Model-assisted Estimation of Forest Resources with Generalized Additive ModelsJean Opsomer, Jay Breidt, Gretchen Moisen, Göran KauermannAugust 9, 20061Outline1. Forest surveys2. Sampling from spatial domain3. Model-assisted estimation4. GAM estimation for forest inventory data5. Variance estimation for systematic samples2Research Project•Collaboration between academic and US Forest Service statisticians•Goal: apply on-going modeling efforts by Forest Service staff to improve efficiency of survey estimators31. Forest Inventory and Analysis (FIA)•Forest Inventory and Analysis is annual survey of all forest lands in US•Multi-phase survey, including field visits phase with approximately 1 plot/6,000 acres•Expensive: $68million in 2004 (nation-wide)4Inference for Surveys?Specific Inference•expensive, high quality•targeted to specific application and/or scientific question •using “custom-built” method (or model) to achieve best possible estimator for particular variable(s) •willing to defend estimates/inference 5Inference for SurveysGeneric Inference•cheap, reasonable quality, good for many purposes •using method appropriate for large number of variables•provide reasonable answers to many possible scientific questions•validity of estimates resistant to model misspecification; model independent6 C o r nC o r nC o r nC o r n F l a k e sF l a k e sF l a k e sF l a k e s N E T W T . 1 2 O Z .Survey Estimation•Classical methods depend only on sampling design (Horvitz-Thompson; Hájek)•Improved methods are still design-based but take advantage of auxiliary information•ratio, regression, post-stratification•model-assisted (Särndal et al, 1992)•calibration (Deville and Särndal, 1992)•nonparametric (Breidt and Opsomer, 2000), nonlinear/generalized (Wu and Sitter, 2001), ...7Current Dataset•2.5 million ha ecological region in Utah•Contains 968 FIA field plots on 5x5km grid•FIA plots embedded in 24,980 remote sensing locations on 1x1km grid8Current Dataset (2)Field Plot Variables‣Forest/non-forest‣Total wood volume‣Tree basal area‣Biomass‣Percent crown cover‣Mean diameter‣...Remote Sensing Variables‣Elevation‣Slope‣Aspect‣Location‣Vegetation Index‣TM spectral bands9Systematic Sampling•Common in natural resource and other spatial surveys•Advantages:•Simple to implement, intuitive•Easy to “nest” within GIS environment•Ensures proportional representation of domains•Optimal for certain stationary processes10Systematic Sampling (2)•Disadvantages•Inflexible, can miss rare features in region•Does not capture spatial relationships at fine scales (modeling)•No design-based variance estimator112. Sampling from Spatial Domain•Phase I sample is systematic from continuous domain •Phase II sample is systematic (discrete) sub-sample of Conditional on , only 25 possible phase II sample12U ⊆ D = [0, L1] × [0, L2]G1G2G1G1Sampling from Spatial Domain (2)•Phase I sample , withuniform random variable on and sampling intervals•Phase II sample , wherediscrete uniform on13u = (u1, u2)G1(u)[0, 1] × [0, 1](δ1, δ2)G2(u, d)d = (d1, d2)[1, 2, . . . , h1] × [1, 2, . . . , h2]j1, j2= 0, 1, . . .}G2(u, d) = {(u1+ d1+ j1h1)δ1, (u2+ d2+ j2h2)δ2) :G1(u) = {(u1+ i1)δ1, (u2+ i2)δ2) : i1, i2= 0, 1, . . .}Population Characteristics•Interested in estimating finite population total for variable on •Total can be “gridded’’ into cells14θzDi1i2θz=!i1i2"Di1i2z(v)dvθz=!Dz(v)dvz(v)= δ1δ2![0,1]×[0,1]"s∈G1(u)z(s)duDSurvey Estimation•Phase I expansion estimator (unfeasible for Phase II variables)•Two-phase expansion estimator•Both unbiased, have exact variance formula 15ˆθ2z(u, d) =!s∈G2(u,d)z(s)1/(δ1δ2h1h2)ˆθ1z(u) =!s∈G1(u)z(s)1/(δ1δ2)3. Model-Assisted Estimation•Variables observed on Phase I can improve precision of survey estimators for Phase II variables•Model-assisted approach provides convenient framework for incorporating auxiliary information within design-based (generic) inference16X(v)Model-Assisted Estimation (2)1. Assume working model2. Fit model onto predict 3. Construct model-assisted estimator17{z(s), X(s) : s ∈ G2(u, d)}Eξ(z(v)) = µ(X(v))ˆθMA,z=!s∈G1(u)ˆµ(s)1/(δ1δ2)+!s∈G2(u,d)z(s) − ˆµ(s)1/(δ1δ2h1h2)ˆµ(s), s ∈ G1(u)Properties of Model-Assisted Estimator•Estimator is approximately design unbiased for large classes of models, with approximate design variance18ˆθMA,zVar(ˆθMA,z) ≈ Var(ˆθ1z(u)) +|D|2n22!1 −1h1h2"E(S2(u))S2(u) =1h1h2− 1h1!d1=1h2!d2=1(td1d2(u) −¯t(u))2td1d2(u) =!s∈G2(u,d)(z(s) − ˆµ(s))Applying Model-Assisted Estimation•In typical survey context, many variables of interest instead of single •Express estimator in the form (automatic for linear estimators)•Survey variables “related to” Phase I variables will benefit from improved efficiency19z(v)ˆθMA,zˆθMA,z=!s∈G2(u,d)w(s)z(s)X(v)4. Estimation for Forest Inventory Data •Forest Service researchers are investigating predictive models for forest characteristics based on remote sensing data•Key variable in this survey: FOREST indicator‣Many other variables not recorded when20IFOR(v)IFOR(v) = 0GAM Variables for FOREST21‣(X,Y)! ! ! ! coordinates (bivariate)‣ELEV90CU ! elevation‣TRASP90! ! ! aspect (transformed)‣SLP90CU!! ! slope‣MRLCOOB5 ! TM satellite band 5‣NDVI ! ! ! vegetation index (TM)‣NLDC7 !! ! vegetation classes (TM)GAM Model for FOREST22•Modelwith logistic link and Phase I variables•Fitted in S-Plus using gam() with lo() smoothers, to obtain prediction forg(·)ˆµFOR(s)s ∈ G1(u)+m6(x6(v)) + x7(v)!β)xk(v)Eξ(IFOR(v)) ≡ µFOR(v) = g(m1(x1(v)) + . . .FOREST Model Components0.20.40.60.8Xs0.20.40.60.8Ys-1.5-1-0.5 00.511.5lo(Xs, Ys, span = 0.5)ELEV90CUs(ELEV90CU, df = 4)1500 2000 2500 3000 3500-3 -1 0 1TRASP90s(TRASP90, df = 4)-150 -100 -50 0-0.8 -0.2 0.4SLP90CUs(SLP90CU, df = 4)0 20 40 60 80-2 -1 0 1 2MRLC00B5s(MRLC00B5, df = 4)0 50 100 150-2 0 2 4NDVIs(NDVI, df = 4)-0.2 0.0 0.2 0.4 0.6 0.8-10 -6 -2 023Other Phase II Variables24‣NVOLTOT! ! total wood volume (cuft/acre)‣BA ! ! ! ! tree basal area (per acre) ‣BIOMASS! ! ! total wood biomass (ton/acre)‣CRCOV! ! ! percent crown cover (%)‣QMDALL ! ! quadratic mean diameter (in)Modeling Other