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

Presentation by

Joseph K. Berry

Adjunct Faculty in Natural Resources, Warner College of Natural Resources, Colorado State UniversityAdjunct Faculty in Geosciences, Department of Geography, University of Denver

Principal, Berry & Associates // Spatial Information Systems

Email: jberry@innovativegis.com — Website: www.innovativegis.com/basis

This PowerPoint with notes and online links to further reading is posted at

www.innovativegis.com/basis/Present/Mapematics_2015/

“They who don’t know, don’t know they don’t know”

A Brown Bag Lunch event hosted by the University of DenverCenter for Statistics and Visualization and Department of Geography & the Environment — September 19, 2014

This presentation describes a comprehensive framework for map analysis and modeling concepts and procedures as direct spatial extensions of traditional mathematics and statistics enabling individuals with

minimal or no GIS background to develop spatial reasoning an problem solving skills—”thinking with maps.” 

…three major considerations will influence map analysis/modeling future development— Mathematical Framework, Data Structure and Educational Approach

This presentation describes a comprehensive framework for map analysis and modeling concepts and procedures as direct spatial extensions of traditional mathematics and statistics enabling individuals with

minimal or no GIS background to develop spatial reasoning an problem solving skills—”thinking with maps.” 

…three major considerations will influence map analysis/modeling future development— Mathematical Framework, Data Structure and Educational Approach

Mapping vs. Analyzing (Processing Mapped Data)

Global Positioning System

(locate and navigate)

Geographic Information Systems

(map and analyze)

GPS/GIS/RS

(Berry)

…GIS is a Technological 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

“Map”

(Descriptive Mapping)

“Analyze”

… 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

(Prescriptive Modeling)

Mega-Technologies… …for the 21st Century

(Biotechnology) (Nanotechnology)

Remote Sensing

(measure and classify)

Descriptive Mapping(Cartographic)

Prescriptive Modeling(Analytic)

Spatial Statistics Operations Spatial Analysis Operations

A Mathematical Structure for Map Analysis/Modeling

Geotechnology RS – GIS – GPS

Grid-basedMap Analysis

Toolbox

(Berry)

Esri Spatial Analyst operations…over 170 individual “tools”

Matrix of Numbers

www.innovativegis.com/basis/BeyondMappingSeries/, Book IV, Topic 9 for more discussion

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”

“Map-ematics”Map Stack

Maps as Data, not Pictures

Vector & Raster — Aggregated & Disaggregated

Qualitative & Quantitative

“Map Variables”

(Continuous, Map Surfaces) Map Analysis/Modeling

Analytical ToolTechnological Tool

Mapping/Geo-Query (Discrete, Spatial Objects)

Geo-registered

Analysis Frame

…organized set of numbers

Spatial Analysis Operations (Geographic Context)

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

(Berry)

GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)

Map StackGrid Layer

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

www.innovativegis.com/basis/BeyondMappingSeries/, Book IV, Topic 9 for more discussion

The integral calculates the

area under the curve for any section of a function. Curve

Map Calculus — Spatial Derivative, Spatial IntegralAdvanced Grid Math — Math, Trig, Logical Functions

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)

Slope draped overMapSurface

0%

65%

Spatial Derivative

…is equivalent to the slope of the tangent plane at a location

SLOPE MapSurface Fitted FOR MapSurface_slope

Fitted PlaneSurface

500’

2500’

MapSurface

Advanced Grid Math

Surface Area

…increases with increasing inclination as a Trig function of the cosine of the slopeangle

S_Area= Fn(Slope)

Spatial Analysis Operations (Math Examples)

Dzxy Elevation

S_area= cellsize / cos(Dzxy Elevation)

ʃ Districts_Average Elevation

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

(Berry)

Seen if new tangent exceeds all previous tangents along the line of sight

Tan = Rise/Run

Rise

Run

Viewshed

Splash

Solid Geometry Connectivity

Spatial Analysis Operations (Distance Examples)

(Berry)

Map Geometry — (Distance, Proximity, Effective Movement, Narrowness)

Plane Geometry Connectivity — (Optimal Path, Optimal Path Density)

Solid Geometry Connectivity — (Viewshed, Visual Exposure)

Proximity

…from a point to everywhere…

HighestWeightedExposure

SumsViewerWeights

Counts # Viewers

270/621= 43% of the entire road network is connected

Visual Exposure

Distance

Shortest straight line between two points…

Travel-TimeSurface

Movement (absolute/relative barriers)

…not necessarily straight lines (movement)

HQ (start)On Road

26.5 minutes

…farthest away by truck

Off RoadAbsolute Barrier

On + Off Road

96.0 minutes

…farthest away by truck, ATV and hiking

Off RoadRelative Barriers

Plane Geometry

Connectivity

…like a raindrop, the “steepest downhillpath” identifies the optimal route(Quickest Path)

Farthest(end)

HQ (start) Truck = 18.8 min

ATV = 14.8 min Hiking = 62.4 min

Spatial Statistics Operations (Numeric Context)

(Berry)

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

GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)

Map StackGrid Layer

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

www.innovativegis.com/basis/BeyondMappingSeries/, Book IV, Topic 9 for more discussion

Histogram

706050403020100 80

In Geographic Space, the typical value forms a horizontal plane implying

the average is everywhere

X= 22.6

Avg=22.6

Spatial Statistics (Linking Data Space with Geographic Space)

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).

Geo-registered Sample Data

Discrete Sample Map

SpatialStatistics

(Berry)

…lots of NE locations exceed Mean + 1Stdev

X + 1StDev= 22.6 + 26.2

= 48.8

Unusually high

values+StDev

Average

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)

Roving Window (weighted average)

Neighboring Parcel

Slope(Percent)

Map Clustering:

Elevation(Feet)

Advanced Classification (Clustering)

Latitude

Longitude X axis = Elevation (0-100 Normalized)Y axis = Slope (0-100 Normalized)

Elevation vs. Slope Scatterplot

Data Space

Slope draped on Elevation

Slo

pe

Elev

Entire MapExtent

Spatially Aggregated CorrelationScalar Value – one value represents the overall non-spatial relationship between the two map surfaces

…where x = Elevation value and y = Slope valueand n = number of value pairs

r =

…1 large data table with 25rows x 25 columns =

625 map values for map wide summary

Cluster 1

Cluster 2

(Berry)

Spatial Statistics Operations (Data Mining Examples)

“data pair” of map values

(Lon,Lat,Value)

+

“data pair” plots here in…

Data Space

r = .432 Aggregated

Geographic Space

+

Geographic Space

…as similar as can be WITHIN a cluster …and as different as can be BETWEEN clusters

Map of the Correlation

Predictive Statistics (Correlation)

Map Correlation:

Slope(Percent)

Elevation(Feet)

Latitude

Longitude

Roving Window

Localized CorrelationMap 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

GIS Evolution

The Early Years

Revisit Analytics(2020s)

GIS Development Cycle (…where we’re heading)

Map Analysis (1990s)

Computer Mapping(1970s)

Spatial dB Mgt (1980s)

The Early Years

Contemporary GIS

GeoWeb(2000s)

Revisit Geo-reference (2010s)

Future Directions

Mapping focus

Data/Structure focus

Analysis focus

…about every decade

Cube(6 squares)

3D Solid(X,Y,Z Data)

Future Directions

Square(4 sides)

2D Planar(X,Y Data)

Cartesian Coordinates

(Berry)

Hexagon(6 sides)

Today

Future

PentagonalDodecahedral(12 pentagons)

Today

Future

Grid-based Map Data Structure (geo-registered matrix of map values)

(Berry)

Conceptual Spreadsheet (73 x 144)

#Rows= 73 #Columns= 144 = 10,512 grid cells

…each 2.50 grid cell is about 140mi x 140mi

18,735mi2…from Lat/Lon

“crosshairs to grid cells”

that contain map

values indicating characteristics or conditions at each

grid location

Lat/Lon

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.

…maximum Lat/Lon decimal degree resolution is

a four-inch squareanywhere in the world

The Latitude/Longitude grid forms a continuous surface for geographic referencing

where each grid cell represents a given portion of the earth’ surface.

300

Grid Lines

90

2.50 Latitude/Longitude Grid (140mi grid cell size)

Coordinate of first grid cell is 900 N 00 E

AnalysisFrame

(grid “cells”)

“Pours” the values into the Analysis Frame —

Georegistered Map Stack

…the bottom line is that…

All spatial topology is inherent in the grid.

Lat/Lon serves as a Universal dB KeyUniversal dB Key for joining data tables based on location

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…

(Berry)

…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 analyticallywith maps”

GIS Education (shifting the current Technical focus to a Pedagogical focus)

TechnologyExperts

“-ists”DomainExperts

“-ologists”

The “-ists” and the “-ologists” Perspectives

(Berry)

SolutionSpace

Together the “-ists” and the “-ologists” frame and develop the Solution for an application.

…understand the “tools” that can be used to display, query and

analyze spatial data

Data and Information focus

…understand the “science” behind spatial relationships that can be

used for decision-making

Knowledge and Wisdom focus

The “-ists” The “-ologists”— and —

GeospatialSpecialists

Why, So What, What If

Spatial Reasoning

Where is What

GIS Expertise

STEMDisciplines

“Public”…under Stakeholder,

Policy & Public auspices

“Policy Makers”

The “-ists” and the “-ologists” Perspectives (toward a much bigger tent)

“Stakeholders”

“Decision Makers”

TechnologyExperts

“-ists”DomainExperts

“-ologists”Solution

Space

Application SpaceGeospatial Technology’s Core

…Decision Makers utilize

the Solution…

(Berry)

Spatial Reasoning

GIS Expertise

We are simultaneously trivializing… …and complicating GIS Technology

…we need to educate people at all levels about the potential,

procedures and pitfalls of quantitative analysis

of mapped data—

SpatialSTEMCurricula

Conclusions/Upshot (from your grandfather's map to mapped data)

(Berry)

– 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.

– 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).

– 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).

– 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.”

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

Computer Mapping (1970 to 1980)

Spatial Database Management Systems (1980 to 1990)

Map Analysis and GIS Modeling (1990 to 2000)

GeoWeb and Mobile Devices (2000 to 2010)

Tomorrow’s GIS arena will be radically different from the past four decades…

In the future, Geospatial Technology will fully exploit its numerical character by extending…

Driving Forces Moving GIS Beyond Mapping

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.

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.

1) Solutions 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 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..

3) The 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 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.

The Bottom Line

“…map analysis quantitative analysis of mapped data”

— not your grandfather’s map

The Bottom Line

“…map analysis quantitative analysis of mapped data”

— not your grandfather’s map

The STEM community will revolutionize how we The STEM community will revolutionize how we conceptualize, utilize and visualizeconceptualize, utilize and visualize spatial relationships… spatial relationships…

… nor his math/stat(Berry)

Online Materials (www.innovativegis.com/Basis/Courses/SpatialSTEM/)

)

This PowerPoint with notes and online links to further reading is posted at

www.innovativegis.com/basis/Present/Mapematics_2015/

Where to Head from Here?

Joseph K. Berry Joseph K. Berry jberry@innovativegis.comjberry@innovativegis.com

eMail Contact

For more papers and presentations on GeotechnologyFor more papers and presentations on Geotechnology

www.innovativegis.comwww.innovativegis.com

For more papers and presentations on GeotechnologyFor more papers and presentations on Geotechnology

www.innovativegis.comwww.innovativegis.com

Website (www.innovativegis.com)

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).

THANK YOU for your kind attention– any final thoughts or questions?