A Blind Spot in the Use of the Weibull Function for Modeling Diameter Distributions

Abstract Diameter distribution models based on probability density functions are integral to many forest growth and yield systems, where they are used to estimate product volumes within diameter classes. The three-parameter Weibull function with a constrained nonnegative lower bound is commonly used because of its flexibility and ease of fitting. This study compared Weibull and reverse Weibull functions with and without a lower bound constraint and left-hand truncation, across three large unthinned plantation cohorts in which 81% of plots had negatively skewed diameter distributions. Near-optimal lower bounds for the unconstrained Weibull function were negative for negatively skewed data, and the left-truncated Weibull using these bounds was 14.2% more accurate than the constrained Weibull, based on the Kolmogorov-Smirnov statistic. The truncated reverse Weibull fit dominant tree distributions 23.7% more accurately than the constrained Weibull, based on a mean absolute difference statistic. This work indicates that a blind spot may have developed in plantation growth modeling systems deploying constrained Weibull functions, and that left-truncation of unconstrained functions could substantially improve model accuracy for negatively skewed distributions.

Deriving Tree Diameter Growth and Probability of Survival Equations from Successive Diameter Distributions

Forest Science ◽

10.1093/forestscience/54.1.31 ◽

2008 ◽

Vol 54 (1) ◽

pp. 31-35

Author(s):

Thomas G. Matney ◽

Emily B. Schultz

Keyword(s):

General Procedure ◽

Growth And Yield ◽

Diameter Distribution ◽

Diameter Growth ◽

Individual Tree ◽

Statistical Probability ◽

Growth And Survival ◽

Tree Diameter ◽

Growth And Yield Models ◽

Abstract Many growth and yield models have used statistical probability distributions to estimate the diameter distribution of a stand at any age. Equations for approximating individual tree diameter growth and survival probabilities from dbh can be derived from these models. A general procedure for determining the functions is discussed and illustrated using a loblolly pine spacing study. The results from the spacing study show that it is possible to define tree diameter growth and survival probability functions from diameter distributions with an accuracy sufficient to obtain a link between the individual tree and diameter growth and yield models.

Stand growth projection: simultaneous estimation of growth and mortality using a single probabilistic function

10.1139/x87-225 ◽

1987 ◽

Vol 17 (11) ◽

pp. 1466-1470 ◽

Cited By ~ 5

Author(s):

K. E. Lowell ◽

R. J. Mitchell

Keyword(s):

Logistic Model ◽

Forest Growth ◽

Growth And Yield ◽

Diameter Distribution ◽

Simultaneous Estimation ◽

Binary Event ◽

Probabilistic Function ◽

Growth Projection ◽

Individual Trees ◽

Stand Characteristics

Logistic regression analysis can be used to estimate the probability of a binary event. In forestry, its use largely has been limited to predicting the probability of mortality of individual trees. However, the potential for broader application in forest growth and yield modelling has largely been overlooked. A logistic model to predict the probability that a tree will attain a specified future diameter can be produced by establishing a series of growth "success" criteria. Given the initial diameter distribution of a forest stand, a future diameter distribution and stand characteristics can be estimated probabilistically by estimating the proportion of stems in each diameter class of the distribution which attains a specified future diameter (the "success" criterion) and the proportion which fails to achieve at least zero growth (i.e., mortality). Using permanent plot data, such a logistic model was calibrated and validated for an oak–hickory forest in southeastern Missouri. Validation indicated that the model performs satisfactorily (estimates are unbiased) for individual trees over a 5-year prediction period, and for stand characteristics over 5-, 10-, 15-, and 20-year prediction periods though precision suffers as prediction period lengthens.

A New Diameter Distribution Model for Unmanaged Loblolly Pine Plantations in East Texas

10.1093/sjaf/30.1.13 ◽

2006 ◽

Vol 30 (1) ◽

pp. 13-20 ◽

Cited By ~ 3

Author(s):

Young-Jin Lee ◽

Dean W. Coble

Keyword(s):

Loblolly Pine ◽

Pine Plantations ◽

Growth And Yield ◽

Distribution Model ◽

Diameter Distribution ◽

Yield Prediction ◽

East Texas ◽

Distribution Models ◽

Parameter Recovery ◽

Diameter Classes

Abstract A parameter recovery procedure for the Weibull distribution function based on four percentile equations was used to develop a diameter distribution yield prediction model for unmanaged loblolly pine (Pinus taeda L.) plantations in East Texas. This model was compared with the diameter distribution models of Lenhart and Knowe, which have been used in East Texas. All three models were evaluated with independent observed data. The model developed in this study performed better than the other two models in prediction of trees per acre and cubic-foot volume per acre (wood and bark, excluding stump) across diameter classes. Lenhart’s model consistently underestimated the larger-diameter classes because it was developed originally with data mostly collected in young plantations. Knowe’s model overestimated volume in sawtimber-sized trees, which could lead to overestimations of volume in older loblolly pine plantations found in East Texas. An example also is provided to show users how to use this new yield prediction system. These results support the recommendation that forest managers should use growth and yield models designed and/or calibrated for the region in which they are implemented.South. J. Appl.For. 30(1):13–20.

A percentile-based estimator for the log-logistic function: Application to forestry

Forestry Studies / Metsanduslikud Uurimused ◽

10.2478/fsmu-2020-0009 ◽

2020 ◽

Vol 72 (1) ◽

pp. 107-120

Author(s):

Friday Nwabueze Ogana

Keyword(s):

Estimation Method ◽

Tectona Grandis ◽

Logistic Function ◽

Growth And Yield ◽

Estimation Methods ◽

Diameter Distribution ◽

Logistic Distribution ◽

Forest Stands ◽

Forest Modelling ◽

AbstractDeveloping a simplified estimation method without compromising the performance of the distribution is germane to forest modelling. Few estimation methods exist for the Log-Logistic distribution and are relatively complex. A simplified estimator for the Log-Logistic parameters will increase its application in diameter distribution yield systems. Therefore, in this study, a percentile-based estimator was applied for the Log-Logistic distribution. The Kolmogorov-Smirnov, Anderson-Darling and Cramer-von Mises statistics were used to evaluate the method in two natural forest stands and two monospecific plantations of Gmelina arborea Roxb. and Tectona grandis Linn. f. in Nigeria. The parameter recovery model (PRM) and parameter prediction model (PPM) were used to predict the diameter distributions of independent stands of G. arborea and T. grandis. The results showed that the percentile estimator did not compromise the quality of fits of the Log-Logistic function across the four forest stands and are comparable to the maximum likelihood estimator. The 25th and 75th, and 40th and 80th were the best sample percentiles for the estimator. The predicted diameter distributions of G. arborea and T. grandis stands from the PRM and PPM were reasonable and compare well with the observed distribution. Thus, either of the models can be incorporated into the growth and yield system of forest stand management.

Calibrating predicted diameter distribution with additional information in growth and yield predictions

10.1139/x02-121 ◽

2003 ◽

Vol 33 (3) ◽

pp. 430-434 ◽

Cited By ~ 4

Author(s):

Annika Kangas ◽

Matti Maltamo

Keyword(s):

Growth Models ◽

Basal Area ◽

Growth And Yield ◽

Diameter Distribution ◽

Individual Tree ◽

Forest Management Planning ◽

Growing Stock ◽

Additional Information ◽

Tree Models ◽

Diameter distribution of the growing stock is essential in many forest management planning problems. The diameter distribution is the basis for predicting, for example, timber assortments of a stand. Usually the predicted diameter distribution is scaled so that the stem number (or basal area) corresponds to the measured value (or predicted future value), but it may be difficult to obtain a distribution that gives correct estimates for all known variables. Diameter distributions that are compatible with all available information can be obtained using an approach adopted from sampling theory, the calibration estimation. In calibration estimation, the original predicted frequencies are modified so that they respect a set of constraints, the calibration equations. In this paper, an example of utilizing diameter distributions in growth and yield predictions is presented. The example is based on individual tree growth models of Scots pine (Pinus sylvestris L.). Calibration estimation was utilized in predicting the diameter distribution at the beginning of the simulation period. Then, trees were picked from the distribution and their development was predicted with individual tree models. In predicting the current stand characteristics, calibrated diameter distributions proved to be efficient. However, in predicting future yields, calibration estimation did not significantly improve the accuracy of the results.

A Comparison of Loblolly Pine Plantation Growth and Yield Models for Inventory Updating

10.1093/sjaf/20.1.15 ◽

1996 ◽

Vol 20 (1) ◽

pp. 15-22 ◽

Cited By ~ 7

Author(s):

James S. Shortt ◽

Harold E. Burkhart

Keyword(s):

Loblolly Pine ◽

Growth And Yield ◽

Diameter Distribution ◽

Distribution Models ◽

Tree Model ◽

Individual Tree ◽

Parameter Recovery ◽

Individual Tree Model ◽

The Individual ◽

Growth And Yield Models

Abstract Four different loblolly pine growth and yield models were evaluated for the purpose of updating forest inventory data. The types of growth and yield models examined were: a whole stand, a diameter distribution-parameter prediction, a diameter distribution-parameter recovery, and an individual tree model. Three different approaches were used to create fitting and validation data sets from permanent plot remeasurement data; each of the four growth and yield models was evaluated at varying projection periods. The periods used were 0, 3, 6, and 9 yr. Evaluations were based solely on the capability of each model to predict merchantable volume. In terms of root mean square error of prediction, the individual tree and whole stand models performed better than the diameter distribution models. At shorter projection periods, the individual tree model performed better than the whole stand model, but the whole stand approach was superior at the 9 yr period. Of the diameter distribution models, the parameter recovery model performed better for shorter periods than the parameter prediction model, but this difference diminished with longer periods. South. J. Appl. For. 20(1):15-22.

Predicting Diameter Distributions for Young Longleaf Pine Plantations in Southwest Georgia

10.1093/sjaf/33.1.25 ◽

2009 ◽

Vol 33 (1) ◽

pp. 25-28 ◽

Cited By ~ 5

Author(s):

Lichun Jiang ◽

John R. Brooks

Keyword(s):

Distribution Function ◽

Goodness Of Fit ◽

Longleaf Pine ◽

Pinus Palustris ◽

Cumulative Distribution ◽

Diameter Distribution ◽

Weibull Function ◽

Recovery Method ◽

Parameter Recovery ◽

Abstract Parameter prediction equations for the Weibull distribution function were developed based on four percentile functions and a parameter recovery method for longleaf pine (Pinus palustris Mill.) in Southwest Georgia. Four percentiles were expressed as functions of stand-level characteristics based on stepwise regression and seemingly unrelated regression. Using a percentile-based parameter recovery method (PCT), estimated diameter distributions were obtained from available stand-level variables. The PCT method was also compared with a cumulative distribution function (CDF) regression method. The PCT method produced consistently better goodness-of-fit statistics than the CDF method. The results indicate that diameter distribution in longleaf pine stands can be successfully characterized with the Weibull function.

Preserving correlation while modelling diameter distributions

10.1139/x03-191 ◽

2004 ◽

Vol 34 (1) ◽

pp. 221-232 ◽

Cited By ~ 18

Author(s):

A Robinson

Keyword(s):

Mixed Effects ◽

Seemingly Unrelated Regression ◽

Mixed Effects Models ◽

Diameter Distribution ◽

Distribution Models ◽

Unrelated Regression ◽

Simultaneous Prediction ◽

Components Analysis ◽

The construction of diameter-distribution models sometimes calls for the simultaneous prediction of population parameters from hierarchical data. Appropriate data for such models have characteristics that should be preserved or accommodated: clustering and contemporaneous correlations. Fitting techniques for such data must allow for these characteristics. Using a case study, I compare two techniques seemingly-unrelated regression (SUR) and principal components analysis (PCA) whilst using mixed-effects models. I adapt and apply a metric that focuses on volume prediction, which is a key application for diameter distributions. The results suggest that using mixed-effects models provides useful insights into environmental variation, and that SUR is more convenient and produces a slightly better fit than PCA. Both techniques are acceptable with regard to regression assumptions.

Dynamics of the diameter distribution after selection cutting in uneven- and even-aged northern hardwood stands: a long-term evaluation

10.1139/cjfr-2019-0204 ◽

2019 ◽

Vol 49 (12) ◽

pp. 1525-1539 ◽

Cited By ~ 1

Author(s):

Sarita Bassil ◽

Ralph D. Nyland ◽

Christel C. Kern ◽

Laura S. Kenefic

Keyword(s):

Temporal Stability ◽

Diameter Distribution ◽

Weibull Function ◽

Northern Hardwood ◽

Selection Cutting ◽

Single Tree ◽

Second Growth ◽

Single Tree Selection ◽

Selection cutting is defined as a tool for uneven-aged silviculture. Dependence on diameter distribution by forestry practitioners for identifying stand conditions has led to misuse of selection-like cuttings in even-aged northern hardwood stands. Our study used several long-term data sets to investigate the temporal stability in numbers of trees per diameter class in uneven-aged northern hardwood stands treated with single-tree selection and in 45-year-old second-growth stands treated with selection-like cuttings. We analyzed data from New York, Michigan, and Wisconsin to determine changes through time in number of trees across 2.5 cm diameter classes, shifts in the shape and scale of the three-parameter Weibull function used to describe the diameter distributions, and dynamics of associated stand attributes. Findings showed that single-tree selection cutting created and sustained stable diameter distributions and uniformity of conditions through consecutive entries in uneven-aged stands. By contrast, these characteristics varied through time in the second-growth stands that had been treated with selection-like cuttings. Analysis also showed that the Weibull shape and scale parameters for stands under selection system migrated towards those of the recommended target diameter distribution in the uneven-aged stands. These parameters diverged from the target with repeated use of selection-like cuttings in the second-growth even-aged stands.

Stand-Level Growth and Yield Component Models for Red Oak–Sweetgum Forests on Mid-South Minor Stream Bottoms

10.1093/sjaf/34.4.161 ◽

2010 ◽

Vol 34 (4) ◽

pp. 161-175 ◽

Cited By ~ 3

Author(s):

Emily B. Schultz ◽

J. Clint Iles ◽

Thomas G. Matney ◽

Andrew W. Ezell ◽

James S. Meadows ◽

...

Keyword(s):

Yield Component ◽

Growth And Yield ◽

Coefficient Of Determination ◽

Diameter Distribution ◽

Red Oak ◽

Average Height ◽

Distribution Models ◽

Yield Model ◽

Validity Assessment ◽

Mean Diameter

Abstract Greater emphasis is being placed on Southern bottomland hardwood management, but relatively few growth and yield prediction systems exist that are based on sufficient measurements. We present the aggregate stand-level expected yield and structural component equations for a red oak (Quercus sectionLobatae)-sweetgum (Liquidambar styraciflua L.) growth and yield model. Measurements from 638 stand-level observations on 258 distinct permanent growth and yield plots collected in 1981, 1988, 1994, and 2006 in minor stream bottoms in Mississippi and Alabama provided data for model development. Equations for average height of dominant and codominant red oaks, trees/ac, arithmetic mean diameter, quadratic mean diameter, and volume were selected on the basis of significance of independent variables, coefficient of determination, index of fit, and biological validity assessment. These models produce expected average yields for combined species or species groups in naturally developing stands and provide an average baseline for individuals managing their lands for the red oak–sweetgum complex. Models will be integrated with log grade volume and diameter distribution models that are in concurrent development to produce a growth and yield system capable of comparing management alternatives on a financial basis.