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Introduction to GIS Modeling Week 9 — Spatial Data Mining GEOG 3110 –University of Denver Presented by Joseph K. Berry W. M. Keck Scholar, Department of Geography, University of Denver Basic Descriptive Statistics and its GIS Expression: Normalizing maps; Mapping spatial dependency Linking Numeric and Geographic Patterns: Map comparison; Similarity maps; Clustering mapped data; Investigating map correlation; Developing prediction models; Assessing prediction results Kicking for the Finish Exercise #9 — to tailor your work to your interests, you can choose to not complete this standard exercise, however in lieu of the exercise you will submit a short paper (4-8 pages) on a GIS modeling topic of your choice. Due 12:00 midnight, Thursday, March 10th. Optional Exercises — you can turn in these exercises for extra credit anytime before 5:00 pm, Tuesday, March 15th. Final Exam Study Questions — covering weeks 7-10, Spatial Statistics and Future Directions; the exam is optional and can only improve your grade. You are encouraged to study together and exchange insights about answering the questions. …at least three-fourths of the exam will be taken directly (verbatim) from the list of study questions. The format will be similar to the last exam, with questions from Terminology, Procedures and Basic Concepts, How Things Work and Mini-Exercises. …study questions for Exam 2 are posted on the class website now; send me an email if a question needs further explanation and I will post the clarification. Final Exam — Covers material from weeks 7 (GIS Modeling), 8 (Surface Modeling), 9 (Spatial Data Mining) and 10 (Future Directions) …exam posted on the class website by 10:00 am, Thursday, March 10th and must be completed by 8:00 am, Monday, March 14th. … at the end of the last class I will be handing out a CD with all of the class material— sort of a “graduation present” that will keep you GIS-ing for years (Berry) An Analytic Framework for GIS Modeling (Last week) Surface Modelling operations involve creating continuous spatial distributions from point sampled data (univariate). (This week) Spatial Data Mining operations involve characterizing numerical patterns and relationships among mapped data (multivariate). See www.innovativegis.com/basis/Download/IJRSpaper/ (Berry) Basic Concepts in Statistics (Standard Normal Curve) See Beyond Mapping III , Topic 7, Linking Data Space and Geographic Space (Berry) Basic Concepts in Statistics (SN_Curve Shape) Kurtosis …shape (positive= peaked; negative= flat) See Beyond Mapping III , Topic 7, Linking Data Space and Geographic Space (Berry) Basic Concepts in Statistics (SN_Curve Shape continued) …Multi-modal …Skewness (positive= right; negative= left) See Beyond Mapping III , Topic 7, Linking Data Space and Geographic Space (Berry) Preprocessing Mapped Data (Preprocessing Types 1-3) Preprocessing involves conversion of raw data into consistent units that accurately represent mapped conditions (4 considerations) Calibration 1 — “tweaking” the values… sort of like a slight turn on a bathroom scale to alter the reading to what you know is your ‘true weight’ Translation 2 — converts map values into appropriate units for analysis, such as feet into meters or bushels per acre (measure of volume) into tons per hectare (measure of mass) Antenna Offset GPS Fix Delay Overlap and Multiple Passes Mass Flow Lag and Mixing Adjustment/Correction 3 — dramatically changes the data, such as post processing GPS coordinates and/or Mass Flow Lag adjustment (Berry) Normalizing Mapped Data (4 th type of preprocessing) Normalization — involves standardization of a data set, usually for comparison among different types of data… “apples and oranges to mixed fruit scale” Goal …Norm_GOAL = (mapValue / 250 ) * 100 0-100 …Norm_0-100 = ((mapValue – min) * 100) / (max – min) + 0 SNV …Norm_SNV = ((mapValue - mean) / stdev) * 100 Norm_GOAL = (Yield_Vol / 250 ) * 100 …generates a standardized map based on a yield goal of 250 bushels/acre. This map can be used in analysis with other goal-normalized maps, even from different crops Key Concept Since normalization involves scalar mathematics (constants), the pattern of the numeric distribution (histogram) and the spatial distribution (map) do not change …same relative distributions See Beyond Mapping III , Topic 18, Understanding Grid-based Data Note: the generalized rescaling equation is… Normalize a data set to a fixed range of Rmin to Rmax = (((X-Dmin) * (Rmax – Rmin)) / (Dmax – Dmin)) + Rmin …where Rmin and Rmax is the minimum and maximum values for the rescaled range, Dmin and Dmax is the minimum and maximum values for the input data and X is any value in the data set to be rescaled. (Berry) Assessing Localized Variation Question 1 – Visual Map Analysis (Spatial and Numeric distributions) The “Scan” operation moves a window around the yield map and calculates the Coefficient of Variation with a 2-cell radius of each location …higher values indicate areas with more localized variability CoffVar= (StDev/Mean) * 100 Scan Yield_Volume Coffvar Within 2 For Yield_Coffvar Where, Coffvar = Stdev/mean *100 (Berry) Data Proximity/Buffer Stratification …proximity to field edge Edge effects “Sweet Spot” (interior) …Stratification partitions the data (numeric) or the project area (spatial) into logical groups— …Proximity map identifies the distance from point, line or polygon features to all other locations …unusually high yield …proximity to high yield areas Far : Close …Yield map > Average + 1Stdev “High Yield” vicinity (Berry) Summarizing Map Regions (template/data) …creates a map summarizing values from a data map (Phosphorous levels) that coincide with the categories of a template map (Soil types) Soil Types BIB Phosphorous levels Soil Type Pavg Ve 15.0 VdC 12.8 BIB 11.2 BIA 14.6 TuC 10.5 HvB 11.3 Overall BIA Pavg = 14.6 …average phosphorous level for each soil type 13.6 15.5 …average P-level for each soil unit (clump first before COMPOSITE) 8.6 (Berry) Comparing Discrete Maps (Multivariate analysis) Thematic Categorization …we often represent continuous spatial data (map surfaces) as a set of discrete polygons Which classified map is correct? How similar are the three maps? Spatial Precision (Where — boundaries) Medium High Low of Points, Lines and Areas (polygons) is a primary concern of GIS, but we are often less concerned with Thematic Accuracy (What — map values) See Beyond Mapping III , Topic 10, Analyzing Map Similarity and Zoning (Berry) Comparing Discrete Maps Two ways to compare Discrete Maps… Coincidence Summary Proximal Alignment 693 …Coincidence Summary generates a cross-tabular listing of the intersection of two maps. Table Interpretation Diagonal (Same) Off-diagonal (Above/Below) Percentages (% Same) Overall Percentage 83% ((475+297+563)/1950)*100= 68% ((631+297+693)/1950)*100= Raster versus Vector See Beyond Mapping III , Topic 10, Analyzing Map Similarity and Zoning (Berry) Comparing Discrete Maps Two ways to compare Discrete Maps… Coincidence Summary Proximal Alignment Question 2 Map2: Med-- 104 + 297 + 225 = 626; (297/626) *100= 47 percent matched 631 + 297 + 693 = 1621; (1621/1950) *100= 83 percent matched Map1 …Coincidence Summary generates a cross-tabular listing of the intersection of two maps. Map2 Map1 Map3 Table Interpretation Diagonal (Same) Off-diagonal (Above/Below) Percentages (% Same) Overall Percentage 475 + 297 + 563 = 1335; (1335/1950) *100= 68 percent matched Map3: Med-- 260 + 297 + 335= 912; (297/912) *100= 36 percent matched See Beyond Mapping III , Topic 10, Analyzing Map Similarity and Zoning 83% ((475+297+563)/1950)*100= 68% ((631+297+693)/1950)*100= Raster versus Vector (Berry) Coincidence Table (idealized conditions) Low Med High Total Low 208 0 0 208/208 = 100% Med 0 208 0 208/208 = 100% 0 0 208 208/208 = 100% High Diagonal elements in the map comparison matrix identify agreement (matches) between two progressive ordinal maps Off-diagonal elements in the map comparison matrix identify disagreement (miss-matches) between two progressive ordinal maps Low Med High Total …ALL MISMATCHES where there is an opposite relationship Overall coincidence = 0% Low 69 70 69 69/208 = 33% Med 70 69 69 69/208 = 33% High 69 69 70 69/208 = 33% Total 69/208 = 33% 69/208 = 33% 208/208 208/208 208/208 624/624 Total = 100% = 100% = 100% = 100% …PERFECT COINCIDENCE where all of the increasing ordinal steps are matched (diagonal) and there is no mismatches (off-diagonal). Overall coincidence is 624/624 = 100% found by the sum of the diagonal elements (matches); the other totals indicate percent agreement by category on each map Low Med High Total 104 104 0/208 = 0% Low 0 …EQUALLY BALANCED matches and mismatches where there is no pattern relationship Med 104 High 104 104 Overall coincidence = 33% partially matched and mismatched Total 0/208 = 0% 69/208 208/624 = 33% = 33% 0 0/208 = 0% 104 0/208 = 0% 0 0/208 = 0% 0/208 = 0% 0/624 = 0% (Berry) Comparing Discrete Maps Two ways to compare Discrete Maps… Coincident Summary Proximal Alignment Proximity_Map1_Category1 * Binary_Map3_Category1 …non-zero values identify changes and how far away …Proximal Alignment isolates a category on one of the maps, generates its proximity, then identifies the proximity values that align with the same category on the other map. Table Interpretation Zeros (Agreement) Values (> Disagreement) PA Index (average) See Beyond Mapping III , Topic 10, Analyzing Map Similarity and Zoning (Berry) Comparing Map Surfaces (Statistical Tests) Three ways to compare Map Surfaces… Statistical Tests Percent Difference Surface Configuration …Statistical Tests compare one set of cell values to that of another based on the differences in the distributions of the data— 1) data sets (partition or coincidence; continuous or sampled) 2) statistical procedure (t-Test, f-Test, etc.) Box-and-whisker graphs See Beyond Mapping III , Topic 10, Analyzing Map Similarity and Zoning (Berry) Comparing Map Surfaces (%Difference) Three ways to compare Map Surfaces… Statistical Tests Percent Difference Surface Configuration Question 3 …Percent Difference capitalizes on the spatial arrangement of the values by comparing the values at each map location— %Difference Map, %Difference Table See Beyond Mapping III , Topic 10, Analyzing Map Similarity and Zoning (Berry) Comparing Map Surfaces (Surface Configuration) Three ways to compare Map Surfaces… Statistical Tests Percent Difference Surface Configuration …Surface Configuration capitalizes on the spatial arrangement of the values by comparing the localized trend in the values — Slope Map, Aspect Map, Surface Configuration Index See Beyond Mapping III , Topic 10, Analyzing Map Similarity and Zoning (Berry) Comparing Map Surfaces (Temporal Difference) 1997_Yield_Volume - 1998_Yield_Volume = Yield_Diff Map Variables… map values within an analysis grid can be mathematically and statistically analyzed …green indicates areas of increased production …yellow indicates minimal change …red indicates decreased production See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships (Berry) Data Analysis (establishing relationships) On-farm studies, such as seed hybrid performance, can be conducted using actual farm conditions… …management action recommendations are based on local relationships instead of Experiment Station research hundreds of miles away …is radically changing research and management practices in agriculture and numerous other fields from business to epidemiology and natural resources (Berry) Spatial Dependency Spatial Variable Dependence — what occurs at a location in geographic space is related to: • the conditions of that variable at nearby locations, termed Spatial Autocorrelation (intra-variable dependence) • the conditions of other variables at that location, termed Spatial Correlation (inter-variable dependence) Map Stack– relationships among maps are investigated by aligning grid maps with a common configuration… #cols/rows, cell size and geo-reference. Data Shishkebab– each map represents a variable, each grid space a case and each value a measurement with all of the rights, privileges, and responsibilities of non-spatial mathematical , numerical and statistical analysis See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships (Berry) Visualizing Spatial Relationships Interpolated Spatial Distribution Phosphorous (P) What spatial relationships do you see? …do relatively high levels of P often occur with high levels of K and N? …how often? …where? See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships (Berry) Identifying Unusually High Measurements …isolate areas with mean + 1 StDev (tail of normal curve) See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships (Berry) Level Slicing …simply multiply the two maps to identify joint coincidence 1*1=1 coincidence (any 0 results in zero) See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships Question 4 (Berry) Multivariate Data Space …sum of a binary progression (1, 2 ,4 8, 16, etc.) provides level slice solutions for many map layers See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships (Berry) Calculating Data Distance …an n-dimensional plot depicts the multivariate distribution; the distance between points determines the relative similarity in data patterns …the closest floating ball is the least similar (largest data distance) from the comparison point See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships (Berry) Identifying Map Similarity Question 5 …the relative data distance between the comparison point’s data pattern and those of all other map locations form a Similarity Index The green tones indicate field locations with fairly similar P, K and N levels; red tones indicate dissimilar areas. See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships (Berry) Clustering Maps for Data Zones Question 6 …a map stack is a spatially organized set of numbers …groups of “floating balls” in data space identify locations in the field with similar data patterns– data zones …fertilization rates vary for the different clusters “on-the-fly” See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships Cyber-Farmer, Circa 1992 Variable Rate Application (Berry) Assessing Clustering Results …Clustering results can be roughly evaluated using basic statistics Average, Standard Deviation, Minimum and Maximum values within each cluster are calculated. Ideally the averages between the two clusters would be radically different and the standard deviations small—large difference between groups and small differences within groups. Standard Statistical Tests of two data sets Box and Whisker Plots to visualize differences See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships (Berry) How Clustering Works (IsoData algorithm) The scatter plot shows Height versus Weight data that might have been collected in your old geometry class The data distance to each weight/height measurement pair is calculated and the point is assigned to the closest arbitrary cluster center The average X,Y coordinates of the assigned students is calculated and used to reposition the cluster centers Repeat data distances, cluster assignments and repositioning until no change in cluster membership (centers do not move) See Beyond Mapping III , Topic 7, Linking Data Space and Geographic Space (Berry) Spatial Data Mining (The Big Picture) …making sense out of a map stack Mapped data that exhibits high spatial dependency create strong prediction functions. As in traditional statistical analysis, spatial relationships can be used to predict outcomes …the difference is that spatial statistics predicts where responses will be high or low See Beyond Mapping III , Topic 16, Characterizing Patterns and Relationships (Berry) An Analytic Framework for GIS Modeling This Week Spatial Data Mining operations involve characterizing numerical patterns and relationships among mapped data. See www.innovativegis.com/basis/Download/IJRSpaper/ (Berry) Regression (conceptual approach) A line is “fitted” in data space that balances the data so the differences from the points to the line (residuals) for all the points are minimized and the sum of the differences is zero… …the equation of the regression line is used to predict the “Dependent” variable (Y axis) using one or more “Independent” variables (X axis) (Berry) Evaluating Prediction Maps (non-spatial) Non-spatial …R-squared value looks at the deviations from the regression line; data patterns about the regression line (Berry) Map Variables The Dependent Map variable is the one that you want to predict… Question 7 …derive from customer data …from a set of existing or easily measured Independent Map variables See Beyond Mapping III , Topic 28, Spatial Data Mining in Geo-Business (Berry) Map Regression Results (Bivariate) Scatter plots and regression equations relating Loan Density to three candidate driving variables (Housing Density, Value and Age) Question 8 Loans= fn( Housing Density ) Loans= fn( Home value ) Loans= fn( Home Age ) The “R-squared index” provides a general measure of how good the predictions ought to be— 40%, 46% indicates a moderately weak predictors; 23% indicates a very weak predictor (R-squared index = 100% indicates a perfect predictor; 0% indicates an equation with no predictive capabilities) See Beyond Mapping III , Topic 28, Spatial Data Mining in Geo-Business (Berry) Generating a Multivariate Regression …a regression equation using all three independent map variables using multiple linear regression is used to generate a prediction map Question 9 See Beyond Mapping III , Topic 28, Spatial Data Mining in Geo-Business (Berry) Evaluating Regression Results (multiple linear) Optional Question 9-1 …a regression equation using all three independent map variables using multiple linear regression is used to generate a prediction map …that is compared to the actual dependent variable data — Error Surface See Beyond Mapping III , Topic 28, Spatial Data Mining in Geo-Business (Berry) Using the Error Map to Stratify One way to improve the predictions, however, is to stratify the data set by breaking it into groups of similar characteristics …and then generating separate regressions …generate a different regression for each of the stratified areas– red, yellow and green …other stratification techniques include indigenous knowledge, level-slicing and clustering See Beyond Mapping III , Topic 28, Spatial Data Mining in Geo-Business (Berry) An Analytic Framework for GIS Modeling This Week Spatial Data Mining operations involve characterizing numerical patterns and relationships among mapped data. See www.innovativegis.com/basis/Download/IJRSpaper/ (Berry) Prescriptive Mapping Four primary types of applied spatial models: Suitability— mapping preferences (e.g., Habitat and Routing) Economic— mapping financial interactions (e.g., Combat Zone and Sales Propensity) Physical— mapping landscape interactions (e.g., Terrain Analysis and Sediment Loading) Mathematical/Statistical— mapping numerical relationships… ― Descriptive math/stat models summarize existing mapped data (e.g., Standard Normal Variable Map for Unusual Conditions and Clustering for Data Zones) ― Predictive math/stat models develop equations relating mapped data (e.g., Map Regression for Equity Loan Prediction and Probability of Product Sales ) ― Prescriptive math/stat models identify management actions based on descriptive/predictive relationships (e.g., Retail Marketing and Precision Ag)… Phosphorous (P) Continuous Actions: Equation defining action(s) Negative linear equation of the form: y = aX P2O5/ If P is 0-4 ppm, then apply 50 lbs P2O5/Acre If P is 4-8 ppm, then apply 30 lbs P2O5/Acre If P is 8-12 ppm, then apply 15 lbs P2O5/Acre If P is >12 ppm, then apply 0 lbs P2O5/Acre 50 50 0 0 30 15 0 P 12 more P 12 more 50 P2O5/ Discrete Actions: If <condition(s)> Then <Action(s)> Negative exponential equation of the form: y = e-x 0 (Berry) 0 Grid-Based Map Analysis Spatial analysis investigates the “contextual” relationships in mapped data… Reclassify— reassigning map values (position; value; size, shape; contiguity) Overlay— map overlay (point-by-point; region-wide) Distance— proximity and connectivity (movement; optimal paths; visibility) Neighbors— ”roving windows” (slope/aspect; diversity; anomaly) Surface modeling maps the spatial distribution and pattern of point data… Density Analysis— count/sum of points within a local window Spatial Interpolation— weighted average of points within a local window Map Generalization— fits mathematical relationship to all of the point data Spatial data mining investigates the “numerical” relationships in mapped data… Descriptive— summary statistics, comparison, classification (e.g., clustering) Predictive— math/stat relationships among map layers (e.g., regression) Prescriptive— appropriate actions (e.g., optimization) (Berry)