2. Presentation by Joseph K. Berry W.M. Keck Scholar in Geosciences, University of Denver Adjunct Faculty in Natural Resources, Colorado State University Principal, Berry & Associates // Spatial Information Systems Email: jberry@innovativegis.com — Website: www.innovativegis.com/basis Map-ematically Messing with Maps Map-ematically Messing with Maps: This presentation describes the idea of spatialSTEM for teaching map analysis and modeling fundamentals within a mathematical/statistical context that resonates with science, technology, engineering and math/stat communities. The premise is that “maps are numbers first, pictures later” and we do mathematical things to mapped data for insight and better understanding of spatial patterns and relationships within decision-making contexts …from Where is What graphical inventories to a Why, So What and What If problem solving environment. WCNR GIS Seminar Series and Geospatial Centroid Colorado State University – January 30, 2012 Extending Traditional Math/Stat to Grid-based Map Analysis and Modeling

3. www.innovativegis.com/basis/Papers/Other/SpatialSTEM/ Map-ematically Messing with Maps Map-ematically Messing with Maps: Extending Traditional Math/Stat to Grid-based Map Analysis and Modeling 12 page white paper Further Reading Teaching Materials 6 page appendix Today’s Introductory Seminar …online links to Handout, PowerPoint and Readings are posted at — www.innovativegis.com/basis/Present/CSU_winter2012/ Leading to a… Possible Mini-Workshop …on teaching SpatialSTEM. Two 2-hour sessions with hands-on experience in the concepts and procedures. One optional 2-hour session developing ‘tailored” exercise. Contact Joe Berry at www.innovativegis.com for more information.www.innovativegis.com

4. Geotechnology (Nanotechnology) Geotechnology (Biotechnology) GPS/GIS/RS Modeling involves analysis of spatial patterns and relationships (map analysis/modeling) Prescriptive Modeling Mapping involves precise placement (delineation) of physical features (graphical inventory) Descriptive Mapping Geotechnology is one of the three "mega technologies" for the 21st century and promises to forever change how we conceptualize, utilize and visualize spatial relationships in scientific research and commercial applications (U.S. Department of Labor) Why So What and What If Global Positioning System (location and navigation) Geographic Information Systems (map and analyze) is Where What (Berry) Remote Sensing (measure and classify) The Spatial Triad Computer Mapping (70s) Spatial Database Management (80s) Today’s focus… Map Analysis (90s) Multimedia Mapping (00s) WOW!!!

5. 1) Maps as Quantitative Data Vector & Raster, Aggregated & Disaggregated Qualitative/ Quantitative & Choropleth/ Isopleth “Map-ematics” Basic GridMath & Map Algebra Advanced GridMath Map Calculus Map Geometry Plane Geometry Solid Geometry Connectivity Unique Map Analytics Basic Descriptive Statistics Basic Classification Unique Map Descriptive Statistics Map Comparison Surface Modeling Advanced Classification Predictive Statistics Geotechnology  RS – GIS – GPS A Mathematical Structure for Map Analysis/Modeling Map Analysis/Modeling Mapping/Geo-Query (Discrete, Spatial Objects) – (Continuous, Map Surfaces) Map Analysis/Modeling Spatial Statistics Operations 3) Spatial Statistics Operations Spatial Analysis Operations 2) Spatial Analysis Operations … traditional math/stat procedures can be extended into geographic space to stimulate students with diverse backgrounds and interests to… “think analytically with maps”

6. Types of Digital Maps Composite Display COVERTYPE Map Water Polygons #11, #12 Meadow Polygon #21 Forest Polygons #31, #32 Vector Vector (Spatial & Attribute Tables) Mapping/Geo-Query (Discrete Spatial Objects)(Continuous Map Surfaces) Map Analysis/Modeling Raster Raster (Grid Matrix) Where Where – Cell Position in the matrix determines location within a continuous, regular grid forming map surfaces What What – Cell Value in the matrix indicates the classification at that location Where Where – a Spatial Table contains X,Y coordinates delineating the location of each point, line and polygon boundary What What – a linked Attribute Table contains text/values indicating the classification of each spatial object : ID#32 X,Y : : : : : ID#32 Forest PP 60 : : : : Spatial Table Attribute Table WHERE WHAT

7. Analysis Frame Col 3, Row 22 Analysis Frame The Analysis Frame provides consistent “parceling” needed for map analysis and extends discrete Points, Lines and Polygons… Col 1, Row 1 Origin Col 1, Row 1 Origin 3D Surface (stair steps) Grid Grid Map A Grid Map consists of a matrix of numbers with a value indicating the characteristic/condition at each grid cell location— Raster (continuous) Grid Data Structure 3D Surface (gradient) …to continuous Map Surfaces Map Stack Map Layers …forming a set of geo-registered Map Layers or “Map Stack” Data listing for a Map Stack Drill-down Contour Lines Intervals Vector (discrete) Points, Lines and Polygons Elevation Contour Interval 2300-2500’

8. Nature of Grid-based Data (hands-on demo) Continuous, regular grid cells (objects) Points, Lines, Polygons and Surfaces : --, --, --, 2438, --, --, --, : Grid Table Analysis Frame Configuration— 100 columns 100 rows 30m cell size Map Display Display Window Shading Manager Grid Analysis …calculate a slope map and drape on the elevation surface

9. Spatial Data Perspectives (What is Where) Numerical Data Perspective Numerical Data Perspective : how numbers are distributed in “Number Space”  Binary: a special type of number where the range is constrained to just two states—such as 1=forested, 0=non-forested  Qualitative: deals with unmeasurable qualities; very few math/stat operations available – Nominal numbers are independent of each other and do not imply ordering – like scattered pieces of wood on the ground 1 2 3 4 5 6 Nominal —Categories  Quantitative: deals with measurable quantities; a wealth of math/stat operations available – Interval numbers have a definite ordering and a constant step – like a typical ladder with consistent spacing between rungs 1 2 3 4 5 6 Interval —Constant Step – Ordinal numbers imply a definite ordering from small to large – like a ladder, however with varying spaces between rungs 1 2 3 4 5 6 Ordinal —Ordered – Ratio numbers has all the properties of interval numbers plus a clear/constant definition of 0.0 – like a ladder with a fixed base. 1 2 3 4 5 6 Ratio —Fixed Zero 0 Spatial Data Perspective Spatial Data Perspective : how numbers are distributed in “Geographic Space”  Choropleth numbers form sharp/unpredictable boundaries in geographic space – e.g., a road “map” Roads —Discrete  Isopleth numbers form continuous and often predictable gradients in geographic space – e.g., an elevation “surface” Elevation —Continuous

10. Basic GridMath and Map Algebra (hands-on exercise) Yield Mapping Yield Mapping: As a combine moves through a field it 1) uses GPS to check its location and then 2) checks the yield monitor at that location to 3) create a continuous map of crop yield variation every few feet. WHERE is WHAT GPS SatellitesYield Monitor Map Algebra Map Algebra: All of the mathematical functions on a typical pocket calculator are available in grid-based map analysis. The operations can be sequenced on map layers to evaluate entire algebraic equations, such as the calculation of a continuous ”percent change” map identifying locations of large increases (green) and decreases (red) in production from year to the next. [%Diff_00-98Yield] Avg= 37.3 % [%Diff_00-98Yield] Avg= 37.3 % [00Yield][98Yield] - = * 100 Grid Math Grid Math: Since modern maps are organized sets of numbers, they can be added, subtracted, multiplied and divided. For example the difference in crop yield on a farmer’s field between two years can be calculated by simply subtracting the two geo-registered maps— [98Yield] Avg= 136.0 bu/ac - = [00Yield] Avg= 182.0 bu/ac [Diff_00-98Yield] Avg= 46.7 bu/ac 109 234 125 Since each map layer contains 3,289 grid cells for the 189 acre field, the computer retrieves two numbers for a grid cell location, subtracts them, and then stores the difference on a new map at that location …repeating the process 3,288 more times to derive a continuous map of the crop difference. 136 182

11. Spatial Analysis Operations (grid math & calculus) Map Calculus Map Calculus — Spatial Derivative, Spatial Integral Basic Grid Math & Map Algebra Basic Grid Math & Map Algebra — + - * /, (([Map1] – [Map2]) / [Map2])*100 Advanced Grid Math Advanced Grid Math — Math, Trig, Logical Functions y = fn(x) The integral calculates the area under the curve for any section of a function. 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. y = e x Spatial Integral Surface COMPOSITE Districts WITH MapSurface Average Average FOR MapSurface_Davg MapSurface_Davg Slope draped over MapSurface 0% 65% Fitted SLOPE MapSurface Fitted FOR MapSurface_slope Surface Spatial Derivative Fitted Plane Advanced Grid Math cosine of …increases with increasing inclination as a function of the cosine of the slope angle Surface Area

12. Spatial Analysis Operations (plane geometry) Map Geometry Map Geometry — Distance, Simple Proximity, Effective Proximity, Narrowness Plane Geometry Connectivity Plane Geometry Connectivity — Optimal Path, Optimal Path Density “Shortest straight line between two points”…S, SL to all other points Simple Proximity …S, not necessarily straight to all other points Effective Proximity Ruler Distance Three distance measurement methods: Analog — align a ruler between two points, then count the number of implied grid spaces and multiply times the map scale a b c c 2 = a 2 + b 2 Mathematical — calculate the grid spaces along the sides of an implied right triangle and solve the Pythagorean Theorem for the hypotenuse …extends the concept of Distance (shortest straight-line between two points) to that of Proximity (set of all shortest straight lines from a point to all surrounding points). In turn the concept of Movement is developed by relaxing the “straight-line” requirement by considering intervening relative and absolute barriers as the wavefront propagates. Algorithmic (splash) — generate a series of waves (ripples) that form increasing concentric rings of one grid space that is equivalent to nailing one end of a ruler and spinning it.

13. Spatial Analysis Operations (solid geometry) Solid Geometry Connectivity Solid Geometry Connectivity — Viewshed, Visual Exposure Unique Map Analytics Unique Map Analytics — Size/Shape/Integrity, Contiguity, Masking, Profile …Line of Sight Exposure…Line of Sight Noise…Line of Sight Buffer Noise-shedVisual ExposureViewshed Viewer Tangent= Rise / Run Rise Run …calculated by a series of “waves” that carry the tangent to beat as the wavefront propagates. A location is seen if the new tangent exceeds all previous tangents along the line of sight— the new tangent then becomes the tangent to beat. …by considering multiple viewer locations the concept of Viewshed (seen, not seen) can be extended to Visual Exposure that counts the number of times a location is seen Like proximity, Visual Connectivity starts somewhere (starter cell) and moves through geographic space by steps (wavefront)— the result is like a lighthouse beam illuminating “line-of-sight” terrain (seen) and leaving non-connected locations in the dark (not seen)… Viewer

14. Spatial Statistics (Patterns and Relationships)  Correlations — what occurs at a specific location in geographic space is related to Spatial Analysis — characterizes “contextual spatial relationships” within and among mapped data Spatial Statistics Spatial Statistics — characterizes “numerical spatial relationships” within and among mapped data Among map layers— the conditions of other map variables at that location (Spatial Correlation) Loan Concentration Home Value Loans= fn(Home Value) R 2 = 46% Home Value Loans  Summaries — basic generalizations of the map values Within a map layer— tabular counts (discrete) and general statistics, such as Median, Average, StDev (continuous) Steepness Among map layers— cross-tabular counts (discrete) summarizing spatial coincidence among map layers Forest & Extreme 4 + 30 = 34 Steepness Cover Within a map layer— the conditions of that map variable at nearby locations (Spatial Autocorrelation) “Nearby things are more alike than distant things” …measure of variation reflected in the pairs of points at various distances Sample Points Among map layers— statistical tests, such as F-test; absolute and percent difference; surface configuration Within a map layer— map normalization involves taking values that span a specific range and representing them in another “standardized “ range, such as 0 to 100 or Standard Normal Variable (((MapValue – Avg) / StDev) *100)  Comparisons — how map values compare to each other …only continuous map variables

15. Surface Modeling (hands-on exercise) Data Space Geographic Space: In Data Space, a standard normal curve can be fitted to the histogram of the map surface data to identify the “typical value” (Average). In Geographic Space, this typical value forms a horizontal plane implying the average is everywhere. In reality, the average is hardly anywhere and the Geographic Distribution denotes where values tend to be higher or lower than the average. Point Sampling: Collecting X,Y coordinates with field samples provides a foothold for generating continuous map surfaces used in map analysis and modeling. Each record contains X,Y coordinates (Where) followed by data values (What) identifying the characteristics/conditions at that location forming a geo-registered database. Surface Modeling: Surface modeling techniques are used to derive a continuous Map Surface from discrete Point Data. This process is analogous to placing a block of modeler's clay over the Point Map’s relative value pillars and smoothing away the excess clay to create a continuous map surface that fills-in the unsampled locations, thereby characterizing the data set’s Geographic Distribution. In the example, Inverse Distance Weighted (IDW) spatial interpolation is used. The procedure calculates the distances from an unsampled location to all sample locations and then uses the inverse of the distance to weight-average, such that nearby sample values influence the average more than distant sample values— repeating the procedure for all locations results in a continuous map surface of the variance in the data set.

16. Discrete Point Map Linking Data Space with Geographic Space Keystone concept is… “Spatial Autocorrelation” In Data Space, a standard normal curve can be fitted to the histogram of the map surface data to identify the “typical value” (Average)– fits a Curve. Data Space Standard Normal Curve Average = 22.9 StDev = 18.7 Numeric Distribution Geographic Space 22.9 + 1StDev (41.6) -1StDev (4.2) …lots of NE locations exceed +1Stdev In Geographic Space, this typical value forms a horizontal plane implying the average is everywhere. X= 22.9 …click anywhere on the map surface and the corresponding histogram pillar is highlighted …click anywhere on the histogram and all map locations in that range are highlighted Continuous Map Surface Geographic Distribution Surface Modeling techniques are used to derive a continuous map surface from discrete point data– fits a Surface to the data. Inverse Distance Weighted (IDW) spatial interpolation assigned distance- weighted average of sample points

17. Spatial Statistics Operations (fundamental procedures) Surface Modeling Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization) …91 total customers in the Roving Window 2D Map 3D Surface 2D Map 3D Surface Basic Descriptive Statistics Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.) Basic Classification Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability) Unique Map Descriptive Statistics Unique Map Descriptive Statistics — (Roving Window Summaries) Map Comparison Map Comparison — (Normalization, Joint Coincidence, Statistical Tests)

18. Spatial Statistics Operations (classification and predictive statistics) Predictive Statistics Predictive Statistics — (Map Correlation/Regression, Data Mining Engines) Advanced Classification Advanced Classification — (Map Similarity, Maximum Likelihood, Clustering) …all other Data Distances are scaled in terms of their relative similarity to the comparison point (0 to 100%) The farthest away point in data space (Least Similar) is set to 0 and the Comparison Point is set to 100% similar… Keystone concept is… “Spatial Correlation”

19. www.innovativegis.com/basis/Papers/Other/SpatialSTEM/ Map-ematically Messing with Maps Map-ematically Messing with Maps: Extending Traditional Math/Stat to Grid-based Map Analysis and Modeling 12 page white paper Further Reading Teaching Materials 6 page appendix Today’s Introductory Seminar …online links to Handout, PowerPoint and Readings are posted at — www.innovativegis.com/basis/Present/CSU_winter2012/ Leading to a… Possible Mini-Workshop …on teaching SpatialSTEM. Two 2-hour sessions with hands-on experience in the concepts and procedures. One optional 2-hour session developing ‘tailored” exercise. Contact Joe Berry at www.innovativegis.com for more information.www.innovativegis.com