424 E Some Fundamental Material from Probability and Statistics
The product p(x|ω
i
)p(ω
i
) is called the joint probability of the “events” x and ω
i
.
It is interpreted strictly as the probability that a pixel occurs at position x and that
the class is ω
i
(this is different from the probability that a pixel occurs at position x
given that we are interested in class ω
i
. The joint probability is written
p(x,ω
i
) = p(x|ω
i
)p(ω
i
) (E.2a)
We can also write
p(ω
i
, x) = p(ω
i
|x)p(x) (E.2b)
where p(ω
i
|x) is the conditional probability that expresses the likelihood that the
class is ω
i
given that we are examining a pixel at position x in multispectral space.
Often this is called the posterior probability of class ω
i
. Again p(ω
i
, x) is the prob-
ability that ω
i
and x exist together, which is the same as p(x,ω
i
). As a consequence,
from (E.2a) and (E.2b)
p(ω
i
|x) = p(x|ω
i
)p(ω
i
)/p(x) (E.3)
which is known as Bayes’ theorem (Freund, 1992).
E.2
The Normal Probability Distribution
E.2.1
The Univariate Case
The class conditional probabilities p(x|ω
i
) in remote sensing are frequently assumed
to belong to a normal probability distribution. In the case of a one dimensional spectral
space this is described by
p(x|ω
i
) = (2π)
−1/2
σ
−1
i
exp
#
−
1
2
(x − m
i
)
2
/σ
2
i
$
(E.4)
in which x is the single spectral variable, m
i
is the mean value of x and σ
i
is its
standard deviation; the square of the standard deviation, σ
2
i
, is called the variance of
the distribution. The mean is referred to also as the expected value of x since, on the
average, it is the value of x that will be observed on many trials. It is computed as the
mean value of a large number of samples of x. The variance of the normal distribution
is found as the expected value of the difference squared of x from its mean. A simple
average of this squared difference gives a biased estimate. An unbiased estimate is
obtained from (Freund, 1992)
σ
2
i
=
1
q
i
− 1
q
i
j=1
(x
j
− m
i
)
2
(E.5)
where q
i
is the number of pixels in class ω
i
and x
j
is the jth sample.