npp.lyx

#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
\textclass article
\use_default_options true
\maintain_unincluded_children false
\language english
\language_package default
\inputencoding auto
\fontencoding global
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100

\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\use_hyperref false
\papersize default
\use_geometry false
\use_amsmath 1
\use_esint 1
\use_mhchem 1
\use_mathdots 1
\cite_engine basic
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\use_refstyle 1
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header

\begin_body

\begin_layout Standard
Some notes/thoughts on the NPP methods.
\end_layout

\begin_layout Subsection*
Importance of the fading record
\end_layout

\begin_layout Standard
We see little evidence of the fading record in our time series since 1960
 based on comparing model fits with and without census data included.
 Our conclusion (I think) is that this is an aggrading (right word?) stand
 and there has been little mortality amongst moderate-large trees so most
 of the biomass increment is accounted for in trees with cores.
 In addition we account for the likely ABI in small trees alive at the end
 of the period and the influence of small trees that died before the rings
 were taken is small because of their size.
 We should also comment on how many dead tree cores we have and if that
 has made a difference.
 
\end_layout

\begin_layout Subsection*
Some extensions
\end_layout

\begin_layout Standard
One might use the census data from beyond just the PalEON plots to better
 inform:
\end_layout

\begin_layout Itemize
relationship of growth with whether a tree dies in subsequent years (bias
 from inferring average growth regardless of tree status)
\end_layout

\begin_layout Itemize
better estimation of population distribution of growth for trees by species
 and size
\end_layout

\begin_layout Itemize
perhaps other population/prior distributions in our model that might be
 informed by the census data?
\end_layout

\begin_layout Itemize
use in scaling up by estimating the spatial heterogeneity of a relevant
 variable across the full census region.
 For example we can estimate variance of AGB across Paleon-size plots within
 the spatial area in which the plots are located.
 This might then substitute for the plot effect variance term.
 For ABI, it's not as clear as we can't get that directly from the census
 data.
 
\end_layout

\begin_layout Subsection*
Allometric uncertainty
\end_layout

\begin_layout Standard
Trees within a small area should be assigned the same site effect when the
 allometric hierarchical model includes a site random effect.
 For a given tree we should use the same tree random effect at different
 times.
\end_layout

\begin_layout Subsection*
Scaling up
\end_layout

\begin_layout Standard
Not clear how important this is to the modelers.
 We might be able to just pass along the individual plot estimates and the
 modelers can treat them as small area estimates against which to calibrate
 the model (somehow).
\end_layout

\begin_layout Standard
However, for ecological analysis and synthesis of results across sites (as
 we've discussed for a synthesis paper), we presumably want an estimate
 we think pertains to an area larger than simply the three plots.
\end_layout

\begin_layout Standard
At a basic level, we can think of empirically estimating a between plot
 variance in ABI based on the three plots.
 This can then be used to infer how different the average of the three plots
 is from an average taken over an entire stand/forest (however defined).
 Alternatively, with plot effects in the statistical model, we can use the
 estimate of the plot effect variance for scaling up.
 The latter is probably best.
 That said, current modeling seems to indicate that there is little plot
 effect.
 
\end_layout

\begin_layout Standard
Perhaps in the data product we report the simple average of the three plots
 with the 
\begin_inset Quotes eld
\end_inset

finite-population
\begin_inset Quotes erd
\end_inset

 variance of that average indicating the uncertainty in ABI in the area
 of the three plots.
 Then we could also report a 
\begin_inset Quotes eld
\end_inset

super-population
\begin_inset Quotes erd
\end_inset

 variance indicating uncertainty if we scale up to a larger area (of course
 the point estimate remains the same).
\end_layout

\begin_layout Subsection*
Accounting for dependence
\end_layout

\begin_layout Standard
[side comment: Mike says they see the ring error/ring variability trading
 off that we do and live with it to some extent and also set fairly tight
 prior on the ring error]
\end_layout

\begin_layout Standard
We need to constrain the observation error because it is an important part
 of the uncertainty estimation for ABI.
 Having replicated cores allows us to estimate the observation error (assuming
 independent errors across cores), whereas if we only have one core, it
 is hard to distinguish observation error from process variability and the
 relative importance of the components plays out as different levels of
 uncertainty in the estimates of ABI.
 By observation error we mean the difference between the true radially-integrate
d increment and the observed increment from one or more cores.
 However, because the increments around the tree must integrate to the true
 increment (inducing negative dependence) and because we expect correlation
 as a function of angle between the cores, we cannot assume independent
 increments.
 If two cores at a given angle are positively correlated then there is higher
 probability that both increment measurements are lower or higher than the
 true increment, i.e.
 that the measurements bracket the truth (more so than with independent
 error), and variance of the average is inflated.
 If the two cores are negatively correlated then the two increment measurements
 are more likely to be on either side of the true increment and the variance
 of the average is smaller than the variance of an average of independent
 measurements.
 
\end_layout

\begin_layout Standard
In general, if the correlation is negative, that implies smaller measurement
 variance and larger process variability while if the correlation is positive
 that implies larger measurement variance and smaller process variability.
 
\end_layout

\begin_layout Standard
Given dispersed cores we expect zero or negative correlation, while if cores
 were placed at less than 90 degrees we would expect positive correlation.
 This suggests that a model that constrains correlations to be zero or negative
 may be reasonable.
 It will be difficult to estimate a complicated model with correlation varying
 by angle in part because for many trees we do not know the angles, so we
 will focus on assuming a constant covariance between any two cores.
\end_layout

\begin_layout Standard
We could simply fit the model with covariance model.
 Unfortunately, adding an unknown covariance into a model with replication
 re-introduces non-identifiability as the model tries to estimate both the
 variance and covariance from the across-core variability.
 We may be able to explore plausible covariance values in light of our prior
 understanding of the problem.
 
\end_layout

\begin_layout Standard
To do so, we consider an approach to isolate the variance and covariance
 and avoid having to estimate the mean for teach tree by time.
 If we knew the true mean, then we could use two (or three) measurements
 to estimate both variance and covariance, but without the mean we lose
 a degree of freedom.
 [need to make language here more precise and more clear] We can consider
 estimation based on differences of measurements as differencing removes
 the unknown mean for the tree.
 We can then write down the variance of the difference and relate that to
 the variance and covariance, allowing us to estimate a linear combination
 of those two quantities.
 We can do this for different angles between cores and comparing two vs
 three-core trees.
 Combined with understanding of how cores are placed on a tree, we can explore
 plausible covariance values.
 
\end_layout

\begin_layout Standard
Note that if we thought we had a reasonable model, the information from
 the method of moments should already be captured in the likelihood.
 However, given the complexity of the model and different sources of data,
 imposing the constraint allows us to inform the variance/covariance linear
 combination directly without the trading off that can go on in the full
 parameter estimation.
 It is cheating a bit in the sense that we use the data twice, but I think
 it may be worthwhile, particularly if the estimation without the constraint
 gives an estimate of the constraint quantity very different from what we
 estimate with method of moments.
 
\end_layout

\begin_layout Standard
Rather than imposing the constraint in the model, we can use the constraint
 as a check on the posterior.
 If the posterior doesn't obey the constraint (perhaps because the model
 learns only indirectly about the hyperparameters), we may wish to simply
 impose a constraint that the covariance is negative since the model thinks
 the measurement variance is small but seems to overestimate the covariance.
 
\end_layout

\begin_layout Standard
Our constraint probably implies negative covariance and small variance;
 otherwise we need to believe positive covariance and larger variance.
 
\end_layout

\begin_layout Standard
[what about outliers affecting our estimation of variances?]
\end_layout

\begin_layout Standard
[we should see how much the size of the ring observation error affects uncertain
ty in the quantities we care about]
\end_layout

\end_body
\end_document