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- [Instructor] In the
second part of the lecture,

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we'll look at the
introduction to geostatistics

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and interpolation concepts.

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I won't lie, we're gonna get pretty close

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to some mathematics here

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but I think we make it
out of the presentation

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without having to look at any formulas

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I'll be offering a
relatively light treatment

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of both of these topics.

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So there's definitely
plenty more to dig in on.

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If you're interested in the mathematics

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or just wanna know more
in general, let me know,

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and I can find some additional
resources for you to review.

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Geostatistics is a statistical approach

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for analyzing spatial and
spacial temporal phenomenon.

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It considers both the location

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and the value of the
observation at that location

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as part of the analysis.

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We can use geostatistics to
describe patterns in our data

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to interpolate values
or fill in the blanks

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between known point locations

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and even to estimate uncertainty.

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Geostatistics has evolved

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to be embraced by a broad
range of disciplines

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for any number of applications.

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When we're talking about
spatial interpolation,

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we consider the prediction of a variable

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at unmeasured location based
on the value of variables

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at measured location.

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So out at sample points
where you've collected data.

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Spatial prediction takes
that one step further

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and includes other variables
in the estimation process.

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Now, part of having data that you can use

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for geostatistical analysis
includes sampling data

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or collecting data in the field,

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or potentially even just sampling data

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from a digital data set that you have.

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A couple of different kinds of sampling

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that we see on this page,
systematic and random.

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Systematic samples are uniformly spaced

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over X and Y intervals.

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They're easy to plan and implement

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and it really removes any
of the subjective judgment

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about where to collect sample data from.

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On the con side,

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it's not necessarily
statistically efficient

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to collect data this way,

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and it certainly doesn't
consider things like travel costs

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or inability to capture
patterns at finer scales

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when using this type of program.

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Random samples on the other
hand are just a random selection

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of X and Y coordinates
usually selected separately.

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On the positive side,

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these are unlikely to match
any pattern in the landscape

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meaning that it's good for
capturing the diversity

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in the landscape itself,

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but on the con side

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it doesn't increase the number of samples

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in areas with high variation,

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so it's likely to miss
some of those patterns

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and it's more challenging
to plan and implement.

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We've also got an option
for cluster sampling

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which kind of combines the
random and systematic selection

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of the cluster centers and the clusters

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that actually surround those
individual point locations.

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The US Forest Service

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uses a random cluster
center point selection

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followed by systematic
designation of the cluster

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of the surrounding points

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for some of their forest sampling
and inventory activities.

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On the plus side, there's
a real strong possibility

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to reduce your travel costs
when sampling in this fashion.

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And then lastly, there's an
adaptive sampling strategy

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where you collect more frequent samples

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in areas with variable information
and less frequent samples

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in uniform areas of the landscape.

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This does require some
assessment prior to field work

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and updates during field work

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which can present some
logistical challenges.

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A few key bits of terminology
I'd like to cover,

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interpolation as you may or may not know

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uses measured values to predict the values

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at unsampled locations.

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We'll look at two different
kinds of interpolation,

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deterministic and geostatistical.

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So deterministic interpolation
uses either a similarity

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or smoothing type function

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and it does not measure
uncertainty or error

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in the output prediction.

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Geostatistical interpretation
on the other hand,

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is a statistical model

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that include spatial autocorrelation.

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These can measure uncertainty
in the output data,

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and are quite powerful

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for any number of different applications.

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We heard a little bit
about the kernel already,

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but just by way of reminder,

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the kernel is a weighted function

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that assigns the weights to
locations based on the proximity

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of that location to an
observation or a sample point.

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Now, spatial autocorrelation
is this assumption that

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the closer two observations
are to one another,

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the more similar their
values are likely to be.

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So depending on the type
of analytical approach

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that we take to interpolation,

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spatial autocorrelation may be required,

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it might be assumed, or it
might be ignored altogether.

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It's important to note that
spatial autocorrelation

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violates this independence assumption

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that you would have otherwise

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in traditional statistical methods.

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For geostatistics, it's
an important assumption

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that makes these calculations possible.

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We can look at a couple of different ways

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for analyzing data patterns,

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Moran's I and the Getis-Ord General G.

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Both of these are statistical measures

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that look at clusters of data.

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In the case of Moran's I,

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it's a measurement of
spatial autocorrelation

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based on feature
locations and their values

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and the algorithm determines
if the data are clustered,

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dispersed or random.

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Like all statistical tests,
this one has a null hypothesis

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that states that the
attribute under consideration

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is randomly distributed.

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You can use Moran's I to
help define both the size

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and the orientation of
search neighborhoods

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that would be used for
interpolation processes.

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Now, the Getis-Ord General G statistic

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measures clusters of high or low values.

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And in this case, it
assumes a null hypothesis

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that there is no spatial
clustering of the feature values.

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For both, depending on the
P value and the Z value

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that you get from your
estimation of these statistics,

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you can determine whether or not

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you can dismiss the null hypothesis

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and state something
definitive about the pattern

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or lack thereof in the
data that you're using.

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Now there's multiple
mathematical approaches

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to spatial interpolation,

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but all of these use
points with known values

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to estimate values at
these unsamped locations.

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Location, both the distance
between the sample location

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and the cell location
and the attribute values

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are considered as part of the process.

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Now, once again, depending
on the approach that you take

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varying weights would be assigned

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to neighboring observations
and you do have some control

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over the way those weights are applied.

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Not surprisingly

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because of the multiple
different approaches

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to performing spatial interpolation,

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the different approaches will
produce different results.

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Deterministic and
geostatistical approaches

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for interrelating
continuous raster surfaces

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happen with control points.

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Now, even with a finite number
of discrete objects in space,

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there are still likely too many to count,

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too many places to go survey

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to consider a completeness to that.

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And because of that, coupled
with potentially costs,

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a lack of access for legal
or practical reasons,

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site locations being dangerous,

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you might be missing samples,

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you might have erroneous samples,

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or you might be re sampling your data.

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These are all reasons why
you might need to perform

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spatial interpolation operations.

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We have both global and
local interpolators.

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On the global front,

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you're looking at trend surface analysis

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and regression models.

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We won't be doing much
of that in this module.

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Instead, we'll focus on local operations

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that use a sample of the known points

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where the user has control over how many

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and how to search for
those point locations.

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And we'll look at both geo statistical

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and deterministic approaches.

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Now control points as I mentioned before,

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are these locations with known values.

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There's a number of observations or values

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or sample locations and their distribution

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influence the accuracy
of the estimated surface.

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In general, you wanna
consider enough control points

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to yield adequate results but no more.

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Same fundamental principle of parsimony

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when constructing other
statistical models.

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And it's up to you to
analyze the properties

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of that surface to understand if or how

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the influence of the weight
changes over space or distance.

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So again, looking for
these patterns in the data

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that you're using to perform
these interpolation functions.

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Now there's two types of interpolators,

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what's known as an exact
or inexact predictor.

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For the exact, the predicted value

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is equal to the measured value.

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That means for each one
of your sample locations,

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the curve that's fit through
your data will perfectly align

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with the value at those sample points.

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In the case of the inexact,

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the predicted values might be different

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than the measured value.

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Oftentimes that means they will avoid

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or not be good at
locating peaks and troughs

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in the landscape.

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So why you search neighborhoods?

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Again, it's this concept
that the further away you are

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from a known point location,

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the less likely there
is to be a relationship

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to the value in that known location.

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So distant points have a limited effect

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on the predictive value at some point,

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and they might even have
a detrimental effect

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on the predicted value itself

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because of the disparity in the values

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from those two locations.

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Lastly, by applying search neighborhoods,

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you're reducing the number of points

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considered for each calculation

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which increases the computational speed.

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Now, in terms of neighborhoods,

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there's many different
kinds that we can specify.

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You might simply say that you
want the closest 10 points

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to the unknown value.

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You might want a
specified number of points

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within some distance and
you could specify both,

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the shape of that distance

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and the number of points you'd
like to locate within there.

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And then lastly, you could
look at something by quadrant

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and specify that you want
at least a minimum number

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of points from each of the quadrants

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and you can rotate those quadrants

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and even combine the quadrants

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with a distance function as well.

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So if we look at one example here

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where I define a 200 meter radius,

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my neighborhood's going
to be circular shape.

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It's going to have four sectors
with a zero degree offset,

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and I want to identify
at least two neighbors

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in each one of my sectors.

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We start by identifying
that location of interest,

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plotting out our radius and the sectors,

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selecting those points within
each one of the sectors

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and then using that
information to make an estimate

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at that unknown value location.

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Now you can change the size and the shape

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of the search neighborhood

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if you're aware of directional trends

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in the location that you're analyzing.

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And we saw an example of that
in the last lecture as well.

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In terms of directional influence,

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global trends affect
all of the measurements

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in a deterministic manner

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and you can represent that
using a mathematical formula.

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It's up to you to understand
the directional signal

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whether one exists and what
direction it's pointing.

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Some examples of this

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include something like a prevailing wind

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or groundwater flow.

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Another concept related to
directional influences anistropy.

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So this is not deterministic.

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This is reflected more by a random process

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with higher auto correlation
in one direction.

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So there is a potential for
a deterministic work around

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to this somewhat funny issue
by modeling the influence

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of that anistropy by changing the size

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and the shape of the search neighborhood

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to see if you can't account
for that in those variations.

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In terms of output surfaces,

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there are a number of different surfaces

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that are produced by these
interpolation methods.

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The prediction surface
is the default output

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of all of the different
interpolation methods.

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Predicted standard error
is the standard deviation

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of the estimated value at each location.

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As the standard error increases,

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the precision of prediction
at the location decreases.

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There's a probability surface,

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the probability that some
threshold value will be exceeded

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at each location.

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And if you have a multinomial
normal distribution

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for your data, you can assume that

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more than 99% of the values are reflected

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within three standard deviations

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of the data.

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You can also produce
quantile output surfaces

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and these are really useful for monitoring

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or modeling best and worst case scenarios.

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Depending on the interpolation method

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that will determine the
possible output surfaces.

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Again, the deterministic approaches

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will not produce any of
the error probability

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surfaces as part of their output.

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And then lastly, we'll touch on accuracy.

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Now this is the measure difference

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between the interpolated
value and the measured value.

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If you want to perform
an accuracy assessment,

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you withhold some validation points

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not to be used in the estimation process,

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and then check against that
once you've estimated a surface.

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You can do this for a subset of the data

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or for all of the data
points in your dataset.

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Alternatively, it's somewhat
more difficult and expensive.

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You could collect a validation dataset.

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So an entire dataset that's set aside

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from the estimation dataset,
that's then used to compare

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against the output from
your model estimation.

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The last thing I'll note here

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before we get into some
examples of interpolation,

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is that no single method
is the best solution

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for all instances.

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This will take some trial and error.

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You need to dig into your data
and understand the contents

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particularly patterns that
might exist in the data

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to be able to better
parameterize your model

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and produce more
realistic output surfaces.

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Well, that's it for key
concepts and terminology.

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Let's take a look at some examples

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in both a deterministic

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and geostatistical
interpolation approaches.