- [Instructor] In the second part of the lecture,
we'll look at the introduction to geostatistics
and interpolation concepts.
I won't lie, we're gonna get pretty close
to some mathematics here
but I think we make it out of the presentation
without having to look at any formulas
I'll be offering a relatively light treatment
of both of these topics.
So there's definitely plenty more to dig in on.
If you're interested in the mathematics
or just wanna know more in general, let me know,
and I can find some additional resources for you to review.
Geostatistics is a statistical approach
for analyzing spatial and spacial temporal phenomenon.
It considers both the location
and the value of the observation at that location
as part of the analysis.
We can use geostatistics to describe patterns in our data
to interpolate values or fill in the blanks
between known point locations
and even to estimate uncertainty.
Geostatistics has evolved
to be embraced by a broad range of disciplines
for any number of applications.
When we're talking about spatial interpolation,
we consider the prediction of a variable
at unmeasured location based on the value of variables
at measured location.
So out at sample points where you've collected data.
Spatial prediction takes that one step further
and includes other variables in the estimation process.
Now, part of having data that you can use
for geostatistical analysis includes sampling data
or collecting data in the field,
or potentially even just sampling data
from a digital data set that you have.
A couple of different kinds of sampling
that we see on this page, systematic and random.
Systematic samples are uniformly spaced
over X and Y intervals.
They're easy to plan and implement
and it really removes any of the subjective judgment
about where to collect sample data from.
On the con side,
it's not necessarily statistically efficient
to collect data this way,
and it certainly doesn't consider things like travel costs
or inability to capture patterns at finer scales
when using this type of program.
Random samples on the other hand are just a random selection
of X and Y coordinates usually selected separately.
On the positive side,
these are unlikely to match any pattern in the landscape
meaning that it's good for capturing the diversity
in the landscape itself,
but on the con side
it doesn't increase the number of samples
in areas with high variation,
so it's likely to miss some of those patterns
and it's more challenging to plan and implement.
We've also got an option for cluster sampling
which kind of combines the random and systematic selection
of the cluster centers and the clusters
that actually surround those individual point locations.
The US Forest Service
uses a random cluster center point selection
followed by systematic designation of the cluster
of the surrounding points
for some of their forest sampling and inventory activities.
On the plus side, there's a real strong possibility
to reduce your travel costs when sampling in this fashion.
And then lastly, there's an adaptive sampling strategy
where you collect more frequent samples
in areas with variable information and less frequent samples
in uniform areas of the landscape.
This does require some assessment prior to field work
and updates during field work
which can present some logistical challenges.
A few key bits of terminology I'd like to cover,
interpolation as you may or may not know
uses measured values to predict the values
at unsampled locations.
We'll look at two different kinds of interpolation,
deterministic and geostatistical.
So deterministic interpolation uses either a similarity
or smoothing type function
and it does not measure uncertainty or error
in the output prediction.
Geostatistical interpretation on the other hand,
is a statistical model
that include spatial autocorrelation.
These can measure uncertainty in the output data,
and are quite powerful
for any number of different applications.
We heard a little bit about the kernel already,
but just by way of reminder,
the kernel is a weighted function
that assigns the weights to locations based on the proximity
of that location to an observation or a sample point.
Now, spatial autocorrelation is this assumption that
the closer two observations are to one another,
the more similar their values are likely to be.
So depending on the type of analytical approach
that we take to interpolation,
spatial autocorrelation may be required,
it might be assumed, or it might be ignored altogether.
It's important to note that spatial autocorrelation
violates this independence assumption
that you would have otherwise
in traditional statistical methods.
For geostatistics, it's an important assumption
that makes these calculations possible.
We can look at a couple of different ways
for analyzing data patterns,
Moran's I and the Getis-Ord General G.
Both of these are statistical measures
that look at clusters of data.
In the case of Moran's I,
it's a measurement of spatial autocorrelation
based on feature locations and their values
and the algorithm determines if the data are clustered,
dispersed or random.
Like all statistical tests, this one has a null hypothesis
that states that the attribute under consideration
is randomly distributed.
You can use Moran's I to help define both the size
and the orientation of search neighborhoods
that would be used for interpolation processes.
Now, the Getis-Ord General G statistic
measures clusters of high or low values.
And in this case, it assumes a null hypothesis
that there is no spatial clustering of the feature values.
For both, depending on the P value and the Z value
that you get from your estimation of these statistics,
you can determine whether or not
you can dismiss the null hypothesis
and state something definitive about the pattern
or lack thereof in the data that you're using.
Now there's multiple mathematical approaches
to spatial interpolation,
but all of these use points with known values
to estimate values at these unsamped locations.
Location, both the distance between the sample location
and the cell location and the attribute values
are considered as part of the process.
Now, once again, depending on the approach that you take
varying weights would be assigned
to neighboring observations and you do have some control
over the way those weights are applied.
Not surprisingly
because of the multiple different approaches
to performing spatial interpolation,
the different approaches will produce different results.
Deterministic and geostatistical approaches
for interrelating continuous raster surfaces
happen with control points.
Now, even with a finite number of discrete objects in space,
there are still likely too many to count,
too many places to go survey
to consider a completeness to that.
And because of that, coupled with potentially costs,
a lack of access for legal or practical reasons,
site locations being dangerous,
you might be missing samples,
you might have erroneous samples,
or you might be re sampling your data.
These are all reasons why you might need to perform
spatial interpolation operations.
We have both global and local interpolators.
On the global front,
you're looking at trend surface analysis
and regression models.
We won't be doing much of that in this module.
Instead, we'll focus on local operations
that use a sample of the known points
where the user has control over how many
and how to search for those point locations.
And we'll look at both geo statistical
and deterministic approaches.
Now control points as I mentioned before,
are these locations with known values.
There's a number of observations or values
or sample locations and their distribution
influence the accuracy of the estimated surface.
In general, you wanna consider enough control points
to yield adequate results but no more.
Same fundamental principle of parsimony
when constructing other statistical models.
And it's up to you to analyze the properties
of that surface to understand if or how
the influence of the weight changes over space or distance.
So again, looking for these patterns in the data
that you're using to perform these interpolation functions.
Now there's two types of interpolators,
what's known as an exact or inexact predictor.
For the exact, the predicted value
is equal to the measured value.
That means for each one of your sample locations,
the curve that's fit through your data will perfectly align
with the value at those sample points.
In the case of the inexact,
the predicted values might be different
than the measured value.
Oftentimes that means they will avoid
or not be good at locating peaks and troughs
in the landscape.
So why you search neighborhoods?
Again, it's this concept that the further away you are
from a known point location,
the less likely there is to be a relationship
to the value in that known location.
So distant points have a limited effect
on the predictive value at some point,
and they might even have a detrimental effect
on the predicted value itself
because of the disparity in the values
from those two locations.
Lastly, by applying search neighborhoods,
you're reducing the number of points
considered for each calculation
which increases the computational speed.
Now, in terms of neighborhoods,
there's many different kinds that we can specify.
You might simply say that you want the closest 10 points
to the unknown value.
You might want a specified number of points
within some distance and you could specify both,
the shape of that distance
and the number of points you'd like to locate within there.
And then lastly, you could look at something by quadrant
and specify that you want at least a minimum number
of points from each of the quadrants
and you can rotate those quadrants
and even combine the quadrants
with a distance function as well.
So if we look at one example here
where I define a 200 meter radius,
my neighborhood's going to be circular shape.
It's going to have four sectors with a zero degree offset,
and I want to identify at least two neighbors
in each one of my sectors.
We start by identifying that location of interest,
plotting out our radius and the sectors,
selecting those points within each one of the sectors
and then using that information to make an estimate
at that unknown value location.
Now you can change the size and the shape
of the search neighborhood
if you're aware of directional trends
in the location that you're analyzing.
And we saw an example of that in the last lecture as well.
In terms of directional influence,
global trends affect all of the measurements
in a deterministic manner
and you can represent that using a mathematical formula.
It's up to you to understand the directional signal
whether one exists and what direction it's pointing.
Some examples of this
include something like a prevailing wind
or groundwater flow.
Another concept related to directional influences anistropy.
So this is not deterministic.
This is reflected more by a random process
with higher auto correlation in one direction.
So there is a potential for a deterministic work around
to this somewhat funny issue by modeling the influence
of that anistropy by changing the size
and the shape of the search neighborhood
to see if you can't account for that in those variations.
In terms of output surfaces,
there are a number of different surfaces
that are produced by these interpolation methods.
The prediction surface is the default output
of all of the different interpolation methods.
Predicted standard error is the standard deviation
of the estimated value at each location.
As the standard error increases,
the precision of prediction at the location decreases.
There's a probability surface,
the probability that some threshold value will be exceeded
at each location.
And if you have a multinomial normal distribution
for your data, you can assume that
more than 99% of the values are reflected
within three standard deviations
of the data.
You can also produce quantile output surfaces
and these are really useful for monitoring
or modeling best and worst case scenarios.
Depending on the interpolation method
that will determine the possible output surfaces.
Again, the deterministic approaches
will not produce any of the error probability
surfaces as part of their output.
And then lastly, we'll touch on accuracy.
Now this is the measure difference
between the interpolated value and the measured value.
If you want to perform an accuracy assessment,
you withhold some validation points
not to be used in the estimation process,
and then check against that once you've estimated a surface.
You can do this for a subset of the data
or for all of the data points in your dataset.
Alternatively, it's somewhat more difficult and expensive.
You could collect a validation dataset.
So an entire dataset that's set aside
from the estimation dataset, that's then used to compare
against the output from your model estimation.
The last thing I'll note here
before we get into some examples of interpolation,
is that no single method is the best solution
for all instances.
This will take some trial and error.
You need to dig into your data and understand the contents
particularly patterns that might exist in the data
to be able to better parameterize your model
and produce more realistic output surfaces.
Well, that's it for key concepts and terminology.
Let's take a look at some examples
in both a deterministic
and geostatistical interpolation approaches.