Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition

290 Supervised Classification Part 2

For the classification of n test data, which a re represented by the i.i.d. sample

X

1

. . . X

n

, the ra ndom variable

Y = X

1

+ X

2

+ . . . X

n

corresponds to the total number of misclassifications. The random variable

describing the misclassification rate is therefore Y/n having mean value

h

1

n

Y i =

1

n

(hX

1

i + . . . + hX

n

i) =

1

n

· nθ = θ. (7.9)

From the independence of the X

i

, i = 1 . . . n, the varia nc e of Y is given by

var(Y ) = var(X

1

) + . . . + var(X

n

) = nθ(1 − θ), (7.10)

so the variance of the misclassificatio n ra te is

σ

2

= var



Y

n



=

1

n

2

var(Y ) =

θ(1 − θ)

n

. (7.11)

For y observed misc lassifications we estimate θ as

ˆ

θ = y/n. Then ...