The Kalman filter is usually applied to a time
series of measurements (Fig. 3). With each new
measurement, a composite estimate is made, whicl
serves as new initial conditions for the next
prediction from the deterministic model (e.g.,
Fig. 3, year 4).
c
<D
O
<5
CL
No
monitoring data
Low precision
monitoring data
Kalman composite
Estimate from
prediction model
disagrees with
measurement
Smaller
prediction error
Composite
estimate
Other
estimate
Model
estimate
Increasing estimate
of prediction error
improves
agreement
Larger
prediction error
Fig. 4 Expected probability densities for two
estimates that disagree. Adaptive filters assume
the estimated variance of model prediction error
is inaccurate, and change this estimate until the
disagreement is within acceptable bounds.
Fig. 3 Kalman estimates and confidence intervals
for percent forest. In this example from
Czaplewski, et al. (1988), intensive forest
inventories were conducted in years 0 and 10;
lower precision monitoring data were gathered in
years 4 and 7.
The Kalman filter is a multivariate estimator
(Maybeck 1979). It can simultaneously estimate
multiple state variables, such as proportions of
different vegetation cover types. Measured rates
of change can be statistically combined with
rates of change predicted from the deterministic
model. The Kalman filter can model correlated
errors among the state variables and rate
coefficients, correlated prediction errors from
the deterministic model, and random errors in
measurement data.
6.1 Verification of the Kalman filter
Two independent estimates disagree or "diverge"
in that neither estimate is likely given the
other (Fig. 4). Contradictory estimates can be
combined, but the resulting composite estimate
can be biased. Discrepancies are can be caused
by biased estimates of the error distribution
(either location or spread) of the measurement at
time t, or the estimate at time t-1 that is
updated to time t using the deterministic
prediction model.
It is possible that bias exists in the current
measurement. For example, calibration equations
are needed to correct for misclassification bias,
as discussed in Section 3. Also, bias might
exist in the estimated variance of errors from
the prediction model; direct estimates of
prediction variance require known differences
between model predictions and the true status of
the system. As an alternative, adaptive filters
modify initial variance estimates until
disagreements are within acceptable bounds (Fig.
4), often using a time series of residuals
(Sorenson 1985). As accuracy of model
predictions increases, the weight placed on model
predictions will increase, and as will accuracy
of the Kalman filter.
7. KALMAN FILTER APPLIED TO SAMPLE UNITS
Consider the following hypothetical example, in
which a 1/4 subsample of the sample units in the
stratum are observed using remote sensing with
imagery acquired at time t in an interpenetrating
design. (Similar examples could be based on
other intensities, such as 1/7, 1/9, 1/12, etc.)
The biased estimate of the status of each sample
unit in the 1/4 subsample is corrected using a
stratum level calibration estimator, as in
Section 3. An estimate of stratum status at time
t is made with the 1/4 subsample using composite
estimation (Section 5). A different 1/4
subsample of sample units is observed using
remote sensing and imagery acquired at time fc+1.
An estimate of stratum status at time fc+1 might
be made using only this second subsample, as
described in Sections 3 and 5.
The estimate for time fc+1 might be improved using
the sample units in the first 1/4 subsample,
which were observed at time fc. However, changes
between times fc+1 and fc have probably occurred in
the status of each sample unit in the first 1/4
subsample. If a model were available to predict
these changes, then estimates from the 1/4
subsample observed at time fc might be combined
with the 1/4 observed at time fc+1 into an
estimate of the stratum status at time fc+1, using
the composite estimator presented in Section 5.
7.1 Updating estimates for one sanple unit
Predicted true status Jft*i (e.g., proportion
forest) of one sample unit at time t+1 is
Xt*i = ft Xt + fb (1-Xt). (34)
Xt is the estimated status of the sample unit at
time t, ft is the estimated conditional
transition probability that a point is truly
forest at time fc+1, given it is was forest at
tine fc, and ft is the estimated conditional
transition probability that a point at time fc+1
is truly forest, given it was other cover at time
fc. Transition probabilities ft and ft are
predicted from the deterministic model. The
predicted status Xt*i of the one sample unit at
375