INTRODUCTION
There is a considerable literature on the problems of distributed detection
and decision in engineering contexts such as Kreidl et
al. (2011), Tsitsiklis (1988) and Tsitsiklis
and Athans (1985). The problem is important because the components of a
distributed detection system may amass more data than they can transmit to a
fusion node and must summarize that data by choice of a message drawn from a
small set. The decentralized or distributed detection problem was first formulated
and studied by Tenney and Sandell (1981) which considers
a “parallel configuration” whereby each sensor makes an observation
and sends a quantized version of that observation to a fusion center. The goal
is to make a decision on the possible hypotheses, based on the messages received
at the fusion center.
The most common architecture in distributed detection is the parallel system
depicted in Fig. 1. It consists of N geographically dispersed
sensors, oneway communication links and a fusion center. Each sensor makes
an observation denoted by of
a random source, quantizes X_{i} into an Mary message U_{i}
= g_{i}(X_{i}) and then transmits
to the fusion center. Upon receipt of U_{1}, U_{1},…, U_{N},
the fusion center makes a global decision U_{0} = D(U_{1}, U_{2},…,
U_{N}) about the nature of the random source.
The optimal design of system, entails choosing quantizers g_{1}, g_{2},…,
g_{N} and a global decision rule D so as to optimize the reliabilities.

Fig. 1: 
Multiple hypotheses parallel distributed detection system,
H: Hypothesis, P: Probability distribution, X: Observation, g: Quantizer,
U: Message 
The messages U_{1}, U_{2},…, U_{N} are all transmitted
to the fusion center which declares hypothesis
to be true, applying a decision rule D.
As was shown by Tsitsiklis (1988), when N tends to
infinity, the error exponents of the absolutely optimal system coincide with
those achieved by the best identical quantizer system. Now, in an Nsensor identical
quantizer system, the quantizer outputs U = g(X) are clearly i.i.d. The optimal
error exponents are then obtained by choosing the mapping g so as to maximize
the appropriate functional such as reliabilities matrix. In case of two hypotheses
both reliabilities corresponding to two possible error probabilities could not
be increased simultaneously, it is an accepted way to fix the value of one of
the reliabilities and try to make the tests sequence get the greatest value
of the remaining reliability.
The need of testing of more than two hypotheses in many scientific and applied
fields has essentially increased recently. The models of multiple hypotheses
optimal testing are studied in some direct such as Ahlswede
and Haroutunian (2006), Hoeffding (1965), Haroutunian
(1990) and Tusnady (1977). The models of the twostage
LAO testing in multiple hypotheses for a pair of families and many families
of Probability Distributions (PDs) are investigated by Hormozi
Nejad and Haroutunian (2012a) and Hormozi Nejad et
al. (2011) and the model of twostage LAO test of distributed detection
for a pair of family of PDs is investigated by Hormozi Nejad
and Haroutunian (2012b). In this paper the problem of distributed detection
of twostage multiple hypotheses LAO testing to detect between hypotheses consisting
of many families of PDs is studied. The matrices of optimal asymptotic interdependencies
of all pairs of the error probability exponents are studied.
Some preliminaries are in coming each sensor observation x takes values in the set X. A deterministic Mary quantizer is a measurable mapping g from the observation space X to the message space U = {1,2,…,M}.
Random Variable (RV) X characterizing the studied object takes values in the
set X and P(X) is the space of all distributions on X. The random source has
S hypothetical PDs that are divided in K disjoint families of PDs. The first
family includes R_{1} PDs P_{1}, P_{2},…, PR_{1},
the second family consists of R_{2} PDs PR_{1}+1, PR_{1}+2,…,
PR_{1}+R_{1} and etc., the Kth family have R_{K} PDs.
The distributions of X under hypotheses
are denoted by .
The distribution of the messages produced by g are denoted by P_{i(g)}
and it is obtainable from P_{i} and g.
Let Nsample x = (x_{1}, x_{2},…, x_{N}) be a
vector of results of N independent observations of the RV X and u = (u_{1},
u_{2},…, u_{N}) be a vector of results of N transmitted
messages to the fusion center. The purpose of the test is using sample u to
detect the actual distribution from given list. The divergence (KullbackLeibler
distance) of PDs P and Q, is defined by Cover and Thomas (1991)
and Haroutunian et al. (2008) as follows:
ONESTAGE MULTIPLE HYPOTHESES LAO TEST OF DISTRIBUTED DETECTION
The procedure of making decision on the base of Nsample is called by the test φ^{N}. The statistician must detect one among S hypotheses. An answer must be defined using vector of results of Nsample u = (u_{1}, u_{2},…, u_{N}).
The probabilities of the erroneous acceptance of hypothesis H_{l} provided that H_{s} is true, are defined as follows:
If the hypothesis H_{s} is true but it is not accepted, then the probability of error is:
Corresponding "reliabilities", are defined as follows for infinite sequence of tests φ:
It follows from Eq. 13 that for every test sequence φ:
The following theorem contains the solution of problem of LAO test φ* construction and existence conditions of the test of elements of matrix E(φ*) of which are positive. For construction of the necessary LAO test for preliminarily given positive values E_{11}, E_{22},…, E_{SS1}, the following subsets of distributions are defined:
Theorem 1: Hormozi Nejad and Haroutunian (2012b):
If all distributions
are different and the positive values E_{11}, E_{22},…,
E_{SS1} are such that the following inequalities hold:
then there exists a LAO sequence of tests, all elements of the reliabilities
matrix E* = {E*_{ls}} of which are positive and are defined in Eq.
46.
When one of the inequalities Eq. 78 is
violated, then at least one element of the matrix E* is equal to zero.
TWOSTAGE MULTIPLE HYPOTHESES LAO TEST OF DISTRIBUTED DETECTION
Now another version of testing will be discussed by supposing N = N_{1}+N_{2}
such that:
and so vectors of messages are as follows:
The twostage test on the base of Nsample denoted by
is the system depicted in Fig. 2. The first stage is to choice
of a family of PDs, it is executed by a nonrandomized test (u)
using the first messages sample u_{1}. The next stage of test is a nonrandomized
test (u_{2},U')
based on another messages sample u_{2} and the outcome of test (u_{1})
that is the first fusion center U'.
First stage of twostage test of distributed detection: The first stage
of decision making consists of using the first messages sample u_{1}
for selection of one family of Pds and it is shown by a test (u_{1}).
Consider for convenience the cumulative numbers:
and the sets of indexes:
Therefore, suppose there are K disjoint families of PDs P_{1}, P_{2},…, P_{K} such that:
Let
be the probability of the erroneous acceptance of the mth family of PDs provided
that the kth family of PDs is true (that is the correct PD is in the kth family):

Fig. 2: 
Twostage multiple hypotheses test of distributed detection
system, H: Hypothesis, P: Probability distribution, X: Observation, g: Quantizer,
U: Message, U', U": Decision rules 
The reliabilities of the infinite sequence of tests φ_{1} are considered
as follows:
For preliminarily given positive values E'_{11}, E'_{22},…, E'_{SS1}, the following subsets of distributions are defined:
Theorem 2: Hormozi Nejad and Haroutunian (2012a):
If all distributions
are different and the positive values E'_{11}, E'_{22},…,
E'_{SS1} are such that the following inequalities hold:
then there exists a LAO sequence of tests, all elements of the reliabilities
matrix E'* = {E'^{*}_{mk}} of which are positive and are defined
in Eq. 911.
When one of the inequalities Eq. 1213
is violated, then at least one element of the matrix E'* is equal to zero.
Second stage of the twostage test of distributed detection: The test
(u_{2},U')
can be defined by using result of the first fusion center U' and the second
messages sample u_{2}.
The probability of the fallacious acceptance of PD P_{l} at the second stage of test, when P_{s} is correct and kth family of PDs is accepted, is:
The probability to reject P_{s} when it is true and kth family of PDs is accepted, is:
Corresponding reliabilities for the second stage of test are:
It follows from Eq. 14 and 15:
Theorem 3: Haroutunian (1990) and Hormozi
Nejad and Haroutunian (2012b): If at the first stage of test the kth
family of PDs is accepted, then for given positive and finite values E"_{ss},
SεD_{k}, s≠C_{k} of the reliabilities matrix, let us
investigate the regions:
and the following values of elements of the future reliabilities matrix E"(φ_{2}*) of the LAO test sequence:
When the following compatibility conditions are valid:
then there exists a LAO sequence of test φ_{2}9, elements of reliabilities
matrix
of which are defined above and are positive.
Even if one of the compatibility conditions is violated, then
has at least one element equal to zero.
Reliabilities of the twostage test of distributed detection: The tool
of making decision according to Nsample denoted
is organized by a pair of LAO tests
and .
Similarly, definitions of error probabilities and reliabilities of twostage
test are as follows:
So error probabilities can be considered as follows:
Using Lemma of types is defined by Cover and Thomas (1991)
and Haroutunian et al. (2008), the following
equality can be created:
According to Eq. 16 and definition of reliabilities are obtained:
Theorem 4: If for different distributions
compatibility conditions of Theorems 2 and 3 are satisfied, then elements of
reliabilities matrix
of the twostage test are defined in Eq. 1719.
When one of the compatibility conditions is violated, then at least one element
of
is equal to zero.
CONCLUSION
The reliabilities of a distributed detection system investigated as the number of sensors tend to infinity. It is assumed that the sensor data are quantized into Mary messages and transmitted to the fusion center for multiple hypotheses testing concerning many families of PDs. The optimal reliabilities in a pair of stages characterized and the compatibility conditions provided for this to happen and description of characteristics of LAO hypotheses testing of distributed detection investigated. The goal was to make a decision on many possible hypotheses, based on the messages received at the fusion centers.
NOTATIONS
N 
= 
Numbers of sensors 
M 
= 
Numbers of messages 
U 
= 
Message 
U', U", U_{0} 
= 
The fusion centre 
D 
= 
Global decision rule 
H 
= 
Hypothesis 
g 
= 
Quantizer 
X 
= 
The space of source data 
U 
= 
The space of messages 
S 
= 
Number of hypotheses 
K 
= 
Number of families 
R_{k} 
= 
Number of PDs in kth family 
P_{k} 
= 
kth probability distribution(PD) 
x = (x_{1}, x_{1},…, x_{N}) 
= 
Vector of observations 
u = (u_{1}, u_{1},…, u_{N}) 
= 
Vector of transmitted messages 
φ, φ_{1}, φ_{2}, Φ 
= 
Tests 
α 
= 
Error probability 
E 
= 
Reliability 
R, D 
= 
Sets 