Evaluation of Alternative Approaches for Predicting Individual Tree

Evaluation of Alternative Approaches for Predicting Individual Tree Volume Increment

ABSTRACT

David W. Hann and Aaron R. Weiskittel The volume increment of individual trees is often inferred from a volume or taper equation and predicted or observed diameter and height increments. Prediction errors can be compounded with this type of approach because of the array of equations used and differences in their accuracy. The consequences of several alternative approaches for indirectly or directly estimating individual tree volume increment were examined using an extensive stem analysis data set of Douglas-fir (Pseudotsuga menziesii [Mirb.] Franco) in southwest Oregon. The data were used to construct new stem volume, taper, and volume increment equations, which were then used to compare predicted and observed 5-year volume increments. The results of this analysis suggest that the indirect prediction of volume increment is sensitive to both the approach used for estimating stem volume and the use of actual versus predicted diameter and height increment, especially diameter increment. In addition, using the indirect method of volume and taper equations was found to have a slightly lower level of accuracy in predicting stem volume increment than the direct method. It was found that the use of local calibration procedures could help to mitigate possible problems with the bias incurred by using predicted rather than actual diameter increment. Keywords: Douglas-fir, diameter increment, height increment, taper

E

stimates of tree volume increment are common outputs from individual tree regional growth and yield models such as the Forest Vegetation Simulator (Wykoff et al. 1982, Crookston and Dixon 2005) and ORGANON (Hester et al. 1989, Hann 2009). These models are widely used to update forest inventories, estimate allowable annual harvests, and evaluate alternative silvicultural regimes. Estimates of tree volume increment have also been commonly used in studies of tree growth efficiency (e.g., Waring et al. 1980 and Mainwaring and Maguire 2004), which have examined the usage of tree volume increment in designing silvicultural prescriptions (e.g., Waring et al. 1981, O’Hara 1996, Gersonde and O’Hara 2005) and in assessing susceptibility to insect attacks (e.g., Larsson et al. 1983, Waring and Pittman 1985). In both cases, tree volume increment is usually estimated indirectly by application of either a volume equation or the integral of a taper equation at the start and end of the growth period and calculation of the difference (Husch et al. 2003). If predicting future volume increment is of interest, then the procedure requires having measurements of the tree attributes used in the volume or taper equation at the start of the growth period and estimates of the change in those attributes over the growth period. In most applications, the attributes of interest are the diameter at breast height (D) and total height (HT) of the tree and their increments. From mathematical expectation (Kmenta 1986), the indirect method of estimating future volume increment will be unbiased if the volume equation is unbiased and the predictor variables at the start and end of the growth period are measured without error. For application in a growth model, the latter requirement will be met if the changes in the predictor variables between the start and end of the growth period are known without any type of measurement

error (Kangas 1996). One way to introduce measurement error is to predict the changes in the predictor variables by auxiliary equations that are less than perfect predictors of those changes (we will call this type of measurement error “prediction error”). Even if these auxiliary equations are unbiased, measurement error theory indicates that the resulting predictions of volume increment may be biased depending on whether the model form for the volume increment equation is linear or nonlinear and on the structure of the prediction error (Kmenta 1986, Fuller 1987, Carroll et al. 1995, Kangas 1998). Therefore, the accuracy of the predictions of volume increment may be reduced by using predicted instead of measured predictor variables [1]. Mathematical expectation also indicates that the amount of variation explained by the indirect method is related to the amount of variation explained by the volume equation used to form the indirect estimate and the size of the covariance between the end and start of growth period estimates of volume (Kmenta 1986). As a result, the choice of the method for estimating volume may also affect the accuracy of indirect method for estimating volume increment. A direct estimator of tree volume increment can avoid prediction error problems because, in at least the first growth period, the tree and plot attributes used as predictor variables are measured rather than predicted. Therefore, the accuracy of the direct method may be better than the indirect method. However, the direct method does have several disadvantages: 1.

Direct measurement of volume increment requires the use of either stem analysis or instruments such as an optical dendrometer. The former can be done only once on a tree, and the latter

Manuscript received June 2, 2009, accepted February 26, 2010. David W. Hann ([email protected]), Department of Forest Engineering, Resources, and Management, Oregon State University, Corvallis, OR 97331. Aaron R. Weiskittel ([email protected]), 5755 Nutting Hall, School of Forest Resources, University of Maine, Orono, ME 04469. We thank Temesgen Hailemariam and two anonymous reviewers for helping to improve a previous version of this article. Copyright © 2010 by the Society of American Foresters.

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2.

3.

is not as precise as the former. Both methods are very time-consuming and therefore expensive. Estimating volume increment from a volume or taper equation applied to repeated measurements of the tree (e.g., Lemmon and Schumacher 1962) increases unexplained variation in the residuals (Kmenta 1986). It also binds the resulting predictions of volume increment to the particular volume or taper equation chosen by the modeler instead of the equation’s user. For both methods of determining the response variable, the user is forced to accept the merchantability standards selected by the modeler.

It is likely that these disadvantages contribute to the popularity of the indirect method for determining volume increment of trees in growth and yield models used to make operational management decisions. The overall goal of this study, therefore, is to provide an empirical assessment of whether or not these theoretical statistical problems with the indirect method are severe enough to justify further research. The following questions arose from this overall goal: 1.

2.

3.

What practical effect does using different models or methods to estimate tree volume have on the accuracy of the indirect method? What practical effect does using predicted rather than measured diameter increment and/or height increment have on the accuracy of the indirect method? How does the accuracy of the indirect method compare with the accuracy of a direct estimator of volume increment?

This study was therefore designed to provide answers to these questions. The different equation forms used in this study were parameterized using a common data set to minimize the effect of potential bias in the equations caused by possible differences between subregions of Douglas-fir. Furthermore, the study used growth predictions for just a single growth period in which the starting attributes were known to minimize the proliferation and accumulation of prediction errors in both the start and end of the growth period that occur in growth predictions made beyond the first growth period (Kangas 1997). Our intent with these actions was to provide results that would most likely represent the best that could be expected given the complexities of the problem.

Methods Data Data for this analysis were a subsample of the data set collected to develop the revised southwest Oregon version of the ORGANON (SWO-ORGANON) growth-and-yield model (Hann 2009). Answering the questions posed in this study required equations for predicting the volume, taper, diameter increment (⌬D), height increment (⌬HT), and volume increment (⌬V) of a selected tree species. The new SWO-ORGANON analyses provided the ⌬D and ⌬HT equations (Hann and Hanus 2002a, b) required for the analysis, leaving the need to develop volume, taper, and ⌬V equations, all of which require data from the stem analysis of felled trees. Of the stem analysis trees measured for SWO-ORGANON, Douglas-fir (Pseudotsuga menziesii [Mirb.] Franco) had the largest sample size, with a total of 817 felled and sectioned trees from 301 plots (Table 1). Details concerning the study area, sampling protocols, measurement techniques, data summary and transformation procedures,

Table 1. Attributes of the sample trees and plots used in this analysis. Attribute Individual tree (n ⫽ 817) DS (in.) DE (in.) ⌬D (in.) HTSabh (ft) HTEabh (ft) ⌬HT (ft) CRS BALS (ft2 ac⫺1) V Sabh (ft3) V Eabh (ft3) ⌬Vabh (ft3) Plot (n ⫽ 301) BAS (ft2 ac⫺1) SI (ft at base age 50 years)

Mean

Standard deviation

Minimum

Maximum

12.8 13.8 1.01 75.3 81.8 6.5 0.55 92.9 58.4 66.4 7.96

9.1 9.1 0.64 41.1 40.1 3.3 0.20 78.8 123.9 129.2 8.51

0.1 0.9 0.06 5.0 9.0 0.5 0.15 0.0 0.001 0.03 0.01

54.8 55.5 3.85 240.4 241.7 17.5 1.0 373.1 1,185.1 1,217.9 47.7

177.7 97.8

78.3 16.0

11.3 47.2

541.9 141.1

DS, diameter at breast height at start of growth period; DE, diameter at breast height at end of growth period; ⌬D, 5-year diameter increment; HTSabh, height above breast height at start of growth period; HTEabh, height above breast height at end of growth period; ⌬HT, 5-year height increment; CRS, crown ratio at start of growth period; BALS, basal area in larger diameter trees at start of growth period; V Sabh, total stem cubic foot volume above breast at start of growth period; V Eabh, total stem cubic foot volume above breast at end of growth period; ⌬Vabh, 5-year total stem cubic foot volume above breast height increment; BAS, total basal area per acre at start of growth period; SI, Douglas-fir site index.

and the predictor variables used to develop the ⌬D and ⌬HT equations can be found in Hann and Hanus (2001, 2002a, b). Trees selected for detailed stem analysis were sectioned at breast height, at either 8.4-ft or 10-ft intervals above breast height, and at the whorl representing the start of the previous 5-year growth interval. At each of the ith section points above the ground, height to that section (hi) and diameters inside bark (dibi) and outside bark (dobi) at that section were measured along both the long and short axes. The 5-year radial increment was also measured at each end of the two dibi measurements. Height at the start of the growth period (HTS) was determined by subtracting the measured 5-year height increment from the measured total height of the felled tree at the end of the growth period (HTE). The dibi at the start of the growth period (dibSi) was determined by subtracting the 5-year radial increments from the dibi measurement taken at the end of the growth period (dibEi). The geometric mean was used to average the long and short axis diameter measurements at both the start and end of the growth periods. The average diameter outside bark at breast height was used as D at the end of the growth period (DE), and the average dibS at breast height was converted to D at the start of the growth period (DS) through application of the bark thickness equations of Larsen and Hann (1985). Data Transformation Only one section at stump height was measured below breast height, which precluded an accurate characterization of stem form below breast height. Therefore, we followed the protocol of Walters et al. (1985) and Hann et al. (1987) and restricted the volume under consideration to the total stem cubic foot volume inside bark that is above breast height (Vabh). We assume that the results of the analysis using Vabh are also applicable to the volume of the entire stem, for which Vabh is the dominant component for all but the smallest trees. Details of the procedures used to determine actual Vabh from the stem analysis measurements can be found in Walters et al. (1985) and Hann et al. (1987). The calculations were done for Vabh at the start and end of the growth period (VSabh and VEabh, respectively), WEST. J. APPL. FOR. 25(3) 2010

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and then actual volume increment above breast height (⌬Vabh) was determined by subtracting the initial volume from the volume at the end of the growth period (VEabh ⫺ VSabh). Total height above breast height (HTabh) was calculated by subtracting 4.5 ft from total height for measurements at the start (HTSabh) and end (HTEabh) of the growth period (HTEabh ⫺ HTSabh). The tree and stand variables needed to develop the ⌬Vabh equation included crown ratio of the tree at the start of the growth period (CRS), the Hann and Scrivani (1987) Douglas-fir site index (SI) of the plot, and the basal area per acre in larger diameter trees at the start of the growth period (BALS). A detailed description of the procedures used to calculate these attributes can be found in Hann and Hanus (2002a). Data Analysis Two volume equations and one taper equation were developed for predicting Vabh to explore the impact of model form and complexity on the indirect method and to make our results as universal as possible. Both of the volume equations used DS and HTSabh to form the predictor variables, and they were selected to cover the extremes in model form complexity from a simple linear equation that incorporated one parameter to a complex nonlinear equation that incorporated six parameters. We also chose to include a taper equation in this analysis because taper equations are commonly integrated over the length of the stem to calculate volume (Avery and Burkhart 2002). All three equations were developed using variables measured at the end of the growth period to avoid the correlation between the starting and ending measurements and to parallel the procedure usually taken to develop volume (and taper) equations. Both volume equations have been developed so that Vabh will be 0 when D and HTabh are jointly 0. The first equation is the constant form factor equation (Husch et al. 2003), which is based on simple geometric forms that use just one parameter to characterize the relationship of D and HTabh to Vabh: Vabh ⫽ a1DE2HTEabh ⫹ ␧Vabh ,

(1)

where ␧Vabh ⫽ random error on Vabh. Equation 1 was included in the analysis to illustrate the capability of a simple equation to predict ⌬V. The second volume equation is a modification of the variable form factor equation presented in Walters et al. (1985) and Hann et al. (1987): Vabh ⫽ b1X1X3DE2HTEabh ⫹ ␧Vabh,

(2)

where X1 ⫽ (HTEabh/DE)X2, X2 ⫽ b2[1.0 ⫺ exp(b3DEb4)], and X3 ⫽ 1.0 ⫺ b5exp(b6[(120.0 ⫺ DE)/100.0]30). The X3 term of Equation 2 corrected an over prediction bias found in trees with D larger than 30 in. Browne (1962) also found that immature Douglas-fir had higher predicted volumes than mature Douglas-fir for the same values of D and HT. He chose to solve the problem by fitting two separate equations to immature and mature trees. We preferred to develop a single equation as a solution so the X3 term was designed to predict a value of 0 for trees under approximately 30 in. D and increase to a value of b5 for trees with a D of 120 in. (approximately the maximum reported D for the species). This behavior required the usage of a sigmoidal model form with a large exponent to keep the predictions at 1 across the first 30 in. of D. After trying a number of alternative exponents in X3, a value of 30 was found to provide the 122

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desired behavior. The need to use a large exponent in a volume equation to achieve intended behavior is not unique to this study. Bruce et al. (1968) used several exponents even larger than 30 in their red alder volume equation. The parameters of both volume equations were estimated using weighted nonlinear regression and a weight of (DE2HTEabh)⫺2. With the exception of b6 in Equation 2, all parameters are different from 0 at P ⫽ 0.05. Only 42 trees in the sample have a D greater than 30 in., which may have contributed to the weakness of the b6 parameter in X3. Setting b6 to 0 would basically eliminate the X3 correction term even though the b5 parameter in the term is highly significant. We therefore chose to retain X3 and the associated b6 parameter to eliminate the bias found in the large trees. After exploring several alternative taper model forms, we chose the Kozak’s (2004) “Model 02” to characterize dibE: c Zi4⫹c4exp共⫺DE/HTE兲⫹c5Xi0.1⫹c6DE⫺1⫹c7HTEQi⫹c8Xi

dibEi ⫽ c0DEc1HTEc2X i 3

, (3)

where Zi ⫽ hi/HTE, Xi ⫽ Qi/[1.0 ⫺ (4.5/HTE)1/3], Qi ⫽ 1.0 ⫺ (Zi)1/3. The parameters of Equation 3 were estimated using nonlinear regression techniques. A model to predict volume increment directly was also developed from the stem analysis data. Given that volume increment is the result of diameter increment and height increment, the tree and stand attributes used to predict volume increment should be similar to those used to predict diameter and height increments. Therefore, a number of alternative model forms that incorporated the attributes used in diameter and height increment equations were examined and the following one proved to be most effective at characterizing 5-year ⌬Vabh:

冉冘 冊 5

⌬Vabh ⫽ exp

fi Xi ⫹ ␧⌬Vabh,

(4)

i⫽0

where X0 ⫽ 1.0, X1 ⫽ ln(DS)/10.0, X2 ⫽ DS/100.0, X3 ⫽ ln[(CRS ⫹ 0.2)/1.2], X4 ⫽ ln(SI ⫺ 4.5)/10.0, X5 ⫽ BALS/(1000.0[ln(DS ⫹ 1.0)]), and ␧⌬Vabh ⫽ random error on ⌬Vabh. ⌬Vabh is predicted to increase, peak, and then decrease as DS increases; increase as CRS and SI increase; and decrease with an increase in BALS. The parameters of Equation 4 were estimated by nonlinear regression. Examination of the residuals about Equation 4 showed no trends across predicted ⌬Vabh or across the tree and plot attributes used in the equation. Three statistics were used to evaluate predictions from Equations 1– 4 and to evaluate the indirect method of predicting ⌬Vabh. In all applications, the residual (␧) of predicted value (y) minus actual value (y) is formed for each tree, and ␧ is used to calculate the average of ␧ (␧៮ ), the mean squared differences (MSD) of ␧. The variance of each actual y (VAR[Y]) and its mean value (Y៮ ) were also calculated. These were then used to calculate the evaluation statistics: RB ⫽ 100.0(␧៮ /Y), RA ⫽ 100.0(MSD1/2/Y), and FIT ⫽ 100.0(1.0) ⫺ MSD/VAR[Y]). RB is a measure of relative bias. If the sign on RB is positive, then the equation is over predicting the attribute on average, and, if negative, it is under predicting the attribute. If RB is 0, then the RA statistic is an indicator of relative precision. The FIT statistic is similar to Kvålseth’s (1985) recommended method of calculating the generalized adjusted coefficient of determination for

linear model forms. Therefore, it indicates how much of the variation in the response variable is reduced by the equation or the indirect approach compared with variance about the mean of the response variable. Therefore, a FIT statistic of 0 or less indicates that the mean of the response variable would be a better estimator than the equation or indirect approach being evaluated. One would like RB and RA to be as close to 0 as possible and FIT to be as close to 100 as possible. In the first application of these statistics, the two volume equations and the taper equation were used to predict Vabh at the end of the growth period. To maintain consistency across equations, calculation of predicted VEabh from the taper equation was done by applying Equation 3 to predict diameter inside bark for each section point from breast height to the top of each tree in the felled tree data set and then applying the same log volume rules as used to calculate the actual Vabh (see Walters et al. 1985 for details). For each of the three approaches, ␧ was calculated as predicted minus observed and the degrees of freedom in the calculation of MSD was reduced by the number of parameters in the particular equation. A parallel approach was used to calculate the statistics for unweighted predictions of ⌬Vabh from Equation 4. The following procedures were used to prepare the indirect estimates of ⌬Vabh to fully explore the questions posed by this study: 1.

2.

3.

4.

Predicted VSabh was calculated from DS and HTSabh using volume Equations 1 and 2 and, in this case, the numerical integration of taper Equation 3 between breast height and the tip of the tree. Numerical integration was used because most applications that calculate ⌬V will be conducted on trees without diameter inside bark measurements along the stem. DE and HTEabh were calculated by adding to the initial observations using either actual or predicted values of ⌬D and ⌬HT, in all possible combinations (i.e., actual ⌬D with actual ⌬HT, actual ⌬D with predicted ⌬HT, predicted ⌬D and actual ⌬HT, and predicted ⌬D with predicted ⌬HT). Predicted VEabh was calculated from all combinations of actual or predicted DE and actual or predicted HTEabh using volume Equations 1 and 2 and the numerical integration of taper Equation 3 from breast height to the tip of the tree. For each combination of volume/taper equation, source of DE, and source of HTEabh, predicted ⌬Vabh was calculated by subtracting predicted VSabh from predicted V Eabh.

These procedures resulted in 12 indirect estimators of predicted ⌬Vabh. For each of these, ␧ was calculated as predicted ⌬Vabh minus observed ⌬Vabh for each tree in the data set, and the three evaluation statistics were determined from the resulting ␧ values. In this usage, the degrees of freedom used in calculating MSD was set to the sum of the parameters in all of the equations needed to compute the ␧ values. Therefore, the resulting RA and FIT statistics have been adjusted for the larger number of parameters used in the indirect approaches.

Results The parameter estimates for the three static equations used to predict V Eabh and the standard errors for the parameters can be found in Table 2. The VEabh evaluation statistics for Equations 1–3 and the evaluation statistics for directly predicting ⌬Vabh using Equation 4 are presented in Table 3. Table 4 contains the evaluation statistics for the indirect methods used to predict ⌬Vabh. Graphs of

Table 2. Parameter estimates and standard errors for the equations developed in the analysis. Equation and Parameter

Estimate

Standard error

1. Volume equation (constant form factor) a1 0.001881061 2. Volume equation (variable form factor) b1 0.000940256 b2 0.428090044 b3 ⫺0.89117095 b4 0.436155457 b5 0.087518489 b6 ⫺22.9435367 3. Stem taper c0 0.889389789 c1 0.942324052 c2 0.0350430748 c3 0.756604248 c4 ⫺1.48426294 c5 1.76264008 c6 1.26458775 c7 0.0117647264 c8 ⫺0.482728513 4. Volume increment f0 ⫺9.36395406 f1 24.2169004 f2 ⫺6.216533068 f3 0.336877259 f4 13.8899413 f5 ⫺6.77974744

0.00000847885 0.0000319749 0.027641954 0.053541807 0.073932143 0.0022774093 16.31927045 0.01540908 0.005028821 0.006433576 0.01050833 0.0408559 0.0379268 0.05372024 0.0003999671 0.01475499 0.3828495 1.002765 0.3982466 0.04899943 0.6545976 0.633746

Table 3. Evaluation statistics for predicting unweighted total stem cubic foot volume above breast height at end of growth period (V Eabh) from volume Equations 1 and 2 and taper Equation 3 and directly predicting 5-year total stem cubic foot volume above breast height increment (⌬Vabh) from Equation 4. The statistics are relative bias (RB), relative accuracy (RA), and an indicator of how well the equations fit the data (FIT), all expressed as percentages. Ideally, RB and RA would be 0, and FIT would be 100. Equation 1. 2. 3. 4.

Volume equation (constant form factor) Volume equation (variable form factor) Stem taper equation Volume increment equation

Attribute predicted

RB

RA

FIT

V Eabh V Eabh V Eabh ⌬Vabh

..... ⫺6.34 ⫺0.53 ⫺0.58 0.20

.(%) . 39.20 21.94 21.84 38.79

..... 96.17 98.80 98.81 87.35

␧ and of lowess curves plotted over predicted ⌬Vabh for the various combinations of volume, ⌬D, and ⌬HT estimation methods are presented in Figure 1. The remaining presentation of results is structured around the three questions that form the objectives of the study. Question 1: What Practical Effect Does Using Different Models or Methods to Estimate Tree Volume Have on the Accuracy of the Indirect Method? Evaluating volume predictions from Equations 1–3, the FIT statistics in Table 3 show that the equations explain from 96% to almost 99% of the variation in VEabh. The underprediction bias ranged from more than 6% for simple Equation 1 to about 0.5% for the other two equations. Even though taper Equation 3 was not fit in a manner to optimize volume prediction, its VEabh evaluation statistics are nearly identical to those of volume Equation 2. The choice of how VEabh was predicted did have an impact on the accuracy of the indirect method. Concentrating on the use of actual ⌬D and actual ⌬HT, there was a range of 8.92% in the RA WEST. J. APPL. FOR. 25(3) 2010

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Table 4. Evaluation statistics for alternative procedures of indirectly predicting 5-year volume increment above breast height. The statistics are relative bias (RB), relative accuracy (RA), and an indicator of how well the equations fit the data (FIT), all expressed as percentages. Ideally, RB and RA would be 0, and FIT would be 100. Source of diameter increment Actual Actual Predicted Predicted

Source of height increment Actual Predicted Actual Predicted

1. Volume equation (constant form factor) RB

RA

2. Volume equation (variable form factor) FIT

RB

RA

3. Stem taper equation FIT

RB

RA

FIT

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(%). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.32 28.26 93.28 0.23 19.85 96.69 ⫺1.20 19.34 96.86 5.73 33.44 90.60 2.15 26.24 94.21 0.62 25.58 94.50 ⫺5.30 39.40 86.95 ⫺6.95 36.02 89.09 ⫺8.45 36.94 88.54 ⫺3.76 43.43 84.14 ⫺4.89 40.50 86.21 ⫺6.48 41.19 85.74

Figure 1. Residuals (predicted ⴚ observed; ft3) and lowess curves plotted over predicted 5-year volume increment above breast height (ft3) using different combinations of observed and predicted ⌬D, ⌬HT, and volume estimation equations (constant form volume Equation 1, variable form volume Equation 2, and stem taper Equation 3).

statistics between the three indirect estimators of ⌬Vabh (Table 4). Simple volume Equation 1 had the largest RB statistic and the poorest RA statistic of the three equations. However, the FIT statis124

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tic shows that Equation 1 still explained more than 93% of the variation in ⌬Vabh. Both volume Equation 2 and taper Equation 3 explained almost 97% of the variation in ⌬Vabh. Bias was larger in

Equation 3 than 2 (Figure 1). The RA values indicate that the overall accuracy of Equation 3 was somewhat better than that of Equation 2. Question 2: What Practical Effect Does Using Predicted Rather Than Measured Diameter Increment and/or Height Increment Have on the Accuracy of the Indirect Method? When both increment attributes are predicted, simple volume Equation 1 still had the poorest accuracy of the three equations, but its underprediction bias, at ⫺3.76%, was the smallest of the three equations (Table 4). Using all predicted values instead of all actual values reduced the FIT statistic of Equation 1 by 9.14%. The FIT statistics indicate that Equations 2 and 3 explained 86.21% and 85.74% of the variation in ⌬Vabh, respectively, a drop of about 11% from when actual increment values are used. The RA and FIT statistical advantages of Equations 2 and 3 over Equation 1 when using actual ⌬D and actual ⌬HT are nearly eroded away when the increments are predicted. Underprediction bias was ⫺4.89% for Equation 2 and ⫺6.48% for Equation 3. The bias appeared to concentrate in larger values of predicted ⌬Vabh (Figure 1). The overall accuracy of Equation 2 was somewhat better than that of Equation 3. Basically, using predicted values of ⌬D and ⌬HT reduced accuracy by a greater amount than using different equations for predicting volume. Examination of the results when using only one predicted increment indicates that the substitution of predicted ⌬D has a larger impact than the substitution of predicted ⌬HT. Substitution of predicted ⌬HT resulted in over prediction bias ranging from 0.62% for Equation 3 to 5.73% for Equation 1, whereas substitution of predicted ⌬D resulted in underprediction bias ranging from ⫺5.30% for Equation 1 to ⫺8.45% for Equation 3. Substitution of predicted ⌬HT resulted in reductions of the FIT statistic that ranged from approximately 2% to 3%, whereas substitution of predicted ⌬D resulted in reductions of approximately 6 – 8% (Table 4). Question 3: How Does the Accuracy of the Indirect Method Compare with the Accuracy of a Direct Estimator of Volume Increment? The RA statistic for directly predicting ⌬Vabh using Equation 4 was about 2% better than the indirect method that used both predicted ⌬D and ⌬HT in Equation 2 or 3 (Tables 3 and 4). Using the same basis for comparison, the FIT statistic for Equation 4 was more than 87%, an improvement of about 1%. Equation 4 over predicted ⌬Vabh by 0.2%, a substantial improvement over the indirect method that used the regional equations.

Discussion Individual tree growth and yield models are widely used to project diameter and height through time by the application of ⌬D and ⌬HT equations. These estimates are then used to indirectly estimate ⌬V. Mathematical expectation and measurement error theory indicate that error can be generated with this type of approach because of the different approaches for estimating stem volume and the use of predicted ⌬D and ⌬HT rather than actual values. Despite the prevalence of inferring ⌬V from predicted ⌬D and predicted ⌬HT, relatively little work has been done on quantifying the consequences of this approach on predictions of ⌬V.

The results of this analysis suggest that the prediction of ⌬V through the indirect method is sensitive both to the approach used for estimating stem volume and to the use of actual versus predicted tree increment values, with the latter being more influential. Compared with a direct prediction of ⌬V, the use of both predicted ⌬D and ⌬HT in the indirect method had slightly more variable estimates of ⌬V. The amount of bias also differed between the indirect and direct methods, with the latter method having less bias than the former method. Several attempts were made to improve the model form used in the direct approach, particularly by trying to include HT and a measure of two-sided competition in it. However, nothing proved superior to Equation 4, which is very similar to the model form used by Hann and Hanus (2002a) for modeling ⌬D. This result is consistent with our experience that modeling Vabh with D alone results in FIT statistics of greater than 90%, implying that the best predictors of ⌬D are likely the same for predicting ⌬Vabh. Equations for predicting ⌬D and ⌬HT use complex model forms that incorporate attributes associated with tree size, tree vigor, site productivity, one-sided competition, and two-sided competition to characterize the relationships (e.g., Hann and Ritchie 1988, Hann and Larsen 1991, Hann and Hanus 2002a, b, Hann et al. 2003). The relationships used in these two dynamic equations are not identical, and they become even more complicated when treatments are added that cause the two equations to react quite differently (e.g., Hann et al. 2003). Using a model form such as Equation 4 to directly predict ⌬Vabh may, therefore, mask the different behaviors found with modeling ⌬D and ⌬HT separately. An alternative approach that recognizes the various equations used in the indirect approach as a system of equations and estimates the parameters through two-stage or, perhaps, three-stage least squares (Kmenta 1986) may provide a more accurate direct predictor of ⌬Vabh than was found with Equation 4. Measurement error theory indicates that reducing the amount of error in the predictor variables could improve the accuracy of the predicted responses (Kmenta 1986, Fuller 1987, Carroll et al. 1995, Kangas 1997, 1998). Our findings show that improving the prediction of ⌬D in particular could lead to substantial improvement in the RA and RB statistics for indirectly predicting ⌬V. One way of doing this is to use the calibration features found in both the Forest Vegetation Simulator (Wykoff et al. 1982, Crookston and Dixon 2005) and ORGANON (Hester et al. 1989, Hann 2009) that calculate a site-specific scaling factor for the ⌬D equation using measured past radial increment for a sample of trees from the stand of interest. For example, the potential of this approach was tested by re-estimating the parameters of the Hann and Hanus (2002a, b) ⌬D and ⌬HT equations using this study’s subsample of trees. When only ⌬D was predicted, the RB statistics improved to a value of ⫹0.05% for Equation 2 and ⫺1.28% for Equation 3, with corresponding small improvements in the RA statistics. When both ⌬D and ⌬HT were predicted, the RB statistics improved to a value of ⫹2.56% for Equation 2 and ⫹1.16% for Equation 3, and the FIT statistics improved slightly to 86.30% for Equation 2 and 86.10% for Equation 3. Therefore, there is evidence that the usage of local calibration procedures could help to mitigate possible problems with bias in the indirect method. Individual stem volume is a widely used measure for trees as it integrates their size, shape, and potential value. Indirectly predicting volume increment, however, often requires the use of multiple equations that can differ in their accuracy. This study evaluated several WEST. J. APPL. FOR. 25(3) 2010

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alternative methods for estimating individual tree volume increment for a single growth period and quantified the consequences of different assumptions. Despite differences in the variability of the estimates, all of the approaches examined in this present analysis were basically unbiased except at the highest predicted values of volume increment (Figure 1). This suggests that the common approach of indirectly inferring volume increment from diameter and height increments is likely not a great source of error in most growth and yield models for the first growth period, given that diameter and height increment equations are well formulated and fit using a sufficient data set. These results do not address the consequences of proliferating and accumulating errors for projections of more than one growth period in length (e.g., Gertner and Dzialowy 1984, Kangas 1997, 1998). Endnote [1] Accuracy is defined as “the closeness of a measurement or estimate to the true value” (Avery and Burkhart 2002, Husch et al. 2003) and therefore includes the effects of both bias and variance in the measurement or estimate.

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