Transcript Spatial Statistics - Berry and Associates Spatial Information Systems

SpatialSTEM:
A Mathematical Structure for Understanding, Communicating and Teaching
Fundamental Concepts in Spatial Reasoning, Map Analysis and Modeling
Workshop Session — Spatial Statistics Operations
There is a “map-ematics” that extends traditional math/stat concepts
and procedures for the quantitative analysis of spatial data
This workshop provides a fresh perspective on interdisciplinary instruction
at the college level by combining the philosophy and approach of STEM
with the spatial reasoning and analytical power of grid-based Map Analysis and Modeling
This PowerPoint is posted at
www.innovativegis.com/basis/Courses/SpatialSTEM/
Presented by
Joseph K. Berry
Adjunct Faculty in Geosciences, Department of Geography, University of Denver
Adjunct Faculty in Natural Resources, Warner College of Natural Resources, Colorado State University
Principal, Berry & Associates // Spatial Information Systems
Email: [email protected] — Website: www.innovativegis.com/basis
SpatialSTEM Seminar Review (Session 1—Spatial Analysis operations)
“Technology Tool” vs. “Analysis Tool”
Geotechnology as 21st Century “mega-technology”
Spatial Triad (RS, GIS, GPS)
“Descriptive Mapping” vs. “Prescriptive Analysis”
Nature of Mapped Data
Vector vs. Raster (Grid)
Grid Map Layers and Map Stack
Analysis Frame
Overview Seminar
sSTEM_workshop1.ppt
Review
Seminar
Session 1
Spatial
Analysis
Grid Math
Hands-on
Exercise
Calculating
Simple and
Effective
Proximity
Viewsheds
Visual Exposure
Viewshed
Visual Exposure
and Noise Buffers
“Contextual”
Relationships
Hands-on
Exercise
Variable-width
Buffers
Viewsheds and
Visual Exposure
(Berry)
Overview of Map Analysis Approaches
(Spatial Analysis and Spatial Statistics)
Spatial Analysis
Traditional GIS
Elevation
(Surface)
Forest Inventory
Map
• Points, Lines, Polygons
• Cells, Surfaces
• Discrete Objects
• Continuous Geographic Space
• Contextual Spatial Relationships
• Mapping and Geo-query
…last session
Traditional Statistics
Spatial Statistics
Spatial
Distribution
(Surface)
Minimum= 5.4 ppm
Maximum= 103.0 ppm
Mean= 22.4 ppm
StDEV= 15.5
• Mean, StDev (Normal Curve)
• Map of Variance (gradient)
• Central Tendency
• Spatial Distribution
• Typical Response (scalar)
• Numerical Spatial Relationships
(Berry)
Spatial Data Perspectives (numerically defining the What in “Where is What”)
Numerical Data Perspective: how numbers are distributed in “Number Space”
 Qualitative: deals with unmeasurable qualities
(very few math/stat operations available)
– Nominal numbers are independent of each other and do not
imply ordering – like scattered pieces of wood on the ground
– Ordinal numbers imply a definite ordering from small to large
– like a ladder, however with varying spaces between rungs
 Quantitative: deals with measurable quantities
4
5
2
Nominal
—Categories
4
1
3
6
5
6
Ordinal
—Ordered
3
2
1
(a wealth of math/stat operations available)
– Interval numbers have a definite ordering and a constant step –
like a typical ladder with consistent spacing between rungs
– Ratio numbers has all the properties of interval numbers plus a
clear/constant definition of 0.0 – like a ladder with a fixed base.
6
5
4
3
2
1
Interval
—Constant
Step
Ratio
—Fixed
Zero
6
5
4
3
2
1
0
 Binary: a special type of number where the range is constrained to just two states— such as 1=forested, 0=non-forested
Spatial Data Perspective: how numbers are distributed in “Geographic Space”
 Choropleth numbers form sharp/unpredictable boundaries
in geographic space – e.g., a road “map”
Elevation
—Continuous gradient
Roads
—Discrete Groupings
 Isopleth numbers form continuous and often predictable
gradients in geographic space – e.g., an elevation “surface”
(Berry)
Desktop Mapping
(GeoExploration) vs.
Map Analysis (GeoScience)
“Maps are numbers first, pictures later” — “Quantitative analysis of mapped data”
Desktop Mapping graphically links generalized statistics to discrete spatial
objects (Points, Lines, Polygons)—spatially aggregated summaries (GeoExploration)
Desktop Mapping
Map Analysis
X, Y, Value
Data Space
Field
Data
Geographic Space
Standard Normal Curve
Point
Sampled
Data
(Numeric Distribution)
Average = 22.0
StDev = 18.7
(Geographic Distribution)
40.7 …not a problem
Discrete
Spatial Object
22.0
Spatially
Generalized
High Pocket
Continuous
Spatial Distribution
Spatially
Detailed
Discovery of sub-area…
Adjacent
Parcels
Map Analysis
map-ematically relates patterns within and among continuous spatial
distributions (Map Surfaces)— spatially disaggregated analysis (GeoScience)
(Berry)
Exercise 2-1)
Linking Data Space and Geographic Space (Calculate)
Data, Map and Stat Views …the following “hands-on” experience
“links” field data values with discrete point map values and nonspatial statistical metrics
Install MapCalc
…if you haven’t.
Calculate Command
Avg
Below
> 1SD above
Above
Continuous Surface Views …the following “hands-on”
experience “links” the spatial distribution of field data with its
non-spatial statistical metrics
(Berry)
Spatial Statistics Operations
(Numerical Context)
GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)
Grid Layer
Map Stack
Spatial Statistics seeks to map the spatial variation in a data set instead of focusing on
a single typical response (central tendency) ignoring the data’s spatial distribution/pattern,
and thereby provides a mathematical/statistical framework for analyzing and modeling the
Numerical Spatial Relationships
within and among grid map layers
Map Analysis Toolbox
(Berry)
Statistical Perspective:
Basic Descriptive Statistics (Min, Max, Median, Mean, StDev, etc.)
Basic Classification (Reclassify, Contouring, Normalization)
Map Comparison (Joint Coincidence, Statistical Tests)
Unique Map Statistics (Roving Window and Regional Summaries)
Surface Modeling (Density Analysis, Spatial Interpolation)
Advanced Classification (Map Similarity, Maximum Likelihood, Clustering)
Predictive Statistics (Map Correlation/Regression, Data Mining Engines)
Creating a Crime Risk Density Surface (Density Analysis)
Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.)
Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability)
Map Comparison — (Normalization, Joint Coincidence, Statistical Tests)
Unique Map Descriptive Statistics — (Roving Window and Regional Summaries)
Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization)
Geo-Coding
Crime
Incident
Reports
Vector to Raster
Crime
Incident
Locations
Roving Window
Grid
Incident
Counts
Density
Surface
Totals
Reclassify
Classified
Crime
Risk
Geo-coding identifies geographic
coordinates from street addresses
Grid Incident Counts
Calculates the total number of reported crimes
within a roving window– Density Surface Totals
the number of incidences
(points) within in each
grid cell
91
3D surface plot
2D display of discrete Grid Incident Counts
2D perspective display
of crime density contours
Classified Crime Risk Map
(Berry)
Spatial Interpolation
(iteratively smoothing the spatial variability)
Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.)
Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability)
Map Comparison — (Normalization, Joint Coincidence, Statistical Tests)
Unique Map Descriptive Statistics — (Roving Window and Regional Summaries)
Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization)
The “iterative smoothing” process is similar to slapping a big chunk of modeler’s clay over the “data
spikes,” then taking a knife and cutting away the excess to leave a continuous surface that
encapsulates the peaks and valleys implied in the original field samples
…repeated
smoothing slowly
“erodes” the data
surface to a flat
plane
= AVERAGE
(digital slide show SStat2)
(Berry)
Surface Modeling
Point Sampling:
Collecting X,Y coordinates with field
samples provides a foothold for
generating continuous map surfaces
used in map analysis and modeling.
(Spatial Interpolation)
Surface Modeling:
Surface Modeling techniques are used to derive a continuous Map Surface from
discrete Point Data. This process is analogous to placing a block of modeler's clay
over the Point Map’s relative value pillars and smoothing away the excess clay to
create a continuous map surface that fills-in the non-sampled locations, thereby
characterizing the data set’s Geographic Distribution.
Each record contains X,Y coordinates
(Where) followed by Data Values (What)
identifying the characteristics/conditions
at that location— a geo-registered dB
29.4
Window Configuration
…size, shape and
weighting procedure
For example, Inverse Distance
Weighted (IDW) spatial
interpolation calculates the
distances from an non-sampled
location to all sample locations
and then uses the inverse of the distance to weight-average, such that nearby
sample values influence the average more than distant sample values—
repeating the procedure for all locations results in a continuous map surface of
the spatial variation in the data set, provided there is sufficient—
Spatial Autocorrelation
meaning that “what occurs at a location is
related to the conditions at nearby locations” (intra-variable dependence)
Exercise 2-2)
Surface Modeling
(Density Analysis and IDW Interpolation)
Customer Density
Unusually High Density
Density Analysis …the following “hands-on” experience creares a
density surface (total customers within 6 cells) and isolates pockets
of unusually high density
Period2
Period2
Big Increase
Spatial Interpolation (IDW) …the following “hands-on”
experience demonstrates spatially interpolating a discrete
point map into a continuous map surface that characterizes
the spatial distribution of the data
(Berry)
Spatial Statistics Operations
Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.)
Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability)
Map Comparison — (Normalization, Joint Coincidence, Statistical Tests)
Unique Map Descriptive Statistics — (Roving Window and Regional Summaries)
Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization)
Advanced Classification — (Map Similarity, Maximum Likelihood, Clustering)
The farthest away point in data space
(Least Similar) is set to 0 and locations
identical to the Comparison Point are set
to 100 % similar (same numerical pattern)
…all other Data Distances are
scaled in terms of their relative
similarity to the comparison
point (0 to 100% similar)
Map Stack
…there are two types of Spatial Dependency—
Spatial Autocorrelation— within a map layer
Spatial Correlation— among map layers
Spatial Correlation
meaning that “what occurs at a location
is related to the conditions of other map variables at that location”
(inter-variable dependence)
(Berry)
Exercise 2-3)
Characterizing Map Similarity (Home Age, Value and Housing Density)
RELATE command
Map Similarity …the following
“hands-on” experience calculates a
similarity map for a location based
on the relative comparison of data
patterns for each grid cell in a map
stack to that of a comparison data
pattern
Least
Similarity
Surface
(Berry)
Predictive Spatial Statistics
Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.)
Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability)
Map Comparison — (Normalization, Joint Coincidence, Statistical Tests)
Unique Map Descriptive Statistics — (Roving Window and Regional
Summaries)
Spatial DBMS export grid
…pass map layers to
any Statistics or
to dB with each cell a
Surface Modeling — (Density Analysis, Spatial Interpolation, Maplayers
Generalization)
Data Mining package
record & each layer a field
Advanced Classification — (Map Similarity, Maximum Likelihood, Clustering)
Predictive Statistics — (Map Correlation/Regression, Data Mining Engines)
Dependent Map variable
is what you want to predict…
Univariate Linear Regressions
Multivariate Linear Regression
Predicted – Actual = Error
…substantially under-estimates
(2/3 of the error within 5.26 and 16.94)
…can use error to generate Error Ranges
for calculating new regression equations
…from a set of easily measured
Independent Map variables
Actual
Predicted
Error Surface
Stratified Error
(Berry)
Exercise 2-4)
Building Predictive Models (map regression)
Loan Accounts
Loan Concentration
Deriving a Dependent Variable Map …density analysis is
used to derive a map surface indicating loan concentration
Univariate Map Correlation and Regression
…coincidence of dependent and independent map layers
(one at a time)
Multivariate Map Regression
…coincidence of dependent and
independent map layers (all at once)
Multivariate Regression
Univariate Regression
(Berry)
Example “Hands-on” Exercises (Session 3 homework)
www.innovativegis.com/basis/Senarios/
Identifying Campground Suitability
…preferences for locating a campground—
• gentle slopes
• near roads
• near water
• good views of water
• westerly oriented.
Identifying Similar Data Patterns
… creates a map of
“data groupings” that are
as “similar as possible
within the group and “as
dissimilar as possible”
among the groupings
given a set of map layers
(spatial clustering).
(Berry)