Spatial Analysis Part 3

Analyzing Raster Data• Types of functions

– Local • These functions happen on a cell-by-cell basis• Example = map algebra

– Neighborhood (a.k.a. Focal)• The values for an output cell are derived using neighboring cells• This happens in “window” or “moving window” analyses• Example = filters (e.g., spatial enhancement)

– Zonal• The values of output cells are derived using cells in pre-defined zones• Often these zones are vector objects• Example = zonal statistics

– Global• The values of the output cells are derived using all cells• Example = cost paths

Neighborhood Functions

In neighborhood operations, we look at a neighborhood of cells around the cell of interest to arrive at a new value.

We create a new raster layer with these new values.

A 3x3 neighborhood

Neighborhood Operations

An input layer

Cell ofInterest

• Neighborhoods of any size can be used• 3x3 neighborhoods work for all but outer edge cells

Neighborhood Operations

• The neighborhood is often called:– A window– A filter– A kernel

– They can be applied to:• Raw data (e.g., imagery pixels)• Classified data (nominal landcover classes)

A 3x3 neighborhood

• The mean for all pixels in the neighborhood is calculated.

• The result is placed in the center cell in the new raster layer.

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Neighborhood Operation: Mean Filter

Landsat TM 543 False Color Image of Tarboro, NC

Normal Image Smoothing Filter

Smoothed Image

Neighborhood Operations• Why might we use a filter like this?

• Suppose you have a nominal dataset (e.g., a landcover classification)

• Sometimes classifications are ‘speckled’.– Usually a few misclassified pixels within a tract of

correctly-classified landcover– We want to reclassify those pixels as the surrounding

landcover type

Neighborhood Operation: Majority Filter

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•The majority value (the value that appears most often, also called a mode filter):

Neighborhood Operation - Variance

• We may want to know the variability in nearby landcover for each raster pixel– To find cultivated areas - usually less variability than natural area– To find where areas where eco-zones meet

• The variance of a 3x3 filter on, for instance, an NIR (near infra red) satellite image band will help find such areas.

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The Mean Operation Revisited•In the mean operation, each cell in the neighborhood is used in the same way:

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Edge Enhancement•Cells can be treated differently within a kernel:

This is an edge enhancement filter (discussed below).

Edge Enhancement Filter-1 -1 -1

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•Why is this an edge enhancement filter? It enhances edges. Let’s look at the kernel’s behavior at and away from edges:

Away from edge (in areas with uniform landcover) At edges (between areas with differing landcover)

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Filter Result:

Edge Enhancement Filter

Edge Enhancement

Sharpening FilterNormal Image

Edge enhancement filters sharpen images.

Density fields

• Neighborhood Operation can create a density surface from discrete point data

A typical kernel function

The result of applying a 150km-wide kernel to points distributed over California

Kernel Function Example

Kernel Function Example

Kernel width is 16 km instead of 150 km. This shows the S. California part of the database.

A More Familiar Example

• This is a neighborhood statistic applied to the “Midwest” shapefile from our lab #4

• In this case I used a circular kernel (radius 3 cells, cell size = 0.25 degrees) and summed the count value for all points

Kernel Size

•The smoothness of the resulting field depends on the width of the kernel

•Wide kernels produce smooth surfaces

•Narrow kernels produce bumpy surfaces

Zonal Functions

Zonal Functions• Zonal statistics provide a summary of what is

going on in an area (i.e., a zone) by including all the raster cell values that are within that zone

• Zones can be defined using raster or vector data

• Summary statistics include: – Mean, min, max, range, count, standard deviation,

etc.

• ArcGIS will also treat lines and points as zones

Example Zones• Raster Zones:

– Each state has a different value

– All cells in each state have that value

• Vector Zones: – Each state has

associated attributes– Attributes include

name, etc.

Zonal Statistics

• Example:– How much of each landcover type is in a county?– Zonal attributes will count the pixels of each cover

type within the land parcel polygon

• Example 2:– What are the average, minimum, and maximum slope

values for a hiking trail?– Zonal attributes will include all pixels that intersect the

hiking trail line feature

Spatial Autocorrelation

Spatial Autocorrelation• Tobler’s Law – "Everything is related to everything else,

but near things are more related to each other" – Waldo Tobler.

• Spatial Autocorrelation is, conceptually as well as empirically, the two-dimensional equivalent of redundancy.

• It measures the extent to which the occurrence of an event in an areal unit constrains, or makes more probable, the occurrence of an event in a neighboring areal unit.

• We won’t get very deep into this topic, but I want you to at least hear the term.

Arthur J. Lembo, Jr., Cornell University www.geography.hunter.cuny.edu/~afrei/gtech702_fall03_webpages/notes_spatial_autocorrelation.htm

Spatial Autocorrelation• Spatial autocorrelation is the correlation of a variable

with itself through space

• Spatial autocorrelation occurs when the pattern of a variable is related to the spatial distribution of that variable

• If there is any systematic pattern in the spatial distribution of a variable, it is said to be spatially autocorrelated– If nearby or neighboring areas are more alike, this is positive

spatial autocorrelation– Negative autocorrelation describes patterns in which

neighboring areas are unlike– Random patterns exhibit no spatial autocorrelation

www.css.cornell.edu/courses/620/lecture9.ppt

Spatial Autocorrelation• Spatial autocorrelation is problematic

because typically we want independent samples when we do statistics and data points close together in space can have very similar characteristics

• Spatial autocorrelation can indicate that we are missing important variables from our analysis (i.e., the data points may be clustered in space for some other reason)

www.css.cornell.edu/courses/620/lecture9.ppt

Spatial Autocorrelation Example

• Imagine someone is conducting a survey about the political interests of UNC students

• If the survey taker only asked people in this building, could she/he claim that the answers were representative of UNC students as a whole?

Spatial Autocorrelation• We measure spatial autocorrelation using

statistics including:– Moran’s I– Geary’s C

• These statistics basically tell us how autocorrelated the data are

• If the data turn out to be spatially autocorrelated, we must account for this in our analysis

Odds and Ends

• Keep (or start) working on your projects

• FYI for the presentations, simpler = better in terms of graphics / movies / etc.

• If you really want to do things like this test them first on MY computer