Lecture 20 Point referenced data (pt. 2) Colin Rundel 04/05/2017 1
Loa Loa Example 2
Loa Loa 3
Data ## 8 ## 0.502 503 11 163 4.88 11.4 16 7 7 0.436 217 909 3 66 5.36 8.93 116 6 6 ## 0.415 109 3 8 8.07 5.10 0.481 min9901 <dbl>, stdev9901 <dbl> ## # ## # ... with 187 more rows, and 3 more variables: max9901 <dbl>, 0.487 268 4 57 6.00 9.31 104 10 ## 10 751 5.90 4 30 5.59 9.02 112 9 9 ## 0.373 103 0 83 167 8.18 loaloa = tbl_df (PrevMap::loaloa) %>% setNames (., tolower ( names (.))) ## 2 2 ## 0.439 108 0 162 5.74 8.04 214 1 1 <dbl> 8.00 <int> <int> <int> <dbl> <dbl> <int> <int> ## row villcode longitude latitude no_exam no_inf elevation mean9901 ## ## # A tibble: 197 x 11 loaloa 215 5.68 212 ## 5 5 ## 0.432 104 5 62 5.92 8.10 219 4 4 0.491 167 783 5 88 5.35 8.91 118 3 3 ## 0.426 99 1 4
Spatial Distribution 5 12 ° N no_inf/no_exam 0.5 10 ° N 0.4 0.3 0.2 8 ° N 0.1 latitude 0.0 6 ° N no_exam 100 200 4 ° N 300 400 2 ° N 8 ° E 10 ° E 12 ° E 14 ° E 16 ° E longitude
Normalized Difference Vegetation Index (NDVI) 6 12˚N 11˚N 10˚N 9˚N 8˚N Latitude 7˚N 6˚N 5˚N 4˚N 3˚N 2˚N 8˚E 10˚E 12˚E 14˚E 16˚E Longitude -0.2 0 0.2 0.4 0.6 0.8 1 USGS LandDAAC MODIS version_005 WAF NDVI
Paper / Data summary Loa loa risk: decision making under uncertainty . Annals of Tropical Medicine and Parasitology, 101, 499-509. • no_exam and no_inf - Collected between 1991 and 2001 by NGOs (original paper mentions 168 villages and 21,938 observations) • elevation - USGS gtopo30 (1km resolution) • mean9901 to stdev9901 - aggregated data from 1999 to 2001 from the Flemish Institute for Technological Research (1 km resolution) 7 Original paper - Diggle, et. al. (2007). Spatial modelling and prediction of
Diggle’s Model log ( 𝑞(𝑡) 1 − 𝑞(𝑡)) = 𝛽 + 𝑔 1 ( ELEVATION (𝑡)) + 𝑔 2 ( MAX.NDVI (𝑡)) + 𝑔 3 ( SD.NDVI (𝑡)) + 𝑥(𝑡) where 𝑥(𝑡) ∼ 𝒪(0, Σ) 8 {Σ} 𝑗𝑘 = 𝜏 2 exp (−𝑒 𝜚)
EDA 9 0 logit_prop −1 −2 −3 −4 −5 0 500 1000 1500 elevation 0 logit_prop −1 −2 −3 −4 −5 0.7 0.8 0.9 max9901 0 logit_prop −1 −2 −3 −4 −5 0.12 0.15 0.18 0.21 stdev9901
Diggle’s EDA 10 t of
Model EDA 5.793e-01 ## elevation:elev_factor(0,1000] *** ## stdev9901 *** ## (Intercept) ## < 2e-16 9.711 5.626e+00 ## elevation:elev_factor(1000,1300] . ## max9901:max_factor(0.8,1] < 2e-16 8.749 6.299e-01 5.511e+00 ## max9901:max_factor(0,0.8] -7.588 3.25e-14 1.887e-04 ## elevation:elev_factor(1300,2000] -1.432e-03 *** ## elevation:elev_factor(1300,2000] *** 1.855 Null deviance: 4208.2 ## Number of Fisher Scoring iterations: 5 ## ## AIC: 2804.6 degrees of freedom on 190 ## Residual deviance: 2042.3 degrees of freedom on 196 ## ## max9901:max_factor(0,0.8] ## ## (Dispersion parameter for binomial family taken to be 1) ## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## Signif. codes: ## --- *** ## max9901:max_factor(0.8,1] *** 0.0636 8.792e-05 loaloa = loaloa %>% stdev9901, family = binomial, data = loaloa, weights = loaloa$no_exam) Max 3Q Median 1Q Min ## ## Deviance Residuals: ## ## -2.5887 ## glm(formula = no_inf/no_exam ~ elevation:elev_factor + max9901:max_factor + ## Call: ## summary (g) data=loaloa, family=binomial, weights=loaloa$no_exam) = cut (max9901, breaks= c (0,0.8,1))) max_factor mutate (elev_factor = cut (elevation, breaks= c (0,1000,1300,2000), dig.lab=5), ## -7.1434 -0.8993 1.631e-04 8.781e+00 ## elevation:elev_factor(1000,1300] < 2e-16 18.358 8.749e-05 1.606e-03 ## elevation:elev_factor(0,1000] 7.288 3.14e-13 1.205e+00 ## stdev9901 1.6375 < 2e-16 4.825e-01 -17.291 -8.343e+00 ## (Intercept) Estimate Std. Error z value Pr(>|z|) ## ## Coefficients: ## 10.9052 11 g = glm (no_inf/no_exam ~ elevation:elev_factor + max9901:max_factor + stdev9901,
Fit geom_abline (slope = 1, intercept = 0) loaloa = loaloa %>% 12 geom_point () + ggplot (loaloa, aes (x=no_inf/no_exam, y=glm_pred)) + mutate (glm_pred = predict (g, type=”response”)) 0.4 0.3 glm_pred 0.2 0.1 0.0 0.0 0.2 0.4 no_inf/no_exam
13 Data 7 ° N no_inf/no_exam 0.5 6 ° N latitude 0.4 5 ° N 0.3 4 ° N 0.2 3 ° N 0.1 2 ° N 0.0 8 ° E 10 ° E 12 ° E 14 ° E 16 ° E longitude GLM Prediction 7 ° N glm_pred 6 ° N 0.4 latitude 5 ° N 0.3 4 ° N 0.2 3 ° N 0.1 2 ° N 8 ° E 10 ° E 12 ° E 14 ° E 16 ° E longitude
Spatial Structure ## variog: computing omnidirectional variogram geoR:: variog (coords = cbind (loaloa$longitude, loaloa$latitude), 14 uvec = seq (0, 4, length.out = 50)) %>% plot () data = loaloa$prop - loaloa$glm_pred, 0.015 semivariance 0.010 0.005 0.000 0 1 2 3 4 distance
spBayes GLM Model spg = spBayes:: spGLM ( no_inf/no_exam ~ elevation:elev_factor + max9901:max_factor + stdev9901, data=loaloa, family=”binomial”, weights=loaloa$no_exam, coords= cbind (loaloa$longitude, loaloa$latitude), cov.model=”exponential”, n.samples=20000, starting= list (beta= rep (0,7), phi=3, sigma.sq=1, w=0), priors= list (phi.unif= c (0.1, 10), sigma.sq.ig= c (2, 2)), amcmc= list (n.batch=1000, batch.length=20, accept.rate=0.43)) save (spg, loaloa, file=”loaloa.Rdata”) 15
spg$p.beta.theta.samples %>% 4.44632 -0.00581 -0.02900 0.00004 max9901:max_factor(0,0.8] 4.87762 3.99492 -2.93030 15.63246 max9901:max_factor(0.8,1] 5.08690 -2.18626 elevation:elev_factor(1300,2000] 14.89011 sigma.sq 0.38088 0.34626 0.12793 0.88673 phi 6.22996 5.18205 0.69584 18.67107 -0.00814 0.00169 post_summary () %>% stdev9901 knitr:: kable (digits=5) param post_mean post_med post_lower post_upper (Intercept) -12.69885 -11.61326 -21.65388 -6.96361 9.24231 -0.00359 9.15244 -14.48649 29.76058 elevation:elev_factor(0,1000] 0.00048 0.00077 -0.00474 0.00291 elevation:elev_factor(1000,1300] -0.00048 -0.00032 16
Prediction 17 0.004 0.003 pred_spg_mean 0.002 0.001 0.000 0.0 0.2 0.4 prop
spBayes GLM Model - Fixed? spg_fix = spBayes:: spGLM ( data=loaloa, family=”binomial”, weights=loaloa$no_exam, coords= cbind (loaloa$longitude, loaloa$latitude), cov.model=”exponential”, n.samples=20000, starting= list (beta= rep (0,7), phi=3, sigma.sq=1, w=0), priors= list (phi.unif= c (0.1, 10), sigma.sq.ig= c (2, 2)), amcmc= list (n.batch=1000, batch.length=20, accept.rate=0.43) ) save (spg_fix, loaloa, file=”loaloa_fix.Rdata”) 18 no_inf ~ elevation:elev_factor + max9901:max_factor + stdev9901,
param -0.26884 -0.00285 -0.00127 max9901:max_factor(0,0.8] 0.88041 0.90550 -1.03795 3.63477 max9901:max_factor(0.8,1] 1.28673 1.13796 3.83860 -0.00204 sigma.sq 1.47552 1.39146 0.43359 3.05883 phi 2.22372 2.09524 0.86456 4.14663 -0.00200 elevation:elev_factor(1300,2000] post_mean -0.41947 post_med post_lower post_upper (Intercept) -2.66090 -2.13138 -6.31576 -0.80487 stdev9901 -0.12840 -5.86766 0.00020 8.58835 elevation:elev_factor(0,1000] 0.00023 0.00024 -0.00051 0.00086 elevation:elev_factor(1000,1300] -0.00054 -0.00055 -0.00128 19
Fit 20 0.4 pred_spg_mean 0.2 0.0 0.0 0.2 0.4 no_inf/no_exam
Diggle’s Predictive Surface 21 FIG. 2. Point estimates of the prevalence of Loa loa microfilaraemia, over-laid with the prevalences observed in field studies.
Exceedance Probability - Posterior Summary 22 Village 40 Village 339 20 15 10 5 village 0 Village 40 density Village 339 Village 116 Village 110 Village 116 20 Village 110 15 10 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 p
Exceedance Probability Predictive Surface 23 Published by Maney Publishing (c) W S Maney & Son Ltd FIG. 4. A probability contour map, indicating the probability that the prevalence of Loa loa microfilaraemia in each area exceeds 20%, over-laid with the prevalences observed in field studies.
Spatial Assignment of Migratory Birds 24
Background Using intrinsic markers (genetic and isotopic signals) for the purpose of inferring migratory connectivity. • Existing methods are too coarse for most applications • Large amounts of data are available ( >150,000 feather samples from >500 species) • Genetic assignment methods are based on Wasser, et al. (2004) • Isotopic assignment methods are based on Wunder, et al. (2005) 25
Data - DNA microsatellites and 𝜀 2 H Hermit Thrush ( Catharus guttatus ) • 138 individuals • 14 locations • 6 loci • 9-27 alleles / locus Wilson’s Warbler ( Wilsonia pusilla ) • 163 individuals • 8 locations • 9 loci • 15-31 alleles / locus 26
Sampling Locations 27 AK2 Ak Hud AK1 Rup BC QCI Al MB Log Ont Sea OR MI Or CT CA UT Co SF AZ2 AZ1 Hermit Thrush Wilson's Warbler
Allele Frequency Model For the allele 𝑗 , from locus 𝑚 , at location 𝑙 exp (Θ 𝑗𝑚𝑙 ) 𝚰 𝑗𝑚 |𝜷, 𝝂 ∼ 𝒪(𝝂 𝑗𝑚 , 𝚻) 28 𝐳 ⋅𝑚𝑙 |𝚰 ∼ 𝒪 (∑ 𝑗 𝑧 𝑗𝑚𝑙 , 𝐠 ⋅𝑚𝑙 ) 𝑔 𝑗𝑚𝑙 = ∑ 𝑗 exp (Θ 𝑗𝑚𝑙 ) {Σ} 𝑗𝑘 = 𝜏 2 exp ( − ({𝑒} 𝑗𝑘 𝑠) 𝜔 ) + 𝜏 2 𝑜 1 𝑗=𝑘
Predictions by Allele (Locus 3) 29
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