2. 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 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 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 Denver Center 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

3. Mapping vs. Ana lyzing (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 21 st Century (Biotechnology)(Nanotechnology) Remote Sensing (measure and classify) Descriptive Mapping (Cartographic) Prescriptive Modeling (Analytic)

4. Spatial Statistics Operations Spatial Analysis Operations A Mathematical Structure for Map Analysis/Modeling Geotechnology  RS – GIS – GPS Grid-based Map Analysis Toolbox (Berry) Esri Spatial Analyst operations …over 170 individual “tools”  Matrix of Numbers www.innovativegis.com/basis/BeyondMappingSeries/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 Tool Technological Tool Mapping/Geo-Query (Discrete, Spatial Objects) Geo-registered Analysis Frame …organized set of numbers

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

6. The integral calculates the area under the curve for any section of a function. Curve Map Calculus — Spatial Derivative, Spatial Integral Advanced Grid Math — Math, Trig, Logical Functions Spatial Integral Surface COMPOSITE Districts WITH MapSurface Average FOR MapSurface_Davg MapSurface_D avg …summarizes the values on a surface for specified map areas (Total= volume under the surface) 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’ MapSurface Advanced Grid Math Surface Area …increases with increasing inclination as a Trig function of the cosine of the slope angle S_Area= Fn(Slope) Spatial Analysis Operations (Math Examples) D z xy Elevation S_area= cellsize / cos(D z xy 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)

7. 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… Highest Weighted Exposure Sums Viewer Weights   Counts # Viewers 270/621= 43% of the entire road network is connected Visual Exposure Distance Shortest straight line between two points… Travel-Time Surface Movement (absolute/relative barriers) …not necessarily straight lines (movement) HQ (start) On Road 26.5 minutes …farthest away by truck Off Road Absolute Barrier On + Off Road 96.0 minutes …farthest away by truck, ATV and hiking Off Road Relative Barriers 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

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

9. Histogram 7060504030 20 100 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 Spatial Statistics (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

10. 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 Slope Elev 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 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 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

11. 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 Pentagonal Dodecahedral (12 pentagons) Today Future

12. 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 2.5 0 grid cell is about 140mi x 140mi 18,735mi 2 …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 square anywhere 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. 30 0 Grid Lines 9090 2.5 0 Latitude/Longitude Grid (140mi grid cell size) Coordinate of first grid cell is 90 0 N 0 0 E Analysis Frame (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. Universal dB Key Lat/Lon serves as a Universal dB Key for joining data tables based on location

13. 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 analytically with maps” GIS Education (shifting the current Technical focus to a Pedagogical focus)

14. Technology Experts “-ists” Domain Experts “-ologists” The “-ists” and the “-ologists” Perspectives (Berry) Solution Space 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 — Geospatial Specialists Why, So What, What If Spatial Reasoning Where is What GIS Expertise STEM Disciplines

15. “Public” …under Stakeholder, Policy & Public auspices “Policy Makers” The “-ists” and the “-ologists” Perspectives (toward a much bigger tent) “Stakeholders” “Decision Makers” Technology Experts “-ists” Domain Experts “-ologists” Solution Space Application Space Geospatial 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— SpatialSTEM Curricula

16. 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 discussionhttp://www.innovativegis.com/basis/BeyondMappingSeries/BeyondMapping_I/Epilog/BM_I_Epilog.htm 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…

17. 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 conceptualize, utilize and visualize spatial relationships… … nor his math/stat (Berry)

18. 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.com jberry@innovativegis.com eMail Contact For more papers and presentations on Geotechnology For more papers and presentations on Geotechnology www.innovativegis.com For more papers and presentations on Geotechnology For more papers and presentations on Geotechnology www.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?