Geographically Weighted Polynomial Regression: Selection of the
Optimal Bandwidth and the Optimal Polynomial Degrees and Its
Application to Water Quality Index Modelling
Toha Saifudin
1
, Fatmawati
2
and Nur Chamidah
1
1
Department of Statistics, Airlangga University, Surabaya, Indonesia
2
Department of Mathematics, Airlangga University, Surabaya, Indonesia
Keywords: Spatial Modelling, Varying Coefficients, Nonlinear Relationships, Model Comparison, Water Quality Index.
Abstract: In this paper we introduce geographically weighted polynomial regression (GWPolR) model as a
generalization of GWR model. It is an alternative solution to overcome the existence of nonlinear relationships
between response variable and one or more explanatory variables involved in spatial modelling. This study
aims to provide a procedure for finding the optimal bandwidth and polynomial degrees in the GWPolR
technique. This procedure is applied to Water Quality Index (WQI) modelling based on several factors.
Because GWR method does not account for nonlinearity relationships of the spatial data type, we hypothesize
that a GWPolR model will help us better understand how the factors are related to WQI patterns. Both types
of models were applied to examine the relationship between WQI and various explanatory variables in 33
provinces of Indonesia. The goal was to determine which approach yielded a better predictive model. Based
on three explanatory variables, i.e. percentage of untreated waste, population density, and number of micro
industries, the GWR produced a spatial precision, i.e. R
2
, of 35.28%. GWPolR efforts increased the value
explained by explanatory variables with better spatial precision (R
2
= 50.12%). The results of GWPolR
approach provide more complete understanding of how each explanatory variable is related to WQI, which
should allow improved planning of explanatory management strategies.
1 INTRODUCTION
The geographically weighted regression (GWR)
model has been one of the useful methods in spatial
analysis (Fotheringham et al., 2002). The GWR
technique has been studied both in theory and
application. In the scope of theory, many authors have
studied the GWR technique, for example: Brunsdon
et al. (1999), and Fotheringham et al. (1998; 2002). In
application, the GWR technique has been also widely
applied to different areas, for example: in climatology
(Al-Ahmadi& Al-Ahmadi, 2013; Brunsdon et al.,
2001; and Wang et al., 2012), in econometric (Mittal
et al., 2004; Lu et al., 2014), in social field
(Fotheringham et al., 2001; Han & Gorman, 2013).
From those studies, some procedures relating to the
GWR model have been established.
The GWR coefficients are spatially varying.
However, it is important to remember that the GWR
model is an expansion of global linear regression
(GLR) model, so the response variable in each
location is fitted as a linear function of a set of
explanatory variables. It may be unrealistic in some
real-life situations. There are many possibilities of
nonlinearity cases in the relationships between one or
more explanatory variables and the response. In a
health study, the relationship between age and child
weight tend to be nonlinear. In economic study, the
relationship between advertising finance and the
revenue is commonly nonlinear. In application of
spatial analysis, Chamidah et al. (2014) inspected the
vulnerability modelling of dengue hemorrhagic fever
(DHF) disease in Surabaya based on geographically
regression. The results obtained have not been
satisfactory. The existence of nonlinear relationships
between one or more explanatory variables and the
DHF level is suspected to be the cause. In other
example, Chiang et al. (2015) showed that the
influence of the convenience factor (access to public
facilities) is nonlinear over the housing prices in
Taipei, Taiwan. If the nonlinear relationships are
present in the real situation, then the linear approach
may be unrealistic. Therefore, some approach models
Saifudin, T., Fatmawati, . and Chamidah, N.
Geographically Weighted Polynomial Regression:Selection of the Optimal Bandwidth and the Optimal Polynomial Degrees and Its Application to Water Quality Index Modelling.
DOI: 10.5220/0008517700930100
In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 93-100
ISBN: 978-989-758-407-7
Copyright
c
2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
93
which accommodate the actual pattern of the real data
are required to improve the basic GWR model.
Some expansions of GWR have been proposed in
recent years. One of the important expansions is
geographically weighted generalized linear models
(GWGLM) covering geographically weighted
poisson regression (GWPR) and geographically
weighted logistic regression (GWLR). Although
there was GWGLM, an extension of GWR which
accommodates response in continuous variable and
has nonlinear relationships with the explanatory
variables has not been found. Thus, an extension of
the GWR model which can overcome the problems
described above is needed.
As a solution to the above problem, we introduce
a generalization of the basic GWR model by using
polynomial function approach. Then, it is called the
geographically weighted polynomial regression
(GWPolR) model. Here, we provide an analytical
formula for the coefficient estimator which still
depends on a bandwidth and several polynomial
degrees. Next, the purpose of this paper is to provide
a procedure for selecting the optimal bandwidth and
the optimal polynomial degree of each explanatory
variable involved in the model. Then, as an example
of application we provide a modelling of water
quality index in Indonesia.
2 GEOGRAPHICALLY
WEIGHTED POLYNOMIAL
REGRESSION
In this section, we will introduce the GWPolR model
and a procedure to get its optimal estimator.
2.1 Model and Weighted Least Squares
Estimation
We briefly review the basic GWR model from
Brunsdon et al. (1996; 1999) and Fotheringham et al.
(1998; 2002). The GWR model is in the form of
,
(1)
where
is distributed
.
We expand the linear relationship in equation (1)
by using polynomial function approach as follows
(2)
where
. In a matrix form, it can be written
as
(3)
where
(4)
and
. (5)
For a given location (
), we can estimate
by minimizing the weighted least square
function as follows
, (6)
with respect to each element of
. An
explicit expression of the solution is
, (7)
where
,
, (8)
and
, (9)
is a diagonal weighting matrix with the elements
for .
By taking
to be each of the designed
locations
, , we can obtain the
vector of the fitted values for the response y at n
designed locations as
G, (10)
where
ICMIs 2018 - International Conference on Mathematics and Islam
94
G
, (11)
is called a hat matrix of GWPolR model. Based on the
hat matrix G, the residual vector is
G
, (12)
and the residual sum of squares(RSS
pol
) is
RSS
, (13)
where I is identity matrix of order n.
2.2 Spatial Weighting Functions
In spatial analysis, the weighting matrix
contains of different emphases on different
observations in producing the estimated parameters.
The first approach of the weighting matrix at location
i can be expressed as
(14)
The weighting function written above has a
discontinuity problem. One method to solve the
problem is to specify
as a continuous function
of
. One preference might be
(15)
where h denotes the bandwidth. It is the most
common choice in practice (Fotheringham et al.,
2002).
An alternative weighting function can be created
by the bisquare kernel weighting function as follows
(16)
2.3 Cross-Validation Criterion for
Choosing the Optimal Model
The estimator of GWPolR depends on the weighting
function selected and on the polynomial degree of
each explanatory variable. So, the bandwidth h and
the number of the polynomial degrees should be
determined.
The problem is how to select the optimal
bandwidth and the optimal number of polynomial
degree of each explanatory variable. Cross-validation
(CV) approach is a solution to this problem
(Fotheringham et al, 2002). Here, we adopt such
procedure by adding polynomial degrees for
explanatory variables which should be selected. So,
we have
(17)
as an objective function, where
is the fitted value of
under bandwidth h and degrees
of polynomial
with the observation at
location
omitted from the process of
estimation. Select
,
as the optimal
values, such that the equation (17) is minimum.
2.4 An Algorithm for Finding
Bandwidth and Polynomial Degrees
in Optimal Condition
Here, we provide an algorithm to select the optimal
bandwidth and the optimal number of polynomial
degrees based on the CV criterion as follows:
1) Determine the number of explanatory variables
involved in the model, denoted by p.
2) Specify the maximum polynomial degree for
each explanatory variable, denoted by
for
3) Find all arrays of numbers obtained from the
existing polynomial degrees for all explanatory
variables. Let s be the number of arrays, then
.
4) Find the minimum CV value of GWPolR
modelling in each array.
5) Find the smallest CV value among the minimum
CV values generated from the entire arrays.
6) Select the bandwidth and polynomial degrees
that yield the smallest CV value obtained in step
5 as the optimal solution.
To explain the above algorithm, we will give the
following illustration. For example, based on a given
spatial dataset we use three explanatory variables in
the model, so we have Then, we specify the
maximum polynomial degree for each explanatory
variable, for instance we set
,
and
for the 1
st
, 2
nd
and 3
rd
explanatory variables,
respectively. Based on the setting, we have several
arrays of polynomial degrees.
These arrays are listed in Table 1.
Geographically Weighted Polynomial Regression:Selection of the Optimal Bandwidth and the Optimal Polynomial Degrees and Its
Application to Water Quality Index Modelling
95
Table 1: All possible arrays of polynomial degrees for the
setting of 𝑑
1
2, 𝑑
2
3 and 𝑑
3
2.
Number
Array
1
(1, 1, 1)
2
(1, 1, 2)
3
(1, 2, 1)
4
(1, 2, 2)
5
(1, 3, 1)
6
(1, 3, 2)
7
(2, 1, 1)
8
(2, 1, 2)
9
(2, 2, 1)
10
(2, 2, 2)
11
(2, 3, 1)
12
(2, 3, 2)
An array represents the polynomial degree of the
1
st
, 2
nd
and 3
rd
explanatory variables, respectively. For
example, in number 3 we have an array of (1, 2,
1) which means that the polynomial degree of the 1
st
,
2
nd
and 3
rd
explanatory variables are 1, 2, and 1,
respectively. This leads to a model as follows
. (18)
Using the model in (18), we select the minimum
CV value based on GWPolR estimation procedure.
The same selection is conducted on other arrays. So,
we have 12 minimum CV values. Then, we select the
smallest CV value among the existing 12 minimum
CV values. In the last step, we take the bandwidth and
polynomial degrees corresponding to the smallest CV
value obtained.
3 AN EMPIRICAL EXAMPLE:
WATER QUALITY INDEX
MODELLING
In this section, we will apply the algorithm explained
above to water quality index modelling. In addition,
we also provided modelling results to such data using
global linear regression (GLR) and GWR for
comparison.
3.1 Water Quality Index Dataset
Some researchers have stated that WQI is influenced
by many factors, including untreated waste (Dhawde
et al., 2018), population density (Opaminola and
Jessie, 2015; Liyanage and Yamada, 2017), and the
number of micro industries (Bhutiani et al., 2018).
The explanation from these researchers led the
authors to include the variables in the example here.
The involved variables in this study are water
quality index (WQI) as the response variable, and
three explanatory variables, i.e. percentage of
untreated waste (PUW), population density (PD), and
number of micro industries (NMI). Data was
provided by Ministry of Environment and Forestry
Republic of Indonesia (Kementerian Lingkungan
Hidup, 2015) and Statistics of Indonesia (BPS, 2016;
2017a; 2017b). Observation units are 33 provinces of
Indonesia in 2014.
Logically in public opinion, WQI variable has
contrary relationship with each explanatory variable
here. If the value of each explanatory variable
increases, it will cause in decreasing of the WQI
value. It means that the parameter of each explanatory
variable should have negative sign. The conformity
of the estimator signs will also be considered in the
comparison of models.
3.2 Global Linear Regression Results
Firstly, WQI is fitted by using GLR model. Based on
three explanatory variables declared above, the
simultaneous test for parameters yields a p-value of
0.007. It means that the parameters affect to the
response variable simultaneously. Here, we have four
global parameters including the intercept.
Furthermore, the results of GLR estimation on the
WQI dataset is listed in Table 2.
Table 2: Summary of global linear regression results on the
WQI dataset.
Predictor
Coef
SE Coef
T
p-value
Intercept
36.92
16.67
2.21
0.035
PUW
0.2228
0.2020
1.10
0.279
PD
-0.0015
0.0004
-3.45
0.002
NMI
-0.0079
0.0064
-1.24
0.226
From table 2, based on partial test with
significance level of 0.05 we know that only PD
variable affect significantly to the WQI (p-value <
0.05), whereas PUW and NMI variables are not
significant (p-value > 0.05). We suspect that PUW
and NMI variables may not have linear relationships
with the response, but they may have nonlinear
relationships. Here we don’t continue with
remodeling that uses significant explanatory
variables, but we let the results. We would like to see
ICMIs 2018 - International Conference on Mathematics and Islam
96
the comparison of results with other modelling
(especially using polynomial approach) involving the
same explanatory variables. In other words, if a
model yielded better results while we didn't apply the
same variables to all models in a comparison, we
would not know if the improvement was due to the
modelling approach or the explanatory variables that
was used to build each model.
To clarify our allegation, we examine the
misspecification function using Ramsey RESET. For
this test, we hypothesize
H0: there is not misspecification function in global
linear regression
H1: there is misspecification function in global
linear regression
By using syntax of resettest() in R statistical
software with optional input power=2 (for testing
the existence of polynomial degree of 2), we find that
the p-value of Ramsey RESET test is 0.0415. If we
take a significance level of 0.05, we decide to reject
the null hypothesis and conclude that there is
misspecification function in GLR modelling. Then,
this conclusion becomes an early reason for the need
to extend the linear modelling with polynomial
modelling on WQI dataset.
3.3 Geographically Weighted
Regression Results
This WQI dataset was drawn based on spatial area,
therefore proceeding with GWR model was
warranted. The GWR model in this study was
implemented using the following model:
. (19)
Based on this model and using weighting function in
equation (15), we found that the
of GWR was
0.3528 with a bandwidth of 31.3 (or equals to
3,484.378 km) when the minimum CV was
10232.400. Our preferred measure of model fit, RSS,
gave a value of 1,253.488. Here, we have several four
parameters that vary in 33 locations. The summary of
the GWR estimators is listed in Table 3.
We can see in Table 3 that the estimated
parameter of PD and NMI variables have negative
sign in every quartile, which is suitable with the
public opinion mentioned above. On the other hand,
the estimated parameter of PUW does not have
negative value at all. It is a contradiction to the public
opinion. We will try to correct this case by using
GWPolR modelling in the next subsection.
Table 3: Summary of geographically weighted regression
results on the WQI dataset.
Coef. of
Predictor
Min
1st
Quartil
Med
3rd
Quartil
Max
Intercept
30.900
33.490
35.4000
38.160
43.6300
PUW
0.13850
0.20720
0.24160
0.26540
0.29780
PD
-0.00161
-0.00159
-0.00158
-0.00156
-0.00153
NMI
-0.00797
-0.00794
-0.00791
-0.00789
-0.00775
3.4 Geographically Weighted
Polynomial Regression Results
Because of the presence of nonlinear relationships in
WQI dataset, we suspected that a GWPolR model
would help us better understand how explanatory
variables were related to WQI patterns. In this
subsection, we use an algorithm for selecting the
optimal bandwidth and polynomial degrees explained
above.
Based on WQI dataset we involved three
explanatory variables in the model, so we have
Then, we specified the maximum polynomial
degree for each explanatory variable. Here, we set the
same value, i.e.
for the maximum
polynomial degree of PUW, PD, and NMI variables.
Based on the setting, we had number of
arrays of polynomial degrees. Further, we
selected the minimum CV value based on GWPolR
estimation procedure in each array. The minimum CV
value and the accordingly optimal bandwidth are
listed in Table 4.
The smallest CV value among eight minimum CV
values was 4878.902. It is found in the row of number
six, according to the optimal bandwidth of 9 (or
equals to 1,001.898 km) and array of (2, 1, 2). This
array means that the optimal polynomial degree for
PUW, PD, and NMI variables are 2, 1, and 2,
respectively. So, the GWPolR model under optimal
condition in this study was
. (20)
Based on model in equation (20) and weighting
function in equation (15) we found that the
of
GWPolR was 0.5012.A goodness indicator of model
fit, RSS, gave a value of 966.1458. Here, we have
several six parameters that vary in 33 locations.
Furthermore, the summary of the GWPolR estimators
is listed in Table 5.
Geographically Weighted Polynomial Regression:Selection of the Optimal Bandwidth and the Optimal Polynomial Degrees and Its
Application to Water Quality Index Modelling
97
Table 4: Optimal bandwidth and minimum CV for each
array of polynomial degrees on the WQI dataset.
Number
Array
Opt h
Minimum CV
1
(1, 1, 1)
31
10253.855
2
(1, 1, 2)
42
11032.556
3
(1, 2, 1)
1
95336.878
4
(1, 2, 2)
1
95336.878
5
(2, 1, 1)
42
20854.561
6
(2, 1, 2)
9
4878.902
7
(2, 2, 1)
1
95336.878
8
(2, 2, 2)
1
95336.878
Table 5: Summary of geographically weighted polynomial
regression results on the WQI dataset.
Coef. Of
Predictor
Min.
1st Quartile
Median
3rd Quartile
Max.
Intercept
-35.0200
64.4100
203.2000
273.0000
322.5000
PUW
-7.1840
-5.9930
-4.1800
-0.4828
2.4000
PUW^2
-0.01593
0.00331
0.02799
0.03972
0.04789
PD
-0.00182
-0.00179
-0.00174
-0.00147
-0.00122
NMI
-0.04314
-0.04153
-0.00371
-0.02613
-0.01414
NMI^2
0.00001
0.00002
0.00004
0.000046
0.000049
From Table 5, the estimated parameter for the first
degree of explanatory variables involved in the model
have negative sign in almost of all positions. It means
that the estimated parameters of GWPolR are
according to the public opinion explained above. In
addition, another information can be obtained from
the GWPolR modelling. The estimated parameters of
the second-degree polynomial variables have positive
sign in almost of all positions. It interprets that the
PUW and NMI variables have an accelerating effect
to the WQI decrease.
3.5 A Comparison on Water Quality
Index Modelling
In this subsection, we make comparison about the
results obtained under the optimal condition of the
GWPolR, GWR, and GLR modelling on the WQI
dataset. The comparison is based on some goodness
of fit indicators (including CV, RSS, and R
2
) and
several other criteria. The results are presented in
Table 6.
We know that the minimum CV is a criterion for
finding the best fitted model. From Table 6, the CV
value of the GWPolR model is much lower than that
of the GWR model.
Table 6: The comparison of the GWPolR, GWR, and GLR
modelling on the WQI dataset.
Indicator
Model
GLR
GWR
GWPolR
Optimal h
-
31.3
9.0
Minimum
CV
-
10232.400
4878.902
RSS
1289.380
1253.488
966.145
R
2
33.40%
35.28%
50.12%
Conformity
to public
opinion
No
No
Yes
Number of
parameters
Simple
(Parsimony)
Complex
More
complex
Table 6 also presents that the RSS of GWPolR is the
lowest among the RSS of involved models here. In
view of GWPolR model, there are RSS decrease of
323.235 and 287.343 from GLR and GWR,
respectively. The coefficient of determination (R
2
) of
the GWPolR model can capture the largest amount
(50.12%) of variance of water quality index based on
the explanatory variables. There are R
2
increase of
16.72 and 14.84 from GLR and GWR, respectively.
These are relatively strong evidence of an
improvement in the model fit to the data. It means that
GWPolR model is the best model among the studied
models here based on goodness of fit indicators.
Furthermore, the estimated parameters of
GWPolR confirm to the public opinion about the
relation of WQI and the explanatory variables. The
other two models give inappropriate results. In
addition, here GWPolR has a parameter for the 2
nd
polynomial degree which interprets that the speed of
response change caused by an explanatory variable is
not constant. In other word, there is an acceleration
here. So, the GWPolR modelling has more complete
interpretation. On the other hand, The GWPolR
model has several more complex parameters. In
modelling, people often prefer to use a model with a
small number of parameters to be easily interpreted.
Therefore, a statistical test especially inter spatial
modelling is needed to examine whether a GWPolR
is significantly better than a GWR or not. If GWPolR
is proven to be significantly better than GWR even
though the number of parameters is more complex,
then GWPolR should be selected.
ICMIs 2018 - International Conference on Mathematics and Islam
98
4 CONCLUSIONS
The algorithm written in this paper yields a
bandwidth and some polynomial degrees for the
GWPolR model in optimal condition. It means that
the result is the best condition based on the used
criterion. However, computational programming
based on this algorithm still takes a long time. The
time will be longer when the number of variables
involved in model increases or the maximum degree
of polynomial is set greater. An efficient algorithm in
term of execution time is needed even though the
results may be only sub-optimal, for example, genetic
algorithm or neural network. Based on the goodness
of fit criteria(inter alia: CV, RSS, and R
2
) and
consideration of conformity with public opinion, we
can empirically conclude that the GWPolR model is
the best model among some models used here on
WQI dataset. Nevertheless, statistical tests between
spatial modelling need to be developed to determine
whether a GWPolR is significantly better than a
GWR or not.
ACKNOWLEDGEMENTS
The authors would like to thank Directorate General
of Science and Technology Resources and Higher
Education of The Ministry of Research, Technology
and Higher Education of Indonesia for scholarship
that be received by the first author throughout
undergoes doctoral program. The authors also thank
to Ministry of Environment and Forestry of Indonesia
and Statistics of Indonesia for kindly providing the
data.
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