CSIS Discussion Paper No. 136R

Configuration of sample points for the reduction of multicollinearity

in regression models with distance variables
Yukio Sadahiro* and Yan Wang**June 2015*Center for Spatial Information Science, The University of Tokyo**Department of Urban Engineering, The University of Tokyo

Center for Spatial Information Science, The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa-shi, Chiba 277-8568, Japan
E-mail: sada@csis.u-tokyo.ac.jp

Abstract
Keywords: sample points, multicollinearity, distance variables, regression models
Regression models often suffer from multicollinearity that greatly reduces the reliability of estimatedcoefficients and hinders an appropriate understanding of the role of independent variables. It occurs inregional science especially when independent variables include the distances from urban facilities. This
paper proposes a new method for deriving the configuration of sample points that reduces
multicollinearity in regression models with distance variables. Multicollinearity is evaluated by the
maximum absolute correlation coefficient between distance variables. A spatial optimization technique isutilized to calculate the optimal configuration of sample points. The method permits us not only to locate
sample points appropriately but also to evaluate the location of facilities from which the distance ismeasured in terms of the correlation between distance variables in a systematic way. Numericalexperiments and empirical applications are performed to test the validity of the method. The results
support the technical soundness of the proposed method, and provided some useful implications for thedesign of sample location.

1. Introduction
Regression models often suffer from multicollinearity, a state of high correlation between two
or more independent variables. Multicollinearity greatly reduces the reliability of estimated coefficients,
and consequently, hinders an appropriate understanding of the role of independent variables. Spatial
analysis often utilizes the distances from urban facilities as independent variables that can be highly
correlated with each other. Hedonic housing price models consider the distance from housing units to
CBD, schools, urban parks, and hazardous waste sites to evaluate the spatial environment around eachhouse (Berry (1976); Li and Brown (1980); Bender and Hwang (1985); Harrison Jr and Rubinfeld (1978);Ihlanfeldt and Taylor (2004); Noonan, Krupka, and Baden (2007)). A close relationship between obesity
and neighborhood shopping environment is represented as a regression model that utilizes theaccessibility to supermarkets, farmers' markets and local food stores as independent variables (Morland,Diez Roux, and Wing (2006); Rundle et al. (2009); Jilcott Pitts et al. (2013)). Spatial distribution of air
quality is often represented as a regression model that includes the distance from chemical plants, power
plants, smelters, and airports (Gordon and Gorham (1963); Riga-Karandinos and Karandinos (1998); Yu
et al. (2004)). This paper calls these facilities landmarks hereafter. Since distance variables are measured
on the same two-dimensional space, they are inevitably correlated at least to some extent with each other.
A wide variety of methods have been developed to handle with multicollinearity in regression
analysis (Farrar and Glauber (1967); Asteriou and Hall (2007); Chatterjee and Hadi (2013)). A simple
method is to remove independent variables that are highly correlated with others. Numerical measures areavailable for this purpose including correlation coefficient, determinant of correlation matrix, VIF, andmore sophisticated indices (Weissfeld and Sereika (1991); Kovàcs, Petres, and Tóth (2005); Curto andPinto (2007); Dormann et al. (2013)). Ridge regression, principal component regression, and theirextensions are also useful for mitigate the multicollinearity among distance variables (Mansfield, Webster,and Gunst (1977); Vigneau et al. (1997); Kashid and Kulkarni (2002); Ni (2011); Chen (2012); Miller(2012)).
The above methods implicitly assume that the observation data used in regression models havealready been prepared. On the other hand, if data are not yet collected, or a plenty of data are available
from which we can select a subset of observations, we can reduce the multicollinearity among distancevariables by carefully choosing the location of sample points. The former case often happens in physicalgeography and environmental science, where analysts collect their own data on the quality of air andwater, soil and vegetation, and the distribution of animal species by a field survey. The latter includes
spatial data of real estate such as multiple listing system in the US, land value data in Germany, and landand property survey data in Japan.
To derive an appropriate location of sample points, Heikkila (1988) evaluates themulticollinearity among distance variables using numerical experiments under hypotheticalcircumstances. The paper assumes a unit interval and a unit square on which either two or three landmarks

are located. Given a population distribution from which sample data are drawn, the paper evaluates thelocation of landmarks in terms of the correlation between distance variables. The results provide usefulimplications for a desirable location of sample domain in relation to the location of landmarks. Focusing
on the case of two landmarks on a one-dimensional unit interval, Dewhurst (1993) also discusses the
choice of sample domain. Instead of distance correlation, the paper utilizes the variances of estimatedparameters to evaluate the degree of multicollinearity. Numerical simulations yield the location of sample
domain that minimizes the variances and hence maximizes the stability of model estimation.
Following the line of Heikkila (1988) and Dewhurst (1993), this paper discusses more
extensively the configuration of sample points that reduces the multicollinearity among distance variables.
Our focus is on the case of two-dimensional space where two or more landmarks are located. We propose
a mathematical method for deriving the optimal location of sample points that minimizes the correlation
between distance variables. The method permits us not only to locate sample points appropriately but also
to evaluate the location of landmarks in terms of the correlation between distance variables in a systematicway.
Section 2 proposes a method for locating sample points that reduces the multicollinearity among
distance variables as small as possible. Description of the method is accompanied with numericalexperiments and empirical applications that test the validity of the method and provide findings useful forthe design of sample location. Section 3 summarizes the conclusions with discussion.

2. Method and applications
Suppose a spatial phenomenon represented as a continuous function y(x) defined over a twodimensional space, such as land price, ground temperature, and elevation. The function is measured at N
sample points. Let Pi, pi, and yi be ith sample point, its location, and the function value observed at Pi,
respectively, where i∈N ={1, 2, ..., N}. Function y(x) is determined by both spatial and aspatial factors,
the former of which include the distance to landmarks. Let Fj and zj be jth landmark and its location,
respectively (j∈L ={1, 2, ..., L}). The distance between location p and zj is denoted by d(p, zj). Aspatial
factors are represented by functions f1(x), ..., fM(x).
We build a regression model that explains y(x) by its determinants based on the data observedat sample points:

yi = α 0 + α1 f1 ( pi ) + ... + α M f M ( pi ) + β1d ( pi , z1 ) + ... + β L d ( pi , z L ) + ε ( pi )(1)
where αk's and βk's are the parameters to be estimated (k∈L) and ε(pi) is an error term. We omit aspatial
factors in the model for the present to focus on the multicollinearity among distance variables:

yi = β 0 + β1d ( pi , z1 ) + β 2 d ( pi , z 2 ) + ... + β N d ( pi , z L ) + ε ( pi )(2)where βk's are the parameters to be estimated (k∈L). The multicollinearity among distance variables isunavoidable since the landmarks are distributed on the same two-dimensional space.
Our objective is to locate sample points with keeping the multicollinearity among distancevariables as small as possible. This requires us to evaluate the multicollinearity in an objective way, forthe purpose of which various methods are available at present. A simple measure is the absolute value ofcorrelation coefficient between two variables. A rule of thumb tells us that the correlation coefficient
smaller than 0.5 is acceptable. The variance inflation factor (VIF) is another popular index that assesses
multicollinearity in terms of the stability of model estimation. Further sophisticated approaches have also
been developed in the literature that evaluate and mitigate the multicollinearity (Belsley, Kuh, and Welsch(1980); Spanos and McGuirk (2002); Kovàcs, Petres, and Tóth (2005); Curto and Pinto (2007); Dormann
et al. (2013)).
Among those methods, we initially choose the maximum absolute correlation coefficient
between distance variables as the measure of multicollinearity because of its simplicity and tractability.
We derive the optimal configuration of sample points that minimizes the maximum absolute correlation
coefficient denoted by max{|rjk|} (j, k∈L, j≠k), where rjk is the correlation coefficient between d(p, zj) andd(p, zk) observed in sample points. We should note, however, that other measures of multicollinearity arealso applicable to the method proposed in the following with a slight modification as discussed later.

2.1 Two landmarks
We start with the optimal configuration of sample points when y(x) depends only on the distanceto two landmarks, i.e., L=2. This case is rather trivial since we can easily keep r12 at zero as follows. We
locate the first sample point P1 arbitrarily and draw the circle of radius d(p1, z1) centered at F1. Placing
other sample points on the circle, we can change d(p, z2) with keeping d(p, z1). The correlation coefficient
r12 becomes zero so that we can estimate β2 without being disturbed by the correlation between d(p, z1)
and d(p, z2). We can similarly estimate β1 by using a circle of radius d(p1, z2) centered at F2.
This method, unfortunately, limits the location of sample points only on two circles. Such atight restriction is practically undesirable because the clustering of sample points often increases thecorrelation between spatial and aspatial variables in regression models, which also greatly reduces thestability and reliability of model estimation. We thus propose an alternative location of sample points thatspreads over a wider area.
Suppose XY coordinate system as shown in Figure 1a, where the location of two landmarks F1and F2 are denoted as z1=(-c, 0) and z2=(c, 0), respectively. We also consider another space whose XY
axes indicate d(p, z1) and d(p, z2) as shown in Figure 1b. We call the latter distance space to distinguishfrom real space shown in Figure 1a. Any location in the real space is projected in light and dark grayregions in Figure 1b, which is bounded by three lines y=x+2c, y=x-2c, and y=-x+2c.

y

E1

D1

F1

-3c

-c

O

F2

D2

c

3c

x

G
E2

(a)

y=x+2cd(p, z2)
D1

4c

2c

y=x-2cF1G’

y=-x+2c

F2

0

2c

D2

4c

d(p, z1)(b)

Figure 1 The relationship between the real space and the distance space defined by d(p, z1) and d(p, z2).
(a) The real space. Landmarks F1 and F2 and circles of radius 2c centered at F1 and F2 are drawn. (b)The distance space. Light and dark gray regions indicate the possible relationship between d(p, z1) andd(p, z2) in the distance space. Dark gray square corresponds to the dark gray region in Figure 1a. No
correlation appears between the two distances if sample points are distributed symmetrically in bothvertical and horizontal directions.

The correlation coefficient r12 becomes zero if sample points are distributed symmetrically inboth vertical and horizontal directions in the distance space. Clearly, numerous distributions satisfy thiscondition. Among such distributions, we take a uniform distribution in the dark gray square denoted by
G' in Figure 1b that corresponds to the dark gray region G in Figure 1a. We choose this distribution since
sample points are spread relatively widely around the landmarks. Suppose a uniform function in G' whoseintegral in G is equal to one. The function corresponds to the continuous function defined by

f12 ( p; F1 , F2 ) =L ( p; F1 , F2 )2 {d ( z1 , z 2 )} d ( p, z1 ) d ( p, z 2 )
2

,

(3)
where L(p; F1, F2) is the length of perpendicular line from location p to F1F2 (see Sadahiro and Wang(2015) for the derivation of the equation). We call this function the density function of sample points,
which is illustrated in Figure 2. The density is high around the circle whose diameter is F1F2. We denotethe circle as C12 indicated by the white broken line in the figure. Two distances d(p, z1) and d(p, z2) are
not highly correlated around the circle. The density function f12(p; F1, F2) decreases as p approaches the
X-axis since the correlation between d(p, z1) and d(p, z2) is highly positive around D1F1 and F2D2 while
highly negative around F1F2.

y

E1

D1

F1

F2

D2

O

x

C12
G
E2

Figure 2 Density function f12(p; F1, F2) defined by Equation (3). Darker shades indicate larger values.
We can reduce r12 by locating sample points according to f12(p; F1, F2). We approximate a point
distribution that follows a probability distribution on a two-dimensional as follows. We first divide theregion in which we locate sample points by a square lattice of a high resolution, say, 10000 or more cells,
and calculate the probability of a sample point located in each cell by using numerical integration (Davisand Rabinowitz (2007)). Let λi be the probability of a sample point located in ith cell. Cumulative
distribution function of ith cell, which is denoted by Λi, is given byi −1

Λ i =∑ λ j .
j =1

(4)

We then generate a random number x between zero and one, and locate a sample point in ith cell thatsatisfiesΛ i ≤ x < Λ i +1 .
(5)
Repeating this process N times, we can approximate a point distribution that follows f12(p; F1, F2). The
below is a computational algorithm of this procedure.

1. Prepare a square lattice of K cells.
2. Set i=1.
3.

Calculate λi, the probability of a sample point located in ith cell.

4.

i=i+1.

5. Repeat the steps 3-4 until i=K.
6. Set i=1.
7.

Calculate the cumulative distribution function of ith cell defined by Equation (4).

8.

i=i+1.

9. Repeat the steps 7-8 until i=K.
10. Set i=1.
11.

Generate a random number x between zero and one.

12.

Locate a sample point in ith cell that satisfies Equation (5).

13.

i=i+1.

14. Repeat the steps 11-13 until i=N.

To test the effectiveness of the density function derived above, we perform a numericalexperiment. We distribute 1,000 points according to f12(p; F1, F2) in G (Dark gray region in Figure 1) andcalculate the correlation coefficient r12. Repeating this process for 10,000 times, we obtain the probability
distribution of r12. We also calculate the probability distribution of r12 when sample points are randomly
distributed in G for comparison purposes.
Figure 3 shows the result of this numerical experiment. The figure clearly indicates that f12(p;F1, F2) greatly reduces r12 compared with the random distribution. The probability of r12 falling between-0.1 and 0.1 is only four percent in the random distribution while it is 95 percent when sample points
follow f12(p; F1, F2). Density function f12(p; F1, F2) sounds safe enough for locating sample points in thesense that no serious correlation occurs between distance variables.

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 3 Probability distributions of correlation coefficient r12. Black bars indicate the probability
distribution when sample points are distributed according to f12(p; F1, F2), while white bars indicate theprobability distribution when sample points are randomly distributed.

The above procedure of locating sample points is effective when sample points can be locatedonly in a specific region, which is often called a sample domain. The density function of sample points insample domain H can be calculated as follows. We first project H from the real space onto the distancespace by calculating the distance from each vertex of H to two landmarks and connecting the vertices inthe distance space to obtain polygon H'. We then choose a subdomain HS' inside H' in which sample points
can be distributed symmetrically in both d(p, z1) and d(p, z2) directions. We finally determine theconfiguration of sample points in HS' and project it back onto the real space.
We can choose subdomain HS' in various ways. A simple method is to draw a small circle or anaxis-parallel square inside HS'. To locate sample points more widely, we can employ computationalalgorithms that calculate the largest axis-parallel rectangle inside a polygon (Daniels, Milenkovic, andRoth (1997); Boland and Urrutia (2001)). The below is a computational algorithm of the above procedure.

1. Approximate H by a polygon of vertices V1, ..., VK.
2. Set i=1.
3.

Calculate the distance from Vi to landmarks F1 and F2.

4.

Locate the point corresponding to Vi in the distance space.

5.

i=i+1.

6. Repeat the steps 3-5 until i=K.
7. Connect the obtained points in the distance space to generate polygon H'.
8. Choose a rectangular subdomain HS' inside H'.
9. Locate N sample points symmetrically in H'. Let z1 and z2 be the coordinates of ith point.
10. Set i=1.

11.

Draw two circles of radius z1 and z2 centered at F1 and F2, respectively, in the real space.

12.

Locate a sample point at an intersection of the circles.

13.

i=i+1.

14. Repeat the steps 11-14 until i=N.

If sample points are already prepared, we extract a subset of points according to f12(p; F1, F2).
We first calculate f12(p; F1, F2) based on a square lattice as mentioned earlier. We then choose sample
points in each cell in proportion to its assigned probability. If a cell does not contain enough number ofpoints, we leave the cell and examine the next cell. We finally choose sample points from the cells thatstill have enough points in proportion to f12(p; F1, F2).
2.2 More than two landmarks
We then consider regression models whose independent variables contain the distances from
more than two landmarks. Unfortunately, it is almost impossible to avoid the correlation between morethan two distance variables simultaneously. We thus aim to reduce the correlation as small as possible,
i.e., to minimize the maximum absolute correlation coefficient between distance variables, by carefully
choosing sample location. We formulate the derivation of the optimal configuration of sample points as a
spatial optimization problem:

Problem CM (Correlation Minimization):{ } ( j, k ∈ L, k ≠ j )minimize max rjk
pi ,i∈N

To solve Problem CM, we initially distribute sample points according to the following density
function:g (p ) =2∑∑ f jk ( p; Fj , Fk ) .
L ( L − 1) j k ≠ j(6)This function is the average of density function fjk(p; Fj, Fk) for all the pair of landmarks. We choose thisfunction in expectation that g(p) reduces rjk of every pair of j and k equally and thus yields a smallmax{|rjk|}. We determine the initial location of sample points in the way similar to that discussed in theprevious subsection. We then move every sample point from P1 to PN in turn by the steepest descentmethod (Hamacher and Drezner (2002); Avriel (2003); Snyman (2005); Fletcher (2013)). The method
moves each point in the direction that decreases the objective function most rapidly. We repeat this processuntil the movement of sample points converges within a predetermined threshold. The obtained max{|rjk|}
is denoted by rmax.

Using this method, we perform a numerical experiment. We locate L landmarks randomly in acircle of radius 1.0 and derive an optimal configuration of sample points. We repeat this process 1000
times to obtain the probability distribution of rmax. Table 1 describes the summary statistics of the
distribution. The mean of rmax increases monotonically with L, which implies that correlation between
distance variables becomes more unavoidable. If we accept rmax smaller than 0.5, we can always optimize
the configuration of sample points when L≤3 since Prob(rmax<0.5) is 100 percent. The probability
Prob(rmax<0.5) decreases to 62.3 percent when L=4, and then 8.7 percent when L=5. A serious gap
between L=4 and L=5 cases suggests that it is safe to consider less than five landmarks simultaneously interms of multicollinearity in regression analysis.

Table 1 Summary statistics of rmax under a random configuration. Prob(rmax<0.5) indicates theprobability of rmax being smaller than 0.5.
L

2

Mean
0.0000Standard deviation 0.0000
Prob(rmax<0.5)
100.0

3

4

5

6

7

0.27380.2041100.0

0.52630.128462.3

0.71820.15078.7

0.81550.10700.0

0.88900.07220.0

In practice, we often have to limit the number of distance variables equal or less than three dueto the high correlation between the variables. Considering this, we can regard the proposed method
effective enough since it always assures rmax<0.5 when L=3, and permits four landmarks with a probabilityof 62%. We even recommend the method when 5≤L since we can reduce rmax at an acceptable level insome situations.
The above procedure is also applicable when sample points are already prepared. We determine
the initial set of sample points according to g(p). We then optimize the combination of sample points tominimize the objective function of Problem CM. Since it is a combinatorial optimization problem, we use
a heuristic algorithm such as simulated annealing, tabu search, and genetic algorithms to reach the final
result (Nemhauser and Wolsey (1988); Wolsey (1998); Karlof (2005)).

2.3 Extensions
This subsection briefly discusses several extensions of the proposed method. Due to space
limitations, we will examine them more thoroughly elsewhere in the near future.
Problem CM utilizes max{|rjk|} as its objective function. The problem, however, can utilize
other measures such as VIF, tolerance value, and eigenvalue instead of max{|rjk|} as a measure ofmulticollinearity. Replacing the objective function by a different measure, we can optimize theconfiguration of sample points from a different perspective. We can even incorporate principal component
regression by replacing rmax with that obtained after principal component analysis. The optimal location

of sample points under the use of principal component regression is highly probable to reduce rmax to 0.5
even when 5≤L.
We can also incorporate aspatial variables into our framework as follows. Aspatial variables areoften correlated with distance variables under the existence of spatial autocorrelation, which typicallyemerges in the clusters of sample points. A practical solution is thus to add the minimum distance between
sample points as a constraint in Problem CM. We may use the nearest neighbor distance between points
for a certain significance level that are randomly distributed in sample region (Pielou (1977); Ripley(2005)). If we can build a mathematical model that represents the spatial distribution of aspatial variables,
we can directly incorporate the model into the optimization procedure. The model gives the correlation
coefficients between spatial and aspatial variables and those between aspatial variables. Replacing {|rjk|}
with the set of all the absolute correlation coefficients, we solve Problem CM to derive the optimal
location of sample points that minimize the correlation between all the variables.

2.4 Three landmarks
The numerical experiment in the previous subsection showed that we can keep the correlation
between distance variables at a reasonable level when L≤4. This subsection discusses further in detail the
relationship between the location of landmarks and the maximum correlation rmax with a focus on the caseof L=3.
Clusters of sample points are undesirable as mentioned earlier. Since careful consideration isindispensable on the spatial pattern of sample points, we introduce two measures to evaluate the degreeof spatial clustering. One is standard distance defined by∑ {d ( p , p )}

2

i

SD =

C

i

,

M

(7)
where pC is the mean center of sample points (Bachi (1962); Rogerson and Yamada (2008); Burt, Barber,and Rigby (2009)). Standard distance SD evaluates the overall degree of spatial dispersion of sample points.
It becomes large when sample points are widely dispersed. Another measure is relative nearest neighbordistance defined by∑ min {d ( p , p )}
i

RD =

i

l

N
SD

l

.
(8)

It is the ratio of the average nearest neighbor distance of sample points to SD. Relative nearest neighbordistance becomes small when points form local clusters, while it is large if points are widely scattered.

The two measures complement with each other by evaluating the degree of spatial clustering fromdifferent perspectives.
Using three measures rmax, SD, and RD, we discuss the relationship between the location oflandmarks and the maximum correlation through numerical experiments. Three landmarks are arranged
in two ways. They form an isosceles triangle in Case 1, while a sheared triangle in Case 2. Landmarks F1and F2 are located at (-1, 0) and (1, 0) in both cases, respectively. The third landmark F3 is located on the
y-axis and horizontal line y = 3 as shown in Figure 4, respectively. The location of F3 is denoted as

 3


3 
m, 3  in the two cases, respectively. Parameter m changes from 0 to 75 inm  and 
 0,
 15 
 15
both cases. In Case 1, triangle ∆F1F2F3 becomes an equilateral triangle when m=15. In Case 2, ∆F1F2F3
is an equilateral triangle when m=0, and gradually becomes a sheared triangle with an increase in m.
y

y

F3
F3 0,

3
m, 3
15

3
m
15

y= 3

x

F1 (-1, 0)

F2 (1, 0)(a)xF1 (-1, 0)

F2 (1, 0)(b)

Figure 4 The location of three landmarks F1, F2, and F3. (a) Case 1: The landmarks form an isosceles
triangle, (b) Case 2: The landmarks form a sheared triangle.
We first discuss Case 1, where ∆F1F2F3 forms an isosceles triangle. Figure 5 shows therelationship between m and numerical measures. As seen in the figure, rmax initially decreases greatly withan increase of m, and stays zero when 7≤m. The correlation between distance variables completely
vanishes when vertex angle ∠F1F3F2 is smaller than 0.567π=102.1°. Figure 6 shows the final
configuration of sample points when m=5, 10, 15, 30, 45, and 75, where Cij is the circle of diameter FiFj
centered at the midpoint of Fi and Fj. While Figure 6a and Figure 6b contain clusters, points are distributed
rather uniformly in Figure 6c, where ∆F1F2F3 forms an equilateral triangle. Points become gathered alonga circle centered at the midpoint of F1 and F2 with an increase in m. This is clearly reflected in Figure 5cwhere RD first increases and then decreases monotonically. Figure 5b shows that standard distance SD
captures the global pattern of sample points successfully, where SD increases monotonically with a spreadof sample points over a wider area.
0.5
Case 2

0.4
0.3

rmax
0.2
0.1
Case 1
0.0
0

15

30

m

45

60

75

(a)
4.0

Case 2

3.0

Case 1

SD

2.0

1.0

0.0
0

15

30

m

45

60

75

45

60

75

(b)
0.00350.0030Case 10.00250.0020RD

Case 20.00150.00100.00050.0000
0

15

30

m

(c)

Figure 5 Measures of the configuration of sample points. (a) The maximum correlation rmax, (b)standard distance SD, (c) relative nearest neighbor distance RD.

(a) m=5(b) m=10(c) m=15(d) m=30(e) m=45

C13

F3

F1

F2
C12

CD
(f) m=75

Figure 6 The final configuration of sample points in Case 1. Gray circle denoted as CD in Figure 6f
indicates the clustering of sample points. Broken lines indicate circles C12 and C13.
To examine the clustering of sample points around a circle in detail, we discuss the relationshipbetween sample location and distance variables in the case of two sample points. Suppose Figure 7a inwhich the first sample point P1 is located. Broken and solid lines indicate arcs centered at F1 and F2 andtheir tangents at P1, respectively. Let us consider the location of second point P2 around the neighborhood

of P1.If we place P2 between the tangents, the area indicated by gray-shades, correlation coefficient r12
becomes positive. Figure 7b depicts the angle of P2 with respect to F1 that yields a positive r12. Black area
in small circles indicates the range of the angle giving positive r12. As seen in the figure, r12 tends to bepositive outside circle C12 while it becomes negative inside C12. If we locate P2 randomly around F1, theprobability of positive r12 is given by

 1
y1
y 
− arctan 1   arctanx1 − cx1 + c  πProb ( r12 > 0 ) =

y1
y1 
 1
1 − π  arctan x − c + arctan x + c 
1
1




( x1 < −c, c < x ).

( −c ≤ x1 ≤ c )(9)Figure 7c show the distribution of Prob(r12>0). Probability Prob(r12>0) is 1/2 on C12, which implies thatpoints clustered around C12 do not exhibit strong correlation between distance variables. We can confirm
in this figure that the correlation r12 is negative inside C12, while it becomes highly positive as sample
points move away from C12 toward its outside.

y

θ
P1
F2

F1
-c

O

F2

F1

x

c
C12
C12

(a)
(c)
y

Q

C

F1

F2

x

(b)

Figure 7 Calculation of the probability distribution of positive r12. Red circles indicate circle C12. (a)The location of the first sample point P1. (b) The angle of P2 with respect to P1 that yields a positive r12.
Black area in small circles indicate the range of the angle giving positive r12. (c) The distribution ofprobability Prob(r12>0). Darker shades indicate that distance variables tend to be highly positively
correlated.

Figure 7b also suggests that a specific form of sample clusters causes a strong correlation, either
positive or negative. Suppose sample points aligned on an extension of a diameter of C12 such as green

line Q in Figure 7b. The line is running through the black area of small circles at every location in the
figure. This indicates that d(p, z1) and d(p, z2) are positively correlated at all these locations, andconsequently, r12 becomes 1.0. In contrast, sample points located on F1F2 yields r12=-1.0. Negative
correlation also occurs when sample points are aligned on a circle centered at the midpoint of F1 and F2
such as circle C in Figure 7b. The circle is passing through the white area of small circles at every location,
which yields a highly negative correlation.
Keeping the above patterns in mind, let us return to Figure 6f. We examine correlation
coefficients r12 and r13 in detail to reveal how the sample points accomplish rmax=0(=r12=r13). Many points
are concentrated along the circle denoted by CD, whose center is located at the midpoint of F1 and F2. We
expect the points to cause a highly negative correlation, which is supported by Figure 8a showing therelationship between d(p, z1) and d(p, z2) on CD. Other points are all dispersed outside C12, which leads
to a positive correlation as confirmed in Figure 7c. The negative and positive correlations cancel out witheach other, and consequently, r12 becomes zero as a whole in Figure 6f. We then turn to the case of r13.
Figure 8b shows the relationship between d(p, z3) and the other distance variables on CD. The figure
exhibits no significant correlation. Other points are distributed equally inside and outside of C13 in Figure
6f so that negative and positive correlations similarly emerge and cancel out with each other. Correlation
r13 also becomes zero, so does rmax.

d(p, z2)d(p, z1)d(p, z2)

10

10

8

8

6
5

6
5

0

5 6

8

10 d(p, z1)

0

5

(a)
d(p, z1)d(p, z2)

10
8.8

10
8.8

6.8

6.8

5

5

5 6.8 8.8 10 d(p, z1)(c)15d(p, z3)15d(p, z3)(b)d(p, z2)

0

10

0

5

10
(d)

Figure 8 The relationship between distance variables of sample points located on circles. Points arelocated on (a)(b) circle CD in Figure 6f, and (c) (d) CD' in Figure 9e. Bold and broken lines in (b) (d)indicate d(p, z1) and d(p, z2), respectively.
We then discuss Case 2, where ∆F1F2F3 forms a sheared triangle. Figure 5a shows that rmax
stays at zero until m reaches at 11, where ∠F1F3F2≈0.287π=51.6°, and then monotonically increases as∆F1F2F3 becomes further sheared. The final configuration of sample points is shown in Figure 9. Points
are distributed uniformly around ∆F1F2F3 when m=0, and then gradually concentrated along a circle
centered at the midpoint between F1 and F2 with an increase of m. This change is reflected in SD and RD
shown in Figure 5. Interestingly enough, a clear concentration along two circles appears in Figure 9b as
indicated by gray lines, where ∆F1F2F3 forms an isosceles triangle. Similar but less clear circles can also
be observed in Figure 6a. Figure 9 suggests that sample points are initially distributed uniformly, thengathered along two circles, and finally clustered along a single circle.

(a) m=0(b) m=15(c) m=30(d) m=45

F3
F1

F2

CD’

(e) m=75

Figure 9 The final configuration of sample points in Case 2. Gray circles indicate the clustering ofsample points.

The above applications suggests sufficient conditions for rij=0. Correlation rij becomes positive
if sample points are either 1) dispersed outside Cij, or 2) aligned on an extension of a diameter of Cij. It
becomes negative when sample points are either 3) dispersed inside Cij, or 4) aligned on a diameter of Cij,or 5) aligned on a circle that is larger than Cij and centered at the midpoints of FiFj. The combination of
one positive and one negative conditions yields rij=0. Figure 6c and Figure 9a are the cases of conditions
1) and 3), where sample points are dispersed equally inside and outside Cij. Figure 6f, Figure 6e, Figure

9d, and Figure 9e are the cases of 1) and 5). Correlation rmax becomes zero only if all the correlations areequal to zero by satisfying some of the above conditions.
What is interesting here is that condition 5) is found only for the shortest edge of ∆F1F2F3 inthe above examples. This is because of the following reason. Suppose that F2F3 is the shortest among thethree edges. Correlation r12 is zero if sample points are concentrated along circle C centered at the
midpoints of F1F2 while other points are distributed outside C12 as seen in Figure 10. A necessary
condition for r23=r31=0 in this case is that the sample points are distributed equally both inside and outside
C23 and C31. However, since C23 is smaller than C, many points are inevitably located outside C23. When
F2F3 is the shortest edge, therefore, a necessary condition for rmax=0 is that sample points are concentrated
along circle C centered at the midpoints of F2F3 while other points are distributed outside C23.

C31
C12

F1

F3

F2

C23

C
Figure 10 The location of sample points, ∆F1F2F3 and its related circles.
We finally discuss a desirable configuration of three landmarks in terms of the correlation
between distance variables. If the concentration of sample points does not cause a strong correlation
between spatial and aspatial variables, we can adopt any configuration discussed in this subsection on theassumption that rmax<0.5 is acceptable. If point clusters are undesirable, it is safe to choose landmarks thatform an almost equilateral triangle such as those in Figure 6c and Figure 9a where sample points exhibit
no significant clusters. If sample points need to be dispersed more widely, we may choose theconfiguration of landmarks such as those in Figure 6e and Figure 9d, though we have to allow local
clusters of sample points to a certain degree.

2.5 Applications to real data
This subsection applies the proposed method to the analysis of real data. We chose two regions,each of which is located in Tochigi and Fukuoka Prefectures, Japan. Source data are land price data in2014 collected at 23380 locations by the Ministry of Land, Infrastructure, Transport and Tourism. We
consider a regression model that describes the land price at each location by the distance to landmarks.
In the first region in Tochigi Prefecture we took three cities as landmarks: Utsunomiya, Tochigi,
and Kanuma (Figure 11). Sample domain H is a circle of radius 25 kilometers centered at the centroid of
the three cities. The domain contains 277 sample points, whose rmax is 0.7425 (Figure 11a). To reduce rmax
smaller than 0.5, we located sample points in two different ways: one is to newly locate 277 sample points
in H without any restriction, and the other is to extract 50 percent of the original points, i.e., 138 from 277
points.
Figure 11b is the distribution of sample points located without any restriction. Sample points
are uniformly distributed in H, which is almost the same as the initial location of sample points given by
Equation (3). If we further continue optimization, we could reduce rmax to 0.00. Figure 11c and Figure
11d indicate the points randomly chosen from the original data and those obtained after optimization,
respectively, where rmax reduced from 0.7310 to 0.50. Figure 11d looks reasonable since it contains no
extreme cluster of points. If we further continue optimization, we could decrease rmax to 0.2988. Otherwise,
we may increase sample points to reduce the loss of information with keeping rmax=0.5. This option ismore effective in this case since we could increase sample points up to 80 percent of the original ones,i.e., we could attain rmax=0.5 with the loss of only 20 percent of the original information.

30

30

N

N
20

20

10

10

0

0

-10

-10

-20

-20

-30
-30

-20

-10

0

10

20

30

-30
-30

-20

-10

(a)

0

10

20

30

10

20

30

(b)
30

30

N

N
20

20

10

10

0

0

-10

-10

-20

-20

-30
-30

-20

-10

0

10

20

30

-30
-30

(c)

-20

-10

0

(d)

Figure 11 The location of sample points and landmarks in Tochigi Prefecture, indicated by black andred points, respectively. Brown lines are railway lines. (a) The actual location in 2014, (b) sample points
located without any restriction, (c) sample points extracted randomly from their original location, (d)sample points obtained after optimization. The maximum correlation rmax is 0.5 in Figure 11b andFigure 11d.

Sample domain H in the second example is located in Fukuoka Prefecture. (Figure 12). It is a
circle of radius 20 kilometers facing Hakata Bay centered at the centroid of three landmarks, i.e.,downtown area of Fukuoka City, Higashi-Hirao Park that contains football and athletic stadiums, and a
big shopping mall called Torius. Domain H contains 381 sample points, where rmax is 0.8445 (Figure 12a).
We optimize the location of sample points in the same way as that in Tochigi Prefecture.
Figure 12b shows the distribution of sample points located without any restriction for rmax=0.5.
We can see a rather obscure cluster of points around the three landmarks. It would be a circular clustersuch as the one observed in Figure 6, a part of which is substituted by the cluster on the peninsula in
Hakata Bay. This distribution seems practically infeasible since the peninsula is not densely inhabited.
Figure 12c and Figure 12d are the points randomly chosen from the original ones and those obtained after
optimization, respectively, where rmax reduced from 0.6827 to 0.5. We could either further reduce rmax to0.2768, or increase sample points up to 95 percent of the original ones with keeping rmax=0.5. The latteris very effective in a practical sense as we can decrease rmax from 0.8445 to 0.5 by discarding only five
percent of the original information. We can use original data more effectively than in the case of Tochigi
Prefecture because Fukuoka Prefecture has more sample points that assure the flexibility of sample
location.

20

20

N

N

10

10

0

0

-10

-10

-20
-20

-10

0

10

20

-20
-20

-10

(a)

0

10

20

10

20

(b)
20

20

N

N

10

10

0

0

-10

-10

-20
-20

-10

0

(c)

10

20

-20
-20

-10

0

(d)

Figure 12 The location of sample points and landmarks in Fukuoka Prefecture, indicated by black andred points, respectively. Brown lines are railway lines. The darker gray region indicates Hakata bay. (a)The actual location in 2014, (b) sample points located without any restriction, (c) sample points
extracted randomly from their original location, (d) sample points obtained after optimization. The
maximum correlation rmax is 0.5 in Figure 12b and Figure 12d.
We then evaluate the location of sample points in a more realistic context. We built a regression
model that describes the land price by both spatial and aspatial variables. The former consists of thedistance to three landmarks, while the latter includes lot size, building and land use regulations, andsurrounding land use represented by ten variables. We compared two models, one is based on sample
points selected randomly from original data, and the other utilizes the same number of sample points thatis optimized in such a way that rmax is smaller than 0.5. Both models chose independent variables by theforward stepwise method.
Table 2 shows the result of model estimation, where ρ is the proportion of sample points selectedfrom the original ones. Regression models in Tochigi Prefecture describe land price better than those in
Fukuoka Prefecture at any ρ. The former utilize more distance variables, which implies that the distanceto landmarks plays an important role in Tochigi Prefecture. Optimization of sample selection generally
improves the model fitness in both regions; R-square and adjusted R-square increase in nine out of twelve
cases. R-square significantly increases especially when the number of distance variables increase by
optimization such as the cases of ρ=0.50, 0.60, and 0.70. Optimization of sample points decreases thecorrelation between distance variables, which permits regression models to evaluate more distancevariables, and consequently, improves the model fitness.
The above discussion also indicates that the optimization of sample selection is not so effective
when the distance to landmarks is not critical to land price. We confirm this by the result obtained in
Fukuoka Prefecture, where optimization of sample selection are less effective than in Tochigi Prefecture.
Optimization of sample selection is worth trying in any case, but we should keep in mind that it does notalways greatly improve the accuracy of regression models.

Table 2 Summary statistics of estimated models. F value is significant at significance level 0.01 in all
the models.

Tochigi Prefecture
Number of independent variables

ρ

Extraction

R2

Adjusted R2

F

Total

Distance

Random

0.747

0.734

54.906

7

1

Optimal

0.782

0.769

57.9218

2

0.60

Random
Optimal

0.690
0.806

0.676
0.796

50.22081.492

7
8

1
3

0.70

Random
Optimal

0.803
0.822

0.793
0.813

83.09194.152

8
9

1
2

0.80

Random
Optimal

0.785
0.797

0.776
0.788

85.57191.791

9
9

2
2

0.90

Random
Optimal

0.774
0.752

0.767
0.743

102.995
80.582

7
9

1
2

0.95

Random
Optimal

0.772
0.755

0.755
0.746

95.02186.456

9
8

1
2

0.50

Fukuoka Prefecture
Number of independent variables

ρ

Extraction

2

R

Adjusted R

F

Total

Distance

Random

0.544

0.539

111.559

20

Optimal

0.634

0.626

80.0674

1

0.60

Random
Optimal

0.526
0.534

0.518
0.530

61.954129.067

2
4

1
1

0.70

Random
Optimal

0.539
0.577

0.534
0.569

102.055
71.193

3
5

0
1

0.80

Random
Optimal

0.523
0.532

0.518
0.528

110.025114.170

3
3

1
1

0.90

Random
Optimal

0.527
0.466

0.522
0.462

125.743
98.517

3
3

1
1

0.95

Random
Optimal

0.518
0.519

0.514
0.515

128.392
128.426

3
3

1
1

0.50

2

Sample points of land price and real estate data are often located primarily in downtown areas
as seen in the two examples. Such distributions inevitably contain point clusters, which is a primary source

of the correlation between distance variables. If data are already prepared, we should avoid local clusters
by choosing sample points in such a way that they are distributed more uniformly than their original
distribution. When we newly collect sample points, uniform distribution is a simple but effective option
because it yields both positive and negative correlations between distance variables as discussed in
Subsection 2.4.

2.6 Limited location of sample points
This subsection discusses the case where sample points can be located only in a limited region.
This happens either when we choose sample location from a given set of locations or when we locate
sample points in a given sample domain.
In the former case, we solve Problem CM in a discrete space, i.e., choose the locations thatminimizes max{|rjk|} from the given set. Since it is an integer programming problem, we employ aheuristic method to derive the optimal set of sample points.
In the latter case, we add a constraint to Problem CM that limit the location of sample points:

Problem CM' (Correlation Minimization):{ } ( j, k ∈ L, k ≠ j )minimize max rjk
pi ,i∈N

subject to ∀i ∈ N , p i ∈ H ,where H indicates sample domain. Sample points are initially distributed according to the following
density function:g '(p ) =g (p )∫q∈H=g ( q ) dq∑∑ f ( p )
j

k≠ j

∑∑ ∫
j

k≠ j

q∈H

jk

.

f jk ( q ) dq
(10)

We move sample points from their initial location in a similar way as to that used in Subsection 2.2.
We apply this method to a numerical experiment. Three landmarks F1, F2, and F3 are located at(0, 0), (3, 0), and (2, 2), respectively, in rectangular region Ψ bounded by four lines x=-5, x=8, y=-5, andy=7. We derive the optimal location of 1000 sample points in circular sample domain HS of radius l. The
center of HS moves at lattice points of interval 0.05 in Ψ. The radius l varies from 0.5 to 2.5 at the intervalof 0.5.
Figure 13 shows the relationship between rmax and the location of the center of HS when l=0.5,

1.5, and 2.5. The left hand figures show max{|rjk|} when sample points are randomly distributed in HS,
while the right hand figures indicate rmax obtained after optimization. The figures clearly present that the
optimization of sample location drastically reduces the correlation between distance variables.
Let us focus on the right hand figures of Figure 13. The figures show that we can keep rmax
smaller than 0.5 when sample domain HS is close to landmarks. The maximum correlation rmax increases
as HS moves away either inside or outside of ∆F1F2F3. This is consistent with the result shown in Figure
7b where the correlation between distance variables increases with the distance from landmarks. We canalso reduce rmax by employing larger sample domain because it increases the flexibility of sample location.
A careful consideration is necessary when HS is located around the lines F1F2, F3F3, and F1F3. The
maximum correlation rmax is small only when HS is close to landmarks. Otherwise, the neighborhood ofthese lines yields worse result.

F3

F3
F1

F1

F2

F2

(a)

(b)

(c)

(d)

0.9 - 1.0
0.8 - 0.9
0.7 - 0.8
0.6 - 0.7
0.5 - 0.6
0.4 - 0.5
0.3 - 0.4
0.2 - 0.3
0.1 - 0.2
0.0 - 0.1(e)(f)

Figure 13 The relationship between rmax and the location of sample domain HS. The left hand figures
show max{|rjk|}when sample points are randomly distributed in HS, while the right hand figures indicate
rmax obtained after optimization. The radius of HS is (a)(b) 0.5, (c)(d) 1.5, and (e)(f) 2.5.

3. Concluding discussion
This paper developed a method for deriving the configuration of sample points that reduces themulticollinearity among distance variables in regression models. In the case of two landmarks, we derived
the density function of sample points that totally vanishes the correlation between distance variables. We
then proposed a mathematical procedure for locating sample points that minimizes the correlation
between distances to more than two landmarks. To test the validity of the proposed method, we performed
numerical experiments and empirical applications varying in the number of landmarks and sample domain.
The results supported the technical soundness of the method, and provided useful implications for thedesign of sample location.
We finally indicate two limitations of the paper and potential extensions for future research.
First, we should discuss more thoroughly the extensions of the proposed method. Though
Subsection 2.3 briefly describes some directions, we have not yet test their validity through numericalexperiments and real applications. Computational cost and initial configuration of sample points in thoseextensions clearly need a detailed examination from both theoretical and empirical perspectives.
Applications in various situations are also necessary to find an appropriate approach that is theoreticallysound and practically feasible.
Second, the definition and treatment of distance variables need further discussion. This paper
defines the distance as the Euclidean distance in a linear form in regression models. However, othermeasures such as the Manhattan distance, network distance and time distance are often used in regional
science and geography. We can incorporate these distances into regression models in various forms
including square root, inverse, logarithm, and exponential functions. We should extend the proposed
method to treat a wider variety of distance measures.

References12

Asteriou, Dimitrios and Stephen G Hall. 2007. Applied Econometrics: a modern approach using eviews andmicrofit: Palgrave Macmillan New York.

3

Avriel, Mordecai. 2003. Nonlinear programming: analysis and methods: Courier Dover Publications.

4

Bachi, Roberto. 1962. "Standard distance measures and related methods for spatial analysis." Papers in

5
6
7
8

Regional Science 10, 83-132.
Belsley, David A, Edwin Kuh and Roy E Welsch. 1980. Regression diagnostics: Identifying influential dataand sources of collinearity: John Wiley & Sons.
Bender, Bruce and Hae-Shin Hwang. 1985. "Hedonic housing price indices and secondary employment

1
2
3
4
5
6
7

centers." Journal of Urban Economics 17, 90-107.
Berry, Brian J. L. 1976. "Ghetto expansion and single-family housing prices: Chicago, 1968–1972." Journalof Urban Economics 3, 397-423.
Boland, Ralph P and Jorge Urrutia. 2001. "Finding the Largest Axis-Aligned Rectangle in a Polygon in O(nlog n) time," paper presented at Proc. 13th Canad. Conf. Comput. Geom.
Burt, James E, Gerald M Barber and David L Rigby. 2009. Elementary statistics for geographers: Guilford
Press.

8

Chatterjee, Samprit and Ali S Hadi. 2013. Regression analysis by example: John Wiley & Sons.

9

Chen, Gikuang Jeff. 2012. "A simple way to deal with multicollinearity." Journal of Applied Statistics 39,

10
11
12
13
14

1893-1909.
Curto, José Dias and José Castro Pinto. 2007. "New Multicollinearity Indicators in Linear Regression Models."International Statistical Review 75, 114-121.
Daniels, Karen, Victor Milenkovic and Dan Roth. 1997. "Finding the largest area axis-parallel rectangle in apolygon." Computational Geometry 7, 125-148.

15

Davis, Philip J and Philip Rabinowitz. 2007. Methods of numerical integration. New York: Dover.

16

Dewhurst, J.H.L. 1993. Spatial Multicollinearity and Sample Selection in Models with Inverse Distance17

Measures: University of Dundee, Department of Economics and Management.

18

Dormann, Carsten F, Jane Elith, Sven Bacher, Carsten Buchmann, Gudrun Carl, Gabriel Carré, Jaime R García

19Marquéz, Bernd Gruber, Bruno Lafourcade and Pedro J Leitão. 2013. "Collinearity: a review of20

methods to deal with it and a simulation study evaluating their performance." Ecography 36, 027-046.

21

Farrar, Donald E and Robert R Glauber. 1967. "Multicollinearity in regression analysis: the problem revisited."

22

The Review of Economic and Statistics, 92-107.

23

Fletcher, Roger. 2013. Practical methods of optimization: John Wiley & Sons.

24

Gordon, Alan G and Eville Gorham. 1963. "Ecological aspects of air pollution from an iron-sintering plant at

25

Wawa, Ontario." Canadian Journal of Botany 41, 1063-1078.

26

Hamacher, Horst W and Zvi Drezner. 2002. Facility location: applications and theory: Springer.

27

Harrison Jr, David and Daniel L Rubinfeld. 1978. "Hedonic housing prices and the demand for clean air."

28
29
30

Journal of Environmental Economics and Management 5, 81-102.
Heikkila, Eric. 1988. "Multicollinearity in Regression Models with Multiple Distance Measures." Journal of
Regional Science 28, 345-362.

31

Ihlanfeldt, Keith R and Laura O Taylor. 2004. "Externality effects of small-scale hazardous waste sites:

32

evidence from urban commercial property markets." Journal of Environmental Economics and

33

Management 47, 117-139.

34

Jilcott Pitts, Stephanie B, Qiang Wu, Jared T McGuirt, Thomas W Crawford, Thomas C Keyserling and Alice

35

S Ammerman. 2013. "Associations between access to farmers’ markets and supermarkets, shopping

36

patterns, fruit and vegetable consumption and health indicators among women of reproductive age in1eastern North Carolina, USA." Public health nutrition 16, 1944-1952.

2

Karlof, John K. 2005. Integer programming: theory and practice: CRC Press.

3

Kashid, D. N. and S. R. Kulkarni. 2002. "A more genral criterion for subset selection in multiple linear

4
5
6
7
8
9
10

regression." Communications in Statistics - Theory and Methods 31, 795-811.
Kovàcs, Péter, Tibor Petres and László Tóth. 2005. "A New Measure of Multicollinearity in Linear Regression
Models." International Statistical Review 73, 405-412.
Li, Mingche M and H James Brown. 1980. "Micro-neighborhood externalities and hedonic housing prices."
Land economics, 125-141.
Mansfield, Edward R, John T Webster and Richard F Gunst. 1977. "An analytic variable selection techniquefor principal component regression." Applied statistics, 34-40.

11

Miller, Alan. 2012. Subset selection in regression: CRC Press.

12

Morland, Kimberly, Ana V Diez Roux and Steve Wing. 2006. "Supermarkets, other food stores, and obesity:13

the atherosclerosis risk in communities study." Am J Prev Med 30, 333-339.

14

Nemhauser, George L and Laurence A Wolsey. 1988. Integer and combinatorial optimization: Wiley New York.

15

Ni, Liqiang. 2011. "Principal component regression revisited." Statistica Sinica 21, 741.

16

Noonan, Douglas S., Douglas J. Krupka and Brett M. Baden. 2007. "Neighborhood dynamics and price effects

17

of superfund site clean-up." Journal of Regional Science 47, 665-692.

18

Pielou, E. C. 1977. Mathematical Ecology. New York: Wiley.

19

Riga-Karandinos, AN and MG Karandinos. 1998. "Assessment of air pollution from a lignite power plant in20

the plain of Megalopolis (Greece) using as biomonitors three species of lichens; impacts on some21

biochemical parameters of lichens." Science of the total environment 215, 167-183.

22

Ripley, Brian D. 2005. Spatial statistics: John Wiley & Sons.

23

Rogerson, Peter and Ikuho Yamada. 2008. Statistical detection and surveillance of geographic clusters: CRC

24

Press.

25

Rundle, Andrew, Kathryn M Neckerman, Lance Freeman, Gina S Lovasi, Marnie Purciel, James Quinn,

26

Catherine Richards, Neelanjan Sircar and Christopher Weiss. 2009. "Neighborhood food environment

27

and walkability predict obesity in New York City." Environ Health Perspect 117, 442-447.

28

Sadahiro, Yukio and Yan Wang. 2015. "Configuration of sample points for the reduction of multicollinearity in29

regression models with distance variables," Discussion Paper Series. Center for Spatial Information

30

Science, The University of Tokyo.

31
32
33
34

Snyman, Jan. 2005. Practical mathematical optimization: an introduction to basic optimization theory andclassical and new gradient-based algorithms: Springer.
Spanos, Aris and Anya McGuirk. 2002. "The problem of near-multicollinearity revisited: erratic vs systematic
volatility." Journal of Econometrics 108, 365-393.

35

Vigneau, E, MF Devaux, EM Qannari and P Robert. 1997. "Principal component regression, ridge regression

36

and ridge principal component regression in spectroscopy calibration." Journal of chemometrics 11,

12
3

239-249.
Weissfeld, Lisa A and Susan M Sereika. 1991. "A multicollinearity diagnostic for generalized linear models."
Communications in Statistics-Theory and Methods 20, 1183-1198.

4

Wolsey, Laurence A. 1998. Integer programming: Wiley New York.

5

Yu, KN, YP Cheung, T Cheung and Ronald C Henry. 2004. "Identifying the impact of large urban airports on

6

local air quality by nonparametric regression." Atmospheric Environment 38, 4501-4507.