in X. Weights in (24) produce minimus variance
estimates given heterogeneous, correlated errors
and n = 2, and can be expressed in matrix form as
Gc = (1 1' - I) C 1 / [1’ (1 1’ - I) C 1 ],
(1 1’ - I) C 1
(n-1) V Q 1,
(28)
where I is the (n x n) identity matrix. Based on
sui hoc rationale for n > 2, the weights in
equation (28) can be used in estimation equation
(26), where G * Gc, for combining estimates from
many sample units (n > 2) into an unbiased
estimate of the stratum status. If the sample
units are independent and have homogeneous
variance, then s 2 , C = I s 2 , and (28) becomes
(11’ -I) s 2 1 (n-1) 1 1
(n-1) 1’ s 2 1 (n-1) n n.
The mean of the sample units can be expressed as
matrix equation (26), where the (n x 1) weighting
vector is
Gb = 1 / n.
(30)
If the sample units are independent with
homogeneous variance, then the composite estimate
in (26) and (28) is identical to the mean because
Gc from (29) is identical to G* from (30).
5.6 Efficiency of vector composite estimator
When errors are heterogeneous and dependent, the
composite estimator in (26) and (28) is expected
to be more efficient than the mean of the sample
units. From (27) and (28), variance var(Ac) of
the composite estimate Ac of stratum status X is
var(Ajc) - Gc * C Gc.
(31)
The weighting vector Ga for the mean in equation
(30) may be rewritten as
G.
11’ Cl 11’C1-C1+C1
n 1’ C 1 n 1’ C 1
(1 1’ - I) C 1 Cl
= + (32)
n 1’ C 1 n 1’ C 1,
From (27), (28), and (31), variance var(Ak) of
the mean estimate A» of stratum status is
var( A»)
Gc ' C Gc ( n~ 1) 2
+
n 2
1’ C’ C C 1
n 2 (n-1) 2 ,
var( Aic ) ( n-1 ) 2
n 2
1’ C’ C C 1
n 2 (n-1) 2 .
(33)
Since (n-1) 2 //? 2 in (33) will nearly equal to one
for typical sample sizes n, variance var(Jc) of
the composite estimate in (31) will be smaller
than variance var(A*) of the mean estimate in
(33), unless there are large negative covariances
among sample units in covariance matrix C such
that the scalar (l’C’CCl) in (33) is negative.
6. KALMAN FILTER
Misclassification in remote sensing will bias
estimated status of individual sample units. A
calibration model can correct for this bias, but
the calibration model will introduce uncertainty
into our remotely sensed estimate of the status
of each sample unit (Section 3). Additional
uncertainty is introduced by changes in land use,
land management, and vegetation succession in
each sample unit. If a deterministic model could
predict these changes after the remotely sensed
imagery is acquired, then this model can estimate
status of each sample unit over time.
Predictions of the deterministic model can be
incorporated into statistical estimates of
stratum status using the Kalman filter (e.g.,
Gregoire and Walters 1988). Dixon and Howitt
(1979) describe how the Kalman filter can be
applied to sampling with partial replacement in
continuous forest inventories. Czaplewski (1990)
presents a simple tutorial example of estimating
forest cover over time using the Kalman filter.
The Kalman filter is portrayed in Fig. 2. One
unbiased estimate is made at time t (e.g., a
calibrated remotely sensed measurement). The
other unbiased estimate (e.g., a previous,
calibrated remotely sensed measurement) is made
at time t-1, but is updated for expected changes
between times t and t-1 using the deterministic
model. Variance for the updated estimate
includes effects of errors in the previous
estimate that are propagated over time, and
prediction errors from the deterministic model.
Perfect Imperfect
prediction model prediction model
Fig. 2 Probability density for a Kalman estimate
that i8 a composite of measurement data at time t
and a prior estimate at time t-1, which is
updated using a prediction model. Given a
perfect prediction model, only estimation error
at t-1 is propagated to time t. More
realistically, the prediction model is imperfect,
and a prediction error also occurs. The Kalman
filter combines measurements and model
predictions into a composite estimate, weighted
inversely proportional to their variances.
