The critical concept of scale
“…the problem of relating phenomena across scales is
the central problem in biology and in all of science”.
Simon Levin, 1992.
Why be concerned about
scale?
• Characterization of biogeographic
pattern involves:
– Scale of the phenomena
– Scale of its detection through observation
– Scale at which a pattern is statistically analyzed
and communicated.
Cartographic scale
From Lewis Carroll’s Alice in Wonderland
Cartographic scale
Representative fractions
1:250,000
1:50,000
1:2500
Cartographic scale
• Assuming that you have two maps of the same
paper size, which of the two representative
fractions shows the most detail and the smaller
surface area?
1:50,000
1:10,000
Large versus small cartographic
scale
Large cartographic scale
Small cartographic scale
Absolute and relative scaling
• Absolute scale
– Distance is physical, measureable
• Relative scale
– Distance is not a physical distance but a more
intangible construct, often similarity in some
property
Ecological topology
• Geometric
representations
of ecological
entities and
phenomena in
varying degrees
of abstracted
space
Eulerian
topologies
• Connectivity
between objects
– Example:
Networks
Poincareian topologies
• Field-based,
geometric
representation
of the dynamical
system states
– Example: state
space
Operational scale
• Refers to the spatial and temporal
dimensions of an object or a process
Operational scale
• Operational scale may
not be independent of
our observations, and in
fact may be very
dependent upon them.
• Operational scale can be
constrained by our
technology and our
senses.
And so these men of Indostan
Disputed loud and long,
Each in his own opinion
Exceeding stiff and strong,
Though each was partly in the right,
And all were in the wrong!
John Godfrey Saxe
1878
Decreasing (finer) grain, extent is constant
Ecological
scaling
• Extent: spatial (or
temporal) dimension of
an object or process
observed or analyzed
• Grain: level of spatial (or
temporal) resolution at
which an object or
process has been
measured or observed.
Increasing extent, grain is held constant
Which map shows more grain?
How much grain is shown can be a subjective
decision. The map maker has to decide upon
grain given the extent.
Map generalization
• Cartographers decide on the amount of
generalization in a map.
• Unavoidable issue related to cartographic
and ecological scale
• All maps have been generalized because all
the detail on the surface cannot be conveyed
on a paper map.
Haggett’s scale coverage
problem
• Nature has an
immense extent and a
fine grain
Haggett’s scale coverage
problem
• To make even small
descriptions of it, we
have to sample
• Sampling requires:
– Sacrificing grain for
extent
– Sacrificing extent for
grain.
Fallacies of scale
• Individualistic fallacy:
extrapolating to the broad
scale based on observations
conducted at small, local
scales
• Ecological fallacy: making
local-scale characterizations
based on broad-scale
observations.
• To an extent, humans are
going to commit these
fallacies in order to
understand the world
Example: local
microclimates
• Local buffering of climate
change due to
microhabitats
• Cannot infer detail of
local temperatures based
on macroscale (large
extent and coarse grain),
nor can we extrapolate
up from the microscale
(small extent and fine
grains) to the macroscale
Modifiable areal unit problem
• Two components
– Aggregation problem
– Zoning problem
Hierarchy theory
• Landscapes organized into
spatial and temporal domains of
shaping processes
• System of vertical
interconnections
• Higher levels set the context for
the lower levels
Hierarchy theory
Example: how hierarchical theory
informs the assembly of communities
Example: hierarchy theory and
extinction filters
Non-hierarchical controls also
relevant
Examples:
Tilt, insolation, climate,
photosynthesis:
hierarchical
Invasive species,
dispersal, natural
selection: nonhierarchical
Cartesian scale
• Hierarchy theory
exemplifies a
Cartesian scaling of
the world
• Scales are imposed
• Not necessarily
“true” scales
• Often tied to XYZ
coordinate system
• Space as a
container
Constructivist scale
• Scale is produced by
organisms
• There is no fixed and
unchanging hierarchy
of scales imposed
from above.
Constructivist scale
• Boundaries in space and
time shaped by
organisms not just our
observation of them.
• Extents and grain are
also a function of the
organisms rather than
our (human) sensing
alone
The epistemological mode of
scale
• Epistemology: strategy of
belief that informs us of what
we can and cannot know
• Epistemological mode
generates information useful
for ascertaining predictability
• Organisms scale the world to
promote predictability or the
appearance thereof
• Our senses, technology, and
beliefs constrain scales of our
perception and consequently
what we can claim to know.
The ontological mode of scale
• Ontology: what exists,
the categories by which
we label and define the
world
• Other organisms also
scale and act upon their
inferences of
predictability
• Borders and boundaries
are not universally shared
• Introduces
unpredictability – scale
mismatches
The biological mode of scale
• The interaction of the
epistemological and
ontological moments of
scale in organisms
• Epistemological scaling
needed for predictability
• Ontological scaling
produces unpredictability
• Microorganisms to
complex organisms
scale the world to
generate predictability –
they infer and act upon
the extent (boundaries)
and grain (or quality) of
phenomena perceived in
time and space
• Unpredictability arises
from environmental
change and other
organisms that also
infer, act upon, and
modify their scaling of
the environment
The biological mode of scale
• Basis for adaptation and evolution
• How have the extents and grains detected by
our senses and technology been intertwined
with our evolution?
Rules for ecological scale for
humans
1. Patterns are dependent upon the scale of
observation
2. The important explanatory variables change with
scale.
3. Statistical relationships may change as scale
changes.
4. Patterns are generated by processes acting over
various temporal and spatial scales.
Patterns are dependent upon the scale of observation
Patterns are dependent upon the scale of observation
The important explanatory variables change with scale.
The important explanatory variables change with scale.
Patterns are dependent upon the scale of observation
The important explanatory variables change with scale.
Patterns are dependent upon the scale of observation
Statistical relationships may change as scale changes.
Rules for ecological scale
5. Scale can be
used to justify
or refute certain
management
practices and
ideas about
nature
Example: Successional response to
clearcut logging
• Grain and extent of post-logging
sampling determine criteria for judging
response
– Resurvey over large extent and fine grain:
criteria for recovery not likely to be met
– Resurvey over small extent and coarse
grain: criteria for recovery easier to meet
Rules for ecological scale
6. The scales experienced by an organism define what
it sees and responds to. For example, what might
constitute a patchy resource to an insect, could be
perceived by a larger vertebrate as homogeneous.
How to work with scale
• There is no single
correct scale or
level at which to
describe a system.
• This does not mean
that all scales serve
equally well or that
there are not rules
or guidelines.
How to work with scale
• Be aware of the different types of scaling
• Don’t be too anthropocentric
• Employ sampling designs and methods that
are sensitive to multiple scales
–
–
–
–
Nested observations
Power laws
Fractals
Networks
Nested observations
Sp - species
Power laws
• Summarize how relationships
change with changes in extent
• Often expressed on a log-log
plot.
• Y = constant (X)n
• Similar slopes are thought to
have similar structuring
processes (n = slope)
• Examples
•
•
Species-area relationships
(left): S=cAz
Animal metabolic rates and
body mass (next slide)
• However, a valid criticism of
power laws is that they often
lack an explanatory process
Metabolic rate and body mass scales according to this power law:
Metabolic rate = constant (mass).75
(Metabolic rate)
(Mass)
Fractals
• A fractal pattern appears the
same across all scales. It is
scale invariant.
• Fractal patterns can be
expressed as a power law with
a particular range of slope
values.
• Power laws do not necessarily
imply fractal structure.
Networks
• Can represent relationships at a
variety of scales at once.
• Eulerian topology
• Structural properties of networks
provide means of understanding
how they work
– Nodes and links
– Degree centrality and
betweenness
– Weak versus strong links
– Directional versus nondirectional graphs
Random versus scale free networks can reflect different modes of
assembly and activity. These may also have implications for their
geographic distribution
Spatial autocorrelation
• A method to summarize how patterns
change with scale.
• Based on Tobler’s Law: near things are
more likely to be similar than things at a
greater distance apart