GIS in the Sciences ERTH 4750 (38031). Modeling spatial structure from point samples. Xiaogang (Marshall) Ma School of Science Rensselaer Polytechnic Institute Tuesday, Feb 26, 2013. Review of last week’s lecture. Spatial structure: point distribution, postplots , quantile plots
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We had no lecture last week,
here we do some catch-up
Distribution over the space:
Symbol size/color reflects
attribute value: Postplot
Contrasting size or color:
Here we are concerned with the formof the surface
First-order: linear surface
Second-order: paraboloid, bowl (lowest in the middle) or dome (highest in the middle)
3 Local spatial dependence: Point-pairs, h-scatterplots, variogram cloud, experimental variogram
If there is no relation between the values at the separation, the h-scatterplot will be a diffuse cloud with a low correlation coefficient.
The cloud of point pairs becomes more diffuse as the separation distance increases.
Variogram / semivariogram: The most common way to visualize local spatial dependence
where is a geographic point and is its attribute value.
Left: separations to 2 km; Right: separations to 200 m
Clearly, the variogram cloud gives too much information. If there is a relation between separation and semivariance, it is hard to see. The usual way to visualize this is by groupingthe point-pairs into lagsor binsaccording to some separation range, and computing some representative semivariancefor the entire lag.
Empirical variogram: Group the separations into lags (separation bins), then compute the averagesemivariance of all the point-pairs in the bin. Below the empirical variogram by Isaaks & Srivastava (1990):
Compared to Semivariance:
Compared to Semivariance:
with different bin widths
Variogram surface showing isotropy
Variogram surface showing anisotropy
Directional variograms of , Meuse soil samples
> plot(variogram(log(zinc) ~ 1,
+ meuse, alpha = c(30, 120),
+ cutoff = 1600),
+ main = "Directional Variograms,
+ Meuse River, log(zinc)",
+ sub = "Azimuth 30N (left),
+ 120N (right)", pl = T,
+ pch = 20, col = "blue")
Modeling spatial structure from point samples
We often have a set of point observations (with some spatial support around the “point”). From these we usually want to:
There are many mathematical forms for a regional trend. We will see the most common, namely the polynomial trend surface.
In last lecture we talked about forms of trend surfaces, now we see the mathematical models behind them
where is the attribute value and are the two coordinates, e.g. UTM E and N.
In the previous lecture and exercise (“visualization”) we computed and displayed first- and second-order trend surfaces for subsoil clay content in a region of southern Cameroon.
The result of this computation is a fitted linear regression of the attribute(here, clay content) on the predictors, which are the grid coordinates.
where is the number of observation and is the number of coefficients.
* The error of an observed value is the deviation of the observed value from the (unobservable) true function value, while the residual of an observed value is the difference between the observed value and the estimated function value (c. Wikipedia).
There is only a 2% improvement in adjusted R2 (from 0.499 to 0.519)
In summary, there are some processes that operate over an entire region of interest. These can lead to regional trends, where an attribute can be modeled as a function of the geographic coordinates.
The presentation is based on R. Webster and M. Oliver, 2001 Geostatistics for environmental scientists, Chichester etc.: John Wiley & Sons.
This is difficult material, mainly because it is abstract and indeed quite strange at first glance. Some modelers do not accept it at all! Our position is that this model of an essentially unobservable hypothetical process has proven to be very useful in practice.
CDF: cumulative distribution function
The above is a difficult point for most people to grasp. It's always easier to visualize, so the next slide shows four different realizations of the same random field.
How similar are the attribute values at any specific location, among the four diagrams?
How similar are the patterns? Compare the patch sizes, sharpness of transitions between patches, etc.
So there is a regionalized variable, i.e. where every point is the outcome of some random process with spatial dependence.
We now introduce the key insight in local spatial dependence: auto-covariance, i.e. the covariance of a variable with itself. How can this be? Patience, and all will be revealed . . .
Recall from non-spatial statistics: the sample covariance between two variables and observed at points is:
How can we relate these?
We can imagine many functions of semivariance based on separation; but not all of these are:
* possibly with a single fluctuation (hole) . . .
* …or a periodicfluctuation
Before we look at the various authorized model forms, we show one of the simplest models, namely the sphericalmodel, with the various model parameters labeled.
These variogram functions are generally available in geostatistical modeling and interpolation computer programs.
1. linear: same
2. convex: increasing
3. concave: decreasing
These two models are quite similar to the common sphericalmodels (see previous slide); they differ mainly in the shape of the “shoulder” transition to the sill:
Linear−with−sill variogram model
Circular variogram model
Spherical variogram model
Exponential variogram model
Gaussian variogram model
Note effective range (95% of asymptotic sill) for exponential & Gaussian models
Minasny, B., & McBratney, A. B. (2005). The Matern function as a general model for soil variograms. Geoderma, 128, 192-207.
There are two steps to modeling spatial dependence from the experimental variogram:
(Simulations computed with krigemethod of gstatpackage in R.)
96, 48, 24, 12
on 256x256 grid
Ranges 50, 0 (pure nugget), 15, 25 on 256x256 grid
Evidence of both
short range ()
long range ()
Hard to judge the relative value of each point
We now look at two models that were fitted first by eye and then automatically.
By eye: ; total sill
Automatic: ; total sill
The total sill was almost unchanged; gstatraised the nugget and lowered the partial sill of the spherical model a bit; the range was shortened by 51 m.
“Automatic” fitting should always be guided by the analyst. If the variogram is quite consistent across the range, almost any automatic t will be the same. Otherwise, different fitting criteria will give different results.