Www.spatialanalysisonline.com Chapter 4 Part C: Queries, Computations & Map Algebra.

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Chapter 4

Part C: Queries, Computations & Map Algebra

3rd edition www.spatialanalysisonline.com 2

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)


Method Description

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

Equals spatially equal to: a=b

Disjoint spatial disjoint: equivalent to ab=

Intersects spatially intersects: [ab] is equivalent to [not a disjoint(b)]

Touches spatially touches: equivalent to [ab 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 aba and abb]

Within spatially within: within(b) is equivalent to [ab=a and I(a)I(b)]

Contains spatially 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 aba and abb]

Relate spatially relates, tested by checking for intersections between the interior, boundary and exterior of the two components

OGC OpenGIS Simple Features Specification: Spatial Relations




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)



Map algebra – local operations, multi-grid

A

B

C'

New rast er

b. G r id resampl ing

A

B

C

New rast er

a. Mat ching gr ids



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



Grid filtering – Linear (weighted average)

Low pass filters, e.g.

High pass filters, e.g.

Kernels - example

aaa

aba

aaa

aaa

aba

aaa

111

141

111

175170124

18818790

20017698

175170124

18816490

20017698 Weighted average as integer



Grid filtering – Linear – common ‘image processing’ functions Sharpening Blurring Edge detection Embossing Gradient operations



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



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



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



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



Standardisation Direct standardisation

Regional/national comparisons Indirect standardisation

Expected values Excess rates

Statistical standardisation Z-scores Range-based (basic or trimmed)



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



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


Queries, Computations & Map AlgebraPoint density – kernel methods

Uniform

50:50

Box kernel

Box sum

Box sum

histogram



Kernel density – Normal kernel function



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



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



2D kernels – single point

Normal kernel (unbounded) Quartic kernel (bounded)



2D mapped kernel density – Lung cancer cases, Quartic kernel and Normal kernel



3D mapped kernel density – lung cancer cases



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)



Cartograms Density adjustment – areas represent population



Cartograms Density adjustment – alternative approaches



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)

Www.spatialanalysisonline.com Chapter 4 Part C: Queries, Computations & Map Algebra.

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