2. Presented by Joseph K. Berry Adjunct Faculty 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: jberry@innovativegis.com — Website: www.innovativegis.com/basis There is a “map-ematics” that extends traditional math/stat concepts and procedures for the quantitative analysis of spatial data This workshop provides a fresh perspective on interdisciplinary instruction at the college level by combining the philosophy and approach of STEM with the spatial reasoning and analytical power of grid-based Map Analysis and Modeling This PowerPoint is posted at www.innovativegis.com/basis/Courses/SpatialSTEM/ SpatialSTEM: A Mathematical Structure for Understanding, Communicating and Teaching Fundamental Concepts in Spatial Reasoning, Map Analysis and Modeling Workshop Session — Spatial Statistics Operations

3. SpatialSTEM Seminar Review (Session 1—Spatial Analysis operations) (Berry) Calculating Simple and Effective Proximity Variable-width Buffers Hands-onExercise Spatial Analysis “Contextual” Relationships Grid Math Hands-onExercise Viewsheds Visual Exposure Viewshed Visual Exposure and Noise Buffers Viewsheds and Visual Exposure Nature of Mapped Data Vector vs. Raster (Grid) Grid Map Layers and Map Stack Analysis Frame ReviewSeminar Session 1 “Technology Tool” vs. “Analysis Tool” Geotechnology as 21 st Century “mega-technology” Spatial Triad (RS, GIS, GPS) “Descriptive Mapping” vs. “Prescriptive Analysis” Overview Seminar sSTEM_workshop1.ppt

4. Traditional Statistics Mean, StDev (Normal Curve) Central Tendency Typical Response (scalar) Minimum= 5.4 ppm Maximum= 103.0 ppm Mean= 22.4 ppm StDEV= 15.5 Spatial Statistics Map of Variance (gradient) Spatial Distribution Numerical Spatial Relationships Spatial Distribution (Surface) Traditional GIS Points, Lines, Polygons Discrete Objects Mapping and Geo-query Forest Inventory Map Spatial Analysis Cells, Surfaces Continuous Geographic Space Contextual Spatial Relationships Elevation (Surface) …last session (Berry) Overview of Map Analysis Approaches (Spatial Analysis and Spatial Statistics)

5. Spatial Data Perspectives (numerically defining the What in “Where is What”) Spatial Data Perspective : how numbers are distributed in “Geographic Space” Numerical Data Perspective : how numbers are distributed in “Number Space”  Qualitative: deals with unmeasurable qualities (very few math/stat operations available)  Quantitative: deals with measurable quantities (a wealth of 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 – 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 – 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 – 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  Choropleth numbers form sharp/unpredictable boundaries in geographic space – e.g., a road “map” Roads —Discrete Groupings  Isopleth numbers form continuous and often predictable gradients in geographic space – e.g., an elevation “surface” Elevation —Continuous gradient  Binary: a special type of number where the range is constrained to just two states— such as 1=forested, 0=non-forested (Berry)

6. Map Analysis Geographic Space (Geographic Distribution) Map Analysis map-ematically relates patterns within and among continuous spatial distributions (Map Surfaces)— spatially disaggregated analysis (GeoScience) — “Quantitative analysis of mapped data” Continuous Spatial Distribution Discrete Spatial Object Adjacent Parcels High Pocket Discovery of sub-area… Desktop Mapping (GeoExploration) vs. Map Analysis (GeoScience) Average = 22.0 StDev = 18.7 Desktop Mapping Data Space Field Data Standard Normal Curve Desktop Mapping graphically links generalized statistics to discrete spatial objects (Points, Lines, Polygons)—spatially aggregated summaries (GeoExploration) X, Y, Value Point Sampled Data (Numeric Distribution) “Maps are numbers first, pictures later” 22.0 Spatially Generalized Spatially Detailed 40.7 …not a problem (Berry)

7. Exercise 2-1) Linking Data Space and Geographic Space (Calculate) Calculate Command BelowAbove Avg> 1SD above Continuous Surface Views …the following “hands-on” experience “links” the spatial distribution of field data with its non-spatial statistical metrics Data, Map and Stat Views …the following “hands-on” experience “links” field data values with discrete point map values and non- spatial statistical metrics Install MapCalc …if you haven’t. (Berry)

8. Spatial Statistics Operations (Numerical Context) (Berry) GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if) 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: Spatial Statistics seeks to map the spatial variation in a data set instead of focusing on a single typical response (central tendency) ignoring the data’s spatial distribution/pattern, and thereby provides a mathematical/statistical framework for analyzing and modeling the Numerical Spatial Relationships within and among grid map layers Map Analysis Toolbox Map StackGrid Layer

9. 2D display of discrete Grid Incident Counts Grid Incident Counts the number of incidences (points) within in each grid cell Classified Crime Risk Classified Crime Risk Reclassify Classified Crime Risk Map Calculates the total number of reported crimes within a roving window– Density Surface Totals 2D perspective display of crime density contours 3D surface plot 91 Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization) Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.) Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability) Unique Map Descriptive Statistics — (Roving Window and Regional Summaries) Map Comparison — (Normalization, Joint Coincidence, Statistical Tests) Creating a Crime Risk Density Surface (Density Analysis) Density Surface Totals Density Surface Totals Roving WindowVector to Raster Grid Incident Counts Grid Incident Counts Crime Incident Reports Crime Incident Locations Geo-Coding Geo-coding identifies geographic coordinates from street addresses (Berry)

10. Spatial Interpolation (iteratively smoothing the spatial variability) The “iterative smoothing” process is similar to slapping a big chunk of modeler’s clay over the “data spikes,” then taking a knife and cutting away the excess to leave a continuous surface that encapsulates the peaks and valleys implied in the original field samples …repeated smoothing slowly “erodes” the data surface to a flat plane = AVERAGE (digital slide show SStat2)SStat2 Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization) Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.) Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability) Unique Map Descriptive Statistics — (Roving Window and Regional Summaries) Map Comparison — (Normalization, Joint Coincidence, Statistical Tests) (Berry)

11. Surface Modeling (Spatial Interpolation) 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— a geo-registered dB For example, Inverse Distance Weighted (IDW) spatial interpolation calculates the distances from an non-sampled 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 spatial variation in the data set, provided there is sufficient— 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 non-sampled locations, thereby characterizing the data set’s Geographic Distribution. 29.4 Window Configuration …size, shape and weighting procedure Spatial Autocorrelation meaning that “what occurs at a location is related to the conditions at nearby locations” (intra-variable dependence)

12. Exercise 2-2) Surface Modeling (Density Analysis and IDW Interpolation) (Berry) Density Analysis …the following “hands-on” experience creares a density surface (total customers within 6 cells) and isolates pockets of unusually high density Unusually High DensityCustomer Density Spatial Interpolation (IDW) …the following “hands-on” experience demonstrates spatially interpolating a discrete point map into a continuous map surface that characterizes the spatial distribution of the data Big Increase Period2

13. Spatial Statistics Operations Advanced Classification — (Map Similarity, Maximum Likelihood, Clustering) Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.) Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability) Unique Map Descriptive Statistics — (Roving Window and Regional Summaries) Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization) Map Comparison — (Normalization, Joint Coincidence, Statistical Tests) …all other Data Distances are scaled in terms of their relative similarity to the comparison point (0 to 100% similar) (Berry) The farthest away point in data space (Least Similar) is set to 0 and locations identical to the Comparison Point are set to 100 % similar (same numerical pattern) Map Stack Spatial Correlation meaning that “what occurs at a location is related to the conditions of other map variables at that location” (inter-variable dependence) …there are two types of Spatial Dependency — Spatial Autocorrelation— within a map layer Spatial Correlation— among map layers

14. Exercise 2-3) Characterizing Map Similarity (Home Age, Value and Housing Density) (Berry) Map Similarity …the following “hands-on” experience calculates a similarity map for a location based on the relative comparison of data patterns for each grid cell in a map stack to that of a comparison data pattern RELATE command Similarity Surface Least

15. Predictive Spatial Statistics Predictive Statistics — (Map Correlation/Regression, Data Mining Engines) Advanced Classification — (Map Similarity, Maximum Likelihood, Clustering) Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.) Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability) Unique Map Descriptive Statistics — (Roving Window and Regional Summaries) Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization) Map Comparison — (Normalization, Joint Coincidence, Statistical Tests) Predicted – Actual = Error …substantially under-estimates (2/3 of the error within 5.26 and 16.94) Error Surface Univariate Linear Regressions PredictedActual Multivariate Linear Regression …can use error to generate Error Ranges for calculating new regression equations Stratified Error Spatial DBMS export grid layers to dB with each cell a record & each layer a field …pass map layers to any Statistics or Data Mining package Dependent Map variable is what you want to predict… …from a set of easily measured Independent Map variables (Berry)

16. Exercise 2-4) Building Predictive Models (map regression) (Berry) Deriving a Dependent Variable Map …density analysis is used to derive a map surface indicating loan concentration Loan AccountsLoan Concentration Univariate Map Correlation and Regression …coincidence of dependent and independent map layers (one at a time) Univariate Regression Multivariate Map Regression …coincidence of dependent and independent map layers (all at once) Multivariate Regression

17. Example “Hands-on” Exercises (Session 3 homework) (Berry) www.innovativegis.com/basis/Senarios/ Identifying Campground Suitability …preferences for locating a campground— gentle slopes near roads near water good views of water westerly oriented. Identifying Similar Data Patterns … creates a map of “data groupings” that are as “similar as possible within the group and “as dissimilar as possible” among the groupings given a set of map layers (spatial clustering).