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Constrained Epsilon-Minimax Test for Simultaneous
Detection and Classication
Lionel Fillatre
To cite this version:
Lionel Fillatre. Constrained Epsilon-Minimax Test for Simultaneous Detection and Classication.
IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2011, 57
(12), pp.8055-8071. �10.1109/TIT.2011.2170114�. �hal-00749180�
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 12, DECEMBER 2011 8055
Constrained Epsilon-Minimax Test for Simultaneous
Detection and Classification
Lionel Fillatre
Abstract—A constrained epsilon-minimax test is proposed to de-
tect and classify nonorthogonal vectors in Gaussian noise, with a
general covariance matrix, and in presence of linear interferences.
This test is epsilon-minimax in the sense that it has a small loss of
optimality with respect to the purely theoretical and incalculable
constrained minimax test which minimizes the maximum classifi-
cation error probability subject to a constraint on the false alarm
probability. This loss is even more negligible as the signal-to-noise
ratio is large. Furthermore, it is also an epsilon-equalizer test since
its classification error probabilities are equalized up to a negligible
difference. When the signal-to-noise ratio is sufficiently large, an
asymptotically equivalent test with a very simple form is proposed.
This equivalent test coincides with the generalized likelihood ratio
test when the vectors to classify are strongly separated in term of
Euclidean distance. Numerical experiments on active user identifi-
cation in a multiuser system confirm the theoretical findings.
Index Terms—Constrained minimax test, generalized likelihood
ratio test, linear nuisance parameters, multiple hypothesis testing,
statistical classification, user activity detection.
I. INTRODUCTION
T
HE problem of detecting and classifying a vector in noisy
measurements under uncertainty of vector presence often
appears in engineering applications. This problem has many
applications including radar and sonar signal processing [1],
image processing [2], speech segmentation [3], [4], integrity
monitoring of navigation systems [5], quantitative nondestruc-
tive testing [6], network monitoring [7] and digital communi-
cation [8] among others. This paper deals with the following
detection and classification problem. It is assumed that a mea-
surement vector
consists of either a vector of interest
(for example, a target or an anomaly) plus an unknown nuisance
vector in additive Gaussian noise, or just an unknown nuisance
vector in additive Gaussian noise. If present, the vector of in-
terest must be detected and classified. The unknown nuisance
vector belongs to the nuisance parameter subspace spanned by
the columns of a known
matrix . Hence, the observa-
tion model has the form
(1)
Manuscript received November 29, 2009; revised June 08, 2011; accepted
June 28, 2011. Date of current version December 07, 2011. This work was sup-
ported in part by the French National Agency of Research under Grant ANR-08-
SECU-013-02.
The author is with the ICD, LM2S, Université de Technologie de Troyes
(UTT), UMR STMR, CNRS 6279, BP 2060, 10010, Troyes, France (e-mail:
lionel.fillatre@utt.fr).
Communicated by M. Lops, Associate Editor for Detection and Estimation.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIT.2011.2170114
where both the nuisance parameter vector
and the
vector
are unknown and deterministic. The zero-mean
Gaussian noise vector
has the known positive definite general
covariance matrix
. The vector belongs to the known set of
different vectors , also called
the vector constellation. The relation between the dimension
of the observed vector and the number of non-null vectors
is arbitrary but it is assumed that
.
Three objectives are aimed to be achieved: i) vector detection
which is to decide if
for a given false alarm probability
(probability to declare an alarm when the observation vector is
anomaly-safe), ii) vector classification which is to specify the
actual index
of the vector and iii) insensitivity to the nui-
sance parameters which consists of taking the simultaneous de-
tection/classification decision independently from the value of
the unknown vector
.
A. Relation to Previous Work
From the statistical point of view, this problem of simulta-
neous detection/classification can be viewed as an hypotheses
testing problem between several composite hypotheses [9], [10].
The goal is to design a statistical test which achieves the above
mentioned three objectives according to a prefixed criterion of
optimality.
The first approach to the design of statistical detection and
classification tests is the uncoupled design strategy where detec-
tion performance is optimized under the false alarm constraint
and the classification is gated by this optimal detection. On the
one hand, the classical Neyman-Pearson criterion of vector de-
tection [10] states that it is desirable to minimize the probability
to miss the target subject to a constraint on the false alarm prob-
ability. On the other hand, in terms of target classification, it
is desirable to minimize the probabilities to badly classify the
target. All the above mentioned probabilities generally vary as
a function of both the vector
and the nuisance parameter .
Hence, the uniform minimization of these probabilities with re-
spect to
and is in general impossible. There is no guarantee
that the global performance of this uncoupled strategy will be
acceptable. Consequently, a different approach must be taken,
namely the coupled design strategies for detection and classifi-
cation. These strategies have been studied by only a few authors.
Pioneering works include the papers [11]–[14]. The common
ground in each of these studies is the Bayesian point of view,
i.e., prior probabilities are assigned to all the parameters so that
average performance can be optimized. The problem of simulta-
neous detection and classification using a combination of a gen-
eralized likelihood ratio test and a maximum-likelihood classi-
fier is studied in [15], [16]. This strategy is optimal only in some
cases.
0018-9448/$26.00 © 2011 IEEE
8056 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 12, DECEMBER 2011
Contrary to a purely Bayesian criterion which needs a com-
plete statistical description of the problem, the minimax crite-
rion is well adapted to detection problems where some param-
eters are deterministic and unknown (typically the nuisance pa-
rameter
) and the appearance probability of each vector is
unknown. This criterion consists of minimizing the largest prob-
ability to make a decision error (typically a miss detection or a
classification error). In [17], the generalized-likelihood ratio test
approach is extended to multiple composite hypothesis testing,
by breaking the problem into a sequence of binary composite
hypothesis tests. In some cases, sufficient conditions for min-
imax optimality of this strategy are provided. Lastly, a general
framework to design minimax tests with a prefixed level of false
alarm, namely the constrained minimax tests, between multiple
hypotheses composed of a finite number of parameters is estab-
lished in [18].
It must be mentioned that some interesting papers like [19],
[20] study the asymptotic performance of Bayesian tests be-
tween multiple hypotheses in absence of constraints on the false
alarm probability. When the number of observations is very
large, these papers show that the error probabilities depend only
on the Kullback-Leibler information between the hypotheses.
These results are not yet extended to the case of multiple hy-
potheses testing with a constraint on the false alarm probability.
B. Motivation of the Study
The design of the optimal constrained minimax test mainly
depends on three major points: 1) the geometric complexity
of the vector constellation, 2) the covariance matrix of the
Gaussian noise and 3) the presence of nuisance parameters. To
underline the importance of these points, the three following
cases must be distinguished.
In the simplest case, the vectors
are orthogonal and have
the same norm (the least complex vector constellation), the co-
variance matrix
is the identity matrix (possibly multiplied by
a known scalar) and there is no nuisance parameter
. The op-
timal solution of the problem is given in [9], [21]: this is the
so-called
-slippage problem.
In a more difficult case, the vectors
are not orthogonal
and/or they have different norms (the most complex vector con-
stellation), the Gaussian noise has a general covariance matrix
(not necessarily diagonal) and there is no nuisance parameter
. The theoretical optimal solution is given by the constrained
minimax test [18] but it is generally intractable. This optimal
test compares the maximum of weighted likelihood ratios to
a threshold to take its decision. Although the existence of the
optimal solution is established, this solution depends on some
unknown coefficients, namely the optimal weights and the
threshold. Furthermore, the presence of a general covariance
matrix
plays a significant role in the calculation of the
optimal weights, even if the vector constellation is simple.
In fact, it is always possible to get a diagonal covariance
matrix after prefiltering but this operation may involve that
the vector constellation becomes more complex. For example,
orthogonal vectors of interest may be no longer orthogonal
after prefiltering. For all these reasons, it is often impossible
to easily calculate the optimal weights and, even, to reduce
the number of weights to be determined by using invariance
principles [10], [22]. The optimization problem to be solved for
calculating the optimal weights is highly nonlinear and leads to
a combinatorial explosion.
Example 1 (Slippage Problem With Unstructured Noise):
This example is directly related to the important problem of
detecting outliers in multivariate normal data [23]. Let
be a
Gaussian random vector with zero mean and a known general
covariance matrix under hypothesis . Under hypothesis
, the th component of has the known mean .
Hence, the vector
to detect and classify corresponds to
where denotes the vector with a 1 in the th coordinate and
0’s elsewhere. Since the covariance is known but it differs from
the identity matrix, it is no longer possible to use the famous
principle of invariance to solve such a slippage problem [9].
The optimal solution is not known up to now. Example 6 shows
that the results proposed in this paper can be used to solve this
slippage problem.
In the most difficult case, the vectors ’s are not orthog-
onal and/or they have different norms, the Gaussian noise has
a general covariance matrix
and there is an unknown nui-
sance parameter
. To our knowledge, the optimal constrained
minimax test is unknown in this case. The main reasons which
explain this lack of results are the followings. First, the presence
of linear nuisance parameters certainly complicates the mutual
geometry between the vectors. Next, there is an unavoidable an-
tagonism between the detection and classification performances
of the test. For example, to get small classification errors, it is
necessary to accept a loss of sensibility for the probability of de-
tection. The tradeoff between these two requirements is essen-
tially based on the worst case of detection and the worst case of
classification which are generally difficult to identify. Finally, as
underlined in [24]–[27], the analytic calculation of the miss de-
tection probability and the classification error probabilities are
intractable, which makes difficult the derivation of an optimal
test.
Example 2 (Integrity Monitoring of Navigation Systems): Let
be a Gaussian random vector with the known covariance ma-
trix . Under hypothesis , its mean is where is the user
unknown parameters and
is a matrix describing the measure-
ment system [5]. Under hypothesis
is contaminated by
a scalar error with intensity
. The common solution, namely
the parity space approach, involves two steps. First, the user un-
known parameters are eliminated by projecting
on the null-
space of
. This null-space is called the parity space in the ana-
lytical redundancy literature [28]. Next, the error is detected and
classified (isolated) directly in the parity space. Unfortunately,
the first step may generate some linear dependencies between
the possible error signatures in the parity space. In this case, the
problem is not theoretically solved. Example 7 shows that this
paper proposes a solution to this problem.
Example 3 (New User Identification in a Multiuser System):
In a multiuser system, after chip-matched filtering and chip rate
sampling, the received signal vector under hypothesis
is
modeled as
where the th column of the ma-
trix is the normalized unit energy signature waveform vector
FILLATRE: CONSTRAINED EPSILON-MINIMAX TEST FOR SIMULTANEOUS DETECTION AND CLASSIFICATION 8057
of user
is the diagonal matrix of user amplitudes and
is the vector whose th component is the antipodal symbol,
or 1, transmitted by user [29], [30]. The random vector
has zero mean and the known covariance matrix where
is the identity matrix of size . Under hypothesis , the
vector
is added to , i.e., a new user with the sig-
nature
emits the symbol with the amplitude .Itis
assumed that belongs to a finite set of predefined nonorthog-
onal signatures. The multiple-access interferences
can be
eliminated by using the above mentioned parity space approach
[31]. The goal is to detect the new user arrival and to identify
the waveform
. This is a difficult problem, especially when
the common length
of each user’s signature is shorter than
the total number of simultaneously active users [32]. Section VI
will further elaborate upon this example.
When the optimal statistical test is unknown or intractable,
it is often assumed that the optimal weights are equal (since
it is the least informative
a priori choice) and the threshold is
tuned to satisfy the false alarm constraint. The resulting test is
called the
-ary Generalized Likelihood Ratio Test (MGLRT)
between equally probable hypotheses. It must be noted that the
optimality proof of the MGLRT is still an open problem.
C. Contribution and Organization of the Paper
The first contribution is the design of a constrained
-min-
imax detection/classification test solving the detection/clas-
sification problem in the case of nonorthogonal vectors with
linear nuisance parameters and an additive Gaussian noise with
a known general covariance matrix. This test is based on the
maximum of weighted likelihood ratios, i.e., it is a Bayesian
test associated to some specific weights. It is
-optimal (under
mild assumptions) in the sense that it is optimal with a loss
of a small part, say
, of optimality with respect to the purely
theoretical minimax test. This loss of optimality, which is the-
oretically bounded, is unavoidable since the purely theoretical
minimax test is intractable due to the difficulties above men-
tioned. This loss is even more negligible as the signal-to-noise
ratio is large. It is also shown that this test coincides with
a constrained
-equalizer Bayesian test which equalizes the
classification error probabilities over the alternative hypotheses
up to a constant .
Secondly, an algorithm is proposed to compute the optimal
weights of the proposed test with a reasonable numerical com-
plexity. This algorithm is based on a graph, namely the separa-
bility map, describing the mutual geometry between the vectors.
This separability map serves to identify the least separable vec-
tors, making possible the design of the associated constrained
-minimax test. This map is also used to calculate in advance
the asymptotic maximum classification error probability of the
constrained -minimax test as the Signal-to-Noise Ratio (SNR)
tends to infinity. Moreover, in the case of large SNR, an asymp-
totically equivalent test is proposed whose optimal weights have
a very simple form.
Finally, it is shown that the MGLRT is
-optimal when the
mutual geometry between the hypotheses is very simple, i.e.,
when each vector has at most one other vector nearest to it in
term of Euclidean distance. In general, the MGLRT is subop-
timal and the loss of optimality may be significant.
The paper is organized as follows. Section II starts with the
problem statement and introduces the statistical framework that
will be used in this paper, including the presentation of the con-
strained
-minimax criterion. Section III describes the general
methodology to reduce the detection/classification problem
between multiple hypotheses with nuisance parameters to a
detection/classification problem between multiple hypotheses
without nuisance parameters. This reduction is based on the fact
that it is sufficient to design a constrained
-equalizer Bayesian
test to get the constrained
-minimax one. Section IV defines
the separability map and proposes the main theorem of this
paper which establishes the constrained
-equalizer Bayesian
test. The proof of this theorem is given in Appendix A. The
false alarm probability of this test is calculated in Appendix B.
Section V gives an asymptotically equivalent test to the con-
strained
-minimax as the SNR tends to infinity. The derivation
of this test is given in Appendix C. It is also shown that the
MGLRT is asymptotically
-optimal when the mutual geometry
between the hypotheses is very simple. Section VI deals with a
practical problem, namely the identification of a new active user
in a multiuser system, showing the efficiency of the proposed
-optimal test. Finally, Section VII concludes this paper.
II. PROBLEM STATEMENT
This section presents the multiple hypotheses testing problem
which consists in detecting and classifying a vector in the pres-
ence of linear nuisance parameters. A new optimality criterion,
namely the constrained
-minimax criterion, is introduced and
motivated.
A. Multiple Hypotheses Testing
The observation model has the form (1). Without any loss
of generality, it is assumed that the noise vector
follows a
zero-mean Gaussian distribution
. In fact, it is always
possible to multiply (1) on the left by the inverse square-root
matrix
of to obtain the linear Gaussian model with the
vector , the nuisance matrix and a Gaussian noise
having the covariance matrix
. It is also assumed that is a
full-column rank matrix. If the matrix
does not satisfy this
assumption, it suffices to keep the maximum number of linear
independent columns to get a full-column rank matrix spanning
the same linear space. It is then desirable to solve the multiple
Gaussian hypotheses testing problem between the statistical hy-
potheses
(2)
where
. The following condition of
separability is assumed to be satisfied:
(3)
In other words, it is assumed that the intersection of the two
linear manifolds
and (which are parallel to each other)
is an empty set for all
(two parallel linear manifolds with
nonempty intersection are equal). Here, the parameter set
associated to hypothesis is not a singleton, hence, is
called a composite hypothesis [33]. Otherwise, the hypothesis is
simple and it is identified by the absence of an underscore, say
8058 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 12, DECEMBER 2011
. The set of decision strategies for the -ary hypotheses
testing problem (2) is specified by the set of test functions.
Definition 1: A test function
for the mul-
tiple hypotheses
is a -dimensional vector
function defined on such that and
Given , the test function decides the hypothesis
if and only if . In this case, the other components
, are zero. The study of randomized test functions
(a random test function satisfies
for some )
is not considered in this paper since the probability distribution
of
is continuous whatever the true hypothesis. The average
performance of a particular test function
is determined by
the
functions where
stands for the expectation of when follows the distribution
. The false alarm probability function is given by
when . The function for
describes the probability of miss detection. When
is the classification error proba-
bility. The maximum classification error probability for the test
function
is denoted
For a test where is a given superscript, all the above men-
tioned notations are completed by the superscript
. Let be
the set of test functions whose maximum
false alarm probability is less or equal to
B. Constrained Epsilon-Minimax Test
As mentioned in the introduction, the constrained minimax
criterion given in [18] is a very natural criterion for problem
(2). Unfortunately, as underlined in [18], since the geometry of
the set
may be very complex, it is impossible
to infer the structure of the minimax test. Hence, to overcome
this difficulty, it makes sense to consider constrained
-minimax
tests, i.e., tests that approximate optimal minimax test with a
small loss, say
, of optimality.
Definition 2: A test function is a con-
strained
-minimax test in the class between the hypotheses
if the following conditions are fulfilled :
i)
;
ii) There exists a positive function
satisfying
as such that
for any other test function .
Obviously, Definition 2 assumes that the positive constant
is
(very) small. In some cases, it is possible to get
(see
Section VI) but, generally,
because of the vector con-
stellation complexity. Contrary to a purely constrained minimax
test, the design of a constrained
-minimax test tolerates small
errors on the classification error probabilities. Hence, it becomes
possible to use lower and upper bounds on these probabilities in
order to evaluate the statistical performances of the test. As un-
derlined in [25], the exact calculation of these probabilities is
generally intractable.
III. E
PSILON-MINIMAX TEST FOR COMPOSITE HYPOTHESES
This section introduces the constrained
-equalizer Bayesian
test of level . Proposition 1 shows that such a test is neces-
sarily a constrained
-minimax one. The first step to design the
constrained
-equalizer Bayesian test between composite hy-
potheses in presence of nuisance parameters consists in elim-
inating these unknown parameters. Proposition 2 shows that
this elimination, based on the nuisance parameters rejection,
leads to a reduced decision problem between simple statistical
hypotheses.
A. Constrained Epsilon-Equalizer Test
Let us recall the definition of the constrained Bayesian test
before introducing the definition of the constrained
-equalizer
test. Let
be a probability distribution over called the a priori
distribution. For all
, this distribution induces
some a priori distributions on the linear manifolds and
some a priori probabilities
such that
(4)
Let
be the probability density function (pdf) of the ob-
servation vector
following the distribution . To each
hypothesis is associated the weighted pdf (see de-
tails in [10]) defined by
Let be the weighted log-likelihood ratio defined by
(5)
for
. The constrained Bayesian test function
of level associated to is given by
(6)
and for
(7)