13 research outputs found
Computing reliable error estimations for integral equations with discontinuities
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On the Challenge of Fitting Tree Size Distributions in Ecology
<div><p>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.</p> </div
Effect of errors on <i>Akaike weights</i> for the correct determination of the underlying distribution.
<p>In each row virtual data sets of sample size = 500 which originate from the three truncated distributions (power-law, negative exponential and Weibull distribution) are evaluated. Weights are calculated supposing these distributions (power-law, negative exponential and Weibull distribution) with (a)–(c) <i>multinomial MLE</i> and (d)–(f) <i>Gaussian MLE</i>. The highest <i>Akaike weight</i> determines the best fit of a frequency distribution to the data. (a)–(c) Effect of binning of virtual data sets with used bin width (x-axis in cm) on <i>Akaike weights</i>. (d)–(f) Effect of random measurement errors added to the virtual data sets on <i>Akaike weights</i>, whereby errors are Gaussian distributed with mean cm and assumed standard deviation (x-axis in cm). Solid lines represent the mean of <i>Akaike weights</i> and shaded areas show the standard deviation (of (a)–(c) 1000 values and (d)–(f) 250 values).</p
Analyses of virtual data including different levels of measurement errors.
<p>We evaluate 1,000 virtual data sets of sample size = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. An error value generated from a Gaussian distribution with mean cm and an assumed standard deviation (x-axis in cm) is added to each virtual data point before applying <i>standard MLE</i>. (a) Effect of random measurement errors on parameter estimates of the three investigated distributions. (b)–(d) Effect of random measurement errors on <i>Akaike weights</i> supposing three distributions (power-law, negative exponential and Weibull distribution) for (b) power-law distributed virtual data, (c) negative exponentially distributed virtual data and (d) Weibull distributed virtual data. The highest <i>Akaike weight</i> determines the best fit of a frequency distribution to the data. Solid lines represent the mean values and shaded areas show the standard deviation (of 1,000 calculated values).</p
Scheme of the evaluation procedure of virtual data sets.
<p>Scheme of the evaluation procedure of virtual data sets.</p
Effects of binning and random measurement errors on parameter estimation using different MLE methods.
<p>(a) MLE including binning (<i>multinomial MLE</i>) and (b) MLE accounting for measurement errors (<i>Gaussian MLE</i>). We evaluate virtual data sets of sample size = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Solid lines represent the mean estimates and shaded areas show the standard deviation (of (a) 1000 values and (b) 250 values). (a) Effect of binning on parameter estimates. Virtual data are classified into classes of certain bin width (x-axis in cm). (b) Effect of random measurement errors on parameter estimates. An error value generated from a Gaussian distribution with mean cm and an assumed standard deviation (x-axis in cm) is added to each virtual data value.</p
Presentation of the three assumed truncated frequency distributions used in our investigations.
<p>Presentation of the three assumed truncated frequency distributions used in our investigations.</p
Analyses of binned virtual data using different bin widths.
<p>We evaluate 1,000 virtual data sets of sample size = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Virtual data are classified into classes of certain bin width (x-axis in cm) before applying <i>standard MLE</i>. (a) Effect of binning on parameter estimates of the three investigated distributions. (b)–(d) Effect of binning on <i>Akaike weights</i> supposing three distributions (power-law, negative exponential and Weibull distribution) for (b) power-law distributed virtual data, (c) negative exponentially distributed virtual data and (d) Weibull distributed virtual data. The highest <i>Akaike weight</i> determines the best fit of a frequency distribution to the data. Solid lines represent the mean values and shaded areas show the standard deviation (of 1,000 calculated values).</p
Outline of tree size measurements in forests.
<p>(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.</p