VTechWorks Repository :: Browsing by Author "Reynolds, Marion R. Jr."

Browsing by Author "Reynolds, Marion R. Jr."

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Adaptive Fourier Analysis For Unequally-Spaced Time Series Data
Liang, Hong (Virginia Tech, 2002-04-16)
Fourier analysis, Walsh-Fourier analysis, and wavelet analysis have often been used in time series analysis. Fourier analysis can be used to detect periodic components that have sinusoidal shape; however, it might be misleading when the periodic components are not sinusoidal. Walsh-Fourier analysis is suitable for revealing the rectangular trends of time series. The flaw of the Walsh-Fourier analysis is that Walsh functions are not periodic. The resulting Walsh-Fourier analysis is more difficult to interpret than classical Fourier analysis. Wavelet analysis is very useful in analyzing and describing time series with gradual frequency changes. Wavelet analysis also has a shortcoming by giving no exact meaning to the concept of frequency because wavelets are not periodic functions. In addition, all three analysis methods above require equally-spaced time series observations. In this dissertation, by using a sequence of periodic step functions, a new analysis method, adaptive Fourier analysis, and its theory are developed. These can be applied to time series data where patterns may take general periodic shapes that include sinusoids as special cases. Most importantly, the resulting adaptive Fourier analysis does not require equally-spaced time series observations.
Adaptive Threshold Method for Monitoring Rates in Public Health Surveillance
Gan, Linmin (Virginia Tech, 2010-04-30)
We examine some of the methodologies implemented by the Centers for Disease Control and Prevention's (CDC) BioSense program. The program uses data from hospitals and public health departments to detect outbreaks using the Early Aberration Reporting System (EARS). The EARS method W2 allows one to monitor syndrome counts (W2count) from each source and the proportion of counts of a particular syndrome relative to the total number of visits (W2rate). We investigate the performance of the W2r method designed using an empiric recurrence interval (RI) in this dissertation research. An adaptive threshold monitoring method is introduced based on fitting sample data to the underlying distributions, then converting the current value to a Z-score through a p-value. We compare the upper thresholds on the Z-scores required to obtain given values of the recurrence interval for different sets of parameter values. We then simulate one-week outbreaks in our data and calculate the proportion of times these methods correctly signal an outbreak using Shewhart and exponentially weighted moving average (EWMA) charts. Our results indicate the adaptive threshold method gives more consistent statistical performance across different parameter sets and amounts of baseline historical data used for computing the statistics. For the power analysis, the EWMA chart is superior to its Shewhart counterpart in nearly all cases, and the adaptive threshold method tends to outperform the W2 rate method. Two modified W2r methods proposed in the dissertation also tend to outperform the W2r method in terms of the RI threshold functions and in the power analysis.
Aids for Unit Planning on the Appalachian National Forests
Burkhart, Harold E.; Leuschner, William A.; Stuck, R. Dean; Porter, John R.; Reynolds, Marion R. Jr. (Virginia Tech. Division of Forestry and Wildlife Resources, 1976)
This report summarizes the results of studies conducted in response to a cooperative agreement between the Southern Region, U.S. Forest Service and the Department of Forestry and Forest Products, Virginia Polytechnic Institute and State University. The objective of the agreement was to improve National Forest management planning techniques. The agreement covered the period July 1, 1973 to June 30, 1975. Literature citations are given for those who desire additional detail.
The analysis of longitudinal ordinal data
Schabenberger, Oliver (Virginia Tech, 1995)
Longitudinal data, in which a series of observations is collected on a typically large number of experimental units is one of the most frequent and important sources of quantitative information in forestry. The dependencies among repeated observations for an experimental unit must be accounted for in order to validate statistical estimation and inference in modeling efforts. The recent advances in statistical theory for correlated data created a body of theory which will become of increasing importance as analysts realize the limitations of traditional methods that ignore these dependencies. Longitudinal data fosters research questions that focus on the individual experimental unit rather than the population as in classical cross-sectional data analysis. Mixed model techniques have emerged as powerful tools to address research problems of this kind and are treated extensively in this dissertation. Over the last years interest in modeling quantal responses that take on only a countable, discrete number of possible values has also increased throughout the discipline. The theory of generalized linear models provides the groundwork to embody quantal response models into the toolbox of applied analysts. The focus of this dissertation is to combine modern analytical tools for longitudinal data with regression methods for quantal responses. Special emphasis is placed on ordinal and binary data because of their prevalence in ecological, biological, and environmental statistics. The first chapters review the literature and introduce necessary theory. The second part of this dissertation consists of a case study in which binary and ordinal fusiform rust response on loblolly and slash pine is modeled in a longitudinal data base provided by the East Texas Pine Plantation Research Project.
Analysis of multispecies microcosm experiments
Mercante, Donald Eugene (Virginia Tech, 1990)
Traditionally, single species toxicity tests have been the primary tool for assessment of hazard of toxic substances in aquatic ecosystems. These tests are inadequate for accurately reflecting the impact of toxicants on the community structure inherent in ecosystems. Multispecies microcosm experiments are gaining widespread acceptance as an important vehicle in understanding the nature and magnitude of effects for more complex systems. Microcosm experiments are complex and costly to conduct. Consequently, sample sizes are typically small (8-20 replicates). In addition, these experiments are difficult to analyze due to their multivariate and repeated measures nature. Working under the constraint of small sample sizes, we develop inferential as well as diagnostic methods that detect and measure community changes as a result of an intervention (i.e. toxicant), and assess the importance of individual species. A multi-factorial simulation analysis is used to compare several methods. The Multi-Response Permutation Procedure (MRPP) and a regression method incorporating a correlation structure are found to be the most powerful procedures for detecting treatment differences. The MRPP is particularly suited to experiments with replication and when the response variable may not be normally distributed. The regression model for dissimilarity data has the advantage of enabling direct estimation of many parameters not possible with the MRPP as well as the magnitude of treatment effects. A stepwise dependent variable selection algorithm with a selection criterion based on a conditional p-value argument is proposed and applied to a real data set. It is seen to have advantages over other methods for assessing species importance.
Applications of Control Charts in Medicine and Epidemiology
Sego, Landon Hugh (Virginia Tech, 2006-04-05)
We consider two applications of control charts in health care. The first involves the comparison of four methods designed to detect an increase in the incidence rate of a rare health event, such as a congenital malformation. A number of methods have been proposed: among these are the Sets method, two modifications of the Sets method, and the CUSUM method based on the Poisson distribution. Many of the previously published comparisons of these methods used unrealistic assumptions or ignored implicit assumptions which led to misleading conclusions. We consider the situation where data are observed as a sequence of Bernoulli trials and propose the Bernoulli CUSUM chart as a desirable method for the surveillance of rare health events. We compare the steady-state average run length performance of the Sets methods and its modifications to the Bernoulli CUSUM chart under a wide variety of circumstances. Except in a very few instances we find that the Bernoulli CUSUM chart performs better than the Sets method and its modifications for the extensive number of cases considered. The second application area involves monitoring clinical outcomes, which requires accounting for the fact that each patient has a different risk of death prior to undergoing a health care procedure. We propose a risk-adjusted survival time CUSUM chart (RAST CUSUM) for monitoring clinical outcomes where the primary endpoint is a continuous, time-to-event variable that is right censored. Risk adjustment is accomplished using accelerated failure time regression models. We compare the average run length performance of the RAST CUSUM chart to the risk-adjusted Bernoulli CUSUM chart, using data from cardiac surgeries to motivate the details of the comparison. The comparisons show that the RAST CUSUM chart is more efficient at detecting deterioration in the quality of a clinical procedure than the risk-adjusted Bernoulli CUSUM chart, especially when the fraction of censored observations is not too high. We address details regarding the implementation of a prospective monitoring scheme using the RAST CUSUM chart.
Applications of the Chinese Remainder Theorem to the construction and analysis of confounding systems and randomized fractional replicates for mixed factorial experiments
Huang, Won-Chin Liao (Virginia Polytechnic Institute and State University, 1989)
A well-known theorem in "Number Theory", the Chinese Remainder Theorem, was first utilized by Paul K. Lin in constructing confounding systems for mixed factorial experiments. This study extends the use of the theorem to cover cases when more than one component from some of the symmetrical factorials are confounded, and to include cases where the number of levels of factors are not all relative prime. The second part of this study concerns the randomized fractional replicates, a procedure which selects confounded subsets with pre-assigned probabilities. This procedure provides full information on a specific set of parameters of interest while making no assumption of zero nuisance parameters. Estimation procedures in general symmetrical as well as asymmetrical factorial systems are studied under a ”fully orthogonalized" model. The type-g estimator, investigated under the generalized inverse operator, and the class of linear estimators of parameters of interest and their variance-covariance matrices are given. The unbiasedness of these estimators can be obtained only under the condition that each subset of treatment combinations is selected with equal probability. This work is concluded with simulation studies to compare the classical and the randomization procedures. The results indicate that when information about the nuisance parameters is not available, randomization procedure guards against a bad choice of design.
Change Detection and Analysis of Data with Heterogeneous Structures
Chu, Shuyu (Virginia Tech, 2017-07-28)
Heterogeneous data with different characteristics are ubiquitous in the modern digital world. For example, the observations collected from a process may change on its mean or variance. In numerous applications, data are often of mixed types including both discrete and continuous variables. Heterogeneity also commonly arises in data when underlying models vary across different segments. Besides, the underlying pattern of data may change in different dimensions, such as in time and space. The diversity of heterogeneous data structures makes statistical modeling and analysis challenging. Detection of change-points in heterogeneous data has attracted great attention from a variety of application areas, such as quality control in manufacturing, protest event detection in social science, purchase likelihood prediction in business analytics, and organ state change in the biomedical engineering. However, due to the extraordinary diversity of the heterogeneous data structures and complexity of the underlying dynamic patterns, the change-detection and analysis of such data is quite challenging. This dissertation aims to develop novel statistical modeling methodologies to analyze four types of heterogeneous data and to find change-points efficiently. The proposed approaches have been applied to solve real-world problems and can be potentially applied to a broad range of areas.
A comparison of alternative methods to the shewhart-type control chart
Hall, Deborah A. (Virginia Tech, 1989-01-05)
A control chart that simultaneously tracks the mean and variance of a normally distributed variable with no compensation effect is defined in this work. This joint control chart is compared to five other charts: an Î§ chart, an s² chart, a Reynolds and Ghosh chart, a Repko process capability plot, and a t-statistic chart. The criterion for comparison is the probability of a Type II sampling error. Several out-of-control cases are examined. In the case of Repko, an equation is defined to compute the Type II error probability. The results indicate that the Reynolds and Ghosh statistic is powerful for cases when the variance shifts out of control. The Î§ chart is powerful when the mean shifts with moderate changes in the variance. The joint chart is powerful for moderate changes in the mean and variance.
Construction and Analysis of Linear Trend-Free Factorial Designs Under a General Cost Structure
Kim, Kiho (Virginia Tech, 1997-07-28)
When experimental units exhibit a smooth trend over time or in space, random allocation of treatments may no longer be appropriate. Instead, systematic run orders may have to be used to reduce or eliminate the effects of such a trend. The resulting designs are referred to as trend-free designs. We consider here, in particular, linear trend-free designs for factorial treatment structures such that estimates of main effects and two-factor interactions are trend-free. In addition to trend-freeness we incorporate a general cost structure and propose methods of constructing optimal or near-optimal full or fractional factorial designs. Building upon the generalized foldover scheme (GFS) introduced by Coster and Cheng (1988) we develop a procedure of selection of foldover vectors (SFV) which is a construction method for an appropriate generator matrix. The final optimal or near-optimal design can then be developed from this generator matrix. To achieve a reduction in the amount of work, i.e., a reduction of the large number of possible generator matrices, and to make this whole process easier to use by a practitioner, we introduce the systematic selection of foldover vectors (SSFV). This method does not always produce optimal designs but in all cases practical compromise designs. The cost structure for factorial designs can be modeled according to the number of level changes for the various factors. In general, if cost needs to be kept to a minimum, factor level changes will have to be kept at a minimum. This introduces a covariance structure for the observations from such an experiment. We consider the consequences of this covariance structure with respect to the analysis of trend-free factorial designs. We formulate an appropriate underlying mixed linear model and propose an AIC-based method using simulation studies, which leads to a useful practical linear model as compared to the theoretical model, because the theoretical model is not always feasible. Overall, we show that estimation of main effects and two-factor interactions, trend-freeness, and minimum cost cannot always be achieved simultaneously. As a consequence, compromise designs have to be considered, which satisfy requirements as much as possible and are practical at the same time. The proposed methods achieve this aim.
Construction and properties of Box-Behnken designs
Jo, Jinnam (Virginia Tech, 1992-10-05)
Box-Behnken designs are used to estimate parameters in a second-order response surface model (Box and Behnken, 1960). These designs are formed by combining ideas from incomplete block designs (BIBD or PBIBD) and factorial experiments, specifically 2^k full or 2^k-1 fractional factorials. In this dissertation, a more general mathematical formulation of the Box-Behnken method is provided, a general expression for the coefficient matrix in the least squares analysis for estimating the parameters in the second order model is derived, and the properties of Box-Behnken designs with respect to the estimability of all parameters in a second-order model are investigated when 2^kfull factorials are used. The results show that for all pure quadratic coefficients to be estimable, the PBIB(m) design has to be chosen such that its incidence matrix is of full rank, and for all mixed quadratic coefficients to be estimable the PBIB(m) design has to be chosen such that the parameters λ₁, λ₂, ...,λ_m are all greater than zero. In order to reduce the number of experimental points the use of 2^k-1 fractional factorials instead of 2^k full factorials is being considered. Of particular interest and importance are separate considerations of fractions of resolutions III, IV, and V. The construction of Box-Behnken designs using such fractions is described and the properties of the designs concerning estimability of regression coefficients are investigated. Using designs obtained from resolution V factorials have the same properties as those using full factorials. Resolutions III and IV designs may lead to non-estimability of certain coefficients and to correlated estimators. The final topic is concerned with Box-Behnken designs in which treatments are applied to experimental units sequentially in time or space and in which there may exist a linear trend effect. For this situation, one wants to find appropriate run orders for obtaining a linear trend-free Box-Behnken design to remove a linear trend effect so that a simple technique, analysis of variance, instead of a more complicated technique, analysis of covariance, to remove a linear trend effect can be used. Construction methods for linear trend-free Box-Behnken designs are introduced for different values of block size (for the underlying PBIB design) k. For k= 2 or 3, it may not always be possible to find linear trend-free Box-Behnken designs. However, for k ≥ 4 linear trend-free Box-Behnken designs can always be constructed.
Contributions to experimental design for quality control
Kim, Sang Ik (Virginia Polytechnic Institute and State University, 1988)
A parameter design introduced by Taguchi provides a new quality control method which can reduce cost-effectively the product variation due to various uncontrollable noise factors such as product deterioration, manufacturing imperfections, and environmental factors under which a product is actually used. This experimental design technique identifies the optimal setting of the control factors which is least sensitive to the noise factors. Taguchi’s method utilizes orthogonal arrays which allow the investigation of main effects only, under the assumption that interaction effects are negligible. In this paper new techniques are developed to investigate two-factor interactions for 2^t and 3^t factorial parameter designs. The major objective is to be able to identify influential two-factor interactions and take those into account in properly assessing the optimal setting of the control factors. For 2^t factorial parameter designs, we develop some new designs for the control factors by using a partially balanced array. These designs are characterized by a small number of runs and some balancedness property of the variance-covariance matrix of the estimates of main effects and two-factor interactions. Methods of analyzing the new designs are also developed. For 3^t factorial parameter designs, a detection procedure consisting of two stages is developed by using a sequential method in order to reduce the number of runs needed to detect influential two-factor interactions. In this paper, an extension of the parameter design to several quality characteristics is also developed by devising suitable statistics to be analyzed, depending on whether a proper loss function can be specified or not.
Control charts based on residuals for monitoring processes with correlated observations
Lu, Chao-Wen (Virginia Tech, 1993-08-05)
In statistical process control, it is usually assumed that observations on the process output at different times are lID. However, for many processes the observations are correlated and control charts for monitoring these processes have recently received much attention. For monitoring the process level, this study evaluates the properties of control charts, such as the EWMA chart and the CUSUM chart, based on the residuals from the forecast values of an ARMA model. It is assumed that the process mean is a ftrst order autoregressive (AR(l)) model and the observations are the mean plus a random error. Properties of these charts are evaluated using a Markov chain approach or an integral equation approach. The performance of control charts based on the residuals is compared to the performance of control charts based on the original observations. A combined chart using forecasts and residuals as the control statistics as well as a combined chart using the EWMA of observations and the EWMA of residuals as the control statistics are also studied by simulation. It is found that no universally "good" chart exists among all the charts investigated in this study. In addition, for monitoring the process variance, two kinds of EWMA chart based on residuals are studied and compared.
Control Charts with Missing Observations
Wilson, Sara R. (Virginia Tech, 2009-04-03)
Traditional control charts for process monitoring are based on taking samples from the process at regular time intervals. However, it is often possible in practice for observations, and even entire samples, to be missing. This dissertation investigates missing observations in Exponentially Weighted Moving Average (EWMA) and Multivariate EWMA (MEWMA) control charts. The standardized sample mean is used since this adjusts the sample mean for the fact that part of the sample may be missing. It also allows for constant control limits even though the sample size varies randomly. When complete samples are missing, the weights between samples should also be adjusted. In the univariate case, three approaches for adjusting the weights of the EWMA control statistic are investigated: (1) ignoring missing samples; (2) adding the weights from previous consecutive missing samples to the current sample; and (3) increasing the weights of non-missing samples in proportion, so that the weights sum to one. Integral equation and Markov chain methods are developed to find and compare the statistical properties of these charts. The EI chart, which adjusts the weights by ignoring the missing samples, has the best overall performance. The multivariate case in which information on some of the variables is missing is also examined using MEWMA charts. Two methods for adjusting the weights of the MEWMA control statistic are investigated and compared using simulation: (1) ignoring all the data at a sampling point if the data for at least one variable is missing; and (2) using the previous EWMA value for any variable for which all the data are missing. Both of these methods are examined when the in-control covariance matrix is adjusted at each sampling point to account for missing observations, and when it is not adjusted. The MS control chart, which uses the previous value of the EWMA statistic for a variable if all of the data for that variable is missing at a sampling point, provides the best overall performance. The in-control covariance matrix needs to be adjusted at each sampling point, unless the variables are independent or only weakly correlated.
Design and analysis for a two level factorial experiment in the presence of dispersion effects
Mays, Darcy P. (Virginia Tech, 1993)
Standard response surface methodology experimental designs for estimating location models involve the assumption of homogeneous variance throughout the design region. However, with heterogeneity of variance these standard designs are not optimal. Using the D and Q-optimality criteria, this dissertation proposes a two-stage experimental design procedure that gives more efficient designs than the standard designs when heterogeneous variance exists. Several multiple variable location models, with and without interactions, are considered. For each the first stage estimates the heterogeneous variance structure, while the second stage then augments the first stage to produce a D or Q-optimal design for fitting the location model under the estimated variance structure. However, there is a potential instability of the variance estimates in the first stage that can lower the efficiency of the two-stage procedure. This problem can be addressed and the efficiency of the procedure enhanced if certain mild assumptions concerning the variance structure are made and formulated as a prior distribution to produce a Bayes estimator. With homogeneous variance, designs are analyzed using ordinary least squares. However, with heterogeneous variance the correct analysis is to use weighted least squares. This dissertation also examines the effects that analysis by weighted least squares can have and compares this procedure to the proposed two-stage procedure.
The Design of GLR Control Charts for Process Monitoring
Xu, Liaosa (Virginia Tech, 2013-02-27)
Generalized likelihood ratio (GLR) control charts are investigated for two types of statistical process monitoring (SPC) problems. The first part of this dissertation considers the problem of monitoring a normally distributed process variable when a special cause may produce a time varying linear drift in the mean. The design and application of a GLR control chart for drift detection is investigated. The GLR drift chart does not require specification of any tuning parameters by the practitioner, and has the advantage that, at the time of the signal, estimates of both the change point and the drift rate are immediately available. An equation is provided to accurately approximate the control limit. The performance of the GLR drift chart is compared to other control charts such as a standard CUSUM chart and a CUSCORE chart designed for drift detection. We also compare the GLR chart designed for drift detection to the GLR chart designed for sustained shift detection since both of them require only a control limit to be specified. In terms of the expected time for detection and in terms of the bias and mean squared error of the change-point estimators, the GLR drift chart has better performance for a wide range of drift rates relative to the GLR shift chart when the out-of-control process is truly a linear drift. The second part of the dissertation considers the problem of monitoring a linear functional relationship between a response variable and one or more explanatory variables (a linear profile). The design and application of GLR control charts for this problem are investigated. The likelihood ratio test of the GLR chart is generalized over the regression coefficients, the variance of the error term, and the possible change-point. The performance of the GLR chart is compared to various existing control charts. We show that the overall performance of the GLR chart is much better than other options in detecting a wide range of shift sizes. The existing control charts designed for certain shifts that may be of particular interest have several chart parameters that need to be specified by the user, which makes the design of such control charts more difficult. The GLR chart is very simple to design, as it is invariant to the choice of design matrix and the values of in-control parameters. Therefore there is only one design parameter (the control limit) that needs to be specified. Especially, the GLR chart can be constructed based on the sample size of n=1 at each sampling point, whereas other charts cannot be applied. Another advantage of the GLR chart is its built-in diagnostic aids that provide estimates of both the change-point and the values of linear profile parameters.
Diagnostics after a Signal from Control Charts in a Normal Process
Lou, Jianying (Virginia Tech, 2008-09-04)
Control charts are fundamental SPC tools for process monitoring. When a control chart or combination of charts signals, knowing the change point, which distributional parameter changed, and/or the change size helps to identify the cause of the change, remove it from the process or adjust the process back in control correctly and immediately. In this study, we proposed using maximum likelihood (ML) estimation of the current process parameters and their ML confidence intervals after a signal to identify and estimate the changed parameters. The performance of this ML diagnostic procedure is evaluated for several different charts or chart combinations for the cases of sample sizes and , and compared to the traditional approaches to diagnostics. None of the ML and the traditional estimators performs well for all patterns of shifts, but the ML estimator has the best overall performance. The ML confidence interval diagnostics are overall better at determining which parameter has shifted than the traditional diagnostics based on which chart signals. The performance of the generalized likelihood ratio (GLR) chart in shift detection and in ML diagnostics is comparable to the best EWMA chart combination. With the application of the ML diagnostics naturally following a GLR chart compared to the traditional control charts, the studies of a GLR chart during process monitoring can be further deepened in the future.
Dimensionally Compatible System of Equations for Tree and Stand Volume, Basal Area, and Growth
Sharma, Mahadev (Virginia Tech, 1999-10-28)
A dimensionally compatible system of equations for stand basal area, volume, and basal area and volume growth was derived using dimensional analysis. These equations are analytically and numerically consistent with dimensionally compatible individual tree volume and taper equations and share parameters with them. Parameters for the system can be estimated by fitting individual tree taper and volume equations or by fitting stand level basal area and volume equations. In either case the parameters are nearly identical. Therefore, parameters for the system can be estimated at the tree or stand level without changing the results. Data from a thinning study in loblolly pine (Pinus taeda L.) plantations established on cutover site-prepared lands were used to estimate the parameters. However, the developed system of equations is general and can be applied to other tree species in other locales.
Discrete Small Sample Asymptotics
Kathman, Steven Jay Jr. (Virginia Tech, 1999-12-07)
Random variables defined on the natural numbers may often be approximated by Poisson variables. Just as normal approximations may be improved by saddlepoint methods, Poisson approximations may be substantially improved by tilting, expansion, and other related methods. This work will develop and examine the use of these methods, as well as present examples where such methods may be needed.
Dominance/Suppression Competitive Relationships in Loblolly Pine (Pinus taeda L.) Plantations
Dyer, Michael E. (Virginia Tech, 1997-11-13)
Data from three long-term field studies with loblolly pine (Pinus taeda L.) plantations were used to examine inequality (Gini coefficient) trends in diameter and the relationship between diameter relative growth rate (r) and initial size. Analysis with two spacing studies shows inequality increases with increasing density. For a given initial density, inequality initially decreases and then begins to increase as trees compete for resources. The slope of the linear relationship between r and relative size also increases with increasing density. The slope is initially negative and switches to positive as competition intensifies. The switch in the slope of the r/size relationship occurs when the crown projection area exceeds 1.05 or when the crown ratio falls below 0.75. These results are consistent with the resource pre-emptive or dominance/suppression theory of intra-specific competition. The r/size trends are not evident when calculations are based on class means as opposed to individual trees. The slope of the r/size relationship is a function of stand height, density, and to a lesser extent, site quality. Density reduction through mid-rotation thinning tends to decrease the slope coefficient. The r/size trends are used to develop a disaggregation model to distribute stand-level basal area growth over an initial tree list. This approach compares well with two other disaggregation models but tends to over predict growth on the largest trees.

Browsing by Author "Reynolds, Marion R. Jr."

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