Traditional Statistics Mean, StDev (Normal Curve) Mean, StDev (Normal Curve) Central Tendency Central Tendency Typical Response (scalar) Typical Response (scalar) Minimum= 5.4 ppm Maximum= ppm Mean= 22.4 ppm StDEV= 15.5 Spatial Statistics Map of the Variance (gradient) Map of the Variance (gradient) Spatial Distribution Spatial Distribution Numerical Spatial Relationships Numerical Spatial Relationships Spatial Distribution (Surface) Traditional GIS Points, Lines, Polygons Points, Lines, Polygons Discrete Objects Discrete Objects Mapping and Geo-query Mapping and Geo-query Forest Inventory Map Grid-based Map Analysis (Spatial Analysis/Statistics) Spatial Analysis Cells, Surfaces Cells, Surfaces Continuous Geographic Space Continuous Geographic Space Contextual Spatial Relationships Contextual Spatial Relationships Erosion Potential (Surface)

Grid-Based Map Analysis Spatial analysis investigates the “contextual” relationships in mapped data… Reclassify — reassigning map values (position; value; size, shape; contiguity) Reclassify — reassigning map values (position; value; size, shape; contiguity) Overlay — map overlay (point-by-point; region-wide; map-wide) Overlay — map overlay (point-by-point; region-wide; map-wide) Distance — proximity and connectivity (movement; optimal paths; visibility) Distance — proximity and connectivity (movement; optimal paths; visibility) Neighbors — ”roving windows” (slope/aspect; diversity; anomaly) Neighbors — ”roving windows” (slope/aspect; diversity; anomaly) (Berry) Data Mining investigates the “numerical” relationships in mapped data… Descriptive — aggregate statistics (e.g., average/stdev, similarity, clustering) Descriptive — aggregate statistics (e.g., average/stdev, similarity, clustering) Predictive — relationships among maps (e.g., regression) Predictive — relationships among maps (e.g., regression) Prescriptive — appropriate actions (e.g., optimization) Prescriptive — appropriate actions (e.g., optimization) Surface modeling maps the spatial distribution and pattern of point data… Map Generalization — characterizes spatial trends (e.g., titled plane) Map Generalization — characterizes spatial trends (e.g., titled plane) Spatial Interpolation — deriving spatial distributions (e.g., IDW, Krig) Spatial Interpolation — deriving spatial distributions (e.g., IDW, Krig) Other — roving window/facets (e.g., density surface; tessellation) Other — roving window/facets (e.g., density surface; tessellation) Spatial Statistics

Classes of Spatial Analysis Operators (Berry) …all spatial analysis involves generating new map values (numbers) as a mathematical or statistical function of the values on another map layer(s) (See |MapCalc\CrossReference for a cross reference of MapCalc operations and ESRI Grid/Spatial Analysis and EDAS)

Reclassifying Maps (Berry)

Overlaying Maps (Berry)

Evaluating Habitat Suitability (Berry) Manual Map Overlay (Binary) Ranking Overlay (Binary Sum) Rating Overlay (Rating Average) Generating maps of animal habitat… (digital slide show Hugag2) Hugag2 The Hugag is a curious beast with strong preferences for terrain configuration:  Prefers low elevations (severe nose bleeds at higher altitudes)  Prefers gentle slopes (fear of falling over and unable to get up)  Prefers southerly aspects (a place in the sun)

Covertype Water Mask 0= No, 1= Yes Habitat Rating 0= No, 1 to 9 Good Constraint Map CombinedMap Habitat Rating Bad 1 to 9 Good (Times 1) (1) (1) Conveying Suitability Model Logic (Short Exercise #3) (Berry) InterpretedMaps gentle slopes Slope Preference Bad 1 to 9 Good Aspect Preference Bad 1 to 9 Good Elevation Preference Bad 1 to 9 Good southerly aspects lower elevations Derived Maps Slope Aspect Base Maps Elevation FactJudgment CalibrateAlgorithmWeight Reclassify Overlay …while Reclassify and Overlay operations aren’t very exciting, they are frequently used

Habitat Rating Bad 1 to 9 Good gentle slopes Slope Preference Bad 1 to 9 Good Aspect Preference Bad 1 to 9 Good Elevation Preference Bad 1 to 9 Good southerly aspects lower elevations Slope Aspect Extending Model Criteria (Berry) Elevation Additional criteria can be added… forests ForestPreference Bad 1 to 9 Good Forest Proximity Forests —Hugags would prefer to be in/near forested areas water WaterPreference Bad 1 to 9 Good Water Proximity Water —Hugags would prefer to be near water —Hugags are 10 times more concerned with slope, forest and water criteria than aspect and elevation (Times 10) (10) (10) (1) (1)

Reclassifying Maps Overlaying Maps Reclassify & Overlay Operations (MapCalc) (Berry)

Reclassify & Overlay Techniques (Full Exercise #3) Spatial analysis … Spatial analysis … use MapCalc to implement the following MapCalc – SIZE Covertype FOR Covertype_size – CLUMP Covertype AT 1 Diagonally FOR Covertype_clumps – SIZE Covertype_clumps FOR Covertype_clump_size – CONFIGURE Covertype_clumps Edges FOR Covertype_clumps_edges – CONFIGURE Covertype_clumps Convexity FOR Covertype_clumps_shape “RENUMBER Slope / Aspect / Elevation FOR S_Pref / A_pref / E_pref” from Hugag_Habitat.scr script “RENUMBER Slope / Aspect / Elevation FOR S_Pref / A_pref / E_pref” from Hugag_Habitat.scr script – COMPUTE S_Pref Times A_Pref Times E_Pref FOR Binary_model – COMPUTE S_Pref Plus A_Pref Plus E_Pref FOR Ranking_model – CROSSTAB Covertype WITH Water Simply – CALCULATE (Covertype * 10) + Water FOR CW_codes – COMPOSITE Covertype WITH Slope Average FOR Covertype_avgSlope – RENUMBER Covertype ASSIGNING 0 TO 2 THRU 3 FOR OpenWater_binary – COMPUTE OpenWater_binary Times Slope FOR OpenWater_slope (Berry)

Establishing Distance and Connectivity (Berry)

(digital slide show DIST2) DIST2 (Berry)

Spatial Analysis (Short Exercise #4a) (Berry)(Berry) Simple Proximity to Roads …far from Roads SPREAD Roads TO 100 Simply FOR Road_prox FOR Road_prox Distance Operators — simple/and effective proximity Effective Proximity to Roads wFriction sFriction Friction Relative Barrier— terrain steepness Absolute Barrier— water Impassable Difficulty Impedance to Movement SPREAD Roads TO 100 Simply THRU Friction FOR Road_hikingprox FOR Road_hikingprox

Distance/Connectivity Techniques (Full Exercise #4a) Spatial analysis … Spatial analysis … use MapCalc to implement the following MapCalc (Berry) – SPREAD Housing TO 20 FOR Housing_simpleprox – SPREAD Roads TO 20 FOR Roads_simpleprox – RENUMBER Covertype ASSIGNING 0 TO 1 ASSIGNING 3 TO 2 ASSIGNING 7 TO 3 ASSIGNING 3 TO 2 ASSIGNING 7 TO 3 FOR C_friction FOR C_friction – SPREAD Roads TO 75 THRU C_friction FOR Road_hikingprox – RENUMBER Locations ASSIGNING 0 TO 2 THRU 5 FOR Ranch – SPREAD Ranch TO 35 Simply FOR Ranch_simpleprox – RENUMBER Roads ASSIGNING 1 TO 1 THRU 43 FOR R_friction – COVER C_friction WITH R_friction FOR CR_friction – SPREAD Ranch TO 75 THRU CR_friction FOR Ranch_hikingprox – RENUMBER Locations ASSIGNING 0 TO 1 ASSIGNING 0 TO 3 THRU 5 FOR Cabin – STREAM Cabin OVER Ranch_hikingprox FOR Path – COMPUTE Ranch_hikingprox times path FOR Path_hikingprox

Generating an Effective Travel-time Buffer (Berry) a) superimposition of an analysis grid over the area of interest b) “burns” the store location into its corresponding grid cell c) “burns "primary and residential streets are identified d) travel-time buffer derived from the two grid layers (Store and Streets) (Store and Streets)

Travel-Time Waves (Berry) Travel-time is computed as a series of increasing waves moving away from a starting location that are constrained by the streets… …creates an Accumulation Surface identifying travel-time to every location considering absolute (streets) and relative (speed) barriers to movement (digital slide show TTime2) TTime2

Travel-Time Connectivity (Berry) …increasing distance from a point forms bowl-shaped accumulation surface …steepest downhill path identifies the optimal path– wave front that got there first. …SPREADing from multiple locations identifies catchment areas– locations closest to starting locations …what do you think the ridges represent?

Accumulation Surface Analysis (Berry) …increasing distance from a point forms bowl-shaped accumulation surface Simple distance – symmetrical bowl; constant slope Absolute barrier – abrupt pillars; constant slope Relative barrier – gradual humps with changing slope depending on relative impedance friction) …subtracting two accumulation surfaces identifies relative advantage Zero – equidistant Zero – equidistant Sign – which store has advantage Sign – which store has advantage Magnitude – strength of advantage Magnitude – strength of advantage …what would get if you added the two surfaces?

CustomerDatabase(non-spatial) …GeoCoding plots customers address on the streets map Vector (point) Latitude, Longitude, C, R CustomerDatabase(spatial) (Berry) Analysis Frame as Primary Key (Column, Row) Raster (cell) Analysis Frame …V to R Conversion plots customers location in the analysis frame (grid) …can append any GIS derived information (Col,Row) …Append Col, Row, Lat, Lon of cell location to customer records

Variable-Width Buffers (Simple/uphill proximity) Simple Buffer– “as-the-crow-flies” proximity to the road; no absolute or relative barriers are considered Clipped Buffer– simple proximity for just the land areas Uphill Buffer– simple proximity to the road for just the areas that are uphill from the road; absolute barrier (uphill only– absolutely no downhill steps) (Berry)

Establishing Visual Connectivity (Viewshed) …like SPREAD, RADIATE starts somewhere (starter cell) and moves through geographic space by steps (wave front) assigning a 1 (seen) to locations with tangents larger than the previous ones to locations with tangents larger than the previous ones (Berry) Seen if new tangent exceeds all previous tangents along the line of sight— At At Thru ) Thru ) Onto Onto Radiate – analogous to a searchlight casting its beam light onto the landscape Simply – viewshed Completely – number of “viewers” that see each location Weighted – viewer cell value is added

Calculating Visual Exposure (# Times Seen) Visual exposure identifies how many times each map location is seen from a set of viewer locations (Berry)

Visual Exposure from Extended Features A visual exposure map identifies how many times each location is seen from an “extended eyeball” composed of numerous viewer locations (road network) (Berry) Simply – viewshed Completely – number of “viewers” that see each location Weighted – viewer cell value is added

Weighted Visual Exposure (Sum of Viewer Weights) Different road types are weighted by the relative number of cars per unit of time …the total “number of cars” replaces the “number of times seen” for each grid location (Berry) Simply – viewshed Completely – number of “viewers” that see each location Weighted – viewer cell value is added

Spatial Analysis (Short Exercise #4b) (Berry)(Berry) Viewshed from Roads …not seen RADIATE Roads OVER Elevation AT 1 TO 100 Simply FOR Road_viewshed Visual Exposure from Roads …seen a lot RADIATE Roads OVER Elevation TO 100 AT 1 Completely FOR Road_VExposure Visual Exposure Operators — viewshed and visual exposure Roads Elevation RADIATE Road_classes OVER Elevation TO 100 AT 1 Weighted FOR Road_wVExposure # Cars Road_classes

Real-World Visual Analysis Weighted visual exposure map for an ongoing visual assessment in a national recreation area— the project developed visual vulnerability maps from the reservoir in the center of the park and a major highway running through the park. In addition, aesthetic maps were generated based on visual exposure to pretty and ugly places in the park (Senior Honors Thesis by University of Denver Geography student Chris Martin, 2003) (Berry)

Variable-Width Buffers (Line-of-sight) Line-of-Sight Buffer– identifies all land locations (clipped) within 250m that can be seen from the road… 250m “viewshed” of the road 250m “viewshed” of the road (Berry) Line-of-Sight Exposure– notes the number of time each location in the buffer is seen Line-of-Sight Noise– locations hidden behind a ridge or farther away from a source (road) greatly decrease noise levels.

Visual Analysis Techniques (Full Exercise #4b) Spatial analysis … Spatial analysis … use MapCalc to implement the following MapCalc – RADIATE Ranch OVER ELEVATION TO 35 AT 5 SIMPLY FOR Ranch_viewshed AT 5 SIMPLY FOR Ranch_viewshed – RADIATE Roads OVER ELEVATION TO 35 AT 5 SIMPLY FOR Roads_viewshed AT 5 SIMPLY FOR Roads_viewshed – RADIATE Roads OVER ELEVATION TO 35 AT 5 COMPLETELY FOR Roads_VExposure AT 5 COMPLETELY FOR Roads_VExposure – RADIATE Housing OVER ELEVATION TO 35 WEIGHTED FOR housing_WeightedVE WEIGHTED FOR housing_WeightedVE (Berry) – SLICE Housing_WeightedVE INTO 4 FOR Housing_VE_Index FOR Housing_VE_Index – SLICE Roads_VExposure INTO 4 FOR Roads_VE_Index FOR Roads_VE_Index – ANALYZE Housing_VE_Index WITH Roads_VE_Index Mean WITH Roads_VE_Index Mean FOR RH_VE_Index_avg FOR RH_VE_Index_avg

Characterizing Neighborhoods (Berry)

Characterizing Terrain Steepness (Slope) (Berry) Min= 0.00 Max= Avg= Min= 0.00 Max= Avg= Min= 0.00 Max= Avg= 3.56 Min= 0.00 Max= Avg= Slope is can be calculated several ways– by calculating the best fitting plane to all nine elevation values, or by selecting the maximum, minimum or average of the eight individual slopes

Creating a Housing Density Map (Scan Total) The TOTAL number of houses within 500 meters is calculated for each map location (Berry)

Creating a Cover Type Diversity Map (Scan Diversity) …a DIVERSITY map indicates the number of different map values that occur within a window… e.g., cover types. As the window is enlarged, the diversity increases. (Berry)

Spatial Analysis (Short Exercise #5) (Berry)(Berry) Neighbor Operators — summarizing nearby values SCAN Covertype Diversity WITHIN 4 CIRCLE FOR Covertype_diversity FOR Covertype_diversity Covertype_diversity Most diverse Covertype SCAN Houses Total 0.0 WITHIN 6 CIRCLE FOR Housing_density FOR Housing_density Housing_density Highest density Houses SCAN Slopemap CoffVar WITHIN 2 CIRCLE FOR Roughness FOR Roughness Terrain_roughness Mostrough Slopemap

Characterizing “Edginess” A simple EDGINESS model for the meadow involves assigning 1 to the meadow (Renumber) and then calculating the total values within a 3x3 window for just the meadow area (Around) (Berry) 2 Very Edgy 8 Not Edgy

Landscape Analysis (Berry)

Neighbor Techniques (Full Exercise #5) – SCAN Covertype DIVERSITY WITHIN 2 FOR Covertype_diversity2 FOR Covertype_diversity2 – SCAN Covertype DIVERSITY WITHIN 4 FOR Covertype_diversity4 FOR Covertype_diversity4 – SCAN COVERTYPE DIVERSITY WITHIN 4 AROUND ROADS FOR Covertype_diversity4_roads AROUND ROADS FOR Covertype_diversity4_roads – SCAN ELEVATION AVERAGE WITHIN 4 FOR Elevation_smooth4 – COMPUTE ELEVATION MINUS Elevation_smooth4 FOR Convex_concave_terrain – SCAN SLOPE COFFVAR WITHIN 1 FOR Slope_roughness – SLOPE SLOPE FOR What? – RENUMBER Ranch_hikingprox ASSIGNING PMAP_NULL TO 100 FOR Ranch_hikingprox_mask – SLOPE Ranch_hikingprox_masked FOR What_else? – ORIENT Ranch_hikingprox_masked FOR You_have_to_be_kiding! (Berry) Spatial analysis … Spatial analysis … use MapCalc to implement the following MapCalc

…but before we leave Spatial Analysis (operations for reclassify, overlay, distance and neighbors) to tackle Spatial Statistics, any… Questions? Questions? (Berry)