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Grid-based Map Analysis (Spatial Analysis/Statistics)
Traditional GIS
Spatial Analysis
Erosion
Potential
(Surface)
Forest Inventory
Map
• Points, Lines, Polygons
• Cells, Surfaces
• Discrete Objects
• Continuous Geographic Space
• Mapping and Geo-query
• Contextual Spatial Relationships
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 the Variance (gradient)
• Central Tendency
• Spatial Distribution
• Typical Response (scalar)
• Numerical Spatial Relationships
Grid-Based Map Analysis
Spatial analysis investigates the “contextual” relationships in mapped data…
 Reclassify— reassigning map values (position; value; size, shape; contiguity)
 Overlay— map overlay (point-by-point; region-wide; map-wide)
 Distance— proximity and connectivity (movement; optimal paths; visibility)
 Neighbors— ”roving windows” (slope/aspect; diversity; anomaly)
Spatial Statistics
Surface modeling maps the “spatial distribution” and pattern of point data…
 Map Generalization— characterizes spatial trends (e.g., titled plane)
 Spatial Interpolation— deriving spatial distributions (e.g., IDW, Krig)
 Other— roving window/facets (e.g., density surface; tessellation)
Data Mining investigates the “numerical” relationships in mapped data…
 Descriptive— aggregate statistics (e.g., average/stdev, similarity, clustering)
 Predictive— relationships among maps (e.g., regression)
 Prescriptive— appropriate actions (e.g., optimization)
(Berry)
Point Density Analysis
Point Density analysis identifies the number of customers with
a specified distance of each grid location
Roving Window (count)
(Berry)
Identifying Unusually High Density
Pockets of unusually high customer density are identified as more
than one standard deviation above the mean
(Berry)
Surface Modeling (Density Surface)
Hugag Density Surface
Roving Window
Total number of counts within 6-cell radius
Hugag
Counts
Avg- 17.5
StDev= 15.0
Hugag
2 Hugags
every 30 min
for 30 days
Discrete
Map Surface
Hugag Activity draped over Elevation
(Short Exercise #6)
Density Surface Modeling
“Counts” the number of occurrences within a
specified “roving window” reach— higher values
indicate concentrations of occurrence
Continuous
Map Surface
Most of the
activity is in the NE
…from discrete observations to continuous spatial distribution
(Berry)
Spatial Interpolation (Smoothing the Variability)
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)
Inverse Distance Weighted Approach
Tobler’s First Law of Geography— nearby things are more alike than distant things
1/DPower
(Berry)
Spatial Autocorrelation (Kriging)
Tobler’s First Law of Geography— nearby things are more alike than distant things
Variogram— plot of sample data similarity as a function of distance between samples
…Kriging uses regional variable theory based on an underlying variogram to develop
custom weights based on trends in the sample data (proximity and direction)
…uses Variogram Equation instead of a fixed 1/DPower Geometric Equation
(Berry)
Surface Modeling Methods (Surfer)
Spatial Interpolation
Inverse Distance to a Power— weighted average of samples in the summary
window such that the influence of a sample point declines with “simple” distance
Modified Shepard’s Method— uses an inverse distance “least squares” method that
reduces the “bull’s-eye” effect around sample points
Radial Basis Function— uses non-linear functions of “simple” distance to determine
summary weights
Kriging— summary of samples based on distance and angular trends in the data
Natural Neighbor—weighted average of neighboring samples where the weights are
proportional to the “borrowed area” from the surrounding points (based on differences
in Thiessen polygon sets)
Minimum Curvature— analogous to fitting a thin, elastic plate through each sample
point using a minimum amount of bending (Spatial Interpolation)
Geometric facets
Nearest Neighbor— assigns the value of the nearest sample point
Triangulation— identifies the “optimal” set of triangles connecting
all of the sample points (Geometric Facets)
Map Generalization
Polynomial Regression— fits an equation to the entire set
of sample points (Map Generalization)
Thiessen Polygons
(Berry)
Spatial Interpolation
Spatial
Interpolation is
similar to
throwing a
blanket over the
“data spikes” to
conforming to the
geographic
pattern of the
data.
…all interpolation algorithms assume that…
1) “nearby things are more alike than distant things” (spatial autocorrelation),
2) appropriate sampling intensity (ample number of samples), and a
3) suitable sampling pattern
…the interpolated surfaces “map the spatial variation” in the data samples
(Berry)
Comparing Spatial Interpolation Results
Comparison of the IDW
interpolated surface to the
whole field average shows
LARGE differences in
localized estimates
Comparison of the IDW
and Krig interpolated
surfaces shows small
differences in in localized
estimates
(Berry)
Surface Modeling (Full Exercise #6)
Spatial Interpolation
Use Surfer to interpolate a continuous surface…
…and generate contour and solid surface plots
Density Surface Derivation (Use MapCalc to derive a customer density surface)
 SCAN Total_Customers TOTAL WITHIN 6 FOR Customer_density6
 RENUMBER Customer_density6 ASSIGNING 0 TO 0 THRU 33.7
ASSIGNING 1 TO 33.7 THRU 1000 FOR Customer_highDensity
Use MapCalc to create a density surface (total count)
(Berry)
Spatial Interpolation Techniques
Characterizes the spatial distribution by fitting a mathematical
equation to localized portions of the data (roving window)
Spatial Interpolation techniques use “roving windows” to
summarize sample values within a specified reach of each map
location. Window shape/size and summary technique result in
different interpolation surfaces for a given set of field data
…no single techniques is best for all data.
AVG= 23 everywhere
Inverse Distance Weighted (IDW)
technique weights the samples
such that values farther away
contribute less to the average
…1/Distance Power
(Berry)
Spatial Interpolation (Evaluating performance)
Assessing Interpolation Results (Residual Analysis)
(Berry)
…the best map is the
one that has the “best
guesses”
AVG= 23
Spatial Interpolation (Spatially characterizing error)
A Map of Error (Residual Map)
…shows you where your estimates are likely good/bad
(Berry)
Spatial Dependency (Spatial Autocorrelation & Correlation)
Spatial Variable Dependence — what occurs at a location
in geographic space is related to:
• the conditions of that variable at nearby locations, termed
Spatial Autocorrelation (intra-variable dependence for Surface Modeling)
Basis for…
…understanding relationships
within a single map layer
Surface
Modeling
• the conditions of other variables at
that location, termed Spatial Correlation
(inter-variable dependence for Spatial Data Mining)
Basis for…
Spatial Data
Mining
…understanding relationships
among map layers
(Berry)
Grid-Based Map Analysis
Spatial analysis investigates the “contextual” relationships in mapped data…
 Reclassify— reassigning map values (position; value; size, shape; contiguity)
 Overlay— map overlay (point-by-point; region-wide; map-wide)
 Distance— proximity and connectivity (movement; optimal paths; visibility)
 Neighbors— ”roving windows” (slope/aspect; diversity; anomaly)
Spatial Statistics
Surface modeling maps the “spatial distribution” and pattern of point data…
 Map Generalization— characterizes spatial trends (e.g., titled plane)
 Spatial Interpolation— deriving spatial distributions (e.g., IDW, Krig)
 Other— roving window/facets (e.g., density surface; tessellation)
Data Mining investigates the “numerical” relationships in mapped data…
 Descriptive— aggregate statistics (e.g., average/stdev, similarity, clustering)
 Predictive— relationships among maps (e.g., regression)
 Prescriptive— appropriate actions (e.g., optimization)
(Berry)
Visualizing Spatial Relationships
Interpolated Spatial Distribution
Phosphorous (P)
What spatial
relationships do you
see?
…do relatively high levels
of P often occur with high
levels of K and N?
…how often?
…where?
(Berry)
Identifying Unusually High Measurements
…isolate areas with mean + 1 StDev (tail of normal curve)
(Berry)
Level Slicing
…simply multiply the two maps to identify joint coincidence
1*1=1 coincidence (any 0 results in zero)
(Berry)
Multivariate Data Space
…sum of a binary progression (1, 2 ,4 8, 16, etc.) provides
level slice solutions for many map layers
(Berry)
Calculating Data Distance
…an n-dimensional plot depicts the multivariate distribution—
the distance between points determines the relative similarity in data patterns
…the closest floating ball is the least similar (largest data distance) from the comparison point
(Berry)
Identifying Map Similarity
…the relative data distance between the comparison point’s data pattern
and those of all other map locations form a Similarity Index
…the green tones indicate field locations with fairly similar P, K and N levels; red tones indicate dissimilar areas
(Berry)
Clustering Maps for Data Zones
…a map stack is a spatially organized set of numbers
…groups of “floating balls” in data space
identify locations in the field with similar data
patterns– data zones
…fertilization rates vary for the
different clusters “on-the-fly”
(Cyber-Farmer, Circa 1990)
Variable Rate Application
(Berry)
Evaluating Clustering Results
…graphical and statistics procedures
assess how “distinct” clusters are—
Clustering Performance
…distinct in
K and N
(less), but
not distinct
in P
(Berry)
Spatial Data Mining (Full Exercise #7)

Spatial statistics …
Similarity Map
use MapCalc to implement derive
relationships among P, K, N and
Yield in a farmer’s field
Cluster Map
(Short Exercise #7)
Regional Average
Composite
Descriptive
Prescriptive
Scatter Plot
Univariate Regression
Multivariate Regression
(Berry)
The Precision Ag Process (Fertility example)
As a combine moves through a field it 1) uses GPS to check its location then
2) checks the yield at that location to 3) create a continuous map of the
yield variation every few feet. This map is
Steps 1) – 3)
4) combined with soil, terrain and other maps to
derive 5) a “Prescription Map” that is used to
6) adjust fertilization levels every few feet
in the field (variable rate application).
On-the-Fly
Yield Map
Step 4)
Map Analysis
Farm dB
Zone 3
Cyber-Farmer, Circa 1992
Zone 2
Zone 1
Prescription Map
Variable Rate Application
Step 5)
Step 6)
(Berry)
Spatial Data Mining
Precision Farming is just one example of applying spatial
statistics and data mining techniques
Mapped data that
exhibits high spatial
dependency create
strong prediction
functions. As in
traditional statistical
analysis, spatial
relationships can be
used to predict
outcomes…
Geo-business SDM
…the difference is
that spatial statistics
predicts where
responses will be
high or low
(Berry)
Data Analysis Perspectives (Review)
Traditional Analysis
Map Analysis
(Data Space — Non-spatial Statistics)
(Geographic Space — Spatial Statistics)
Field Data
Standard Normal Curve
fit to the data
Spatially
Interpolated data
Central Tendency
Typical
How Typical
22.0
28.2
Average = 22.0
StDev = 18.7
Discrete
Spatial Object
Continuous
Spatial Distribution
(Generalized)
(Detailed)
Identifies the Central Tendency
Maps the Variance
(Berry)
Grid-Based Map Analysis (Review)
Spatial Analysis investigates the “contextual” relationships in mapped data…
 Reclassify— reassigning map values (position; value; size, shape; contiguity)
 Overlay— map overlay (point-by-point; region-wide; map-wide)
 Distance— proximity and connectivity (movement; optimal paths; visibility)
 Neighbors— ”roving windows” (slope/aspect; diversity; anomaly)
Spatial Analysis
Spatial Statistics
Surface Modeling maps the “spatial distribution” and pattern of point data…
 Map Generalization— characterizes spatial trends (e.g., titled plane)
 Spatial Interpolation— deriving spatial distribution (e.g., IDW, Krig)
 Other— roving window (e.g., density surface; tessellation)
Data Mining investigates the “numerical” relationships in mapped data…
 Descriptive— aggregate statistics (e.g., average/stdev, similarity)
 Predictive— relationships among maps (e.g., regression)
 Prescriptive— appropriate actions (e.g., optimization)
(Berry)
More Map Analysis Experience (MapCalc & Surfer)
Tutorial Exercises
Workshop Exercises
MapCalc Tutorial
Short & Full
Exercise Sets
Surfer Tutorial
Workshop CD
See Default.htm
Complete Experience
NEW BOOK — see the description of the Map Analysis book (Berry, 2007;
GeoTec Media) at… www.innovativegis.com/basis
…develops a structured view of the important concepts, considerations and
procedures involved in grid-based map analysis.
…the companion CD contains further readings and software for hands-on
experience with the material presented.
(Berry)
…but before we leave Spatial Statistics (Surface Modeling and Spatial
Data Mining—descriptive, predictive and prescriptive) to tackle GIS
Modeling, any…
Questions?
Questions?
(Berry)