Definition of Spatial AnalysisTypes of Spatial Analysis3. TransformationsNeighborhood OperationsNeighborhood Operation - MeanNeighborhood Operation - VarianceNeighborhood Operation - MajorityThe Mean Operation RevisitedA More Complex Neighborhood OperationSpatial EnhancementsDensity Estimation and PotentialThe Kernel FunctionKernel Function ExampleKernel Function Example4. Descriptive SummariesMeasures of Central Tendency - MeanThe CentroidDispersionMeasures of Dispersion – Variance, Standard Deviation, Z-scoresPoint Pattern AnalysisThe Nearest Neighbor IndexFragmentation StatisticsFrag. Stats. Example5. OptimizationApplications of the MATOptimizing Point LocationsLocation-Allocation ProblemsRouting ProblemsOptimum PathsLeast-Cost Path Example6. Hypothesis TestingDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Definition of Spatial Analysis• A method of analysis is spatial if the results depend on the locations of the objects being analyzed• i.e. if you move the objects and the results change, or the results are not invariant under relocation, spatial analysis is being applied• To conduct a spatial analysis requires both attributes and locations of objects• Conveniently, GIS has been designed to store both … we usually assemble geographic information in a GIS so that we might analyze itDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Types of Spatial Analysis•There are literally thousands of spatial analysis techniques, with new ones developed all the time• We will consider six categories of spatial analyses, each having a distinct conceptual basis:1. Queries and reasoning2. Measurements3. Transformations4. Descriptive summaries5. Optimization6. Hypothesis testingChapter 13Chapter 14David Tenenbaum – GEOG 070 – UNC-CH Spring 20053. Transformations• The category encompassing transformations of spatial data includes many analytical approaches that can be applied using either the vector or raster spatial data models, or combining bothtogether• Transformations create new attributes and objects, based on some simple rules:• They involve geometric construction or calculation• They may also create new fields, either from existing fields or from discrete objectsDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005•In raster overlay analysis, we compared each cell in a raster layer with another cell in the same position on another layer•In neighborhood operations, we look at a neighborhood of cells around the cell of interest to arrive at a new value:Neighborhood OperationsA 3x3 neighborhoodAn input layerCell ofInterest•Neighborhoods can be of any possible size; we can use a 3x3 neighborhood for any cell except on the edge of the layerDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Neighborhood Operation - Mean•One neighborhood operation is to calculate the mean for all pixels in the neighborhood and put the result in the center of the neighborhood. This is why a neighborhood size is often an odd number (3x3, 5x5, 7x7, …) because these have a well defined center for the result value:223455215726497388381914622345521572649738838191466419183883794627512554322InputLayerResultLayer3 4 5David Tenenbaum – GEOG 070 – UNC-CH Spring 2005Neighborhood Operation - Variance•Rather than calculating a mean using the 9 values from a 3x3 neighborhood, we could instead calculate variance•This operates in the same fashion; we collect the values in the neighborhood, calculate the statistic, and write the result in the center of the neighborhood in the result layer:64191838837946275125543226419183883794627512554322ResultLayer6419183883794627512554322InputLayer2.756 5.75David Tenenbaum – GEOG 070 – UNC-CH Spring 2005Neighborhood Operation - Majority•Another operation we might use on a neighborhood is to find the majority value (the value that appears most often, the mode):242211338313322723253332224221133831332272325333222422113883133227232533322InputLayerResultLayer2 3 3•Under what circumstances might we find this to be a useful operation to apply to a raster layer?David Tenenbaum – GEOG 070 – UNC-CH Spring 2005The Mean Operation Revisited1/9 1/9 1/91/9 1/9 1/91/9 1/9 1/9•In the mean operation, each cell in the neighborhood is used in the same way:641918388379462751255432264191838837946275125543226419183883794627512554322InputLayerResultLayer3 4 5David Tenenbaum – GEOG 070 – UNC-CH Spring 2005-1 -1 -1-1 9 -1-1 -1 -1A More Complex Neighborhood Operation•In more complex operations, the cells in the neighborhood can be treated differently:This is an edge enhancement filter 641918388379462751255432264191838837946275125543226419183883794627512554322InputLayerResultLayer-7 -26 5David Tenenbaum – GEOG 070 – UNC-CH Spring 2005Spatial Enhancements• Filters - used to emphasize or de-emphasize spatial information• Low-pass filter -emphasize large area changes and de-emphasize local detail• High-pass filter -emphasize local detail and de-emphasize large area changesDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Landsat TM 543 False Color Image of Tarboro, NCSpatial EnhancementsEdge DetectionSharpening FilterDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Landsat TM 543 False Color Image of Tarboro, NCSpatial EnhancementsSmoothing FilterNormal ImageDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005Density Estimation and Potential• Spatial interpolation is used to fill the gaps in a field (i.e. from a few known values to lots of estimate values)• Density estimation creates a field from discrete objects (i.e. from many known values to a smaller number of values that summarize the known values’ distribution at a certain spatial scale)• the field’s value at any point is an estimate of the density of discrete objects at that point• e.g., estimating a map of population density (a field) from a map of individual people (discrete objects)• You can think of this as applying the neighborhood operation concept by searching for vector objects within a certain distance, and using a kernel functionDavid Tenenbaum – GEOG 070 – UNC-CH Spring 2005The Kernel Function• Each discrete object is replaced by a mathematical function known as a kernel that once again finds objects within a given distance and weights their contribution to the result accordingly• Kernels are summed to obtain a composite surface of density•The smoothness of the resulting field depends on the width of the kernel• narrow kernels produce bumpy