Comparison of beta and weibull functions for modelling basal area diameter distribution in stands of <i>pinus sylvestris</i> and <i>picea abies</i>

Maltamo, Matti; Puumalainen, Janna; Päivinen, Risto

doi:10.1080/02827589509382895

Cited by 113 publications

(135 citation statements)

References 8 publications

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“…The two-parameter Weibull function has been reported to be the most simple and accurate for modelling diameter distributions in birch stands (Betula alba L.) in northwestern Spain (Gorgoso, 2003), in accordance with Maltamo et al (1995) for Pinus sylvestris and Picea abies stands in Finland, Álvarez-González (1997) for Pinus pinaster stands, and Condés (1997) for different species in Spain. These authors discussed the advantage of the use of the two-parameter Weibull probability density function over the three parameter function.…”

Section: Introductionmentioning

confidence: 98%

“…The Weibull probability density function was first used for modelling diameter distributions of pure and even-aged stands (Bailey and Dell, 1973), and since then it has been used in many growth models based on diameter distributions because of its flexibility and simplicity (Rennolls et al, 1985;Maltamo et al, 1995;Kangas and Maltamo, 2000;Zhang et al, 2003;Liu et al, 2004).…”

Section: Introductionmentioning

confidence: 99%

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Modelling diameter distributions of Betula alba L. stands in northwest Spain with the two-parameter Weibull function

et al. 2007

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The diameter distributions of 125 permanent plots installed in birch dominated (Betula alba L.) stands in Galicia were modelled with the two-parameter Weibull distribution. Four different fitting methods were used: that based on percentiles of the distribution, non linear regression, maximum likelihood and the method of moments. The most accurate fit was obtained with the non linear regression (NLR) approach, considering the following statistics in the comparison: bias, mean absolute error (MAE), mean square error (MSE) and number of plots rejected by the Kolmogoroff-Smirnoff (KS) test.The scale parameter (b) and the shape parameter (c) obtained with the most accurate method (non linear regression), were first modelled with simple linear models and then related to commonly measured prediction variables (quadratic mean diameter, dominant height and stand density) with the parameter prediction model (PPM). The parameters fitted with the method of moments were recovered with the parameter recovery model (PRM) from the first and the second moments of the distribution (mean diameter and variance, respectively). Results indicated that both methods were successful in predicting the diameter frequency distributions. The PRM was more accurate than the PPM method.Key words: diameter class model, two-parameter Weibull distribution, fitting methods, parameter modelling. ResumenModelización de las distribuciones diamétricas en masas de Betula alba L. en el noroeste de España con la función Weibull biparamétrica Las distribuciones diamétricas de 125 parcelas permanentes instaladas en masas puras de abedul (Betula alba L.) en Galicia fueron modelizadas con la distribución Weibull de dos parámetros. Se emplearon cuatro métodos de ajuste: basados en percentiles de la distribución, regresión no lineal, máxima verosimilitud y el método de los momentos. Los ajustes más precisos fueron obtenidos con regresión no lineal, considerando los siguientes estadísticos en la comparación de los resultados: sesgo, error medio absoluto, error medio cuadrático y número de parcelas rechazadas por el test de Kolmogoroff-Smirnoff.El parámetro de escala (b) y el parámetro de forma (c) obtenidos con el método más preciso (regresión no lineal), fueron relacionados con variables de masa de frecuente medición (diámetro medio cuadrático, altura dominante y densidad) mediante modelos lineales sencillos aplicando la metodología de predicción de parámetros. Los parámetros ajustados con el método de los momentos fueron recuperados con modelos de recuperación de parámetros a partir del primer y del segundo momento de la distribución (diámetro medio y varianza, respectivamente). Los resultados indicaron que ambos métodos fueron satisfactorios para predecir las distribuciones de frecuencias de diámetros. El mé-todo de recuperación de parámetros fue más preciso que el método de predicción de parámetros.Palabras clave: modelo de clases diamétricas, distribución Weibull biparamétrica, métodos de ajuste, modelización de parámetros.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Modelling diameter distributions of Betula alba L. stands in northwest Spain with the two-parameter Weibull function

et al. 2007

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“…The method of moments for the beta function (MMB) was first used by Loestch et al (1973) and more recently by, e.g., Maltamo et al (1995) and Palahí et al (2007). Parameters of the function are obtained from the first and second moments of the distributions (the arithmetic mean diameter d -and the variance (s 2 ), respectively) with the following expressions:…”

Section: -Methods Of Moments (Mmb)mentioning

confidence: 99%

“…Maltamo et al (1995) found that the two-parameter approach of the Weibull distribution gave better results than the three-parameter approach in the fits by maximum likelihood for modelling basal area diameter distributions in non-truncated stands of Pinus sylvestris and Picea abies in Finland.…”

Section: Diameter Distributions For Pine In Spainmentioning

confidence: 99%

A comparison of estimation methods for fitting Weibull, Johnson’s SB and beta functions to Pinus pinaster, Pinus radiate and Pinus sylvestris stands in northwest Spain

Gorgoso¹,

Rojo

Cámara-Obregón

et al. 2012

For. syst.

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The purpose of this study was to compare the accuracy of the Weibull, Johnson's S B and beta distributions, fitted with some of the most usual methods and with different fixed values for the location parameters, for describing diameter distributions in even-aged stands of Pinus pinaster, Pinus radiata and Pinus sylvestris in northwest Spain. A total of 155 permanent plots in Pinus sylvestris stands throughout Galicia, 183 plots in Pinus pinaster stands throughout Galicia and Asturias and 325 plots in Pinus radiata stands in both regions were measured to describe the diameter distributions. Parameters of the Weibull function were estimated by Moments and Maximum Likelihood approaches, those of Johnson's S B function by Conditional Maximum Likelihood and by Knoebel and Burkhart's method, and those of the beta function with the method based on the moments of the distribution.The beta and the Johnson's S B functions were slightly superior to Weibull function for Pinus pinaster stands; the Johnson's S B and beta functions were more accurate in the best fits for Pinus radiata stands, and the best results of the Weibull and the Johnson's S B functions were slightly superior to beta function for Pinus sylvestris stands. However, the three functions are suitable for this stands with an appropriate value of the location parameter and estimation of parameters method.

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“…Alternatively, the location parameter (a) may be set to zero. The use of this condition restricts the parameters of the Weibull function to two, thus making it easier to model, and providing similar results to the three-parameter Weibull, at least for even-aged, single-species stands [2,68,69]. Thus, to recover parameters b and c the following expressions were used:…”

Section: Diameter Distributionmentioning

confidence: 99%

A growth model for Pinus radiata D. Don stands in north-western Spain

2007

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-A dynamic whole-stand growth model for radiata pine (Pinus radiata D. Don) stands in north-western Spain is presented. In this model, the initial stand conditions at any point in time are defined by three state variables (number of trees per hectare, stand basal area and dominant height), and are used to estimate total or merchantable stand volume for a given projection age. The model uses three transition functions derived with the generalized algebraic difference approach (GADA) to project the corresponding stand state variables at any particular time. These equations were fitted using the base-age-invariant dummy variables method. In addition, the model incorporates a function for predicting initial stand basal area, which can be used to establish the starting point for the simulation. Once the state variables are known for a specific moment, a distribution function is used to estimate the number of trees in each diameter class by recovering the parameters of the Weibull function, using the moments of first and second order of the distribution. By using a generalized height-diameter function to estimate the height of the average tree in each diameter class, combined with a taper function that uses the above predicted diameter and height, it is then possible to estimate total or merchantable stand volume.whole-stand growth model / radiata pine plantations / generalized algebraic difference approach / basal area disaggregation / Galicia Résumé -Un modèle de croissance pour des peuplements de Pinus radiata D. Don du nord ouest de l'Espagne. Un modèle dynamique de croissance de peuplement est présenté pour Pinus radiata D. dans le nord ouest de l'Espagne. Dans ce modèle, les conditions initiales du peuplement en tout point et temps sont définies par trois variables d'état (nombre d'arbres à l'hectare, surface terrière et hauteur dominante) et sont utilisées pour estimer le volume total ou marchand du peuplement pour un âge donné. Le modèle utilise trois fonctions de transition dérivées avec une approche par différence algébrique généralisée (GADA) pour projeter les variables d'état correspondantes du peuplement à n'importe quel moment. Ces équations ont été ajustées en utilisant la méthode des variables indicatrices indépendantes de l'âge. En plus, le modèle incorpore une fonction de prédiction de la surface terrière initiale du peuplement qui peut être utilisée pour établir le point de départ de la simulation. Une fois que les variables d'état sont connues à un instant donné, une fonction de distribution est utilisée pour estimer le nombre d'arbres dans chaque classe de diamètre en récupérant les paramètres de la fonction de Weibull, en utilisant les moments de premier et de second ordre de la distribution. En utilisant une fonction généralisée hauteur-diamètre pour estimer la hauteur de l'arbre moyen de chaque classe de diamètre, combinée avec une fonction qui utilise la prédiction précédente du diamètre et de la hauteur, il est alors possible d'estimer le volume total ou marchand du peuplement. modèl...

show abstract

Comparison of beta and weibull functions for modelling basal area diameter distribution in stands of pinus sylvestris and picea abies

Cited by 113 publications

References 8 publications

Modelling diameter distributions of Betula alba L. stands in northwest Spain with the two-parameter Weibull function

Modelling diameter distributions of Betula alba L. stands in northwest Spain with the two-parameter Weibull function

A comparison of estimation methods for fitting Weibull, Johnson’s SB and beta functions to Pinus pinaster, Pinus radiate and Pinus sylvestris stands in northwest Spain

A growth model for Pinus radiata D. Don stands in north-western Spain

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