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The ability to predict cumulative bole volume to any predefined upper stem diameter on a standing tree is essential for estimating current inventory levels and making informed decisions regarding the management of forest resources. Several types of mathematical models have been developed to predict cumulative bole volume, requiring only the collection of low cost data but high accuracy tree measurements. This paper reports on the comparison of a variable-exponent taper model and a volume-ratio model for estimating inside-bark cumulative bole volume to three predefined upper stem diameters using stem analysis data

Regression analysis was initiated into forestry research more than 70 years ago. One common use of applying ordinary least squares and non-linear least squares techniques was to fit a regression model to predict volume content in the stem of a standing tree (

Due to changes in utilization standards within the forest industry, many different bole-volume equations have been developed. Volume prediction to any merchantability limit has been accomplished in many ways, including: (1) a constrained volume equation for different merchantable limits that differed by a fixed amount (

Construction of taper and volume-ratio equations entails the compilation of longitudinal data on each tree, resulting in a lack of independence between observations and biased estimates of parameters if ordinary least squared techniques are used in constructing the models (

The fixed effects model that includes explanatory variables accounts for a part of the total variation in tree volume content. There is a residual variability component, representing differences in attributes acting at stand or tree level, that are difficult to observe, encompassing heredity and environmental factors, or even measurement errors, introducing a stochastic level of variability in tree volume content.

To properly analyze this type of data, the error structure inherent within the data must be considered in the modeling process. Even though ordinary least squares analysis provides unbiased parameter estimates in the presence of autocorrelation, it does not provide minimum variance estimators (

In addition to being correlated, the data for the construction of taper and volume-ratio models is unbalanced. Each sample tree is measured by short sections and the desired volume (total or merchantable) is determined by summing up the volumes of all the sections in a tree. Often, the number of observations of cumulative bole volume will vary among trees, as short trees tend to have fewer sections than larger trees. The observed sample will be unbalanced by having unequal numbers of observations per subject (

Two general methods have been proposed to fit continuous, unbalanced, multilevel longitudinal data. The first proposed by

Mixed-effects models, consisting of both fixed and random-effects parameters have the advantage to allow for modeling of the variance-covariance matrix of correlated data (

Despite its economic and ecologic importance, this species has never been the subject of serious growth and yield studies, making it difficult to estimate current inventory levels and to account for the amount of volume harvested (

The primary objectives of this study were to: (1) fit both a variable-exponent taper equation and a total volume-ratio model to

The study area is a region of approximately 1800 km^{2} in the north central portion of Cordillera Central, Dominican Republic. Five natural stands of

To validate the models, the original data set was randomly divided into two parts. Eighty percent of the observations (

Sample trees identified from the three ecological zones were felled and measured for diameter outside- and inside-bark at predetermined points along a stem; stump height (0.3 m above ground), breast height (1.3 m above ground), and then in equal intervals of 1.0 m thereafter to a 4-cm upper stem diameter. At each point, outside-bark diameter measurements were taken with diameter tape. Measurements were carried out using diameter tapes and bark gauge. Volume was computed for each one meter long bole section using Smalian’s formula (

The development of both the taper and volume-ratio models required the use of multiple measurements of diameters up the stem in each individual sampled tree; therefore the nature of the data violated the assumption of independence and absence of autocorrelation (

where _{ij} represents the _{i} is the corresponding diameter at 1.30 m outside bark (cm) for the _{0} _{1} _{2} _{1} _{2} _{3} _{4} _{5} _{1} _{2} _{g}^{2}); _{ij} are the residual errors assumed to be iid _{ε}^{2}); and independent of _{1}, _{2}; _{i} is the corresponding total tree height (m) for the _{ij} is the relative height above the ground (h/H) at the _{ij} ahead (_{ij} is the height above the ground (m) at the

Previous studies (

Using a general notation, the nonlinear mixed-effects modeling approach for the taper model equation can be expressed as (

where _{ij} is a vector of regressor variables and

Random effects are denoted by the vector _{i}, modelers of heterogeneity of observations between trees. The _{i} are zero mean random variables with variance covariance matrix as expressed in _{ij} are diameters inside bark observed on a subject tree _{ij} are within three errors with mean 0 and possibly correlated. The extent to which trees vary around the population average response is expressed by the variability of the _{i}’s. It is assumed that _{i}’s and the _{ij}’s are normally distributed with (

Variations in tree taper can be expressed using matrix notation as follows (

_{1}, _{2}, _{3}, _{4}, _{5}, and choosing the one providing the best fit according to Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (BIC) values. This permitted for the effects of correlation among observations of longitudinal data to be accounted for (

To develop the volume-ratio model, we followed the combined modeling approaches first reported by

The mixed effects model for estimating the cumulative bole volume was specified as follows (

where _{id j} is the cumulative volume of the _{j}; _{i} is the diameter (cm) at 1.30 m outside bark for the _{i} is the total height (m) for the _{j} is the upper diameter at _{ij} _{j } _{i}); _{0} _{1} _{2} _{3} _{4} _{1i} _{2i} are fixed and random parameters to be estimated; _{ij} is the within-cluster error term for the

In the model represented by _{1i } _{i}^{2}), _{2i } _{i}^{2}), cov(_{1i} _{2i}) = _{12}; _{k } _{ij }~ _{ij} _{ik}] = 0 for all distinct pairs. The _{1i} models random slopes in the total volume equation and the _{2i} models the rate of change and point of inflection in the ratio term (

The inclusion of the random effects in the total volume component is due to the variability in size which is reflected in total volume variations and the random effects in the ratio term are included to reflect the variability in shape of the stem profiles (_{0 }+ (_{1 }+_{1i}) _{i}^{2} _{i}, several mathematical models were preliminarily examined. The combined variable equation, as it is known in quantitative silviculture, provided a straightforward model capable of relating total volume to easily measurable tree explanatory variables; although the constant variance assumption was expected to be violated due to the fact that a direct measure of stem content was used as the dependent variable in a regression model (stem volume variability of larger trees is usually greater than the variability of smaller trees).

The volume proportion component is always positive and tends to 1 as _{j} tends to 0. This portion of the model in

The conditional distribution _{(y/β)}(_{idj} / ~_{i0 } _{idj}, ^{2}) where (

and (

The distribution of the _{i} is as follows (

where _{i } _{j}] = 0.

PROC NLMIXED in SAS (

Fitting the models by nonlinear least squares procedures requires initial estimates of parameters to be input at the start of the iteration procedure. In the case of the taper model, we used the values obtained from the linearized version of the taper equation which has a mathematical value described by the following formula (

where

To locate the position of the random parameters (_{1} and _{2}) within the non-linear taper model, every parameter (_{0}, _{1}, _{2}, _{1}, _{2}, _{3}, _{4}, and _{5}) was tested individually as being random. The final model was selected based on the achievement of best fit as indicated by the lowest AIC, BIC and -2LL values. For the volume-ratio model, starting values for the parameters (_{0}, _{1}, _{2} and _{3}) were chosen as the converged iterates from the RMLE fit base on linearization. The positions of the random effects (_{1} and _{2}) within the model were chosen following the procedure used with the taper model.

Once the taper and volume ratio models were obtained, the fixed-effects parameters from the best linear unbiased predictions were used to predict a mean merchantable volume inside-bark up to three pre-established upper stem diameters (14, 8 and 4 cm), which defined merchantability standards in the study region. Linear interpolation (

The better model was selected based on its ability to accurately predict merchantable volume to the selected upper stem diameters using an independent verification data set. In the case of the volume-ratio model, best linear unbiased predicted merchantable volumes were directly obtained from SAS output for the selected upper stem diameters. For the taper model, the fixed-effect parameters from the best linear unbiased prediction were incorporated using a procedure developed by (

To calculate tree merchantable volume (m^{3}) to specified upper stem diameters with the taper model, an iteration procedure was applied using estimated coefficients for _{0 }- _{1} < 0.00000001 was obtained, where _{0} represent an initial value for relative height of diameters up the stem and _{1} represents relative height of the defined diameter of the presumed utilization standard. A 4.0 cm top diameter inside bark was assumed. To compute merchantable height and merchantable length, a stump height of 0.30 m was assumed. Then merchantable length was divided into 10 sections of equal length and the height above ground from the middle and the top of each section was computed. Prediction of diameter inside bark at the middle and top of each section was accomplished using the taper equation. The final product of the program allowed computing the merchantable volume of the tree to 4.0 cm top dib and calculating merchantable volume in m^{3} using Newton’s formula (

Using the observed and calibrated volumes, goodness-of-fit statistics were used to assess the predictive capability of the selected models (^{2}), average bias (

where _{i} and _{i} are the observed and estimated volumes for the ^{2} is the coefficient of determination;

Both the mean and percent prediction errors give an average measure of the prediction bias, in absolute and relative terms respectively.

In the DR, three main products are derived from

The fixed effects parameters of both the taper and the volume-ratio model were used on the validation data set to generate estimates, and to serve as reference for comparing the fitting of both models, including two random parameters. The results in

The coefficients _{1} and _{5} of the taper equation were not statistically significant at α = 0.05 and removed from the original model thus; the final taper model was (

After testing every parameter of the non-linear taper model individually as being random, the best fit (lowest AIC, BIC and -2LL values) was achieved when _{2} and _{3} were set as random. Fixed parameter estimates for model in ^{2}, while within tree variance as reflected by the random effects variance components var(_{1}) and var(_{2}) is in the order of 0.01 and 0.004 cm^{2}.

Residuals from the model fitting portion plotted against the predicted diameters inside bark values showed no conspicuous pattern in their distribution. The inclusions of the random effects in the model, which account for a large portion of the variability, have effectively removed the autocorrelation among the observations. The predicted diameters (inside bark) by the taper model were compare to the observed values on the basis of bias and precision (

In terms of precision, as measured by the root-mean-square error as a percentage of the mean observed diameter, the corresponding percentages from relative height 0.0 to 0.9 were 10.7, 6.0, 4.8, 5.5, 6.6, 6.5, 8.3, 11.2, 16.4 and 20.6 respectively. These results show low precision at the bottom and top of the stem, being worse at the tip.

For the volume-ratio model (

In terms of precision, as estimated by

Observed cumulative volume as percentage and superimposed to the corresponding predicted cumulative volume percentages by the fixed effect parameters model and the model including random parameters at different relative heights on three contrasting trees (small, medium and large) of the verification data set were studied. The fixed effect model approximated better the cumulative observed volume between 0.1 and 0.3 relative heights in all three trees. From 0.35 to 0.7 the fixed plus random effects model approximates better the observed cumulative volume percentage and from 0.7 to 0.1 relative heights, at the tip, both models predict similar cumulative volume values.

Cumulative bole volume predictions by the taper and volume-ratio models on verification data set were estimated to three predefined upper stem diameters normally used by forest industries in the study region, employing both the taper and the volume-ratio models. For “saw logs” that upper stem diameter is 14 cm, for “posts” it is 8 cm and for “bars” 4 cm. The predictions were carried out considering both the fixed and the random part of the models. The predicted cumulative volumes were then compared to the corresponding observed cumulative volumes by means of residual analysis. The results are shown in

When comparing the volume-ratio model to the taper model for their ability to predict merchantable volume to varying merchantability limits using independent data, the taper model proved slightly superior to the volume-ratio model according to most goodness-of-fit statistics. Only for the “bar” product (4 cm upper diameter) the volume-ratio model was slightly better than the taper model in terms of bias and MAD.

The selection of the random parameters in the taper models was based on the values of AIC, BIC and -2LL according to the model validation results. The lowest values for the fit statistics above were obtained when the first random coefficient _{1} was added to the intercept term _{0} _{i}^{1} _{2}^{Di}, and the second random coefficient _{2} was added to the fix parameter _{3} accompanying the _{1 }and _{2 }was low (0.3) indicating that variation has been accounted by current covariates. _{0} parameter) and another for _{1} to obtain the best results for several candidate variable-exponent taper equations for lodgepole pine in Alberta, Canada.

Several authors have reported the need to incorporate surrogates of stem form into taper models (

The location term

Our taper model had somewhat large prediction bias in the lower portion of the bole (21 % at a relative height Z = 0.1). Higher up the bole, the bias percentage decreases substantially, to values between 5 and 9 %. These results agree with the findings of

Previous studies (

The random effects in the volume-ratio model also take into account the inter-tree variation through the marginal covariance structure. As with the taper model, the random parameters allow individualization of the model fit to each subject tree, accounting for within-tree covariances. The random parameters in the volume-ratio model explain variability in size (total volume) and shape of the volume profile. In this model, the random _{1i }term models random slopes in the total volume component of the equation and _{2i }models the rate of change of the tree profile.

The improvement of the model fit by the inclusion of the random effect parameters can be verified by the value of the model fit statistics (

In the present work, we fitted both a variable-exponent taper equation and a total volume-ratio model to

Unbiased and accurate individual tree volume prediction is useful in determining stand volume in management inventories. Based on this standpoint, the prediction performance by the two models are very similar. However, whenever merchantability standards should change, the taper model would be more flexible to adapt to the new standards and to predict stand volume estimates.

The main concern of these models to predict merchantable volume in

Nonetheless, it is important to point out that, despite the inconsistent results obtained, the methodology proposed in the present work could be useful. The improvement of the predictions by both models (

We fitted a variable-exponent taper equation and a total volume-ratio model to

Due to the fact that the country is mostly mountainous and soils are very fragile for agriculture production, it is clear that forest production should probably be the most important economic activity in the study region and in the country. The soils in the study area are generally poor and shallow, with great extensions of terrain severely degraded and eroded due to deforestation and migratory agriculture. These areas were once populated by our endemic species

To maintain and increase

The productive capacity of the humid zone calls for dedicating its fragile soils to the production of wood, especially

This investigation was carried out thanks to contributions from FONDOCYT, a research grant from the Dominican Republic government administered by the

^{th}edn). McGraw Hill, New York, USA, pp. 453.

^{rd}edn). John Wiley and Sons, New York, USA, pp. 428.

^{th}edn). Wiley, New York, USA, pp. 456.

^{th}edn). SAS Institute Inc., Cary, NC, USA. pp. 157.

Residual values (diameter, cm) from the fitting of the taper model to the validation data set at different relative heights.

Bias performance of the volume-ratio model as percentage in the validation data set at different relative heights.

Descriptive statistics for the estimation and verification data sets of

Datasets | Variables | Mean | St. Dev. | Min | Max |
---|---|---|---|---|---|

Estimation(n=149) | Diameter at breast height (cm) | 29.8 | 8.31 | 11.5 | 53.5 |

Total height (m) | 21.9 | 5.51 | 8.6 | 35 | |

Total volume (m^{3}) |
0.6 | 0.45 | 0.03 | 2.49 | |

Cumulative volume (m^{3}) to 4 cm top diameter |
0.59 | 0.45 | 0.03 | 2.49 | |

Cumulative volume (m^{3}) to 8 cm top diameter |
0.58 | 0.45 | 0.01 | 2.48 | |

Cumulative volume (m^{3}) to 14 cm top diameter |
0.54 | 0.46 | 0 | 2.46 | |

Verification(n=39) | Diameter at breast height (cm) | 30.32 | 9.13 | 9.2 | 53.1 |

Total height (m) | 20.99 | 4.33 | 7.9 | 29.4 | |

Total volume (m^{3}) |
0.55 | 0.42 | 0.03 | 2.08 | |

Cumulative volume (m^{3}) to 4 cm top diameter |
0.55 | 0.42 | 0.03 | 2.08 | |

Cumulative volume (m^{3}) to 8 cm top diameter |
0.54 | 0.42 | 0.02 | 2.08 | |

Cumulative volume (m^{3}) to 14 cm top diameter |
0.49 | 0.43 | 0.01 | 2.04 |

Akaike’s Information Criterion (AIC), Baye’s Information Criterion (BIC) and minus twice log likelihoods (-2 LL) values for the taper and volume ratio fixed effect models and corresponding models including two random parameters each.

Model | Random effects | AIC | BIC | -2 LL |
---|---|---|---|---|

Taper | None | 14608 | 14689 | 14582 |

g_{1} and g_{2} |
12197 | 12225 | 12179 | |

Volume-ratio | None | -5931 | -5875 | -5949 |

u_{1i} and u_{2i} |
-14638 | -14617 | -14652 |

Parameter estimates, corresponding standard error and

Parameter | Estimate | Standard error | |
---|---|---|---|

a_{0} |
1.058 | 0.201 | **** |

a_{1} |
0.894 | 0.082 | **** |

a_{2} |
1.001 | 0.003 | **** |

b_{1} |
Non-significant | ||

b_{2} |
0.024 | 0.006 | **** |

b_{3} |
-1.093 | 0.099 | **** |

b_{4} |
0.709 | 0.052 | **** |

b_{5} |
Non-significant | ||

Residual variance | 1.443 | 0.035 | **** |

Var(g_{1}) |
0.017 | 0.002 | **** |

Var(g_{2}) |
0.004 | 0.001 | ** |

Parameter estimates, corresponding standard error and p-values for the volume-ratio model (see

Parameter | Estimate | Standard error | Pr > |t| |
---|---|---|---|

b_{0} |
0.018 | 0.004 | **** |

b_{1} |
0.029 | 0.001 | **** |

b_{2} |
10.289 | 0.437 | **** |

b_{3} |
5.931 | 0.038 | **** |

Residual variance | 0.001 | 0.000 | **** |

Var(u_{1}) |
0.000 | 0.000 | **** |

Var(u_{2}) |
12.933 | 1.861 | **** |

Goodness-of-fit statistics from the residual analysis performed on the validation data set for cumulative bole volume predictions to three predefined upper stem diameters of 14 cm, 8 cm and 4 cm by the taper and volume-ratio models.

Goodness- of-fitstatistics | Upper stem diameter | |||||
---|---|---|---|---|---|---|

14 cm | 8 cm | 4 cm | ||||

Taper | Volume ratio | Taper | Volume ratio | Taper | Volume ratio | |

MSE | 0.014 | 0.022 | 0.014 | 0.021 | 0.014 | 0.02 |

RMSE | 0.09 | 0.115 | 0.09 | 0.11 | 0.09 | 0.11 |

RMSE % | 15.65 | 19.97 | 14.34 | 17.69 | 14.276 | 17.408 |

Bias | 0.021 | -0.042 | 0.032 | -0.037 | 0.036 | -0.34 |

Bias % | 3.69 | 7.24 | 5.13 | 5.873 | 5.648 | 5.297 |

MAD | 0.049 | 0.054 | 0.051 | 0.05 | 0.052 | 0.05 |

R^{2} |
0.93 | 0.917 | 0.928 | 0.92 | 0.926 | 0.92 |