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Progress in Human Geography 20,4
(1996)
pp.
540-551
GE,
spatial analysis and spatial
statistics
David
J.
Unwin
Department of Geography, Birkbeck College, University of London, 7-l
5
Gresse Street, London
Wl
P
1
PA, UK
I
Introduction
Spatial analysis will be applied to an ever increasing number of application areas.
GIS
data manipulation tools
will become ever more sophisticated and easier to use. They are already today being included in office software
packages such as spreadsheets. We are rapidly approaching the time when every desktop PC will be able to
perform spatial analysis.
The above quotation comes from a draft of
G12000:
towards a European
geographic
infor-
mation infrastructure,
published during the past year by DG XIII of the European Com-
mission (EGII, 1995), which attempts to lay the foundations of a European infrastruc-
ture for spatial information. Its content may well be a surprise to readers whose
knowledge of spatial analysis ended with a practical class on the nearest neighbour
statistic taken, probably unwillingly, as part of their degree studies and who thought
that it had disappeared into history. Largely as a result of the growth of GIS, spatial
analysis is back on the research agenda and in this year’s review I will attempt to give
a flavour of current work in the field.
At the outset, it is best to be clear what we mean by the term. In the GIS literature,
and especially in system manuals and brochures, the view seems to be that spatial
analysis is simply the general ability to manipulate spatial data using a familiar set of
largely deterministic functions which includes basic spatial queries, buffering, overlay
using simple map algebra and the calculation of derivatives on surfaces such as slope
and aspect. This type of work can be called
spatial data manipulation
and, since it is
precisely this ability to handle spatial data
spatially
that differentiates a GIS from any
other database management system, it is essential to any information system claiming
to be geographical. It is also what differentiates a true GIS from computer-aided design
or mapping packages. In
spatial statistical analysis
knowledge of a process is used to
predict the spatial patterns that might result, and the likelihood of any observed pattern
0
Arnold 1996
David
J.
Unwin 541
being a result of this process is then established by an analysis of one or more of its
realizations. In contrast, exploratory
spatial data analysis
examines an observed distri-
bution and attempts to infer the process that produced it. The objective is usually to find
patterns in data that are meaningful in relation to the investigators’ existing domain
knowledge. Both are different from
spatial
modelling
in which the objective is to produce
realistic mathematical models of the type used in, for example, retailing (Birkin et al.,
1996) and the environmental sciences (Goodchild, Parks et
al.,
1993; Goodchild, Steyaert
et al., 1996) that deterministically predict spatial pattern. In this year’s review, I will
be concerned mostly with exploratory spatial data analysis (ESDA) where there has
been a major renaissance brought about by widespread access to very powerful com-
puter workstations and, equally critically, to often very large, structured data of the
type which are common in any GIS. This renaissance of spatial analysis has been driven
by several academic changes.
First, as we saw in last year’s review (Unwin, 1995), users of geographical infor-
mation systems have begun to ask questions about the reliability of the results obtained
from simple spatial manipulations of geographic data, such as interpolation and map
overlay, and have begun to realize the importance of a statistical approach. As a result,
there has been a series of discussions of the role of spatial analysis in GIS and the
relationships between the two (see Goodchild, 1987; Anselin, 1989; Fotheringham, 1992;
Goodchild et
al.,
1992; Anselin et al., 1993; Fotheringham and Rogerson, 1993; 1994;
Bailey and Gatrell, 1995; Fischer et al.,1996). The greater part of this debate has
addressed technical questions about how to couple GIS with the required statistical
‘functionality’. With the notable exception of Anselin et al. (1993), rather less attention
has been directed to what should be coupled and why.
Secondly, work by statisticians has developed a substantial body of statistical theory
about spatial data to which GIS users can turn but which did not exist in geography’s
so-called ‘quantitative revolution’ (see, for example, Diggle, 1983; Ripley, 1981; 1988;
Upton and Fingleton, 1985; 1989; Haining, 1990; Cressie, 1991; Walden and Guttorp,
1992). The statistical view is characterized by the notion that spatially distributed infor-
mation can be regarded as the outcome of some stochastic process operating in the
plane. If we can postulate the nature of the process in mathematical terms, we can
deduce its spatial outcomes and examine whether or not an observed pattern is a plaus-
ible realization of it. As Harvey forcefully pointed out many years ago (Harvey, 1966),
a much more difficult alternative is to identify the process and model it appropriately
from the evidence of a single mapped realization, yet this is frequently what is required.
Very few of these new methods of analysis are as yet implemented in existing GIS but
specialized software is now readily available (Anselin, 1990; Rowlingson and Diggle,
1991; Diggle and Rowlingson, 1993). Recently INFOMAP (Bailey, 1990) has become
available at very low cost and includes analysis methods such as density estimation,
kriging and K-function computation, all of which were developed during the 1980s
specifically to handle spatial data (Bailey and Gatrell, 1995).
Thirdly, now that spatial data are easily obtained at extremely high spatial resolution,
and computing and mapping the results are unproblematic, there has been developed
a series of strategies for exploratory spatial data analysis often using visualization
(Unwin, D.J., 1994). This is a data and cartography driven approach to spatial analysis,
concerned with the recognition and description of spatial patterns and their represen-
tation on maps. The methods employed vary from allegedly simple statistics dating
from the 1950s (notably the spatial autocorrelation tests of Cliff and Ord, 1973), through
542 GIS,
spatial analysis and spatial statistics
the direct use of visualization (Hasslett, Wills
et al.,
1990; Haslett, Bradley
et al.,
1991;
Hearnshaw and Unwin, 1994) to automatic machines (Openshaw, 1994) and artificial
life forms for pattern detection (Openshaw, 1995).
In the remainder of this review I will examine some of the recent approaches to
exploratory spatial data analysis. First, I will look at the meaning of the central idea
of
pattern
in a data-rich GIS environment. This leads naturally to a discussion of a class
of
local
statistics
that are rapidly gaining acceptance, and in turn leads to a consideration
that is central to geography, the definition of what is, and what is not, in the locality
of some place. Much of the review is concerned with the influence of developments in
statistical analysis on GIS, but the review concludes by an examination of the influence
of GIS on the development of spatial statistics,in particular the notion of
GLSable
methods
of analysis.
II Spatial pattern, projection and process
Pattern is that characteristic of the spatial arrangement of objects given by their spacing
in relation to each other. It should not be confused with the idea of dispersion, which
is relative to some defining area, or with density, which is the average number of
objects in a given area. Patterns might consist of clusters of points, a more-regular-
than-random arrangement, trends across real and statistical surfaces and so on. Given
Tobler’s ‘first law of geography’, that near places are more likely to be related than
distant ones, it is hardly surprising that most geographical patterns of interest involve
groupings of similar values in clusters.
In almost all the work by statisticians, and in most of ‘quantitative geography’, the
approach taken has been to map objects of interest using their location on the planet’s
surface as measured by plane Cartesian co-ordinates based on a map projection. This
is, of course, a reasonable assumption for small areas of interest and it allows the use
of simple geometry to find distances and areas. If large areas are studied, then referenc-
ing can be a latitude/longitude pair and in a GIS environment calculations of the real
distances and areas are only slightly more difficult. There is a danger that standard
functions in proprietary GIS used, for example, in interpolation or the estimation of
distance functions do not recognize the need to compute the great circle distances.
What is often not realized in the modelling and search for spatial pattern is that the
concept must also contain some
projectional
component. Patterns come and go according
to how we project the data. A simple example is the view we get of the hemispherical
night sky on which we see a projection of objects distributed in at least three dimen-
sions. Provide an extra dimension of time/space and the well-known nonrandom pat-
terns studied by astrologers, such as The Plough, disappear. Opposite operations are
also well known, perhaps the simplest being the often-cited correlation between the
number of storks and births in India. Project these same data with a third axis which
locates them in time and the reason for the pattern becomes obvious.
The search for the ‘correct’ number of dimensions in data is (was?) the subject of
methods of ‘factor’ analysis and its more modern, computer-based alternative, called
‘projection pursuit’. Within appropriate software it is a relatively easy matter for non-
standard projections to be calculated and these can provide much more useful frame-
works in which to display data and look for pattern. For example, Dorling (1992; 1994)
has used the projection given by an area cartogram based on small area population
David
J.
Unwin 543
totals to show detailed variations in the social geography of Britain. Similarly, patterns
revealed in plots of data on to empirical projections defined using multidimensional
scaling have been exploited in a number of studies by Gatrell (1979; 1983; 1991).
The second component of a pattern is that given by the process which generated it.
Such a process could be totally deterministic, with a single unique outcome at each
location. The temptation to think that this inevitably results in patterns which always
appear simple should be resisted. There is now a substantial literature using cellular
automata models to show how simple rules of spatial behaviour, combined with a
discrete dynamical systems approach, can generate extremely complex patterns (see
Von Neumann, 1966; Toffoli and Margolus, 1987). The approach was introduced into
geography by Couclelis (1985) and has been developed in a GIS content by Camara
and Castro (1996) and Sanders (1996). By their very nature, these models are readily
implemented in the framework of a raster GIS, and there is a strong link between them
and the world of fractals, nonlinear dynamics and ‘chaos’ theory.
It is, however, more usual to think of spatial processes as being stochastic, in which
the outcome at each location or area represents a probabilistic (random) selection from
some underlying generating distribution. The patterning in the spatial phenomena that
results usually arises as a result of two types of variation, called first and second-order
effects. First-order effects relate to variations in the mean value, or intensity, of the
process over space. The process is spatially stationary if this intensity is constant over
the study area.Second-order effects involve relationships between objects in the
study area.
III
Detecting spatial pattern
Clearly, this statistical notion of pattern depends upon our having some standard
against which to judge the spatial arrangement of data and the usually adopted one
is that of complete spatial randomness (CSR
-
Diggle, 1983). In this approach pattern
is equated with spatial homogeneity and inhomogeneity resulting from departures
from CSR. In the first case, the underlying distribution function of the postulated pro-
cess remains unchanged over space. In the second it changes in some way from place
to place. Although most work has been done using point processes, the same notion
of CSR can be applied to data describing line, area and surface objects
Unwin,
1981).
A number of methods have been developed to enable patterns in spatial data to be
described.
1 Global statistics
Global statistics attempt to characterize the patterning across an entire region. In ecol-
ogy, geography and geology, a popular example is the Clark and Evans nearest neigh-
bour index, calculated as the ratio of the observed mean distance from each point object
to its nearest neighbour to the value expected from a CSR process with the same inten-
sity. In effect this collapses the pattern on to a single dimension given by the X-value.
A similar projection on to two axes is given by more recently developed methods, such
as those which use the G,
F
and
K
functions to test for CSR in a point pattern (see
Bailey and Gatrell, 1995). The G function assembles the complete cumulative probability
544 GIS,
spatial analysis and spatial statistics
distribution of point event to point event nearest neighbour distances as a function of
distance,
d:
G(d) =
#
(w,
<
d)
/
n
In which # means ‘the number of’ and w are the nearest neighbour distances. Similarly,
the F function does the same thing for distances between a randomly selected point in
the region and the point events:
F(x)
= #
(xi
<
x)
/
m
in which
m
equals the number of point samples. These functions can be estimated from
an observed point pattern and the general strategy is to compare the shape of the actual
curve with that given by a CSR process with the same average intensity. Both are useful
for examining small scales of pattern and both are a definite advance on the uncritical
use of the simple nearest neighbour value. To examine pattern over a wide range of
scales use is made of the
K
function, defined as:
X
. K
(d)
= E
(#
events within distance d of an arbitrary event)
in which
h
is the average intensity. For a recent, accessible overview of these statistics
and their usefulness in the analysis of point patterns in a GIS environment, see Gatrell
et al.
(1996). For area enumerated data a number of approaches have been adopted,
ranging from simple spatial moving averages and smoothing by median polish to the
computation of correlograms to show the spatial autocorrelation at differing adjacency
and distance lags (Cliff and Ord, 1973). For visualizing such data the problem of low
observed numbers, particularly in spatial epidemiology, has been addressed by the
careful use of Poisson probability mapping (Langford,
19941,
and the equally difficult
problem of correction for variations in the ‘weight of evidence’ that arises naturally
from the use of differently sized spatial units has been solved by the application of
Bayes’ theorem to derive appropriately weighted estimates (Marshall, 1991). Finally,
for spatially continuous data, methods from geostatistics, notably use of the correlog-
ram and
(semi)variogram
have become virtually standard (see Isaaks and Srivastava,
1989; Cressie, 1991). A particularly useful display is the semi-variogram cloud devised
by Haslett and others (Haslett
et
al.,
1991).
Global statistics have severe limitations for work in spatial data analysis. First, the
assumption is usually that the pattern
-
and hence the process
-
is stable, or
sfaficmauy,
over space. As GIS systems have enabled researchers to use either larger study regions
or, equivalently, data sets at much finer spatial resolution, so this assumption seems
more and more unrealistic. Basic geographical theory shows that such spatial homogen-
eity over large areas of the earth’s surface or at fine resolution is extremely unlikely
(Fotheringham et al., 1996). What may well happen is that large areas of uninteresting
spatial variation swamp those of real interest. Secondly, almost all these global meas-
ures are subject to potentially severe edge effects in their calculation and, thirdly, the
expectations are frequently subject to the so-called modifiable area unit (MAU) problem
(see, for example, Fotheringham and Wong, 1991; Fotheringham
et al.,
1995). Taking
the Clark and Evans X-index as example, its value depends to a large extent on the
area chosen over which to study and hence the assumed mean
intensify
of the process.
David I. Unwin 545
Many years ago, in an attempt to show that within drumlin fields the distribution of
these landforms is spatially random, I misguidedly computed the Clark and Evans
nearest neighbour statistic (my collaborator is blameless, see Smalley and Unwin, 1968).
It should be abundantly clear that, simply by redefining the study area, I could have
produced almost any R-index. As it was by total
(mis)chance
I seemed to hit on the
scale of analysis that produced values close to the magical random expectation.
Similar, much more subtle dependencies occur in virtually all the work that we do.
In the example given, the use of a measure of pattern based on an inter-event distance is
reasonable (although there will be edge effect problems) and the technical fault would
nowadays easily be solved by randomization, but the same MAU problem is present
in most of the tests for spatial autocorrelation where it appears first in the zonal aggre-
gate values used and, secondly, in their conversion into standard scores based on some
arbitrary global mean, but this problem is seldom mentioned by practitioners. Finally,
and as a consequence of the considerations outlined above, where formal statistical
tests are employed, the assumptions employed are almost invariably broken.
2 Visualization
In response to these problems, much exploratory spatial data analysis has turned to
visualization as a means of pattern detection, the notion being that the eye/ brain sys-
tem, when given sufficient help, is capable of a high degree of sophisticated pattern
recognition. This is the philosophy of SPIDER/REGARD (Haslett et
nl.,
1990), cdv
(Dykes, 1995) and a system based on
XLispStat
(Brunsdon and Charlton, 1995). The
philosophy is that we use data display as a means of analysis in its own right and the
problem becomes one of designing appropriate and useful types of display.
As outlined in my 1994 report (Unwin, D.J., 1994), pure visualization has its adher-
ents and critics. First, it is well known in the literature on cartographic communication
that apparently quite minor changes to a map can greatly change how it is viewed, a
good example being the choice of class intervals in choropleth mapping. More subtle,
but none the less important examples occur in any contour mapping and in almost all
the use of colour coding. Very few visualization practitioners, at least in the Anglo-
Saxon world, would agree with Bertin’s notion of the monosemic (single sign) map,
preferring instead to think of maps as polysemic (capable of many interpretations)
products of frequently fallible cartographers. Secondly, it is also well known that the
eye /brain frequently synthesizes a pattern where, strictly, the data are random. Similar
effects have been seen where test groups produce different maps of the same numbers
according to the information they are given about the phenomenon being mapped.
3 Local statistics
A third strategy is to harness the power of simple statistical summary of the type
employed when global statistics are used to define pattern with the less formal, but
equally less demanding, process of visualization by mapping what have been termed
IOCQ[
statistics. Typically, in using local statistics we attempt to learn more about each
individual datum relating to a point, line or area object in the data set by comparing
it in some way to the values for its neighbouring objects. Several local statistics have
been suggested and their use illustrated.
Getis and Ord (1992) define a G-function (which is not the same as the function of
546 GIS,
spatial analysis and spatial statistics
the same
name
used in point pattern analysis, see above) which gives an index of
spatial clustering of a set of observations over a defined neighbourhood:
Gi
(d)
=
C
Zl'b
(d).X,
/
C
Xi
In this, x is the regional variable and W(d) is a symmetric 0
/
1 matrix of weights with
Is
for all the areas defined to be within distance, d, of the given area,
i.
All other
elements are zero, including the link of
i
to itself which means that the value for the
area is not considered. Readers who have struggled to compute by hand one or other
of the global statistics mentioned above will appreciate that the computation of G is
hardly possible without the computer and access to a
GlS
type of data structure from
which to determine the
W(d).
In effect, the vector of values for each region,
G(d),
shows how locally anomalous the
region is with increasing distance for the given variable, x. Each area object in the data
has its associated G-function which can be mapped for given
d,
or plotted as a function
of distance. A restriction on this statistic is that as defined it is only useful if the variable
x has a natural origin. It is thus inappropriate for the study of change variables or
variables that have negative values.
Ord and Getis (1995) and, more recently, Bao and Henry (1996) develop a distri-
bution theory for this statistic under the hypothesis of a random allocation of values
which enables a standardization to give Z(G). In practical applications, where the num-
ber of zones in a neighbourhood is very low, the exclusion of the point- itself can give
awkward problems, and a variant, G”,is sometimes calculated in the same way but
including the zone’s own value. As with the standard form, so it is possible to calculate
a standard score as Z(G”). These statistics are used to detect possible nonstationarity
in data, where clusterings of similar values are found in specific subregions of the
area studied.
As an alternative to the G statistics, Anselin (1993) has shown that the spatial autocor-
relation coefficients Moran’s I and Geary’s contiguity ratio C can be decomposed into
local values. The local form of Moran’s I is a product of the zone value and the average
in the surrounding zones:
1,
(d)
=
Zi.
C
W;j
(dl.2,
In making these calculations, the observations,
z,
are in standardized form, the
W(d)
matrix is row-standardized, and the summation is for all
j
not equal to
i.
Bao and Henry
(1996) show that under the conditional assumption of nonrandom observation at
i,
this
local Moran is a linear transformation of the G statistic. Finally, and in fact rather differ-
ent from G and
1,
Anselin (1995) outlines a local variant of Geary’s contiguity ratio as:
C;(d) =
C
wj,
(d).(z,
-z/Y
In its use of the differences between the location value and its neighbours, this local
form of C is in many ways similar to the gradient operators used in image processing.
The idea of a local statistic is not new and similar calculations have been routine in
image processing for decades. In the restricted world of the raster GIS, Tomlin’s analyti-
cal operations of the focal type are a second example (Tomlin, 1990). Recent work by
Moore (1996) has shown how operations from image processing can be generalized
David
J.
Unwin 547
within a GIS to apply to irregular grids of polygonal areas or
wsels
(resolution
elements). Similarly, once we have the computer power, almost any of the classical
statistics can be calculated as a local value (for example, the mean, standard deviation
and correlation). This idea has recently been exploited by Fotheringham
rf
al. (1996)
who compute maps of how the estimated parameters of a regression model vary spati-
ally over a fine raster of grid cells. Maps of these estimates provide additional infor-
mation about the spatial stationarity of the model of the process which is complimen-
tary to more conventional maps of residuals over a global regression.
IV Defining localities
No matter what the statistic, a key question that must be addressed in these operations
is the definition of what is meant by local. Depending on the definition adopted every
location in the database will generate different statistics, and, as can be seen from the
formal definitions of the local forms of
1,
G and C, use is made of a weights matrix
W(d) representing all possible 0.5
y1*(y1-
1) interobject pairs. If each element,
IO,,,
is given
the value 0 or 1 this forms an adjacency matrix which contains all the information
needed to define the concept of local by adjacency. It is easily shown that successive
powers of this matrix (with zeros down the diagonal, see Garner and Street, 1978)
give the objects that are adjacent but two steps away, and so on, permitting an easy
generalization of the notion of local. An early article describing the use of these
extended neighbourhoods is Lebart (1969). For an example of the information content
of powers of this adjacency matrix, see Unwin (1981: 87-93). In addition to the simple
fact of adjacency (O/l
1,
the same matrix can be used to record the strength of the
adjacency by modification of the
w,,
to record, for example, the length of common
boundary.
Alternatively, metric distances can be used to define local in any one of several ways
to provide a
kernel
around the location of interest (see Bailey and Gatrell, 1995: 261-
62; Moore, 1996). The simplest option is to set the relevant
wj,
equal to 0 or 1 according
to whether or not the zone centroids are within some distance,
d,
of each other. Alterna-
tively, use can be made of inverse distance weighting according to some function as in
statistical density estimation (Silverman, 1986) or spatial interpolation using algorithms
based on the old SYMAP scheme in what Fotheringham et al. (1996) call spatial
regression. Although one could use all objects in the database in such a weighting, in
practice attention is usually confined to a restricted kernel width but it may be that
this restriction is unnecessary in these days of very high-performance machines. A third
alternative is to use an adaptive kernel which is responsive to the local data density
or to optimize the distance used in some way. An obvious approach is to use the range,
as deduced from estimation of the semi-variogram, as the kernel width, but there is
scope here for experimentation with other criteria based, for example, on kernel widths
which maximize or minimize the local variation (i.e., the local geography). In the long
run, it may well be that the most valuable information is contained in the behaviour
of these local statistics as the kernel width is expanded. Preliminary work by Wood
(1996) using local regression results to provide multiscale characterizations of landform
from digital elevation matrices suggests that this scale dependence contains useful geo-
graphical information.
548 GIS,
spatial analysis and spatial statistics
V
Conclusion: putting spatial statistics into GIS
It should be clear that, influenced by GIS, the availability of very large, high spatial
resolution data, and access to extremely powerful computer power, spatial data analy-
sis has already changed greatly and will continue to change as methods which recog-
nize the existence of today’s data and computer-rich environment are developed.
A concept that has been developed by Openshaw and his colleagues (see Openshaw
and Clarke, 1996) is that of
GlSable
statistics (see Table 1). By this they mean analytical
and other approaches that are suited to a world in which computer power, very large
data sets and the availability of GIS should be taken for granted. The concept is a useful
one, since, as they point out, it helps define a research agenda for developing new
methods and draws attention to the fundamentally unsatisfactory, even unsound, nat-
ure of the traditional methods of statistical analysis when applied to spatial data. For
example, almost all the spatial statistical methods I included in Introductory
spatiul
anuly-
sis (Unwin, 1981) should have very little place in our current research environment.
Almost without exception they can be replaced by more recent techniques or, as is
Openshaw’s predilection, by a variety of compute intensive procedures.
Already, there is a long list of possible GISable statistical functions that might be
added to improve the functionality for spatial statistical analysis of existing GIS. We
all have our special favourites but, on the basis of the methods I generally find myself
persuading MSc and PhD students to adopt, my own list would include the following:
All the methods for the analysis of point events developed by statisticians in the 40
years since that dreadful ‘nearest neighbour’ statistic was first proposed.
The spatial form of density estimation in the manner of Silverman (1986) and as
demonstrated by Gatrell (1994).
Generalized linear modelling tools, in the manner of GLIM (Aitkin et al., 19891, to
be used for calibrating various forms of favourability functions in map overlay
(Bonham-Carter, 1991).
A series of local statistical indicators of spatial association and inhomogeneity of the
type outlined above.
The ability easily to change and visualize W(d).
Table
1
Ten rules for developing
‘GISable’
statistical analysis
Rule
1
Rule 2
Rule 3
Rule 4
Rule 5
Rule 6
Rule 7
Rule 8
Rule 9
Rule
10
GIS
methods should be useful in an applied sense
A GlSable spatial analysis method should be able to handle large and very large N values
Useful GlSable analysis and modelling tools are study region independent
GIS
relevant methods need to be sensitive to the special nature of spatial information
The results should be mappable
GlSable spatial analysis is generic
GlSable spatial analysis methods should be useful and valuable
Interfacing issues are initially irrelevant and subsequently a problem for others to solve
Ease of use and understandability are very important
ClSable
analysis should be safe technology
Source: Adapted from Openshaw and Clarke (1996).
David
J.
Unwin 549
l
Good exploratory visualization tools of the type proposed by Densham (1994) and
offered by REGARD (Unwin, A.R., 1994) and cdv (Dykes, 1996).
There may well be others, but the general nature of my list should be clear. It attempts
to update what most geographers think of as spatial statistical analysis and thus correct
the evident inability of many to take advantage of almost all the developments in stat-
istics, spatial statistical analysis and computing since the 1960s.
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