Download map projection and global-scale spatial cognition

MAP PROJECTION AND GLOBAL-SCALE SPATIAL
COGNITION:
ISSUES IN PSYCHOPHYSICAL ESTIMATION
Sarah E Battersby
University of California at Santa Barbara
[email protected]
Is cognition of geographic space affected by the source from which the knowledge is acquired (i.e., map, navigation,
verbal description, etc.)? Studies have shown that it is and that these effects may be the result of the source, source
content, or sensory mode, amongst other potential causes. If cognition of geographic space is affected by the source
from which the information is acquired then it should also be affected by the distortions inherent in the information in
the source. Spatial information for global-scale space can be acquired through limited means – small-scale
(cartographic) representations (i.e., maps) being the most common and most practical. In fact, maps are the only way in
which the entire Earth can be viewed at one time. This paper summarizes results of two studies on global-scale spatial
cognition with respect to distortion in map projections and standard effects of psychophysical estimation.
INTRODUCTION
How do people learn and remember information about their environment? In small geographic areas (e.g., single
rooms, buildings, or neighborhoods) there many ways that this can be done, for instance through direct environmental
experience (wayfinding), verbal descriptions, or static pictorial representations (e.g., a “photo tour” or map). Studies
have shown that each of these methods develops a slightly different configuration of the environment in the
individual’s cognitive map. Cognitive scientists have studied the distortions in these cognitive maps from numerous
different perspectives and at several scales ranging from desktop space to global-scale. While there are endless
measures that need to be controlled for in studies for these relatively small areas, from a geographic and cartographic
perspective they can be managed. For instance, it is possible to ensure that the study area is unfamiliar to all subjects in
the experiment or that the level of cartographic abstraction or the symbology used in the reference maps is appropriate
for the task and is understood by all subjects. When examining cognition of very large, or global-scale, areas,
however, there is a particular cartographic problem that is difficult to control for – the distortion caused by map
projections. In global-scale spatial cognition this can play a serious role in altering an individuals perception of the
world, and can lead to gross modifications of “reality” in terms of measuring properties of a cognitive map accurately.
The specific effects of these modifications are the topic of my study and will be discussed in this paper. In particular
the role of cartography and map projections in studying global-scale spatial cognition will be discussed. While there
are many properties of cognitive maps that can be measured this work concentrates on the property of area. Area is a
nice property to study in spatial cognition work because it is easily understood and can be relatively simply calculated
without any training.
Since measuring cognitive maps requires externalization of properties of the map (i.e.,
estimating the area of a feature) psychophysical estimation methods are used in these studies. This paper aims to frame
the role of map projections versus common errors that would be found in any standard psychophysical estimation
exercise.
MAP PROJECTIONS
Obviously there is distortion in map projections, and when studying issues in spatial cognition and cognitive maps we
need to pay proper attention to the possible effects of these distortions on forming mental representations of space. As
quantity of distortion varies at different scales the concern that is expressed about the distortion affecting data gathered
about cognitive maps should vary. For instance, distortion at the country or even continent scale may be negligible,
however at the global-scale it is undeniably important. This can be seen easily in Figure 1 showing the differences in
national- (United States) and global-scale data in three different projections. While there is noticeable distortion in the
national-scale data, there aren’t any distortions that would be easily measurable when examining the externalization of
a cognitive map – it could very easily just appear as expected random variations in the data. At the global scale,
however, the distortion can be tremendous and thus the differences could lead to significant distortions of a cognitive
map.
Figure 1. Three different projections of the United States (top figure) and
countries of the world (lower figure).
BACKGROUND
Cognitive maps
Formation of cognitive maps is a gradual developmental process that involves the accumulation and integration of
different bits of spatial information over time in order to form a usable cognitive map of an area. The spatial
knowledge used to form a cognitive map can be acquired from a number of different sources (e.g., maps, wayfinding,
verbal descriptions, etc.), and each of these sources provides different types of spatial knowledge and different types of
errors/distortions for an area. Procedural knowledge derived from navigation through an area should differ from the
survey knowledge gained from study of a map of the area. For instance, Thorndyke and Hayes-Roth (1982) studied
ability to estimate Euclidean distance, route distance, orientation, simulated orientation, and location of points in an
office building after subjects had learned the area either from navigation or map study. While the authors found
navigation-based knowledge to be more robust, it takes a substantially longer time to develop; map learning provided
somewhat “instant” survey knowledge in comparison. With formation of cognitive maps at different scales the
methods that are available to learn about the area will differ.
Montello (1993) divides space into four different scales: figural, vista, environmental, and geographic (later miniscule
was added and geographic was changed to gigantic). Across these scales, the sources used in acquiring spatial
information will differ. Miniscule space is too small to see without aid (e.g., microscope), figural space is smaller than
the body, vista space can be perceived from one location without substantial locomotion, environmental space requires
considerable locomotion to perceive fully, and gigantic space is too big to see without aid. At the geographic scale, the
scale of this research, the only source material that can be used to obtain a picture of the entire area studied is a map –
even using a globe does not provide an uninterrupted view (the globe must be rotated and mentally knitted together in
order to have a complete view). With the distortion in the source material it is only logical that it would affect the
overall accuracy of the resulting cognitive map.
Area estimation
There have been numerous studies on estimation of area of geometric features. These studies have primarily focused
on regular geometric shapes, rather than the irregular shapes that geographic features often take. Since this work is
focused on cognitive geographic research we will only review in detail the few relevant works interested in cognition of
large geographic areas. First, however, we will briefly review psychophysical estimation methods, as they are relevant
to the data gathering and analyses in the studies reported here.
Psychophysical estimation
Psychophysics is a field conceived of by G.T. Fechner, beginning with the 1860 publication of his book Elemente der
Psychophysik (Fechner 1860). The goal of psychophysics is to identify the relationship(s) between the physical
magnitude of a stimulus and the perceived psychological magnitude as measured by an individual. While Fechner was
primarily interested in measuring what he considered to be “lower mental activities” such as sensation, the field of
psychophysics is also interested in measurement of other, “higher” mental activities such as distance or area
estimations. For obtaining area estimates there are only a few methods – psychophysical ordinal, interval, or ratio
scaling. Ordinal measurements provide the least information and contain basic information about the rank order of
areas, indicating which feature is larger or smaller than other features without specifically indicating how much larger
or smaller the feature is. Interval measures provide quantitative information about the difference in area between
features, but there is no absolute zero-point in the measurements so the difference cannot be described as a proportion
of any other distance. Ratio measurements have an absolute zero-point and thus provide a more meaningful
measurement of area. As the work in this study is based on ratio-scaled measurements this is the only measurement
method that will be examined in detail. For a more in-depth discussion of measurement methods see Golledge (1976).
Stevens’ levels of measurement divide data into four categories: nominal, ordinal, interval, and ratio. This scheme for
measurement is based on the definition of measurement as the “assignment of numbers to objects according to a rule.”
(Stevens 1946:677). Ratio data are characterized by an absolute zero-point and an arbitrary interval (the distance to be
called “one unit”). These properties of the data allow for more substantial quantitative analysis to be done with the
data. Because of this characteristic, ratio data is commonly gathered for analysis of psychophysical data. There are
several ways in which ratio data can be gathered. These methods are generally classified into four techniques:
magnitude production, magnitude estimation, ratio production, and ratio estimation. For area estimation, the focus of
this work, magnitude estimation is the most useful method for measurement, so this is the only one of the four that will
be discussed in detail.
With respect to magnitude estimation the estimates are generally made in terms of familiar units such as square miles or
kilometers or with respect to a standard measure (modulus) with which the subjects are familiar. For estimation of
area the estimation is typically done using a relative measure based on a given standard measure (e.g., if rectangle A is
100 units, how many units is rectangle B?) as this provides a controlled standard size to be used in estimation, as
opposed to relying on the individual’s notion of a “mile” or other unit of measure.
Many studies have found that magnitude estimation data (ratio) is often well fit by a power function. A power function
states that the estimated (cognitive) measure of a distance () or area is equal to the actual measure () raised to the
power of some exponent times a constant scaling factor (k); it is represented by the following equation:
  k n
For the purposes of this paper, we will be discussing the relationship of area estimates of geographic features as fit by
power functions, and as they relate to distortion in map projections. First we will review existing studies of geographic
area estimation.
Literature review
Kerst and Howard (1978) examined the effect of memory on psychophysical estimation of geographic features. This is
in contrast to many previous psychophysical area estimation studies that had used regular geometric shapes in
controlled sizes. In Kerst and Howard’s work subjects were asked to estimate the size of states or countries relative to
the area of a reference stimulus - Pennsylvania for the subjects estimating state size and France for the subjects
estimating country size. Each of the two stimuli were assigned the same value of 100 units of area. For both country
and state data a power function fit to the log mean estimates and log actual areas provided a good fit for the data, with
exponents of n=0.4 for states (r=0.96) and n=0.31 (r=0.88) for countries. While these exponents were consistent
with the general trend for estimates of area from memory they were much lower than those generally seen in perceptual
experiments. These results cannot be directly compared to estimations of the area of regular geometric features
because in this case no subject was shown any stimuli. In any experiment using regular shapes or unfamiliar areas a
stimulus must be shown before an estimate of area can be given.
This study can be examined in terms of Brown and Siegler’s (1993) research on area estimation. While they did not
calculate a power function exponent, a reanalysis of their data indicates an exponent of 0.47. Even with the reanalysis
it is difficult to put Kerst and Howard and Brown and Siegler’s work on the same ground due to differences in the
methods used to obtain the subjects’ estimates. While Kerst and Howard asked for estimates relative to a reference
feature (Pennsylvania or France equals 100 units) Brown and Siegler asked for estimates in square miles and only gave
the area of the entire United States (including Alaska and Hawaii) for reference. Not only might it be particularly
difficult to calculate area relative to a stimulus that is over 3.6 million square miles, but in calculating the subjects still
had to remember that the reference area given included two states not attached to the main body of the United States
and that one of these states is extremely large. It is difficult to imagine that the estimates obtained were of the highest
quality possible in this type of experiment.
Returning to Kerst and Howard’s (1978) work, as they found the power function exponent for estimation from memory
to be substantially lower that those generally seen in perceptual estimation exercises using regular geometric shapes generally about n=0.8, they continued on to conduct a second, perceptual, study. In order to compare the results for
perceptual estimation they conducted a second study in which one group of subjects was allowed to view a map for 7
minutes before estimating the area of the study regions, and a second group of subjects were allowed to use a map
while estimating the areas.
In both of these two conditions the power function exponents increased substantially to n=0.79 (r = 0.99) for the
perceptual condition and n=0.60 (r=0.97) for the memory condition. While these results are more in line with existing
psychophysical estimation tasks there is a question of geographic error that was not addressed in the work. Kerst and
Howard did not state what map projection was being shown to the subjects. While the projection could have been
equal area, which would provide a valid cue to the actual area of the regions, it could also have very easily been a nonequal area projection and have provided inaccurate cues for area estimation. Without this information about potential
distortion due to projection is it difficult to fully analyze the results in terms of how accurately the subjects were able to
estimate the area of regions perceptually.
Neither of these studies analyzed their results with consideration to the distortion that could have been introduced
through viewing non-equal area map projections. In the next section we will discuss a series of experiments that we
have conducted that examine more closely the relationship between area estimation for global-scale areas, distortion in
map projections, and standard trends in psychophysical estimation.
METHODS
Both studies discussed in this paper used the same set of twenty-six test regions. Regions were selected from each of
four latitude “bands” – 0-20 degrees, 20-40 degrees, 40-60 degrees, and over 60 degrees. For lack of a better method,
the centroid was calculated for the countries of the world (and Alaska) and the latitude of the centroid was used to
classify each country into its latitude band. Within each of these bands a series of regions were selected to provide a
range of areas. An attempt was made to ensure that the regions selected were likely to be familiar to an average
undergraduate non-geography major audience. The twenty six regions selected are as follows: Greenland, Brazil,
Austria, South Africa, North Korea, Denmark, Australia, Antarctica, Spain, Sweden, Vietnam, Greece, Peru, India,
Mexico, Japan, Switzerland, Ethiopia, Italy, New Zealand, Alaska, China, Russia, Norway, Guatemala, and Venezuela
(Figure 2).
Figure 2. Regions selected for estimation tasks.
Tasks – estimation from memory
Each subject was presented with the names of the twenty-six regions in one of ten random orders. To help them make
their estimates in the instructions they were told to assume that the area of the conterminous United States (or the
“lower 48”) is 1,000 units and to base their estimates for the regions on that value. An example was given to ensure
that the instruction was clear. In addition the subjects were asked to rate their familiarity with each region as well as
provide some minor background information (gender, age, and place of birth).
Subjects. Subjects (n=194) were recruited from an undergraduate regional geography class. The subjects were run as
a group during a class meeting. Participation was optional and students were given extra credit for participating.
Results. Perhaps one of the most telling statistics is that in general the subjects didn’t do as poorly as might have been
expected. In fact, the correlation between estimated area and actual area was 0.82 (calculated individually for each
subject and averaged across subjects).
While it would be almost ideal to be able to identify a strong “Mercator
effect” (increasing distortion with increasing absolute latitude) or another common projection effect, there is little to
report in that arena. The correlation between absolute latitude and relative overestimation (log(estimated area / actual
area)) was a mere 0.18. Also of note is that the correlation between actual area and relative overestimation was –0.53.
If the estimates for the individual regions are examined more closely there is a definite visible trend of overestimating
the small regions and underestimating the large regions. With limited relationships between map projection distortion
and estimation appearing, but with a stronger trend of following typical psychophysical estimation patterns of this type
it is logical to explore other factors that could be affecting the estimations. The most reasonable first choice is to
examine the fit of the data with a power function.
A power function was fit to the log estimates and log actual areas for each subject individually. When the data was
collapsed across all subjects the average coefficient was n=0.45 (range 0.06 – 0.92). For reference, Brown and Siegler
(1993) found an exponent of n=0.47. When considering that these result is based on estimates from memory there are
many confounding factors that could have affected the data – for instance, general confusion about regions – perhaps
confusion between the similar sounding Austria and Australia or mis-identifying locations in the cognitive map, etc.
In light of this and the fact that the exponent found is substantially lower than those that are typically found in
perceptual studies (e.g., 0.8 in Teghtsoonian 1965), a second study with perceptual data is merited.
Tasks – estimation from perception
Study 1 was intended to explore the distortions that are introduced into area estimates when the estimates are made
using distorted and undistorted reference materials. Acquiring perceptual estimates will serve two purposes –
examination of effects of projection on estimates from memory and perception, and examination of how knowledge of
projection distortion is used in estimation. Estimates based on projected and un-projected (globe) source data would
allow for controlling of external (non-projection related) factors that might cause distortion in area estimates, as only
the projected data would be distorted. Additionally, perceptual estimates from different sources with varying patterns
of distortion will also provide cues as to how distortion is recognized in sources and how it impacts estimates.
Tasks. Each subject was presented with the names of the twenty-six regions in one of ten random orders. To help
them make their estimates each subjects was given one of three references (globe, Mercator projection map, or a
Sinusoidal projection map) to use when making their estimates. As was the case in the previous experiment, in the
instructions they were told to assume that the area of the conterminous United States (or the “lower 48”) is 1,000 units
and to base their estimates for the regions on that value. Some of the results from this study will be discussed with
respect to a third study completed by every subject following completion of the estimation exercise. In the third study
subjects were shown a series of six map projections (three equivalent, three non-equivalent) and asked (1) does this
projection preserve area, and (2) if not, please mark the areas in which the projection is the most distorted.
Materials. For the globe condition a 12” diameter (1:42,000,000) globe was used. With the exception of graticule,
country/state boundaries, and arrows indicating the name and location of the regions being tested there were no other
markings on the globe. The Mercator and Sinusoidal projection maps were both printed at a scale of 1:42,000,000 and
contained no markings other than graticule, country/state boundaries, and labels for the regions being tested.
Subjects. Subjects (n=58) were recruited from an introductory human geography class and an introductory “maps and
charts” class. The experiment was given outside of class hours and students were offered extra credit for participation.
Results. When subjects were given a reference to use while estimating area of the regions they were very accurate
with respect to the reference given – but not with respect to actual area when a non-equivalent projection was the
reference. The correlations between estimated and actual area for subjects in the globe and Sinusoidal projection
conditions (both equal-area) were both 0.93. This would be expected – people are quite good at estimating the area of
the feature directly in front of them. The subjects in the Mercator condition were almost as accurate at estimating the
displayed area, showing a correlation between estimated and projected area of 0.82 – bear in mind the relationship
between actual area and Mercator projected area is only 0.53. The fact that the correlation between estimated area and
Mercator area was lower than for the two equal area conditions is an indicator of the fact that there was likely some
compensation for projection distortion being done by the subjects in the Mercator condition when they were making
their estimates. Furthering this notion that the subjects were able to compensate to some extent for the projection
distortion is the inverse (though weak) relationship between the log of the relative estimation and latitude in the
Mercator condition, -0.38. This would seem to indicate that the subjects were underestimating the areas at higher
latitudes, as would be expected when there is knowledge of the distortion patterns in the Mercator projection. As even
more support for this idea, in the third study in which subjects indicated whether or not various projections were equalarea the Mercator condition subjects universally indicated that the Mercator projection was not an equal area projection
and a majority of the subjects indicated that the areas of greatest distortion were in the high absolute latitudes.
Unfortunately the results seem to indicate that while the subjects know that there is distortion in the high absolute
latitudes in the Mercator projection they do not have the requisite knowledge to accurately estimate how much
distortion there really is – which likely is the basis for the relatively high (but not as high as with the equal-area
conditions) correlation between estimated and projected area.
With respect to how well the data is fit by a power function, the results, once again are what would be typically found
in perceptual psychophysical estimation exercises. For the globe and Sinusoidal conditions the exponents were both
n=0.67 and for the Mercator condition the exponent was n=0.69. Not only do these values fit with other
psychophysical studies they are also well described by Kerst and Howard’s (1978) finding that the memory exponent
was the square of the perceptual exponent.
DISCUSSION
To date, little of the published spatial cognition research has focused on global-scale representations. While many
spaces can be experienced either directly (e.g., navigation) or indirectly (e.g., textual or verbal description), it would be
difficult, if not impossible, to experience global-scale space directly. Individuals’ interaction with global-scale space
generally comes in the form of a globe, global-scale flat map, or a series of maps representing large components of the
area (e.g., an atlas). Each of these sources provides a different perspective on the data and shows the data with a
different type and amount of distortion. Understanding how people interpret and store geographic information
appearing in a variety of different forms and with a variety of different distortions is important to cognitive science,
geographic information science, and geography education. In cognitive science research, systematic distortions in
memories for space (e.g., Tversky 1981; Friedman and Brown 2000), and properties of space (e.g., Brown and Siegler
1993) have been studied, but the role of distortion in the source data has not been explicitly studied.
At this point is seems that the results are indicating that the subjects are using a relatively accurate input source and
modifying their estimates as would be expected according to known psychophysical estimation procedures. All in all,
it could be said that from memory subjects are either not basing their estimates on projections or that they are not using
a memory of non-equal area projections. Based on subjects’ relative accuracy when able to use an equal or non-equal
area projection where the estimates matched the projected area very well, there must be some compensation for
distortion that happens somewhere in the storage or decoding of the information.
Expansion of the perceptual study is planned to increase the number of subjects used in the analysis, as well as formal
analysis of data that has been gathered to measure subjects’ projection knowledge. More formal analysis of all of the
data will be done looking at the relationships between factors such as region familiarity, distance from the
conterminous United States, and self-rated and experimenter determined projection knowledge. In addition, future
work could be done to determine the effect of measurement method on the estimates, for instance having the subjects
perform mapping exercises or selecting the appropriate size of the region from a series of provided images. Of course,
all of this may lead to the conclusion that the distortion introduced by map projections doesn’t have a measurable effect
on global-scale spatial cognition – as the preliminary results of these analysis may be leading to – in this case perhaps
we will at least be able to show that distortion of global-scale space is not input related, but it related to some process in
storage or decoding of the spatial information and further research to explain the introduction of errors at these stages
can be conducted. Additionally, if there are relatively few indicators of what is happening between input, storage, and
decoding to cause the distortion in the subject’s cognitive map more research can be done to examine how the
projection process is actually understood and what this means for distortions that may have been introduced into
memory.
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