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A Map-ematical Framework for Quantitative Analysis of Mapped Data

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

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A Map-ematical Framework for Quantitative Analysis of Mapped Data

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  1. 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 for map analysis and modeling concepts and proceduresas 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 proceduresas 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 PowerPoint with notes and online links to further reading is posted at www.innovativegis.com/basis/Present/Mapematics_2015/ Presentation byJoseph 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

  2. Mapping vs. Analyzing (Processing Mapped Data) • …GISis a Technological Tool involving — • Mappingthat creates a spatial representation of an area • Displaythat generates visual renderings of a mapped area • Geo-querythat 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 “Map” “Analyze” Geographic Information Systems (map and analyze) Descriptive Mapping (Cartographic) Prescriptive Modeling (Analytic) (Descriptive Mapping) (Prescriptive Modeling) Global Positioning System (locate and navigate) Remote Sensing (measure and classify) GPS/GIS/RS (Biotechnology) (Nanotechnology) Mega-Technologies… …for the 21st Century (Berry)

  3. A Mathematical Structure for Map Analysis/Modeling GeotechnologyRS – GIS – GPS Technological Tool Analytical Tool (Continuous, Map Surfaces) Map Analysis/Modeling Mapping/Geo-Query(Discrete, Spatial Objects) Geo-registered Analysis Frame Map Stack “Map-ematics”  Matrix of Numbers Maps as Data, not Pictures Vector & Raster — Aggregated & Disaggregated Qualitative &Quantitative …organized set of numbers “Map Variables” Grid-based Map Analysis Toolbox Spatial Analysis Operations 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)

  4. Spatial Analysis Operations(Geographic Context) GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if) Map Stack Grid Layer Spatial Analysis Spatial Analysisextends 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 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) Map Analysis Toolbox Unique spatial operations www.innovativegis.com/basis/BeyondMappingSeries/, Book IV, Topic 9 for more discussion (Berry)

  5. Spatial Analysis Operations(Math Examples) Spatial Derivative Advanced Grid Math —Math, Trig, Logical Functions MapSurface 2500’ …is equivalent to the slope of thetangent plane at a location Map Calculus —Spatial Derivative, Spatial Integral 500’ The derivativeis the instantaneous “rate of change” of a function and is equivalent to the slope of thetangent line at a point Surface Fitted Plane Slope draped over MapSurface Curve SLOPE MapSurfaceFitted FOR MapSurface_slope 65% Spatial Integral Dzxy Elevation 0% ʃ Districts_AverageElevation …summarizes the values on a surface for specified map areas (Total= volume under the surface) Advanced Grid Math Surface Area COMPOSITE Districts WITH MapSurface Average FOR MapSurface_Davg …increases with increasing inclination as a Trig function of the cosine of the slope angle MapSurface_Davg S_Area= Fn(Slope) S_area= cellsize / cos(Dzxy Elevation) Surface Theintegral calculates the area under the curve for any section of a function. Curve (Berry)

  6. Spatial Analysis Operations(Distance Examples) 96.0minutes …farthest away by truck, ATV and hiking Map Geometry —(Distance, Proximity, Effective Movement, Narrowness) Plane Geometry Connectivity —(Optimal Path, Optimal Path Density) Solid Geometry Connectivity—(Viewshed, Visual Exposure) HQ(start) Distance Proximity Movement(absolute/relative barriers) Off Road Relative Barriers On Road 26.5minutes …farthest away by truck Off Road Absolute Barrier On + Off Road Travel-Time Surface Farthest (end) Plane Geometry Connectivity HQ (start) Shortest straight line between two points… …from a point to everywhere… …not necessarily straight lines (movement) Truck = 18.8 min ATV = 14.8 min Hiking = 62.4 min …like a raindrop, the “steepest downhill path” identifies the optimal route (Quickest Path) Solid Geometry Connectivity Rise Visual Exposure Run Tan = Rise/Run • Counts • # Viewers Seen if new tangent exceeds all previous tangents along the line of sight Sums Viewer Weights  Splash Highest Weighted Exposure 270/621= 43% of the entire road network is connected Viewshed (Berry)

  7. Spatial Statistics Operations(Numeric Context) GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if) Map Stack Grid Layer Spatial Statistics Spatial Statisticsseeks 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 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) Map Analysis Toolbox Unique spatial operations www.innovativegis.com/basis/BeyondMappingSeries/, Book IV, Topic 9 for more discussion (Berry)

  8. Spatial Statistics (Linking Data Space with Geographic Space) Roving Window (weighted average) Spatial Distribution Geo-registered Sample Data Spatial Statistics Continuous Map Surface Discrete Sample Map Non-Spatial Statistics 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 …lots of NE locations exceed Mean + 1Stdev In Geographic Space, the typical value forms a horizontal plane implying the average is everywhere StDev = 26.2 (48.8) X + 1StDev = 22.6 + 26.2 = 48.8 Histogram In Data Space, a standard normal curve can be fitted to the data to identify the “typical value” (average) X= 22.6 Avg= 22.6 80 0 30 40 50 60 70 10 20 Unusually high values Numeric Distribution +StDev Neighboring Parcel Average (Berry)

  9. Spatial Statistics Operations(Data Mining Examples) Map Clustering: Elevation vs. Slope Scatterplot Cluster 2 “data pair” of map values (Lon,Lat,Value) …as similar as can be WITHIN a cluster …and as different as can be BETWEEN clusters “data pair” plots here in… Data Space Elevation (Feet) Geographic Space Slope draped on Elevation + + Cluster 1 Slope Slope (Percent) Elev Latitude X axis = Elevation (0-100 Normalized) Y axis = Slope (0-100 Normalized) Geographic Space Longitude Advanced Classification (Clustering) Data Space Map Correlation: Spatially Aggregated Correlation Scalar Value– one value represents the overall non-spatial relationship between the two map surfaces Roving Window r = .432 Aggregated …1 large data table with 25rows x 25 columns = 625 map values for map wide summary Map of the Correlation Entire Map Extent Elevation (Feet) 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 Slope (Percent) Latitude Longitude Localized Correlation Map Variable– continuous quantitative surface represents the localized spatial relationship between the two map surfaces Predictive Statistics (Correlation) (Berry)

  10. Revisit Analytics (2020s) GeoWeb (2000s) Revisit Geo-reference (2010s) Map Analysis (1990s) Mapping focus Data/Structure focus Analysis focus GIS Development Cycle(…where we’re heading) Future Directions GIS Evolution 2D Planar (X,Y Data) 3D Solid (X,Y,Z Data) Future Directions Cartesian Coordinates Today Today Square (4 sides) Cube (6 squares) Contemporary GIS Future Future Spatial dB Mgt (1980s) Pentagonal Dodecahedral (12 pentagons) Hexagon (6 sides) The Early Years …about every decade The Early Years Computer Mapping (1970s) (Berry)

  11. Grid-based Map Data Structure(geo-registered matrix of map values) 2.50 Latitude/Longitude Grid (140mi grid cell size) 90 “Pours” the values into the Analysis Frame — Georegistered Map Stack Analysis Frame (grid “cells”) 300 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 #Rows= 73 #Columns= 144 = 10,512 grid cells Conceptual Spreadsheet(73 x 144) The Latitude/Longitude grid forms a continuous surface for geographic referencing where eachgrid cell represents a given portion of the earth’ surface. Lat/Lon …each 2.50grid cell is about 140mi x 140mi 18,735mi2 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. …from Lat/Lon “crosshairs to grid cells” that contain map valuesindicating characteristics or conditions at each grid location …maximum Lat/Lon decimal degree resolution is a four-inch square anywhere in the world …the bottom line is that… All spatial topology is inherent in the grid. (Berry)

  12. 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— “WhereisWhat”graphical inventories, to a “Why, So What and What If” problem solving environment— “thinking analytically with maps” (Berry)

  13. The “-ists” and the “-ologists” Perspectives Together the “-ists” and the “-ologists” frame and develop the Solution for an application. The“-ists” The“-ologists” — and — …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 Application Space Geospatial Technology’s Core Geospatial Specialists STEM Disciplines “-ists” “-ologists” Solution Space Technology Experts Domain Experts GIS Expertise Spatial Reasoning Where is What Why, So What, What If (Berry)

  14. The “-ists” and the “-ologists” Perspectives (toward a much bigger tent) …Decision Makers utilize the Solution… …under Stakeholder, Policy & Public auspices “Public” “Policy Makers” “Stakeholders” …we need to educate people at all levels about the potential, procedures and pitfalls of quantitative analysis of mapped data— SpatialSTEM Curricula “Decision Makers” Application Space Geospatial Technology’s Core “-ists” “-ologists” …and complicating GIS Technology We are simultaneously trivializing… Solution Space Technology Experts Domain Experts Spatial Reasoning GIS Expertise (Berry)

  15. 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. – 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.” Spatial Database Management Systems(1980 to 1990) Map Analysis and GIS Modeling(1990 to 2000) GeoWeb and Mobile Devices(2000 to 2010) 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.htmfor more discussion (Berry)

  16. Driving Forces Moving GIS Beyond Mapping The STEM community will revolutionize how we conceptualize, utilize and visualizespatial relationships… 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.. 2) Grid-based map analysis and modeling involving Spatial Analysisand Spatial Statisticsare in large part simply spatial extensions of traditional mathematical and statisticalconcepts and procedures. 3) The recognition by the STEM communitythatspatial relationships exist and are quantifiableand the recognition by the GIS communitythatquantitative analysis of maps is a realityshould 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 … nor his math/stat (Berry)

  17. Where to Head from Here? Online Materials (www.innovativegis.com/Basis/Courses/SpatialSTEM/) ) Website (www.innovativegis.com) 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 YOUfor your kind attention– any final thoughts or questions?

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