CSIS Discussion Paper No. 10
Accuracy of Areal Weighting Interpolation:
Effects of Geometrical Properties of Zonal Systems
Yukio Sadahiro
FEBRUARY, 1999

Center for Spatial Information Science and Department of Urban Engineering
University of Tokyo
7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan

February 24, 1999
Accuracy of Areal Weighting Interpolation:
Effects of Geometrical Properties of Zonal Systems

Keywords: areal interpolation, accuracy, geometrical properties
Short title: accuracy of areal weighting interpolation
Yukio Sadahiro
Center for Spatial Information Science and Department of Urban Engineering
University of Tokyo
7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Phone: +81-3-3812-2111 (ext. 6273)
Fax: +81-3-5800-6965
E-mail: sada@ua.t.u-tokyo.ac.jp

Accuracy of Areal Weighting Interpolation:
Effects of Geometrical Properties of Zonal Systems
Abstract
This paper analyzes the accuracy of count data transferred through the areal
weighting interpolation with respect to the geometrical properties of zonal systems used
for aggregating spatial data. A stochastic model is employed to measure the estimation
error caused in data transfer between incompatible zonal systems. The relationshipbetween estimation error and the geometrical properties of zones is approximately
represented in analytical forms. The major results are as follows: 1) the perimeter of thetarget zone and the area and perimeter of the source zones are crucial to the accuracy of
the areal weighting interpolation; 2) estimation error increases in proportion to the squareroot of the perimeter of the target zone; 3) concerning the lattice system, estimation error
is proportional to the square root of the perimeter and the biquadratic root of the area ofthe cell, and inversely proportional to the biquadratic root of the number of cells; 4) thehexagonal lattice yields the most accurate estimates among all lattices.

1. Introduction
The areal weighting interpolation is a data transfer procedure between incompatible
zonal systems. There are diverse zonal systems used for aggregating and reporting spatial
data, say, census tracts, administrative districts, school districts, and so forth. Since they
are often geographically incompatible, integration of spatial data requires data transfer
between zonal systems. This process is called areal interpolation, and the areal weighting
interpolation is one of the most popular interpolation methods in GIS (Markoff andShapiro, 1973; Lam, 1983; Flowerdew and Green, 1991).
Assuming a uniform distribution of spatial objects, the areal weighting interpolation
divides the count of spatial objects according to area in each zone, and sums up thecounts in another incompatible zone. Obviously, obtained value is erroneous to someextent because the uniformity assumption does not hold in general (Goodchild et al.,
1993; Fisher and Langford, 1995). One method to improve the accuracy of estimates is touse source data based on smaller geographical zones. For instance, census data
aggregated across census tracts are more desirable than those aggregated across higher
levels of administrative districts, say, towns or counties. On the other hand, detailed data
are generally expensive or hard to obtain, and they need much space for storage and highperformance processors for manipulation. Consequently, GIS users have to choosesource data balancing the data handling cost and the expected accuracy of estimates.
Such data choice requires us to understand the relationship between the source data
and the accuracy of estimates. Sadahiro (1999) proposes a method for analyzing theaccuracy of count data estimated by the areal weighting interpolation, and investigates itin relation to the geometrical properties of zones, that is, their shape and size. The
obtained results seem quite reasonable: source data consisting of small and convex zones
give accurate estimates. In that paper, however, the accuracy measure is represented in anintegral form, so that the computational cost reduces the accessibility to the analyzing
method and results. For instance, it is not clear what properties are crucial to estimation
accuracy. It is also difficult to answer simple questions such as "to what extent theestimation accuracy improves if the number of the source zones increases by twice?"
The motivation for the study described in this paper is to acquire more practicalrepresentations of the accuracy of the areal weighting interpolation. The accuracy measure
is approximately represented in analytical forms on the basis of the stochastic areal
weighting interpolation model developed by Sadahiro (1999). This allows us tounderstand what geometrical properties of zones are crucial to estimation accuracy, andhow they affect the accuracy of the areal weighting interpolation.
2. Areal weighting interpolation model

1

This paper follows an approach taken by Sadahiro (1999): a stochastic model of theareal weighting interpolation. This section outlines the model and accuracy measureproposed in that paper.
Let S0 be a region of area A0 whose shape can cover a plane by its lattice, say, atriangle, a square, or a parallelogram. The region S0 consists of K zones S1, S2, ..., SK,
which represent spatial units used for data aggregation such as census tracts (Figure 1).
We call them source zones in this paper. The area and perimeter of Si are denoted by Aiand Li , respectively. This zonal system is referred to as the source zonal system Ω.
Figure 1 An example of the source zonal system Ω.
In the region S 0 , N points (say, households) are independently distributed
according to the uniform distribution. The location of point j is denoted by yj This paper
limits the case to the uniform distribution in order to focus on the effects of thegeometrical properties of zones on estimation accuracy. We then assume that S 0 issurrounded by its copies, and the copies have the same zonal system and point
distribution as those of S0 (Figure 2). This assumption is called periodic continuation
which often used in spatial statistics (Ripley 1981, Stoyan and Stoyan 1994).
Figure 2 Periodic continuation assumption.
Let us then consider a target zone T that represents a region in which the number ofpoints needs to be estimated. The area and perimeter of T are denoted by A T and L T,
respectively. We assume that T is fairly larger than the source zones, or to be exact, that Tis large enough that the incircle of T is larger than the circumcircle of S i 's. This
assumption seems reasonable in areal interpolation, because if T is not larger than Si ' sthen it will lead to considerably imprecise interpolation.
The target zone T is dropped in such a way that it intersects S 0. If T does notcompletely lie in S0, we replace the portion of T outside S0 by its corresponding figure asshown in Figure 3. All possible positions of T appear randomly. The location of T isrepresented by an indicator function:
1 if x ∈ T
1T ( x ) = .
(1)
0 otherwise
Figure 3 Transformation of T.
In the above stochastic model, the number of points in T and its estimator given by2

the areal weighting interpolation are written as
M = ∑1T y j( )j(2)andM̂ = ∑∫x ∈Si

1T ( x )dx∑1 (y ) ,i

Aii(3)

jj

respectively, where1 if x ∈ Si.
1i ( x ) = 
0 otherwise
Estimation accuracy is measured by the mean square error (MSE) defined by2
ˆ − M .
MSE[Ω] = E  M
Substituting equations (2) and (3) into equation (5), we obtain
N
11MSE[Ω] =m(T ; x − t )dxdt  AT −∑∫∫A0 
2πA0 i Ai t ∈Si x ∈Si


()

(4)

(5)

(6)

(for details, see Sadahiro 1999), where m(T; l) is the measure of the set of all figurescongruent to T containing two points separated by a distance l.
3. Estimation error and geometrical properties of zones
Since equation (6) contains an integral term, calculation of MSE[Ω] requires
numerical integration. Because of its computational cost, the accessibility to the measure
is fairly reduced and thus it is not clear from equation (6) what geometrical properties ofzones are crucial to estimation accuracy. We hence derive analytical representations of themeasure using some approximations.
Instead of MSE[Ω], this paper adopts the root mean square error (RMSE) as theaccuracy measure which is given by()2
ˆ −M RMSE[Ω] = E  M=N1
1m(T ; x − t )dxdt  AT −∑∫∫A0 
2πA0 i Ai t ∈Si x ∈SiTo replace the integral term in equation (7), we employ an approximationm(T ; l ) ≈ 2πAT − 2 LT l.

(7)

(8)

(Stoyan and Stoyan, 1994; Sadahiro, 1998). Substitution of equation (8) into equation(7) yields
λLT
RMSE[Ω] ≈(9)
∑ Ai Di ,πA0 iwhere λ is the density of points (=N/A0) and Dj is the mean distance between two points

3that are randomly distributed in Sj . The distance Dj is given in an analytical form if Sj hasa simple shape. For instance, if Sj is a circle of radius r,
128
Di =r.
(10)45π
For a rectangle of sides a, b, we have1
Di =2( a 5 + b 5 ) − 2( a 4 − 3a 2 b 2 + b 4 ) a 2 + b 2
30 a 2 b 2(11)a2 + b2 + ba 2 + b 2 + a 
44+5a b log+ 5ab logab
(Ghosh, 1951). If Sj has more complicated shape, an approximate expression obtained
through an empirical study is useful:Di ≈ 0.13 Li(12){(Koshizuka, 1978). Substituting equation (12) into equation (9) we obtain
0.13λLTRMSE[Ω] ≈∑i Ai Li .
πA0(13)This equation indicates that the accuracy of the areal weighting interpolation depends on
the perimeter of the target zone and the area and perimeter of the source zones. In otherwords, the effects of the geometrical properties of zones can be described by theseparameters.
Having obtained analytical representations of the estimation error, we now discussin detail how the geometrical properties of the source and target zones affect estimationaccuracy.
3.1 Geometrical properties of the target zone
Let us first consider the effects of the target zone T. From equation (9) we noticethat estimation error RMSE[Ω] increases in proportion to the square root of the perimeterof the target zone T. This implies that elongated forms of T yield less precise estimates
than convex forms. Suppose, for instance, a rectangular target zone of v/h ratio γ (γ≥1).
Then equation (9) becomes
RMSE[Ω] ≈A2λ(1 + γ ) T ∑ Ai Di .
πA0γ i(14)Figure 4 shows the relationship between RMSE[Ω] and the v/h ratio of the rectangulartarget zone.
Figure 4 The relationship between RMSE[Ω] and the v/h ratio of the rectangular targetzone. The area of the target zone is fixed. The value of RMSE[Ω] is standardized so that
RMSE[Ω]=1.0 for the square target zone.

4

Fixing the shape of the target zone, we obtain
RMSE[Ω] ∝ 4 AT .

(15)

This implies that RMSE[Ω] is proportional to the biquadratic root of the area of the targetzone (Figure 5). Comparing Figures 4 and 5, we find that the size of the target zone isstill more influential than its shape on estimation accuracy.
Figure 5 The relationship between RMSE[Ω] and the area of the target zone. The shapeof the target zone is fixed. The value of RMSE[Ω] is standardized so that RMSE[Ω]=1.0
for AT=1.0.
3.2 Geometrical properties of the source zones
We then turn to the effects of the source zones. As we mentioned earlier, they arefully described by equation (13). However, it is somewhat difficult to understandintuitively from the equation how the geometrical properties of the source zones affect
RMSE[Ω]. We hence consider the lattice as the zonal system, though the lattice system is
not so popular in geography. Consideration on the lattice would make the effects of thesource zones more clear and easy to understand.
Let S be the fundamental cell of the lattice. The area and perimeter of S are denotedby AS and LS, respectively. Equations (13) then becomes
0.13λ
RMSE[Ω] ≈LT LS .
(16)πEquation (16) indicates that the source and target zones are approximately symmetric withrespect to the effects on RMSE[Ω]. Therefore, the relationships shown in Figures 4 and5 equally hold for the source zones: estimation error is proportional to the square root of
the perimeter of the cell; if the shape of the cell is fixed, the error increases in proportion
to the biquadratic root of its area, that is,RMSE[Ω] ∝ 4 AS .
(17)
The cell size is even more influential than its shape, which suggests that we only have topay attention to the size of zones when we choose source data. These results areconsistent with those obtained in Cockings et al. (1997) and Sadahiro (1999).
Equation (16) also shows that the lattice system of high convexity yields better
estimates. Consequently, we can say that the hexagonal lattice is the most desirable
among all lattices, which is also compatible with the result obtained in Sadahiro (1999).
We finally examine how the number of cells K affects the accuracy of arealinterpolation. Fixing the shape of the cell, we have
1RMSE[Ω] ∝ 4 .
(18)
K5This implies that estimation error decreases in proportion to the biquadratic root of thenumber of cells (Figure 6).
Figure 6 The relationship between RMSE[Ω] and the number of cells in the latticesystem. The shape of the fundamental cell of the lattice is fixed. The value of RMSE[Ω]is standardized so that RMSE[Ω]=1.0 for K=1.
4. Conclusions
In this paper we have analyzed the accuracy of count data estimated by the areal
weighting interpolation. Using approximate expressions, we obtained analyticalrepresentations of estimation accuracy shown as equations (9), (13), and (16). Sincecomputation of equations (13) and (16) requires only the area and perimeter of the sourceand target zones, RMSE[Ω] can easily be obtained in commercial GIS. From theseequations we obtained the following results:
1) The perimeter of the target zone and the area and perimeter of the sourcezones are crucial to the accuracy of the areal weighting interpolation(equation (13)).
2) Estimation error increases in proportion to the square root of the perimeterof the target zone (equation (9)).
3) Concerning the lattice system, estimation error is proportional to thesquare root of the perimeter (equation(16)) and the biquadratic root of thearea (equation(17)) of the cell. Moreover, it is in inverse proportion to thebiquadratic root of the number of cells (equation (18)).
4) The hexagonal lattice yields the most accurate estimates among all lattices
through the areal weighting interpolation (equation (16)).

6References
Cockings, S., Fisher, P. F., and Langford, M., 1997, Parameterization andVisualization of the Errors in Areal Interpolation. Geographical Analysis, 29, 314328.
Fisher, P. F. and Langford, M., 1995, Modelling the Errors in Areal Interpolation
between Zonal Systems by Monte Carlo Simulation. Environment and Planning A,27, 211-224.
Flowerdew, R. and Green, M., 1991, Data Integration: Statistical Methods forTransferring Data between Zonal Systems. In Handling Geographical Information:
Methodology and Potential Applications, edited by I. Masser and M. Blakemore(New York: Longman), 38-54.
Ghosh, B., 1951, Random Distances within a Rectangle and between Two Rectangles.
Bulletin of Calcutta Mathematical Society, 43, 17-24.
Goodchild, M. F., Anselin, L., and Deichmann, U., 1993, A Framework for the Areal
Interpolation of Socioeconomic Data. Environment and Planning A, 25, 383-397.
Koshizuka, T., 1978, On the Random Distance within an Area. Journal of theOperations Research Society of Japan, 21, 302-319 (in Japanese).
Lam, N. N-S., 1983, Spatial Interpolation Methods: a Review. American Cartographer,
10, 129-149.
Markoff, J. and Shapiro, G. 1973, The Linkage of Data Describing Overlapping
Geographical Units. Historical Methods Newsletter, 7, 34-46.
Ripley, B. D., 1981, Spatial Statistics (New York: John Wiley).
Sadahiro, Y. (1998). "Accuracy Count Data Estimated by the Point-in-Polygon Method."
Discussion Paper Series 77E, Department of Urban Engineering, University ofTokyo.
Sadahiro, Y. (1999). "Accuracy of Count Data Transferred through the Areal Weighting
Interpolation Method." International Journal of Geographical Information Science,to appear.
Stoyan, D. and Stoyan, H., 1994, Fractals, Random Shapes and Point Fields (NewYork: John Wiley).

7

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