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\begin_body

\begin_layout Title
Statistical modeling of biomass increment
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Statistical modeling of biomass increment
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\begin_layout Author
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Chris Paciorek
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Chris Paciorek
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Department of Statistics; University of California, Berkeley
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\begin_layout Date
June 2015
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\begin_layout BeginFrame
Model overview - data
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\begin_layout Itemize
Modeling Lyford 13 plots plus HF census data for 1960-2013
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Two data sources and therefore two likelihood terms: census DBH and ring
 increment
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Trees not in census in a year are assumed < 5 cm dbh
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\begin_layout BeginFrame
Model overview - growth process
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DBH in a year (end of season) is equal to DBH in previous year plus increment
 that year
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Unknown true increment assumed to come from a distribution
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\begin_deeper
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Overall mean increment for all trees varies by year (year effect)
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Each tree also has a mean increment common to all years (tree effect)
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Can have increment vary by taxon and with tree size (random taxon effects
 + regression on size)
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Playing with having increment be autocorrelated over time via AR process
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Account for when in a season DBH is measured by scaling relative to inferred
 seasonal growth curve
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Estimated increment is sum across all trees
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Assume no missing trees given census
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Model details
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measured log DBH assumed to follow a 
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 distribution with 3 degrees of freedom, centered on true log DBH
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log increment assumed to follow a normal distribution centered on true log
 increment, 
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\begin_layout Itemize
\begin_inset Formula $\log X_{it}\sim t_{\nu}(\alpha_{i}+\gamma_{t}+\beta(D_{i,t-1}-30)I(D_{i,t-1}>30),\sigma_{i}^{2})$
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 (increment increases with size only for size > 30 cm)
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\begin_inset Formula $\alpha_{i}\sim N(\alpha_{\mbox{taxon}(i)},\sigma_{\mbox{taxa}}^{2})$
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\begin_inset Formula $\gamma_{t}\sim N(0,\sigma_{t}^{2})$
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Not indicated above, but also can include residual increment process as
 AR(1) process with AR parameter 
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, estimated in model.
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Some results
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Assessment of model fit and raw data has revealed some anomalies in both
 increment and DBH data and inconsistencies in HF and Neil/Dan's data; some
 of this resolved based on consultation with Neil/Dan
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Initial fits to all individual trees in Neil/Dan plots: see tree_plots.pdf
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Still to be explored
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\begin_layout Itemize
Can we better pin down distinction between DBH error and increment error
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Increment error may be strongly autocorrelated (because of use of two cores
 only)
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Not clear how to incorporate long-term trends in increment (or if needed)
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Do increments decline as tree nears death?
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Need to deal with uncertain year of death
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Need to get allometry information
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Would be good to assess impact of not having census data - loss of some
 larger trees that are decayed and miss many small trees
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How get overall stand estimate from having three plots? 
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How incorporate info from nested rings?
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