A Map-ematical Framework for Quantitative Analysis of Mapped Data
…Map Analysis and GIS Modeling for Understanding and Communicating
Spatial Patterns and Relationships within STEM Discipline Contexts
A Brown Bag Lunch event hosted by the University of Denver
Center for Statistics and Visualization and Department of Geography & the Environment — September 19, 2014
“They who don’t know, don’t know they don’t know”
This
presentation
describes
a comprehensive
framework
map
analysis
and
modeling
concepts
and
This
presentation
describes
a comprehensive
framework
forfor
map
analysis
and
modeling
concepts
and
procedures
direct
spatial
extensions
traditional
mathematics
and
statistics
enabling
individuals
with
procedures
asas
direct
spatial
extensions
ofof
traditional
mathematics
and
statistics
enabling
individuals
with
minimal
GIS
background
develop
spatial
reasoning
problem
solving
skills—”thinking
with
maps.”
minimal
oror
nono
GIS
background
toto
develop
spatial
reasoning
anan
problem
solving
skills—”thinking
with
maps.”
…three
major
considerations
influence
map
analysis/modeling
future
development—
…three
major
considerations
willwill
influence
map
analysis/modeling
future
development—
Mathematical
Framework,
Data
Structure
and
Educational
Approach
Mathematical
Framework,
Data
Structure
and
Educational
Approach
This PowerPoint with notes and online links to further reading is posted at
www.innovativegis.com/basis/Present/Mapematics_2015/
Presentation by
Joseph K. Berry
Adjunct Faculty in Natural Resources, Warner College of Natural Resources, Colorado State University
Adjunct Faculty in Geosciences, Department of Geography, University of Denver
Principal, Berry & Associates // Spatial Information Systems
Email: jberry@innovativegis.com — Website: www.innovativegis.com/basis
Mapping vs. Analyzing
…GIS is a Technological
(Processing Mapped Data)
Tool involving —
−Mapping that creates a spatial representation of an area
−Display that generates visual renderings of a mapped area
−Geo-query that searches for map locations having a specified classification, condition or characteristic
…and an Analytical
Tool involving —
−Spatial Mathematics that applies scalar mathematical formulae to account for geometric positioning,
scaling, measurement and transformations of mapped data
−Spatial Analysis that investigates the contextual relationships within and among mapped data layers
−Spatial Statistics that investigates the numerical relationships within and among mapped data layers
“Analyze”
“Map”
Descriptive Mapping
(Descriptive Mapping)
(Cartographic)
Geographic Information
Systems
Prescriptive
Modeling
(Prescriptive
Modeling)
(Analytic)
(map and analyze)
Remote
Sensing
Global Positioning
System
(locate and navigate)
(Biotechnology)
Mega-Technologies…
GPS/GIS/RS
(measure and classify)
(Nanotechnology)
…for the 21st Century
(Berry)
A Mathematical Structure for Map Analysis/Modeling
Technological Tool
Mapping/Geo-Query
Geotechnology
(Discrete, Spatial Objects)
 RS – GIS – GPS
Analytical Tool
(Continuous, Map Surfaces)
Map Analysis/Modeling
Geo-registered
Analysis Frame  Matrix
Map Stack
“Map-ematics”
of Numbers
Maps as Data, not Pictures
Vector & Raster — Aggregated & Disaggregated
Qualitative & Quantitative
…organized set of numbers
“Map Variables”
Grid-based
Spatial Analysis
Operations
Map Analysis
Toolbox
Spatial Statistics
Operations
A Map-ematical
Framework
Traditional math/stat procedures
can be extended into
geographic space to support
Quantitative Analysis
of Mapped Data
“…thinking analytically
with maps”
Esri Spatial Analyst operations
…over 170 individual “tools”
www.innovativegis.com/basis/BeyondMappingSeries/, Book IV, Topic 9 for more discussion
(Berry)
Spatial Analysis Operations (Geographic Context)
GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)
Grid Layer
Map Stack
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
Unique spatial
operations
Mathematical Perspective:
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)
www.innovativegis.com/basis/BeyondMappingSeries/, Book IV, Topic 9 for more discussion
(Berry)
Spatial Analysis Operations (Math Examples)
Advanced Grid Math — Math, Trig, Logical Functions
Map Calculus — Spatial Derivative, Spatial Integral
Spatial Derivative
MapSurface
2500’
…is equivalent to the slope
of the tangent plane at a
location
Slope draped over
MapSurface
500’
Surface
Fitted Plane
65%
SLOPE MapSurface Fitted
FOR MapSurface_slope
0%
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
ʃ Districts_Average Elevation
Spatial Integral
Advanced Grid Math
…summarizes the values on a
surface for specified map areas
(Total= volume under the surface)
Surface Area
S_Area=
Fn(Slope)
…increases with
increasing inclination
as a Trig function of
the cosine of
the slope
angle
COMPOSITE Districts WITH MapSurface
Average FOR MapSurface_Davg
MapSurface_Davg
S_area= cellsize / cos(Dzxy Elevation)
The integral calculates the
area under the curve for any
section of a function.
Surface
Curve
(Berry)
Spatial Analysis Operations (Distance Examples)
96.0 minutes
Map Geometry — (Distance, Proximity, Effective Movement, Narrowness)
Plane Geometry Connectivity — (Optimal Path, Optimal Path Density)
Solid Geometry Connectivity — (Viewshed, Visual Exposure)
Distance
Proximity
Movement
…farthest away by truck,
ATV and hiking
(absolute/relative barriers)
Off Road
Relative Barriers
HQ (start)
On Road
26.5 minutes
Off Road
Absolute Barrier
…farthest away
by truck
On + Off Road
Travel-Time
Surface
Farthest
(end)
Shortest straight line
between two points…
…from a point to
everywhere…
…not necessarily straight
lines (movement)
Connectivity
HQ
Truck = 18.8 min
ATV = 14.8 min
Hiking = 62.4 min
(start)
…like a raindrop, the
“steepest downhill
path” identifies the
optimal route
Solid Geometry Connectivity
Rise
Run
Plane Geometry
Visual Exposure
(Quickest Path)
Tan = Rise/Run
Seen if new tangent exceeds
all previous tangents
along the line of sight
 Counts
# Viewers
Sums
Viewer
Weights 
Splash
270/621= 43% of the entire
Viewshed
road network is connected
Highest
Weighted
Exposure
(Berry)
Spatial Statistics Operations (Numeric 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 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
Unique spatial
operations
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)
www.innovativegis.com/basis/BeyondMappingSeries/, Book IV, Topic 9 for more discussion
(Berry)
Spatial Statistics (Linking Data Space with Geographic Space)
Roving Window (weighted average)
Geo-registered Sample Data
Spatial Distribution
Spatial
Statistics
Discrete Sample Map
Non-Spatial Statistics
Continuous Map Surface
Surface Modeling techniques are used to derive a continuous map surface
from discrete point data– fits a Surface to the data (maps the variation).
Standard Normal Curve
Average = 22.6
In Geographic Space, the typical value
forms a horizontal plane implying
the average is everywhere
StDev =
26.2
Histogram
(48.8)
10
20
30
40
50
Numeric Distribution
(Berry)
X + 1StDev
= 22.6 + 26.2
=
In Data Space, a
standard normal curve can
be fitted to the data to identify
the “typical value” (average)
0
…lots of NE locations
exceed Mean + 1Stdev
60
70
80
Unusually
high
values
48.8
X= 22.6
Avg=
22.6
+StDev
Average
Neighboring
Parcel
Spatial Statistics Operations (Data Mining Examples)
Map Clustering:
Elevation vs. Slope Scatterplot
“data pair”
of map values
“data pair”
plots here in…
Cluster 2
(Lon,Lat,Value)
Data
Space
…as similar as can be WITHIN
a cluster …and as different as
can be BETWEEN clusters
Elevation
(Feet)
Geographic
Space
Slope
+
Slope
(Percent)
Slope draped
on Elevation
Elev
X axis = Elevation (0-100 Normalized)
Y axis = Slope (0-100 Normalized)
Geographic Space
Data Space
Advanced Classification (Clustering)
Map Correlation:
+
Cluster 1
Spatially Aggregated Correlation
Scalar Value – one value represents the overall non-spatial relationship
between the two map surfaces
Roving Window
…1 large data table
Entire Map
Extent
Elevation
(Feet)
Slope
(Percent)
with 25rows x 25 columns =
625 map values for map wide summary
r=
…where x = Elevation value and y = Slope value
and n = number of value pairs
…625 small data tables
within 5 cell reach =
81map values for localized summary
Localized Correlation
Predictive Statistics (Correlation)
(Berry)
Map Variable – continuous quantitative
surface represents the localized spatial
relationship between the two map surfaces
r = .432 Aggregated
Map of the Correlation
GIS Development Cycle (…where we’re heading)
Future Directions
GIS Evolution
Revisit Analytics
(2020s)
Future Directions
2D Planar
3D Solid
(X,Y Data)
(X,Y,Z Data)
Cartesian Coordinates
GeoWeb
(2000s)
Today
Today
Revisit Geo-reference
(2010s)
Square
(4 sides)
Cube
(6 squares)
Contemporary GIS
Spatial dB Mgt (1980s)
Map Analysis
Future
(1990s)
Hexagon
(6 sides)
…about every decade
Future
Pentagonal
Dodecahedral
(12 pentagons)
The Early Years
Mapping focus
Data/Structure focus
Analysis focus
Computer Mapping
(1970s)
(Berry)
Grid-based Map Data Structure
90
2.50 Latitude/Longitude Grid
(140mi grid cell size)
“Pours” the values into
the Analysis Frame —
Analysis
Frame
300
(geo-registered matrix of map values)
Georegistered
(grid “cells”)
Map Stack
Grid Lines
Lat/Lon serves as a Universal
dB Key
for joining data tables based on location
Coordinate of first grid cell is 900 N 00 E
The Latitude/Longitude grid forms a
continuous surface for geographic referencing
where each grid cell represents a given
portion of the earth’ surface.
The easiest way to conceptualize a grid map is as
an Excel spreadsheet with each cell in the table
corresponding to a Lat/Lon grid space (location)
and each value in a cell representing the
characteristic or condition (information) of a
mapped variable occurring at that location.
…the bottom line is that…
All spatial topology is inherent in the grid.
#Rows= 73 #Columns= 144 = 10,512 grid cells
Conceptual Spreadsheet (73 x 144)
Lat/Lon
…each 2.50 grid cell is
about 140mi x 140mi
18,735mi2
…maximum Lat/Lon
decimal degree resolution
is a four-inch square
anywhere in the world
…from Lat/Lon
“crosshairs
to grid cells”
that contain map
values indicating
characteristics or
conditions at each
grid location
(Berry)
GIS Education (shifting the current Technical focus to a Pedagogical focus)
The lion’s share of the growth has been GIS’s ever expanding capabilities as a “technical tool” for
corralling vast amounts of spatial data and providing near instantaneous access to remote sensing
images, GPS navigation, interactive maps, asset management records, geo-queries and awesome displays.
However, GIS as an “analytical tool” hasn’t experienced the same meteoric rise—
in fact it can be argued that the analytic side of GIS has somewhat stalled… partly because of…
…but modern digital
“maps are numbers first,
pictures later”
and we do mathematical and
statistical things to map variables
that moves GIS from—
“Where is What” graphical
inventories, to a
“Why, So What and What If”
problem solving environment—
“thinking analytically
with maps”
(Berry)
The “-ists” and the “-ologists” Perspectives
Together the “-ists” and the “-ologists” frame and develop the Solution for an application.
The
“-ists”
— and —
The
“-ologists”
…understand the “tools” that can
be used to display, query and
analyze spatial data
…understand the “science” behind
spatial relationships that can be
used for decision-making
Data and Information focus
Knowledge and Wisdom focus
Application Space
Geospatial Technology’s Core
Geospatial
Specialists
STEM
Disciplines
“-ists”
Technology
Experts
Solution
Space
“-ologists”
Domain
Experts
GIS Expertise
Spatial Reasoning
Where is What
Why, So What, What If
(Berry)
The “-ists” and the “-ologists” Perspectives (toward a much bigger tent)
…Decision Makers utilize
the Solution…
…under Stakeholder,
Policy & Public
auspices
“Public”
“Policy Makers”
…we need to educate
people at all levels
about the potential,
procedures and pitfalls
of quantitative analysis
of mapped data—
“Stakeholders”
“Decision Makers”
SpatialSTEM
Curricula
Application Space
Geospatial Technology’s Core
“-ists”
Technology
Experts
Solution
Space
“-ologists”
Domain
Experts
Spatial
Reasoning
GIS
Expertise
…and complicating GIS Technology
We are simultaneously trivializing…
(Berry)
Conclusions/Upshot
(from your grandfather's map to mapped data)
Tomorrow’s GIS arena will be radically different from the past four decades…
Computer Mapping (1970 to 1980) – automated the cartographic process where points,
lines and areas (spatial objects) defining geographic features on a map are represented as
an organized set of X,Y coordinates.
Spatial Database Management Systems
(1980 to 1990)– linked computer mapping capabilities with
traditional database management capabilities by assigning an ID# to each spatial object that serves as a
common database key between a spatial table (Where) and an attribute table (What).
Map Analysis and GIS Modeling
(1990 to 2000) – developed a comprehensive theory of map analysis
where spatial information is represented numerically as continuous spatial distributions (raster), rather than in
graphic fashion as discrete spatial objects (vector) identified by inked lines on a map. These digital maps are frequently
conceptualized as a set of "floating maps" with a common registration, allowing the computer to "look" down and across the stack of digital
maps to characterize spatial relationships of the mapped data that can be summarized (database queries) or mathematically manipulated
(analytic processing).
GeoWeb and Mobile Devices
(2000 to 2010) – the Internet has moved maps and mapping from a “down the
hall and to the right” specialist’s domain, to everyone’s desktop, notebook and mobile device. While the bulk of
these applications involve navigation, mapping and geo-query (technological), they have fully established the digital
map beachhead that sees “maps as data, not just images.”
In the future, Geospatial Technology will fully exploit its numerical character by extending…
1) Scientific
Use of Spatial Data – SpatialSTEM education will infuse science with “analytical tools” for
unlocking radically new understandings of spatial patterns and relationships in their research.
2) Spatial
Solutions to Devices – “map-ematical solutions” will be directly tied to automated devices through
continued coupling of the Spatial Triad (RS, GIS, GPS) and robotics.
3) Spatial
Reasoning and Dialog – GIS models will enable decision-makers to interactively investigate and
better communicate “Why, So What and What If” of the probable spatial outcomes/impacts of critical decisions.
See http://www.innovativegis.com/basis/BeyondMappingSeries/BeyondMapping_I/Epilog/BM_I_Epilog.htm for more discussion
(Berry)
Driving Forces Moving GIS Beyond Mapping
The STEM community will revolutionize how we
conceptualize, utilize and visualize spatial relationships…
to complex spatial problems need to engage “domain expertise”
through GIS– outreach to other disciplines to establish spatial reasoning skills
needed for effective solutions that integrate a
multitude of disciplinary and general public perspectives..
1) Solutions
2) Grid-based
map analysis and modeling involving Spatial Analysis and
Spatial Statistics are in large part simply spatial extensions of traditional
mathematical and statistical concepts and procedures.
recognition by the STEM community that spatial relationships exist and
are quantifiable and the recognition by the GIS community that quantitative
analysis of maps is a reality should be the glue that binds the two perspectives–
a common coherent and comprehensive SpatialSTEM approach.
3) The
The Bottom Line
“…map analysis  quantitative analysis of mapped data”
— not your grandfather’s map … nor his math/stat
(Berry)
Where to Head from Here?
Website (www.innovativegis.com)
Online Materials
(www.innovativegis.com/Basis/Courses/SpatialSTEM/)
For more papers and presentations
on Geotechnology
)
www.innovativegis.com
This PowerPoint with notes and online links to further reading is posted at
www.innovativegis.com/basis/Present/Mapematics_2015/
Beyond Mapping Compilation Series (www.innovativegis.com/basis/BeyondMappingSeries/)
…nearly 1000 pages and more than 750 figures
in the Series provide a comprehensive and
longitudinal perspective of the underlying
concepts, considerations, issues and
evolutionary development of modern
geotechnology (RS, GIS, GPS).
eMail Contact
Joseph K. Berry
jberry@innovativegis.com
THANK YOU for your kind attention– any final thoughts or questions?