1. Joseph K. Berry, SpatialSTEM
A Mathematical/Statistical Framework for
Understanding and Communicating Map Analysis and Modeling
Premise: There is a “map-ematics” that extends traditional math/stat concepts
and procedures for the quantitative analysis of map variables (spatial data) 
Premise: There is a “map-ematics” that extends traditional math/stat concepts
and procedures for the quantitative analysis of map variables (spatial data) 
This presentation 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
The PowerPoint slide sets with notes supporting this workshop is posted online at—
Overview (.5 hour). This presentation is open to all GIS and non-GIS personnel interested in an overview of map analysis and modeling that should be of general interest to all NGA analysts, GIS specialists, scientists, management and staff interested in better understanding the potential and pitfalls of quantitative map analysis. The presentation thesis is that there is a mathematical structure for understanding and communicating map analysis operations that provides a common foothold in understanding the concepts, procedures and considerations that resonate with both non-GIS and GIS communities. Similarly, the final session on Future Directions are appropriate for general audiences.
Spatial Analysis (1.5 hours). Modern digital maps are “numbers first, pictures later.” In map-ematical processing, these data can be conceptualized as a set of "floating maps" with a common registration that enables the computer to "look" down and across the stack of map layers to spear or corral sets of numbers for processing. Spatial Analysis involves quantitative analysis of the “spatial context” of mapped data, such as add, subtract, multiply, divide, exponentiation, root, log, cosine, differentiate and even integrate maps. In addition, the spatial coincidence and juxtaposition of values among and within map layers create new mathematical operations, such as effective distance, optimal path routing, visual exposure density and landscape diversity, shape and pattern.
Spatial Statistics (1.5 hour). Spatial Statistics involves quantitative analysis of the “numerical context” of mapped data, such as characterizing the geographic distribution, relative comparisons, map similarity or correlation within and among data layers. Spatial Analysis and Spatial Statistics form a map-ematics that uses sequential processing of analytical operators to develop complex map analyses and models. Its approach is similar to traditional statistics except the variables are entire sets of geo-registered mapped data.
Future Directions (.5 hour). Most GIS technology has deep roots in manual mapping and geo-query procedures involving discrete spatial objects— continuous mapped data promises a future that moves well beyond mapping. The current cycle of innovation is focused on hexagonal/dodecahedral grid representation and implementation of a latitude/longitude-based universal spatial database key which are poised to change how we conceptualize, visualize, process and analyze spatial data.
This PowerPoint with notes and online links to further reading is posted at
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
— Website:
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2. (Nanotechnology) Geotechnology (Biotechnology)
Joseph K. Berry, SpatialSTEM
Geotechnology is one of the three "mega technologies" for the 21st century and promises to forever change how we conceptualize, utilize and visualize
spatial relationships in scientific research and commercial applications (U.S. Department of Labor)
Geographic Information
Systems (map and analyze)
Remote Sensing
(measure and classify)
Global Positioning
System (location and navigation)
GPS/GIS/RS
The Spatial Triad
(Top) The top portion of the figure relates Geotechnology to “spatial information” in a broad stroke similar to Biotechnology’s use of “biological systems” and Nanotechnology’s use of “control of matter.”
(Middle) The middle portion identifies the three related technologies for mapping and analyzing spatial information on the surface of the earth— GPS, GIS and RS, known as the “spatial triad.”
(Bottom) The bottom portion identifies the two dominant application arenas that emphasize Descriptive Mapping (Where is What) and Prescriptive Modeling (Why , So What and What If) that focuses on understanding spatial patterns and relationships.
What is most important to keep in mind is that Geotechnology, like Bio- and Nanotechnology, is greater than the sum of its parts—GPS, GIS and RS. While these individual mapping technologies provide the enabling capabilities, it is the application environments themselves that propel geotechnology to mega status.
The SpatialSTEM approach for teaching map analysis and modeling fundamentals within a mathematical/statistical context provides a quantitative framework for spatial reasoning that resonates with science, technology, engineering and math/stat communities. The premise is that “maps are numbers first and foremost, pictures later” and we do mathematical things to mapped data for insight and better understanding of spatial patterns and relationships within decision-making contexts.
Slide 2, Geotechnology – Overview of Spatial Analysis and Statistics; Is it Soup Yet? ; What’s in a Name?; Melding the Minds of the “-ists” and “-ologists”
Computer Mapping (70s)
Spatial Database Management (80s)
Map Analysis (90s)
Multimedia Mapping (00s) is
Where What
Analytical Tool
Technological Tool
Mapping involves precise placement (delineation) of physical features
(graphical inventory)
Descriptive
Mapping
Modeling involves analysis of spatial
patterns and relationships
(map analysis/modeling)
Prescriptive
Modeling
Why So What and What If
(Berry)
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3. Joseph K. Berry, SpatialSTEM
A Mathematical Structure for Map Analysis/Modeling
Joseph K. Berry, SpatialSTEM
Geotechnology  RS – GIS – GPS
Mapping/Geo-Query (Discrete, Spatial Objects) (Continuous, Map Surfaces) Map Analysis/Modeling
Technological Tool
Analytical Tool
Maps as Data, not Pictures
Vector & Raster — Aggregated & Disaggregated
Qualitative & Quantitative
“Map-ematics”
…organized set of numbers
Map Stack
Geo-registered
Analysis Frame
 Matrix
of Numbers
Spatial Statistics Operations
Spatial Analysis Operations
Reclassify and Overlay
Distance and Neighbors
GISer’s Perspective:
Surface Modeling
Spatial Data Mining
Grid-based
Map Analysis
Toolbox
(Top) Of the spatial triad, the GIS component contains most of the spatial analysis and statistics operations for investigating the relationships within and among mapped data. While some of the basic analytics can be accomplished in vector-based software, most Map Analysis/Modeling involves grid-based software due to 1) the continuous nature of the data (map surfaces instead of discrete irregular spatial objects defined as separate point, line or polygon features), 2) the inherent consistency of the analysis frame, 3) the implicit topology relating all of the grid cells, and 4) the advanced mathematical/statistical operations available.
(Middle) Grid-based Spatial Analysis is analogous to traditional mathematics as it uses “cyclical processing of nested parentheticals.” Equation variables are sequentially defined (map layers retrieved), processed by an analytical algorithm (e.g., subtract one map layer from another on a cell-by-cell basis), store the intermediate solution (e.g., difference map) in the map stack. The cyclical process is continued (e.g., , retrieve divide store solution; retrieve multiply times100 store solution) to evaluate a “spatial equation,” such as the percent difference equation of %Diff= ((old – new) / old) …however in map algebra, the variables represent entire map layers defined by thousands of spatially organized numbers (geo-registered matrices) with the answer forming a map of the percent difference identifying where differences are high and where they are low throughout a project area.
Whereas Spatial Analysis operations are extensions of traditional scalar mathematics analyzing “geographic context,” Spatial Statistics extends traditional scalar statistics by analyzing the “numerical context” within and among map layers. The procedures account for the “spatial distribution” of map variables , as well as the “numerical distribution” used in central tendency analysis seeking to characterize the typical value– spatial statistics seeks to account for the spatial variation, such as generating cluster maps that identify where groupings of similar data patterns occur.
(Bottom) The SpatialSTEM approach and framework cross-references a wealth of existing grid-based map analysis operations used in GIS and RS (most available since the 1970s) with the seven classes of fundamental mathematical operations and the seven classes of fundamental statistical operations as shown in the lower portion of the slide. It is hoped that this recasting of map analysis/modeling procedures into the common framework/language of quantitative data analysis helps STEM disciplines better understand modern digital “maps as data, not just pictures” and the existence of a “map-ematics” that resonates with and extends non-spatial analysis and modeling techniques they currently employ– “thinking with maps” through quantitative spatial reasoning.
Slide 3, A Mathematical Structure for Map Analysis/Modeling – Moving Mapping to Map Analysis; Use Map-ematical Framework for GIS Modeling; Getting the Numbers Right
Mathematician’s Perspective:
Basic GridMath & Map Algebra
Advanced GridMath
Map Calculus
Map Geometry
Plane Geometry Connectivity
Solid Geometry Connectivity
Unique Map Analytics
Statistician’s Perspective:
Basic Descriptive Statistics
Basic Classification
Map Comparison
Unique Map Statistics
Surface Modeling
Advanced Classification
Predictive Statistics
The SpatialSTEM Framework
Traditional math/stat procedures can be extended into geographic space to stimulate those with diverse backgrounds and interests for…
“thinking analytically
with maps”
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4. Joseph K. Berry, SpatialSTEM
Spatial Analysis Operations (Geographic Context)
GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)
Spatial Analysis
Map Stack
Grid Layer
Spatial Analysis extends the basic set of discrete map features (points, lines and polygons) to
map surfaces that represent continuous geographic space as a set of contiguous grid cells (matrix),
thereby providing a Mathematical Framework for map analysis and modeling of the
Contextual Spatial Relationships within and among grid map layers
Map Analysis Toolbox
Basic GridMath & Map Algebra ( * / )
Advanced GridMath (Math, Trig, Logical Functions)
Map Calculus (Spatial Derivative, Spatial Integral)
Map Geometry (Euclidian Proximity, Effective Proximity, Narrowness)
Plane Geometry Connectivity (Optimal Path, Optimal Path Density)
Solid Geometry Connectivity (Viewshed, Visual Exposure)
Unique Map Analytics (Contiguity, Size/Shape/Integrity, Masking, Profile)
Mathematical Perspective:
Unique spatial
operations
(Top) There are two broad types of digital maps– Vector and Raster. Mapping and geo-query use a “vector” data structure that is akin to manual mapping in which discrete spatial objects (vector points, lines and polygons) form a collection of independent, irregular features to characterize geographic space (vis., a jigsaw puzzle). Grid-based map analysis and modeling procedures, on the other hand, operate on continuous map variables (raster map surfaces) composed of thousands of map values stored in geo- registered matrices.
Spatial analysis can be thought of as an extension of traditional mathematics involving the “contextual” relationships within and among mapped data layers. It focuses on geographic associations and connections, such as relative positioning, configurations and patterns among map locations.
From a GIS Perspective, the spatial analysis operators can be classified as Focal, Local and Zonal (Tomlin) based on how the algorithms work or by Reclassify, Overlay, Distance and Neighbors (Berry) based on the spatial information generated.
(Bottom) The basic spatial analysis operations can be categorized to align with traditional mathematical concepts and operations. The first three groupings are associated with basic math, algebra and calculus, the middle three involve spatial expressions of distance, proximity, movement and connectivity, with the last category identifying unique spatial operators that do not have corollaries in traditional mathematics (red check).
Slide 4, Spatial Analysis Operations (Geographic Context) – Simultaneously Trivializing and Complicating GIS; SpatialSTEM Has Deep Mathematical Roots; Understanding Grid-based Data; Suitability Modeling
(Berry)
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5. Joseph K. Berry, SpatialSTEM
Spatial Analysis Operations (Math Examples)
Joseph K. Berry, SpatialSTEM
Map Calculus — Spatial Derivative, Spatial Integral
Advanced Grid Math — Math, Trig, Logical Functions
Slope draped over
MapSurface 0%
65%
Spatial Derivative
…is equivalent to the slope of the tangent plane at a location
SLOPE MapSurface Fitted
FOR MapSurface_slope
Fitted Plane
Surface
500’
2500’
Curve
The derivative is the instantaneous “rate of change” of a function and is equivalent to the slope of the tangent line at a point
Dzxy Elevation
Advanced Grid Math
Surface Area
…increases with increasing inclination as a Trig function of the cosine of
the slope
angle
S_Area=
Fn(Slope)
ʃ Districts_Average Elevation
Spatial Integral
Surface
COMPOSITE Districts WITH MapSurface
Average FOR MapSurface_Davg
MapSurface_Davg
…summarizes the values on a surface for specified map areas (Total= volume under the surface)
(Top) Grid Math and Map Algebra can be expanded into other mathematical arenas. The Calculus Derivative traditionally identifies a measure of how a mathematical function changes as its input changes by assessing the slope along a curve in 2-dimensional abstract space— calculated as the slope of the tangent line at any location along the curve. The spatial equivalent calculates a “slope map” depicting the rate of change in a continuous map variable in 3-dimensional geographic space— calculated as the slope of a “best fitted plane” at any location on the map surface.
(Middle) Advanced Grid Math includes most of the buttons on a scientific calculator to include trigonometric functions. For example, taking the cosine of a slope map of an elevation surface expressed in degrees and multiplying it times the planimetric surface area of a grid cell calculates the increased real-word surface area of the “inclined plane” at each grid location.
(Bottom) The Calculus Integral is identified as the area of a region under a curve expressing a mathematical function. The spatial integral counterpart summarizes map surface values within specified regions (volume under the surface). The data summaries are not limited to just a total but can be extended to most statistical metrics. For example, the average map surface value can be calculated for each district in a project area. Similarly, the coefficient of variation ((Stdev / Average) * 100) can be calculated to assess data dispersion about the average.
 Slide 5, Spatial Analysis Operations (Math Examples) – Map-ematically Messing with Mapped Data; Characterizing Micro-terrain Features; Reclassifying and Overlaying Maps ; Use Map-ematical Framework for GIS Modeling
S_area= cellsize / cos(Dzxy Elevation)
The integral calculates the area under the curve for any section of a function.
Curve
(Berry)
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6. Joseph K. Berry, SpatialSTEM
Spatial Analysis Operations (Distance Examples)
Joseph K. Berry, SpatialSTEM
Map Geometry — (Euclidian Proximity, Effective Proximity, Narrowness)
Plane Geometry Connectivity — (Optimal Path, Optimal Path Density)
Solid Geometry Connectivity — (Viewshed, Visual Exposure)
On + Off Road
96.0 minutes
…farthest away by truck, ATV and hiking
Off Road
Relative Barriers
Distance
Shortest straight line between two points…
Euclidean Proximity
…from a point to
everywhere…
Travel-Time
Surface
Effective Proximity
…not necessarily straight lines (movement)
HQ (start)
On Road
26.5 minutes
…farthest away
by truck
Off Road
Absolute Barrier
Plane Geometry
Connectivity
…like a raindrop, the
“steepest downhill
path” identifies the optimal route
(Quickest Path)
Farthest
(end) HQ
(start)
Truck = 18.8 min
ATV = 14.8 min
Hiking = 62.4 min
Seen if new tangent exceeds
all previous tangents
along the line of sight
Tan = Rise/Run
Rise
Run
Viewshed
Splash
Solid Geometry Connectivity
(Top) Traditional geometry defines Distance as “the shortest straight line between two points” and routinely measures it using a ruler or calculates it using the Pythagorean Theorem. Map Geometry extends the concept of distance to simple Euclidean Proximity by relaxing the requirement of just “two points” for distances to all locations surrounding a point or other map feature, such as a road. Concentric equidistance zones are established around a location or set of locations (“Splash” algorithm). This procedure is similar to the wave pattern generated when a rock is thrown into a still pond. Each ring indicates one “unit farther away”— increasing distance as the wave moves away. A further extension involves Effective Proximity, which relaxes “straight line” to consider intervening absolute and relative barriers to movement. For example, movement by truck might be constrained to roads (absolute barrier) but off-road travel by ATV or foot is affected by terrain conditions expressed as impedance to travel (relative barriers). The result is that the “shortest but not necessarily straight distance” is assigned to each grid location.
(Middle) Because a straight line connection cannot be assumed, optimal path routines in Plane Geometry Connectivity (2D space) are needed to identify the actual shortest route as the “steepest downhill path over the travel-time surface”– the twisting/turning route the wave-front propagation took to get there first. In Least Cost Path routing, the optimal path identifies the minimal cost (most preferred) route between two points considering discrete cost assigned to each intervening map location.
(Bottom) Solid Geometry Connectivity (3D space) involves line-of-sight connections that identify visual exposure among locations. The Viewshed algorithm uses a series of expanding rings (splash algorithm) to determine relative elevation differences from the viewer position to all other map locations. The ratio of the elevation difference (Rise) indicated as striped boxes to the distance away (Run) is used to determine visual connectivity (rise/run= tangent). Whenever the ratio exceeds the previous tangent ratio, the location is marked as seen (red); when it fails it is marked as not seen (grey). A Visual Exposure density surface is generated by noting the number of times each grid location is seen from a set of viewer locations. A Weighted Visual Exposure surface is generated by summing the weights assigned to each viewer location as an indicator of its relative importance.
Slide 6, Spatial Analysis Operations (Distance Examples) – Bending Our Understanding of Distance; Calculating Effective Distance and Connectivity; E911 for the Backcountry; Routing and Optimal Paths; Deriving and Using Travel-Time Maps; Deriving and Using Visual Exposure Maps; Creating Variable-Width Buffers; Applying Surface Analysis
Counts
# Viewers
270/621= 43% of the entire road network is connected
Visual Exposure
Highest
Weighted
Exposure
Sums
Viewer
Weights 
(Berry)
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7. Joseph K. Berry, SpatialSTEM
Spatial Statistics Operations (Numeric Context)
GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)
Spatial Statistics
Map Stack
Grid Layer
Spatial Statistics seeks to map the variation in a data set instead of focusing on
a single typical response (central tendency), thereby providing
a Statistical Framework for map analysis and modeling of the
Numerical Spatial Relationships within and among grid map layers
Map Analysis Toolbox
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)
Statistical Perspective:
Unique spatial
operations
(Top) There are two broad types of digital maps– Vector and Raster. Mapping and geo-query use a vector data structure that is akin to manual mapping in which discrete spatial objects (vector points, lines and polygons) form a collection of independent, irregular features to characterize geographic space (viz., a jigsaw puzzle). Grid-based map analysis and modeling procedures, however, operate on continuous map variables (raster map surfaces) composed of thousands of map values stored in geo-registered matrices.
Spatial statistics can be thought of as an extension of traditional statistics involving the “numerical” relationships within and among mapped data layers. It focuses on 1) mapping the variation inherent in a data set rather than characterizing its central tendency and 2) summarizing the coincidence and correlation of the spatial distributions.
From a GIS Perspective, two dominant classes of spatial statistics are often identified— Surface Modeling that derives a continuous spatial distribution of a map variable from point sampled data and Spatial Data Mining that investigates numerical relationships among map variables.
(Bottom) The basic spatial statistic operations can be categorized to align with traditional non-spatial statistical concepts and operations. The first three groupings are associated with general descriptive statistics, the middle two involve unique spatial statistics operations that do not have corollaries in traditional statistics (red checks) and the final two identify classification and predictive statistics procedures.
 Slide 7, Spatial Statistics Operations (Numeric Context) – Infusing Spatial Character into Statistics; Paint by Numbers Outside the Traditional Statistics Box; Use Spatial Statistics to Map Abnormal Averages
(Berry)
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8. Joseph K. Berry, SpatialSTEM
Spatial Statistics (Linking Data Space with Geographic Space)
Geo-registered Sample Data
Continuous Map Surface
Spatial Distribution
Surface Modeling techniques are used to derive a continuous map surface from discrete point data– fits a Surface to the data (maps the variation).
Roving Window (weighted average)
Discrete Sample Map
Spatial
Statistics
Standard Normal Curve
Average = 22.6
Numeric Distribution
StDev =
26.2
(48.8)
Non-Spatial Statistics
In Data Space, a
standard normal curve can
be fitted to the data to identify
the “typical value” (average)
Histogram 70 60 50 40 30 20 10 80
In Geographic Space, the typical value forms a horizontal plane implying
the average is everywhere to
form a horizontal plane
X= 22.6
…lots of NE locations exceed Mean + 1Stdev
X + 1StDev =
= 48.8
Unusually high values
+StDev
Average
(Left side) Non-spatial statistics fits a standard normal curve in “data space” to assess the central tendency of the data as its average and standard deviation. In processing, the geographic coordinates are ignored and the typical value and its dispersion are assumed to be uniformly (or randomly) distributed in “geographic space.”
(Top right) The discrete point map locates each sample point on the XY coordinate plane and extends these points to their relative values (higher values in the NE; lowest in the NW). A roving window is moved throughout the area that weight-averages the point data as an inverse function of distance— closer samples are more influential than distant samples. The effect is to fit a continuous map surface that represents the geographic distribution of the data in a manner that is analogous to fitting a SNV curve to characterize the data’s numeric distribution. Underlying this process is the nature of the sampled data which must be numerically quantitative (measurable as continuous numbers) and geographically isopleth (numbers form continuous gradients in space).
(Lower-right ) The numerical distribution and the geographic distribution views of the data are directly linked. In geographic space, the “typical value” (average) forms a horizontal plane implying that the average is everywhere. In reality, the average is hardly anywhere and the geographic distribution denotes where values tend to be higher or lower than the average. In data space, a histogram represents the relative occurrence of each map value. By clicking anywhere on the map, the corresponding histogram interval is highlighted; conversely, clicking anywhere on the histogram highlights all of the corresponding map values within the interval. By selecting all locations with values greater than + 1SD, areas of unusually high values are located— a technique requiring the direct linkage of both numerical and geographic distributions.
Slide 8, Spatial Statistics Operations (Linking Data Space with Geographic Space) – Spatial Interpolation Procedures and Assessment; Linking Data Space and Geographic Space; Babies and Bath Water; Making Space for Mapped Data
(Berry)
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9. Joseph K. Berry, SpatialSTEM
Spatial Statistics Operations (Data Mining Examples)
Joseph K. Berry, SpatialSTEM
Box and Whisker
Slope
(Percent)
Map Clustering:
Elevation
(Feet)
X axis = Elevation (0-100 Normalized)
Y axis = Slope (0-100 Normalized)
Elevation vs. Slope Scatterplot
Data Space
Slope draped
on Elevation
Slope
Elev
Data Pairs +
Plots here in…
Data
Space
Cluster 1
Cluster 2
Geographic Space +
Geographic
Space
Advanced Classification (Clustering)
Advanced Classification (Clustering)
Entire Map
Extent
Spatially Aggregated Correlation
Scalar Value – one value represents the overall non-spatial relationship between the two map surfaces
…where x = Elevation value and y = Slope value
and n = number of value pairs
r =
…1 large data table
with 25rows x 25 columns =
625 map values for map wide summary
Map Correlation:
Slope
(Percent)
Elevation
(Feet)
Roving Window
Localized Correlation
Map Variable – continuous quantitative surface represents the localized spatial relationship between the two map surfaces
…625 small data tables
within 5 cell reach =
81map values for localized summary
r = .432 Aggregated
As overview, Spatial Variable Dependence can be defined as what occurs at a location in geographic space is related to 1) the conditions of that variable at nearby locations, termed Spatial Autocorrelation (intra-variable dependence) used in surface modeling to generate a continuous map surface from discrete point data, or 2) the conditions of other map layer variables at that location, termed Spatial Correlation (inter-variable dependence) used in spatial data mining to characterize spatial relationships among map layers.
(Top) Map clustering uses spatial correlation to identify the location of inherent groupings of two or more map layers. For example, pairs of elevation and slope values are partitioned in data space into groups (called clusters) so that the value pairs in the same cluster are more similar to each other (minimal differences) than to those in other clusters while at the same time the difference between the clusters are maximized. The results is the optimal partitioning of data into the specified number of clusters considering two or more map surface layers.
(Bottom) Map correlation assesses the degree of dependency among the same maps of elevation and slope. Spatially “aggregated” correlation involves solving the standard correlation equation for the entire set of paired values to represent the overall relationship as a single metric. Like the statistical average, this value is assumed to be uniformly (or randomly) distributed in “geographic space” forming a horizontal plane.
“Localized” spatial correlation, on the other hand, maps the degree of dependency between the two map variables by successively solving the standard correlation equation within a roving window to generate a continuous map surface. The result is a map representing the geographic distribution of the spatial dependency throughout a project area indicating where the two map variables are highly correlated (both positively, red tones; and negatively, green tones) and where they have minimal correlation (yellow tones).
“Localized T-statistic” (Map Comparison) is generated by using a “roving window” to collect surrounding difference values that are evaluated using the standard “paired” T-test equation…
TStatistic = dMean / ( dStdev / Sqrt(n) )
…for a T-statistic map whose values are compared to the critical T-test metric to create a T-test map that identify those locations that have a significant statistical difference.
The “bottom-line” is that map variables can be substituted for scalar values in traditional equations to generate spatial solutions for most (all?) mathematical and statistical procedures– map analysis and modeling are direct extensions of traditional techniques..
 Slide 9, Spatial Statistics Operations (Data Mining Examples) – Characterizing Patterns and Relationships; Analyzing Map Similarity and Zoning; Discover the “Miracle” in Mapping Data Clusters
Localized T_test
Map Comparison
(Statistical Tests)
T-statistic
Map
Predictive Statistics (Correlation)
Predictive Statistics (Correlation)
(Berry)
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10. Joseph K. Berry, SpatialSTEM
PowerPoint and Online Book Chapter on SpatialSTEM
Joseph K. Berry, SpatialSTEM
This PowerPoint with notes and online links to further reading is posted at…
Part 1) Overview
Part 2) Spatial Analysis
(contextual relationships)
Part 3) Spatial Statistics
(numerical relationships)
A series of Beyond Mapping columns in GeoWorld compiled into Topic 30, “SpatialSTEM: A Math/Stat Framework for Grid-based Map Analysis and Modeling” in the online book Beyond Mapping III posted at…
Online book chapter…
This PowerPoint with notes is posted online at—
There are numerous other online reference papers and teaching materials including PowerPoint slide sets and software for “hands-on” exercises in SpatialSTEM concepts, grid-based map analysis and modeling operations, considerations and procedures …see
Further Reading
Making a Case for SpatialSTEM — a 15-page white paper describing a framework for grid-based
map analysis and modeling concepts and procedures
SpatialSTEM: Extending Traditional Mathematics and Statistics to Grid-based Map Analysis and Modeling — white paper describing an innovative approach for teaching map analysis and modeling fundamentals within a mathematical/statistical context
Further SpatialSTEM Readings — a comprehensive appendix to the SpatialSTEM “extended readings” with URL links to over 125 additional readings on the grid-based map analysis/modeling concepts, terminology, considerations and procedures described in the papers on SpatialSTEM
Part 4) Future Directions
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