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SpatialSTEM: A Mathematical Structure for Understanding, Communicating and Teaching Fundamental Concepts in Spatial Reasoning, Map Analysis and Modeling Workshop Session — Spatial Statistics Operations 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/ 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: [email protected] — Website: www.innovativegis.com/basis SpatialSTEM Seminar Review (Session 1—Spatial Analysis operations) “Technology Tool” vs. “Analysis Tool” Geotechnology as 21st Century “mega-technology” Spatial Triad (RS, GIS, GPS) “Descriptive Mapping” vs. “Prescriptive Analysis” Nature of Mapped Data Vector vs. Raster (Grid) Grid Map Layers and Map Stack Analysis Frame Overview Seminar sSTEM_workshop1.ppt Review Seminar Session 1 Spatial Analysis Grid Math Hands-on Exercise Calculating Simple and Effective Proximity Viewsheds Visual Exposure Viewshed Visual Exposure and Noise Buffers “Contextual” Relationships Hands-on Exercise Variable-width Buffers Viewsheds and Visual Exposure (Berry) Overview of Map Analysis Approaches (Spatial Analysis and Spatial Statistics) Spatial Analysis Traditional GIS Elevation (Surface) Forest Inventory Map • Points, Lines, Polygons • Cells, Surfaces • Discrete Objects • Continuous Geographic Space • Contextual Spatial Relationships • Mapping and Geo-query …last session Traditional Statistics Spatial Statistics Spatial Distribution (Surface) Minimum= 5.4 ppm Maximum= 103.0 ppm Mean= 22.4 ppm StDEV= 15.5 • Mean, StDev (Normal Curve) • Map of Variance (gradient) • Central Tendency • Spatial Distribution • Typical Response (scalar) • Numerical Spatial Relationships (Berry) Spatial Data Perspectives (numerically defining the What in “Where is What”) Numerical Data Perspective: how numbers are distributed in “Number Space” 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 – Ordinal numbers imply a definite ordering from small to large – like a ladder, however with varying spaces between rungs Quantitative: deals with measurable quantities 4 5 2 Nominal —Categories 4 1 3 6 5 6 Ordinal —Ordered 3 2 1 (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 – Ratio numbers has all the properties of interval numbers plus a clear/constant definition of 0.0 – like a ladder with a fixed base. 6 5 4 3 2 1 Interval —Constant Step Ratio —Fixed Zero 6 5 4 3 2 1 0 Binary: a special type of number where the range is constrained to just two states— such as 1=forested, 0=non-forested Spatial Data Perspective: how numbers are distributed in “Geographic Space” Choropleth numbers form sharp/unpredictable boundaries in geographic space – e.g., a road “map” Elevation —Continuous gradient Roads —Discrete Groupings Isopleth numbers form continuous and often predictable gradients in geographic space – e.g., an elevation “surface” (Berry) Desktop Mapping (GeoExploration) vs. Map Analysis (GeoScience) “Maps are numbers first, pictures later” — “Quantitative analysis of mapped data” Desktop Mapping graphically links generalized statistics to discrete spatial objects (Points, Lines, Polygons)—spatially aggregated summaries (GeoExploration) Desktop Mapping Map Analysis X, Y, Value Data Space Field Data Geographic Space Standard Normal Curve Point Sampled Data (Numeric Distribution) Average = 22.0 StDev = 18.7 (Geographic Distribution) 40.7 …not a problem Discrete Spatial Object 22.0 Spatially Generalized High Pocket Continuous Spatial Distribution Spatially Detailed Discovery of sub-area… Adjacent Parcels Map Analysis map-ematically relates patterns within and among continuous spatial distributions (Map Surfaces)— spatially disaggregated analysis (GeoScience) (Berry) Exercise 2-1) Linking Data Space and Geographic Space (Calculate) Data, Map and Stat Views …the following “hands-on” experience “links” field data values with discrete point map values and nonspatial statistical metrics Install MapCalc …if you haven’t. Calculate Command Avg Below > 1SD above Above Continuous Surface Views …the following “hands-on” experience “links” the spatial distribution of field data with its non-spatial statistical metrics (Berry) Spatial Statistics Operations (Numerical Context) GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if) Grid Layer Map Stack 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 (Berry) 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) Creating a Crime Risk Density Surface (Density Analysis) Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.) Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability) Map Comparison — (Normalization, Joint Coincidence, Statistical Tests) Unique Map Descriptive Statistics — (Roving Window and Regional Summaries) Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization) Geo-Coding Crime Incident Reports Vector to Raster Crime Incident Locations Roving Window Grid Incident Counts Density Surface Totals Reclassify Classified Crime Risk Geo-coding identifies geographic coordinates from street addresses Grid Incident Counts Calculates the total number of reported crimes within a roving window– Density Surface Totals the number of incidences (points) within in each grid cell 91 3D surface plot 2D display of discrete Grid Incident Counts 2D perspective display of crime density contours Classified Crime Risk Map (Berry) Spatial Interpolation (iteratively smoothing the spatial variability) Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.) Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability) Map Comparison — (Normalization, Joint Coincidence, Statistical Tests) Unique Map Descriptive Statistics — (Roving Window and Regional Summaries) Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization) 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) (Berry) Surface Modeling Point Sampling: Collecting X,Y coordinates with field samples provides a foothold for generating continuous map surfaces used in map analysis and modeling. (Spatial Interpolation) 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. Each record contains X,Y coordinates (Where) followed by Data Values (What) identifying the characteristics/conditions at that location— a geo-registered dB 29.4 Window Configuration …size, shape and weighting procedure 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— Spatial Autocorrelation meaning that “what occurs at a location is related to the conditions at nearby locations” (intra-variable dependence) Exercise 2-2) Surface Modeling (Density Analysis and IDW Interpolation) Customer Density Unusually High Density Density Analysis …the following “hands-on” experience creares a density surface (total customers within 6 cells) and isolates pockets of unusually high density Period2 Period2 Big Increase 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 (Berry) Spatial Statistics Operations Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.) Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability) Map Comparison — (Normalization, Joint Coincidence, Statistical Tests) Unique Map Descriptive Statistics — (Roving Window and Regional Summaries) Surface Modeling — (Density Analysis, Spatial Interpolation, Map Generalization) Advanced Classification — (Map Similarity, Maximum Likelihood, Clustering) 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) …all other Data Distances are scaled in terms of their relative similarity to the comparison point (0 to 100% similar) Map Stack …there are two types of Spatial Dependency— Spatial Autocorrelation— within a map layer Spatial Correlation— among map layers Spatial Correlation meaning that “what occurs at a location is related to the conditions of other map variables at that location” (inter-variable dependence) (Berry) Exercise 2-3) Characterizing Map Similarity (Home Age, Value and Housing Density) RELATE command 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 Least Similarity Surface (Berry) Predictive Spatial Statistics Basic Descriptive Statistics — (Min, Max, Median, Mean, StDev, etc.) Basic Classification — (Reclassify, Binary/Ranking/Rating Suitability) Map Comparison — (Normalization, Joint Coincidence, Statistical Tests) Unique Map Descriptive Statistics — (Roving Window and Regional Summaries) Spatial DBMS export grid …pass map layers to any Statistics or to dB with each cell a Surface Modeling — (Density Analysis, Spatial Interpolation, Maplayers Generalization) Data Mining package record & each layer a field Advanced Classification — (Map Similarity, Maximum Likelihood, Clustering) Predictive Statistics — (Map Correlation/Regression, Data Mining Engines) Dependent Map variable is what you want to predict… Univariate Linear Regressions Multivariate Linear Regression Predicted – Actual = Error …substantially under-estimates (2/3 of the error within 5.26 and 16.94) …can use error to generate Error Ranges for calculating new regression equations …from a set of easily measured Independent Map variables Actual Predicted Error Surface Stratified Error (Berry) Exercise 2-4) Building Predictive Models (map regression) Loan Accounts Loan Concentration Deriving a Dependent Variable Map …density analysis is used to derive a map surface indicating loan concentration Univariate Map Correlation and Regression …coincidence of dependent and independent map layers (one at a time) Multivariate Map Regression …coincidence of dependent and independent map layers (all at once) Multivariate Regression Univariate Regression (Berry) Example “Hands-on” Exercises (Session 3 homework) 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). (Berry)