Transcript Uncertainty in a GIS - UBC Department of Geography

Uncertainty in a GIS
Brian Klinkenberg
Geography 516
ToC
• Spatial variability
• Data uncertainty I: measurement and sampling
errors
• Representational uncertainty I: objects vs fields
• Representational uncertainty II: cartographic
conventions
• Data uncertainty II: MAUP and S.A.
• Rule uncertainty
• Managing uncertainty: error propagation and MC
simulation
Why an issue?
• Basic search for the truth in science
• Safeguard against litigation
• Jurisdictional requirements for mandatory
data quality reports
• Protect individual and agency reputations
• Touches everyone involved in GIS, from
data producers to software and hardware
vendors, to end users
Definition
• Uncertainty: our imperfect and inexact
knowledge of the world
• Data: we are unsure of what exactly we
observe or measure in society or nature
• Rule: we are unsure of the conclusions we
can draw from even perfect data (how we
reason with the observations)
Spatial variability
• Just about everything varies over space
(spatial dependence).
• Why?
• Therefore, an estimation of uncertainty is
important. The estimate can be:
– Descriptive
– Quantitative:
• sensitivity analysis
• Confidence limits (statistical or error propagation)
Spatial variability
• The causes of natural phenomena vary in
space (continental drift > mountains >
rainy regions / rain shadows)
• Therefore, so do the natural resources that
depend on them; as do socio-economic
processes.
• E.g., Soil ≈ f(parent material, climate,
organisms, topography, time, ….)
Spatial gradients
• Four components:
– Non-spatially structured environmental
variation
– Spatially-structured environmental variation
– Spatial variation of the variable of interest that
is not shared by the environmental variables
– Unexplained, non-spatial variation
Data uncertainty I
Measurement and sampling errors
• H0: The true value of a parameter is
unknown.
• Measurement errors:
– Limited precision of our measuring devices
– The typical concern of standard statistics
– Errors are independent, normally distributed,
exactly characterizable
– Can characterize through repeated sampling
and statistical characterization (I,ND)
Data uncertainty I
– Mistakes
– Systematic and random equipment problems
– Data collection methods
– Observer diligence and bias
– Mismatch of data collected by different
methods
• In most GIS analyses those types of errors
are usually insignificant relative to:
Data uncertainty I
• Sampling uncertainty:
– We almost always use a sample to collect data
– To collect accurate samples requires extensive
knowledge of the object (and its relations to other
objects), knowledge we typically don’t have until after
we sample
– Sampling strategies: random, stratified, systematic
(effects of spatial autocorrelation?)
– Use GIS to predetermine sampling strategy (sampling
framework)
– Can quantify through repeated sampling, but $ and
time often preclude such efforts
Data uncertainty I
• In addition to the standard sampling issues, we should
also consider:
– Positional accuracy (x, y, z)
– Semantic accuracy (does everyone agree what the term means)
– If several variables are correlated (i.e., not independent), it is not
sufficient to describe their univariate distribution. Instead, we
must determine their multivariate distribution, which in general
involves the computation of a variance-covariance matrix. This is
much more difficult to establish than univariate distributions.
– The covariance structure is important because if we compute a
function of several variables, the uncertainty of the result
depends not only on the individual variances of the variables, but
also on their covariances.
Representation I
Objects and Fields
• Issue: GIS fosters the combination of
disparate data sets, each of which may
have a very different uncertainty structure
associated with it.
• How best to represent the data
(uncertainty) so that the results best reflect
the overall uncertainty?
Representation I
• Points – not an issue, although scale can
create problems
• Areas: object or field representation?
– Implies different sampling strategies and data
conversion issues (raster <> vector)
• Object: representing the map unit
– a single representative value
• Use mean / “expected value”
• Loss of knowledge of variability within map unit, or
of quality of estimate
Representation I
– Using a range of values (classes; e.g., 0-3% slope)
• Useful for sensitivity analyses
• What is the expected value?
– Using a statistical distribution
• Should be able to completely describe the data values and
their probability of being encountered
• Problem: theoretical or observational justification
– Using a non-parametric distribution
• Based on repeated sampling of object
• Makes no assumptions about distribution
• Less statistical power
Representation I
• Field: representing ‘continuous’ space
• Requires sampling scheme
• Compounded by rule uncertainty
associated with spatially interpolating the
points (lines & areas) into the ‘field’
(inverse-distance weighted, geostatistics,
trend surface analysis, etc.)
• Storage considerations (raster, TIN, ?)
Representation II
Cartographic presentation
• Cartographic conventions / abstraction
– Simplification, classification, induction,
symbolization
• Simplification: how to retain the character
of the feature (fractals)
– The effect of scale on the symbolization (most
methods actually perform coordinate
reduction, not entity simplification [coordinates
are valued over topology]—e.g., Douglas and
Poiker algorithm)
Representation II
• As abstraction increases, rules change
from quantitative (simple scale reduction
by coordinate elimination) to qualitative
(feature selection, exaggeration)
• As the scale is reduced, features evolve
from areal (wide rivers) to linear, from
areal to point (cultural features), from
being ‘to scale’ to gross exaggeration
(roads)
Representation II
• Typical problems:
• Consider the Arctic, with its multitude of small
lakes. An area may be 50% water, but as the
scale is reduced the size of the individual lakes
is such that they would not be represented on
the map. That would lead to the impression that
the area is devoid of water. What to do?
• Consider the coast of BC, with its many fjords.
As the map scale is reduced, most fjords have to
be removed; otherwise no details would be
visible.
• Some cart notes on the subject.
Data uncertainty II
MAUP and S.A.
• The modifiable areal unit problem and spatial
autocorrelation create data uncertainty.
• Gerrymandering example of MAUP:
Liberals →
Strongly
conservative
Strongly
conservative
Liberals →
Weakly liberal
Weakly liberal
Alternative
groupings
↑
↑
Conservatives
Conservatives
Data uncertainty II
• Spatial autocorrelation can create ‘false’
gradients in data:
– In a ‘true’ gradient the value z observed at any
location (i,j) can be expressed as a function of
its geographic coordinates x & y plus an error
term that is independent from location to
location
• Zij = b0 + b1xij + b2yij + εij
Data uncertainty II
– In a true gradient, the error terms at
neighbouring points are not correlated with
one another.
– A true gradient violates the stationarity
assumption of most spatial-analysis methods
because the expected value varies from place
to place. It should be removed from the data
before proceeding (e.g., trend surface
analysis).
Data uncertainty II
• In a “false” gradient, the observed trend is
caused by spatial autocorrelation.
– There is no change in the expected value
throughout the area, although the observed
value at each locality is partly determined by
neighbouring values:
• zij = b0 + ∑f(zij) + εij
• where ∑f(zij) represents the sum of the effects of
points located within some distance d from the
value zij that we are trying to describe.
Data uncertainty II
• If the gradient is considered false, then it
should not be removed, since it is part of
the process that is being studied (a result
of the dynamics of the objects themselves,
as opposed to the gradient imposed on the
objects by a true gradient).
Spatial correlation
• Spatial correlation can be:
– Spurious
– Interpolative (e.g., DEMs created using
interpolation routines, smoothed)
– True (arising from causal interactions among
nearby sample locations)
– Induced (induced by a dependent variable
through a causal relation with another
spatially autocorrelated variable)
Rule uncertainty
• Rule uncertainty should be distinguished from
errors arising through processing (e.g., map
projections and straight lines) (orthophoto
example)
• Even if all data values were known without error,
the combination of variables to a result may be
‘uncertain’ in various senses:
– The true form of a function is unknown (e.g.,
logarithmic vs. polynomial yield response)
– In expert judgment, the result is uncertain (all facts
being ‘perfectly’ provided, the expert still can’t give an
unambiguous answer).
Rule uncertainty
• Spatial interpolation is a perfect example
of rule uncertainty, since there are so
many different methods that can be used,
each with many different parameters, and
there is no unambiguous means by which
we can decide the ‘best’ method to use
(example and another )
Managing uncertainty
• Error propagation
• Monte Carlo simulation
• Fuzzy logic and continuous classification
Managing uncertainty
• Error propagation:
– If data values are described by probability
distributions, and the combination of these is by a
continuous function, then we can apply error
propagation methods that allow for the determination
of precise confidence limits in the results.
– However, very few GIS analyses are amenable to
such methods (e.g., a land evaluation based on
predicted yields of an indicator crop, this yield being
predicted by a multiple regression equation from a set
of land characteristic values, where each land
characteristic has a probability distribution).
Managing uncertainty
• GIS issues: propagation is strongly affected by
correlations between variables—this may vary
spatially.
• General rules from error propagation analyses:
– multiplicative indices should be avoided, if possible;
– the number of factors should be as small as possible;
– identify the largest sources of error and try to control
them
Managing uncertainty
• Monte Carlo simulation
• In many instances we cannot fully analyze
errors:
– The functional form is unknown
– The function’s parameters are unknown
– The function has no total differential.
Managing uncertainty
• The basic concept:
– Set up the model (function)
– Randomly vary the input variables according to their
probably distribution (accounting for covariance if
known)
– Compute the model and record the results
– Do so for n times in order to get a frequency
distribution of the results, from which we can compute
the expected value and variance.
– Example showing how a shortest path (drainage
channel) varies as we randomly vary the elevations.
Managing uncertainty
• Fuzzy logic and continuous classification
– When the concept being classified is not
precisely defined, the techniques of fuzzy
logic may be applicable. This allows us to
compute and express results when the (e.g.)
land characteristics are not precisely
measured, but instead are expressed by wellunderstood linguistic terms that can be
quantified in some fashion.
– Example: forest decision making