 ; q ð0Þ 2 ; .; q ð0Þ J 0 (1) for t ¼ 1 to T do (2) sample q ðtþ1Þ 1 wp q1 y; q ðtÞ 2 ; .; q ðtÞ J (3) sample q ðtþ1Þ 2 wp q2 y; q ðtþ1Þ 1 ; q ðtÞ 3 ; .; q ðtÞ J (4) . (5) sample q ðtþ1Þ J wp qJ y; q ðtþ1Þ 1 ; .; q ðtþ1Þ J1 (6) qðtþ1Þ ) q ðtþ1Þ 1 ; q ðtþ1Þ 2 ; .; q ðtþ1Þ J 0 (7) end for Here, when T/N, the empirical distribution of the sample series q ðtÞ 1 ; q ðtÞ 2 ; .; q ðtÞ J 0 ; t ¼ 1; .; T, converges on a joint distribution (poste- rior distribution). Now, by extracting the sample series q ðtÞ j ; t ¼ 1; .; T for a given parameter qj, we can calculate a point estimate and credible interval. In practice, the average and standard deviation are obtained for the part that excludes the period that depends on the initial value, termed the burn-in period. Where sampling from the conditional posterior distribution cannot be done simply, the MH algorithm is used. The MH algorithm is a method of sampling from a proposal distribution q(q*jq(t1) ), which approximates in lieu of a conditional posterior distribution, where the generation of random numbers is difficult. In instances where the proposal distribution is a symmetric distribution such as a normal distribution, since 26 Hajime Seya and Yoshiki Yamagata
- 39. q q qðt1Þ ¼ q qðt1Þ q is established, the simpler Metropolis algo- rithm can be used. The Metropolis algorithm can be described as follows: (0) Set the initial value: qð0Þ ¼ q ð0Þ 1 ; q ð0Þ 2 ; .; q ð0Þ J 0 (1) for t ¼ 1 to T do (2) sample q from proposal q q qðt1Þ (3) compute ratio a ¼ pðq jyÞ pðqðt1Þ jyÞ ¼ exp h ln p q y ln p qðt1Þ y i (4) if a 1,set q(t) ¼ q* else if a 1; set qðtÞ ¼ q with probability: a qðt1Þ with probability: ð1 aÞ (5) end if (6) end for Since we usually wish to sample many high-probability points in the probability density function, the movement to increase the value of the posterior distribution is accepted with a probability of 1. However, if we reject all movements that decrease the value of the posterior probability distribution, we will be unable to move from the position with the highest probability. Hence we also accept this move with an acceptance ratio a. The ratio should be about 0.2e0.4 (e.g., Brooks et al., 2011, p.424); however, care is required, as this depends on the dimensions of q. The normal distribution q q qðt1Þ ¼ N q qðt1Þ ; E0 is often used as the proposal distribution. However, because it is difficult to find a proposal distribution that closely approximates a posterior distribution where there are multiple dimensions, it is more practical to assume a one-dimensional proposal distribution qj for each parameter, like the Gibbs sampler. For this purpose, the random walk process can be used. Using the random walk process, the proposal distribution q q j q ðt1Þ j relating to a given parameter qj can be expressed thus: q q j q ðt1Þ j ¼ N q j q ðt1Þ j ; x2 ; (2.5.5) where x2 is an important parameter that determines the acceptance ratio, and is adjusted within the algorithm such that it approaches the appropriate acceptance ratio. For instance if we wish the acceptance ratio to be about 0.4, if the acceptance ratio falls below 0.3, x2 is multiplied by 0.9, and if it Mathematical preparation 27
- 40. exceeds 0.5, is multiplied by 1.1, during the burn-in period (Han and Lee, 2013). Where it is desirable that the proposal distribution be symmetrical, this kind of random walk process can be used; however, there are many instances where the normal distribution does not match the proposal distribution, such as when there is a positive limit on the state space that the parameter can take (Banerjee et al., 2014). In this case, it is necessary for the proposal distribution to use the asymmetric q q qðt1Þ . We use the MH algorithm where it is not symmetrical, which is the generalization of the Metropolis algorithm. The acceptance ratio in the MH algorithm is given by: a ¼ pðq jyÞq qðt1Þ q p qðt1Þ y q q qðt1Þ : (2.5.6) For details concerning the MCMC method, refer to Brooks et al. (2011). 2.5.3 Bayesian estimation of the classical linear regression model In this section, we explain Bayesian estimation, taking the CLR model as an example, as follows: y wN Xb; s2 εI : (2.5.7) The likelihood of this regression model is given by p y b; s2 ε ¼ Y N i¼1 1 ffiffiffiffiffiffiffiffiffiffi 2ps2 ε p exp ðyi XibÞ2 2s2 ε # f s2 ε N=2 exp ðy XbÞ0 ðy XbÞ 2s2 ε (2.5.8) Here, with y Xb ¼ y Xb b þ Xb b Xb ¼ b ε þ X b b b and X0b ε ¼ 0, we obtain ðy XbÞ0 ðy XbÞ ¼ h b ε þ X b b b i0h b ε þ X b b b i ¼ b ε0 b ε þ b b b 0 X0 X b b b ¼ ðN KÞs2 þ b b b 0 X0 X b b b ; (2.5.9) 28 Hajime Seya and Yoshiki Yamagata
- 41. where b b ¼ ðX0XÞ1 X0y, s2 ¼ N K 1 y Xb b 0 y Xb b . By substituting Eq. (2.5.9) into Eq. (2.5.8) we obtain p y b; s2 ε f s2 ε n=2 exp 2 6 4 ðN KÞs2 þ b b b 0 X0X b b b 2s2 ε 3 7 5 (2.5.10) The posterior distribution can be obtained by combining this likelihood with the prior distribution. Useful prior distributions, known as conjugate prior distributions, are often used. That is, if the posterior distributions are in the same probability distribution family as the prior probability distribution, the prior is called as conjugate. Now, for the prior distributions of b and s2 ε, we assume p b; s2 ε ¼ p b s2 ε p s2 ε : (2.5.11) Then, we assume that the conditional prior probability of b when given s2 εis the normal distribution b s2 εwN b0; s2 εE1 0 , and that the prior distri- bution of s2 ε follows the inverse gamma distribution s2 εwIGða0=2; b0=2Þ. Their density functions are given by p b s2 ε ¼ 1 2ps2 ε k=2 E0 1=2 exp ðb b0Þ0 E0ðb b0Þ 2s2 ε ; (2.5.12) p s2 ε ¼ ðb0=2Þða0=2Þ Gða0=2Þ s2 ε ða0=2þ1Þ exp b0 2s2 ε ; (2.5.13) respectively. According to Bayes’ theorem, because the posterior probability density function becomes p b; s2 εjy fp y b; s2 ε p b s2 ε p s2 ε , through a simple calculation, the conditional posterior distributions for each parameter are obtained as b s2 ε; ywN e b; s2 ε e E ; (2.5.14) s2 εjywIG e a=2;e b . 2 ; (2.5.15) where e b ¼ ðX0X þ E0Þ1 X0Xb b þ E0b0 e E ¼ ðX0X þ E0Þ1 e a ¼ a0 þ N, and e b ¼ b0 þ N K s2 þ b0 b b 0 h X0X 1 þ E1 0 i1 b0 b b . Mathematical preparation 29
- 42. Therefore, it is sufficient to perform the Gibbs Sampler based on these conditional distributions. Note that Bayesian estimation is a shrinkage estimator, in the sense that the OLS estimator is corrected to the direction toward average component in the prior distribution. In these explanations, we assumed that p b; s2 ε ¼ p b s2 ε p s2 ε . Next, we assume an independent prior distribution as p b; s2 ε ¼ pðbÞp s2 ε ; (2.5.16) and we set bwN _ b; _ E ; (2.5.17) s2 εwIG _ a=2; € b . 2 ; (2.5.18) Then, the conditional posterior distribution is given as b s2 ε; ywN € b; € E ; (2.5.19) s2 ε b; ywIG € a=2; € b . 2 ; (2.5.20) where € b ¼ s2 ε X0X þ _ E1 1 s2 ε X0y þ _ E1 _ b , € E¼ s2 ε X0X þ _ E1 1 , € a ¼ _ a þ N, and € b ¼ _ b þ y XbÞ0 y Xb . Lastly, with p b; s2 ε ¼ pðbÞp s2 ε , we set noninformative priors as pðbÞf1; (2.5.21) p s2 ε f 1 s2 ε : (2.5.22) With these priors, the conditional posterior distribution is given by b s2 ε; ywN b b; s2 εðX0 XÞ 1 ; (2.5.23) s2 ε ywIG ðN KÞ=2; ðN KÞs2 2 : (2.5.24) References Abadir, K., Magnus, J., 2002. Notation in econometrics: a proposal for a standard. The Econometrics Journal 5, 76e90. Banerjee, S., Carlin, B.P., Gelfand, A.E., 2014. Hierarchical Modeling and Analysis for Spatial Data, second ed. Chapman Hall/CRC, Boca Raton. Brooks, S., Gelman, A., Jones, G., Meng, X.L. (Eds.), 2011. Handbook of Markov Chain Monte Carlo. Chapman and Hall/CRC, Boca Raton. 30 Hajime Seya and Yoshiki Yamagata
- 43. Congdon, P., 2010. Applied Bayesian Hierarchical Methods. Chapman and Hall/CRC, London. Hayashi, F., 2000. Econometrics. Princeton Univ Pr. Gelfand, A.E., Smith, A.F.M., 1990. Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association 85 (410), 398e409. Han, X., Lee, L.F., 2013. Bayesian estimation and model selection for spatial Durbin error model with finite distributed lags. Regional Science and Urban Economics 43 (5), 816e837. Hansen, L.P., 1982. Large sample properties of generalized methods of moments estimators. Econometrica 50 (4), 1029e1054. Hastie, T., Tibshirani, R., 1990. Generalized Additive Models. Chapman Hall/CRC, London. Hayashi, F., 2000. Econometrics. Princeton University Press, Princeton. LeSage, J.P., Pace, R.K., 2009. Introduction to Spatial Econometrics. Chapman Hall/ CRC, Boca Raton. Rue, H., Martino, S., 2007. Approximate Bayesian inference for hierarchical Gaussian Markov random field models. Journal of statistical planning and inference 137 (10), 3177e3192. Ruppert, D., Wand, M.P., Carroll, R.J., 2003. Semiparametric Regression (Cambridge Series in Statistical and Probabilistic Mathematics). Cambridge University Press, Cambridge. Seya, H., Tsutsumi, M., Yoshida, Y., Kawaguchi, Y., 2011. Empirical comparison of the various spatial prediction models: in spatial econometrics, spatial statistics, and semiparametric statistics. Procedia-Social and Behavioral Sciences 21, 120e129. Wood, S.N., 2017. Generalized Additive Models: An Introduction With R, second ed. Chapman Hall/CRC, Boca Raton. Mathematical preparation 31
- 44. CHAPTER THREE Global and local indicators of spatial associations Hajime Seya Departments of Civil Engineering, Kobe University, Kobe, Hyogo, Japan Contents 3.1 Spatial weight matrix 33 3.1.1 Definition of the spatial weight matrix 34 3.1.2 Specification of the spatial weight matrix 37 3.1.3 Standardization of the spatial weight matrix 38 3.2 Testing for spatial autocorrelation 39 3.2.1 Testing for global spatial autocorrelation 39 3.2.2 Testing for local spatial autocorrelation 43 3.2.2.1 Local Moran statistic 44 3.2.2.2 Local Geary statistic 45 3.2.2.3 Gi and Gi* statistics 45 3.2.3 Examples 47 3.2.3.1 Japanese income data: an application of local Moran 47 3.2.3.2 Japanese population data: an application of local Geary 49 3.3 Testing for spatial heterogeneity 51 3.3.1 Testing for global spatial heterogeneity 51 3.3.2 Testing for local spatial heterogeneity: Hi statistic 51 References 53 3.1 Spatial weight matrix In Chapter 2, we explained that bias may occur in standard errors of regression coefficient estimates when spatial autocorrelation and/or spatial heterogeneity exists in the residuals of the regression model. To confront this problem, we can use spatial econometric models. A spatial weight matrix, which we describe in the following subsections, is a central tool in spatial econometrics. Spatial Analysis Using Big Data ISBN: 978-0-12-813127-5 https://doi.org/10.1016/B978-0-12-813127-5.00003-5 © 2020 Elsevier Inc. All rights reserved. 33 j
- 45. 3.1.1 Definition of the spatial weight matrix The spatial weight matrix is a convenient and easy-to-understand tool for addressing spatial autocorrelation among data. Here, the data may be observed values or residuals obtained from a regression model. First, we introduce the concept of spatial weight matrix with the following definition. The spatial weight matrix W of N N describes the relationship between the data observed at i and j. Let Si denote the neighborhood (label) set consisting of areas/points with dependency on area/point i. If there is a dependency between the data yi and yj observed at i and j (i:e:; j ˛ Si), the element of W is given as wijs0; if there is no dependency (i:e:; j ; Si), then it is given as wij ¼ 0. Let us take an example used in Seya and Tsutsumi (2014). We consider a simple case of five areas {1, 2, 3, 4, 5} as shown in Fig. 3.1.1. The coordinates of the central point of each area are given by s1; s2; s3; s4; s5, respectively. Here, for instance, in the case of area 1, if only area 2 is dependent, the neigh- borhood set of area 1, say S1 , becomes {2} and (w12s0), and the remaining {3, 4, 5} are excluded from the neighborhood set (w1j ¼ 0; for j ¼ 3; 4; 5). Because a neighborhood set can be considered for each of the areas {1, 2, 3, 4, 5}, we consider a matrix W with i rows and j columns, and provide its elements as wij. To avoid it explaining itself, the elements of the diagonal matrix are usually 0. Different from time series data, where unidirectional effects from old data / new data can be found, there is no clear order in cross-sectional Figure 3.1.1 Virtual area. 34 Hajime Seya
- 46. spatial data. Hence we typically assume that the relationship is bidirectional, and it can be modeled by setting {wijs0 and wjis0}. Of course we can also assume a unidirectional effect in an adhoc manner, for instance, as from large economy to small economy (Seya et al., 2012). So many ways of providing W can be considered, but those that are typically used in empirical research are as follows.1 • Contiguity-based W: it is 1 if the zone boundaries are in contact (shared) and 0 if not in contact (symmetric matrix). 0 1 0 1 0 1 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 0 = W . Affecting area 1 2 3 4 5 Affected area 1 2 3 4 5 (3.1.1) • k nearest neighbor (kNN)-based W: point j might be less than or equal to kth nearest neighbor of point i, the weight is 1;otherwise, the weight is 0 (asymmetric matrix). For example, if k ¼ 2, the following matrix can be obtained. W ¼ 0 B B B B B B @ 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 C C C C C C A (3.1.2) Notably, the W obtained using kNN is not necessarily a symmetrical matrix because i is not necessarily included in the area neighborhood k unit of area j, even if j is included in the area neighborhood k unit of area i. One more point to note is that, in the case of k ¼ 3, {2, 4} is included in the neighborhood set S5 of area 5, but it is not obvious which of {1, 3} should be included. Therefore, it is necessary to exogenously select one side, or make the selection including both (as weight 1/2, if needed). 1 The typical spatial weight matrices listed here can be easily implemented using the R spdep package, GeoDa, etc. Global and local indicators of spatial associations 35
- 47. • Inverse-distance-based W without a distance cutoff: W ¼ 0 B B B B B B @ 0 1=2 1=2 1=ð2:83Þ 1=ð4:12Þ 1=2 0 1=ð2:83Þ 1=2 1=ð2:24Þ 1=2 1=ð2:83Þ 0 1=2 1=ð4:12Þ 1=ð2:83Þ 1=2 1=2 0 1=ð2:24Þ 1=ð4:12Þ 1=ð2:24Þ 1=ð4:12Þ 1=ð2:24Þ 0 1 C C C C C C A z 0 B B B B B B @ 0 0:50 0:50 0:35 0:24 0:50 0 0:35 0:50 0:45 0:50 0:35 0 0:50 0:24 0:35 0:50 0:50 0 0:45 0:24 0:45 0:24 0:45 0 1 C C C C C C A (3.1.3) It is W that uses the inverse of the (typically Euclidean) distance, which does not set a cutoff that takes 0 when exceeding a certain distance. And it is a symmetrical matrix (in case of Euclidean). Here, we set wij ¼ (1/dij)a with a ¼ 1, but in empirical analysis, a ¼ 2 is also used in an analogy of gravity models. However, in recent years, when using a dense matrix in whichwijs 0 for almost all elements, it is suggested that spatial process may be overly smoothed and spatial correlation parameters are systematically underesti- mated in the spatial econometric model. Hence it is necessary to pay attention when applying it (see Smith, 2009; Arbia et al., 2019). • Inverse-distance-based W with a distance cutoff It is W using the inverse of the (typically Euclidean) distance with a cutoff that sets the weight to 0 if it exceeds a certain distance and it is a symmetrical matrix (in case of Euclidean). For example, if we set the cutoff to 2.5 km (i.e., 0.40), the following matrix is obtained. W ¼ 0 B B B B B B @ 0 0:50 0:50 0 0 0:50 0 0 0:50 0:45 0:50 0 0 0:50 0 0 0:50 0:50 0 0:45 0 0:45 0 0:45 0 1 C C C C C C A (3.1.4) Instead of the Euclidean distance, LeSage and Polasek (2008) use a traffic network distance. In the traffic network distance, W need not to be symmetric, if one-way roads exist. 36 Hajime Seya
- 48. 3.1.2 Specification of the spatial weight matrix The choice of weight matrix W affects both estimations and inferences (e.g., Florax and Folmer, 1992; Griffith, 1996; Stakhovych and Bijmolt, 2009). However, currently, there are few guidelines for selecting the correct W (Anselin, 2002). Following Stakhovych and Bijmolt (2009), we classified the means of providing W into the following three types: [A] Completely exogenous [B] Determining from the data [C] Estimating [A] is a typical method provided by whether area boundaries are in contact (contiguity-based W) as previously mentioned, as well as the inverse distance, Delaunay triangulation, and so forth (LeSage, 1999). For [B], Aldstadt and Getis (2006) proposed an algorithm termed AMOEBA (a multidirectional optimum ecotope-based algorithm),2 in which the geometric form of spatial clusters are identified, similar to a region growing algorithm of image segmentation. The distance need not be limited to the geographic one. We can consider other types of distances, including social economic distances (e.g., difference of gross domestic product [GDP]), migration flow (Conway and Rork, 2004), technological similarity (Lychagin et al., 2016), among others. These methods may also be categorized to [B]. However, we need to note that methods for parameter estimations and inferences of spatial econometric models have been developed for exogenous W. If W is endogenous, we need to rely on recently proposed parameter estimation methods for endogenous W (e.g., Qu and Lee, 2015; Zhou et al., 2016). Kostov (2010) noted that the reason why W based on geograph- ical distance is often used is that the exogeneity is automatically satisfied. There are very few studies classified as [C]. Fern andez-V azquez et al. (2009) tried to estimate W by generalized maximum entropy and generalized cross-entropy techniques. In a panel setting when the number of time series variables can grow faster than the number of time points for data, studies employed (graphical) LASSO to estimate W (Ahrens and Bhattacharjee, 2015; Moscone et al., 2017; Lam and Souza, 2019). Because research regarding the estimation of the elements of W in [C] is still under development, it is common to select the most adequate W from prepared candidates. We could mention Kostov (2010) based on boosting and LeSage and Pace (2009), who attempted a model selection using 2 GIS software that implements AMOEBA is available at the website of the first author (http://www.acsu. buffalo.edu/wgeojared/tools.htm). It can also be implemented using the AMOEBA package of R. Global and local indicators of spatial associations 37
- 49. posterior model probabilities based on a Bayesian approach.3 The latter Bayesian approach is useful for the selection of both X and W (LeSage and Pace, 2009). Kelejian (2008) and Kelejian and Piras (2011) extended the J-test, which is a representative method of nonnested model selection, to the spatial econometric model. Meanwhile, Mur and Angulo (2009) noted the usefulness of information criterion (comprehensibility and clarity of results in the sense of selecting the unique best model). Similarly, Stakhovych and Bijmolt (2009) stated that using information criterion (e.g., Akaike’s informa- tion criterion [AIC]) increases the possibility of correct model specification. Seya et al. (2013) attempted to simultaneously select W and X to minimize the AIC by applying a technique termed transdimensional simulated annealing that combines reversible jump Markov chain Monte Carlo and simulated annealing. Other attempts include not model selections but model ensembles. We can mention those based on Bayesian model averaging (LeSage and Parent, 2007) and weighted average least squares (WALS) (Seya et al., 2014).4 3.1.3 Standardization of the spatial weight matrix The spatial lag model (SLM), described in Chapter 6, is a representative spatial econometric model. For the maximum likelihood estimation of the parameters of this model, calculation of the term ðI rWÞ1 is necessary during the process of estimation (here r is a parameter that indicates the degree of spatial autocorrelation). In many previous studies, jrj 1 is assumed in the analogy of time series models, but the singular point of ðI rWÞ is not necessarily outside the range of ð 1; 1Þ. Therefore, some standardization is typically performed on W to guarantee the existence of the inverse matrix. The most widely used standardization is row- standardization (also termed row-normalization), in which the elements of each row of W sum to unity. With the contiguity-based W (Eq. 3.1.1), row-standardization may be performed as W ¼ 0 B B B B B B @ 0 1=2 1=2 0 0 1=3 0 0 1=3 1=3 1=2 0 0 1=2 0 0 1=3 1=3 0 1=3 0 1=2 0 1=2 0 1 C C C C C C A (3.1.5) 3 The Spatial Econometrics Toolbox of Matlab is available at the website of the first author. 4 See Magnus and De Luca (2016) for details about the WALS. 38 Hajime Seya
- 50. Other documents randomly have different content
- 54. The Project Gutenberg eBook of Busy Ben and Idle Isaac
- 55. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Busy Ben and Idle Isaac Author: Unknown Release date: June 5, 2019 [eBook #59676] Language: English Credits: Produced by hekula03, Robert Tonsing, and the Online Distributed Proofreading Team at http://www.pgdp.net (This book was produced from images made available by the HathiTrust Digital Library.) *** START OF THE PROJECT GUTENBERG EBOOK BUSY BEN AND IDLE ISAAC ***
- 56. DEAN SON’S PENNY BOOKS. 11 NURSE ROCKBABY’S PRETTY STORY BOOKS. BUSY BEN, AND IDLE ISAAC. LONDON, DEAN SON, THREADNEEDLE STREET. BUSY BEN AND IDLE ISAAC.
- 57. IN a very pretty village, there once lived two boys, named Benjamin and Isaac. Benjamin was always at work, doing something; but Isaac liked best to take his ease, and did not care, like Benjamin, to be tidy and clean; but went ragged and dirty: so, at last, they were called by the neighbours, Busy Ben, and Idle Isaac. One morning, as Busy Ben was passing through the village he saw Isaac idling about, “What! not doing any thing?” said Ben. “I have
- 58. been lying in the sun,” replied Isaac, “and now I am so hot, I have taken off my jacket, to cool myself.” Busy Ben saw that he had, and he saw also what a torn shirt he had under it. “And I,” said Ben, “have been working in the garden, and watering the beds. Is it not much better to be at work, than lying in the sun?” “You may think so,” said Idle Isaac, yawning, “but I do not; besides, my father can afford to keep me without,—he has plenty of work.” “But he may not always have plenty of work,” answered Ben: “and even if he has, it is better to learn to keep ourselves.”
- 59. Such was the way these two boys talked to each other; but Isaac did not improve, he still idled his time away; true, he would now and then take a spade in hand, but he never did any work with it, but stood still or, sauntered about, staring first at one thing and then at another. Ben, on the contrary, by being always busy, grew up strong healthy, and clever; and, though still a boy, was soon able to maintain himself. The farmers round about were always glad of his help to tend their sheep, for Ben was never above being industrious, and though he often worked hard, Busy Ben was happier, and more cheerful, than Idle Isaac.
- 60. And so it will always be: for “From honest labour many a blessing springs, And health, and wealth, and happiness, it brings.” When Ben was fifteen years old, a gentleman, who was a ship- builder, came to the village, and was so much pleased by what he heard of Busy Ben, that he apprenticed him. Ben was attentive to what he was taught, and soon became a clever workman,—for, by trying to find out the meaning of what he was doing, he was soon able to finish the work in a much better manner than he would otherwise have done.
- 61. And thus Busy Ben passed many years; the more he got on, the harder he worked; and the cleverer he became, the more pains he took to improve. At last, by industry and frugality, he was able to set up in business for himself, and still keeping on as he began, he became a rich man. Idle Isaac had grown up, too, but was as lazy a man as he had been idle when a boy. As usual, he was never seen at work,—but was often seen idling away his time,—or in the company of boys, equally idle as himself.
- 62. Never having learned to be useful to anybody else, he was now of no use to himself, and was often without a meal, and always in shabby clothes;—for no one cared to assist an idle man, who had brought on his own troubles. One day, whilst sitting in a public-house, as he often did, reading the newspaper, he saw in it that Busy Ben had built a ship, which was the talk of all London; and that the Queen, hearing how Busy Ben got on, from a poor boy, to be what he was, had knighted him. And this was all owing to his good conduct and industry. Idle Isaac now began to wish he had minded what Busy Ben had said to him when a boy: he was at this time very poor, so he knew
- 63. he must do something;—but never having been taught a trade— what that something was, we shall soon see. Ben, by this time had a wife and two children, a little boy and girl; and they had a pet dog and cat: and, one morning, there was a great yelping and mewing between them, for the man who usually brought their meat was a long while past his time.
- 64. “O, here he is,” at last cried one of the children; “but, papa, it is not the same man that came before.” And who do you think the new cats’ meat man was?—Idle Isaac!— yes, it was indeed him! Ben rang for his servant, and desired him to ask Isaac into the hall. When Idle Isaac came in, and saw Sir Benjamin, he was very sad, for he well knew that he had passed the time in idleness in which he might, had he been industrious and saving, have gained a good home for himself. “I am properly punished,” said Idle Isaac, “for my want of industry; and you are justly made happy for your application.” “It is never too late to mend,” said Ben; “but it is of little use to work hard, unless you save up a part of what you earn.” “I am sure you
- 65. speak truth,” answered Isaac, “for you have earned for yourself a fine house, and wealth, whilst I have only a poor hovel, and a barrow.” “It is never too late to mend,” said Ben again, “and if you will promise to save up a part of what you earn, and put it into the savings’ bank, I will put as much money to it every half year, that you may have something to live on in your old age.” Save for old age while you may; Sunshine lasts not all the day. So Isaac promised, as he wheeled away his barrow, that he would try.—And I hope all my little readers, who may be like Idle Isaac, will take pattern by BUSY BEN.
- 66. NURSE ROCKBABY’S PRETTY STORY BOOKS. 12 Sorts. ——oo——
- 67. 1 New Historical Alphabet 2 The Ramble; and what was seen in it 3 Hans Dolan, and his Cat 4 Easy Reading and Pretty Pictures 5 The Brother and Sister 6 Little Rhymes for Little Readers 7 Greedy Peter 8 An Entertaining Walk with Mamma 9 The Little Merchant 10 Story of the Princess Fairlocks 11 Busy Ben and Idle Isaac 12 A Little Child’s Little Book of Goodness and Happiness ——oo—— Nurse Rockbaby’s Pretty Stories, COLOURED ENGRAVINGS, 12 Sorts, 2d. each. Transcriber’s Notes: Silently corrected typographical errors.
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