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<j 1 X ik(m<j - I- L Kj)
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<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).
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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