Model-assisted Estimation of Forest Resources with ... - Forest Service

Model-assisted Estimation of Forest Resources with Generalized Additive Models Jean D. Opsomer, F. Jay Breidt, Gretchen G. Moisen, and G¨oran Kauermann∗ March 26, 2003

Abstract Multi-phase surveys are often conducted in forest inventory, with the goal of estimating forested area and tree characteristics over large regions. This article describes how design-based estimation of such quantities, based on information gathered during ground visits of sampled plots, can be made more precise by incorporating auxiliary information available from remote sensing. The relationship between the ground visit measurements and the remote sensing variables is modelled using generalized additive models. Nonparametric estimators for these models are discussed and applied to forest data collected in the mountains of northern Utah in the United States. Model-assisted estimators that utilize the nonparametric regression fits are proposed for these data. The design context of this study is two-phase systematic sampling from a spatial continuum, under which properties of model-assisted estimators are derived. Difficulties with the standard variance estimation approach, which assumes simple random sampling in each phase, are described. An alternative assessment of estimator performance based on simulation is implemented. The simulation provides strong evidence that using the model predictions in a model-assisted survey estimation procedure results in substantial efficiency improvements over current estimation approaches. KEY WORDS: multi-phase survey estimation, nonparametric regression, local scoring, calibration, systematic sampling, variance estimation. ∗

Jean D. Opsomer is Associate Professor, Department of Statistics, Iowa State University, Ames, IA 50011; F. Jay Breidt is Professor, Department of Statistics, Colorado State University, Fort Collins, CO 80523; Gretchen G. Moisen is Research Forester, USDA Forest Service, Rocky Mountain Research Station, 507 25th Street, Ogden, UT 84401; G¨ oran Kauermann is Professor, Department of Economics, University of Bielefeld, 33501 Bielefeld, Germany. This work was supported in part by USDA Forest Service Rocky Mountain Research Station RJVA 02-JV-11222007-004 and 01-JV-11222007-307, and National Science Foundation grants DMS-0204531 and DMS-0204642.

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1

Introduction

Accurate estimation of forest resources over large geographic areas is of significant interest to forest managers and forestry scientists. Nationwide forest surveys of the U.S. are conducted by the U.S. Department of Agriculture Forest Service Forest Inventory and Analysis (FIA) program (U. S. Department of Agriculture Forest Service (1992), Frayer and Furnival (1999), Gillespie (1999)). In these surveys, design-based estimates of quantities like total tree volume, growth and mortality, or area by forest type are produced on a regular basis. In the current article, we consider the estimation of such quantities within a 2.5 million ha ecological province (Bailey et al. 1994) that includes the Wasatch and Uinta Mountain Ranges of northern Utah. Forests in the area consist of pinyon-juniper, oak, and maple generally in the lower elevations, and lodgepole pine, ponderosa pine, aspen, and spruce-fir generally in the higher elevations. Many forest types intermix and swap elevation zones according to other topographic variables like aspect and slope. In addition to its ecological diversity, the area hosts numerous large ownerships including National Forests, Indian Reservations, National Parks and Monuments, state land holdings, and private lands. Each owner group faces different land management issues requiring precise forest resource information. Figure 1 displays the region of interest and the sample points collected in the early 1990’s for the survey we will consider here. While this article will focus on this particular example, the approach proposed here can be applied in other natural resource estimation problems. Currently, forest survey data are being collected through a two-phase systematic sampling procedure. In phase 1, remote sensing data and geographical information system (GIS) coverage information are extracted on an intensive sample grid. Phase 2 consists of a fieldvisited subset of the phase 1 grid. During these field visits, several hundred variables are collected, ranging from individual tree characteristics and size measurements to complex ratings on scales of ecological health. Once the data are collected, estimates of population totals and related quantities need to be calculated and tabulated for the overall region, as well as for a variety of domains defined by political subdivisions, types of forest, ownership category, etc. There are literally

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thousands of estimates in the core tables put out by the FIA, with an even larger number of potential “custom estimates” that can be requested by data users. It is desirable for these estimates to be internally consistent, in the sense that the estimate of a sum of subdomain totals equals the sum of the subdomain total estimates. We refer to this estimation context as the problem of generic inference: making sensible estimates for a large number of quantities in a straightforward and internally consistent way. This can be contrasted with specific inference, in which the statistician responsible for producing estimates is studying a small number of variables and is able to build custom models for the dataset at hand. In the generic inference context, the statistician has neither time nor resources to conduct detailed analyses of all response variables. Therefore, the only practical way to produce estimates is often through design-based estimation, in which survey weights are constructed and applied to all variables and domains of interest. These weights are derived from the sampling design, but are adjusted based on ancillary information available for the sampled universe and/or collected as part of the survey. The ancillary information is used to calibrate the survey weights (making them sum to known universe quantities), and to improve the efficiency of the survey estimators. Once the weights are computed, users of the data can easily produce estimates for any variable of interest. Subdomain analyses are also simplified because the linear form of the estimators guarantees internal consistency. A large number of techniques are available to adjust survey weights based on auxiliary information. The use of auxiliary information in surveys dates back at least to Laplace (see Cochran, 1978), who employed a ratio estimator. The earliest references to regression in surveys include Jessen (1942) and Cochran (1942). Typically, auxiliary information is incorporated into the survey inference through parametric, linear models, leading to the familiar ratio and regression estimators (e.g., Cochran, 1977), post-stratification estimators (Holt and Smith, 1979), best linear unbiased estimators (Brewer, 1963; Royall, 1970), generalized regression estimators (Cassel et al. 1977; S¨arndal, 1980; Robinson and S¨arndal, 1983), and related estimators (Wright, 1983; Isaki and Fuller, 1982). Fuller (2002) is an excellent review. Recent advances in the use of auxiliary information include nonlinear estimation (Wu and Sitter, 2001), nonparametric survey regression estimation (Kuo (1988), Dorfman (1992), 3

Dorfman and Hall (1993), Chambers et al. (1993), Breidt and Opsomer (2000)), and the calibration point of view (Deville and S¨arndal, 1992). The approach currently used at the FIA is based on two-phase post-stratification (Chojnacky, 1998). Photo-interpreted vegetation cover type and ownership are used to divide the region of interest into homogeneous subsets, and the survey weights are calibrated to the phase 1 counts in each of the subsets. While this relatively simple estimator is more efficient than the two-phase expansion estimator, the increasing availability of a variety of inexpensive auxiliary information derived from remote sensing sources creates a tremendous opportunity, both to reduce costs and to further improve precision on forest survey estimates. This opportunity is all the more pressing because scientists within the Forest Service and other institutions have been using remote sensing and other GIS data to develop predictive and analytical models describing forest characteristics. This has been done in the specific inference context, in which significant effort is directed toward finding appropriate models for a small number of important variables. Because of the multivariate nature of the data and the complicated relationships among variables, nonparametric and semiparametric models have often been found to be a good compromise between model specification and flexibility. While these modelling efforts have led to improved understanding of the relationships between key forestry variables and remotely sensed information, so far this has not been reflected in corresponding improvements in forest survey estimates. The ultimate objective of this article is to explain how the results from the specific inferential efforts by forestry specialists can be used to improve the quality of their generic inference outputs as well. Model-assisted survey estimation (S¨arndal et al. 1992) is a well-known approach for incorporating auxiliary information in design-based survey estimation. It assumes the existence of a “superpopulation model” between the auxiliary variables and the variable of interest for the population to be sampled. This model is used to “assist” in the estimation of population quantities of interest in the sense that the estimators are quite efficient if the model is correctly specified, but maintain desirable properties such as design consistency and approximate design unbiasedness even if the model is misspecified (Robinson and S¨arndal, 1983). This is in contrast to purely model-based estimation, for which model misspecification 4

can lead to biased or inconsistent estimators. This is a critical distinction for generic inference, since any assumed model is unlikely to be equally appropriate across all the variables for which estimates need to be constructed. While model-assisted estimation has the potential to improve the precision of survey estimators when appropriate auxiliary information is available, it typically requires that these models be linear or at least have a known parametric shape. Breidt and Opsomer (2000) introduced local polynomial regression estimation, a survey estimation approach combining the modelling flexibility of nonparametric regression with model-assisted estimation. In this article, we describe how this approach can be extended to estimation for survey data from a two-phase design and with generalized nonparametric regression models. In the Utah mountains application as in many other forestry surveys, sampling is systematic, so that no direct design-based estimator of the variance is available. As we will discuss, the frequently used simple random sampling approximation results in a variance estimator with very poor practical behavior, so that reliance on this approximation for evaluating the reliability of survey estimators should be done with some care. We will argue that this traditional approach should at least be supplemented by other means of assessing variability. In this paper, we describe an approach based on simulating the population to be sampled and calculating the variance across repeated systematic samples from that population. In Section 2.1, we explain the two-phase systematic sampling design for the continuous spatial domain of interest, and in Section 2.2, we describe model-assisted survey estimation in this context. In Section 2.3, we incorporate additive and generalized additive models into this estimation framework. We discuss the specific models used for prediction for the northern Utah mountains forest inventory in Section 3.1, and show the results of applying nonparametric model-assisted estimation methods to the Utah data in Section 3.2. Section 3.3 discusses the issue of variance estimation for systematic sampling, Section 3.4 provides the evaluation of the methodology via simulation, and Section 4 concludes.

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2

Methodology

2.1

Two-phase systematic sampling from a spatial domain

The northern Utah mountains data were collected in two phases on a regularly spaced grid (see Figure 1). We describe the design properties of model-assisted estimation in this context by considering a rectangular spatial domain D = [0, L1 ]×[0, L2 ], where Lk = n1k δk = n2k hk δk , with njk and hk positive integers and δk positive real numbers. Here, δk represents the “grid spacing” in dimension k, and hk the sub-sampling rate on dimension k for the phase two sample. Then, n1 = n11 n12 is the phase one sample size and n2 = n21 n22 is the phase two sample size. An irregular spatial domain is handled by intersecting it with the rectangle D. The two-phase systematic sampling design is implemented as follows. Let uk represent independent Uniform(0, 1) random variables and dk independent Discrete Uniform{1, 2, . . . , hk } random variables, with the uk , dk independent of each other. Given u = (u1 , u2 ), the phase one sample is the randomly-located lattice {Li (u)} = {((u1 + i1 − 1)δ1 , (u2 + i2 − 1)δ2 )} for i = (i1 , i2 ) ∈ {1, . . . , n11 } × {1, . . . , n12 }. Given d = (d1 , d2 ), the phase two sample is the random sub-lattice {`j (u, d)} = {((u1 + d1 + (j1 − 1)h1 − 1)δ1 , (u2 + d2 + (j2 − 1)h2 − 1)δ2 )} for j = (j1 , j2 ) ∈ {1, . . . , n21 } × {1, . . . , n22 }. Note that ∪d {`j (u, d)} = {Li (u)}.

2.2

Model-assisted estimation

To motivate the model-assisted approach which we use, we begin with a discussion of the twophase difference estimator. Let z(v) denote the study variable of interest, defined for v ∈ D but observed only for v ∈ {`j (u, d)}, and let z 0 (v) represent a different variable that is known for all v ∈ {Li (u)}. Note that neither z(·) nor z 0 (·) is assumed stochastic, and in particular

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neither depends on the random vectors u, d. Define Di = [(i1 − 1)δ1 , i1 δ1 ] × [(i2 − 1)δ2 , i2 δ2 ]. Then the population total Z

θ :=

z(v) dv =

XZ

D

i

X z(Li (u))

Z

=

z(v) dv

Di

[0,1]×[0,1]

i

1/(δ1 δ2 )

du

(1)

can be estimated with the two-phase difference estimator X z 0 (Li (u))

θˆ :=

i

1/(δ1 δ2 ) (

XX

=

d0

j

X z(`j (u, d)) − z 0 (`j (u, d))

+

1/(h1 δ1 h2 δ2 )

j

z 0 (`j (u, d0 )) z(`j (u, d0 )) − z 0 (`j (u, d0 )) 1{d=d0 } + , 1/(δ1 δ2 ) 1/(δ1 δ2 ) 1/(h1 h2 ) )

(2)

with 1{d=d0 } = 1 if d = d0 and 0 otherwise, where the summation over d0 is over all the possible values for the random pair (d1 , d2 ). Since the indicator 1{d=d0 } has expectation 1/(h1 h2 ), we have that E(θˆ | u) =

X X z(`j (u, d0 )) d0

j

1/(δ1 δ2 )

=

X z(Li (u)) i

1/(δ1 δ2 )

,

(3)

from which it is immediate that ˆ = E(θ)

Z

E(θˆ | u) du = θ.

(4)

[0,1]×[0,1]

Also by standard results on systematic sampling from a finite population, we have that 



Var θˆ | u = where |D| = 2

R D

|D|2 1 1− S 2 (u), 2 (n21 n22 ) h1 h2 



(5)

dv,

S (u) =

2 d td (u)

P

− ( d td (u))2 /(h1 h2 ) , for d ∈ {1, . . . , h1 } × {1, . . . , h2 } h1 h2 − 1 P

and td (u) =

X



z(`j (u, d)) − z 0 (`j (u, d)) .

j

7

(6)

Therefore, the estimator θˆ is design-unbiased regardless of the relationship between z and z 0 , with design variance given by 







ˆ = Var E(θˆ | u) + E Var(θˆ | u) Var(θ) =

Z



E(θˆ | u) − θ

2

[0,1]×[0,1]

|D|2 1 du + 1 − E(S 2 (u)). 2 (n21 n22 ) h1 h2 



(7)

The first component of the variance does not depend on the choice of z 0 , but the second component of the variance will be small if z 0 is a good predictor of z. In the following result, (4) and (7) are combined to show that θˆ is design consistent under an asymptotic formulation in which the sampling density in D increases (“infill asymptotics”), assuming integrability conditions on z and z 0 . This result is similar to consistency results obtained in design-based stereology (Arnau and Cruz-Orive, 1996), but the two-phase structure is novel. Result 1 If z(·) and z 0 (·) are bounded and continuous almost everywhere on D, then θˆ converges in mean square to θ as njk → ∞ with D fixed. Proof: The estimator θˆ is unbiased by (4), so it suffices to show that its variance goes to zero. By hypothesis, both z and z0 are Riemann integrable on D, so that from (3) E[θˆ | u] = θ

lim

n11 ,n12 →∞

and from (6) Z td (u) = (z(v) − z 0 (v)) dv. lim |D| n21 ,n22 →∞ n21 n22 D

Since z and z 0 are bounded, we have that n11

lim ,n →∞ 12

Z



E[θˆ | u] − θ

2

du =

[0,1]×[0,1]

Z



[0,1]×[0,1] n11

lim E[θˆ | u] − θ ,n →∞

2

du = 0

12

and E[S 2 (u)] |D| = |D|2 E lim n21 ,n22 →∞ (n21 n22 )2 2

"

S 2 (u) lim = 0, n21 ,n22 →∞ (n n )2 21 22 #

so that mean square consistency follows. In the absence of useful information from the first-phase sample, the simple expansion estimator θˆexp =

X z(`j (u, d)) j

1/(h1 δ1 h2 δ2 ) 8

(8)

obtained from (2) with z 0 ≡ 0 can be used. In most cases of two-phase sampling, however, relatively inexpensive auxiliary information X(Li (u)) is collected at each phase one site. This information can be used to construct predictors of z guided by a superpopulation model E[z(v) | X(v)] = µ(X(v)).

(9)

Typically, µ(·) is estimated from regression of {z(`j (u, d))} on {X(`j (u, d))}, and the resulting µ ˆ(X(`j (u, d))) ≡ µ ˆj (u, d) is then substituted into (2) to form the model-assisted estimator (

θˆma =

XX d0

j

µ ˆj (u, d0 ) z(`j (u, d0 )) − µ ˆj (u, d0 ) 1{d=d0 } + . 1/(δ1 δ2 ) 1/(δ1 δ2 ) 1/(h1 h2 ) )

(10)

Unlike z 0 (·), µ ˆ(·) usually does depend on u and d so the unbiasedness argument in (4) and the variance expression in (7) no longer hold exactly. However, under mild conditions which we do not explore further here, the model-assisted estimator should follow the traditional model-assisted paradigm and remain asymptotically design-unbiased and consistent, with approximate variance given by Z

Var(θˆma ) =



[0,1]×[0,1]

E(θˆma | u) − θ

2

1 |D|2 1− E(Se2 (u)) 2 (n21 n22 ) h1 h2 

du +



(11)

where P 2 2 ˆ ˆ t (u) − t (u) /(h1 h2 ) d d d d

P

Se2 (u) =

h1 h2 − 1

, for d ∈ {1, . . . , h1 } × {1, . . . , h2 }

and tˆd (u) =

X

(z(`j (u, d)) − µ ˜(`j (u, d))) ,

j

and µ ˜(·) is obtained from the (hypothetical) regression of {z(Li (u))} on {X(Li (u))}. It is now clear why a model can improve the efficiency of the estimator. If the model fits the data well, the variance of the residuals z(Li (u)) − µ ˜(Li (u)) can be expected to be smaller than the variance of the z(Li (u)). If the model fits poorly, the residual variance should be equally large or even potentially larger than the study variable’s variance. Hence, the efficiency gains of the model-assisted estimator depend on the selection of a good model for µ(·) in (9).

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Traditionally, µ(·) is assumed to be linear, in which case θˆma is known as the generalized regression estimator (see S¨arndal et al. 1992). The post-stratified estimator for θ can be considered as a special case of the generalized regression estimator, in which the auxiliary variables are categorical. By classifying the phase two sample into a small number of poststrata based on the phase one information, this estimator is commonly used in forest resource monitoring as a relatively simple way to incorporate auxiliary information in the estimation. See S¨arndal et al. (1992, Section 7.6) for explicit expressions for the post-stratified estimator and its variance. Recently, Breidt and Opsomer (2000) considered nonparametric models, fitted by local polynomial regression, as a more flexible alternative to linear and parametric models. A wide range of other nonparametric regression methods are available, including other kernel-based methods, spline methods and orthogonal decomposition-based approaches (see e.g. Opsomer (2002) for an overview), and in principle any of these can be used to produce predictions µ ˆj . So far, applications of these methods to survey estimation have been limited to element sampling. The estimator θˆma has some additional desirable properties if the regression method is linear, in the sense that µ ˆi (u, d) =

P

j

ωij z(`j (u, d)) for a set of smoothing weights ωij that

do not depend on the {z(`j (u, d))}. This linearity holds for many generalized regression estimators, including the post-stratification estimator. In this case, θˆma can be written as P a linear combination of the sample observations θˆma = j ωj z(`j (d, u)), with weights {ωj }

independent of the z(`j (u, d)). These regression weights are ideal for generic inference, as they can be used for any variables collected in the same survey, and to the extent that such variables also follow model (9), they will also benefit from the efficiency gain.

2.3

Model-assisted estimation using generalized additive models

Suppose now that µ(X(v)) is the generalized additive model µ(X(v)) = E(z(v) | X(v)) = g(m1 (X 1 (v)) + . . . + mr (X r (r)))

(12)

for some known link function g(·) and unknown smooth functions mk (·), k = 1, . . . , r, where the X k (v) are known subsets of the vector X(v). Given a set of estimated func10

tions m ˆ k (·), k = 1, . . . , r, model predictions µ ˆi = g(m ˆ 1 (X 1 (Li (u))) + . . . + m ˆ r (X r (Li (u)))) are readily calculated, for instance using the gam() local scoring estimation routines (Hastie and Tibshirani, 1990) implemented in S-Plus. When the link function g(·) is the identity link, model (12) is referred to as an additive model and the resulting estimators are linear, in the sense that they can be written as a linear combination of the observations. If g(·) is not the identity link, however, local scoring estimators are not linear and the resulting estimator θˆgam is no longer a linear combination of the {z(`j (u, d))}, so that weights are not available. In Section 3.2, we discuss an approach for obtaining weights from a generalized additive model for the forestry application.

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Application to Forest Inventory

3.1

Generalized additive models for the forest inventory data

We now discuss generalized additive models for the Utah mountains forest inventory. Field data used in this study were collected on a 5 km sample grid (Figure 1). On the 968 phase 2 sample plots, numerous forest site variables and individual tree measurements were collected, including a binary classification (FOREST) of the plot into “forest” or “non-forest”. The FOREST variable is critical in the inventory because many other response variables are defined to be zero on non-forested sites. In this study, we consider five additional variables, all of which follow this definitional constraint: NVOLTOT, total wood volume in cubic ft per acre; BA, tree basal area per acre; BIOMASS, total wood biomass in tons per acre; CRCOV, percent crown cover; and QMDALL, quadratic mean diameter in inches. In addition to the field plot data, remotely sensed information was extracted on the 5 km field plot locations as well as on an intensified 1 km grid (24,980 points), which will represent the phase 1 data. The ancillary variables used in our models came from three sources: 1. Digital elevation models produced by the U.S. Defense Mapping Agency, which provided elevation (ELEV90CU), transformed aspect (TRASP90) and slope (SLP90CU). 2. 30-m resolution Thematic Mapper (TM) imagery, from which we extracted the vegeta11

tion cover type from the U.S. National Land Cover dataset (Vogelmann et al. 2001) collapsed to seven vegetation classes (NLCD7). Also, letting MRLC00Bk denote the kth TM spectral band, we used MRLC00B5 by itself and we computed a Normalized Difference Vegetation Index (NDVI) as (MRLC00B4−MRLC00B3)/(MRLC00B4+MRLC00B3). 3. Spatial coordinates (Xs and Ys). Moisen and Edwards (1999), Frescino et al. (2001) and Moisen and Frescino (2002) developed parametric and nonparametric models relating remotely sensed data to forest attributes observed during field visits. Taking a similar approach here, we model the response variable FOREST as a nonparametric function of the ancillary predictor variables mentioned above through a generalized additive model with a logit link function g(·). The model (12) was fitted using gam() in S-Plus. Component functions were obtained through loess smoothers with local polynomials of degree 1 and a relatively large smoothing parameter (see Opsomer (2002) for an explanation of the loess smoothing method and smoothing parameter selection). Predictor variables ELEV90CU, TRASP90, SLP90CU, MRLC00B5, and NDVI entered the model as univariate smooth terms, while Xs with Ys contributed as a bivariate smooth function, and NLCD7 entered as a categorical variable in the model. The plots of the smooth contributing terms in the FOREST model are shown in Figure 2.

3.2

Model-assisted estimation for the forest inventory

We calculate the following estimators for FOREST: 1. EXP, the expansion estimator in (8), 2. PS, a two-phase post-stratified estimator with the seven categories of variable NLCD7 as post-strata, representing a common choice in FIA, 3. REG, a model-assisted estimator from (10), with parametric regression on the dummy variables for NLCD7 plus linear terms for ELEV90CU, TRASP90, SLP90CU, MRLC00B5, NDVI, and Xs and Ys spatial coordinates,

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4. GAM, the gam-assisted estimator from (10) with the model described in the subsection above fitted via local scoring. For the remaining response variables, we take the traditional large-scale survey point of view in which estimation is performed through the use of survey weights, as explained in Section 1. To obtain the operational advantages of weights, along with the efficiency gains of the gam, we consider a regression model that treats the gam fits for FOREST as an auxiliary variable in the estimation of the remaining variables. One possible way to do this is to simply treat the FOREST fits as an auxiliary variable in a linear model specification, and compute the weights of a (linear) regression estimator. In that way, any variables correlated to the model-fitted probability of the presence of forest will be estimated more efficiently. In this case however, the special structure of the relationship between the presence/absence of forest and the other variables suggests a more appropriate model. For every phase 1 site, we use the available auxiliary information to construct the indicator that is one when the GAM-predicted probability of forest is greater than the empirical proportion of forest in phase 2. A regression model consisting entirely of interactions between this forest indicator and other covariates is then constructed. The covariates include dummy variables for NLCD7 (with non-forest categories collapsed to ensure full rank), plus linear terms for ELEV90CU, TRASP90, SLP90CU, MRLC00B5, NDVI, and Xs and Ys spatial coordinates. Note that this regression model predicts zero for the response variable at any site for which FOREST is predicted to be zero. A standard model-assisted linear regression estimator is then built from this regression model. We refer to this regression-interaction model-assisted estimator as REGI. Treating the gam-predicted probabilities as fixed with respect to the design, the estimator REGI can be written as a linear combination of the response variables, and weights are obtained. Because only the interaction variables were included in the model, these weights will be calibrated to the totals on the part of phase 1 classified as likely to be forest, and will only be approximately calibrated on all of phase 1. Table 1 shows the estimates for all six variables, as well as the estimated standard deviations. Following standard FIA practice, these estimated standard deviations assume simple random sampling with replacement in phase 1 and without replacement in phase 2. These 13

empirical results suggest that the GAM estimator and the related regression estimator with interactions (REGI) dominate the simple expansion estimator, the post-stratification estimator, and the regression estimator.

3.3

Variance estimation under systematic sampling

The estimated efficiencies in Table 1 are somewhat suspect, however, because they rely on asymptotic variance approximations, and they act as if the actual systematic samples were in fact drawn via simple random sampling. This last point is potentially serious when the number of possible systematic samples is small, as in this 1-in-25 systematic subsample. To illustrate this problem, consider the ideal circumstance in which the trend in the variable of interest can be completely removed by covariates, and the residuals are iid normal (0, σ 2 ) random variables. Condition on phase 1 and let H = h1 h2 denote the total number of systematic phase 2 samples. Let n2 denote the phase 2 sample size. Then, as shown in Theorem 8.5 of Cochran (1977), the average (over all possible realizations of the normal residuals) systematic sampling variance in (5) is equal to the average of the simple random sampling variance estimator, 2

P 2 2  P {z(` (u, d)) − µ ˆ } − {z(` (u, d)) − µ ˆ } /n2 j j j j 1 j j

|D| 1− (n21 n22 )2 h1 h2 

n2 − 1

.

(13)

But such unbiasedness is not so interesting for a given realization of the population. Indeed, consider F =

systematic sampling variance in (5) . simple random sampling variance estimator in (13)

Under the assumptions above, it is immediate that this ratio is F-distributed with H − 1 numerator degrees of freedom and n2 − 1 denominator degrees of freedom. As n2 → ∞, ( )1/2  2(1 + H/n ) 2 , F = 1 + Op 

H −1

so that, at least in this simple case, the simple random sampling variance estimator is inconsistent unless H tends to infinity. In the northern Utah mountains data set, n2 = 968 but H = 25. The quartiles of the corresponding F distribution are 0.792 and 1.181. Thus, 14

in about half of the possible realizations of the population, the simple random sampling variance estimator will be off by ±20% or more. The 0.025 quantile is 0.514 and the 0.975 quantile is 1.655, so departures on the order of ±50% are easily possible. This problem of variance estimation is basically intractable given only the sample, since it amounts to a sample of size one. Indeed, all of the variance estimators for systematic sampling given in Wolter (1985, Section 7.2.1) will perform poorly, as they all are forced to rely on within-systematic-sample variation to approximate between-systematic-sample variation. We therefore consider a simulation-based alternative in the following subsection.

3.4

Simulations

Because the variance estimators are so unreliable in this context, we undertake a numerical experiment to assess the efficiencies of the various estimators. Our approach is to construct a population that closely mimics the one we are sampling from, draw repeated systematic samples from that population, and calculate variances based on these repeated samples. While any conclusions drawn from this procedure will of course depend on how well the chosen population model corresponds to the true population, we believe that it has the potential to provide a more reliable measure of the true variability of the estimators than the simple random sampling approximation. We begin by fitting large, parametric models to each of the variables studied in Table 1. The first model is a logistic regression for the forest/non-forest indicator. It includes six dummy variables for the categories of NLCD7, fourth-order polynomials for ELEV90CU, TRASP90, SLP90CU, MRLC00B5, NDVI and the two spatial coordinates, as well as a first-order interaction term for the spatial coordinates. The models for the remaining study variables contain similar terms, and are fitted to the positive responses after suitable transformation (typically square root). Using these fitted models, we simulate populations of study variables on all of the phase 1 sites. In this experiment, we condition on phase 1 because its percentage contribution to the empirical variances of the estimators was found to be small (around 5-7%), as shown in Table 1, and its contribution is common to all the estimators. The binary variable is 15

simulated with unequal probability Bernoulli random variables, and the remaining response variables are simulated on the transformed scale with Gaussian noise, then mapped back to the original scale. These response variables are set to zero wherever the simulated FOREST variable is zero. We draw all 25 possible systematic phase 2 samples from phase 1, compute the estimates for each sample, and then compute averages and variances over these 25 samples. Note that these 25 represent the entire conditional randomization distribution of the estimators, so that empirical means and variances are exactly the conditional expectation and conditional variance, given phase 1. The results are given in Table 2. The expectations of the estimators for the simulated populations are comparable to the corresponding estimates for the actual populations in Table 1, and the expectations of the estimated standard errors for the simulated populations are comparable to the corresponding estimates for the actual populations. These comparisons suggest that the simulated populations reproduce at least the second-order moment structure of the real data fairly well. As expected, the expansion estimator is exactly unbiased, and the remaining estimators are all essentially unbiased (percent relative biases no more than 0.25% in all cases) due to their model-assisted structure. The PS estimator, which is the Forest Service standard, is better than the EXP estimator in all cases, but even the simple regression estimator REG usually offers gains over both the expansion estimator and the PS estimator. The GAM estimator is much more efficient than its competitors for the FOREST variable, and the regression estimator with GAM-dependent interaction terms (REGI) is more efficient than its competitors for all of the other variables. The efficiency gains estimated through this simulation procedure are quite different from those estimated using the simple random sampling approximation, with those for FOREST, BA and QMDALL larger but those for NVOLTOT, BIOMASS and CRCOV smaller. This is readily explained by the results in the last column of Table 2, which shows that the simple random sampling variance estimator performs very poorly in this context, behaving somewhat like the hypothetical F random variable described in Section 3.2. Overall, these simulation results suggest that though the efficiency gains reported in Table 1 may be unre16

liable due to the lack of good variance estimators, there are in fact real gains obtained with the GAM-assisted and related regression estimators.

4

Conclusion

Auxiliary information from remote sensing or other sources is becoming increasingly available to organizations involved in natural resource surveys. Scientists in these organizations are already developing detailed prediction models for many variables of interest, but they have tended not to use these prediction models in their survey estimation procedures. In this article, we have explained how nonparametric model-assisted estimation techniques can be used to incorporate the results of such modelling efforts in the production of survey estimates, even in the case of fairly complex models and multi-phase designs. We have provided some theoretical justification for gam-assisted survey inference in the context of two phases of systematic sampling from a spatial domain. The gam-assisted methodology was applied in a survey of forest resources in the mountains of northern Utah, a region important for its ecological and land-use diversity. Theoretical properties of this approach in complex surveys deserve further investigation. Important open issues include model selection and the selection of the smoothing parameters for the nonparametric regression fitting algorithms, since this affects both the estimates of the quantities of interest and their estimated variances. In the course of this research, the unsatisfactory behavior of the traditional estimator of the design variance under systematic sampling became apparent, and we used a simulation-based alternative to evaluate our proposed estimation procedure. Future research into simulation-based variance estimation in this context, including choice of models and robustness to their selection, certainly appears warranted.

References Arnau, X. G. and L. M. Cruz-Orive (1996). Consistency in systematic sampling. Advances in Applied Probability 28, 982–992. 17

Bailey, R. G., P. E. Avers, T. King, and W. H. McNab (Eds.) (1994). Ecoregions and Subregions of the United States (map). Washington, DC: U.S. Geological Survey. Scale 1:7,500,000, colored, accompanied by a supplementary table of map unit descriptions, prepared for the U.S. Department of Agriculture, Forest Service. Breidt, F. J. and J. D. Opsomer (2000). Local polynomial regression estimators in survey sampling. Annals of Statistics 28, 1026–1053. Brewer, K. R. W. (1963). Ratio estimation and finite populations: Some results deducible from the assumption of an underlying stochastic process (Corr: 66V8 p37). The Australian Journal of Statistics 5, 93–105. Cassel, C. M., C. E. S¨arndal, and J. H. Wretman (1977). Foundations of Inference in Survey Sampling. New York: Wiley. Chambers, R. L., A. H. Dorfman, and T. E. Wehrly (1993). Bias robust estimation in finite populations using nonparametric calibration. Journal of the American Statistical Association 88, 268–277. Chojnacky, D. C. (1998). Double sampling for stratification: A forest inventory application in the interior west. Research Paper RMRS-RP-7, U. S. Department of Agriculture, Forest Service, Rocky Mountain Research Station, Ogden, UT. Cochran, W. G. (1942). Sampling theory when the sampling units are of unequal size. Journal of the American Statistical Association 37, 199–212. Cochran, W. G. (1977). Sampling Techniques (3rd ed.). New York: John Wiley & Sons. Cochran, W. G. (1978). Laplace’s ratio estimator. In H. A. David (Ed.), Contributions to survey sampling and applied statistics, pp. 3–10. New York: Academic Press. Deville, J.-C. and C.-E. S¨arndal (1992). Calibration estimators in survey sampling. Journal of the American Statistical Association 87, 376–382. Dorfman, A. H. (1992). Non-parametric regression for estimating totals in finite populations. In ASA Proceedings of the Section on Survey Research Methods, pp. 622–625. American Statistical Association (Alexandria, VA). 18

Dorfman, A. H. and P. Hall (1993). Estimators of the finite population distribution function using nonparametric regression. The Annals of Statistics 21, 1452–1475. Frayer, W. E. and G. M. Furnival (1999). Forest survey sampling designs: A history. Journal of Forestry 97, 4–8. Frescino, T., T.C. Edwards, Jr, and G. Moisen (2001). Modelling spatially explicit structural attributes using generalized additive models. Journal of Vegetation Science 12, 15–26. Fuller, W. A. (2002). Regression estimation for survey samples. Survey Methodology 28, 5–23. Gillespie, A. J. R. (1999). Rationale for a national annual forest inventory program. Journal of Forestry 97, 16–20. Hastie, T. J. and R. J. Tibshirani (1990). Generalized Additive Models. Washington, D. C.: Chapman and Hall. Holt, D. and T. M. F. Smith (1979). Post-stratification. Journal of the Royal Statistical Society, Series A 142, 33–46. Isaki, C. and W. Fuller (1982). Survey design under the regression superpopulation model. Journal of the American Statistical Association 77, 89–96. Jessen, R. J. (1942). Statistical investigation of a sample survey for obtaining farm facts. Research Bulletin 304, Iowa Agriculture Experiment Station. Kuo, L. (1988). Classical and prediction approaches to estimating distribution functions from survey data. In ASA Proceedings of the Section on Survey Research Methods, pp. 280–285. American Statistical Association (Alexandria, VA). Moisen, G. and T. Edwards (1999). Use of generalized linear models and digital data in a forest inventory of Utah. Journal of Agricultural, Biological and Environmental Statistics 4, 372–390. Moisen, G. G. and T. S. Frescino (2002). Comparing five modelling techniques for predicting forest characteristics. Ecological Modelling 157, 209–225. 19

Opsomer, J. D. (2002). Nonparametric regression model. In A. H. El-Shaarawi and W. W. Piegorsch (Eds.), Encyclopedia of Environmetrics, Volume 3, pp. 421–427. Chichester, UK: Wiley & Sons. Robinson, P. M. and C.-E. S¨arndal (1983). Asymptotic properties of the generalized regression estimator in probability sampling. Sankhya, Series B 45, 240–248. Royall, R. M. (1970). On finite population sampling theory under certain linear regression models. Biometrika 57, 377–387. S¨arndal, C.-E. (1980). On π-inverse weighting versus best linear unbiased weighting in probability sampling. Biometrika 67, 639–650. S¨arndal, C.-E., B. Swensson, and J. Wretman (1992). Model Assisted Survey Sampling. New York: Springer-Verlag. U. S. Department of Agriculture Forest Service (1992). Forest Service resource inventories: An overview. Technical report, Washington, DC. Vogelmann, J. E., S. M. Howard, L. Yang, C. R. Larson, B. K. Wylie, and N. V. Driel (2001). Completion of the 1990s National Land Cover data set for the conterminous United States from Landsat Thematic Mapper data and ancillary data sources. Photogrammetric Engineering and Remote Sensing 67, 650–662. Wolter, K. M. (1985). Introduction to Variance Estimation. New York: Springer-Verlag Inc. Wright, R. L. (1983). Finite population sampling with multivariate auxiliary information. Journal of the American Statistical Association 78, 879–884. Wu, C. and R. R. Sitter (2001). A model-calibration approach to using complete auxiliary information from survey data. Journal of the American Statistical Association 96, 185– 193.

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Table 1: Estimation results for the northern Utah mountains data. Estimators are expansion (EXP), post-stratification (PS), regression on linear terms of the continuous auxiliaries, plus dummies for the categorical variable (REG); generalized additive model on the same variables (GAM); and linear regression on the same terms, but with all terms interacted with the indicator that the GAM-predicted probability of forest is greater than the empirical proportion of forest (REGI).

Study Variable

Estimator

Estimated Mean

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)

EXP PS REG GAM EXP PS REG REGI EXP PS REG REGI EXP PS REG REGI EXP PS REG REGI EXP PS REG REGI

0.51 0.54 0.54 0.54 845.81 877.41 877.67 853.85 45.19 47.12 47.29 46.01 13.51 14.01 14.00 13.60 21.02 22.03 22.18 21.64 3.77 3.95 3.96 3.89

Estimated Standard Error

Percent Variance from Phase 1

Est. Relative Efficiency of GAM/REGI

0.02 0.01 0.01 0.01 44.07 39.10 35.35 32.98 2.01 1.77 1.63 1.54 0.69 0.60 0.54 0.49 0.86 0.77 0.68 0.65 0.15 0.14 0.14 0.14

3.73 4.93 5.80 6.82 3.73 4.74 5.80 6.66 3.73 4.77 5.63 6.33 3.73 4.86 6.17 7.32 3.73 4.65 5.93 6.46 3.73 4.37 4.66 4.70

1.83 1.38 1.18

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1.79 1.41 1.15 1.70 1.33 1.12 1.96 1.51 1.19 1.73 1.39 1.09 1.26 1.08 1.01

Table 2: Results from all 25 possible systematic sub-samples from the phase 1 sample for the simulated populations. Estimators are expansion (EXP), post-stratification (PS), regression on linear terms of the continuous auxiliaries, plus dummies for the categorical variable (REG); generalized additive model on the same variables (GAM); and linear regression on the same terms, but with all terms interacted with the indicator that the GAM-predicted probability of forest is greater than the empirical proportion of forest (REGI).

Simulated Variable

Estimator

Expectation of Estimator

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)

EXP PS REG GAM EXP PS REG REGI EXP PS REG REGI EXP PS REG REGI EXP PS REG REGI EXP PS REG REGI

0.54 0.54 0.54 0.54 892.59 892.51 892.52 893.77 47.83 47.84 47.85 47.91 15.32 15.33 15.33 15.35 22.88 22.88 22.89 22.93 4.10 4.10 4.10 4.11

Expectation of Estimated Std. Error

Percent Relative Bias

Relative Efficiency of GAM/REGI

Percent Bias of Variance Estimator

0.02 0.01 0.01 0.01 40.81 36.01 31.68 28.10 1.96 1.72 1.51 1.32 0.69 0.61 0.53 0.46 0.88 0.76 0.66 0.55 0.15 0.13 0.12 0.10

0.00 0.01 0.02 0.12 0.00 -0.01 -0.01 0.13 0.00 0.03 0.04 0.17 0.00 0.02 0.06 0.17 0.00 0.02 0.05 0.25 0.00 0.00 0.03 0.12

4.51 3.13 1.92

12.62 9.33 24.77 -31.01 -23.71 -32.11 -43.88 -52.71 19.35 19.81 20.97 8.93 17.48 34.31 -1.32 -14.61 -4.04 -20.93 -31.38 -44.01 5.92 23.70 50.32 13.27

22

1.31 1.14 1.07 2.02 1.55 1.19 1.67 1.12 1.14 1.49 1.36 1.17 2.55 1.79 1.20

Figure Captions • Figure 1: Representation of the study region in northern Utah. Each triangle represents a field-visited phase 2 plot. Each dot in the magnified section represents a remotelysensed phase 1 plot. See Section 3.1 for an explanation of the phase 1 and phase 2 plots. • Figure 2: GAM model fits for binary indicator of forest/non-forest (FOREST).

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