Van Kootwijk c.a. (1990) these referenees arc

discussed and six methods for predicting

ground cover composition from pixel values

are compared. Only the corrected inverse

regression method (IRe) is presented in this

paper. For a better understanding, first the

generalised least squares (GLS) method is

discussed.

GLS-prediclion is directly based on the

theoretically expected relation (1). It is assum

ed that the deviations r,, = y^-E^) are indepen

dent drawings from a normal distribution with

mean 0 and variance o^. In order to guarantee

that jpredictions respect the unit-sum constraint

^k"=i x ik = l the no-intercept model (1) is rewrit

ten as a model with intercept for p = K-l

cover fractions:

E(yij) = 1 X ikM-kj = m
or:

E(yij) = a j + X£ =1 Xj k (3 K j (2)

The excluded class can be chosen arbitrarily

(here the last class, K, has been chosen), and

the predicted value for this class is simply the

complement x ik .

It is useful to write the model in matrix

notation as

Y = al’ + XB + R (3)

where Y is the n*q matrix with pixel values

for n pixels, X is the n*p matrix with pixel

compositions, a is vector of length q, B is a

p*q matrix, 1 is a vector (here of length n)

with all elements equal to 1, and R is the n*q

matrix of residuals, the rows of which have an

expected covariance matrix £.

The use of this model in multivariate

calibration is standard (e.g. Brown 1982): first

ly, ordinary least squares analysis applied to

the training data provides estimates for a, B

and the residuals R. From the latter an esti

mate for £ is easily obtained. Then predictions

for a set of m pixels with pixel values collec

ted in a m*q matrix Y are made as

X’gls = (Br 1 B’)- 1 Br 1 (Y’-al’) (4)

This predictor is known as the generalised

least squares (GLS) estimator or the classical

estimator. It can be derived by considering a

and B as known and X as a matrix of un

known parameters (hence the usual designation

"estimator" instead of "predictor"). The estimat

ed covariance matrix of the predictions can be

shown to be (BSr'B’)' 1 . It can further be noted

that the predictor qnly exists provided that

p
the number of cover classes should not exceed

the number of spectral bands by more than 1.

Prediction method IR (for Inverse Regres

sion) is computationally simple, as it consists

of calculating the ordinary least squares regres

sion equations for predicting the columns of X

from the columns of Y:

X’ 1R = pF + QY’ (5)

where p is a vector of length K with regressi

on constants, and Q is a K*q matrix with each

row containing the regression coefficients for

one cover class. In many calibration situations

X is fixed, and only the variation in the pixel

values y, is considered stochastic. Inverse re

gression may then seem unnatural because it

would involve a model specifying stochasticity

for x, at fixed values yj. However, equation (5)

can also be derived without this consideration.

The inverse regression predictor is the optimal

linear predictor if we consider the mean

squared error of prediction (MSEP), if only

means and covariances of x, in the training set

are equal to those in the population of all

pixels to be predicted (Lwin and Maritz 1982).

Moreover, the inverse predictor can be shown

to be MSEP-superior to the GLS predictor if

only the means in the training set are repre

sentative for the population and the spreaa of

the cover percentages is not smaller than that

in the population (Sundberg 1985). Of course

according to standard regression theory it

would also be the optimal linear predictor if

round cover data were collected at pixels

aving prespecified pixel value combinations.

Both linear predictors, GLS and IR, give

predicted cover fractions that sum to 1 over

classes, but individual values may be outside

the interval |0,1 ]. Three approaches have been

considered to remedy this: (1) a posterior

correction procedure, (2) linear modelling on

another, unrestricted scale, and (3) generalised

linear modelling. All three approaches make

the predictors in effect nonlinear functions of

the pixel values.

A simple posterior correction procedure

consists of setting negative predictions to 0

followed by rescaling to sum 1. This has been

applied to the results of the inverse regression

method, and we shall refer to the correspon

ding predictor as IRc. The other procedures

are dealt with in Van Kootwijk e.a. (1990).

TRAINING SET FORMATION

A major problem when compiling a training

set of individual mixels with continuous varying

amounts of heather and grass is to find the

position in the image corresponding to the

position in the field where data were collected.

In the following, the area on the ground cor

responding with a pixel will be called a

ground element. Exact delineation in the field

of the ground element corresponding to a

single pixel is impossible, mainly because of

the limited accuracy of the geometric correcti

on and due to the point spread function of

the sensor.

In hcathland vegetation, where spatial varia

tion is high, selecting pixels that correspond to

measured ground elements is critical: shifts of

one pixel yield significantly different training

sets, regression coefficients and cover estima

tes. To overcome this problem, training pixels

and ground elements were arranged in linear

arrays of about 40 elements. In the field, for

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