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Engelmann spruce site index models: a comparison of model functions and parameterizations.

Nigh G - PLoS ONE (2015)

Bottom Line: Engelmann spruce (Picea engelmannii Parry ex Engelm.) is a high-elevation species found in western Canada and western USA.The model parameterizations that were tested are indicator variables, mixed-effects, GADA, and g-GADA.Model parameterization had more of an influence on the fit than did model formulation, with the indicator variable method providing the best fit, followed by the mixed-effects modelling (9% increase in the variance for the Chapman-Richards and Schumacher formulations over the indicator variable parameterization), g-GADA (optimal approach) (335% increase in the variance), and the GADA/g-GADA (with the GADA parameterization) (346% increase in the variance).

View Article: PubMed Central - PubMed

Affiliation: Forest Analysis and Inventory Branch, British Columbia Ministry of Forests, Lands and Natural Resource Operations, Victoria, British Columbia, Canada.

ABSTRACT
Engelmann spruce (Picea engelmannii Parry ex Engelm.) is a high-elevation species found in western Canada and western USA. As this species becomes increasingly targeted for harvesting, better height growth information is required for good management of this species. This project was initiated to fill this need. The objective of the project was threefold: develop a site index model for Engelmann spruce; compare the fits and modelling and application issues between three model formulations and four parameterizations; and more closely examine the grounded-Generalized Algebraic Difference Approach (g-GADA) model parameterization. The model fitting data consisted of 84 stem analyzed Engelmann spruce site trees sampled across the Engelmann Spruce - Subalpine Fir biogeoclimatic zone. The fitted models were based on the Chapman-Richards function, a modified Hossfeld IV function, and the Schumacher function. The model parameterizations that were tested are indicator variables, mixed-effects, GADA, and g-GADA. Model evaluation was based on the finite-sample corrected version of Akaike's Information Criteria and the estimated variance. Model parameterization had more of an influence on the fit than did model formulation, with the indicator variable method providing the best fit, followed by the mixed-effects modelling (9% increase in the variance for the Chapman-Richards and Schumacher formulations over the indicator variable parameterization), g-GADA (optimal approach) (335% increase in the variance), and the GADA/g-GADA (with the GADA parameterization) (346% increase in the variance). Factors related to the application of the model must be considered when selecting the model for use as the best fitting methods have the most barriers in their application in terms of data and software requirements.

No MeSH data available.


Related in: MedlinePlus

Relationship between local parameters when fit using the indicator variable parameterization (dots) for the Chapman-Richards model (part a), modified Hossfeld IV model (parts b and c), and the Schumacher model (part d).Also shown are the imposed relationships (lines) for the g-GADA models 14 (part a), 15 (part b and c), and 16 (part d).
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pone.0124079.g003: Relationship between local parameters when fit using the indicator variable parameterization (dots) for the Chapman-Richards model (part a), modified Hossfeld IV model (parts b and c), and the Schumacher model (part d).Also shown are the imposed relationships (lines) for the g-GADA models 14 (part a), 15 (part b and c), and 16 (part d).

Mentions: The calibration of the g-GADA model using a nonlinear fitting algorithm removes the need to algebraically rearrange the model so that the local parameter is expressed as a function of the other parameters and variables, as is done in the GADA parameterization. This increases the flexibility in the functions that can be used to express the relationships between parameters as compared to the GADA parameterization. To improve the fit of the models by taking advantage of this flexibility, graphs of the local parameter values from fitting the models with the indicator variable parameterization were used to guide the selection of the functions relating the local parameters for the optimal g-GADA analysis. Fig 3 shows the relationships between the estimates of the local parameters obtained from the indicator variable parameterization as well as the relationship between the two parameters implied by the fitted optimal g-GADA models 14–16. There are a few outcomes of note. For the HIV model, parameter b1 was global for the indicator variable parameterization. However, for the optimal g-GADA parameterization, all three parameters varied across trees. For the SCH model, parameter c2 was global for the indicator variable parameterization but it varied along with parameter c0 for the optimal g-GADA method. A g-GADA model was also fit with c2 being global, but this resulted in a slightly poorer fit. In all cases (see Fig 3), the relationship between local parameters that was implied by the fitting of the optimal g-GADA model was different from that implied by the indicator variables fits. For the HIV model, these differences were quite significant (Fig 3 parts b and c). A possible explanation for this is that with the indicator variable parameterization, the parameters are independent from each other and hence are free to take whatever value that optimizes the fit for each tree. The g-GADA parameterization forces a relationship between parameters across all trees. Unless this relationship holds exactly for all trees, there are going to be trade-offs in the fit as the parameter estimates vary to obtain an overall best fit. This issue is likely more severe for the HIV model because the parameter values have such widely different magnitudes so that a small change in one parameter has a large effect on another parameter, and also because all three parameters vary by subject. This issue was also noted by Stewart et al. [6].


Engelmann spruce site index models: a comparison of model functions and parameterizations.

Nigh G - PLoS ONE (2015)

Relationship between local parameters when fit using the indicator variable parameterization (dots) for the Chapman-Richards model (part a), modified Hossfeld IV model (parts b and c), and the Schumacher model (part d).Also shown are the imposed relationships (lines) for the g-GADA models 14 (part a), 15 (part b and c), and 16 (part d).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4390286&req=5

pone.0124079.g003: Relationship between local parameters when fit using the indicator variable parameterization (dots) for the Chapman-Richards model (part a), modified Hossfeld IV model (parts b and c), and the Schumacher model (part d).Also shown are the imposed relationships (lines) for the g-GADA models 14 (part a), 15 (part b and c), and 16 (part d).
Mentions: The calibration of the g-GADA model using a nonlinear fitting algorithm removes the need to algebraically rearrange the model so that the local parameter is expressed as a function of the other parameters and variables, as is done in the GADA parameterization. This increases the flexibility in the functions that can be used to express the relationships between parameters as compared to the GADA parameterization. To improve the fit of the models by taking advantage of this flexibility, graphs of the local parameter values from fitting the models with the indicator variable parameterization were used to guide the selection of the functions relating the local parameters for the optimal g-GADA analysis. Fig 3 shows the relationships between the estimates of the local parameters obtained from the indicator variable parameterization as well as the relationship between the two parameters implied by the fitted optimal g-GADA models 14–16. There are a few outcomes of note. For the HIV model, parameter b1 was global for the indicator variable parameterization. However, for the optimal g-GADA parameterization, all three parameters varied across trees. For the SCH model, parameter c2 was global for the indicator variable parameterization but it varied along with parameter c0 for the optimal g-GADA method. A g-GADA model was also fit with c2 being global, but this resulted in a slightly poorer fit. In all cases (see Fig 3), the relationship between local parameters that was implied by the fitting of the optimal g-GADA model was different from that implied by the indicator variables fits. For the HIV model, these differences were quite significant (Fig 3 parts b and c). A possible explanation for this is that with the indicator variable parameterization, the parameters are independent from each other and hence are free to take whatever value that optimizes the fit for each tree. The g-GADA parameterization forces a relationship between parameters across all trees. Unless this relationship holds exactly for all trees, there are going to be trade-offs in the fit as the parameter estimates vary to obtain an overall best fit. This issue is likely more severe for the HIV model because the parameter values have such widely different magnitudes so that a small change in one parameter has a large effect on another parameter, and also because all three parameters vary by subject. This issue was also noted by Stewart et al. [6].

Bottom Line: Engelmann spruce (Picea engelmannii Parry ex Engelm.) is a high-elevation species found in western Canada and western USA.The model parameterizations that were tested are indicator variables, mixed-effects, GADA, and g-GADA.Model parameterization had more of an influence on the fit than did model formulation, with the indicator variable method providing the best fit, followed by the mixed-effects modelling (9% increase in the variance for the Chapman-Richards and Schumacher formulations over the indicator variable parameterization), g-GADA (optimal approach) (335% increase in the variance), and the GADA/g-GADA (with the GADA parameterization) (346% increase in the variance).

View Article: PubMed Central - PubMed

Affiliation: Forest Analysis and Inventory Branch, British Columbia Ministry of Forests, Lands and Natural Resource Operations, Victoria, British Columbia, Canada.

ABSTRACT
Engelmann spruce (Picea engelmannii Parry ex Engelm.) is a high-elevation species found in western Canada and western USA. As this species becomes increasingly targeted for harvesting, better height growth information is required for good management of this species. This project was initiated to fill this need. The objective of the project was threefold: develop a site index model for Engelmann spruce; compare the fits and modelling and application issues between three model formulations and four parameterizations; and more closely examine the grounded-Generalized Algebraic Difference Approach (g-GADA) model parameterization. The model fitting data consisted of 84 stem analyzed Engelmann spruce site trees sampled across the Engelmann Spruce - Subalpine Fir biogeoclimatic zone. The fitted models were based on the Chapman-Richards function, a modified Hossfeld IV function, and the Schumacher function. The model parameterizations that were tested are indicator variables, mixed-effects, GADA, and g-GADA. Model evaluation was based on the finite-sample corrected version of Akaike's Information Criteria and the estimated variance. Model parameterization had more of an influence on the fit than did model formulation, with the indicator variable method providing the best fit, followed by the mixed-effects modelling (9% increase in the variance for the Chapman-Richards and Schumacher formulations over the indicator variable parameterization), g-GADA (optimal approach) (335% increase in the variance), and the GADA/g-GADA (with the GADA parameterization) (346% increase in the variance). Factors related to the application of the model must be considered when selecting the model for use as the best fitting methods have the most barriers in their application in terms of data and software requirements.

No MeSH data available.


Related in: MedlinePlus