3. MISCLASSIFICATION IN REMOTE SENSING 
(5) 
A portion of deviation will be caused by 
errors in measuring the status (e.g., proportion 
forested) of sample unit i. 
3.1 Misclassification error model 
Let a measurement or calibration model for the 
unknown true status Xi of sample unit i be 
Xi = Ht Yi + Ho (1-Fi), (2) 
where Fi is the imperfect remotely sensed 
estimate of proportion forest in sample unit i, 
and Ht is the known conditional probability that 
a point is truly forest given that it is 
classified as forest by the remote sensing 
process. Similarly, (1-Fi) is the remotely 
sensed proportion measurement of other cover, and 
Ho is the conditional probability that a point is 
truly forest given it is classified as other 
cover by the remote sensing process. When 
classification accuracy of the remote sensing 
process is high, Ht will nearly equal 1, and Ho 
will nearly equal 0. 
3.2 Misclassification bias 
The remotely sensed estimate Yi is a biased 
estimate of true status Xi of sample unit i if 
classification errors occur. Solving (2) for Fi , 
Yi = (Xi - Ho) / (Ht -Ho). (3) 
The remotely sensed estimate Yi in equation (3) 
will not equal the true status Ai unless Ht 
equals 1 and Ho equals 0, i.e., perfect 
classification accuracy. 
3.3 Estimation of calibration model 
In practice, the values of Ht and Ho are not 
perfectly known. Rather, Ht and Ho are assumed 
the same for all sample units in the stratum, and 
their values are estimated using a finite sample 
of reference points for which the remotely sensed 
and true status are known. For example, 
reference points might be available for M 
systematically located 0.4 ha forest inventory 
plots, which are measured in the field by USDA 
Forest Service crews, where the field 
classification is considered to be without error. 
Under certain conditions, the location of these 
field plots can be accurately registered to 
remotely sensed images, so that both remotely 
sensed and true classifications are available for 
a small sample of point plots. This would 
provide the necessary sample of reference points 
to make estimates (Ht and Ho) of the true 
conditional probabilities (Ht and Ho). 
Consider the statistical sampling model: 
Ht - Ht + Jf , Ho - Ho + Jo , (4) 
where Jt and Jo are random variables that equal 
the differences between the true and estimated 
conditional probabilities. Ht might be estimated 
from the Mt 0.4 ha Forest Service plots 
classified as forest using remote sensing 
equation (5): 
Ht = [(Wf)i + (Ht) 2 + ... + (Ht)nt]/Mt, 
where (Ht)i = 1 if 0.4 ha Forest Service plot i 
is truly forest given it is classified as forest 
using the remote sensing procedure, and (Ht)i = 0 
otherwise. Similarly, Ho might be estimated from 
the Mo 0.4 ha Forest Service plots that are 
classified as other cover using the remote 
sensing procedure, 
Ho = [(/fc)l + (Ho) 2 + ... + (Ho)no]/Mo, (6) 
where (Ho)i = 1 if 0.4 ha plot i is truly forest 
given it is classified as other cover using 
remote sensing, and (Ho)i = 0 otherwise. 
The following is an estimate (Xi ) of the status 
of sample unit i, using the estimated conditional 
probabilities (Ht and Ho) of correct and 
incorrect remotely sensed classifications from 
(5) and (6), and the known remotely sensed status 
}'i (Tenenbein 1972): 
Xi = Ht Yi + Ho (l-Yi). (7) 
3.4 Variance of calibrated estimate 
From equations (2), (4), and (7), the unbiased 
estimate of the true status Xi of sample unit 
i, given the imperfect remotely sensed 
measurement Yi of the same sample unit, is 
Xi = (Ht + Jt) Yi + (Ho + Jo) (1-tt), 
= [Ht Yi + Ho (1 - Fi ) ] + [Jt Yi + Jo (1 - Fi) ], 
= Xi + [Jf Fi + Jo (1-Fi ) ]. (8) 
The estimate Xi of the status of sample unit i in 
(8) contains uncertainty propagated from the 
imperfect model for classification error. Since 
the estimated conditional probabilities (Ht,Jt) 
are assumed unbiased, E[Jf] = E[Jo] = 0, and 
measurement Fi is a known nonrandom constant, 
then the variance of the estimate in (8) is 
var(Ai) = [Jf Fi + Jo (1-Fi ) ] 2 , 
= E[Jf 2 ] Fi 2 + E [ Jo 2 ] (1-Fi) 2 , 
= var(Ht) Fi 2 + var(/jb) (1-Fi) 2 . (9) 
If it i8 assumed that there are no registration 
errors between field points and the remote 
sensing imagery, then the random errors Jf and Jo 
are caused solely by sampling error. The 
sampling variances var (Ht) and var (Ho) can be 
estimated from the simple randomized sample of 
M=Mt+H> plots using the binomial distribution: 
\&r(Ht) - 
Ht 
(l -Ht) 
/ 
Mt, 
(10) 
var (Ho) = 
Ho 
(1 -Ho) 
/ 
Mo. 
(11) 
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