Model-assisted Estimation of Forest Resources with Generalized Additive Models

Transcription

1 Model-assisted Estimation of Forest Resources with Generalized Additive Models Jean Opsomer, Jay Breidt, Gretchen Moisen, Göran Kauermann August 9,

2 Outline 1. Forest surveys 2. Sampling from spatial domain 3. Model-assisted estimation 4. GAM estimation for forest inventory data 5. Variance estimation for systematic samples 2

3 Research Project Collaboration between academic and US Forest Service statisticians Goal: apply on-going modeling efforts by Forest Service staff to improve efficiency of survey estimators 3

4 1. 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) 4

5 Inference 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 5

6 Inference for Surveys Generic 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 independent C o r n F l a k e s N E T W T. 1 2 O Z. 6

7 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),... 7

8 Current 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 grid 8

9 Current Dataset (2) Remote Sensing Variables Elevation Slope Aspect Location Vegetation Index TM spectral bands Field Plot Variables Forest/non-forest Total wood volume Tree basal area Biomass Percent crown cover Mean diameter... 9

10 Systematic 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 processes 10

11 Systematic Sampling (2) Disadvantages Inflexible, can miss rare features in region Does not capture spatial relationships at fine scales (modeling) No design-based variance estimator 11

12 2. Sampling from Spatial Domain Phase I sample is systematic from continuous domain U D = [0, L 1 ] [0, L 2 ] Phase II sample is G 1 G 2 systematic (discrete) sub-sample of G 1 G 1 Conditional on, only 25 possible phase II sample 12

13 Sampling from Spatial Domain (2) Phase I sample G 1 (u), with u = (u 1, u 2 ) uniform random variable on [0, 1] [0, 1] and sampling intervals(δ 1, δ 2 ) G 1 (u) = {(u 1 + i 1 )δ 1, (u 2 + i 2 )δ 2 ) : i 1, i 2 = 0, 1,...} Phase II sample G 2 (u, d), where d = (d 1, d 2 ) discrete uniform on [1, 2,..., h 1 ] [1, 2,..., h 2 ] G 2 (u, d) = {(u 1 + d 1 + j 1 h 1 )δ 1, (u 2 + d 2 + j 2 h 2 )δ 2 ) : j 1, j 2 = 0, 1,...} 13

14 Population Characteristics Interested in estimating finite population total for variable z(v) on D θ z = D z(v)dv Total θ z can be gridded into cells D i1 i 2 θ z = i 1 i 2 D i1i2 z(v)dv = δ 1 δ 2 [0,1] [0,1] 14 s G 1 (u) z(s)du

15 Survey Estimation Phase I expansion estimator ˆθ 1z (u) = z(s) s G 1 (u) 1/(δ 1 δ 2 ) (unfeasible for Phase II variables) Two-phase expansion estimator ˆθ 2z (u, d) = s G 2 (u,d) z(s) 1/(δ 1 δ 2 h 1 h 2 ) Both unbiased, have exact variance formula 15

16 3. Model-Assisted Estimation Variables X(v) 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) inference 16

17 Model-Assisted Estimation (2) 1. Assume working model E ξ (z(v)) = µ(x(v)) 2. Fit model on {z(s), X(s) : s G 2 (u, d)} to predict ˆµ(s), s G 1 (u) 3. Construct model-assisted estimator ˆθ MA,z = s G 1 (u) ˆµ(s) 1/(δ 1 δ 2 ) + s G 2 (u,d) z(s) ˆµ(s) 1/(δ 1 δ 2 h 1 h 2 ) 17

18 Properties of Model- Assisted Estimator Estimator ˆθ MA,z is approximately design unbiased for large classes of models, with approximate design variance Var(ˆθ MA,z ) Var(ˆθ 1z (u)) + D 2 n 2 2 ( 1 1 h 1 h 2 ) E(S 2 (u)) S 2 1 (u) = h 1 h 2 1 t d1 d (u) = 2 h 1 d 1 =1 h 2 d 2 =1 (t d1 d 2 (u) t(u)) 2 (z(s) ˆµ(s)) s G 2 (u,d) 18

19 Applying Model- Assisted Estimation In typical survey context, many variables of interest instead of single z(v) Express estimator ˆθ MA,z = ˆθ MA,z in the form w(s)z(s) s G 2 (u,d) (automatic for linear estimators) Survey variables related to Phase I variables X(v) will benefit from improved efficiency 19

20 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 I FOR (v) Many other variables not recorded when I FOR (v) = 0 20

21 GAM Variables for FOREST (X,Y) ELEV90CU TRASP90 SLP90CU coordinates (bivariate) elevation aspect (transformed) slope MRLCOOB5 TM satellite band 5 NDVI NLDC7 vegetation index (TM) vegetation classes (TM) 21

22 GAM Model for FOREST Model E ξ (I FOR (v)) µ FOR (v) = g(m 1 (x 1 (v)) m 6 (x 6 (v)) + x 7 (v) β) with g( ) logistic link and x k (v) Phase I variables Fitted in S-Plus using gam() with lo() smoothers, to obtain prediction ˆµ FOR (s) for s G 1 (u) 22

23 FOREST Model Components lo(xs, Ys, span = 0.5) Ys Xs s(elev90cu, df = 4) ELEV90CU s(trasp90, df = 4) s(slp90cu, df = 4) TRASP SLP90CU s(mrlc00b5, df = 4) s(ndvi, df = 4) MRLC00B NDVI

24 Other Phase II Variables NVOLTOT BA BIOMASS total wood volume (cuft/acre) tree basal area (per acre) total wood biomass (ton/acre) CRCOV percent crown cover (%) QMDALL quadratic mean diameter (in) 24

25 Modeling Other Variables Approaches considered: 1. Classical model-assisted 2. Model-assisted with FOREST prediction as auxiliary variable 3. Model-assisted with FOREST prediction indicator 25

26 We didn t do Classical model-assisted E ξ (z(v)) = m 1 (x 1 (v)) m 6 (x 6 (v)) with +x 7 (v) β m 1 ( ),..., m 6 ( ) parametric or nonparametric 2. Model-assisted with FOREST prediction as auxiliary variable E ξ (z(v)) = m 1 (x 1 (v)) m 6 (x 6 (v)) +x 7 (v) β + ˆµ FOR (v)γ 26

27 Selected Method Construct indicator for FOREST prediction { 1 if ˆµFOR (v) ˆθ 2,FOR (u, d)/ D Î FOR (v) = 0 otherwise and FOREST-interaction Phase I variables x k F (v) = ÎFOR(v)x k (v) Working model E ξ (z(v) µ(x(v)) = X F (v)β model predicts 0 whenever 27 Î FOR (v) = 0

28 Selected Method (2) Estimator constructed as ˆθ MA,z = s G 1 (u) ˆµ(s) 1/(δ 1 δ 2 ) + s G 2 (u,d) z(s) ˆµ(s) 1/(δ 1 δ 2 h 1 h 2 ) = w(s)z(s) s G 2 (u,d) Only approximately linear, due to presence of Î FOR (v) on RHS 28

29 Calibration Properties Weights w(s) calibrated for Phase I totals of auxiliary variables x k F (v) x k F (s) w(s)x k F (s) = s G 2 (u,d) s G 1 (u) 1/(δ 1 δ 2 ) and approximately for ˆµ FOR (v) and x k (v) Estimators for domains U h D improved over simpler estimators, by incorporating forest/ non-forest prediction locally 29

30 Comparing Estimators 1. EXP: expansion estimator ˆθ 2z (u, d) 2. PS: Model-assisted using NLDC7 categories only (=post-stratified, current FIA method) 3. REG: Model-assisted with linear model using Phase I variables 4. GAM/REGI: Model-assisted with linear model using Phase I variables interacted with GAM forest/non-forest prediction 30

31 Results Estimated Est. Relative Study Estimated Standard Efficiency of Variable Estimator Mean Error GAM/REGI FOREST EXP (forest/ PS non-forest REG binary) GAM NVOLTOT EXP (total wood PS volume in REG cuft/acre) REGI BA EXP (tree basal PS area per REG acre) REGI BIOMASS EXP (total wood PS biomass in REG tons/acre) REGI CRCOV EXP (percent PS crown REG cover) REGI QMDALL EXP (quadratic PS mean diameter REG in inches) REGI

32 5. Variance Estimation for Systematic Samples No design-based variance estimator for systematic sampling in each phase, sample contains only one ˆθ 1z (u) = of all possible grids in population/phase s G 1 (u) θ z = [0,1] [0,1] z(s) 1/(δ 1 δ 2 ) s G 1 (u) ˆθ 2z (u, d) = z(s) 1/(δ 1 δ 2 ) du s G 2 (u,d) z(s) 1/(δ 1 δ 2 h 1 h 2 ) 32

33 Simple Random Sampling Approximation Efficiency comparison relied on simple random sampling approximation for variance estimation For large numbers of possible samples and stationary populations, approximation is good on average Deviations can be significant for individual samples 33

34 Simple Random Sampling Approximation (2) Stationarity is reasonable for modelassisted estimators (model removes trend) But: only 25 possible samples in Phase II Approximate variance relies on asymptotic arguments 34

35 Alternative Approach to Assess Efficiency Gains Ignore Phase I variance component Var(ˆθ 1z (u)): identical across all estimators Generate synthetic population and compute exact Phase II variance over 25 samples avoid both asymptotic and simple random sampling approximations depends on appropriateness of model 35

36 Synthetic Population Higher order polynomial models fitted to sample data logistic model for FOREST remaining variables fitted only on locations with I FOR (v) = 1 Predict variables for Phase I Sample means of variables approximately match those of the original population 36

37 Approximation Bias Simulated Variable FOREST (forest/ non-forest binary) NVOLTOT (total wood volume in cuft/acre) BA (tree basal area per acre) BIOMASS (total wood biomass in tons/acre) CRCOV (percent crown cover) QMDALL (quadratic mean diameter in inches) Estimator EXP PS REG GAM EXP PS REG REGI EXP PS REG REGI EXP PS REG REGI EXP PS REG REGI EXP PS REG REGI Percent Bias of Variance Estimator

38 Relative Efficiency Relative Simulated Efficiency of Variable Estimator GAM/REGI FOREST EXP 4.51 (forest/ PS 3.13 non-forest binary) GAM REG 1.92 NVOLTOT binary) GAM EXP 1.31 (total wood PS 1.14 volume in REG 1.07 cuft/acre) BA REGI EXP 2.02 (tree basal PS 1.55 area per REG 1.19 BIOMASS acre) REGI EXP 1.67 (total wood PS 1.12 biomass in REG 1.14 tons/acre) CRCOV REGI EXP 1.49 (percent PS 1.36 cover) crown REGI 1.17 QMDALL cover) REGI EXP 2.55 (quadratic PS 1.79 meanindiameter inches) REGI 1.20

39 6. Conclusions: modelassisted estimation Model-assisted framework provides flexible approach to incorporate sophisticated models in survey estimation Nonparametric models make it possible to capture complex patterns in forest resource data Use on-going spatial modeling efforts by forestry researchers to improve tabular data 39

40 6. Conclusions: systematic sampling Systematic sampling is popular in natural resource surveys, but does not allow for a design-based variance estimator Synthetic population approach provided ad hoc solution in this case On-going research: predict design-based variance under nonparametric model (Li, 2006) 40

41 Almost-final version of this paper available at Contact info: