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

Proof of Theorem 7.1 425

It is easy to see that the estimate

ˆ

b, Equation (A.14), can be calculated

as the slope of the first principal component vector u = (U − 1, u

2

)

⊤

of the

cova riance matrix

s =



s

xx

s

xy

s

yx

s

yy



,

that is,

ˆ

b = u

2

/u

1

.

Thus orthogonal linear regres sion on o ne independent variable is equivalent to

principal components analysis. This is the basis for the IDL procedure shown

in Listing A.1, which performs orthogonal regression on the input arrays X

and Y. The routine is used in some of the ENVI/IDL extensions described

in Appendix C. The Python counterpart orthoregress() can be found in the

module auxil.py.

A.5 Proof of Theorem 7.1

We need