headed and how we get there Presentation by Joseph K. Berry Keck Scholar 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 Email: [email protected] — Website: www.innovativegis.com/basis This PowerPoint with notes and online links to further reading is posted at www.innovativegis.com/basis/Present/Arkansas_Sep2013/ 2013 Arkansas GIS Symposium The State of GIS | September 9th - 13th 2013 Presentation Premise: There is a “map-ematics” that extends traditional math/stat concepts and procedures for the quantitative analysis of map variables (mapped data) “They who don’t know, don’t know they don’t know”

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… Making a Case for SpatialSTEM (Setting the Stage) (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”

and navigate) Remote Sensing (measure and classify) 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)

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 Stack Grid 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

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_Davg …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) Dz xy Elevation S_area= cellsize / cos(Dz 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)

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 Stack Grid 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

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 Histogram 70 60 50 40 30 20 10 0 80 In Geographic Space, the typical value forms a horizontal plane implying the average is everywhere to form a horizontal plane X= 22.6 (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)

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 + “data pair” plots here in… Data Space Predictive Statistics (Correlation) 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 Geographic Space + Geographic Space …as similar as can be WITHIN a cluster …and as different as can be BETWEEN clusters Map of the Correlation

2.50 Latitude/Longitude Grid (140mi grid cell size) Coordinate of first grid cell is 900 N 00 E Analysis Frame (grid “cells”) (Berry) …the bottom line is that… All spatial topology is inherent in the grid. Conceptual Spreadsheet (73 x 144) #Rows= 73 #Columns= 144 …each 2.50 grid cell is about 140mi x 140mi 18,735mi …from Lat/Lon “crosshairs to grid cells” that contain map values indicating characteristics or conditions at each 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.

Space Each column (field) represents a single map layer with the values in the rows indicating the characteristic or condition at each grid cell location (record) “What” …Spatially Keyed data in the cloud are downloaded and configured to the Analysis Frame defining the Map Stack Grid-based Map Data (moving Lat/Lon from crosshairs to grid cells) Lat/Lon as a Universal Spatial Key Once a set of mapped data is stamped with its Lat/Lon “Spatial Key,” it can be linked to any other database table with spatially tagged records without the explicit storage of a fully expanded grid layer— all of the spatial relationships are implicit in the relative Lat/Lon positioning. (Berry) Conceptual Organization Spreadsheet 30m Elevation (99 columns x 99 rows) Wyoming’s Bighorn Mts. Spatially Keyed data in the cloud Lat/Lon serves as a Universal dB Key for joining data tables based on location Keystone Concept Each of the conceptual grid map spreadsheets (matrices) can be converted to interlaced RDBMS format with a long string of numbers forming the data field (map layer) and the records (values) identifying the information at each of the individual grid cell locations. 2D Matrix 1D Field

Spatially Aware Database (XY, Value) Where (XY) …Lat/Lon coordinates identify earth position of a dB record 5-step Process to Unlocking Universal Spatial Data Step 1. User identifies the geographic extent of the analysis window. Step 2. User specifies the cell size of the analysis window. Longitude Latitude Step 3. Computer determines the Lat/Lon ranges defining each grid cell (cutoffs) and the centroid location. …defines the Analysis Frame 3 13 6 9 1 3 10 0 Step 4. Computer determines the appropriate grid cell for each database record that falls within the analysis frame’s geographic extent based on its Lat/Lon coordinates the repeat for all selected dB records. Step 5. Computer summarizes the values if more than one value “falls” into an individual grid cell-- result is a “Grid Map Layer” for inclusion in a map stack for subsequent map analysis. Shish Kebab of numbers …but Lat/Lon grid cells are only square at the equator— so is the entire idea a bust? (Berry) Hint: spatial resolution is key

vector-based principles and procedures and/or a growing knowledge of grid-based quantitative data analysis and modeling. Discrete Graphic Patterns “Visual Interpretation” (Summaries) Vector-based Map Continuous Spatial Distributions “Quantitative Analysis” (Models) Grid-based Map Surface Data Providers/GIS Specialists create spatially consistent mapped data and maintain GIS databases; very knowledgeable in GPS, RS and Spatial Mathematics; good knowledge in database development, processing, and geoweb procedures; limited knowledge in quantitative data analysis and modeling. General Users sporadically utilize established spatial databases, applications and custom models; minimal knowledge of geospatial principles and procedures. Power Users routinely use geotechnology; very knowledgeable in application field; strong skills in GIS processing procedures; some knowledge in geospatial principles. “Spatial Objects” “Spatial Gradients” Developers/Modelers develop general applications and advanced models; very knowledgeable in vector-based principles and procedures and/or a growing knowledge of grid-based quantitative data analysis and modeling. SpatialSTEM Disciplines …newly developing niche The Bottom of the Bottom Line “…map-ematics quantitative analysis of mapped data” — not your grandfather’s map, nor his math/stat The recognition by the GIS community that quantitative analysis of maps is a reality and the recognition by the STEM community that spatial relationships exist and are quantifiable should be the glue that binds the two perspectives– a common coherent and comprehensive SpatialSTEM approach.

) www.innovativegis.com/Basis/Courses/SpatialSTEM/ Handout, PowerPoint and Online References …also see www.innovativegis.com/basis, online book Beyond Mapping III This PowerPoint with notes and online links to further reading is posted at www.innovativegis.com/basis/Present/Arkansas_Sep2013/ Website (www.innovativegis.com) Beyond Mapping III, Topic 30 Joseph K. Berry — [email protected]