thoughts_aug2016.lyx

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\begin_layout Standard
Some notes/thoughts on the NPP methods.
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[see also paper notes (but have looked at 4/6/17 notes]
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\begin_layout Standard
emphasize use for model assimilation - focus on ABI estimation at plot level;
 concern about fading record and that we want to see if we can use data
 w/o census data; have section on paleon design
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\begin_layout Standard
stat stuff: replication, when measured in year, imputation for missing trees/plo
t-level inference, scaling up, allometric uncertainty
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\begin_layout Standard
our dbh errors (on cm scale) are big relative to Clark et al.
 2007
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\begin_layout Standard
Dawson, Paciorek, Dietze, McLachlan [Pederson, Dye, Woods, Bishop, Moore,
 Barker, other HF census folks?]
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\begin_layout Subsection*
Importance of the fading record
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\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 by trees with cores.
 The influence of small trees that died before the cores could be taken
 (or that simply were not cored) 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.
 
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\begin_layout Standard
Desire to core dead trees but hard in practice.
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Mention that we project live trees back in time to before they entered the
 measured size pool of trees.
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\begin_layout Subsection*
Paleon design
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\begin_layout Standard
An observation is a plot with multiple trees.
 We know there is strong dependence between trees (competition, correlated
 exogenous effects), hence we collect data and do modeling at the plot level.
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\begin_layout Subsection*
Some extensions
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\begin_layout Standard
One might use census data to better inform:
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\begin_layout Itemize
relationship of growth with whether a tree dies in subsequent years (bias
 from inferring average growth regardless of tree status); otherwise our
 only data on this comes from the few dead trees with cores (and note that
 rings for years closest to death most likely to be degraded)
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\begin_layout Itemize
better estimation of population (prior) distribution of growth for trees
 by species and size
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\begin_layout Itemize
other population (prior) distributions that might be of use?
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\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, whose estimation
 is difficult with only three plots per site.
 For ABI, it's not as clear as we can't get that directly from the census
 data.
 
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Not suggesting we do all this for the paper, just some brainstorming that
 might be discussed in the Discussion.
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\begin_layout Subsection*
Allometric uncertainty
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\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.
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To what extent does the Pecan allometry model provide site vs.
 tree effects? Are we mimicing this by simple random selection from a set
 of tree-specific allometries? We only want separate trees to share the
 same effect to the extent that there are site/geographic variations.
 Is geography/region simply a proxy for site in the Pecan allometry modeling?
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Are we simply doing multiple runs, each with different allometric sampling?
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\begin_layout Subsection*
Scaling up
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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).
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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.
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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.
 
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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 within 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 unless there is some sort of adjustment
 made based on information that the three plots are different in some way
 from the stand as a whole).
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\begin_layout Subsection*
Accounting for dependence
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\begin_layout Standard
The increment error variance and the increment process variance parameters
 trade off in models such as the one above and related models such as Clark
 et al.
 (2007).
 This makes sense - year to year variation in increments could be explained
 statistically either as error in increments or true variability in increments
 from year to year.
 To address such identifiability issues, Clark et al.
 chose to constrain the priors on the increment and dbh error variance using
 information on these errors from replicated data to construct informed
 priors.
 We take the approach of directly using the replicates in the model.
 However directly modeling individual increments rather than average increment
 raises the question of correlation in error between increments.
 
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[summary - we avoid informative priors but don't expect to learn a lot about
 certain parameters because of identifiability issues - we constrain covariance
 and allow replication to tell us about variance in light of this, but only
 difference of variance and covariance directly identified from sufficient
 statistics in the data.
 In principle dbh data could help additionally constrain by comparing increment
 + previous dbh with current dbh but errors in dbh measurement and lack
 of yearly data degrade this additional constraint.]
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The identifiability story is:
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\begin_layout Enumerate
need measurement unc.
 vs.
 process variability to properly characterize uncertainty in ABI - how much
 does it vary year to year?
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\begin_layout Enumerate
replication in cores can help us characterize process variability -- unless
 errors across cores are highly positively correlated and variance is large
 (which would I think be ruled out by increment data matching census) then
 the variability from year to year is probably mostly process
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\begin_layout Enumerate
however core data are not uncorrelated - at the very least because truth
 is the integral, so an 'error' at one core must be balanced by errors elsewhere
 (basically a sum constraint)
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\begin_layout Enumerate
correlation probably varies by angle, so even having three cores won't fully
 get at mean, variance, covariance (and we don't always know the angles)
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\begin_layout Enumerate
by differencing we can remove mean effect and isolate variance+covariance;
 can possibly use this as a constraint; but still have non-identifiability
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\begin_layout Enumerate
census data can help constrain the mean effect and could help infer positive
 covariance if census and cores disagree (but note temporal correlation
 in error not accounted for)
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\begin_layout Enumerate
at least having positive corr allowed for allows the truth to be systematically
 different than the mean increment, though we don't model the temporal autocorre
lation; but if we have a tree effect can that stand in for the autocorr?
 only if we interpret it as error, which I don't think we are; could we
 add a tree-level error random effect? if we do, how identify its variance
 relative to dbh error?
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\begin_layout Enumerate
alternatively think of the 'tree' as a pseudo-tree represented by its cores;
 if those cores were elsewhere on the tree, we'd get a different ABI but
 the cores do represent growth for a part of the tree; the difference relative
 to full tree is in some sense just reflected in the tree to tree variability
 around the plot; except that DBH is for the true tree; if we did use notion
 of a pseudo-tree than by definition I think the separate cores are conditionall
y independent given their pseudo-tree effect
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\begin_layout Enumerate
census-core match consistent with (a) nopositive core correlation or with
 positive core correlation but no temporal corr; (b) census-core mismatch
 consistent the positive core and temporal correlation; but in (a) we don't
 really believe in no temporal corr so must not have pos core correlation;
 but we're not modeling temporal corr so model is not constrained in that
 way
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\begin_layout Enumerate
just model cores as independent? discuss why they are not but say can't
 disentangle covariance, variance, temporal corr in incr error and dbh error?;
 say that replication and census allow us to get better handle on incr error
 vs process variability
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\begin_layout Enumerate
in order to have process var be low and incr error be high (given we see
 big year-to-year changes) if replicates show similar behavior, we'd need
 big positive corr in incr error and big incr error variance; and we'd need
 to have the errors in different years bounce around a lot; this seems unlikely;
 our model does allow for this (in fact we don't enforce temporal autocorrelatio
n) but not favored by data; our model does allow pos corr though lack of
 temporal correlation disfavors systematic offset of avg increment from
 truth
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\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]
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\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 do not 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 bracket the true increment and the variance of the average
 is smaller than the variance of an average of independent measurements.
 
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\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 a measurement error covariance.
 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.
 To do so, we consider an approach to isolate the variance and covariance
 and avoid having to estimate the mean for each 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 based on the method of moments.
 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 on the linear combination 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 it could
 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.
 
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\begin_layout Standard
However, rather than imposing the constraint in the model, we can instead
 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.
 Our estimate of the constraint based on the method of moments probably
 implies negative covariance and small variance; otherwise we need to believe
 positive covariance and larger variance.
 
\end_layout

\begin_layout Standard
The field sampling was designed to place cores sufficiently far apart radially
 to capture different growth patterns (e.g., 90 degrees).
 This suggests either little correlation or perhaps negative correlation.
 Negative correlation implies less marginal variance as high marginal variance
 means the obs tend to be far from the true mean.
 Consider that the variance of the contrast of replicates is half the marginal
 variance minus the covariance.
 So for a given degree of variability amongst replicates, this can occur
 either with high variance and high positive covariance or lower variance
 and either zero or negative covariance (which goes along with even lower
 variance).
 
\end_layout

\begin_layout Standard
Positive error temporal correlation allows for mismatch.
 To degree there is mismatch, if don't have that temporal correlation it
 will push the core correlation up to allow mismatch even in face of temporal
 averaging out.
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\begin_layout Standard
Some match of dbh and increment should assure the model that correlation
 can't be too positive
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\begin_layout Standard
[what about outliers affecting our estimation of variances?]
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\begin_layout Standard
[we should see how much the size of the ring observation error affects uncertain
ty in the quantities we care about]
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\begin_layout Standard
From census data, we have an independent estimate of true increment (i.e.,
 the mean), albeit aggregated over multiple years.
 The hope is that this information then allows us to estimate both variance
 and covariance.
 E.g., if sum over years of core-based increment isn't consistent with census,
 that suggests positively-correlated increment error; but complication is
 the correlation of the increment error over time.
 Presumably this is positive and could account for mismatch? I think one
 needs both positive core correlation and positive time correlation to give
 a systematic offset of cores vs census.
 If just have positive core correlation, then the year to year errors should
 average out.
 If don't have positive core correlation then each year's average should
 not be systematically different than truth, and again the year to year
 errors should average out.
\end_layout

\begin_layout Standard
We are way of overinterpreting the apparent information in the posterior
 relative to the prior or the scientific implications of the posteriors
 given the identifiability concerns and the emergence of the posterior based
 on simultaneous estimation and trading off of various variance terms for
 both core and dbh data and the variety of error and process variances fit
 in the model, and likely sensitivity to model structure and likely shortcomings
 of the model (e.g., lack of temporal correlation in the ring width errors
 over time).
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\begin_layout Subsection*
Issues to discuss
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\begin_layout Standard
mortality-growth
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\begin_layout Standard
does msmt error uncertainty matter
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\begin_layout Standard
correlation in ring msmt error over time - how account for this?
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\begin_layout Section*
Old notes
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\begin_layout Standard
dependence not working w/o dbh data
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npp thinking:
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sig_x_obs constrained by dbh data becuase summed increment should
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equal dbh change; but aggregated data can't tell what is sig_x_obs
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distinguished from fact that sig_x_obs is correlated across years
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no dbh data means even less ability to constrain sig_x_obs
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\begin_layout Standard
if have replicates then within-year variability tells us about
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\begin_layout Standard
sig_x_obs, but there is non-zero covariance and we need to estimate
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\begin_layout Standard
that too; 3 cores may be critical because otherwise only have 2df
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and 2 parameters
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still a concern that we don't accoutn for temporal correlation (high)
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of errors but perhaps that only affects overall level and the pattern of
\end_layout

\begin_layout Standard
wiggles is still useful
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