Suborna Shekhor Ahmed Department of Forest Resources Management Faculty of Forestry, UBC Western Mensurationists Conference Missoula, MT June 20 to 22,

Suborna Shekhor Ahmed

Department of Forest Resources Management Faculty of Forestry, UBC

Western Mensurationists ConferenceMissoula, MT

June 20 to 22, 2010

Modeling Tree Mortality for Large Regions Using Combined Estimators and

Meta-Analysis Approaches

Objectives

[email protected] Modelling Tree Mortality Using Meta Modelling Slide-2

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.

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.

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.

sp '

From this, the annual probability of mortality is:

sm pp 1

)( 110 kk xxFx

L

s Fxp

)exp(1

1

Published Mortality (or Survival) Models for Aspen


ReferenceStudy

location Model

Yao et al. (2001)

Alberta mixed wood forests

Lacerte et al. (2006)

Ontario

Senecal et al. (2004)

Quebec’s boreal forest

)022964.0097591.0696387.0

490443.7001175.098991.0716708.1(

)exp(1

1

2

2

BAHA

DBH

BAHA

SPISC

DINDBHDBHFx

Fxp

SW

L

s

)6.2857.119

00952.0(

)exp(1

1

2

1

DINBAL

DBHFx

Fxps

)5603.17566.02052.0(

)exp(1

11

growthPositionCanopyFx

Fxpm

: white spruce (Picea glauca) 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.

swSC

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:

jk

r

jjkcombinedk w ˆˆ

1

: 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.

combinedk̂Where,

jk̂

jkw

jkw



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,

rw jk /1

: weight between 0 and 1 applied to the estimated parameters for local scale model j; r : number of local scale models.

jkw



Based on Cochran (Inverse Variances):

Weight the parameters by the inverse of their variances, based on Cochran (1977) and extending to r >2:

where indicates variance of a particular parameter estimate.

r

jjk

jkjkw

1

)ˆvar(/1

)ˆvar(/1

)ˆvar( jk



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.



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.

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.


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.

Natural Regions of Alberta

Map source: http://www.royalalbertamuseum.ca/vexhibit/eggs/vexhome/regdesc.htm


http://www.royalalbertamuseum.ca/vexhibit/eggs/vexhome/regdesc.htm

Results: Fitted Model


The probability of survival model for the entire data set and for each region is:

)(

)exp(1

1

43210 SpOtherBALDINDBHFx

Fxp

L

s

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: percentage of basal area per ha of all trees that were not aspen.

sp

Results: Statistics for Tree- and Plot-Level Variables


Data set Number

of Plots

Number

of trees

Variable Mean Minimum Maximum

All data

combined269 20835

DBH 23.84 9.2 75.40

DIN 0.17 -0.50 2.28

L 9.25 3.0 30.0

Region 1 220 17611

DBH 24.43 9.2 75.40

DIN 0.17 -0.50 2.28

L 9.24 3.0 30.0

Region 2 49 3224

DBH 20.27 9.2 53.9

DIN 0.16 -0.47 1.06

L 9.41 3.0 19.0



Data set Number

of Plots

Number

of trees


All data

combined269 20835

DBH 23.84 9.2 75.40

DIN 0.17 -0.50 2.28

L 9.25 3.0 30.0

Region 1 220 17611

DBH 24.43 9.2 75.40

DIN 0.17 -0.50 2.28

L 9.24 3.0 30.0

Region 2 49 3224

DBH 20.27 9.2 53.9

DIN 0.16 -0.47 1.06

L 9.41 3.0 19.0



Data set Number

of Plots

Number

of trees


All data

combined269 20835

DBH 23.84 9.2 75.40

DIN 0.17 -0.50 2.28

L 9.25 3.0 30.0

Region 1 220 17611

DBH 24.43 9.2 75.40

DIN 0.17 -0.50 2.28

L 9.24 3.0 30.0

Region 2 49 3224

DBH 20.27 9.2 53.9

DIN 0.16 -0.47 1.06

L 9.41 3.0 19.0

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


0

1

2

4

Parameter/

Fit

Statistics

All data

combinedRegion 1 Region 2

(Intercept)2.728438 (0.0747)

2.610003 (0.0779)

3.851521 (0.2615)

(DBH)0.017387 (0.00205)

0.017897(0.00215)

0.018665 (0.00801)

(DIN)10.77865 (0.2220)

10.83272 (0.2382)

10.37064 (0.6273)

(BAL)-0.02488 (0.00146)

-0.02222 (0.00152)

-0.0404 (0.00566)

(SpOther)0.002286

(0.000533)0.002675 (0.000614)

-0.00768 (0.00246)

-2logL 11056 9894 1128

3



0

1

2

4

Parameter/

Fit

Statistics

All data


(Intercept)2.728438 (0.0747)

2.610003 (0.0779)

3.851521 (0.2615)

(DBH)0.017387 (0.00205)

0.017897(0.00215)

0.018665 (0.00801)

(DIN)10.77865 (0.2220)

10.83272 (0.2382)

10.37064 (0.6273)

(BAL)-0.02488 (0.00146)

-0.02222 (0.00152)

-0.0404 (0.00566)

(SpOther)0.002286

(0.000533)0.002675 (0.000614)

-0.00768 (0.00246)

-2logL 11056 9894 1128

3



0

1

2

4

Parameter/

Fit

Statistics

All data


(Intercept)2.728438 (0.0747)

2.610003 (0.0779)

3.851521 (0.2615)

(DBH)0.017387 (0.00205)

0.017897(0.00215)

0.018665 (0.00801)

(DIN)10.77865 (0.2220)

10.83272 (0.2382)

10.37064 (0.6273)

(BAL)-0.02488 (0.00146)

-0.02222 (0.00152)

-0.0404 (0.00566)

(SpOther)0.002286

(0.000533)0.002675 (0.000614)

-0.00768 (0.00246)

-2logL 11056 9894 1128

3

Results: Weights of Parameters Using Meta-Regression for Aspen


Method RegionWeights of parameters

Equal weight

Region 1 0.5

Region 2 0.5

Inverse Variances

Region 1 0.92 0.93 0.87 0.93 0.94

Region 2 0.08 0.07 0.13 0.07 0.06

MLRegion 1 0.85

Region 2 0.15

ML(Differential

Weight)

Region 1 0.91 0.00 1.00 0.97 0.88

Region 2 0.09 1.00 0.00 0.03 0.12

Stein RuleRegion 1 0.85

Region 2 0.15

0 1 2 3 4




Equal weight

Region 1 0.5

Region 2 0.5

Inverse Variances

Region 1 0.92 0.93 0.87 0.93 0.94

Region 2 0.08 0.07 0.13 0.07 0.06

MLRegion 1 0.85

Region 2 0.15

ML(Differential

Weight)

Region 1 0.91 0.00 1.00 0.97 0.88

Region 2 0.09 1.00 0.00 0.03 0.12


Region 2 0.15

0 1 2 3 4




Equal weight

Region 1 0.5

Region 2 0.5

Inverse Variances

Region 1 0.92 0.93 0.87 0.93 0.94

Region 2 0.08 0.07 0.13 0.07 0.06

MLRegion 1 0.85

Region 2 0.15

ML(Differential

Weight)

Region 1 0.91 0.00 1.00 0.97 0.88

Region 2 0.09 1.00 0.00 0.03 0.12


Region 2 0.15

0 1 2 3 4

Results: Estimated Parameters Using Meta-Regression for Aspen


MethodCombined Parameter Estimates

Equal weight

3.2308 0.0183 10.6017 -0.0313 -0.0025

Inverse Variances

2.7112 0.01794 10.7745 -0.0234 0.0021

ML 2.7986 0.018 10.7625 -0.025 0.0011

ML(Differential

Weight)

2.7214 0.0187 10.8327 -0.0228 0.0015

Stein Rule 2.7948 0.018 10.7639 -0.0249 0.0011

0 1 2 3 4




Equal weight

3.2308 0.0183 10.6017 -0.0313 -0.0025

Inverse Variances

2.7112 0.01794 10.7745 -0.0234 0.0021

ML 2.7986 0.018 10.7625 -0.025 0.0011

ML(Differential

Weight)2.7214 0.0187 10.8327 -0.0228 0.0015

Stein Rule 2.7948 0.018 10.7639 -0.0249 0.0011

0 1 2 3 4




Equal weight

3.2308 0.0183 10.6017 -0.0313 -0.0025

Inverse Variances

2.7112 0.01794 10.7745 -0.0234 0.0021

ML 2.7986 0.018 10.7625 -0.025 0.0011

ML(Differential

Weight)

2.7214 0.0187 10.8327 -0.0228 0.0015

Stein Rule 2.7948 0.018 10.7639 -0.0249 0.0011

0 1 2 3 4

Results: Predicted Annual Probability of Survival and Likelihood Using

Meta-Regression and All Data Combined for Aspen


Method

Predicted

-2logLLive Dead

Mean 25%

(percentile)

75%

(percentile)Mean 25%

(percentile)

75%

(percentile)

Equal weight 0.89 0.86 0.98 0.70 0.55 0.89 11074

Inverse Variances 0.88 0.83 0.98 0.67 0.51 0.86 10996

ML 0.88 0.84 0.98 0.67 0.51 0.87 10994ML

(Differential Weight)

0.88 0.84 0.98 0.68 0.52 0.87 10990

Stein Rule 0.88 0.84 0.98 0.67 0.51 0.87 10994All data

combined 0.88 0.83 0.98 0.66 0.50 0.86 11056

sp




Method

Predicted

-2logLLive Dead

Mean 25%

(percentile)

75%


(percentile)

75%

(percentile)

Equal weight 0.89 0.86 0.98 0.70 0.55 0.89 11074


ML 0.88 0.84 0.98 0.67 0.51 0.87 10994ML


0.88 0.84 0.98 0.68 0.52 0.87 10990


combined 0.88 0.83 0.98 0.66 0.50 0.86 11056

sp




Method

Predicted

-2logLLive Dead

Mean 25%

(percentile)

75%


(percentile)

75%

(percentile)

Equal weight 0.89 0.86 0.98 0.70 0.55 0.89 11074


ML 0.88 0.84 0.98 0.67 0.51 0.87 10994ML


0.88 0.84 0.98 0.68 0.52 0.87 10990


combined 0.88 0.83 0.98 0.66 0.50 0.86 11056

sp

Discussion


The simplest meta-regression approach would be to combine existing models using equal weights which resulted in the highest -2logL.

Maximum likelihood (differential weight ) to find weights resulted in lower -2logL value than other meta-regression approaches.

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.

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.

Acknowledgements


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

Approaches


Suborna Shekhor Ahmed Department of Forest Resources Management Faculty of Forestry, UBC Western Mensurationists Conference Missoula, MT June 20 to 22,

Documents

modeling tree mortality

local mortality models

published mortality

meta modelling slide

meta modelling approaches

survival models

number of local scale

existing local scale