Direct Variance-Covariance Modeling as an Alternative to the

For. Sci. 60(4):652– 662 http://dx.doi.org/10.5849/forsci.13-019 Copyright © 2014 Society of American Foresters

FUNDAMENTAL RESEARCH

biometrics

Direct Variance-Covariance Modeling as an Alternative to the Traditional Guide Curve Approach for Prediction of Dominant Heights Mingliang Wang, Michael B. Kane, Bruce E. Borders, and Dehai Zhao A new method for developing dominant height prediction models, which is closely related to certain earlier guide curve-based methods, is derived from linear prediction theory by directly modeling variances and covariances of height remeasurements without involving any local parameters. This method relies on population-averaged mean height-age curve and covariances (correlations and variances/standard deviations) between heights at different ages for height projection given prior observations for a new subject. We constructed a new height model consisting of the means, correlations, and variances approximated, respectively, by the Chapman-Richards model, first-order continuous autoregressive model, and the power of the mean function. The new model was compared with four difference models derived from the (generalized) algebraic difference approach, by evaluating their projection accuracy using a second-rotation loblolly pine (Pinus taeda L.) data set collected from a well-designed experiment. Both kinds of models produced very comparable predictions, but the new guide model can be preferred for height projection using earlier (e.g., before age 10) measurements. Keywords: site index model, guide curve, linear prediction, correlation and regression

T

he height growth of dominant trees has been commonly used as a measure of site quality in pure even-aged stands because it is relatively independent of stocking over a fairly wide range of stand density (Carmean 1972, Monserud 1984). Various height models have been developed over the past few decades. Clutter et al. (1983) summarized three general methods of constructing site index curves: the guide curve method, the difference equation method, and the parameter prediction method (PPM). For some variants of the three methods and other additional methods, one may refer to Weiskittel et al. (2011) or Burkhart and Tome (2012). The guide curve method of Bruce (1926) was initially based on the graphical procedure known as anamorphosis (Bruce 1923). In this graphical method, single pairs of dominant height-age (H-A) measurements are taken on a large number of temporary sample plots (TSPs), an average H-A curve is determined typically by grouping ages into age classes and then calculating average heights by age class, and a set of anamorphic curves are constructed by scaling the average H-A guide curve to go through a range of site

indices at a predetermined base age. With the availability of computing facilities for regression analysis, the guide curve can be obtained by fitting a base model given in the general form as H ⫽ f 共 A; ␪兲 ⫹ ␧

(1)

where f denotes a general function, H is dominant height, A is age, ␪ is a vector of parameters, common to all of the plots, and ␧ is the error term. The fitted guide curve is commonly used to construct a family of anamorphic curves by multiplicative scaling. Given repeatedly measured heights taken on permanent sample plots (PSPs) or obtained by stem analysis, the guide curve method can still be applied (Clutter et al. 1983), usually assuming that the errors are independent and identically normally distributed. Both the difference equation method and PPM, which require longitudinal height measurements, can be used to develop anamorphic or polymorphic site index models. The PPM models are generally polymorphic with multiple asymptotes (e.g., Monserud 1984). A major criticism of the PPM is that its resulting models are base-age dependent, because site index for each plot is

Manuscript received February 12, 2013; accepted September 26, 2013; published online October 31, 2013. Affiliations: Mingliang Wang ([email protected]), Warnell School of Forestry and Natural Resources, University of Georgia, Athens, GA. Michael B. Kane ([email protected]), University of Georgia. Bruce E. Borders ([email protected]), University of Georgia. Dehai Zhao ([email protected]), University of Georgia. Acknowledgments: We would like to express our gratitude to the University of Georgia’s Warnell School of Forestry and Natural Resources and the Plantation Management Research Cooperative for providing the study data. We are grateful to the associate editor and two anonymous reviewers for their constructive comments and suggestions. 652

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predetermined, either observed or estimated at an arbitrarily chosen base age, and enters the base model (in relation to usually two varying base parameters) as a predictor variable. To avoid base-age dependence, Bailey and Clutter (1974) introduced base-age invariance as a property of site index curves. They presented a method to derive base-age invariant models, known as the “algebraic difference approach” (ADA) in forestry (Borders et al. 1984, Cieszewski and Bailey 2000). This method, which is equivalent to the “difference equation method” of Clutter et al. (1983) if we ignore the data structure used for model fitting (Wang et al. 2007), recognizes one parameter in the base model as site-specific (local), while leaving the others global (common). That is, H ⫽ f 共 A; ␹ , ␤兲 ⫹ ␧

(2)

where ␤ is a vector of global parameters common to all of the subjects and ␹ indicates a site-specific (local) parameter. The ADA-derived models are, in general, either anamorphic or polymorphic but with single asymptotes. To develop polymorphic models with multiple asymptotes, Cieszewski and Bailey (2000) introduced the generalized algebraic difference approach (GADA). This approach allows several base parameters, usually two in practice including generally an asymptote parameter and one of the others, to be site-specific, and the local parameters are related to each other in such a way that after some algebraic manipulation only one parameter varies in the resulting site index model. The two varying parameters GADA model can be expressed in general form as H ⫽ f 共 A; ␹ 1 , ␹ 2 , ␤兲 ⫹ ␧

(3)

␹ 1 ⫽ f 1 共 ␹ 兲, ␹ 2 ⫽ f 2 共 ␹ 兲 where ␹1 and ␹2 indicate two local parameters, both related to the nominal site quality measure ␹ (set to be site-specific but not well defined) through functions f1 and f2. Cieszewski and colleagues (Cieszewski and Bailey 2000, Cieszewski 2002, 2004, Cieszewski and Strub 2008) formulated a variety of GADA models. Wang et al. (2008b) suggested that the asymptote parameter ␹1 (⬅␹) and one of the other parameters ␹2 had a power functional dependence which can adapt the PPM base-age-specific models to be base-age invariant. More recently, mixed-effects models have been increasingly used in H-A modeling (Lappi and Bailey 1988, Fang and Bailey 2001, Hall and Bailey 2001, Huang et al. 2009, Meng and Huang 2010, Wang et al. 2011). Mixed-effects modeling essentially shares the same logic as ADA/GADA models in recognizing the varying (local) parameters, although treating them as random effects in contrast to fixed individual effects via dummy variables that appear in the ADA/GADA models (for details, see Wang et al. 2008a). Given remeasured dominant heights, all of the above methods (guide curve, PPM, ADA/GADA or difference equation method, and mixed models) can be used to develop height models. In the model prediction stage for height projection given prior height measurements taken on new subjects, all of the methods are in a “conˆ 2 ⫽ f(H1, A1, A2) where H1 ditional” sense [in the general form of H is previous measurements, A1 and A2 are tree ages, and H2 is heights to be projected], because at least one H-A pair is needed to estimate/predict the local parameter(s) for the new subjects. This is to determine the scaling factor (implying an asymptotic parameter in most cases) for the guide curve, to estimate the local parameter identified in the difference equation/ADA/GADA model, to esti-

mate the site index required by a PPM model, or to predict random effects via linear prediction theory commonly known as “best linear unbiased prediction” (BLUP) (Goldberger 1962, see also Robinson 1991) for mixed models. In the model-fitting stage, however, the four primary methods differ in the way they handle the heteroscedasticity and autocorrelation that are usually associated with longitudinal measurements. Specifically, the guide curve method focuses on the single mean H-A curve, which is a population-averaged (PA) or marginal modeling approach. The other three primary methods take into account between-subject variation in heights by involving local parameters, which can be viewed as a subject-specific (SS) or conditional (given plot or tree) height modeling approach in the essence of Zeger et al. (1988). In other words, roughly speaking, the guide curve-fitting method ignores the appropriate variance-covariance structure of longitudinal height responses to tree ages, whereas all the other methods model the marginal errors (deviations from the PA/ marginal mean H-A model) indirectly by assuming varying local parameters, with or without further considering within-subject variation (mostly serial correlation) after between-subject variations have been accounted for. Note that for mixed models in particular, the indirectly modeled marginal variances and covariances, mostly through the use of random effects, are required to make predictions of future observations for a new subject given prior observations by BLUP. In this regard, mixed modeling of heights was introduced in Lappi and Bailey (1988), whereby BLUP gives the new BLUP predictions rather than the traditional methods (can be viewed as a least squares kind, see Wang et al. 2008a). To the best of our knowledge, within the context of dominant height (site index) modeling, we are not aware of any methods in the literature that directly portray the variance-covariance structure of marginal errors (deviations from the PA/marginal mean H-A curve) with an implication for height projection. Therefore, in this study, we attempted to directly model variances and covariances of dominant heights at different stand ages in addition to means and then used linear prediction theory to make height projections based on the fitted mean and covariance models and prior observation(s) for new plots. We evaluated the predictive performance of this fairly simple yet intuitively appealing method compared with that of four models arising from the difference equation method/ADA/GADA in a case study, using dominant height data repeatedly measured in permanent plots of a second-rotation loblolly pine experiment. Comparison of the proposed method with mixed models for height projection is still in progress and will be reported separately because it is lengthy and because ADA/GADA models have been used extensively in plantation management. This new method can be regarded as an alternative to the traditional guide curve method, because the PA/marginal mean H-A curve plays an indispensable role in model projection. It is closely related to some earlier methods of Osborne and Schumacher (1935), Tveite (1969), and Heger (1968), which will be elaborated on in the Discussion section.

Data The data set is from an improved planting stock/vegetation control study of loblolly pine. This designed study was established in 1986 by the Plantation Management Research Cooperative of the University of Georgia at 16 locations in the Coastal Plain region of Forest Science • August 2014

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Table 1.

Summary statistics of loblolly pine dominant heights for the fit and validation data sets. Fit data set

Age

n

6 yr 9 yr 12 yr 15 yr 18 yr 21 yr 25 yr

58 58 57 58 51 38 32

Mean

SD

Validation data set

Minimum

Maximum

n

Mean

. . . . . . . . . . . . . . . . .(m) . . . . . . . . . . . . . . . . .

Figure 1.

5.5 8.6 11.9 14.7 17.0 18.6 21.3

0.9 1.3 1.5 1.5 1.9 2.3 2.4

3.5 4.8 8.8 11.3 13.1 12.9 14.7

7.9 11.7 15.0 18.1 21.2 22.7 26.3

Minimum

Maximum

. . . . . . . . . . . . . . . . .(m) . . . . . . . . . . . . . . . . . 52 52 52 52 48 35 22

5.8 8.9 12.3 15.0 17.5 18.7 21.0

1.0 1.4 1.7 1.8 2.1 2.0 2.8

4.1 6.7 9.2 10.8 11.1 11.7 12.4

8.5 12.3 15.5 18.6 22.2 22.0 25.0

Observed dominant height trends (fit data).

Georgia and northern Florida and at 15 locations in the Piedmont region of South Carolina, Georgia, and Alabama. The objective of the study was to evaluate the effects of first-generation genetic improvement alone and in combination with vegetation control on yields of loblolly pine plantations. The three levels of genetically improved planting stock were unimproved, single family improved, and bulk lot improved. The two levels of vegetation control were none (other than that provided by the operational site preparation treatment applied by the cooperator before planting) and complete control of all competing vegetation. Plots were randomly assigned to each of the six 2 ⫻ 3 factorial treatment combinations. Each plot was about 0.16 ha in size with a centrally located measurement plot of about 0.08 ha in size. Seedlings were hand-lifted and planted in January 1987 at a density of 1,730 –1,850 trees/ha. Every third pine tree on the measurement plot was measured for total height to the nearest foot after the 3rd, 6th, 9th, 12th, 15th, 18th, 21th, and 25th growing seasons. Previous studies (Martin and Shiver 2002, Harrison and Shiver 2005, Kane and Harrison 2008) have shown that both vegetation control and genetically improved stock significantly increased dominant height, and there were no significant differences between single family and bulk lot. Therefore, in this study, for simplicity, we chose the data from plots with single family improved or bulk lot improved planting stock and no vegetation control, resulting in a total of 110 experimental plots. 654

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Forest Science • August 2014

Dominant height is defined as the average measured height of the trees with crown class identified as dominant/codominant. Measurements from the 6th growing season and forward were used. The 3rd growth season data were not used because at that time trees were of very young age, and dominant trees could not be well-differentiated from the others. The 110 experimental plots were randomly split into two data sets, one for model fitting and the other for validation (Table 1 and Figure 1). By age 25, however, there are only 32 plots in the model-fitting data set and 22 in the model validation data set. All the other plots were inactive by age 25, either destroyed or inaccessible due to property changes.

Methods Linear Prediction Theory

A growth series y may be partitioned into two parts, y1 (the observed components) and y2 (the unobserved components). Consider a general model as y⬅

冋 册

冋 册 冋 册 冋 册 y1 y2

⫽

␮1 ␮2

⫹

e1 e2

(4)

␮1 ␮2 is the mean vector, and e1 and e2 are error terms. The variance-covariance matrix ⌺ of y (or equivalently the errors) takes the following general form where ␮ ⬅

冋

⌺11 ⌺12 ⌺⫽ ⌺ 21 ⌺22

册

(5)

Given known ␮ and ⌺, the best linear predictor (see, e.g., Harville 1985) of y2 given y1 is yˆ2 ⫽ ␮2 ⫹

冘冘

⫺1 11

21

共y1 ⫺ ␮1兲

(6)

If only ⌺ is known, we usually use the generalized least squares estimation to obtain a best linear unbiased estimate of ␮ and substitute the estimate ␮ ˆ into the best linear predictor to have the BLUP as follows ˆ2 ⫹ yˆ2 ⫽ ␮

冘冘 21

⫺1 11

ˆ 1兲 共y1 ⫺ ␮

(7)

This predictor was originally attributed to Goldberger (1962). Additional information can be found in Harville (1985), Rao and Toutenburg (1999), or Christensen (2011). In practice, however, both ␮ and ⌺ are usually unknown, and they must be estimated from a previously collected sample of y and associated covariates X. Hence,

冘冘

ˆ 2 ⫹ ˆ 21 ˆ ⫺1 ˆ yˆ2 ⫽ ␮ 11 共y1 ⫺ ␮1兲

(8)

In this case, the predictor is no longer the BLUP, but an estimate of BLUP, i.e., the estimated or empirical BLUP (EBLUP). One may notice the symbolic similarity between the (estimated) BLUP of Equation 7 or 8 and multivariate normal induced regression formulations. In fact, by assuming a multivariate normal of y, the (estimated) BLUP of y2 given y1 is actually the mathematical expectation of y2 conditional on y1 (e.g., Harville in discussion of Robinson 1991, Christensen 2011), i.e., yˆ2 ⫽ E(y2兩y1), leading to the well-known correlation-based regression and the best predictor of y2 on y1. For the special case of predicting future observation (y2) given one single prior observation (y1), we have EBLUP of y2 as

冘冘

ˆ 2 ⫹ ˆ 21 ˆ ⫺1 ˆ yˆ2 ⫽ ␮ 11 共y1 ⫺ ␮1兲

␴ˆ ␴ˆ ˆ 2 ⫹ 212 共y1 ⫺ ␮ ˆ 1兲 ⫽ ␮ ˆ 2 ⫹ ␳ˆ 21 2共y1 ⫺ ␮ ˆ 1兲 ⫽␮ ␴ˆ 1 ␴ˆ 1

(9)

where ␴ˆ 21 and ␳ˆ 21 are, respectively, the estimated covariance and correlation between y2 and y1 and ␴ˆ 1 and ␴ˆ 2 are the estimated SDs of y2 and y1, respectively. Height Prediction in General

We focus on predicting height observation (H2), given one single prior observation (H1) and reexpressing Equation 9 as _ _ ␴ˆ 2 Hˆ 2 ⫽ H 2 ⫹ ␳ˆ 21 ˆ 共H 1 ⫺ H 1 兲 ␴1

Mean Modeling The base model (guide curve) we used is the Chapman-Richards (CR) (Richards 1959) equation, which has been widely used in height modeling (Garcia 1983, Rennolls 1995), given as H ij ⫽ a共1 ⫺ e ⫺bA ij兲 c ⫹ ␧ ij

(11)

(12)

where (a, b, c) are parameters, Hij is the jth (j ⫽ 1, 2, …, ni) observed height for the ith (i ⫽ 1, 2, …, n) plot, Aij is the age, ␧ij is the error term, and ni is the number of height observations for the ith plot, and n is the number of plots. Variance-Covariance Modeling For each growth series, the error vector ␧i ⫽ (␧i1, ␧i2, ␧i3,…, ␧ini)T (i ⫽ 1, 2, …, n) is assumed to be normally distributed with zero means and an ni by ni positive-definite within-subject covariance matrix ⌺i, i.e., ␧i ⬃ N(0,⌺i), independent from each other. The set of parameters in ⌺i typically do not depend on i. Both within-plot temporal correlation of remeasurements and between-plot heteroscedasticity can be taken into account in parameterizing the variance-covariance matrix ⌺i. The first-order continuous autoregressive error structure, CAR(1), is chosen to account for within-plot serial correlation, largely because it is intuitively appealing and has been widely used (Monserud 1984, Gregoire 1987, Gregoire et al. 1995, NordLarsen 2006, Fortin et al. 2007, Huang et al. 2009, Wang et al. 2011) and because our height measurements are unevenly spaced over time. It is defined as Corr共␧ij , ␧ik 兲 ⫽ ␳兩A ij⫺A ik兩

(13)

where ␳ is the autocorrelation parameter and Aij and Aik are, respectively, tree ages of the jth and kth observations of the ith plot. The CAR(1) model is otherwise known as the spatial power or spatial exponential (for positive ␳) error model when dealing with irregularly spaced time series measurements (see SAS MIXED procedure; SAS Institute, Inc. 2004). We did not use other higher-order autocorrelation models, e.g., CAR(2), because the gain in model fits in terms of log-likelihood or Akaike information criterion was very slight although statistically significant, and, more importantly, because we focused on height projection given only one single pair of H-A measurements and CAR(2) necessitates two pairs for projection. The unequal variances at differing stand ages (see Figure 1) are modeled by the power of the mean function (e.g., Davidian and Carroll 1987) that often works well in accounting for nonconstant variation across the range of predictor(s) (e.g., Fortin et al. 2007) Var共␧ij 兲 ⫽ ␴2 兩f 共 Aij ; ␪兲兩2 ␣

(10)

_ _ where H1 and H2 are PA/marginal height means at (nominal) ages 1 and 2, respectively. Note that in expression 10, no smoothing models for mean and variance-covariance are involved. Replacing the means, correlation, and SDs by an estimated mean model fˆ (A) and an estimated covariance model [including correlation ␳ˆ (A1, A2) and SD ␴ˆ (A)], we have

␴ˆ 共 A 2 兲 Hˆ 2 ⫽ fˆ 共 A 2 兲 ⫹ ␳ˆ 共 A 1 , A 2 兲 ˆ 共H ⫺ fˆ 共 A 1 兲兲 ␴ 共 A 1兲 1

The Alternative Guide Curve Prediction

(14)

where ␴ and ␣ are variance parameters. Combining the CAR(1) autocorrelation model with the power of the mean variance model, the within-plot covariance is given Cov共␧ij , ␧ik 兲 ⫽ ␯i 共 j, k 兲 ⫽ ␴2 共 f 共 Aij ; ␪兲兲␣ 共 f 共 Aik ; ␪兲兲␣ ␳兩A ij⫺A ik兩 (15) Height Projection By substituting the means, autocorrelation, and variances estimated by models 12 through 14 into the general prediction model Forest Science • August 2014

655

Table 2.

(Generalized) algebraic difference approach models compared in this study.

No. CR-a

H2 ⫽ H 1

冉

⫺bA 2

1⫺e 1 ⫺ e⫺bA 1

再 冋

CR-b

H2 ⫽ a 1 ⫺

CR-GADA

H2 ⫽ H 1

冉

LL-GADA

再

册 冎 A 2 /A 1

冊

c 1 ⫹c 2 / ␹ i

冉

1 ⫺ e⫺bA 2 Hˆ2 ⫽ H1 1 ⫺ e⫺bA 1

冢 冣

再 冋

c

Hˆ2 ⫽ a 1 ⫺

⫹ ␧

冎

冑共ln H1 ⫺ c1L兲2 ⫺ 4c2L

再

Hˆ2 ⫽ H1

冑共H1 ⫺ ␣兲2 ⫹ 4␤H1A⫺c 1

册 冎

冊

c 1 ⫹c 2 / ␹ i

c

and L ⫽ ln共1 ⫺ e⫺bA 1兲

⫹ ␧

1 H ⫺ ␣ ⫹ 2 1

c

A 2 /A 1

冉

1⫹

where ␹i ⫽

冊

1 ⫺ 共H1 /a兲1/c

1 ⫺ e⫺bA 2 Hˆ2 ⫽ H1 1 ⫺ e⫺bA 1

⫹ ␧

1 ln H1 ⫺ c1 L ⫹ 2

␤ ⫺c A ␹i 1 H 2 ⫽ H1 ␤ 1 ⫹ A⫺c ␹i 2

Projection model

⫹ ␧

1 ⫺ 共H1 /a兲1/c

1 ⫺ e⫺bA 2 1 ⫺ e⫺bA 1

where ␹i ⫽

冊

Difference form c

冎

冢 冣 ␤ ⫺c A ␹i 1 ␤ 1 ⫹ A⫺c ␹i 2 1⫹

H (H1, H2) is dominant height measurement at age A (A1, A2); a, b, c, c1, c2, ␣, and ␤ are common parameters; and ␹i indicates local parameters.

11, we obtain the EBLUP of H2 at age A2, given a known measurement H1 taken at age A1 ˆ 2 ⫽ f 共 A 2 ; ␪ˆ兲 ⫹ ␳ˆ 兩A 2⫺A 1兩 H

共 f 共 A2 ; ␪ˆ兲兲␣ˆ 共H1 ⫺ f 共 A1 ; ␪ˆ兲兲 共 f 共 A1 ; ␪ˆ)兲␣ˆ

(16)

A special case of 16 with ␳ ⫽ 1 was also considered in model comparison, motivated by Osborne and Schumacher (1935) and Tveite (1969) (see Discussion).

For the four ADA/GADA models, we prefer their difference formulations and use the forward nonoverlapping difference data to fit models. This type of difference data fitting of ADA or GADA models has been used consistently (e.g., Borders et al. 1984, Cao 1993, Amaro et al. 1998) and there appears to be little difference in the predictive ability of the resulting models in comparison with those fitted by other methods (e.g., dummy variable fitting) (Wang et al. 2007, Weiskittel et al. 2009). Parameter Estimation

Comparison of Alternative Guide Curve and ADA/GADA Models

We compared the alternative guide model with four models developed by the difference equation method/ADA/GADA, listed in Table 2 in “difference form.” The two ADA models are derived from the CR base model, denoted in Table 2 as CR-a and CR-b, respectively, by assigning parameter a or b to be site-specific and “differencing” them out. CR-a is an anamorphic model, whereas CR-b is polymorphic (with single asymptotes). The two GADA models, developed from the CR and the log-logistic (LL) (e.g., Monserud 1984) base models, respectively, are termed CR-GADA and LLGADA, each capable of producing site index curves that are polymorphic with multiple asymptotes. CR-GADA results from allowing parameters a and c of the base model 12 to be local parameters and related each other as a ⫽ e ␹, c ⫽ c 1 ⫹ c 2␹

a ⫹␧ 1 ⫹ bA ij⫺c

Model Evaluation

Note that in the model-fitting stage, the guide curve model and the four ADA/GADA models are not comparable with respect to fit statistics. Evaluation of the models will be performed using only validation data, by predicting heights given measurements on initial ages. The performance of all the five projection models can be evaluated based on the following three criteria: mean difference (MD), root mean squared error (RMSE), and modeling efficiency (R2, akin to the coefficient of determination; see Vanclay and Skovsgaard 1997)

(17)

where c1 and c2 are derived common parameters of the model and ␹ indicates a site-specific parameter. The LL base model is given as H ij ⫽

The guide curve model, consisting of the mean model 12 and the covariance model 15, necessitates generalized least squares parameter estimation. Ordinary least squares estimation is used to fit the four ADA/GADA models, based on the forward nonoverlapping difference data.

(18)

冘冘

Hij ⫺ Hˆ ij 共n1 ⫹ n2 ⫹ . . . ⫹ nn 兲 i⫽1 j⫽1 n

MD ⫽

RMSE ⫽

ni

冑

冘 冘 共n ⫹共Hn ⫺⫹ .H. .兲⫹ n 兲 n

ni

ij

i⫽1 j⫽1

LL-GADA is developed by allowing parameters a and b to be local and related to each other as a ⫽ ␣ ⫹ ␹, b ⫽ ␤/␹

(19)

where ␣ and ␤ are derived common parameters. The two GADA models were developed by Cieszewski (2002, 2004) and used recently (e.g., Dieguez-Aranda et al. 2006, Nord-Larsen 2006). 656

Forest Science • August 2014

(20)

1

ˆ

2

ij

2

冘 冘共H ⫺ Hˆ 兲 n

ni

ij

R2 ⫽ 1 ⫺

2

ij

i⫽1 j⫽1

冘冘 n

(21)

n

ni

i⫽1 j⫽1

_ 共H ij ⫺ H兲 2

(22)

Table 3. Common parameter estimates, SEs, t values, and P values for the guide curve and four ADA/GADA site index models. Model

Parameter

Estimate

a b c ␳ ␣ b c a c b c1 c2 ␤ c

28.1224 0.0697 1.5291 0.9685* 0.5216* 0.0715 1.5183 28.3530 1.5237 0.0732 ⫺5.9460 24.8898 2460.37 1.4850

CR guide

CR-a CR-b CR-GADA LL-GADA

t value

Pr ⬎ 兩t兩

1.0938 0.0050 0.0548

25.71 14.08 27.88

⬍0.0001 ⬍0.0001 ⬍0.0001

0.0046 0.0545 0.8565 0.0558 0.0047 3.0074 10.0001 148.036 0.0413

15.47 27.87 33.10 27.33 15.49 ⫺1.98 2.49 16.62 35.96

⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 0.0490 0.0134 ⬍0.0001 ⬍0.0001

SE

* Statistical significance by likelihood ratio tests but no SE and t value given by S-Plus fitting.

where n is the number of plots, ni is the number of H-A pairs for the ˆ ij are the jth _observed and predicted heights, ith plot, Hij and H respectively, for the ith plot, and H is the overall height mean. The first criterion measures bias of the model prediction, whereas the last two measure precision in absolute and relative scales. Two kinds of evaluations are performed. The first is to choose one single starting age (fixed initial age) from each of the seven ages (6, 9, 12, 15, 18, 21, and 25 years) available from our loblolly pine measurements and to project heights at the other ages. In doing so, both forward and backward projections will be involved except for the first starting age 6 (all forward projection) and last starting age 25 (all backward projection). Although backward projection is of no operational use in practice, it does help to see the overall predictive ability of a model. Second, we focus on forward projection with varying starting ages in two situations, the first to consider only nonoverlapping projection (i.e., from age 1 to age 2 and age 2 to age 3, but not age 1 to age 3) and the second to all possible forward projections. The nonoverlapping forward projection serves to evaluate short-term model prediction, because projection intervals, the same as the measurement intervals, are 3 years before and at age 21 and 4 years thereafter. In contrast, both all possible forward projections and fixed initial age projections involve short-term as well as long-term (maximum projection interval of 19 years) predictions.

Results Estimates of Common Parameters

Based on the fit data, the guide model and the four ADA/GADA models were fitted using the S-Plus gnls function and SAS Proc NLIN, respectively. The estimate of parameter ␣ of model LLGADA was found to be not significantly different from 0, and, therefore, parameter ␣ was dropped. Parameter estimates of the common parameters are presented in Table 3. All models fit the data well, and no obvious residual trends were found. Figure 2 presents both raw and studentized residual plots obtained from fitting the guide model. Height Prediction

Results of height prediction evaluations conducted on the validation data in the cases of fixed initial age forward and/or backward projection and varying initial age forward projection are presented in Tables 4 and 5, respectively. In general, no obvious bias in height prediction was observed. Scatter plot examples of height predictions

for fixed initial age projections given age 12 observations, varying initial age nonoverlapping forward and all possible forward projections, all using the guide model projection, are provided in Figures 3, 4, and 5, respectively. For fixed initial age prediction (Table 4), the CR guide model achieved the best predictive performance in terms of RMSE or R2 at initial ages of 6, 9, 12, and 25. For the other three initial ages, the best predictions were obtained by LL-GADA for age 15 and by CR-a for ages 18 and 21. In general, both the CR guide, including the special case of ␳ ⫽ 1, and the four ADA/GADA models obtained very comparable projection results, except that models CR-a and LL-GADA at initial ages of 6 and 9 and CR-b and CR-GADA at initial age 25 produced apparently worse predictions than the others. Of special note, the better prediction performance by the CR guide model given initial ages 6 and 9 should be interesting to forest modelers, because it involves mostly forward projection using measurements taken at earlier ages. For varying-initial age predictions (Table 5), similar results were obtained for the CR guide and all four ADA/GADA models for nonoverlapping forward projections with RMSEs all of about 0.7 m and for the CR guide and CR-b and CR-GADA for all possible forward projections with RMSE of about 1.4 –1.5m. The smaller RMSEs (or higher R2) values obtained in nonoverlapping forward projections in contrast to all possible forward projections were expected, because in this case study nonoverlapping forward projection intervals were mostly 3 or 4 years, whereas all possible forward involved other time intervals with a maximum of 19 years. Projection accuracy decreases with increasing prediction intervals, indicated clearly in Figure 3. This finding is also self-evident by comparing Figure 4, for which prediction errors are generally within 2 m, and Figure 5, for which prediction errors reach about 7.5 m at maximum.

Discussion Links to Certain Previous Site Index Modeling Methods

The graphical guide curve method (Bruce 1926) is limited to producing anamorphic site index curves. Osborne and Schumacher (1935) proposed an alternative method capable of constructing polymorphic curves, by using the average H-A guide curve as well as height SDs at sorted age classes. This method can be expressed as _ ␴ˆ i Hˆ i ⫽ H i ⫹ ˆ 共S ⫺ S៮ 兲 ␴s

(23)

_ _ ␴ˆ 2 Hˆ 2 ⫽ H 2 ⫹ ˆ 共H 1 ⫺ H 1 兲 ␴1

(24)

_ where Hi is the mean height at a given age class, S៮ is the mean site index, ␴៮ i and ␴៮ s are SDs at the given age i and index age, respectively, ˆ i is the height at age class i corresponding to site index S. For and H the purpose of model projection of H2 given H1, we have straightforward from expression 23 that

The method of Osborne and Schumacher was originally based on TSP data. Tveite (1969) applied this method, termed the deviation method, to PSP data, yielding results that were very similar to those of other methods of constructing site index curves including essentially the GADA model of Strand (1964). Note that 24 is a special case of the general model 10 with ␳ ⫽ 1. This is the reason that we have considered the special case of the new guide curve method of 16 Forest Science • August 2014

657

Figure 2. curve.

Scatter plots of raw (top) and studentized (bottom) residuals versus predicted dominant heights resulted from the fitted guide

with ␳ ⫽ 1 in our model comparison exercise. The very comparable results we obtained for the new guide curve method (and the special case with ␳ ⫽ 1) and the two GADA models is in agreement with Tveite (1969). Heger (1968) developed site index curves by using a set of linear regression models (H ⫽ a ⫹ bS ⫹ e) fitted to each age class i, that is, Hˆ i ⫽ a i ⫹ b i S

(25)

where ai and bi are estimated regression coefficients specific to age class. The linear regression model 25 is simply an alternative expression of height prediction (given site index) in the general EBLUP form of 10 by recognizing the fact that the two regression coefficients are estimated as _ ␴ˆ i ␴ˆ ៮ b i ⫽ ␳ˆ i i a i ⫽ H i ⫺ ␳ˆ i ˆ S, ␴s ␴ˆ s

(26)

where ␳ˆ i is the correlation between heights (given age class i) and site indices. That is, 658

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_ ␴ˆ i ៮ Hˆ i ⫽ H i ⫹ ␳ˆ i ˆ 共S ⫺ S兲 ␴s

(27)

We have greatly enjoyed observing the connections between expression 27, i.e., the linear regression method of Heger (1968), expression 23, i.e., the method of Osborne and Schumacher (1935), and the alternative guide curve method as suggested in this study. That is, the EBLUP-type method of Heger (1968) includes the method of Osborne and Schumacher (1935) as a special case with ␳ ⫽ 1. In both methods, means and SDs are calculated for each sorted age (class) without involving any smoothing functions, and, in contrast, the proposed alternative guide curve method can be viewed as a modeled generalization of Osborne and Schumacher (1935) by considering autocorrelation as well as that of Heger (1968) by using smooth functions for means, correlations, and SDs. We note that simply from a correlation point of view, height prediction models of the general form 10 or 27 clearly indicate the decreasing projection accuracy with increasing time lags, because

Table 4. Evaluation statistics of height projections given fixed initial age (A0) observations. A0 6 yr

9 yr

12 yr

15 yr

18 yr

21 yr

25 yr

Table 5.

2

Model

n

MD

RMSE

R

CR guide CR guide, ␳ ⫽ 1 CR-a CR-b CR-GADA LL-GADA CR guide CR guide, ␳ ⫽ 1 CR-a CR-b CR-GADA LL-GADA CR guide CR guide, ␳ ⫽ 1 CR-a CR-b CR-GADA LL-GADA CR guide CR guide, ␳ ⫽ 1 CR-a CR-b CR-GADA LL-GADA CR guide CR guide, ␳ ⫽ 1 CR-a CR-b CR-GADA LL-GADA CR guide CR guide, ␳ ⫽ 1 CR-a CR-b CR-GADA LL-GADA CR guide CR guide, ␳ ⫽ 1 CR-a CR-b CR-GADA LL-GADA

261 261 261 261 261 261 261 261 261 261 261 261 261 261 261 261 261 261 261 261 261 261 261 261 249 249 249 249 249 249 197 197 197 197 197 197 132 132 132 132 132 132

⫺0.11 ⫺0.27 ⫺0.35 ⫺0.24 ⫺0.21 ⫺0.33 0.28 0.28 0.42 0.32 0.31 0.42 ⫺0.02 ⫺0.07 ⫺0.01 ⫺0.03 ⫺0.02 0.03 ⫺0.04 ⫺0.09 ⫺0.05 ⫺0.09 ⫺0.08 ⫺0.05 ⫺0.02 ⫺0.10 ⫺0.08 ⫺0.15 ⫺0.15 ⫺0.12 0.24 0.26 0.16 0.22 0.22 0.16 ⫺0.04 ⫺0.05 ⫺0.18 ⫺0.21 ⫺0.20 ⫺0.18

1.53 1.65 2.17 1.64 1.58 1.89 1.40 1.52 1.84 1.48 1.46 1.66 1.12 1.17 1.27 1.13 1.14 1.19 0.93 0.92 0.90 0.91 0.94 0.89 0.86 0.90 0.83 0.92 0.96 0.85 0.95 1.07 0.93 1.08 1.13 0.97 1.15 1.52 1.24 1.71 1.70 1.37

0.8720 0.8505 0.7421 0.8529 0.8631 0.8052 0.9265 0.9139 0.8737 0.9176 0.9204 0.8965 0.9594 0.9553 0.9474 0.9588 0.9576 0.9537 0.9714 0.9718 0.9733 0.9725 0.9705 0.9737 0.9723 0.9698 0.9743 0.9688 0.9660 0.9734 0.9655 0.9563 0.9672 0.9558 0.9515 0.9641 0.9409 0.8970 0.9312 0.8707 0.8717 0.9164

Evaluation statistics of forward only height projections. Model

n

MD

RMSE

R2

. . . . .(m) . . . . . Nonoverlapping forward projection CR guide CR guide, ␳ ⫽ 1 CR-a CR-b CR-GADA LL-GADA All possible forward projection CR guide CR guide, ␳ ⫽ 1 CR-a CR-b CR-GADA LL-GADA

261 261 261 261 261 261

⫺0.01 ⫺0.05 ⫺0.03 ⫺0.02 ⫺0.02 ⫺0.03

0.69 0.68 0.72 0.69 0.68 0.70

0.9736 0.9750 0.9715 0.9743 0.9748 0.9732

811 811 811 811 811 811

0.01 ⫺0.05 0.00 ⫺0.01 0.01 0.02

1.41 1.49 1.81 1.47 1.44 1.63

0.8684 0.8533 0.7825 0.8575 0.8632 0.8246

correlations between the paired heights die off with long time intervals. However, letting the correlation parameter be 1 (perfect linear correlation) did not markedly affect prediction accuracy (Tables 4 and 5) and even improved short-term projection (Table 5, nonoverlapping forward), for which we were unable to provide a tenable explanation.

Developing Polymorphic Curves by the Old Guide Curve Method

The traditional guide curve method (especially based on TSP data) is typically criticized because the average H-A curve may be biased due to unequal representation of sites across the full range of ages or age classes and because the resulting proportional curves imply a constant relative rate of height growth for all sites, which is probably not accurate in depicting dominant height growth patterns (e.g., Monserud 1985). We argue that the first criticism is essentially a sampling problem and holds for the other site index modeling methods, although using repeated measurements and other modeling methods may alleviate the data requirement with respect to site-age distribution. There is no doubt that the second criticism holds for the original guide curve method of Bruce (1926) due to graphical limitation. Nonetheless, the modification by Osborne and Schumacher (1935) of expression 23 can be used to construct polymorphic curves. Requiring a large number of TSP data, Brickell (1968) elaborated on Osborne and Schumacher (1935) to result in a PPM kind of site index models by generating a family of percentile-defined pseudo growth series, fitting them to a common model form and relating parameter estimates to site indices. The two improved methods, however, were not used widely in practice. When a model-fitted mean guide curve is available, either anamorphic or polymorphic curves can be developed, which is seldom recognized in site index modeling exercises. That is, multiplicative scaling of the fitted guide curve to construct anamorphic curves is, in most cases, equivalent to varying the asymptotic parameter of the base model with site quality. If we choose the other parameter to depend on site index, polymorphic (with single asymptotes) models will be obtained. From a model development point of view, however, this kind of guide curve method of developing site index models does not recognize SS height variation in the model-fitting stage, compared with the difference equation method/ADA per Bailey and Clutter (1974). In our experience and that of Cao (1993), anamorphic or polymorphic models developed as such are generally inferior to those resulting from the difference equation method/ADA regarding height projection. By the discussion above, we can see two lines for developing site index models that diverge from the traditional guide curve method. One is that the single PA mean curve evolves into a family of SS curves by identifying local parameters to vary with sites, including PPM, difference equation/ADA/GADA, and mixed models, among others. The other can be viewed as a correlation-regression type by maintaining the single mean curve along with correlations and variances without involving any varying parameters, in line with Osborne and Schumacher (1935), Tveite (1969), Heger (1968), and the new guide curve method described in this study. We believe each of the two general approaches to height modeling has pros and cons, and the new alternative method offers a choice one may consider in site index modeling. Autocorrelation in Site Index Modeling

In our direct modeling of error variance-covariance, within-subject autocorrelation is meant to be of a marginal sense, because the within-subject errors are defined as the deviations from the one single PA/marginal H-A curve. The autocorrelation defined as such has played a pivotal role in our PA approach of modeling exercise for height projection. Of special note, our study is in line with the study of Fortin et al. (2007), who achieved improved basal area prediction Forest Science • August 2014

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Figure 3.

Scatter plots of projection errors versus projected dominant heights given age 12 observations by the guide model projection.

Figure 4.

Scatter plots of projection errors versus projected dominant heights by the guide model nonoverlapping forward projection.

using prior measurements by linear prediction theory in conjunction with an error submodel that took into account both autocorrelation and heteroscedasticity. Meng et al. (2012) obtained similar results. Unlike H-A models, in both studies, between-subject (plot) basal area variation was accounted for to a large degree by using plot/stand covariates but without individual plot-specific effects, a typical two-stage procedure for dealing with repeated measurements of stand basal area or volume from PSPs (West et al. 1984). In contrast, with the SS approach of site index modeling (e.g., PPM, difference equation method or ADA/GADA, and mixed models), within-subject errors are defined as the deviations from individual H-A curves, and thereby serial correlation is necessarily of a conditional sense (depending on the local parameters). Conditional autocorrelation has long been a concern in site index modeling (e.g., Borders et al. 1984, Monserud 1984, Huang 1997, Beau660

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mont et al. 1999, Fang and Bailey 2001, Dieguez-Aranda et al. 2006, Nord-Larsen 2006) and in other modeling exercises (e.g., Sullivan and Clutter 1972, West et al. 1984, Gregoire 1987, Gregoire et al. 1995, Fortin et al. 2008). However, this kind of autocorrelation problem has been largely ignored in site index modeling using traditional PPM, the difference equation method or ADA/GADA, or recently developed mixed models (e.g., Borders et al. 1984, Monserud 1984, Lappi and Bailey 1988, Huang 1997, Hall and Bailey 2001) because knowledge of autocorrelation is of no use in practice in the traditional least squares kind of model projection (Monserud 1984, Huang 1997) compared with the mixed-effects models via EBLUP. Even with mixed-effect modeling and EBLUP for projection, conditional autocorrelation may improve height projection accuracy (Wang et al. 2011) or may not (Meng and Huang 2010). Conditional autocorrelation should not be

Figure 5.

Scatter plots of projection errors versus projected dominant heights by the guide model all possible forward projection.

confused with between-subject heterogeneity within the context of mixed-effects modeling, which may be interpreted as the marginal within-subject autocorrelation due to observations from the same subject sharing the same random effects (see Hall and Bailey 2001). Within the context of mixed-effects modeling, it is generally accepted that random effects should be included to account for between-subject variation (e.g., Lappi and Bailey 1988, Hall and Bailey 2001). We believe that with any SS method of site index modeling, the conditional within-subject autocorrelation should be a minor issue, because between-subject variation (accounted for by involving local parameters) is the main source of dominant height variability.

Conclusion We have proposed a new guide curve method for developing height projection models. This method can be derived from linear prediction theory by directly modeling variances and covariances of marginal errors (deviations from the PA marginal means), and by assuming normal distributions, it can be viewed as a correlation-regression type of height prediction given prior observation(s). Simply put, by means of average heights, correlations between heights at differing ages, and SDs of heights given ages, height projection can be readily made. The new method is closely related to the earlier methods of Osborne and Schumacher (1935), Tveite (1969), and Heger (1968) in developing site index models. We have compared the new guide curve model with four ADA/GADA models with respect to height projection and obtained very comparable results.

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