34.2gong-Clarifying the Standard Deviational Ellipse-¹¨½¨ÐÂ

8/2/2019 34.2gong-Clarifying the Standard Deviational Ellipse-

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Jianxin Gong is an associate professor of geography, Nanjing Normal University, China.E-mail:[email protected]

Geographical Analysis,Vol. 34, No. 2 (April 2002) The Ohio State UniversitySubmitted: 9/29/00. Revised version accepted: 9/10/01

Jianxin Gong

Clarifying the Standard Deviational Ellipse

For a set of geographical units in the Cartesian coordinate system, the locus of thestandard deviation of the x coordinates of the set forms a closed curve as the system isrotated about the origin. This curve, often referred to as standard deviational ellipse(SDE), is not in fact an ellipse. The actual shape of the curve has remained unclear

since the issue was mentioned initially by Lefever in 1926. In the present paper thisclosed curve, referred to as standard deviation curve (SDC), is clarified mathemat-

ically, and some of its applications in spatial analysis are discussed.The shape of SDC changes from a single circle to double circles when the distribu-

tion of the set of geographical units changes from an even condition to a straight line.The shape of SDC is determined explicitly by the ratio of its minor axis to its majoraxis. This ratio, therefore, is a useful index to show to what extent the distribution of a

set of geographical units is circular, or linear. In addition, the size and radius of SDCcan be used to indicate the distribution density of geographical units. The major axisof SDC, whose angle is determined explicitly for the first time, indicates the major ori-

entation of geographical units.A program has been developed to apply SDC to spatial analysis (mean center,

major orientation, distribution density, circular condition, etc.). The program is avail-able from [email protected]. It is written in the MapBasic language, and runs

underMapInfo.

More than seventy years ago there were two interesting articles published byLefever (1926) and Furfey (1927) in The American Journal of Sociology. Both dis-cussed what Lefever called standard deviational ellipse and its application in spatialanalysis for a set of geographical units regarded as point set in two-dimensional space.Standard deviational ellipse, or SDE for short, is not at all an ellipse as its name im-

plies. This fact was clarified by Furfey in 1927. Despite this, SDE has been widely in-troduced and applied as an ellipse in later studies. To the present, it has remainedunclear what the actual curve of SDE is.

The main focus of this paper is to clarify mathematically the actual shape of thiscurve (hereafter known as SDE), and to discuss its applications in spatial analysis. Inthe first of four parts, this paper reviews related studies of others in the field. Thesecond part introduces and proves two mathematical theorems, and clarifies the ac-tual shape of the SDE curve based on these theorems. The third part discusses in de-tail the applications of the actual curve in spatial analysis, including its role in


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determining distribution density and circular condition, and in showing the major ori-entation of geographical units. The fourth and final part gives a conclusion.

Review of Previous Studies on SDE

Lefevers (1926) procedure to determine SDE can be summarized as follows (Fig-ure 1):For convenience, first of all, move the origin of the Cartesian coordinate system to

the mean center of the set ofn units studied. Here

{(xi,yi);i 1,2,,n} are the coordinates of the units in the coordinate systemX Y(Figure 1).

Then calculate the standard deviation, x0, of thex coordinates of the units.

(1)

where {(x0,i,y0,i);i 1,2,,n} are the coordinates of the units in the transformed co-ordinate systemX0 Y0.

Finally, rotate the coordinate systemX0 Y0 about the new origin (x,y) by angle

(02) and calculate the standard deviation, x, of thex coordinates again.

(2)

x y ii

n

ii

n

i i ii

n

i

n

a nx

nx y x y

/ ,

, , , ,cos sin sin ,

2

1

12

2

1

202 2

102

0 011

x ii

n

n x01

0

2

1

, ,

x n x y n yiin

ii

n

1 1

1 1

,

156 / Geographical Analysis

FIG. 1. Coordinate Systems


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where {(x,i ,y,i);i 1,2,,n} are the coordinates of the units in the rotated coordi-nate systemX Y, and y is the standard deviation of they coordinates.

The locus ofx (02) forms a closed curve. Lefever (1926) claimed that theclosed curve was an ellipse, and named it standard deviational ellipse. He also sug-

gested that the major axis of the ellipse indicates spatial orientation, the area of theellipse indicates spatial dispersion, and the ratio of the number of the units withinthe ellipse to the total number of units indicates the relative dispersion of geographi-cal units.

However, Furfey (1927) pointed out that by changing to Cartesian coordinates,equation (2) becomes

(x 20 y20)

2 2x0x

20 2y0y

20 2r0x0y0x0y0, (3)

which clearly is not an ellipse. Here r0 is the correlation coefficient between thex andy coordinates in the coordinate systemX0 Y0. However, Furfey (1927) did not dis-

cuss equation (3) any further except for mentioning three special cases (y0 x0,r0 0; y0 2x0, r0 0.5, and y0 2x0, r0 1; referring to Figure 3, curves a, c,and d).

Later, Caprio (1970) argued four special cases of equation (2), depicting them as acircle, an ellipse, a collapsing ellipse, and double circles. Yuill (1971) and Ebdon(1977) applied SDE in some regions to describe the distribution density of geograph-ical units according to Lefevers (1926) proposal. Smith (1989) mentioned SDE whenhe introduced some methods for tourism research. More recently, Levine, Kim, andNitz (1995) tried to explain the spatial pattern of vehicle crashes by means of SDE.

Wong (1999) suggested a spatial segregation index based on SDE. All of these reportsapplied Lefevers SDE as if it were actually an ellipse without any discussion about its

mathematical foundation.Review of Studies on Distribution Density

To describe the amount of scatter of geographical units, Furfey (1927) defined thefollowing index:

(4)

It has been suggested that the smaller Sd is, the greater the distribution density orconcentration of geographical units is. Sd, which Bachi (1957) later dubs the standarddistance has been introduced rather widely (for example, Burt and Barber 1996).

Unfortunately, Sd is hardly a useful index for comparing the distribution density orconcentration among the sets with varying numbers of points. For example, if a pointset is spread symmetrically on a circle, its Sd stays equal to the circles radius whetherthere are three points or three thousand. It is obvious, however, that a set with fewerpoints is less dense than one with more points (Gong 1994). The population of a largecity has a larger Sd than the population of a small town, but generally the former is moredensely populated than the latter (Smith 1989). By using Sd as an index of distributiondensity, Bachi (1963) obtained one such unreasonable conclusion: that is, in Francefrom 1801 to 1954 the population in urban areas was slightly more concentrated than in

rural areas, while in the United States from 1870 to 1950 the opposite was true.By modifying Sd, Gong (1994) developed a new distribution density indicator,

Pcb,m. The indicator can be applied to geographical units in one-, two-, and three-dimensional spaces.

Sn

x x y yd i ii

n

x y

10 22 2

1

2 2{( ) ( ) } ( ) .

Jianxin Gong / 157


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(5)

where,dij is the distance between uniti and unitj (i,j 1,2,,n); G andb can be anypositive real number and natural number, respectively; andm 1, 2, 3 indicates thedimensions of space within which then units studied exist.

Whenmb 2, equation (5) becomes

(6)

Pcb,m is acquired by comparing the moment of distances among the points studiedwith that of a set of same number of points uniformly spread in space. Pcb,m solves theproblem of Sd being strongly influenced by the number of points studied. It can,therefore, be applied to different point sets, regardless of the number of points ineach. Pc2,2 has previously been applied to classify rural villages in Japan (Gong, Kita-mura, and Kobayasi 1994a, 1994b).

1. THE ACTUAL CURVE OF SDE

1.1 Theorems

THEOREM 1: If the Cartesian coordinate system, in which npoints exist, is rotated

to the angle, , that satisfies the condition

(7)

where

then

(8)

(9)

(10)

where ris the correlation coefficient between the x andycoordinates of the points inthe rotated coordinate systemX Y.

PROOF: First of all, equation (7) means .sin cos2 22 2 2 2

b

a b

a

a band

r

a b

a b

x x x y

x y x x y

0

01

2

01

2

2 2 2 2 2 2

2 2 2 2 2 2 2

0 0

2 0 0

,

max{ ; ( , ]} ( ) ,

min{ ; ( , ]} ( ) ,/

a b r a bx y x y

1

20

0 0 0 0

2 20( ), , ,

tan a b a

b

2 2

Pc GS

nd

2 2

2

, .

Pc G

d

nb m

ijb

j

n

i

n

b

m

, ,

11

2



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1. Note that although 2x in equation (2) is similar to the eigenvalue in principal component analysis(PCA) in the case of two variables, they are not, in fact, identical. First of all, there is a formulaic difference

in that 2x has a denominator ofn in contrast ton1 for the eigenvalue in PCA. Secondly, x and y inTheorem 1 are the extreme values ofx.The first eigenvalue in PCA is similar to x as a maximum value,but the second eigenvalue has little significance. In addition, the condition ab 0 in Theorem 1 has beenleft out of most, and quite possibly all discussion on PCA (for example, Okuno et al. 1981).

When the coordinate system is rotated to the angle, , shown in equation (7), we thenhave

(11)

This means r 0.From equation (2) and equation (7), on the other hand, we have

(12)

Equation (9) and equation (10) are then immediately derived based on equation(12). Thus ends the proof of Theorem 1.

Theorem 1 means that for any set of geographical units, if the Cartesian coordinatesystem is rotated properly, the correlation coefficient between thex andy coordinatesbecomes zero, and the standard deviation of the x coordinates is maximized, while

that of they coordinates is minimized.1

THEOREM 2: If a point set is spread evenly on concentric circles, and on each circlethere are more than two points (Figure 2a), then for any angle ,

2x constant , (13)

r 0 . (14)

PROOF: Letdi be the radius of the circlei on whichni 2 points are evenly spread;let i be the angle betweenX-axis and the line from the origin to any one of the points

on circlei; and let i 2/ni,i 1,2,,m, n (Figure 2b). Herem is the

number of concentric circles concerned,n is the number of the points.When the Cartesian coordinate system is rotated to an angle , we have

1

nii

m

x y x y

x y

a b

a b

2 2 2 2

2 2 2 2

2 0 0

0 0

2 21

2

21

2

/ cos sin ( ) ;

cos ( ) ( ) .

r n x y

r

a b

x y i ii

n

y x x y

1

1

22 2

2 2 0

1

2 20

0 0 0 0

, , ;

( )sin cos ;

sin cos .

Jianxin Gong / 159


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(15)

is a constant independent from any angles. Thus equation (13) is proven.

Note that equation (13) also means 2y 2x

. On the other hand, for any angle ,from equation (11), we still have

(16)

Therefore, r 0. equation (14) is proven.Thus ends the proof of Theorem 2.Note that a distribution like that mentioned in Theorem 2 will hereafter be re-

garded as an even condition. Such a point set will also be said to be spread evenly.Theorem 2 means if a set of geographical units is spread evenly, the standard devi-

ation of itsx coordinates will always equal a constant independent from the Cartesiancoordinate system, and the correlation coefficient between the x and y coordinates

will always equal zero.

1.2 Standard Deviation Curve

When the Cartesian coordinate system is rotated to the angle (Figure 3) satisfy-ing equation (7), equation (2) and equation (3) then become the following equation(17) and equation (18) respectively (Theorem 1):

1

2

2 2 02 2( )sin cos .

y x x yr

1

22

1nn di i

i

m

x ii

m

i ik

n

i i

ii

m

i ik

n

i ii

m

nd

n d n n d

i

i

2 2

1 0

12

21

20

1

21

1

1 1

2

{cos( )cos sin( )sin } ;

cos { ( )} (Otsuki 1982) .


FIG. 2. Points Evenly Distributed on Concentric Circles


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(17)

(18)

where 2max max{2x; (0,]},2min min{2x; (0,]}. 2max and 2min form

respectively the major and minor axes of the discussed curve.When all units are spread evenly, according to Theorem 2, max min, the dis-

cussed curve forms a circle: x2 y

2 2max. When the units are spread along a

straight line, min 0, the discussed curve forms double circles intersecting at the

origin: (x max/2)2 y2 (max/2)2. The features of the discussed curve are sum-marized in Table 1 and their relevant shapes are shown in Figure 3.

As mentioned above, the discussed curve is not an ellipse. For this reason, thispaper will hereafter call it standard deviation curve, or SDC for short. The actual re-lation between SDC and an ellipse with major axis 2max and minor axis 2min isshown in Figure 4. The equation of the tangent line of the ellipse at a point of tan-gency (x,y) can be written as (Mathematical Handbook Editing Team 1979):

(19)

Therefore, the distance from the origin to the tangent line is the radius of SDC, ()[equation (17)]. It then follows that SDC can be drawn from an ellipse, and vice

versa.

y xsin cos cos sin . max min2 2 2 2 0

x

x y x y

( ) cos sin ,

( ) ,

max min

max min

2 2 2 2

2 2 2 2 2 2 2

Jianxin Gong / 161

FIG. 3. Standard Deviation Curve


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TABLE 1

Features of Standard Deviation Curve

condition feature Location of the feature remarks r

a: max min circle radius max min r 0

b: max 2min extreme values x 0 y2 2min r1

extreme valuesx 0 y

2 2min

c: max 2minr1

point of 5/12 /2,inflection /2 7/12

d: min 0 double circles rmax 1

Notation: ris the correlation coefficient betweenx coordinates andy coordinates.

radiusmax

2

sin( )

( )2

2 2 2

4 4

2

2

max max min

max min

y

2

4

2 24

max

max min( )x

2

2 2 2

2 2

2

4

max max min

max min

( )

( )

( ) ( ) cos sinx y x y x 2 2 2 2 2 2 2 2 2 2 2 max min max min, or

FIG. 4. Relation between Ellipse and Standard Deviation Curve


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2. APPLICATIONS OF SDC

2.1 Circularity Index

There are some indices to quantitatively describe the geometric form of a closed

geographical region. For example, Ebdon (1977) once introduced five shape indicesfor showing how circular a geographical region is. Wentz (2000) defined three indicesas a set to evaluate the edge, elongation, and perforation of a geographical region. Itseems, however, that no index has been developed to describe the distribution shapeof a set of geographical units regarded as a point set.

The ratio

(20)

is a useful index to show to what extent the distribution of a set of geographical unitsis circular, or linear. Here, max and min are the maximum and minimum values, re-spectively, of the standard deviation of the x coordinates [equation (9) and equation(10)]. First, SDC describes fully the prolongation (or density) of a point set in all di-rections. On the other hand, SDC is completely determined by the ratio ofmin tomax. When the distribution of a set of geographical units changes from an even con-dition to a straight line, the relevant ratio, min/max, changes from 1 to 0. In other

words, the larger the ratio, the more circular the distribution is. Likewise, the smallerthe ratio, the more linear the distribution is.

In terms of circularity of distribution of a set of geographical units, SDC gives thesame results as the ratio shown in equation (20). However, whereas SDC is a visuallygraphical tool, ratio min/max is a precise numerical index. Figure 5 shows the distrib-ution of residences in two rural settlements. Settlement a is obviously more circularthan settlement b, as their SDCs imply. The ratio min/max of each settlement a and bare 0.72 and 0.39, respectively.

It is worthy to note, however, that SDC does not describe the shape itself of a set ofgeographical units, as Furfey (1927) argued.

2.2 DISTRIBUTION DENSITY

Since SDC is determined explicitly bymin/max, both x ()/n and x ()/n1/2 are

similar to x() (0 2). Heren is the number of units studied.

0 1

min

max

Jianxin Gong / 163

FIG. 5. Application of Standard Deviation Curve to Rural Settlements


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The area enclosed by the curve x()/n1/2 (0 2) is

(21)

That is, the area can be used to indicate the distribution density of a set of geograph-ical units. The smaller the area is, the denser the distribution appears. Residences insettlement a (Figure 5), for instance, are spread more densely than those in settle-ment b. The area ofx()/n

1/2 (0 2) for settlement a is 2.91, smaller than 4.36for settlement b.

Note that an ellipse also changes when its axes change. However, an ellipsechanges from a circle to two lines, the latter of which has no area at all. Therefore, anellipse is not suitable to indicate the distribution density of a set of geographical units.

On the other hand, from equation (5), ifm

1 andb

2, then (remembering thatG can be any positive real number):

(22)

Here Pc2,1() is the distribution density ofx coordinates along an axis rotated to anglein one-dimensional space. That is, the radius of curve x()/n (0 ) can beused to indicate the distribution density of a set of geographical units along the orien-tation of angle . The longer the radius, the more likely it is that units are spreadsparsely along that radius. Units have the smallest density along the major orientation(the major axis of SDC), OX max (Figure 3), whose angle satisfies equation (7).

Note that Pc2,1() and Pc2,2 have the following relation.

(23)

2.3 Major Orientation

The major axis of SDC indicates the major orientation of the set of geographicalunits studied, as Lefever (1926) suggested. Current methods to determine the angleof the major orientation, however, are either inexplicit (as in Lefever 1926) or incor-rect (as in Ebdon 1977).

From Theorem 1, it is evident that when ab 0, tan . Here

is the angle of the major orientation.

On the other hand, when ab 0, or ab 0, [equation

(12)] becomes a constant equal to the average of the squares of the distances of allpoints from the origin. Thus SDC becomes a circle, and has no orientation.

To sum up what has been mentioned above, the angle of the major orientation iscalculated as

20 0

2 2 x x y

a b a

b

2 2

Pc

Pc dGn2 2

2 10

2

,

, ( )

Pc Gn

x2 1

2

20, ( )

( ), .

d tdt

n

Pc Gxn

2

5

200

2 2 22 2

( )/,( ) ( ), . max min

equation



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(24)

where a andb are the same as in Theorem 1.

3. CONCLUSION

For a set of geographical units in a Cartesian coordinate system, the locus of thestandard deviation of thex coordinates of the units forms a closed curve as the systemis rotated about the origin. This closed curve, referred to as standard deviation curve(SDC), is not an ellipse as previously thought. The present study has made the fol-lowing achievements.

1. Proofs of Two TheoremsTheorem 1 shows that for any point set, if the Cartesian coordinate system is ro-

tated properly, the correlation coefficient betweenx andy coordinates of the set be-comes zero, and the standard deviation of coordinates of one axis is maximized whilethat of the other is minimized.

Theorem 2 shows if a point set is spread evenly, the standard deviation of itsx co-ordinates will always equal a constant independent from the Cartesian coordinatesystem. Furthermore, the correlation coefficient between thex andy coordinates willalways equal zero.

2. Clarification of the Standard Deviation Curve (SDC)

Using the theorems mentioned above, SDC can by simply expressed as

where, max and min refer respectively to the maximum and minimum values of thestandard deviation of thex coordinates, which are calculated by

SDC describes fully the prolongation (or density) of a set of geographical units inall directions, and is determined explicitly by the ratio ofmin to max. 2max and 2minform the major axis and minor axis of SDC respectively. Detailed features of SDC aresummarized in Table 1 and Figure 3.

3. Clarification of the Relation between SDC and an EllipseA radius of SDC is equal to the distance from the origin to a tangent line of the el-

lipse with the same minor and major axes as SDC (referring to Figure 4). Therefore,SDC can be drawn from an ellipse, and vice versa.

max min2 2 2 2 2 2 2 2 2 2

2 20

12

12

1

2

0 0 0 0

0 0 0 0

a b a b

a b r

x y x y

x y x y

( ), ( ) ;

( ), .

x

x y x y

( ) cos sin ,

( ) ,

max min

max min

2 2 2 2

2 2 2 2 2 2 2

or

arctan ,

,

a b a

ba b

a b

2 2

0

0

no solution

Jianxin Gong / 165


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4. Creation of a Useful Circularity Index

This index is acquired by the ratio of two axes of SDC:

When the distribution of a set of geographical units changes from an even condi-tion to a straight line, the ratio min/max changes from 1 to 0. In other words, thelarger the ratio, the more circular the distribution is; the smaller the ratio, the morelinear the distribution is. There had previously been no reasonable circularity index ofthis kind.

5. Proof of Applicability of SDC in Describing Distribution Density of GeographicalUnits

It is proven that the radius of the curve x()/n, calculated by

and the area of the curve x()/n1/2, calculated by

indicate the distribution density of a set of geographical units in one- and two-dimen-sional spaces, respectively. Heren is the number of geographical units. Both x()/nand x()/n1/2 are similar to x().

6. An Equation for Calculating the Major Orientation of Geographical Units

The major axis of SDC indicates the major orientation of the set of geographicalunits studied, as Lefever (1926) suggested. The two theorems mentioned above pro-

vide a way to determine the angle of the major orientation explicitly for the firsttime. That is,

Note that SDC can be drawn about the mean center of geographical units, thoughit need not necessarily be. The author has developed a program to apply SDC to spa-tial analysis (mean center, major orientation, distribution density, circular condition,etc.) which is available from [email protected]. The program is written in theMapBasic language, and runs under MapInfo.

arctan ,

, .

a b a

ba b

a b

2 2

0

0

no solution

22 2

n( ) ,max min

max min2 2 2 2cos sin

n

0 1 minmax .



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LITERATURE CITED

Bachi, R. (1957). Statistical Analysis of Geographical Series. Bulletin de lInstitut international de Statis-tique 36, 22940.

______ (1963). Standard Distance Measures and Related Methods for Spatial Analysis. Papers, the Re-gional Science Association 10, 83132.

Burt, J. E., and G. M. Barber (1996). Elementary Statistics for Geographers, 2d ed. New York: The Guil-ford Press.

Caprio, R. J. (1970). Centrography and Geostatistics. The Professional Geographers 22, 159.Ebdon, D. (1977). Statistics in Geography: A Practical Approach. Oxford: Basil Blackwell.Furfey, P. H. (1927). A Note on LEFEVERs Standard Deviation Ellipse.American Journal of Sociology

33, 948.Gong, J. X. (1994). Scatteration Measure of Objects in Space. Human Geography 46(5), 45573. (in

Japanese)Gong, J. X., T. Kitamura, and S. Kobayasi (1994a). On Classification of Villages by Using Uniform Point

Set.Journal of Rural Planning Association 13(3), 815. (in Japanese)______ (1994b). Simple Calculation of Classification Indices of Villages by means of Mesh Data.Journal

of Rural Planning Association 13(3), 1622. (in Japanese)Lefever, D. W.(1926). Measuring Geographic Concentration by means of the Standard Deviation El-

lipse.American Journal of Sociology 32, 8894.Levine, N., K. E. Kim, and L. H. Nitz (1995). Spatial Analysis of Honolulu Motor Vehicle Crashes: I. Spa-

tial Patterns.Accident Analysis and Prevention 27(5), 67585.Mathematical Handbook Editing Team (1979). Mathematical Handbook. Beijing: Peoples Educational

Press. (in Chinese)Okuno, T., H. Kume, T. Haga, and T. Yoshizawa (1981). Multivariate Analysis. Tokyo: Nichikagiren Press.

(in Japanese)Otsuki, Y. (1982). Mathematical Formulas Collection. Tokyo: Maruzen Press.(in Japanese)Smith, S.L.J. (1989). Tourism Analysis. England: Longman.Wentz, E. A.(2000). A Shape Definition for Geographic Applications Based on Edge, Elongation, and

Perforation. Geographical Analysis 32(2), 95112. Wong, D.W.S. (1999). Geostatistics as Measures of Spatial Segregation. Urban Geography 27(7),

63547.Yuill, R. S.(1971). The Standard Deviational Ellipse: An Updated Tool for Spatial Description. Ge-

ografiska Annaler53B(1), 2839.

Jianxin Gong / 167

Documents

34.2gong-Clarifying the Standard Deviational Ellipse-¹¨½¨ÐÂ