Spatial estimation

How is spatial estimation

commonly used?

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Spatial estimation: Outline

Introduction

Sampling

Spatial interpolation methods

Spatial prediction methods

Core area mapping

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Spatial estimation: Intro

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What are examples of each?

Spatial Interpolation

Prediction of variables at unmeasured locations based on sampling of same variables at known locations

Spatial Prediction

Prediction of a variable at unmeasured locations based in part on other variables

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Spatial estimation: Intro

Core area delineation:

Delineate areas of high utilization, density, intensity, probability of occurrence

Core areas may consist of a home range, center of criminal activity, or center of business activity

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Spatial estimation: Sampling

We control location and number of samples

With additional samples the law of diminishing returns

Diminishing returns not usually a problem

Funding limits samples

Few established guidelines for determining number of samples needed for interpolation

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Interpolating ETr for irrigation management

For agricultural water management or climate change studies weather station placement is of paramount importance

If the sampling scheme includes anomalous areas then the resulting interpolation could be very biased

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Sampling

What is each sampling scheme called?

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Spatial estimation: Systematic sampling

Not appropriate if there is greater interest in specific areas within the study area

Travel to each point can be difficult or costly

Systematic can be problematic

Roads are laid out systematically

Geologic features can vary systematically

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Spatial estimation: Random Sampling

Random samples are less likely to correspond to systematic landscape variation

Little chance of inaccurate predictions due to biased sampling

Random sampling can cause more sampling than necessary in uniform areas and undersample variable areas

Difficult to implement over large areas

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Spatial estimation: Cluster sampling

Cluster centers located randomly or systematically

Reduce travel time between sample locations

Used for mineral exploration and by U.S. Forest Service to conduct forest productivity and condition surveys

Good for natural resources applications requiring significant off-road travel

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Spatial estimation: Adaptive samplingHigher sampling densities where the feature of interest is more variable

Increases sampling efficiency

In some cases it is obvious where an attribute is variable (elevation)

Otherwise sampling intensity decisions are made on the fly or after lab results following preliminary sampling

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Other aspects of sampling design

Other sampling design strategies are appropriate for specific applications

Sampling can be performed with or without replacement

This is only a problem if linear or area features are the sampling units

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Spatial Interpolation Methods

Combine sampled values and their positions to estimate values at unsampled points

Spatial interpolation method variables:

Observation weights and how we choose these weights

Use all or only some observations

Use different mathematical functions

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Sample dataset: What is unrealistic?

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Thiessen polygon interpolation

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Nearest neighbor / Thiessen polygon

Assign value of unmeasured location to nearest sample point value

Polygon output

Abrupt transition between polygon edges

Exact interpolator

Interpolated surface equals sampled values at each sample point

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Fixed radius Local averaging

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Fixed radius Local averaging

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Fixed radius interpolation

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Fixed radius Local averaging

Moving average of samples within radius

Radius is centered sequentially at each raster cell center

No value assigned if no samples fall in search radius

Neighboring cell values usually similar

Smooths sample data

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Fixed radius Local averaging

Extreme values are maintained only if no other values are present in the search radius

The search radius chosen strongly affects the resultant surface

Not exact interpolator

Too large of a search radius will smooth the surface too much

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Inverse distance weighting (IDW)

Estimates values at unknown locations based on sample values and distance to nearby points

Closest points have strongest influence

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n and i for IDW

n = distance exponent

i = number of sample points used to estimate a raster cell center value

What effects do n and i have on interpolated surface?

What does this equation become when d approaches zero?

zj =

zidniji

∑1dniji

∑

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IDW

An exponent (n) can be applied to the distance to influence the rate at which the influence of a point wanes with distance

IDW is an exact interpolator

As distance becomes very small, its inverse approaches infinity, eliminating the influence of any other points

Creates smooth raster surfaces

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IDW

Larger exponent (n):

Increases the influence on closer points

Higher peaks, lower valleys, steeper gradients near sampled points

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IDW

• Lower exponent (n) reduces the influence of proximal points, smoothing resultant surface

Spatial interpolation: Splines

Fit polynomial function to measured values:

Minimizes surface curvature—the sum of the squares of the second derivative terms

Smooth surface that passes through input points

Interpolated surface is differentiable

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Spatial prediction

Statistical fitting process using coordinate locations and independent variables

Waldo Tobler: “everything in the universe is related to everything else, but closer things are more related.”

Spatial autocorrelation is the tendency of nearby values to be more similar than distant values

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Spatial prediction: Autocorrelation

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Moran’s I measures spatial autocorrelation

0 low correlation

1 positive autocorrelation

-1 anti-correlated

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Spatial cross-correlation

In the absence of spatial autocorrelation and cross correlation what is the best estimate of values at unmeasured locations?

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Spatial prediction: Correlation

The degree of spatial autocorrelation can vary over space

Spatial prediction methods incorporate knowledge of the correlation structure of the data into predicting unmeasured locations

Cross-correlation is the tendency for two variables to change in concert

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Spatial prediction: Regression

Predict output variable distribution as a function of several inputs

Combine measurements of temperature with elevation, latitude, and longitude to predict temperature in a statistical model

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Spatial prediction: Trend surface

Not exact predictors

Most accurate when fitting smoothly varying surfaces

Mean daily temperature

No Bulls-Eyes

Not appropriate for modeling complex surfaces (elevation in a mountainous region)

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Spatial prediction: Trend surface

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Spatial prediction: Kriging

Developed in early 1900s by D.G. Krige and Georges Matheron for mining

They aimed to develop a spatial prediction method that:

accounts for trends, spatial autocorrelation, and random (stochastic) variation

estimates local prediction variance

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Spatial prediction: Kriging and Co-Kriging

Based on regionalized variable theory

Model components:

Spatial trend

Local spatial autocorrelation

Random (stochastic) variation

Weights are chosen to be statistically optimal

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Kriging

h: lag distance

lag distance is the distance between sample points

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Kriging

Lag tolerance is a distance between points that is considered small so that pairs of points can be grouped together for spatial covariance calculations

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Semi-variogram anatomy

Nugget: Semivariance when autocorrelation is highest, at a lag distance of zero

Sill: Point where variogram levels off

Inherent or background variation when there is little spatial autocorrelation

Range: Lag distance where the sill is reached

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Semi-variogram

Semi-variance measures spatial autocorrelation for a given lag distance

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Semi-variogram fit to data

Semivariance is smaller at small lag distances and plateaus as lag distance increases

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Determining kriging weights

Optimal weights sum to zero and are calculated to minimize variance using a system of equations

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Kriging

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Co-kriging

Kriging with another correlated predictive variable in addition to the variable of interest

Good for situations where one variable is widely or easily measured and measurements of the other variable are more scarce or expensive

Temperature (primary), Elevation (secondary)

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Spatial estimation: Interpolation accuracy

Difference between measured and interpolated values when several sample points are withheld

No single method is more accurate than any other for all applications

Each has advantages and disadvantages

Test several methods before choosing one

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Spatial prediction: Interpolation accuracy

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Spatial prediction: Interpolation accuracy

RMSE is very common

Squaring errors magnifies the effects of outliers possibly leading to an overly pessimistic error estimate in some cases

Mean absolute error is less common

Less sensitive to outliers

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Interpolation accuracy

How accurately does the estimated surface match the actual surface?

Generally, use the most accurate model

A slightly less accurate model may be preferable if it is simpler or easier to implement

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Cross-validation

Accuracy assessment requires collection of observed and predicted values for a set of points

Cross-validation is the process of fitting a surface iteratively each time withholding one point to derive an accuracy estimate

A statistical package such as R can be used for advanced geostatistical analysis

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Spatial estimation: Sampling

How can we determine if sampling is too sparse to interpolate a surface in between measurement locations?

Why might it be beneficial to have two measurement locations very nearby?

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Why might we develop a core area map?

Core area is a primary area of activity for an organism (endangered species), humans (criminals), objects, or resources

Delineate area features from a set of point or line observations

Additional resources or protections may be devoted to this area or habitat requirements can be inferred

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What limitations exist in using mean circles?Mean center is the average x, y coordinate

Circles of maximum distance, standard error, or mean distance can define variability around the center

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Disadvantages of mean circles

Distributions may exhibit irregular shapes

Outliers can bias both the mean center and parametric mean circles

Core area can be over-estimated

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Spatial estimation: Convex Hulls

Identify irregularly shaped core areas

Smallest polygon created by edges that:

completely enclose a set of points

all exterior angles between edges >=180

Outliers will cause convex hulls to overestimate core areas

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Concave and Convex hulls

Simple

No subjectivity

Ignore clustering

Information on frequency and density is lost

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Convex Hull algorithm

Identify an extreme point

Largest or smallest x or y

Find the point with the largest angle of deflection

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Characteristic Hull Polygons

Delaunay triangulation to develop a set of minimum spanning triangles

Remove largest area or longest perimeter subset

Reduces influence of outliers

Allows for holes within a core area

Depends on threshold value chosen

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Kernel Density Mapping can identify core areas

Continuous density surface estimated from a set of points

Mathematically flexible

Easy to implement

Robust to outliers and irregular shapes

Incorporates clustered samples

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Kernel density

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Kernel density: bandwidth effects

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Kernel density

Collect samples

User selects bandwidth (h) that defines spread

Narrower bandwidths create higher, narrower peaks

Choose kernel density function and sum

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Kernel density with Gaussian density distrib.

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Kernel density

Density or probability of occurrence of the variable

Crime density

Habitat utilization density

Defect density

Core areas delineated from kernel density depend on bandwidth selection

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Bandwidth effects on core area delineation

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Gaussian density

distribution by

bandwidth

Core area

Time-Geographic Density Estimation

Home range analysis

Combines a sequential set of control (GPS) points with maximum velocity information to estimate spatial occurrence probability

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Time-Geographic Density Estimation

A geoellipse describes the furthest an object could have reached given known end points and known maximum velocity

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What controls ellipse shape?

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Controls on Ellipse Shape

As more time passes ellipse becomes larger

Ellipse will approach a straight line as the distance between points approaches the maximum set by the maximum velocity

Ellipses approach circular shape if object moves short distance over long time interval

Densities highest where points cluster or paths intersect

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From b to d maximum velocity increasesThis figure assumes uniform distribution

We could assume a linear decay function, that occurrence probability is highest near the connecting line

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Primary sources

http://www.paulbolstad.net/gisbook.html

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