Model-Based Time-Series Analysis 

of FIA Panel Data Absent 

Re-Measurements

Raymond L. Czaplewski and Mike T. Thompson

United States Department of Agriculture / Forest Service

Rocky Mountain Research Station

Research Paper RMRS-RP-102WWW

May 2013
Czaplewski, Raymond L.; Thompson, Mike T. 2013. Model-based time-series 

analysis of FIA panel data absent re-measurements. Res. Pap. RMRS-RP102WWW. Fort Collins, CO: U.S. Department of Agriculture, Forest Service, 

Rocky Mountain Research Station. 13 p.

Abstract

An epidemic of lodgepole pine (Pinus contorta) mortality from the mountain pine 

beetle (Dendroctonus ponderosae) has swept across the Interior West. Aerial 

surveys monitor the areal extent of the epidemic, but only Forest Inventory 

and Analysis (FIA) field data support a detailed assessment at the tree level. 

Dynamics of the lodgepole pine population occur at a more rapid rate than the 

FIA 10-year re-measurement cycle. A model-based approach links populationlevel estimates from each annual FIA panel estimate. A simple multivariate 

model predicts the statewide rates of annual change among live uninfected 

trees, live infected trees, mortality trees, and standing dead trees. A multivariate 

weighted sum of panel estimates and model predictions of the same attributes 

improve estimates for each year. Biological structure incorporated into the 

model improves logical consistency among the various categories of tree-level 

estimates and smooths annual fluctuations caused by random sampling error. 

We present concepts in simple terms and illustrate results with FIA data from 

2002 to 2008.

Authors

Raymond L. Czaplewski, Emeritus Scientist, Interior West Forest Inventory and 

Analysis Program, Fort Collins, Colorado

Mike T. Thompson, Forester, Interior West Forest Inventory and Analysis 

Program, Ogden, Utah

Available only online at http://www.fs.fed.us/rm/pubs/rmrs_rp102.html.
Contents

Introduction...................................................................................1

Monitoring Design.........................................................................1

Design-Based Annual Panel Estimates........................................3

Model-Based Time-Series Estimates ...........................................5

Uncertainty Metric.........................................................................7

Results and Discussion ................................................................7

Conclusions................................................................................12

Acknowledgments ......................................................................13

Literature Cited...........................................................................13

USDA Forest Service RMRS-RP-102WWW. 2013. 1

Introduction

A regional epidemic of tree mortality from mountain pine beetles 

(Dendroctonus ponderosae) is sweeping through the montane forests of the 

western United States. The Forest Inventory and Analysis (FIA) program designed an annual system, in part, to monitor such episodic changes (Gillespie 

1999). We present one case study that compares conventional analyses of 

annual FIA data to a model-based statistical approach. The objective is to 

compute efficient and reliable estimates of the state of the lodgepole pine 

(Pinus contorta) tree population in Colorado each year from 2002 to 2008.

The model-based estimator considered here is the Kalman filter (e.g., 

Maybeck 1979). Multitudes of engineering applications rely on this mathematical tool, with the common GPS system being one everyday example. 

Other mature applications abound in electrical engineering, econometrics, 

physical oceanography, and atmospheric sciences.

The Kalman filter estimates the multivariate “state” of a system over time. 

An example is the number of lodgepole pine trees in the State of Colorado 

within various categories of live trees, mortality trees, and standing dead 

trees (Czaplewski and others 2012). The Kalman filter separates a timeseries (e.g., 2002 to 2008) into a sequence of discrete time intervals. In the 

case of FIA, that interval is one year. Initial conditions are established with 

the best available vector estimate of the multivariate state of the system at 

the beginning of the year. A simple linear model predicts the state-transition 

from the beginning to the end of the year (e.g., Table 1). The Kalman filter 

estimate at the end of the year is the weighted vector sum of the predicted 

state-vector and an independent measurement vector. A multivariate estimate from a single annual FIA panel is an example of the latter. The vector of 

weights, which are computed as known constants, minimizes the estimated 

covariance matrix for the vector sum. The best estimate of the state-vector at 

the end of the time-increment serves as initial conditions for the next timeincrement. This sequential estimation cycle repeats until the full time-series 

is complete. Czaplewski and Thompson (2009) provided a tutorial example.

Monitoring Design

In the western United States, FIA measures a 10 percent interpenetrating subsample (i.e., “annual panel”) of all Phase-2 field plots each year (Patterson 

and Reams 2005). It requires 10 years to measure the full sample of FIA 

field plots. During these 10 years, there are no re-measured plots to estimate 

components of change at the tree or plot levels. Re-measurements of plots 

begin during the 11th year, and all plots are fully re-measured by the 20th 

year. Unlike periodic inventories produced by FIA between 1950 and 2000, 

which measured all plots in a large region once every 10 to 20 years, the 

annual design produces new design-based sample survey estimates each and 

every year. However, each annual estimate uses the 10 percent sub-sample, 

which introduces relatively large sampling error and inferential uncertainty.

To increase sample size and reduce uncertainty from random sampling 

error, FIA uses the “temporally indifferent” estimator, which is a moving 
2 USDA Forest Service RMRS-RP-102WWW. 2013.

Table 1. Multivariate state-transition model used in Kalman filter to predict the state of the Colorado lodgepole pine forest 

at the end of a one-year time-increment (i.e., time t) using the best estimate of that population at time t-1 as initial 

conditions. Our best estimates for each parameter ci appear in Table 2. 

State variables

Xt Number of live trees in Colorado with no insect damage at time t

Yt Number of live trees in Colorado with insect damage at time t

(Z1)

t Number of mortality trees in Colorado that died between times t-1 and t

(Z2)

t Number of mortality trees in Colorado that died between times t-2 and t-1

(Z3)

t Number of mortality trees in Colorado that died between times t-3 and t-2

(Z4)

t Number of mortality trees in Colorado that died between times t-4 and t-3

(Z5)

t Number of mortality trees in Colorado that died between times t-5 and t-4

(Z6+)

t Number of standing dead trees in Colorado that died before time t-6

At Area of lodgepole pine forest in Colorado at time t

t = 2002, 2003, 2004, 2005, 2006, 2007, 2008

Model Components of change 

Xt = + Xt-1 Live trees with no insect damage at time t-1 

+ (c5 × Xt-1) Annual regeneration rate 

- (c6 × Xt-1) Annual non-insect mortality rate 

+ (c3 × Yt-1) Infected live trees at time t-1 that are uninfected live trees at t 

- [(X2002-Xt-1) × c1] × Yt-1 Uninfected live trees at time t-1 that are infected live trees at t 

Yt = + [(X2002-Xt-1) × c1] × Yt-1 Density-dependent increase live uninfected trees at time t-1 infected at time t

+ (1 - c2 - c3) × Yt-1 Infected live trees at time t-1 that remain infected live trees at t 

- (c2 × Yt-1) Infected live trees at time t-1 that are mortality trees at time t 

- (c3 × Yt-1) Infected live trees at time t-1 that are uninfected live trees at time t 

+ (c4 × Xt-1) Annual non-insect mortality rate 

(Z1)

t

 = + (c2 × Yt-1) Infected live trees at time t-1 that are mortality trees at time t 

- (Z1)

t-1 1st-year mortality trees at time t-1 that are 2nd-year mortality trees at time t 

(Z2)

t

 = +(Z1)

t-1- (Z2)t-1 2nd-year mortality trees at time t that were 1st-year mortality trees at time t-1 

(Z3)

t

 = +(Z2)

t-1- (Z3)t-1 3rd-year mortality trees at time t that were 2nd-year mortality trees at time t-1

(Z4)

t

 = +(Z3)

t-1- (Z4)t-1 4th-year mortality trees at time t that were 3rd-year mortality trees at time t-1

(Z5)

t

 = +(Z4)

t-1- (Z5)t-1 5th-year mortality trees at time t that were 4th-year mortality trees at time t-1

(Z6+)

t

 = +(Z6+)

t-1 Standing dead trees at time t-1 (mortality trees 6 years and older)

+ (Z5)

t-1 5th-year mortality trees at time t-1 that become standing dead trees at t 

- c6 (Z6+)

t-1 Standing dead trees at time t-1 that are fallen coarse woody debris at time t 

At = At-1 No change in lodgepole pine forest area over time
USDA Forest Service RMRS-RP-102WWW. 2013. 3

average that essentially treats plots measured during the preceding five 

years as though they were measured during a single year (Patterson and 

Reams 2005). This estimator is design-unbiased if estimation of the fiveyear moving average is the analysis objective, although that metric is 

rarely an analyst’s first choice. The temporally indifferent estimator is 

design-unbiased for conditions during a single year if there are no net 

changes in the population during these five years, although that situation requires strong assumptions by the analyst that cannot be rigorously 

tested. Regardless, this assumption is implausible during epidemic tree 

mortality, and the temporally indifferent estimator is biased whenever applied to rapidly changing populations (Patterson and Reams 2005).

Implementation of the FIA annual design in Colorado began during 2002. 

The Colorado State Forest Service and FIA measure approximately 1100 

forest and non-forest Phase-2 plots each year across the State of Colorado. 

Between 2002 and 2008, about 60 plots per year had one or more tallied 

lodgepole pine trees, either live or dead. Less than half of these plots were 

affected by the bark beetle epidemic.

Thompson (2009) analyzed these data with conventional statistical 

methods. He treated annual panels as independent samples to estimate 

population-level conifer mortality in Colorado. Statistical tests rejected 

the null hypothesis that there was no change in mean number of mortality 

trees per acre during the 2002 to 2008 time periods. The most pronounced 

change during these six years occurred in the estimated number of insectkilled lodgepole pine, especially after 2005.

Design-Based Annual Panel Estimates

Figure 1 illustrates the multivariate trends in individual panel estimates. 

These fully estimate the multivariate state of the lodgepole pine system 

in Colorado between 2002 and 2008. Each annual panel supports independent estimates for a single year of acres that contain lodgepole pine 

trees and numbers of live trees, mortality trees, and standing dead trees. 

Transformations of those estimates further characterize the statewide 

lodgepole pine population as follows. The estimated mortality rate of infected live trees is the estimated number of first-year mortality trees in 

year t divided by the number of infected live trees in the preceding year 

t-1, although this metric is not exactly a mortality rate because live infected trees can survive for multiple years. The annual rate of net change 

in infected live trees is the number of infected live trees in year t divided 

by the number of infected live trees in the preceding year t-1. The overall 

rate of change (Figure 2) is estimated with a simple linear regression with 

measurement year as the independent variable; the dependent variable includes the seven annual population estimates (2002 to 2008) for numbers 

of live uninfected trees.

As expected, the estimated numbers of infected live trees and mortality 

trees increased rapidly over time (Figure 1). However, other estimates, if 

taken literally, describe unexpectedly volatile fluctuations. The estimated 

area of lodgepole pine forest and the estimated number of uninfected live 
4 USDA Forest Service RMRS-RP-102WWW. 2013.

Figure 1. Annual panel estimates for the lodgepole pine population in the State of Colorado with individual 

annual panel data. Medians of the bootstrap distribution appear as the white center lines. The dark gray region 

portrays the range between the 0.16 and 0.84 quantiles of the bootstrap distribution. This asymmetrical region 

approximately corresponds to ±1 standard deviation units if the bootstrap distribution were Gaussian, while the 

light gray region represents the 0.02 and 0.98 quantiles, roughly corresponding to ±2 standard deviation units.

Figure 2. Overall rate of change in live trees with no insect 

damage estimated with a simple linear regression and the 

seven annual population estimates for 2002 through 2008. 

Spread of the estimated change during those seven years 

represents the bootstrap distribution. The temporally indifferent 

estimates indicate no change in number of uninfected live 

trees, in spite of the well documented epidemic of lodgepole 

pine tree mortality since 2002. Patterson and Reams (2005) 

anticipated this problem when they stated that the temporally 

indifferent estimator “dampens trends by obscuring annual 

fluctuations” and “makes changes appear smaller than they 

are.” Unlike the temporally indifferent estimator, the FIA panel 

estimators and the model-based estimator both estimate an 

approximate 20 percent decrease in uninfected live trees 

between 2002 and 2008.
USDA Forest Service RMRS-RP-102WWW. 2013. 5

trees and standing dead trees can increase or decrease 30 percent or more 

from one year to the next. In 2005, the estimated number of infected live 

trees decreased markedly from the overall time trend, but the estimated 

number of mortality trees in 2006 did not exhibit a similar response. The 

estimated annual rates of change in infected live trees varied widely from 

a 50 percent decrease between 2004 and 2005 to a three-fold increase between 2002 and 2003 and from 2005 to 2006. The estimated mortality 

rates varied between 10 and 50 percent from one year to the next. While 

these latter estimates might be plausible, they might be misleading because ratios between estimates from two different years have relatively 

large estimation errors.

Most analysts would comfortably attribute these unexpected volatilities to 

random sampling error. This is most likely because an analyst’s conceptual 

model of a statewide tree population envisions forest attributes that do 

not fluctuate wildly each year. This same conceptual model might further 

suggest that variables in Figure 1 should be related to each other through 

obvious demographic processes. For example, the number of mortality 

trees one year should be correlated with the number of infected live trees 

in the preceding year, and the number of standing dead trees should be 

related to the number of mortality trees in previous years. The next section describes one such model that we incorporated into the time-series 

analysis.

Model-Based Time-Series Estimates

Czaplewski and Thompson (2009) introduced the Kalman filter as an efficient and intuitive model-based estimator for time-series of annual FIA 

panel data. The model predicts annual changes in the statewide area of 

lodgepole forest and number of live trees, live infected trees, and mortality trees (Table 1). The analyst uses professional judgment to set the 

feasible range for each model parameter and selects the final set of model 

parameters aided by informal goodness-of-fit statistics. The results in our 

case are given in Table 2. While model parameters are treated as known 

constants in the Kalman filter, the model includes an additive vector of 

random prediction errors that accounts for imperfect model predictions 

(Maybeck 1979).

The Kalman filter estimate for year t is the weighted vector sum of model 

predictions at time t, which can be relatively precise but biased, and the 

independent design-based vector estimates from the FIA annual panel at 

time t, which are design unbiased but can be imprecise. The vector of 

weights is computed from the difference between the estimated covariance 

matrix for model prediction errors at time t and the covariance matrix for 

random sampling errors in the FIA panel estimate at time t. This computation is derived with the minimum variance optimality criterion, but it can 

also be viewed as a maximum likelihood estimator if the error distributions 

are multivariate Gaussian (Maybeck 1979). In addition, the Kalman filter 

qualifies as an empirical Bayes estimator (Maybeck 1979). More weight 

is placed on estimates with smaller variances, either model predictions or 

design-based panel estimates. Viewed differently, unbiased but imprecise 
6 USDA Forest Service RMRS-RP-102WWW. 2013.

estimates from each annual FIA panel help the simple demographic model 

more accurately predict the true population dynamics, which are rarely 

simple.

The exact mortality year for a tree is difficult to infer in the field. To reduce 

aberrations caused by nonrandom measurement errors, panel estimates for 

mortality trees that died during years t-2 and t-5 are merged into a single 

measure. Also, mortality trees that died between times t and t-1 are not 

separated by cause (i.e., insect damage or other causes). This is mathematically feasible because dimensions of the measurement vector from 

annual panel estimates need not conform to dimensions of the state-vector 

(Maybeck 1979). An indicator matrix of zeros and ones relates this abbreviated vector of panel estimates to the full state-vector in Table 1.

Bar-Shalom and others (2001) documented the LDL' square-root implementation of the Kalman filter, and all matrix computations in our study 

use this algorithm. The LDL' algorithm is an algebraic transformation of 

the matrix formulae in the standard Kalman filter. This algorithm substantively reduces numerical round-off error that is a common hazard 

in applications of the digital Kalman filter. Furthermore, a nonlinear optimization algorithm scaled the estimated covariance matrix for annual 

model prediction errors to reduce sub-optimal weighting of model predictions. The algorithm focuses on the time-series of multivariate residual 

differences between model estimates at time t and corresponding panel 

estimates at time t. There are seven vectors of residuals (2002 to 2008) 

and each measurement vector has six variables: Xt; Yt; (Z1)t; (Z2)t+(Z3)

t

+(Z4)t+(Z5)t; (Z6+)t; and At (Table 1). The estimated covariance matrix 

for model errors is the basis for a linear transformation of these multivariate residuals into a time-series of 42 nominally orthogonal scalar values 

with unit variances (Maybeck 1979). If the true covariance matrix were 

Table 2. Numerical parameters (ci) used with the model-based estimator (Table 1). Ranges are estimated by the analyst through 

professional judgment, and the final parameter set selected by the analyst are aided by informal goodness-of-fit statistics. 

This provides estimates of certain components of change, which are not available without re-measurement data from the 

independent annual panel estimator or with the temporally indifferent estimator. However, the statistical properties of these 

parameter estimates are unknown.

Sets of parameters evaluated Best

Model parameter (Table 1) minimum maximum estimate increment

c1

Baseline annual rate of increase in new infections without density 

dependence 1.9 2.7 0.1 2.1

c2

Proportion of infected live trees that are mortality trees the 

following year 0.4 1.0 0.05 0.4

c3

Proportion of surviving infected live trees that are uninfected the 

following year 0.1 0.7 0.1 0.7

c4

Annual mortality rate for uninfected live trees from all causes 

except insect damage 0.0000 0.0004 0.0001 0.0001

c5

Annual regeneration rate for live uninfected trees (assumes 

background regeneration and mortality rates are equal) = c4 = c4 = c4 = c4

c6

Annual rate of change from standing dead trees to fallen coarse 

woody debris trees (single value assumed in all trials) 0.01 0.01 0 0.01
USDA Forest Service RMRS-RP-102WWW. 2013. 7

known without error, then these 42 transformed residuals are expected to 

be mutually independent with mean equal to 0 and variance equal to 1. 

The optimization algorithm performs a multivariate scaling of the each of 

the seven estimated covariance matrices so that the realized mean squared 

errors of the standardized residuals conform to their expected distribution. The mean of the squared errors simultaneously addresses prediction 

bias (i.e., non-zero mean of residuals) and bias in estimated variance of 

prediction errors. This reduces risk from placing too much weight on potentially biased model predictions and too little weight on unbiased panel 

estimates. However, there is no known theoretical basis to assert that this 

estimator is unbiased.

Uncertainty Metric

Random sampling errors and model prediction errors cause uncertainty in 

population estimates. The standard-deviation statistic is one measure of 

uncertainty. The standard deviation accurately describes the spread of a 

Gaussian distribution. However, small sample sizes for non-negative and 

relatively rare attributes, such as tree mortality, are associated with skewed 

error distributions. Therefore, the bootstrap method (Efron and Tibshirani 

1994) is used to measure uncertainty in the population estimates.

The bootstrap method independently re-samples measured FIA plots 

within each annual panel. Each of the three estimators is applied to each 

bootstrap sample. There were 2500 replicate bootstrap samples used to 

estimate the bootstrap distribution of random errors. The population estimates in Figures 1, 2, and 3 use the median of the bootstrap distribution to 

accommodate extreme but rare outliers in the bootstrap distributions. The 

uncertainty metric is the difference between the 16th and 84th percentiles 

of the bootstrap distribution. If the bootstrap distribution were Gaussian, 

then this uncertainty metric would approximately equal one standard deviation unit.

Results and Discussion

Figures 2 and 3 compare time-series estimates from individual annual panels, the temporally indifferent five-year moving average, and the 

model-based Kalman filter. All three estimators paint a similar picture of 

changes in the statewide population of lodgepole pine between 2002 and 

2008. This is not surprising since all three use the same FIA field data. 

However, there are notable differences among estimators.

The temporally indifferent moving average estimates that there is no change 

in number of uninfected live trees between 2002 and 2008 (Figure 2), in 

spite of the well-documented epidemic of lodgepole pine tree mortality 

since 2002.The FIA panel estimators and the model-based estimator both 

estimate approximately 20 percent decrease in number of uninfected live 

trees between 2002 and 2008, which is far more plausible than estimates 

with the moving average.
8 USDA Forest Service RMRS-RP-102WWW. 2013.

Figure 3. Comparison of results from three alternative estimators. The estimator based on individual annual panels 

(solid lines), which does not merge observations from different panels, tends to be more variable over time with 

higher uncertainty. The estimated time-series from the other two estimators are smoother over time and less 

uncertain. The temporally indifferent five-year moving average estimator (dashed lines) tends to lag behind the 

model-based Kalman filter estimator (dotted lines). Absent knowledge of the true population state, there are no 

definitive criteria available to judge which estimator is the most accurate.
USDA Forest Service RMRS-RP-102WWW. 2013. 9

The time-series estimates from individual annual panels (solid lines in 

Figure 3) fluctuate more from one year to the next than those with other 

estimators. Uncertainty, as gauged by the spread of the bootstrap distribution, is high compared to the other two estimators, both of which use 

multiple annual panels with larger numbers of FIA plots.

Estimates with the temporally indifferent five-year moving average 

(dashed lines in Figure 3) have the least variation from one year to the 

next and tend to have the lowest uncertainty metrics. However, Patterson 

and Reams (2005) cautioned that “older panels potentially create a lag bias 

(with the temporally indifferent estimator) when estimating current conditions.” The uncertainty metric in Figure 3 ignores lag bias, the degree of 

which is unknown, and uncertainty with the temporally indifferent estimator is surely higher than portrayed with the bootstrap distribution.

Uncertainty metrics for the model-based estimates with the Kalman filter 

(dotted lines in Figure 3) are nearly as low as those for the temporally 

indifferent estimator. Model-based estimates more closely track variations 

in the panel estimates, but with less-extreme annual fluctuations. Like the 

temporally indifferent estimator, the Kalman filter “smoothes” variations 

in annual panel estimates. This is particularly true with estimates of live 

trees with no insect damage, first-year mortality trees, annual net mortality rates of infected live trees, and annual rate of increase in infected 

live trees. Unlike the alternative estimators, the model-based approach can 

forecast trends in the absence of FIA panel data, examples of which are the 

estimates for 2009 and 2010 in Figure 3.

Which Estimator Is Best?

The best estimator is expected to produce estimates with the least variance 

and greatest precision (i.e., least uncertainty) with negligible bias relative 

to that uncertainty. But bias and precision are expectations over all possible samples, and we have only one sample in this case study. Statistical 

theory can infer precision and bias for certain estimators, such as the design-based estimator separately applied to each annual panel. That same 

theory can prove the existence of lag bias with the temporally indifferent 

estimator, but the degree of bias is unique to each population and unknown 

in any practical application without strong but unverifiable assumptions. 

Mathematical theory can prove that the model-based estimator is unbiased 

if there is no bias in model predictions, but that too is unverifiable. We 

believe that our implementation of the model-based algorithm effectively 

reduces the magnitude of potential model bias, especially relative to the 

temporally indifferent estimator, but that has not been proven through theory. Therefore, we simply describe the differences we observe among the 

three sets of estimates given the sample survey data in hand. At best, this 

single case study offers weak evidence for judging the merits of alternative 

estimators.

Both the model-based and temporally indifferent estimators have less 

uncertainty than individual panel estimates (Figure 3). However, this empirical metric ignores lag bias and model bias. For example, the temporally 
10 USDA Forest Service RMRS-RP-102WWW. 2013.

indifferent estimator infers that there is no change in number of live trees 

between 2002 and 2008 (Figure 2). This is almost surely an artifact of lag 

bias. The uncertainty metrics in Figure 3 simply do not capture this type 

of uncertainty because the magnitude of bias, if any, remains a mystery. 

However, the model-based algorithm automatically places less weight on 

model predictions if those predictions show evidence of prediction bias 

during analyses of residuals (see the Uncertainty Metric section). This is 

no guarantee that the model-based estimator is unbiased, but the abovementioned estimation method presumably reduces risk. In principal, the 

model-based estimator should lessen or eliminate lag bias and under-estimation of annual changes that are inherent with the temporally indifferent 

estimator.

All three estimators estimate overall net rates of annual change in both 

infected and uninfected live trees (Figure 3), but only the Kalman filter 

estimates the components of those changes (c1, c2, c3, and c4 in Tables 2 

and 3). However, the statistical properties of these estimates are not well 

understood, so little can be generalized about their precision and accuracy. 

In addition, the Kalman filter incorporates a demographic model, which 

forecasts the biological state of the population into the future, including 

the estimated error distribution propagated through the model (Figure 3). 

This all contributes to analysts’ quantitative understanding of annual demographic process within a dynamic tree population.

Future Improvements to Model-Based Approach

In this case study, we considered a relatively small suite of demographic 

models with the Kalman filter, and there is potential for improvement. For 

example, our best model has parameter values equal to the limits of those 

tested (Table 2). These are percent of infected live trees that are mortality 

trees following year (c2 in Table 3) and percent of surviving infected live 

trees that are uninfected following year (c3 in Table 3). Future analyses 

will extend the range of these parameter values in pursuit of more accurate 

estimates.

The Kalman filter algorithm includes a nonlinear optimization routine that 

adjusts weights on all model predictions so that the distribution of the 42 

standardized residuals agrees with its expected mean and variance (see 

the Uncertainty Metric section). If residuals from any single variable or 

measurement year are unexpectedly large, then model predictions from 

all variables for all years receive less weight. This increases overall uncertainty in the model-based estimates (see Figure 3). For example, the annual 

panel estimate for number of live infected trees is strikingly low in 2005, 

but there is no corresponding phenomenon with the number of mortality 

trees in 2006 (Figure 3). This is inconsistent with our conceptual model 

for lodgepole pine population dynamics. Random sampling error could 

cause a spurious anomaly in the time-series estimates. Ten percent of the 

bootstrap samples manifested a similar anomaly in infected live trees with 

2006 to 2008 panel data, and 60 percent of all bootstrap samples had a 

50 percent decrease or two-fold increase in at least one variable between 

two consecutive years. Therefore, an anomaly caused by sampling error 
USDA Forest Service RMRS-RP-102WWW. 2013. 11

is not unusual given the relatively small sample size with annual panels 

in Colorado. Future analysis will consider censoring the 2005 panel estimate for number of infected live trees as part of our model, which should 

reduce the uncertainty in the model-based estimates for all variables over 

all years. (Recall that dimensions of the annual measurement-vector need 

not conform to those of the population state-vector so long as a linear relationship with known constants, e.g., zeros and ones, exists between the 

two vectors.)

A less striking anomaly is the estimated area of Colorado occupied by 

lodgepole pine trees, which varies by ±30 percent from one year to the 

next. This too is inconsistent with our conceptual model of statewide population dynamics. In an attempt to reduce uncertainty in estimates of the 

tree state-variables, future analyses will consider replacing annual estimates of lodgepole pine area with a single time-invariant state-variable for 

the entire 2002 to 2008 time-interval.

Estimates of mortality rate of infected trees and net rate of increase in infected trees exhibit wide fluctuations over time (Figure 3). This might be 

caused by random sampling error rather than actual processes in the lodgepole pine population. These variables are not included in the state-vector 

or state-transition model. Rather, they are separate nonlinear transformations of other estimates. However, these highly variable rates of change 

could be added to the model as additional state-variables in the Kalman 

filter. Expert judgment could constrain their values to a biologically feasible range, limit their annual variability, or serve as a different type of 

independent “measurement.” This would push our model-based approach 

toward the realm of Bayesian estimation, which is another valid perspective on the Kalman filter (Maybeck 1979).

Bootstrap resampling uses all FIA plots, including plots that do not include 

lodgepole pine trees. Uncertainty would likely decrease if we analyzed 

only those plots (regardless of their forest condition) within the potential 

range of lodgepole pine in Colorado. This would require external geospatial data to supplement FIA plots. In the longer-term, geospatial data 

from forest health risk models and forest health aerial surveys and remotely sensed data from earth-observing satellites might further improve 

estimates (Czaplewski and others 2012). This would require expanding 

dimensions of the state-vector and state-transition model, but that would 

be no more challenging than developing the model in Table 1. These geospatial data need not be unbiased or share the same accuracy as the field 

data. Geospatial data merely need to be sufficiently correlated or associated with field measurements in order to improve accuracy of model-based 

population estimates. However, the potential effect of these changes is unknown. Regardless, success would be constrained by the small sample size 

of annual FIA plots within different geospatial categories. Success might 

improve with continuous geospatial variables, such as the remotely sensed 

proportion of mortality trees, rather than simply a dichotomous indicator 

of insect damage or no damage.

We anticipate additional improvements. Large and small trees appear to 

have fundamentally different mortality processes during the pine beetle 
12 USDA Forest Service RMRS-RP-102WWW. 2013.

epidemic, and we will consider partitioning tree variables into two size 

classes. Since infected live trees can survive several years before mortality, we will consider subdividing the state-variable for infected live trees 

into two separate variables: live trees infected during the past year, and 

live trees infected two or more years previously. We will also add FIA 

panel data from 2009 through 2013. This could improve model-based estimates for 2002 to 2008 because estimates of current conditions improve 

estimates of past conditions with the Kalman filter. Future analyses might 

pool annual panel data from a larger ecoregion to increase sample size and 

decrease uncertainty. However, this would compromise the geographic 

specificity of the analyses. Finally, the density dependent parameter for 

rate of increase in infected live trees, i.e., [(X2002-Xt-1) × c1] in Table 1, 

is indexed relative to the estimated number of live uninfected live trees at 

the beginning of the time-series, i.e., X2002. This estimate varies among 

bootstrap iterations, which might contribute unintended variability to the 

bootstrap distributions. The model will be changed so that the X2002 parameter is replaced with a similarly scaled constant across all bootstrap 

iterations.

The current research is strictly an exploratory proof-of-concept. Ultimately, 

any useful model-based approach must be implemented within FIA operational systems. This requires analysis tools that are intuitive, user-friendly, 

and do not need constant support from a mathematical statistician. If future results with this model-based approach are promising, significant 

development remains.

Conclusions

Can model-based time-series analysis of FIA panel data enhance estimates 

of mortality under an interpenetrating design? The annualized FIA inventory was implemented under the assumption that trends in forest attributes 

such as area, tree volume, tree growth, and tree mortality could be assessed 

in a timelier manner than periodic inventories. The penalty for more timely data is increased uncertainty from the small sample sizes inherent with 

independent panel estimates. Many proponents of the annual inventory 

system felt that supplementary data or modeling procedures could significantly reduce this uncertainty. The model-based estimator we described 

reduces conditional uncertainty and improves quantitative understanding 

of population dynamics and components of change during periods of rapid 

change. We envision several improvements to the model-based implementation, such as changes in the state-vector and state-transition model. 

Hopefully, these will further improve results. While these initial results 

appear promising, more years of annual data in Colorado might be needed 

before signatures of mortality events can be fully evaluated with confidence. Estimates derived from the re-measured (paired) plots, which will 

become available from 2012 to 2022, will substantially increase the power 

to infer significant effects related to tree mortality.
USDA Forest Service RMRS-RP-102WWW. 2013. 13

Acknowledgments

We appreciate the helpful reviews by Dr. KaDonna Randolph and Sara 

Goeking. Any remaining errors or omissions remain our sole responsibility.

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