Suborna Shekhor Ahmed

1. Modeling Tree Mortality for Large Regions Using Combined Estimators and Meta-Analysis Approaches Department of Forest Resources Management Faculty of Forestry, UBC Western Mensurationists Conference Missoula, MT June 20 to 22, 2010 SubornaShekhor Ahmed

2. Objectives • Develop mortality models for four target species of the Boreal Forest of Canada (aspen, white spruce, black spruce and jack pine) • Data from Alberta, Ontario and Quebec will be used, along with a combined estimator and local mortality models. • To select a combined estimator, several estimators will be proposed and tested using the PSP data from Alberta. • For this presentation, I will present preliminary results of testing combined estimators using PSP data from Alberta for aspen.

3. Tree Mortality • Tree mortality is an important aspect of stand dynamics and is commonly expressed in terms of loss of volume or basal area per year. • Cause and time of death are very important to model the tree mortality. The following variables are considered for modeling tree mortality: • Diameter at breast height (DBH), • Annual diameter increment during the preceding interval (DIN), • Total basal area per hectare at the beginning of the growth interval (BAHA), • Site productivity index, • Species composition, • Length of the growth interval (L), • Other measures of competition.

4. Generalized Logistic Model For repeated measures where the time interval, L, is irregular, a generalized logistic model has been used to model survival where is the annual probability of survival. s are the unknown parameters with explanatory variables. From this, the annual probability of mortality is:

5. Published Mortality (or Survival) Models for Aspen : white spruce (Piceaglauca) species composition as a percentage of BAHA; SPI: site productivity index; canopy position: an ordinal variable of position of the tree within the canopy; growth: corresponds to the last year of radial growth (millimetres); All other variables are previously defined.

6. Meta Modelling Approaches Combined Estimator • Meta-modeling approaches use observational data to obtain weights for combining existing local scale models may result in improved precision over the naïve approaches. The general approach would be: Where, : kth estimated parameter using the combination of parameter estimates from the r local spatial models; : kth estimated parameter for local scale model j; : weight between 0 and 1 applied to the kth estimated parameters for local scale model j; r: number of local scale models. Sum of the over all regions is 1 for each parameter.

7. Meta Modelling Approaches • Native Approach 1: • One of the native approaches is to use all available data to fit a large scale model • Native Approach 2 ( Equal Weights ) : • Giving equal weight to each estimate: • Where, : weight between 0 and 1 applied to the estimated parameters for local scale model j; r : number of local scale models.

8. Meta Modelling Approaches • Based on Cochran (Inverse Variances): • Weight the parameters by the inverse of their variances, based on Cochran (1977) and extending to r >2: • whereindicates variance of a particular parameter estimate.

9. Meta Modelling Approaches Maximum Likelihood Optimal weights are found that meet a maximum likelihood objective function. Options include having the same weights for all parameters versus having differential weights by parameter.

10. Meta Modelling Approaches • Stein Rule Estimator (Shrinkage) Where, : vector of estimated parameter for combined model; : weight between 0 and 1 applied for local scale model j; : vector of estimated parameters for local scale model j. : number of observations in the jth region.

11. Alberta Data • Study includes over 1,700 plots measured up to seven times with a variable number of years between measurements, dispersed over the forested land of Alberta. • Each plot summarized at each measurement period to obtain explanatory variables for modeling for each species and all species combined. • The tree-level variables were then merged with the plot-level variables. • The summarized data were considered as census data at the large spatial scale for this research. • Competition mortality of trees was taken into account. • Plots that have a majority of aspen trees (greater than 30% by basal area per ha in any measurement period) were selected for use.

12. Steps for Meta Modelling Using Aspen Data • Fitted the generalized logistic survival model using all data combined. • Split Alberta data into two regions using township and fitted the model separately. • Implemented the combined estimators.

13. Natural Regions of Alberta Map source: http://www.royalalbertamuseum.ca/vexhibit/eggs/vexhome/regdesc.htm

14. Results: Fitted Model The probability of survival model for the entire data set and for each region is: Where, : probability of survival to the end of the period. DIN: annual diameter increment during the preceding interval. BAL: basal area of all trees larger than the subject tree. L: length of the growth interval. SpOther: percentageof basal area per ha of all trees that were not aspen.

15. Results: Statistics for Tree- and Plot-Level Variables

16. Results: Statistics for Tree- and Plot-Level Variables

17. Results: Statistics for Tree- and Plot-Level Variables

18. Results: Estimated Parameters (Standard Errors in Brackets) and Fit statistics

19. Results: Estimated Parameters (Standard Errors in Brackets) and Fit statistics

20. Results: Estimated Parameters (Standard Errors in Brackets) and Fit statistics

21. Results: Weights of Parameters Using Meta-Regression for Aspen

22. Results: Weights of Parameters Using Meta-Regression for Aspen

23. Results: Weights of Parameters Using Meta-Regression for Aspen

24. Results: Estimated Parameters Using Meta-Regression for Aspen

25. Results: Estimated Parameters Using Meta-Regression for Aspen

26. Results: Estimated Parameters Using Meta-Regression for Aspen

27. Results: Predicted Annual Probability of Survival and Likelihood Using Meta-Regression and All Data Combined for Aspen

28. Results: Predicted Annual Probability of Survival and Likelihood Using Meta-Regression and All Data Combined for Aspen

29. Results: Predicted Annual Probability of Survival and Likelihood Using Meta-Regression and All Data Combined for Aspen

30. Discussion • The simplest meta-regression approach would be to combine existing • models using equal weights whichresulted in the highest -2logL. • Maximum likelihood (differential weight ) to find weights resulted in lower • -2logL value than other meta-regression approaches.

31. Conclusion • Further research will include: • Accuracy testing will be done for the estimators. • One estimator will be selected for use. • Regional models will be fitted for white spruce, black spruce and jack pine using Alberta, Ontario and Quebec PSP data.

32. Acknowledgements • Thanks to Alberta, Ontario and Quebec governments for providing PSP data. • Thanks to my committee members: • Dr. Valerie Lemay, UBC. • Dr. Steen Magnussen, Nrcan. • Dr. Frank Berninger, UQAM. • Dr. Peter Marshall, UBC. • Dr. Andrew Robinson, U. Melbourne. • Thanks to NSERC and ForValueNet for providing the research funding.

33. Modeling Tree Mortality for Large Regions Using Combined Estimators and Meta-Analysis Approaches