Presentation on theme: "Spatial statistics in practice"— Presentation transcript:
1 Spatial statistics in practice Lecture #5: MAPS WITH GAPS-- Small geographic area estimation, kriging, and kernel smoothingSpatial statistics in practiceCenter for Tropical Ecology and Biodiversity, Tunghai University & Fushan Botanical Garden
2 Topics for today’s lecture The E-M algorithmThe spatial E-M algorithmKriging in ArcGISgeographically weighted regression (GWR)approaches to map smoothing
3 THEOREM 1When missing values occur only in a response variable, Y, then the iterative solution to the EM algorithm produces the regression coefficients calculated with only the complete data.PF: Let b denote the vector of regression coefficients that is converged upon. Then if ,
4 THEOREM 2 When missing values occur only in a response variable, Y, then by replacing the missing values with zeroes and intro-ducing a binary 0/-1 indicator variable covariate -Im for each missing value m, such that Im is 0 for all but missing value observation m and 1 for missing value observation m, the estimated regression coefficient bm is equivalent to the point estimate for a new obser-vation, and hence furnishes EM algorithm imputations.PF:Let bm denote the vector of regression coefficients for the missing values, and partition the data matrices such that
5 The EM algorithm solution where:the missing values are replaced by 0 in Y, andIm is an indicator variable for missing value mthat contains n-m 0s and a single 1
6 THEOREM 3 For imputations computed based upon Theorem 2, each standard error of the estimated regression coefficients bm is equivalent to the conventional standard deviation used to construct a prediction interval for a new observation, and as such furnishes the corresponding EM algorithm imputation standard error.PF:
7 What is the set of equations for the following case? 107y4 = ?
11 EM algorithm solution for aggregated georeferenced data: vandalized turnips plots
12 MTB > regress c4 8 c7-c14 Regression Analysis: C4 versus C7, C8, C9, C10, C11, C12, C13, C14 The regression equation is C4 = C C C C C C C C14 Predictor Coef SE Coef T P Constant C7 [I1-I6] C8 [I2-I6] C9 [I3-I6] C10 [I4-I6] C11 [I5-I6] C12 [plot(6,5)] C13 [plot(5,6)] C14 [plot(6,6)]
13 Analysis of Variance for C4 Source DF SS MS F P C5 5 1289. 0 257. 8 8 Analysis of Variance for C Source DF SS MS F P C Error Total Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev (-----*-----) (----*-----) (----*-----) (-----*-----) (-----*-----) (------*-----) Pooled StDev =
14 Residual spatial autocorrelation What does this mean?
16 SAR-based missing data estimation where ym is a missing value (replaced by 0 in Y),Im is an indicator variable for ym, andis the mth column of geographic weightsmatrix W
17 The Jacobian termNOTE: denominator becomes (n-nm)
18 What is the set of equations for the following case? 7Y2 = ?10
19 spatial autoregressive (AR) krigingestimate withsemivariogram modelfit semivariogram model with
20 The pure spatial autocorrelation CAR model NOTE: exactly the same algebraic structure as the kriging equationDispersed missing values:Imputation = the observed mean plus a weighted average of the surrounding residuals
21 Employing rook’s adjacency and a CAR model, what is the equation for the following imputation? 10376y5 = ?495
22 The spatial filter EM algorithm solution where:the missing values are replaced by 0 in Y, andIm is an indicator variable for missing value mthat contains n-m 0s and a single 1
26 Predicted from spatial filter Missing 1992 georeferenced density of milk production in Puerto Rico: constrained (total = 1918)Predicted from 1991 DMILKPredicted from spatial filterPredicted from both235703851,3391,8481,065344468predictionsMoran scatterplot
27 USDA-NASS estimation of Pennsylvania crop production covariatetotalconstraintsmap gaps
39 Cross-validation of spatial filter for observed turnip data
40 Kriging: best linear unbiased spatial interpolator (i.e., predictor) The accompanying table contains a test set of sixteen random samples (#17-32) used to evaluate three maps. The “Actual” column lists the measured values for the test locations identified by “Col, Row” coordinates. The difference between these values and those predicted by the three interpolation techniques form the residuals shown in parentheses. The “Average” column compares the whole field arithmetic mean of 23 (guess 23 everywhere) for each test location.
41 ArcGIS: Geostatistical Wizard density ofGermanworkersanisotropycheck
42 Cross-validation check of krigged values This is one use ofthe missing spatial dataimputation methods.
43 Unclipped krigged surface exponential semivariogram modelvalues increase with darkness of brownextrapolationkrigged (mean response) surfaceprediction error surface
44 Clipped krigged surface krigged (mean response) surfacevalues increase with darkness of brownprediction error surface
45 Detrended population density across China anisotropycheck
46 Cross-validation check of krigged values This is one use ofthe missing spatial dataimputation methods.
47 Unclipped krigged surface exponential semivariogram modelvalues increase with darkness of brownextrapolationkrigged (mean response) surfaceprediction error surface
48 Clipped krigged surface krigged (mean response) surfacevalues increase with darkness of brownprediction error surface
49 THEOREM 4The maximum likelihood estimate for missing georeferenced values described by a spatial autoregressive model specification is equivalent to the best linear unbiased predictor kriging equation of geostatistics.
50 Geographically weighted regression: GWR Spatial filtering enables easier implementation of GWR, as well as proper assessment of its dfsStep #1: compute the eigenvectors of a geographic connectivity matrix, say CStep #2: compute all of the interactions terms XjEk for the P covariates times the K candidate eigenvectors (e.g., with MC > 0.25)Step #3: select from the total set, including the individual eigenvectors, with stepwise regression
51 Step #4: the geographically varying intercept term is given by: Step #5: the geographically varying covariate coefficient is given by factoring Xj out of its appropriate selected interaction terms:
52 A Puerto Rico DEM example Mean elevation (Y) is a function of: standard deviation of elevation (X), eigenvectors E1-E18, and 18 interaction terms (XE)Resultsintercept: 1, E2, E5-E7, E9, E11-E13, E15, E18slope: 1, E4, E6, E9, E10R2 increases from (with X only) to (with geographically varying coefficients)P(S-W) = 0.52 for the final model
56 A summary: what have we learned during the 5 lectures? The nature of data and its information content.What is spatial autocorrelation?Visualizing spatial autocorrelation: Moran scatterplots, semivariogram plots, and maps.Defining and articulating spatial structure: topology and distance perspectives; contagion and hierarchy concepts.Necessary concepts from multivariate statistics.An example of the elusive negative spatial autocorrelation.Some comments about spatial sampling.Implications about space-time data structure.
57 Multivariate grouping, and location-allocation modeling. Lecture #2Multivariate grouping, and location-allocation modeling.Going from the global to the local: variability and heterogeneity.Impacts of spatial autocorrelation on histograms.The LISA and Getis-Ord statistics.Cluster analysis: multivariate analysis, cluster detection, and spider diagrams.An overview of geographic and space-time clusters.Regression diagnostics and geographic clusters
58 Lecture #3Autoregressive specifications and normal curve theory (PROC NLIN).Auto-binomial and auto-Poisson models: the need for MCMC.Relationships between spatial autoregressive and geostatistical modelsSpatial filtering specifications and linear and generalized linear models (PROC GENMOD).Autoregressive specifications and linear mixed models (PROC MIXED).Implications for space-time datasets (PROC NLMIXED)
59 Lecture #4Frequentist versus Bayesian perspectives.Implementing random effects models in GeoBUGS.Spatially structured and unstructured random effects: the CAR, the ICAR, and the spatial filter specificationsLecture #5The E-M algorithmThe spatial E-M algorithmKriging in ArcGISApproaches to map smoothing