A method of classifying multisource data in remote sensing is presented. The proposed method considers each data source as an information source providing a body of evidence, represents statistical evidence by interval-valued probabilities, and uses Dempster's rule to integrate information based on multiple data source. The method is applied to the problems of ground-cover classification of multispectral data combined with digital terrain data such as elevation, slope, and aspect. Then this method is applied to simulated 201-band High Resolution Imaging Spectrometer (HIRIS) data by dividing the dimensionally huge data source into smaller and more manageable pieces based on the global statistical correlation information. It produces higher classification accuracy than the Maximum Likelihood (ML) classification method when the Hughes phenomenon is apparent

Kim, H.

Swain, P. H.

English

NASA Technical Reports Server

N91-15622
A Method for Classification of Muitisource Data using
Interval-Valued Probabilities and Its Application to HIRIS Data
H. Kim and P.H. Swain
School of Electrical Engineering
and
Laboratory for Applications of Remote Sensing
Purdue University
West Lafayette, IN 47907, U.S.A.
Tel: (317)494-0212, FAX: (317)494-0776
E-mail address: swain@ecn.purdue.edu
ABSTRACT
A method of classifying multisource data
in remote sensing is presented. The pro-
posed method considers each data source as
an information source providing a body of
evidence, represents statistical evidence by
interval-valued probabilities, and uses
Dempster's rule to integrate information
based on multiple data sources.
The method is applied to the problems of
ground-cover classification of multispectral
data combined with digital terrain data such
as elevation, slope, and aspect. Then this
method is applied to simulated 201 -band
High Resolution Imaging Spectrometer
(HIRIS) data by dividing the dimensionally
huge data source into smaller and more
manageable pieces based on the global sta-
tistical correlation information. It produces
higher classification accuracy than the
Maximum Likelihood (ML) classification
method when the Hughes phenomenon is
apparent.
1 INTRODUCTION
The importance of utilizing multisource
data in ground-cover classification lies in
the fact that it is generally correct to as-
sume that improvements in terms of classi-
fication accuracy can be achieved at the ex-
pense of additional independent features
provided by separate sensors. However, it
should be recognized that information and
knowledge from most available data sources
in the real world are neither certain nor
complete. We refer to such a body of uncer-
tain, incomplete, and sometimes inconsis-
tent information as "evidential informa-
tion."
The objective of the current research is
to develop a mathematical framework
within which various applications can be
made with multisource data in remote
sensing and geographic information sys-
tems. The methodology described in this
paper has evolved from "evidential reason-
ing," where each data source is considered
as providing a body of evidence with a cer-
tain degree of belief. The degrees of belief
based on the body of evidence are repre-
sented by "interval-valued (IV) probabili-
ties" rather than by conventional point-
valued probabilities so that uncertainty can
be embedded in the measures.
There are three fundamental problems
in the multisource data analysis based on IV
probabilities: (1) how to represent bodies of
evidence by IV probabilities, (2) how to
combine IV probabilities to give an overall
assessment of the combined body of evi-
dence, and (3) how to make decisions based
on IV probabilities.
The paper describes a formal method of
representing statistical evidence by IV
probabilities based on the Likelihood
Principle. In order to integrate informa-
tion obtained from individual data sources,
the method presented in the paper uses
Dempster's rule for combining multiple
bodies of evidence. Although IV probabili-
ties together with Dempster's rule provide
an innovative means for the representation
and combination of evidential information,
they make the decision process rather
complicated. We need more intelligent
strategies for making decisions. This paper
also focuses on the development of decision
rules over IV probabilities.
75
https://ntrs.nasa.gov/search.jsp?R=19910006309 2020-03-19T19:15:18+00:00Z
2 AXIOMATIC DEFINITION OF
IV PROBABILITY
Interval -valued probabilities can be
thought as a generalization of ordinary
point-valued probabilities. The endpoints of
IV probabilities are called the "upper prob-
ability" and the "lower probability."
There have been various works intro-
ducing the concepts of IV probabilities in
the areas of philosophy of science and
statistics [1][2][3][4]. Although the mathe-
matical rationales behind those approaches
are different, there are some properties of
IV probabilities which are commonly re-
quired. The axiomatic approach to IV prob-
abilities is based on those common proper-
ties, so that it can avoid conceptual ambi-
guities.
DEFINITION [5] Suppose O is a finite set of
exhaustive and mutually exclusive events.
Let _ denote a Boolean algebra of the subsets
of O. The IV probability [L, U] is defined by
the set-theoretic functions:
L" 13-* [0, 11 (2.1)
U'I3 --> [0, 1] (2.2)
satisfying the following properties:
I ) U(A) _>L(A) _>0 for any Ae 13 (2.3)
I I ) U(O) = L(O) = 1 (2.4)
III) U(A) + L(A) = 1 for any Ae_ (2.5)
IV) For any A, Be15 and AnB=O,
L(A)+L(B) <_L(AuB) < L(A)+U(B)
< U(AuB) <_U(A)+U(B) (2.6)
Given a system of IV probabilities over 13,
the actual probability measure, P(A), of any
subset A of @ is assumed to lie in the interval
[L, U] such that
L(A) < P(A) < U(A) (2.7)
The degree of uncertainty about the actual
probability of A is represented by the width,
U(A)-L(A), of the interval. In particular,
U(A)=L(A)=P(A) when there is complete
knowledge of the probability of A. In this
case, the IV probability becomes an ordi-
nary additive probability. And L(A)+L(A)=0
when there is absolutely no knowledge of
the probability of A.
The basic probability assignment m de-
fined in Shafer's mathematical theory of
evidence[6] has the following relations with
the IV probabilities:
£(A) = 2 re(B) (2.8)
Beat
re(A) =Z (_I)IA-BI L(B) for all AcO (2.9)
Bff.A
U(A) = Z re(B) (2.10)
BnA,O
3 REPRESENTATION OF STATISTICAL
EVIDENCE BY IV PROBABILITY
When a body of evidence is based on the
outcomes of statistical experiments known
to be governed by any probability model, it
is called "statistical evidence." One of the
basic problems for any theory of IV prob-
abilities is how to represent a given body of
statistical evidence by IV probabilities.
DEFINITION [6] An upper probability
function U is said to be "consonant" if its
focal elements are nested, i.e., if for A i_O
(i=l ..... r) such that m(Ai) > 0 for all i and
r
m(Ai)=l, AicA j for any i < j, where m is the
i=l
basic probability assignment of u.
Suppose the observed data in a statistical
experiment are governed by a probability
model {P0:0eO}, where P0 is a conditional
probability density function on a sample
space X given 0. Our intuitive feeling is that
an observation xe X seems to more likely
belong to those elements of O which assign
the greater chance to x.
Based on the above intuition along with
the consonance assumption of the upper
probability function, Shafer[6] proposed the
linear plausibility function defined as:
76
_(AIx) -
max p0.(x)0'_ A
max pe(x)
_0
for all A_ I] and A#_ (3.1)
The corresponding lower probability func-
tion is given as:
max po.(x)
0"e A
L(AIx) = 1 - max po(x) for all A_ 13 (3.2)
In particular, when the set A is singleton,
say {e'}, the function in Eq.(3.1) gives the
relative likelihood of 0' to the most likely
element in O.
4 DEMPSTER'S RULE FOR
COMBINING IV PROBABILITIES
Dempster's rule is a generalized scheme
of Bayesian inference to aggregate bodies of
evidence provided by multiple information
sources. Let m 1 and m 2 be the basic prob-
ability assignments associated respec-tively
with the belief functions Bel 1 and Bel 2
which are inferred from two entirely dis-
tinct bodies of evidence E 1 and E 2. For all A i,
Bj, and XkCO, Dempster's rule (or Dempster's
orthogonal sum) gives a new belief func-
tion denoted by
Bel= Bel l _ Bel 2 (4.1)
The basic probability assignment associated
with the new belief function is defined as:
m(Xk )=(1"/0"1 2 ml(Ai)'m2(Bj)
AinBj=X k
for any Xk¢ O (4.2)
where k is the measure of conflict between
Bel 1 and Bel 2 defined as:
k= Z ml (Ai)'m2(Sj) (4.3)
AinBj=_
Dempster's rule computes the basic
probability of X k, m(Xk), from the product
of ml(Ai)and m2(Bj) by considering all A i
and Bjwhose intersection is X k. Once m is
computed for every XkCO, the belief func-
tion is obtained by the sum of m's committed
to X k and its subsets. The denominator (l-f0
normalizes the result to compensate for the
measure committed to the empty set so that
the total probability mass has measure one.
Consequently, Dempster's rule discards the
conflict between E 1 and E 2 and carries their
consensus to the new belief function.
Dempster's rule is both commutative and
associative. Therefore, the order or group-
ing of evidence in combination does not
affect the result, and a sequence of infor-
mation sources can be combined either
sequentially or pairwise.
5 DECISION RULES FOR
IV PROBABILITIES [7]
Consider a classification problem where
an arbitrary pattern xe X from an unknown
class is assigned to one of n classes in O. Let
k(0il0 j) be a measure of the "loss" incurred
when the decision 0 i is made and the true
pattern class is in fact 0j, where i, j = 1..... n.
Also, let _(x) denote a decision rule that tells
which class to choose for every pattern x.
We define the "upper expected loss" and the
"lower expected loss" of making a decision
O(x)=0 i as:
n
fi (x) = 2 2L(OilOj) Ux(Oj) (5.1)
j=l
n
hi(x) = Z k(0il0j) Lx(0j) (5.2)
j=l
where U x and L x are respectively the upper
and the lower probabilities for x being
actually from 0j.
The "Bayes-like rule" is the one which
minimizes both the upper and the lower
expected losses, i.e.,
_(x)=0 i if _(x)_<q(x) and l,i(x)_<l,j(x)
for j=l ..... n (5.3)
A problem with the above decision rule is
that there does not always exist 0 which sat-
isfies the condition in Eq.(5.3), which can
lead to ambiguity. In such an ambiguous
77
situation, one may withhold the decision
and wait for a new piece of information.
Otherwise, the ambiguity may be resolved
by resorting to the following rule, so-called
"minimum average expected loss rule":
_i(x)+ (,i(x) /_i(x)+ [.j(x)
O(x)=0 i if _<2 2
for j=l ..... n (5.4)
As an alternative to the Bayes-like rule,
there are two other rules by which a deci-
sion is made according to individual mea-
sures of the interval, that is, either the up-
per expected loss or the lower expected loss:
(A) minimum upper expected loss rule:
d(x)=0 i if _i(x) </_(x) for j=l,.:., n (5.5)
(B) minimum lower expected loss rule:
§(x)=0 i if /.i(x) < [.j(x) for j=l ..... n (5.6)
Although the above two rules always pro-
duce decisions and there is no ambiguous
situation in making a decision according to
the rules, they do not utilize all of the in-
formation represented by the IV probabili-
ties. The performance of these rules are
compared with the minimum average ex-
pected loss rule in the experiments by
applying them to problems of ground-cover
classification based on remotely sensed and
geographic data.
6 EXPERIMENTAL RESULTS
The experiments have been performed
over two different image data sets. In the
experiments, the classification accuracies of
the multisource data (MSD) classification
based on the proposed method were com-
pared with those of Maximum Likelihood
(ML) classifications based on the stacked
vector approach.
Table 1 describes the set of data sources
for the first experiment. The image in this
data set consists of 256 lines by 256 columns
and covers a forestry site around the
Anderson River area of British Columbia,
Canada. Source 1 is ll-band Airborne
Multispectral Scanner (A/B MSS) data.
Sources 2 and 3 are Synthetic Aperture
Radar (SAR) imagery in Shallow mode and
Steep mode, respectively. Sources 4 through
6 provide digital terrain data.
In this experiment, 6 classes were de-
fined as listed in Table 2, and 100 pixels per
class were used for training data, which is
between 4% and 8% of the total pixels of the
classes in the test fields. The training sam-
ples are uniformly distributed over the test
Source
Index
Table 1. Anderson River Data Set.
Data Spectral Input Spectral
Type Region Channel Band(Ixm)
1 A/B
MSS
2 SAIl
Visible
Near IR
Thermal
Shallow
1 .38 - .42
2 .42 - .45
3 .45 - .50
4 .50 - .55
5 .55 - .60
6 .60 - .65
7 .65 - .69
3 SAR Steep
8 .70 - .79
9 .80 - .89
1 0 .92 - 1.10
4 Topo- Elevation
5 graphic Aspect
6 Slope
11 8 - 14
XHV
XHH
LHV
LHH
XHV
XHH
LHV
I.HH
Table 2. Information Classes for Test of
Anderson River Data Set.
Class
Inde_
1
2
3
5
6
:Total
Cover
T_,pes
Douglas Fir 2 (df2)
Douglas Fir 3 (df3)
DF+Other Species 2
(df+os2)
DF+Lodgepole Pine 2
(df+lp2)
Hemlock+Cedar (hc)
Sorest Clearings _fc)
Tree
Sizes
31 - 40m
21 - 30m
31 - 40m
21 - 30m
31 - 40m
No. of % of
?ixels Total
2246 21.72
1501 14.52
1352 13.08
1589 15.37
1587 15.35
2064 19.96
1033_ I00.0
78
fields so that they may be considered as good
representatives of the total samples.
We have observed that some of the
classes defined in Table 2 cannot be assumed
to be normally distributed in the topo-
graphic data. Thus it was decided to adopt a
nonparametric approach such as the
"Nearest Neighbor" (NN) method [8] in
computing probability measures while the
optical and radar data sources were assumed
to have Gaussian probability density func-
tions.
First, the ML classification based on the
stacked vector approach was carried out for
various sets of the data sources, adding one
source at a time to the A/B MSS data in the
order Elevation, SAR-Shallow, SAR-Steep,
Aspect, and Slope. Then the MSD classifica-
tion based on the proposed method was per-
formed using different decision rules.
Tables 3 and 4 compare the results for the
training samples and the test samples, re-
spectively. Even though the compounded
data in the ML classification were treated as
having Gaussian distributions, the ML and
the MSD methods produced similar results
for the training samples. This is not sur-
prising because the ML method uses con-
ventional additive probabilities assuming
that the knowledge concerning the actual
unknown probabilities is complete, which is
reasonable as far as the training samples
are concerned.
Table 3. Results of Classifications over
Training Samples of Anderson River Data.
Decision
Rule
ML
MUEL
MSD MLEL
MAEL
Sources
1 1,4 1, 2,4 1 -411 - 5 1-6
g2.513 ]8.67 91.67 92.00 _2.83 _3.5(
89.83 92.00 92.5093.17_4.32
88.67 91.17 91.3392.33_3.65
88.50 91.00 91.6791.67_3.5C
Comparing the performance of the three
decision rules, the minimum upper expected
loss (MUEL) rule was superior to the other
rules, the minimum lower expected loss
(MLEL) rule and the minimum average ex-
pected loss (MAEL) rule. It is not known in
general which rule is the best. Further
research is needed to determine whether
guidelines can be devised for selection of
the decision rule.
Table 4. Results of Classifications over Test
Samples of Anderson River Data.
Decision
Rule
ML
MUEL
MSD MLEL
MAEL
Sources
1 1, 4 1,2,41 -4 1 -5 1 -6
74.1677.77 79.13 78.9379.80 ;1.0
80.60 82.39 82.6983.02 _4.54
78.45 81.42 81.6782.24_3.65
78.21 80.9582.0581.88_3.1(
In the second experiment, the proposed
method was applied to the classification of
HIRIS data by decomposing the data into
smaller pieces, i.e., subsets of spectral
bands. The data set used in this experiment
is simulated HIRIS data obtained by RSSIM
[9]. RSSIM is a simulation tool for the study
of multispectral remotely sensed images and
associated system parameters. It creates
realistic multispectral images based on de-
tailed models of the ground surface, the at-
mosphere, and the sensor. Table 5 provides
a description of the simulated HIRIS data set.
Figure 1 is a visual representation of the
global statistical correlation coefficient
matrix of the data. The image is produced by
converting the absolute values of coeffi-
cients to gray values between 0 and 255.
Based on the correlation image, the 201
bands were divided into 3 groups in such a
way that intra-correlation is maximized and
inter-correlation is minimized. Table 6 de-
scribes the multisource data set after divi-
sion. Note that the spectral regions of the
input channels in Source 3 coincide with
the water absorption bands.
With 225 training samples (a third of the
total samples) for each class, the ML classi-
fication and the MSD classification using
the minimum upper expected loss rule were
performed over the total samples for vari-
ous sets of the sources, and the results are
listed in Table 7.
The results of the ML method apparently
show effects of the Hughes phenomenon;
79
the accuracy goes down as the dimensional-
ity of the source increases while the num-
ber of training samples is fixed. In particu-
lar, the accuracy decreases by a consider-
able amount when all features are used.
Presence of the Hughes phenomenon causes
the ML method to be particularly sensitive
to a bad source, Source 3 in this case.
Meanwhile, the proposed MSD classification
method always shows robust performance
and gives consistent results.
Table 5. Description of Simulated
Name
Data Type
Spectral Region
Spectral
Resolution
Image Size
Information
Classes
HIRIS Data Set.
Finney County Data Set
201-band HIRIS data simulated by
RSSIM
0.4 - 2.4m m
0.01ram
45 lines x 45 columns (2025
samples)
Winter Wheat, Summer Fallow,
Unknown
Table 6. Divided Sources of HIRIS Data Set.
Source
Index
Source 1
Source 2
Input
Channels
1- 35 T 107 - 141_ 157 - 201
36 - 95
No. of
Features
115
60
Source 3 96- 106 (1.35 - 1.451.tm) 26
142- 156 (1.81 - 1.951am)
Table 7. Results of Classifications over Test
Samples of Simulated HIRIS Data Set.
Sources
S1 $2 $3 S1, $2 All
ML 75.75 75.60 45.83 74.56 65.14
MSD - 77.83 77.63
6 CONCLUSIONS
In this paper we have investigated how
interval-valued probabilities can be used to
represent and aggregate evidential infor-
mation obtained from various data sources.
Overall concepts of interval-valued proba-
bilities have been employed to develop a
new method of classifying multisource data
in remote sensing and geographic infor-
mation systems. The experiments demon-
strate the ability of our method to capture
1 uncertain information based on inexact and
incomplete multiple bodies of evidence. The
basic strategy of this method is to decompose
36 the relatively large size of evidence into
smaller, more manageable pieces, to assess
52 plausibilities and supports based on each
piece, and to combine the assessments by
75 Dempster's rule. In this scheme, we are able
95 to overcome the difficulty of precisely
estimating statistical parameters, and to
107 integrate statistical information as much as
possible.
141
157
201
Figure 1. Global Statistical Correlation
Coefficient Image of Simulated HIRIS Data.
ACKNOWLEDGEMENTS
This research is partially supported by
the National Aeronautics and Space
Administration under Contract No. NAGW-
925.
The SAR/MSS Anderson River data set
was acquired, processed and loaned to
Purdue University by the Canadian Center
80 ORIGINAL PAGE
BLACK AND WHITE PHOTOGRAPH
for Remote Sensing, Department of Energy,
Mines and Resources, of the Government of
Canada.
REFERENCES
[1] B.O. Koopman, "The axioms and algebra
of intuitive probability", The Annals of
Mathematics, vol.41, pp269-292, 1940.
[2] I.J. Good, "Subjective probability as the
measure of a non-measurable set", in
E.Nagel, Suppes and Tarski (eds.), Logic,
Methodology and the Philosophy of
Science, pp319-329, Stanford University
Press, 1962.
[3] C.A.B. Smith, "Consistency in statistical
inference and decision", J. Roy Stat. Soc.,
Ser.B, vol.23, ppl-25, 1961.
[4] A.P. Dempster, "Upper and lower pro-
babilities induced by a multivalued
mapping", Ann. Math. Stat., vol.38,
pp325-339, 1967.
[5] P. Suppes, "The measurement of be-lief",
J. Roy. Stat. Soc., Ser.B, vol.36, pp160-191,
1974.
[6] G. Shafer, A Mathematical Theory of
Evidence, Princeton University Press,
1976.
[7] H. Kim, "A Method of Classification for
Multisource Data in Remote Sensing
based on Interval-Valued Probabilities",
Ph.D. Thesis, School of Electrical
Engineering, Purdue University, West
Lafayette, IN 47907 U.S.A., August 1990.
[8] Fukunaga, K., Intro. to Statistical Pattern
Recognition, Academic Press, 1972.
[9] Kerekes, J.P., and D.A. Landgrebe,
"Modeling, Simulation, and Analysis of
Optical Remote Sensing Systems", TR-EE
89-49, School of Electrical Engineering,
Purdue University, West Lafayette, IN
U.S.A., 1989.
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A method for classification of multisource data using interval-valued probabilities and its application to HIRIS data

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