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

70 Image Statistics

Y (ν) is a random variable representing the νth mea surement of the dependent

variable and R(ν), referred to as the residual error, is a random variable rep-

resenting the measurement uncertainty. The x(ν) are exact. We will assume

that the individual measurements ar e uncorrelated and that they all have the

same variance:

cov(R(ν), R(ν

′

)) =



σ

2

for ν = ν

′

0 otherwise.

(2.83)

The realizations of R(ν) are y(ν) − a − bx(ν), ν = 1 . . . m, from which we

define a least squares goodness-of-fit function

z(a, b) =

m

X

ν=1



y(ν) − a − bx(ν)

σ



2

. (2.84)

If the residuals R(ν) are normally distributed, then we recognize Equation

(2.84) as a re alization ...