Chapter 4 Part C: Queries, Computations & Map Algebra

3 rd editionwww.spatialanalysisonline.com2 Queries, Computations & Map Algebra Queries Non-spatial queries Spatial queries SELECT queries/select by location Other SQL-like queries, e.g. Make table Spatial JOIN and RELATE operations OGC Spatial Relations (See next slide, from 4A)

3 rd editionwww.spatialanalysisonline.com3 MethodDescription Note: a and b are two geometries (one or more geometric objects or features points, line objects, polygons, surfaces including their boundaries); I(x) is the interior of x; dim(x) is the dimension of x, or maximum dimension if x is the result of a relational operation Spatial relations Equalsspatially equal to: a=b Disjoint spatial disjoint: equivalent to a b= Intersects spatially intersects: [a b] is equivalent to [not a disjoint(b)] Touches spatially touches: equivalent to [a b and I(a) I(b)= ]; does not apply if a and b are points Crosses spatially crosses: equivalent to [dim(I(a) I(b))<max{dim(I(a)),dim(I(b))} and a b a and a b b] Within spatially within: within(b) is equivalent to [a b=a and I(a) I(b) ] Containsspatially contains: [a contains(b)] is equivalent to [b within(a)] Overlaps spatially overlaps: equivalent to [dim(I(a) I(b))=dim(I(a))=dim(I(b)) and a b a and a b b] Relatespatially relates, tested by checking for intersections between the interior, boundary and exterior of the two components OGC OpenGIS Simple Features Specification: Spatial Relations Queries, Computations & Map Algebra

3 rd editionwww.spatialanalysisonline.com4 Queries, Computations & Map Algebra Simple computations Field level – attribute processing SQL/Query operations – e.g. Make Table Raster layer operations Basic Map Algebra Single layer operations: local, focal/neighbourhood, zonal, global Multi-layer operations Raster-vector combined operations Algebraic expressions – e.g. C=(A-B)/(A+B)

3 rd editionwww.spatialanalysisonline.com5 Queries, Computations & Map Algebra Map algebra – local operations, multi-grid

3 rd editionwww.spatialanalysisonline.com6 Queries, Computations & Map Algebra Map algebra operations – broader view Resolution, orientation and resampling Classification Algebraic and Statistical operations Proximity/landscape metric operations Surface and hydrological analysis operations Transformation and interpolation Filtering

3 rd editionwww.spatialanalysisonline.com7 Queries, Computations & Map Algebra Grid filtering – Linear (weighted average) Low pass filters, e.g. High pass filters, e.g. Kernels - example Weighted average as integer

3 rd editionwww.spatialanalysisonline.com8 Queries, Computations & Map Algebra Grid filtering – Linear – common image processing functions Sharpening Blurring Edge detection Embossing Gradient operations

3 rd editionwww.spatialanalysisonline.com9 Queries, Computations & Map Algebra Grid filtering – points to note Attribute data range Multi-band image processing Kernel size and shape Single or multi-pass Edge effects User-defined kernels Spatial vs frequency domain filtering

3 rd editionwww.spatialanalysisonline.com10 Queries, Computations & Map Algebra Grid filtering – Non-linear Non-linear local adjustment Local operation, not based on kernel (weighted average) Minimum, Maximum, Variance, IQR Median deviation and thresholding Noise reduction

3 rd editionwww.spatialanalysisonline.com11 Queries, Computations & Map Algebra Grid filtering – Erosion and dilation Erosion – pixel removal/alteration Dilation – pixel addition/alteration Kernels and structural elements Source DEM/image Dilation of source Erosion of source

3 rd editionwww.spatialanalysisonline.com12 Queries, Computations & Map Algebra Ratios, indices and normalisation Spatially extensive variables Spatially intensive variables Normalisation of count data e.g. cars/household, ethnic group counts/1000 population Common types: Averages Proportions/percentages Densities

3 rd editionwww.spatialanalysisonline.com13 Queries, Computations & Map Algebra Standardisation Direct standardisation Regional/national comparisons Indirect standardisation Expected values Excess rates Statistical standardisation Z-scores Range-based (basic or trimmed)

3 rd editionwww.spatialanalysisonline.com14 Queries, Computations & Map Algebra Ratio computations – some issues Division by 0 and missing data handling Normalisation of normalised data Variance instability Divisor selection Appropriateness Availability/accuracy/timeliness Rate selection Raw rates, expected rates, rate smoothing

3 rd editionwww.spatialanalysisonline.com15 Queries, Computations & Map Algebra Point density Density: counts/area (n/A) Occupancy: area/counts (A/n) Zone boundaries/area definition Grid counts (cell shape, size, orientation issues) … need for a boundary free approach … develop ideas from univariate statistics

3 rd editionwww.spatialanalysisonline.com16 Queries, Computations & Map Algebra Point density – kernel methods Uniform 50:50 Box kernel Box sum histogram

3 rd editionwww.spatialanalysisonline.com17 Queries, Computations & Map Algebra Kernel density – Normal kernel function

3 rd editionwww.spatialanalysisonline.com18 Queries, Computations & Map Algebra Kernel density – decisions Functions (finite or infinite) Bandwidth – key criterion (fixed/adaptive) Grid resolution Relative or absolute densities, or probabilities Comparison issues with other density datasets temporal mapped values

3 rd editionwww.spatialanalysisonline.com19 Queries, Computations & Map Algebra Kernel density estimation (KDE) – 2D approach Symmetric functions Finite (e.g. box) or infinite (e.g. Normal) Rotated 1D function 2D function Procedure: Select function and parameters Specify grid resolution (or extent and number of cols/rows) Apply 2D function to each data point and record value at every grid intersection Sum grid intersection values and normalise Map resulting grid

3 rd editionwww.spatialanalysisonline.com20 Queries, Computations & Map Algebra 2D kernels – single point Normal kernel (unbounded) Quartic kernel (bounded)

3 rd editionwww.spatialanalysisonline.com21 Queries, Computations & Map Algebra 2D mapped kernel density – Lung cancer cases, Quartic kernel and Normal kernel

3 rd editionwww.spatialanalysisonline.com22 Queries, Computations & Map Algebra 3D mapped kernel density – lung cancer cases

3 rd editionwww.spatialanalysisonline.com23 Queries, Computations & Map Algebra Kernel density – alternative functions Normal (or Gaussian), Quartic (spherical) (Negative) Exponential, Triangular (conic) Uniform (flat), Epanechnikov (paraboloid/quadratic) Applications Density/probability surface creation/comparisons Case/control analysis Spatio-temporal analysis Hot spot analysis Network analysis (see Okabe et al)

3 rd editionwww.spatialanalysisonline.com24 Queries, Computations & Map Algebra Cartograms Density adjustment – areas represent population

3 rd editionwww.spatialanalysisonline.com25 Queries, Computations & Map Algebra Cartograms Density adjustment – alternative approaches

3 rd editionwww.spatialanalysisonline.com26 Queries, Computations & Map Algebra Line and intersection density Kernel methods applied to linear forms Route systems (and intersections) Line frequency (number per unit area) Length density (length per unit area) Intersection density (intersections per unit of length, per unit area)