Methods

For each plot in each forest type, we calculated the two sums found in the right-hand portion of Eq. 30, using the EF (Eq. 31) to scale each tree to a per hectare basis:

1.6

A variety of statistical techniques is available to determine maximum density levels including ocular estimation (Reineke 1933), ordinary least squares (Chisman and Schumacher 1940; Su 2014), reduced major axis regression (Solomon and Zhang 2002), quantile regression (Ducey and Knapp 2010; Su 2014) and segmented regression

(Zhang et al. 2005). In this paper, we followed Ducey and Knapp (2010) in using quantile regression. Quantile regression is a useful approach when solving an optimization problem of minimizing an asymmetric function of absolute error loss (Koenker and Bassett 1978). Quantile regression is able to provide parameter estimates for linear regressions fit to any portion of a response distribution, including near the upper bounds, without imposing stringent assumptions on the error distribution (Scharf et al. 1998; Cade et al. 1999; Cade and Noon 2003). So, quantile regression has been suggested for estimating boundaries or envelopes in a variety of ecological settings (Scharf et al. 1998; Cade and Noon 2003). Following the concept of TAR (Chisman and Schumacher 1940), RD is set to 1 to estimate each parameter and to determine

maximum stand density index. Rewriting Eq. 30 using Eq. 32 and Eq. 33 we get,

34

0 0 1 1

1

RD b X b X  (34)

For quantile regression, the parameter values were chosen as minimizing the weighted sum of absolute values of errors (ρτ(ɛ)) over all observations, where the weight depended on whether the error was positive or negative. For an individual observation,

 

( )= 0

          (35)

where τ is a predetermined quantile between 0 and 1. The function I(ɛ < 0) was equal to 1 if ɛ < 0, and the function was 0 when ɛ >= 0. So, quantile regression chooses

parameters such that the τ^th quantile of ɛ is zero, and the fraction of observations that fall below their predicted values is τ.

In this study, quantile regression found a relationship such that RD=1 described the τ^th quantile of density within all datasets for each forest type. When τ was set to very close to 1, the parameters would be chosen so that RD=1 described the upper limit of observed density. It should be noted that, because we set RD equaled to 1 and τ was set to be close to 1, the absolute values of errors would be positive values which means I (ɛ

< 0) would always be equal to 0. This characteristic leads to maximizing the sum of the ρτ(ɛ). Thus, in this study, the τ^th quantile becomes 1 - τ for estimating the limit of observed density within the entire dataset. We used quantile regression to fit Eq. 34 with quantiles from 0.50 to 0.99. All analyses were conducted using the “quantreg”

library (Koenker 2004) within the R statistical package (R Development Core Team 2016).

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Observed maximum stand density depends not only on stand biology, but also on sampling methods employed. For example, small plot sizes generally result in higher estimates of stand density than those from larger plot sizes. Also, maximum stand density or normal stocking might occur at quantiles less than 1, and the distribution of the data itself was not adequate to determine which quantile should be used (Ducey and Knapp 2010). So, in order to select a suitable quantile, we hypothesized that all trees have DBH = 25 cm and RD was equal to 1, and we then used the TFRI4 dataset to estimate the coefficients for X0 and X1 for each quantile to solve for N, which is referred to as the implied maximum ASDI,

0 1

N= 1

b b SG

(36)

where SG is an average SG determined from all trees in all plots. The mean SG and SG range for each forest type are presented in Table 3.1. We also used published equations to obtain the number of trees per hectare (N) when QMD was set to 25 cm and this number was regarded as a reference SDI. For pine forests, there were no related

published equations in Taiwan to obtain reference SDI. So, we chose plots where % BA of the desired species were over 80 % to fit Reineke’s model (Eq. 13), and used quantile regression to estimate coefficients. With the fitted equation, we also set QMD equal to 25 cm and solved for N. So, the fitted equation for each quantile was viewed as a method to estimate reference SDI. For a successful mixed-species density measure, its result should be consistent with previous research developed for both single-species stands and well-studied simple mixtures. So, with both implied ASDI and reference SDI, we chose the quantile where implied ASDI and reference SDI intersected. A

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flowchart summarizing the sequence of steps in the method used here is presented in Figure 3.2. The step-by-step procedures were,

Step 1. Calculate X0 (Eq. 32) and X1 (Eq. 33) to scale up our dataset to per ha levels and fit to Ducey’s model (Eq. 30)

Step 2. Set RD = 1 (Eq. 34) and use quantile regression to estimate the coefficients of X0 and X1 for quantiles from 0.50 to 0.99, keeping in mind that the τ^th quantile is fit as the 1 - τ^th quantile.

Step 3. Calculate an average SG by all trees in all plots, and then substitute the coefficients of X0 and X1 and the mean SG into Eq. 36 to estimate implied maximum ASDI for each quantile.

Step 4. Use published equations and diagrams, or fit Reineke’s model (Eq. 13) to “pure”

plots having at least 80 % of the BA constituted by desired species to obtain reference SDI.

Step 5. The best quantile is chosen as the one where implied maximum ASDI intersects reference SDI. The parameter estimates associated with this quantile form the RD formula for the mixed-species forest.

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