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On the challenge of fitting tree size distributions in ecology.

Taubert F, Hartig F, Dobner HJ, Huth A - PLoS ONE (2013)

Bottom Line: We test whether three typical frequency distributions, namely the power-law, negative exponential and Weibull distribution can be precisely identified, and how parameter estimates are biased when observations are additionally either binned or contain measurement error.We show that uncorrected MLE already loses the ability to discern functional form and parameters at relatively small levels of uncertainties.We conclude that it is important to reduce binning of observations, if possible, and to quantify observation accuracy in empirical studies for fitting strongly skewed size distributions.

View Article: PubMed Central - PubMed

Affiliation: Department of Ecological Modelling, Helmholtz Centre for Environmental Research, Leipzig, Saxony, Germany. franziska.taubert@ufz.de

ABSTRACT
Patterns that resemble strongly skewed size distributions are frequently observed in ecology. A typical example represents tree size distributions of stem diameters. Empirical tests of ecological theories predicting their parameters have been conducted, but the results are difficult to interpret because the statistical methods that are applied to fit such decaying size distributions vary. In addition, binning of field data as well as measurement errors might potentially bias parameter estimates. Here, we compare three different methods for parameter estimation--the common maximum likelihood estimation (MLE) and two modified types of MLE correcting for binning of observations or random measurement errors. We test whether three typical frequency distributions, namely the power-law, negative exponential and Weibull distribution can be precisely identified, and how parameter estimates are biased when observations are additionally either binned or contain measurement error. We show that uncorrected MLE already loses the ability to discern functional form and parameters at relatively small levels of uncertainties. The modified MLE methods that consider such uncertainties (either binning or measurement error) are comparatively much more robust. We conclude that it is important to reduce binning of observations, if possible, and to quantify observation accuracy in empirical studies for fitting strongly skewed size distributions. In general, modified MLE methods that correct binning or measurement errors can be applied to ensure reliable results.

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Related in: MedlinePlus

Outline of tree size measurements in forests.(a) In general, the stem diameter of a tree is measured at breast height (1.3 m). Each tree in the area of interest is tagged, recorded and measured. Using a specific class width (here 1 mm) each measured stem diameter is classified in its corresponding class. This results in a number of stems per class and is summarized in a stem size distribution. (b)-(c) Change of the functional form of the stem size distribution of stem diameters under binning or including measurement errors. (b) Change of the stem size distribution using binning of measured stem diameters with bin widths of 1 cm and 5 cm. (c) Change of the stem size distribution adding random measurement errors of standard deviations  cm and  cm to the recorded stem diameters.
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pone-0058036-g001: Outline of tree size measurements in forests.(a) In general, the stem diameter of a tree is measured at breast height (1.3 m). Each tree in the area of interest is tagged, recorded and measured. Using a specific class width (here 1 mm) each measured stem diameter is classified in its corresponding class. This results in a number of stems per class and is summarized in a stem size distribution. (b)-(c) Change of the functional form of the stem size distribution of stem diameters under binning or including measurement errors. (b) Change of the stem size distribution using binning of measured stem diameters with bin widths of 1 cm and 5 cm. (c) Change of the stem size distribution adding random measurement errors of standard deviations cm and cm to the recorded stem diameters.

Mentions: When testing theoretical predictions, we have to consider that field data contain uncertainties. For example, in forest science field data on tree size are typically analysed by constructing a stem size frequency distribution which summarizes the number of trees in different measured stem diameter classes (Fig. 1a). Such a classification of the measured data into diameter classes of a certain width is also called binning of data. Thus, results of analyses depend on the class width, whereby in forestry widths of 5 cm or 10 cm are often used. Besides the influence of binning, uncertainties in field data can also arise from irregularities or errors that occur during the measurement process [20]. Such measurement errors typically lead to a symmetric variation around the true value. Both binning and measurement errors change the functional shape of the analysed frequency distribution (Fig. 1b, 1c).


On the challenge of fitting tree size distributions in ecology.

Taubert F, Hartig F, Dobner HJ, Huth A - PLoS ONE (2013)

Outline of tree size measurements in forests.(a) In general, the stem diameter of a tree is measured at breast height (1.3 m). Each tree in the area of interest is tagged, recorded and measured. Using a specific class width (here 1 mm) each measured stem diameter is classified in its corresponding class. This results in a number of stems per class and is summarized in a stem size distribution. (b)-(c) Change of the functional form of the stem size distribution of stem diameters under binning or including measurement errors. (b) Change of the stem size distribution using binning of measured stem diameters with bin widths of 1 cm and 5 cm. (c) Change of the stem size distribution adding random measurement errors of standard deviations  cm and  cm to the recorded stem diameters.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3585190&req=5

pone-0058036-g001: Outline of tree size measurements in forests.(a) In general, the stem diameter of a tree is measured at breast height (1.3 m). Each tree in the area of interest is tagged, recorded and measured. Using a specific class width (here 1 mm) each measured stem diameter is classified in its corresponding class. This results in a number of stems per class and is summarized in a stem size distribution. (b)-(c) Change of the functional form of the stem size distribution of stem diameters under binning or including measurement errors. (b) Change of the stem size distribution using binning of measured stem diameters with bin widths of 1 cm and 5 cm. (c) Change of the stem size distribution adding random measurement errors of standard deviations cm and cm to the recorded stem diameters.
Mentions: When testing theoretical predictions, we have to consider that field data contain uncertainties. For example, in forest science field data on tree size are typically analysed by constructing a stem size frequency distribution which summarizes the number of trees in different measured stem diameter classes (Fig. 1a). Such a classification of the measured data into diameter classes of a certain width is also called binning of data. Thus, results of analyses depend on the class width, whereby in forestry widths of 5 cm or 10 cm are often used. Besides the influence of binning, uncertainties in field data can also arise from irregularities or errors that occur during the measurement process [20]. Such measurement errors typically lead to a symmetric variation around the true value. Both binning and measurement errors change the functional shape of the analysed frequency distribution (Fig. 1b, 1c).

Bottom Line: We test whether three typical frequency distributions, namely the power-law, negative exponential and Weibull distribution can be precisely identified, and how parameter estimates are biased when observations are additionally either binned or contain measurement error.We show that uncorrected MLE already loses the ability to discern functional form and parameters at relatively small levels of uncertainties.We conclude that it is important to reduce binning of observations, if possible, and to quantify observation accuracy in empirical studies for fitting strongly skewed size distributions.

View Article: PubMed Central - PubMed

Affiliation: Department of Ecological Modelling, Helmholtz Centre for Environmental Research, Leipzig, Saxony, Germany. franziska.taubert@ufz.de

ABSTRACT
Patterns that resemble strongly skewed size distributions are frequently observed in ecology. A typical example represents tree size distributions of stem diameters. Empirical tests of ecological theories predicting their parameters have been conducted, but the results are difficult to interpret because the statistical methods that are applied to fit such decaying size distributions vary. In addition, binning of field data as well as measurement errors might potentially bias parameter estimates. Here, we compare three different methods for parameter estimation--the common maximum likelihood estimation (MLE) and two modified types of MLE correcting for binning of observations or random measurement errors. We test whether three typical frequency distributions, namely the power-law, negative exponential and Weibull distribution can be precisely identified, and how parameter estimates are biased when observations are additionally either binned or contain measurement error. We show that uncorrected MLE already loses the ability to discern functional form and parameters at relatively small levels of uncertainties. The modified MLE methods that consider such uncertainties (either binning or measurement error) are comparatively much more robust. We conclude that it is important to reduce binning of observations, if possible, and to quantify observation accuracy in empirical studies for fitting strongly skewed size distributions. In general, modified MLE methods that correct binning or measurement errors can be applied to ensure reliable results.

Show MeSH
Related in: MedlinePlus