Full text: Proceedings of the Symposium on Global and Environmental Monitoring (Part 1)

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 
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<j 1 X ik(m<j - I- L Kj) 
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<q (otherwise BL'B’is not invertible), that is 
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). 
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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