Proportional Symbol Mapping in R

Abstract

Visualization of spatial data on a map aids not only in data exploration but also in communication to impart spatial conception or ideas to others. Although recent carto-graphic functions in R are rapidly becoming richer, proportional symbol mapping, which is one of the common mapping approaches, has not been packaged thus far. Based on the theories of proportional symbol mapping developed in cartography, the authors developed some functions for proportional symbol mapping using R, including mathematical and perceptual scaling. An example of these functions demonstrated the new expressive power and options available in R, particularly for the visualization of conceptual point data.

Full text

JSS
Journal of Statistical Software
January 2006, Volume 15, Issue 5. http://www.jstatsoft.org/
Proportional Symbol Mapping in R
Susumu Tanimura
Nagasaki University
Chusi Kuroiwa
University of Tokyo
Tsutomu Mizota
Nagasaki University
Abstract
Visualization of spatial data on a map aids not only in data exploration but also in
communication to impart spatial conception or ideas to others. Although recent carto-
graphic functions in R are rapidly becoming richer, proportional symbol mapping, which
is one of the common mapping approaches, has not been packaged thus far. Based on
the theories of proportional symbol mapping developed in cartography, the authors devel-
op ed some functions for proportional symbol mapping using R, including mathematical
and p erceptual scaling. An example of these functions demonstrated the new expressive
p ower and options available in R, particularly for the visualization of conceptual point
data.
Keywords: visualization, cartography, thematic map, propotional symbol, R.
1. Introduction
The visualization of spatial data on a map is crucial to impart spatial conception or ideas
to others or to explore spatial data. This spatial data is usually expressed on a common
thematic map, such as a choropleth, proportional symbol, isarithmic, or dot map.
Each type of spatial data can be expressed using the most suitable type of thematic mapping
method mentioned above for the best representation of data. Therefore it is essential to
carefully select a suitable mapping metho d in order to draw a thematic map. However,
some mapping software occasionally do not support the mapping method selected as the best
choice. Sophisticated mapping software should support many types of mapping so that users
can select one for the best representation of their spatial data.
Recently, cartographic packages in R including maptools, mapproj, rgdal, Rmap, and RAr-
cInfo have been developed more actively and improved. This has facilitated the construction
of thematic maps such as choropleth and dot maps. Despite these recent enhancements in car-
tographic functions in R, prop ortional symbol mapping, which is one of the common mapping
approaches, has not been packaged thus far.

2 Proportional Symbol Mapping in R
Data
Mathematical Scaling
Perceptual Scaling
10 100 500
Figure 1: Contrast between mathematical and perceptual scaling
The objective of this study is to introduce proportional symbol mapping in R. We demonstrate
proportional symbol mapping with new functions written in R, and discuss the feasibility and
limitations of the current code.
2. Proportional symbol mapping
A proportional symbol map — one of the common thematic maps — represents spatial point
data with a symbol, whose area varies in proportion to an attribute variable. The symbol
used could be a circle, square, bar, sphere, cube, or a more complicated symbol such as a
pictographic one.
The methodology of proportional symbol mapping has been discussed in detail by cartogra-
phers. There have been a number of related rep orts, and many theories have been proposed
(Slocum 1999). In particular, the scaling method has been an issue. Two types of scaling
techniques are widely used, namely, mathematical scaling and perceptual scaling.
In this study, we developed a function for prop ortional symbol mapping based on the above-
mentioned theories. A legend function was also developed by considering the discussion on
cartography.
The type of symbol used in the function, however, was limited to a circle because the difference
in the shape of the symbols is not essential for avoiding an inappropriate map expression (see
discussion for more reason).
2.1. Mathematical scaling
The size of the variable in the data proportionally corresponds to the area of a point symbol.
For instance, if a data value is five times another, the area of the point symbol will be five
times as large. The relation is expressed as follows:
πr
2
i
πr
2
max
=
v
i
v
max
,
where r
i
is the radius of the circle to b e drawn; r
max
, radius of the largest circle on the map;

Journal of Statistical Software 3
v
i
, value of the variable for which the circle will be drawn; and v
max
, the maximum value of
the variable.
By solving for r
i
, we obtain
r
i
=

v
i
v
max

0.5
× r
max
. (1)
This formula was implemented in R code as the default scaling for drawing a map.
2.2. Perceptual scaling
It is well known that the perceived area of proportional symbols dose not match their math-
ematical area; rather, we are inclined to underestimate the area of larger symbols. As a
solution to this problem, it is reasonable to modify the area of larger circles in order to match
it with the p erceived area.
Flanney (1971) experimentally derived a power function exponent of 0.5716 to adjust for this
mismatch. For perceptual scaling, we can replace the exponent in (1) with the following
approximation:
r
i
=

v
i
v
max

0.57
× r
max
.
This figure is still widely cited and used for proportional symbol mapping (Slocum 1999).
Figure 1 shows an example of circles drawn with mathematical and perceptual scaling to
understand the extent of the difference between them. Since perceptual scaling adjusts the
area of circles in order to account for underestimation, the area of the larger circle in perceptual
scaling is larger than that in mathematical scaling.
2.3. Legend design
In proportional symbol mapping, two basic legend arrangements are used: nested and linear
(Figure 2). In the nested legend arrangement, a large circle includes a smaller one in sequence,
while in the linear legend arrangement, the circles are aligned vertically or horizontally in the
ascending or descending order. The label texts of these legends can be arranged inside or
outside the circles. The advantages and disadvantages of legend arrangements have been
discussed elsewhere (Slocum 1999; Slocum, McMaster, Kessler, and Howard 2005).
We included simple nested and linear arrangements in the legend function with limited flexi-
bility. The available output of the legend function is shown in Figure 2.
250
160
90
40
10
10
40
90
160
250
Nested-legend arrangement Linear legend arrangement
Figure 2: Nested and linear legend arrangements

4 Proportional Symbol Mapping in R
3. R code
The R code of the proportional symbol mapping function (also available in the accompanying
file ‘ProportionalSymbolMap.R’) is as follows:
ProportionalSymbolMap <- function(map,variable,
type=c("mathematical","perceptual"),
max.size=1500,
symbol.fg=grey(.2), symbol.bg=grey(.5),
legend.loc, legend.breaks,
legend.type=c("nested","linear"),
legend.cex=.6)
{
if(missing(map)) stop("map object is missing.")
if(!inherits(map, "Map")) stop("Map.obj must be of class Map")
if(missing(variable)) stop("variable to be plot is missing")
verts <- Map2points(map)[order(variable, decreasing = TRUE),]
type <- match.arg(type)
switch(type,
mathematical = scale <- sqrt(variable/max(variable))*max.size,
perceptual = scale <- ((variable/max(variable))^0.57)*max.size)
scale <- scale[order(variable, decreasing = TRUE)]
symbols(verts[,1:2],circle=scale/1.1,bg=symbol.bg,fg=symbol.fg,
inches=FALSE,add=TRUE)
if((! missing(legend.loc)) & (! missing(legend.breaks))) {
switch(type,
mathematical = {
legend.r <- sqrt(legend.breaks/max(variable))*max.size
},
perceptual = {
legend.r <- ((legend.breaks/max(variable))^0.57)*max.size
})
legend.type <- match.arg(legend.type)
switch(legend.type,
nested = {
r <- rev(legend.r); b <- rev(legend.breaks)
for (i in 1:length(r)) {
symbols(legend.loc[1],legend.loc[2]+r[i],
circle=r[i]/1.1,inches=FALSE,add=TRUE,
bg=symbol.bg,fg=symbol.fg)
lines(c(legend.loc[1],legend.loc[1]+1.2*r[1]),
rep(legend.loc[2]+2*r[i],2))
text(legend.loc[1]+1.2*r[1]+par("cxy")[1]/2,
legend.loc[2]+2*r[i], b[i], adj=c(0,.5),
cex=legend.cex)
}
},

Journal of Statistical Software 5
linear = {
r <- legend.r
gap <- r[length(r)%/%2]
s <- vector(); for (i in 1:length(r)) s[i] <- sum(r[1:i])
x <- 2*s-r+(0:(length(r)-1))*gap + legend.loc[1]
symbols(x, rep(legend.loc[2],length(r)), circles=r/1.1,
inches=FALSE, bg=symbol.bg, fg=symbol.fg, add=TRUE)
text(x, legend.loc[2]+r+par("cxy")[2]*legend.cex, legend.breaks,
adj=c(.5,1), cex=legend.cex)
})
}
}
The default set in this function shows mathematical scaling, a gray symbol, and the maximum
radius of 1500. Prior to mapping proportional symbols, a base map is essentially required on
the graphic device of R as a “new plot” termed in R. This is because a proportional symbol
without the base map appears vague and may be confusing for the readers; however, in the
future, the function can be expanded to be independent of the base map if the users strongly
request this. The order of the plotting symbol is determined such that a circle overlays a
larger circle.
When both the legend location (legend.loc) and the divided sizes of the legend circles
(legend.breaks) are specified, the legend is drawn; otherwise, it is not drawn. The default
legend is designed in the nested legend arrangement, while the linear legend arrangement is
used when legend.type="linear". Gaps between the legend circles in the linear type are
fixed at the same length as the radius of the medium-sized circle in the legend.
Another problem is how to decide the breaks, i.e., whether to set equal intervals or select the
most representative symbols appearing on the map. The solution to this problem is left to
the user’s decision rather than incorporating it as an automatic process in the function.
A sample output of ProportionalSymbolMap() is shown in Figure 3. From this figure, we
can easily understand the distribution of child population in Nagasaki City.
4. Discussion
We developed the proportional symbol mapping function by using the R language rather than
by the linkage (e.g. import and export) of external mapping software. It is important that
all operations are completely controlled by R commands and limited to the R environment
because such a condition yields immense benefits.
The advantage of proportional symbol mapping is that it displays attribute values for both
the measured point location (true point data) and the representative point of a polygon (con-
ceptual point data). In such case, this mapping method is superior to choropleth mapping
because the latter requires the display of the classification of values, while prop ortional symbol
mapping does not. Therefore, providing the proportional symbol mapping function is benefi-
cial for R users because it provides an appropriate approach to visualize true or conceptual
point data.
The current code supports only circles for symbolization because they have been used most

6 Proportional Symbol Mapping in R
0 10000 20000 30000 40000 50000
−50000 −40000 −30000 −20000 −10000
Easting
Northing
1500
500
100
Child population
Figure 3: An example output obtained with ProportionalSymbolMap(). The circle denotes
child population in census blocks 2000 in Nagasaki City, Japan. The symbols were scaled by
perceptual scaling.
frequently, they are preferred by users, they are visually stable, and they conserve map space
(Slocum 1999). However, the code can still be expanded for other geometric or pictographic
symbols.
Another scaling method termed “range-graded scaling” exists in which the attribute variable
to be represented is grouped into classes and symbolized with a different size of circle (Slocum
1999). In this study, we did not include this scaling method in the R code, because it is feasible
only when data is grouped prior to drawing the symbols and appropriate arguments are spec-
ified. There are three issues in this method: the number of classes, method of classification,
and symbol sizes, although these depend on the users’ mapping policy.
The handling of symbol overlap is also a major issue in proportional symbol mapping. A
small overlap with smaller symbols may result in a vague spatial pattern. On the other hand,
it may be difficult to interpret individual symbols in the event of large overlap due to the
stacking of larger symbols. Thus far, some solutions have been proposed; however, we have
left solution to this problem to the users. With the current code, the users can decide the
extent of the overlap or determine the the symbols to be used, i.e., transparent or opaque
symbols.

Journal of Statistical Software 7
5. Conclusion
The R function described in this paper is an initial function for proportional symbol mapping.
It provides basic but sufficinet functionality for users who would like to express the attribute
values of spatial point data. Consequently, R is equipped with new expressive power and more
options.
Acknowledgments
We are especially grateful to Dr. Akihiko Suyama of the Radiation Effects Research Founda-
tion for his valuable comments and advice.
References
Flanney J (1971). “The Relative Effectiveness of Some Common Graduated Point Symb ols
inthe Presentation of Quantitative Data.” Canadian Cartographer, 8(2), 96–109.
Slocum TA (1999). Thematic Cartography and Visualization. Prentice Hall, New Jersey.
Slocum TA, McMaster RB, Kessler FC, Howard HH (2005). Thematic Cartography and
Geographic Visualization. Pearson Education, Inc., Upper Saddle River, NJ, 2nd edition.
Affiliation:
Susumu Tanimura
Department of Socio-environmental Medicine
Institute of Tropical Medicine
Nagasaki University
1-12-4 Sakamoto, Nagasaki, 852-8523, Japan
E-mail: stanimura-ngs@umin.ac.jp
Journal of Statistical Software http://www.jstatsoft.org/
published by the American Statistical Association http://www.amstat.org/
Volume 15, Issue 5 Submitted: 2005-10-02
January 2006 Accepted: 2006-01-24