EP A/600/ J-94/167
Hexagon Mosaic Maps for Display of Univariate
and Bivariate Geographical Data
Daniel B. Carr, Anthony R. Olsen, and Denis White
ABSTRACT. This paper presents concepts that motivate the use of hexagon mosaic maps and hexagon-based
ray-glyph maps. The phrase "hexagon mosaic map" refers to maps that use hexagons to tessellate major areas
of a map, such as land masses. Hexagon mosaic maps are similar to color-contour (isarithm) maps and show
broad regional patterns. The ray glyph, an oriented line segment with a dot at the base, provides a convenient
symbol for representing information within a hexagon cell. Ray angle encodes the local estimate for the hexagon.
A simple extension adds upper- and lower-confidence bounds as a shaded arc bounded by two rays. Another
extension, the bivariate ray glyph, provides a continuous representation for showing the local correlation of two
variables. The theme of integrating statistical analysis and cartographic methods appears throughout this paper.
Example maps show statistical summaries of acidic deposition data for the eastern United States. These maps
provide useful templates for a wide range of statistical summarization and exploration tasks. Correspondingly,
the concepts in this paper address the incorporation of statistical information, visual appeal, representational
accuracy, and map interpretation.
KEYWORDS: hexagon mosaic maps, ray-glyph maps, comparison plots, bivariate maps, brushing.
Introduction
Maps based on hexagon tessellations are seldom
used, but offer numerous opportunities for rep-
resenting statistical summaries. Two fundamen-
tal maps, the hexagon mosaic map and hexagon-based ray-
glyph map, provide a foundation for many task-specific
variations. For example, the hexagon mosaic map shows
broad regional patterns for a single variable. Variations of
this map bring out distribution features of the variable rep-
resented. Important application variants include compari-
son of two variables through the direct display of differences
and through map overlays. The ray-glyph map provides a
foundation for adding more information to a map. One
map variation shows local estimates and their confidence
intervals. A second ray-glyph variation shows the local as-
sociation of two or more variables. This variation provides
an alternative to two juxtaposed maps (Monmonier 1979)
and color-coded bivariate maps (Olson 1981; Eyton 1984).
A third variation highlights values in specific regions based
on dynamic graphic subset-selection techniques or com-
puted criteria. This paper discusses the relative merits of
these proposed maps and commonly used alternatives.
An acidic deposition study of the United States provides
Daniel B. Carr is an associate professor in the Center for Com-
putational Statistics, George Mason .University, Fairfax, VA 22030.
Anthony R. Olsen is an ecological statistics program leader at
the U.S. EPA Environmental Research Laboratory and Denis
White is a research geographer with ManTech Environmental
Technology Inc. at the U.S. EPA Environmental Research Lab-
oratory, Corvallis, OR 97333.
examples that illustrate the map variations. The study in-
tegrates measurements from several monitoring networks
and addresses the substantial variation in data quality. The
variables in the study include measurements on 19 chem-
ical ion species. The current examples use quarterly sum-'
maries for the years 1982 to 1987 and focus on sulfate and
nitrate deposition. Simpson and Olsen (1990a, 1990b) de-
scribe these data in detail. '
A Context for Hexagon Mosaic Maps
A tessellation is an aggregate of cells that covers space
without overlapping. Only three regular polygons tessel-
late the plane: equilateral triangles, squares, and hexagons.
This paper focuses oh hexagon tessellations. The phrase
"hexagon mosaic map" refers to a map that uses hexagons
to tessellate major areas, such as land masses.
We do not know who created the first hexagon mosaic
map. Since hexagon tessellations occur in nature (i.e., in
bee honeycombs), the extension to game boards, image
processing, and maps seems straightforward. We conjec-
ture that the first use occurred in the context of games. In
image processing, Pfaltz and Rosenfeld (1967) state, "Other
digitized image configurations are possible, for example,
using a hexagonal rather than a rectangular grid, which in
fact seems preferable for some applications." The image-
processing history surely goes further back. While the im-
age-processing literature has devoted significant attention
to hexagon tessellations (Serra 1982), technological conve-
nience for raster devices and computational convenience
have fostered the use of rectangular pixels. Most of today's
image-processing literature addresses square pixels. Thus,
students new to image processing have ample opportunity
Cartography and Geographic Information Systems, Vol. 19, No. 4, 1992, pp. 228-236, 271
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to rediscover hexagon tessellations. This situation likely is
similar in other fields. The purpose here is not to claim
priority for hexagon mosaic maps, which may have been
discovered independently by many different people over
recent decades, but rather to further explore the use of this
structure in cartography.
Hexagons have at least two advantages over squares: vis-
ual appeal and representational accuracy. Carr et al. (1987)
and Carr (1991) discuss the visual appeal. The construction
of maps based on either hexagon or square tessellations
creates visual lines. These visual lines are artifacts of the
construction process and compete with data-generated pat-
terns. The basic claim is that humans, with their sense of
gravitational balance, have a strong response to horizontal
and vertical lines. Thus, the horizontal and vertical lines
generated by square tessellations (in standard orientation)
are particularly distracting and should be avoided.
The strong visual response to horizontal and vertical lines
at different scales might be questioned. At a fine-grain level,
such as using fill patterns based on parallel line screens,
the visual lines should be oriented at 45 degrees from hor-
izontal to facilitate interpretation as value. Castner and Ro-
binson (1969) note that anyone can observe this phenomenon
by examining a half-tone print in a newspaper from differ-
ent orientations. Might not a strong response occur at a
coarser level as well?
Consider an example that addresses symbol congestion
in a scatterplot. Figure la shows a hexagon tessellation of
a scatterplot. The size of a hexagon symbol, as shown in
Figure Ib, represents the relative counts of observations
falling in grid cells or "bins." The largest symbol fills the
highest count cell. Figure Ib is a form of a bivariate density
plot designed to handle large sample sizes (Carr et al. 1987;
Scott 1992). Figure Ic is similar, but it uses a square tes-
sellation with cells having the same area as those in the
hexagon tessellation. Comparison of binned plots to the
original scatterplot shows that the lattice of bin centers dis-
tracts from the pattern of the data. Comparison of the binned
plots against each other suggests that the nonorthogonal
visual lines of the hexagon tessellation are less distracting.
The shape (edges) of the symbols contribute to the dis-
traction in the binned plots. Plotting round symbols in the
square cells helps somewhat (i.e., the sunflower symbols
in Chambers et al. [1983]), but the visual equivalent of put-
ting round pegs in square holes wastes space and calls
attention to the lattice lines. Another improvement shifts the
symbol for each cell toward the center of mass of points in
the cell (Carr 1991). This reduces the visual emphasis on the
lattice lines. The best one can do is ameliorate the visual
artifacts; the hexagon tessellation provides a good starting
place.
Representational accuracy also favors using hexagon cells
over square cells. For bivariate densities with the necessary
derivatives, Scott (1985) has shown that hexagon-based
density estimates have a somewhat smaller integrated mean
square error than square-based estimates. The situation of
approximating a bivariate function using a cell-based step
function is similar. When the bivariate function is reason-
ably smooth, the range of function values over a fixed area
cell generally will be smaller if the cell is as compact as
Figure 1. (a, top) Bivariate points falling in hexagon tessellation
cells of a scatterplot. The bivariate points are sulfate deposition
trends for 1982-87 (x-axis) and nitrate deposition trends for 1982-
87 (y-axis) for sites in the eastern region of the United States.
Binning consists of counting the number of points in each cell.
(b, middle) Hexagon-bin bivariate density plot. The size of the
hexagon symbol represents the number of points in the tessellation
cell. The symbol is scaled so that the symbol representing the
largest number of points exactly fills the tessellation cell, (c, bot-
tom) Square-bin bivariate density plot. The square tessellation
cells of the scatterplot are not shown. Like in Figure Ib, the size
of the square symbol represents the number of points in a square
tessellation cell. The horizontal and vertical lines compete for at-
tention with the trend in the data.
Cartography and Geographic Information Systems
229
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possible. (One possible measure of a cell's compactness is
the dimensionless second moment about the cell center, as
defined by Con way and Sloane J1982]. This measure yields
0.0833, 0.0802, and 0.0796 for squares, hexagons, and cir-
cles, respectively.) Restricting the range of function values
over the cell constrains the integrated mean square error.
Thus, hexagon partitions generally yield better approxi-
mations than square partitions.
The function-approximation arguments in favor of hex-
agons are as not as strong as the visual arguments. One
can construct functions for which hexagon-cell step-func-
tion approximations are not better. When the hexagon-based
approximations are better, Scott's (1985) result suggests that
the improvement is usually slight. Nonetheless, consider-
ations of representational accuracy add further support to
the use of hexagon mosaic maps.
One measure of success of a particular map variation is
its acceptance for routine use by government agencies. A
triangular grid and accompanying hexagonal tessellation
has been proposed for use in the U.S Environmental Pro-
tection Agency's (EPA) Environmental Monitoring and As-
sessment Program (EMAP) (Messer et al. 1991), which
emphasizes probability sampling using a regular geometric
arrangement of samples to achieve spatial coverage. The
use of hexagonal regions is natural both for spatial sam-
pling and data display.
EMAP's design objectives translate into a set of geometric
and cartographic properties for a sampling grid (White et
al. 1992):
1. Equitable spatial coverage of all environmental re-
sources of interest
2. Random positioning to yield a probability sample
3. Equal-area sampling units to enhance precision of
estimates
4. Compact arrangement of sampling units
5. Minimal correlation with any regularly spaced cultural
features
6. A hierarchical structure to facilitate increasing and de-
creasing grid density
7. A realization of the grid on a single planar surface for
the entire domain of application
The grid designed to satisfy these properties is based on
a triangular array of points with a corresponding dual tes-
sellation of regular hexagons established on the plane of
the Lambert azimuthal equal-area map projection. For ap-
plication in EMAP, this sampling grid has been placed on
a hexagonal face of a truncated icosahedron n't to the globe
(Figure 10 in White et al. [1992]). The base density grid for
EMAP consists of points placed about 27 km apart and
tessellation hexagons about 635 km2 in area. This density
represents a compromise between desired resolution and
cost of sampling. The tessellation of these hexagons over
the conterminous United States is shown in Figure 2.
For sampling purposes, the tessellation hexagons may be
considered as strata within which point or area samples
may be taken. The equal-area property assures that all points
or arbitrary areas within the domain of the grid have an
equal probability of being selected for sampling. For analy-
sis and display purposes, the tessellation hexagons provide
a set of equal-area units that minimizes analytical and vis-
ual bias inherent in the use of arbitrary spatial units bounded
by political or other features.
The Hexagon Mosaic Map and Variations
An important task for understanding sulfate deposition over
the United States is to show broad regional patterns. In this
case, the broad regional patterns must be constructed from
irregularly sampled point data. The standard approach for
this task uses a spatial estimation algorithm to obtain es-
timates on a regular grid. The graphical algorithms trans-
form the regular grid estimates into maps. Our approach
used kriging to produce estimates for a hexagonal grid.
(Cressie [1991] discusses spatial smoothing methods, in-
cluding several variants of kriging.) We then transform the
estimates into a hexagon grid-cell choropleth map or hex-
agon mosaic map, as shown in Figure 3 (see page 271).
This map distinguishes different sulfate deposition levels
using color. The colors are ordered in terms of value and
are light enough to allow the state boundaries generally to
be visible.
The use of an equal-area regular grid on a map has inter-
pretational advantages. In Figure 3, each hexagon covers
approximately 2,670 square km. (Figure 3 has been repro-
duced from early studies of acidic deposition and is not an
equal-area map. The area distortion in this Lambert con-
formal conic projection is modest, so the map has not been
revised.) The total area affected by sulfate deposition, say
in the range of 20 to 30 kg per hectare, can be determined
simply by counting hexagons. We use this fact and deter-
mine the class intervals for the United States from the per-
centage of hexagons involved.
The lowest deposition class in the eastern United States
covers 10% of the area and has an upper bound of 11.4 kg/
ha. The next class covers 15% of the area and its upper
bound is 14.8 kg/ha. Thus, 25% of the hexagons have val-
ues at or below 14.8 kg/ha. The cumulative percents used
to obtain boundaries are 10, 25, 50, 75, 90, and 95. The
highest deposition class covers the last 57o of the area and
is shaded with the darkest color. For this application, we
prefer defining classes by the percent area involved. If sul-
fate values determine directly the class intervals, showing
the percent of hexagons involved still provides an imme-
diate area-based interpretation.
Figure 3 itself is a variation on a basic mosaic map, since
the map is presented in two parts. In this application, the
big difference in values between the western and eastern
regions strongly motivates the split. Notice that at least 90%
of the values in the western region fall in the lowest cate-
gory of the eastern region. Thus, splitting the map into two
regions provides better presentation of the distribution of
deposition occurring in the United States.
Alternative methods can show the information rep-
resented in Figure 3, with the leading alternatives being
contour (isarithm) maps and perspective views. Colored
contour maps provide the main alternative. Colored con-
tour maps are similar to mosaic maps in several ways. Both
normally involve an estimation step that produces values
on a regular grid. While contouring algorithms typically
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Figure 2. EMAP 635 km2 nonrandomized sampling grid tessellation hexagons. The grid provides a basis for probabilistic environmental
sampling.
assume a rectangular grid, it is quite possible to base both
contour maps and hexagon mosaic maps on the same tri-
angular grid of estimates. The two maps are fundamentally
the same at this level of construction. Both methods rep-
resent areas and hide details concerning the amount and
placement of the underlying point data. The differences
between the two similar methods appear at the interpre-
tation stage.
The hexagon mosaic map can have interpretation advan-
tages over a contour map, because the regular tessellation
suggests the use of an estimation process and facilitates
thought about confidence intervals. The hexagon edges at
class boundaries imply the estimation lattice that has been
used. In contrast, smooth contour lines give little clue to
this underlying estimation step. The value for each hexa-
gon typically is presumed to be an estimate that represents
the whole hexagon region. The value for the hexagon does
not have to match the value at any particular sampling site
within the hexagon. In contrast, many people interpret
contour lines as precise, and knowledgeable local experts
argue about the placement of a contour line relative to ob-
served values at sampling sites.
Interpretation differences also appear in terms of confi-
dence intervals. Analysts can easily think of the estimated
value for a hexagon as having upper and lower bounds
corresponding to the upper- and lower-confidence sur-
faces. The units for confidence bounds are data units. Prag-
matically, a contour line is a sequence of line segments
whose endpoints are two-dimensional (bivariate) spatial
coordinates. The notion of confidence intervals for contour
lines is not well defined. Some understanding about the
accuracy of a contour line can be obtained by drawing the
corresponding contour lines from the upper- and lower-
confidence surfaces. However, one-to-one correspondence
between the points on the different lines is not guaranteed,
so probabilistic considerations for contour lines do not lead
to the one-dimensional simplicity of traditional confidence
intervals. Consequently, the hexagon mosaic map, al-
though less aesthetically pleasing because of its jagged
boundaries, has an interpretational advantage.
Analysts often use perspective views to obtain a Gestalt
impression of a surface. For rough surfaces, several per-
spective views may be required because part of the surface
can be hidden. (See MacEachren and DiBiase [1991] for
representations covering variation in continuity and ab-
ruptness.) Perspective views typically are poor for identi-
fying the geographic location of peaks and valleys.
Consequently, contour plots and perspective views often
are both shown (Crotch 1983; Tufte 1991) and are best re-
garded as complementary, rather than competing, views.
In general, hexagons help convey the spatial structure of
information. The idea of neighboring regions is clear. The
notion of averaging values from neighboring regions to ob-
tain a smoother surface estimate is straightforward and
provides a reasonable introduction to more complex
smoothing approaches such as kriging.
Cartography and Geographic Information Systems
231
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Comparison Plots
A recurrent graphical task is the comparison of two surfaces
on a map (Lloyd and Steinke 1976,1977; Steinke and Lloyd
1983; Monmonier 1990a). For example, researchers might
want to compare sulfur dioxide (SOj) emission data with
sulfate (SO,) deposition data in the attempt to better un-
derstand the deposition process. However, comparing sur-
faces is nontrivial. The complexity of surfaces and short-
term visual memory limitations complicate the comparison
of juxtaposed surface representations. Superimposed plots
reduce the demand on visual memory, but restrict the form
of the plot. Fishnet (perspective view) plots and translucent
surfaces can be superimposed. However, this leads to a
further difficulty.
Consider the simpler case of comparing two superim-
posed curves. Cleveland and McGill (1984) show that hu-
mans are very poor at assessing the difference between two
superimposed curves. Our visual system tells us the dis-
tance between closest points on the two curves, regardless
of direction, rather than computing the vertical distance
between corresponding points. There is no reason to think
we will do any better at comparing superimposed surface
representations. Consequently, an advantageous approach
computes and represents differences (and relative differ-
ences) directly. Such representations are straightforward
for hexagon mosaic maps, since values are available for
each hexagon.
A more limited way of making comparisons is to overlay
a pair of two-class distributions, one from each surface,
creating a bivariate correlation map. Olson (1981), Carsten-
sen (1982), Lavin and Archer (1984), and Eyton (1984) dis-
cuss this type of map. Figure 4a shows the two classes from
the sulfate-deposition plot. The class-interval break at 20
kg/ha was chosen because the Canadian government is
concerned about values above 20 kg/ha.
The next issue to consider is which class-interval break
to use from the smoothed emission data. (Smoothing fa-
cilitates area-based comparison because the emissions come
from point sources, such as coal-fired plants.) Several plau-
sible answers can be given. Figure 4b shows a class-interval
break that was selected to produce the same areas as the
classes in Figure 4a. Thus, attention can be focused on the
location of the class differences represented in the overlay
in Figure 4c. The visual impression is that the SO2 rises and
mixes with cleaner air as it moves to the northeast and is
eventually.deposited as SO,. The movement toward the
northeast is only a general visual impression and does not
necessarily match meteorological data. The visual impres-
sion of a much smoother deposition surface is very strong
and consistent with our understanding. Use of two-class
overlays provides much less information than the direct
plot of differences, but this procedure can help by focusing
attention on specific aspects of two surfaces.
Showing Trends and Confidence Intervals
Using Ray-Glyph Maps
One purpose of the acidic deposition study was to leam
about trends. For the trend study, seasonally (three-month)
aggregated data for the period 1982 to 1987 are available
for a number of monitoring sites. The aggregated data could
be used to yield 24 seasonal maps. Comparing the sequence
of 24 maps to assess trends would be a difficult visual task,
complicated by the presence of seasonal variation. A direct
approach shows estimated trends at each monitoring site.
min = 5
Figure 4a. Two-class map of sulfate deposition (kg/ha). The class
boundary of 20 kg/ha was chosen based on policy considerations.
min = 0
Figure 4b. Two-class map of sulfur dioxide emissions (kgfha).
The emissions surface has been smoothed somewhat, but the class
boundary line is still rough. The specific class boundary defines
regions with the same areas as the classes in figure 4a.
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Cartography and Geographic Information Systems
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Eastern North America
Figure 4c. Overlay of two two-class maps. Two general impres-
sions are that the SO, boundary is much smoother than the SOt
boundary, and that the SO, area lies more toward the northeast.
Monitoring studies often yield data of varying quality,
so we use nonparametric trend methods to mitigate prob-
lems presented by poor data. In particular, we used Sen's
nonparametric slope estimate and associated confidence in-
tervals (Gilbert 1987). For each site, slope estimation pro-
ceeds by fixing a season and computing all possible pairs
of slopes between years, then listing the slopes from all
four seasons, and finally selecting the median from this list
as Sen's median slope estimate. Confidence intervals for
the estimate are based on order statistics; therefore, they
are typically asymmetrical about the estimate. Computing
the nonparametric slope estimates and confidence intervals
for individual sites is straightforward.
The next task is to show the estimates and confidence
intervals. If we want to portray only the estimates and their
broad regional patterns, we could use a hexagon mosaic
map. However, incorporating local confidence intervals on
the map challenges us to use a different technique. The
large number of sites poses a problem of graphic conges-
tion. A solution is to aggregate the sites into hexagon re-
gions and show only a summary slope and confidence
interval for each region with data. While this introduces
statistical summarization issues that deserve attention, Fig-
ure 5 illustrates the representation concept, and the octa-
gons in the plot locate the centers of hexagon regions (not
drawn) that contain site data.
Figure 5 is a ray-glyph map. The ray is composed of a
line segment and a region-centered dot or polygon at the
base. Ray angle encodes the slope estimate. In this case,
the scaling was chosen so that a horizontal ray represents
a zero-degree slope (no change). A ray straight up rep-
resents an increase of 6.6 kg/ha per year; a ray straight
down represents a decrease of 6.6 kg/ha per year. In Figure
/ Sulfate Deposition Trends
f Period = 1982 to 1987
Units = kg/ha Per Year
- i 6.6
\ S 3.3
~-^ "" V j ',
» \ •- o.o
\ ]__ % -3.3
t -6.6
Figure 5. Ray-glyph and arc map showing sulfate deposition trends
and 90% confidence intervals. The rays and arcs represent sum-
maries for local hexagon-shaped regions. Rays with a negative
slope represent a decrease in deposition. The shaded arcs covering
zero suggest weak evidence concerning nonzero trends.
5, the scale upper limit for the ray is determined by the
90% confidence intervals shown as filled gray arcs; the scale
lower limit is determined by symmetry. This diminishes
the resolution for the estimate. (One observation with a
large confidence interval was omitted for this reason.) On
the plot, the majority of rays point slightly down, sug-
gesting a small decrease in deposition over the six years.
However, the majority of confidence intervals include zero,
and a few might be expected to exclude zero at random
since this is a multiple comparison situation. Thus, evi-
dence for change is weak. Note also that regions without
data and regions with highly variable estimates are quite
evident. The ray-glyph map is effective for representing
local area summaries.
Of other representations that might be considered, framed-
rectangle symbols, as shown in Figure 6, provide a partic-
ularly informative alternative. Cleveland and McGill (1984)
developed the framed-rectangle symbol based on their
studies of perceptual accuracy of extracting the encoded
information. Their studies showed that judging positions
against common nonaligned scales was superior to judging
circle areas, angles, colors, and other representations. Con-
sequently, they added a frame with center ticks to provide
a scale for judging bar height. The framed-rectangle sym-
bols achieve the goal of perceptual accuracy, but other de-
sign issues also are relevant. These include addressing the
Cartography and Geographic Information Systems
233
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Figure 6. Two framed-rectangle plotting symbols. The frame and
tics increase the perceptual accuracy for judging bar heights. The
left bar height can be assessed by comparison with the center tics.
The right bar height can be assessed by the white between the bar
and the top of the frame.
recurrent problem of symbol congestion and deciding on
the relative emphasis between local symbols and the other
information on the map. The horizonal and vertical lines
of the symbol frame draw visual attention and may inter-
rupt the visual flow from symbol to symbol. Perceptual
accuracy of extraction provides just one design criterion.
The ray glyph, with an open octagon as the base, is a
line symbol and is better suited for overplotting than an
area symbol like a bar. The ray glyph can be made to blend
with or stand out from the rest of the map through control
of line thickness, octagon size, and ray length. The octa-
gon, with lines connecting opposite vertices, provides a
visual anchor that improves the perceptual accuracy of ex-
traction of the ray angle. When the glyph must be small, a
polygon with few sides, such as a diamond, can be used
as a visual anchor. In the context of hexagon tessellations,
neighboring octagons also provide local scale against which
angles can be judged. Positive and negative ray slopes can
be assessed by looking at the corresponding octagons on
the right. The ray glyph provides even greater accuracy on
a hexagon mosaic map when ray scale is nested within
classes that are distinguished by different colors. The ray
glyph is well suited for use with hexagon tessellations and
provides the flexibility to address several design objectives.
Bivariate Ray-Glyph Maps
and Graphical Interaction
The representation of bivariate information using maps is
a challenge. The task of interpreting side-by-side univariate
maps has not proved easy, so various researchers have
proposed different methods. Monmonier (1979) discussed
the cartographic cross-classification table, which is an in-
termediate step between juxtaposed univariate maps and
bivariate maps. Carstensen (1982) and Lavin and Archer
(1984) experimented with and developed continuous bi-
variate crosshatching maps. Wainer and Francolini (1980)
and Olson (1981) have investigated the utility of bivariate
color maps. Eyton (1984) has proposed additional tech-
niques. Investigations suggest that bivariate color maps can
be helpful when they have few categories and the colors
are carefully selected. Bivariate ray-glyph maps (Carr 1991)
provide an alternative that represents the variables with
much greater resolution.
Figure 7a shows a bivariate ray-glyph map. Rays pointing
to the right represent sulfate deposition trends and rays
pointing to the left represent nitrate deposition trends.
Dropping the confidence intervals and omitting scale con-
siderations of zero slope and symmetry increase the reso-
lution of sulfate deposition rays, in comparison with
corresponding rays in Figure 5. The two rays of the bivar-
iate ray glyphs generally point down or up together, so the
attributes are positively correlated. With a few exceptions,
the spatial change in the two trends is reasonably smooth.
Eastern North America
; -^ "-, i/^r ' >
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—'~*^f~\ "M 7>y{ ^-*"^ s**^-'-*^ -•'
\ T"<^^rrr^^ ^
V fv V- .1**-. -*C-V^*-^-.' X
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Bivariate Trends
Figure 7a. Highlighted, bivariate ray-glyph map of sulfate and
nitrate deposition trends. Sulfate rays point to the right, and
nitrate rays point to the left. Rays pointing straight up correspond
to maxima, and rays pointing straight down correspond to min-
ima. The scaling is less than optimal for assessing zero trends,
but makes assessing correlations easier. The highlighted points are
determined from Figure 7b.
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Cartography and Geographic Information Systems
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The bivariate ray map seems to be an effective technique
for showing bivariate associations, provided the individual
areas to be represented are not too small.
Having several attribute variables available for mapping
introduces new possibilities. In particular, subsets can be
selected from one view of the data and highlighted in an-
other. This has been done in Figures 7a and 7b. Figure 7b
shows a scatterplot of bivariate slopes. To select a set of
relatively unusual points for highlighting, we computed
the bivariate density for each point and chose the four low-
est bivariate density points. This selection approach can
easily be automated. The points have also been highlighted
on the map (Figure 7a), using larger symbols with thicker
lines. Alternatively, points can be selected through direct
graphical interaction.
The notion of dynamic simultaneous highlighting using
a mouse is often called "brushing" in the literature. Mc-
Donald (1982), Becker and Cleveland (1987), and Stuetzle
(1987) provided early papers on this technique, typically in
the context of attribute variable plots. Can- et al. (1987),
Monmonier (1989, 1990b), Dunn (1989), and Haslett et al.
(1991) have all addressed dynamic subset selection and
highlighting in the geographic context. Brushing is a pow-
erful discovery technique.
Highlighting is important in ray-glyph plots, because
unusual rays may not be immediately obvious. The ideal
highlight allows unusual rays to be found almost instantly.
Julesz and Bergen (1983) discuss the visual phenomenon
of immediate symbol location and make the distinction be-
tween rapid preattentive vision and slow attentive search.
They propose the theory of textons to characterize circum-
CO
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_
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municate with the public. The computing world and the
rise of scientific visualization provides us with unprece-
dented opportunities. The pressing needs of humankind
on a finite planet provide us with unprecedented challenges.
ACKNOWLEDGMENTS
The authors would like to thank Jeanne Simpson, who played an
active role in the data preparation and summarization; Kevin Ad-
ams, who helped develop software for the initial hexagon mosaic
maps; and the reviewers for their helpful comments. The research
described in this article has been funded by the National Science
Foundation under grant no. DMS-9107188 and by EPA through
contracts 68-C8-0006 to ManTech Environmental Technology Inc.
and 68-CO-0021 to Technical Resources Inc. This paper went through
the U.S. EPA's peer and administrative review and was approved
for publication.
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236
Cartography and Geographic Information Systems
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!
Annual 1985-1987 Sulfate Deposition
kg/ha
Max = 17.8
95% -M- 13.3
90% -m~ 11.2
75
6.9
4.0
Min=0.8
kg/ha
Max = 42.7
95% -H- 32.3
90% -§1- 28.6
75%
23.8
17.3
14.7
11.4
Min=5.5
Figure 3, Carr el al. Hexagon mosaic map of sulfate deposition (kg/ha). The map is split to provide greater resolution in both the east and west. Percent of area, found by counting
hexagons, determines the class interval boundaries. The hexagon edges at class boundaries indicate the hexagon cell size. The underlying estimation lattice consists of hexagon cell
centers. The map is similar to a color-contour map, but suggests involvement of an estimation process.
-------
Bivariate "Cross Map"
!
I I BELOUJ the mean for BOTH uariables
B9 RBOUE meon ONLY for Female Officials
I >. I RBOUE mean ONLV for Females Working
HH] RBOUE the mean for BOTH uariables
Femole$ Working.
i..rU'v ' 'M<-•.•.«<:r/itoiK4JlS8|;
Breaks at nationwide means
J
f
.y
-e
tx.
1
IX
t
Figure 4, Monmdnier. Juxtaposed cross map and scatlerplot used in the correlation script's second act. Conventional key does not appear below the map
until the end of the scene, after spiral variation of the category breaks (described in the text).
-------
TECHNICAL REPORT DATA
frtcau md liuruenonj on ttit rtvmt btfon
1. REPORT NO.
EPA/600/J-94/167
2.
3. RI
PB94-160538
4. TITLl ANDSUiTlTLE
Hexagon Mosaic Maps for Display of Univariate
and Bivairiate Geographical Data.
K. REPORT DATE
•. PERFORMING ORGANIZATION CODE
7. AUTHORISI.
br'Carr1, A.R.01senb,
D. White6
•. PERFORMING ORGANIZATION REPORT NO.
I. PERFORMING ORGANIZATION NAME ANO ADDRESS
•George Mason Univ.
"USEPA, ERL-Corvallis, Corvallis, OR
CMETI, Corvallis, OR
10. PROGRAM ELEMENT NO.
ITTCONTRACT/ORANT NO.
12. SPONSORING AGENCY NAME ANO ADDRESS
US Environmental Protection Agency
Environmental Research Laboratory
200 SW 35th Street
Corvallis, OR 97333
13. TYPE Of REPORT ANO PERIOD COVERED
Journal Article
14. SPONSORING AOENCY CODE
EPA/600/02
IS. SUPPLEMENTARY NOTES
1992. Cartography and Geographic Information Systems
19(4) :228-236, 271,271
1C. ABSTRACT
Hexagon mosaic maps and hexagon-based ray glyph maps are presented.
The phrase "hexagon mosaic map" refers to maps that use hexagons to
tessellate major areas of a map such as land masses. Hexagon mosaic
maps are similar to color-contour (isarithm) maps and show broad
regional patterns. The ray glyph, an oriented line segment with a dot
at the base, provides a convenient symbol for representing information
within a hexagon cell. Ray angle encodes the local estimate for the
hexagon. A simple extension adds upper- and lower-confidence bounds as
a shaded arc bounded by two rays. Another extension, the bivariate ray
glyph, provides a continuous representation for showing the local
correlation of two variables. The theme of integrating statistical
analysis and cartographic methods appears throughout this paper.
Example maps show statistical summaries of acidic deposition data for
the Eastern United States. These maps provide useful templates for a
wide range of statistical summarization and exploration tasks.-
Correspondingly, the concepts in this paper address the incorporation
of statistical information, visual appeal, representational accuracy,
and map interpretation.
7.
KEY WORD* ANO DOCUMENT ANALYSIS
DESCRIPTORS
b.lOENTIFIERS/OPEN ENDED TERMS
c. COSATi Field/Group
hexagon mosaic maps, ray-gylph maps,
comparison plots, bivariate maps,
brushing.
, DlSTRHUTlON STATEMENT
Release to Public
"
ea
31. NO. O^ PACES
8
0 SECURITY CLASS (Ttat ftfti
Unclassified
aa. PRICE
•PA
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