Target Identification with Dynamic Hybrid Bayesian

Target Identification with Dynamic Hybrid Bayesian Networks Sampsa K. Hautaniemi, Petri T. Korpisaari and Jukka P.P. Saarinen Digital and Computer Systems Laboratory Tampere University of Technology P.O. Box 553 33101 Tampere Finland Email: [email protected] Tel: +358 3 3653878 Fax: +358 3 3653095

ABSTRACT The continuous growth of data has created a demand for better data fusion algorithms. In this study we have used a method called Bayesian networks to answer the demand. The reason why Bayesian networks are used in wide range of applications is that modelling with Bayesian networks offers easy and straightforward representation for combining a priori knowledge with the observations. Another reason for growing use of the Bayesian networks is that Bayesian networks can combine attributes having different dimensions. In addition to the quite well-known theory of discrete and continuous Bayesian networks, we introduce a reasoning scheme to the hybrid Bayesian networks. The reasoning method used is based on polytree algorithm. Our aim is to show how to apply the hybrid Bayesian networks to identification. Also one method to achieve dynamic features is discussed. We have simulated dynamic hybrid Bayesian networks in order to identify aircraft in noisy environment. Keywords: Bayesian networks, attribute association, type identification.

1. INTRODUCTION Target identification in noisy, or sometimes jammed, environment is a very difficult task. However, identification friend, foe or neutral (IFFN) is one of the main questions in military applications. In general, target identification is a part of system that is designed to track an aircraft. Such a system is referred to as a tracking system. Identification, or hostility, of the target is almost impossible to determine using only its kinematic components (such as position, velocity, acceleration). However, it is known that every aircraft has its own specialities such as the form of the wings, the number of the engines, an ability to use only certain band of frequencies etc. In this study these beforehand known (a priori) quantities are referred as attributes. The problem of using attributes is that dimensions of the attribute observations can be almost anything, e.g. frequency is given in Hertz, IFFN has no dimension, exhaust fumes are measured in kelvins etc. Therefore, the association of attributes is difficult. In addition, the association of somehow combined attributes and kinematic observations is even more difficult. There are proposed only few theories, which can be used in attribute association. Bayesian networks are one of them 1,5. The purpose of this paper is to represent theory of the hybrid Bayesian networks and apply it to an identification system. The order of this paper is as follows. First, we define a few preliminary definitions. Then the theories of discrete and continuous Bayesian networks are briefly introduced. The theory of the hybrid Bayesian network is represented in detail. After theoretical discussion we show the modification needed to Bayesian networks in order to solve identification problem. Simulations of dynamic hybrid Bayesian networks in noisy environment are illustrated in Chapter 7.

2. PRELIMINARY DEFINITIONS Bayesian network is a graph with a few extra properties. Figure 1 illustrates a Bayesian networks, circles with an alphabet are called variables. A variable corresponds to an attribute. The arrows connecting two variables are referred as arcs. Path between any two variables is a chain. A Bayesian network is acyclic, which means that loops are prohibited. Moreover, when arcs are directed (i.e. direction of the causality has been determined) the graph is called directed acyclic graph (DAG). Acyclic networks are a subclass of single connected networks. A network is singly connected if there is no more than one chain between any two variables. An example of a singly and non-singly connected networks is illustrated also in Figure 1. A

A

B

C

B

C

D

D

Singly connected network

Non-singly connected network

Figure 1. A singly connected and non-singly connected network. Some variables have special names. A child variable is a variable that is directly dependent of one or more variables. A parent variable has one or more children. The set descendants refers to one variable’s children and children’s children etc. A variable can be a parent and a child at the same time. If a variable does not have any parent, it is called a root variable. The root variable is the most important variable in the networks, usually it is the attribute we are interested in. A leaf variable is a variable that does not have children. An arc indicates the direction of the causality relation. In addition, it also describes how two variables are related to each other. Causality relation between two variables is referred as conditional probability distributions. In a discrete case the conditional probability distributions are called conditional probability tables (CPTs). The CPT and corresponding a priori distribution define a joint distribution. A joint distribution together with a corresponding DAG constitutes Bayesian networks if the following three conditions hold9: 1.Specified conditional probability distributions determine a joint probability distribution of the variables in the DAG. 2.Specified conditional distributions are indeed the conditional probabilities, relative to that joint distribution, of every variable given its parents. 3.Specified joint distribution together with the DAG satisfies the conditional independence assumptions in a Bayesian network.

3. DISCRETE BAYESIAN NETWORKS The theory of the discrete Bayesian networks is quite well-known. Therefore in this article we define only briefly necessary notions. More detailed descriptions can be found e.g. in 8,9,10. 3.1 Propagation in discrete Bayesian networks In this section we explain how information is propagated throughout the network. We define set W to be a set which contains all instantiated variables. An instantiated variable means the same as an observed attribute. Set W is divided into two subsets: WB- is a set containing all the instantiated variables below B. Set WB+ represents instantiated variables above B.9 Later on evidence from the set WB- is referred with the symbol λ . In a similar manner evidence from the set WB+ is referred with the symbol π . In the following pages we divide λ and π further and describe these quantities formally. The first quan-

tity to consider is λ -value, which denotes a conditional probability of all received evidence from descendants. Eq. 1 tells the same in mathematical terms.   P ( W - b ) ,if B ∉ W B i  . λ( bi) =  1 ,if B ∈ W and b i is the instantiated value   0 ,if B ∈ W and b i is not the instantiated value 

(1)

π -value is an easier concept, because instantiation of the current variable does not affect it: +

π ( b i) = P ( bi WB ) .

(2)

When a particular variable receives new evidence from the parents, the evidence must be propagated forward, if there are any children. To help the conceptualizing of propagation we speak of π -message, when the evidence is propagated toward leafs. π -value is an interior value of one variable and is used to compute the belief of the variable. Like π , λ is also divided to λ message and λ -value. λ -message denotes message toward root. π - and λ -values are used in order to update variable’s belief. A belief is a distribution that is conditioned to all evidence in the network. Therefore we can define the total strength of belief in “X = x” by fusing these two supports and normalizing them: Bel ( x ) = αλ ( x )π ( x ) ,

(3)

where α is a normalization constant. Let us assume that variable B is a child of A and has k possible values. Variable A has m values and D is another parent to B and has n values. Resulting network is represented in Figure 2. A

D

B

Figure 2. A Bayesian network with two parents. As B has two parents, new λ -message to A is defined:  k    λ (a ) = π ( d ) P ( b a , d )λ ( b )  . B j B p i j p i   p=1 i = 1 

n

∑

∑

(4)

π -message from A to B is:     π (a ) =  B j    

1 0

,if A is instantiated, for a j ,if A is instantiated, but not for a j ,

P′ ( a j ) ----------------- ,if A is not instantiated λB ( a j )

(5)

where P’(aj) is the conditional probability table of the variables thus far instantiated. λ value is defined by:   ∏ λC ( b i ) ,if B ∉ W   C ∈ s( B ) λ ( bi ) =  . ,if B ∈ W and b i is the instantiated value  1  ,if B ∈ W and b is not the instantiated value  0 i 

(6)

Now we can determine the π value: m

π ( bi ) =

n

∑ ∑

P ( b i a j, d p )π B ( a j )π B ( d p ) .

(7)

j = 1p = 1

Final step is to calculate the belief of variables thus far instantiated: Bel′ ( b i ) = αλ ( b i )π ( b i ) ,

(8)

where α is a normalization constant.

4. CONTINUOUS BAYESIAN NETWORKS In reality there are many events, which are continuous. Discretization makes more or less inaccurate approximations and sometimes inaccuracy is not an alternative. In this chapter theory of networks containing continuous variables only is presented. Through this chapter we have made following assumptions10: 1.All continuous variables are scalars. 2.All distributions are Gaussian. 3.The network is singly connected. If a variable X has a set of parents U = {U1,U2,...,Un} the conditional distribution of X is given: f ( X U i ) = N ( x ;µ x + b i µ i, σ x ) , where bi’s are coefficients containing all relationships between X and its parents: 2

 1 ( x – (µx + biµi ))  N ( x ;µ x + b i µ i, σ x ) = C ( σ x ) ⋅ exp  – --- ⋅ ----------------------------------------- , σx  2 

1 where C ( σ x ) = ------------------------------ is the normalizing constant, which ensures that ( 2 ⋅ π ⋅ σx )

(9)

∫ N ( x ;µ

x

+ b i µ i, σ x ) = 1 .

x

The power of Gaussian distributions is that combination of the parameters is simple. So, when we are applying Gaussian distributions to Bayesian networks, relationships between a variable (X) and its parents (U1,U2,...,Un) are possible to depict using the standard linear regression model: X = b1U1 + b2U2 +... + bnUn + Qx,

(10)

where b1,b2,...,bn are “weights” or regression coefficients representing the relative contribution of each U to depended variable X. Qx is a noise term, which is Gaussian distributed with zero mean. Q x is also assumed to be uncorrelated with any other noise term. If X have children, they are denoted as set Y1,Y2,...,Ym. Eq. 9 replaces the conditional probability tables which were used in Chapter 3.1 to relate each variable to its parents. After a network’s topology has been determined, there are following quantities to define: 1.the link coefficients (bi, i = 1,..., n) 2.the variances of the noise terms (Qi, i = 1,...,n) 3.means and variances of the root variables ( µ, σ ) A continuous network is completely defined using n + 1 parameters per variable, here n is the amount of continuous parents. As we assumed linearity (Eq. 10), and normality (Eq. 9), all belief distributions are Gaussian. Thus λ- and π -messages are fully specified by the mean and the variance of the corresponding distribution. The belief of the variable X is defined (exact formulas in order to compute σ x and µ x are given further): Bel ( X ) = f ( x W ) = N ( x ;µ x, σ x ) .

(11)

In Eq. 11 W stands for the set off all data observed so far. When belief of the variable X is known, next step is to send message to children and parents. These messages are defined: -

+

+

π Yj ( X ) = f ( x W – Wj ) = N ( x ;µ j , σ j ) ,

(12)

and +

-

-

λ X ( u i ) = f ( x W – W i ) = N ( u i ;µ i, σ i ) .

(13)

Because all densities are Gaussian, only the means and the variances need to be computed and propagated. Recalling the definition of the belief, we can say that a variable’s belief is a product of λ - and π -values. λ - and π -values are defined in Eq. 14 and Eq. 15. 10 σλ =

∑ j

1 ------– σj

–1

,

–

µj -------, – σj

(14)

∑b ⋅ µ

(15)

µλ = σ λ ⋅

∑ j

and σπ = Qx +

∑b

2 i

+

⋅ σi ,

µπ =

i

i

+ i .

i

Positive and negative supports that appear in Eq. 14 and Eq. 15 are defined: –

µ µk -------- + -----πσ σ k π + k≠j µ j = ----------------------------, 1 1 ------+ ----– σπ σ

∑ ∑

k≠j

and

k

+

σj =

1 ------ + σπ

1 ∑ ------σ

k≠j

– k

–1

,

(16)

1 µ i = ----  µ λ – bi

∑b

k≠i

⋅ µk  ,  +

k

∑

1 2 + σ i = ------2  σ λ + Q X + bk ⋅ σk .   bi k≠i

(17)

The belief can be computed using Eq. 14 and Eq. 15: σπ ⋅ µλ + σ λ ⋅ µ π µ X = ---------------------------------------, σπ + σλ

σπ ⋅ σλ σ X = ------------------ . σπ + σλ

(18)

There are three boundary conditions, which must be taken into account when implementing general continuous Bayesian networks2: 1.If variable X is a root variable that has not been instantiated, then π ( x ) is set to be equal with the a priori density function. 2.If variable X is a leaf variable that has not been instantiated, then λ ( x ) = 1. Direct result is that Bel(X) = π ( x ) . 3.If variable is instantiated to value x then λ ( x ) = N ( x ;x, 0 ) regardless of the incoming λ -messages. Therefore Bel(X) = N(x;x,0).

5. HYBRID BAYESIAN NETWORKS Theory of hybrid Bayesian networks is similar to principles discussed in discrete and continuous networks. However, compared to continuous and discrete networks, a hybrid network has following restrictions: 1.A continuous variable can have only continuous children. 2.Continuous variables are characterized with (scalar) Gaussian distributions. 3.A network is singly connected. If a continuous child has at least one discrete parent, the belief of the continuous child is characterized via conditioned Gaussians (CGs). The number of the CGs is equivalent to the number of all possible parent value combinations. CGs are defined a priori, just like CPTs in discrete case. The belief of the continuous child is a combination of the CGs. It is notable that belief is no more Gaussian distributed as in the case, where continuous child has only continuous ancestors. A network with three continuous variables (B,C,D) and one discrete variable (A) is illustrated in Figure 3. In addition, the belief of the variable C is drawn on the right. In this network variable A has three values, which can be determined directly by counting the number of the CGs. A continuous variable, whose ancestor is discrete, has as many CGs as there are CGs in its parents. For example, variable D (Figure 3) has three CGs.

0 .0 5

A

B

0 .0 4 5 C G 2 0 .0 4

0 .0 3 5

0 .0 3

0 .0 2 5

C

0 .0 2 C G 3

0 .0 1 5

C G 1

0 .0 1

0 .0 0 5

0

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

D Figure 3. The belief and CGs of the variable B.

7 0 0

8 0 0

9 0 0

1 0 0 0

5.1 π−propagation in hybrid Bayesian networks When discrete parent A (Figure 3) updates its belief, also continuous child C should update its belief. Updating is done by computing weighted sum of all CGs: n

Bel ( C ) =

∑ p ⋅ N ( µ, σ ) , i

(19)

i

i=1

where p is the belief of the parent variable A, n is the number of the values in A and every Ni(µ,σ) refers to corresponding CG. One can obtain directly that Eq. 19 merely scales the CGs in order to achieve new belief. If a continuous parent sends a message to continuous child, the new belief is computed via CGs. The belief is updated nearly same way as in Chapter 4. A slight difference is that when the belief is updated, the influence of the discrete parent must be taken into account. This is done simply by applying Eq. 19 to the resulting distributions from the continuous updating procedure. 5.2 λ-propagation in hybrid Bayesian networks

λ-propagation is a little bit more difficult than π-propagation. In the beginning of this section we made an assumption that continuous variable does not have discrete children. Therefore discrete variables lies in “clusters”, which are possible to separate from the whole network. We take an advantage of this property and apply so called conditioning algorithm to the hybrid Bayesian network9. Traditionally conditioning algorithm is used to cope with non-singly connected networks7,9. Now we use conditioning algorithm because we want to deliver a π-message to continuous variable’s discrete parents. If we consider the network in Figure 3, and assume that the belief of the variable C has been changed, we are interested how the belief of the variable A changes. That is, we want to know the value of the λ−message i.e. P(A | C = cobs). The conditioning algorithm states9: P ( A C ) = α ⋅ P( C A ) ⋅ P ( A ) ,

(20)

where α is normalizing constant, P(C | A) are values that have been obtained from CGs, when an observation has been occurred and P(A) is the belief of the variable A. In other words Eq. 20 is Bayes’ rule. This notion is in harmony with the theory of discrete Bayesian networks, where λ-propagation is done using Bayes’ rule8. As a conclusion we illustrate graphically separation in Figure 4. Variables A and E are discrete and the rest are continuous. The number of the values in A is two, thus both C and D are characterized with two CGs .

E

discrete part of the network.

A

B

C

D

}

E

A

resulting network, conditioned to A = a1.

B

E

resulting network, conditioned to A = a2.

A

B

C

CG1

C

CG2

D

CG1

D

CG2

Figure 4. Separation of the hybrid Bayesian networks.

The separation of the network results as many continuous (sub)networks as there are value combinations in the discrete parent variables. This separation is done to every discrete variables having continuous children. It is obvious that separation increases computation time. Especially networks containing many discrete variables with multiple values may be slow to execute. Almost all networks, which are used in tracking or identification are non-singly connected. One method to cope with the loops is above-mentioned conditioning algorithm. However, exact details are omitted in this study, but interested reader is advised to read10 .

6. DYNAMIC HYBRID BAYESIAN NETWORKS Bayesian networks are time invariant. That is, the a priori distributions at every moment t are the same. A solution to the time dependency problem is to calculate a posteriori probabilities at the moment t and then enroll them to the next moment t+1. This method is theoretically convenient because the future is conditionally independent of the past given a current state: P(x(t)|x(t-1),...,x(t-n)) = P(x(t)|x(t-1)). This property is called the Markov assumption. 4 Links between time slices include a state evolution model, which describes the transition probabilities between states. Propagation from time slice t - 1 can be considered as propagating the π-message. Thus variables in the time moment t - 1 can be consider as dummy variables; they transfer evidence to the networks, but do not affect otherwise the inference. We also notice that because variables in the moment t - 1 are dummy variables, network in the moment t is singly connected. Schematics of a dynamic Bayesian network is shown in Figure 5.1 A

A

B

C

B

C

D

D

t+1

t Figure 5. Dynamic Bayesian networks.

Hybrid networks are partitioned to continuous and discrete parts. Thus, the parameters describing the beliefs in discrete and continuous variables, are transmitted as π-messages to the next moment. As a summary, we can say that dynamic features can be applied to basic Bayesian networks as follows: 1.Bayesian networks process all the observation at the certain moment. Resulting beliefs are a posteriori probabilities. 2.When changing toward next time slice, the beliefs are transferred as π-messages. 3.Initialization is done otherwise normally, but in all dynamic variables there is one π-message more.

7. SIMULATION OF A DYNAMIC HYBRID BAYESIAN NETWORK Before we proceed to the simulation, we explain how sensors are possible to depict using Bayesian networks. Observations of each sensor are corrupted by clutter, jamming etc. Normally, it is possible to model inaccuracies of the sensors. The result of modelling the sensors is an extra child to every observed attribute characterizing the sensor observing the attribute. When the Bayesian network is discrete, the arc between these variables represents sensor’s mixing matrix (i.e. mathematical model of sensor’s inaccuracies). The difference between attribute variables and sensors variables, is that sensor variables are not updated. Thus, the observations are transmitted to Bayesian networks as λ-messages. This method provides modelling the sensors and coping with multiple observations together with one Bayesian network. In a continuous case the arc is again

weight and the actual observation variable includes sensor’s noise covariance as a variable in the continuous Bayesian network In this scenario there has been assumed that there are four possible aircraft types in the surveillance area. The goal of this scenario is to compute the probability of each target after an observation has been made. The network used in the simulation is illustrated in Figure 6. The properties of the variables are represented in Table 1. Last column tells the probability of transmission, which is the probability that a sensor detects the signal target is sending

A Bobs

B

Dobs

C

Cobs

D

Figure 6. A Bayesian network used in simulation. . Table 1. Properties of the variables.

Meaning

Values

Distributions

Observed

Probability of transmission

discrete

type of target

4

-

No

-

discrete

IFF

2

-

Yes

0.9

C

continuous

a continuous feature

-

1

Yes

0.2

D

continuous

a continuous feature

-

2

Yes

0.45

Variable

Type

A B

In the beginning we assume that type distribution is uniformly distributed. Other a priori information is CPT between variables A and B: 0.75 0.4 0.2 0.9 . CGs between D and B: µ1 = 10, µ2 = 40, σ 1= 3, σ2= 5, a normal distribution in C: µ = 40, 0.25 0.6 0.8 0.1 σ = 6 and weight between D and C: 2. Also noises of continuous variables are known: QC = 2 and QD = 1.2. Noises of the sensor (in continuous case) are QCobs = 2 and QDobs = 1. The CPT between Bobs and B is 0.8 0.1 , i.e. a friend is truly a 0.2 0.9 friend by the probability of 0.8 and an enemy is truly an enemy by probability of 0.9. One should notice that Bayesian networks must obey d-separation10. Therefore observation of the variable C does not affect to variables B and A unless variable D is observed. 7.1 Results of the scenario The duration of one scenario was 20 moments. We executed 1000 times one 20 moments scenario. The results of identification are in Figure 7. The reason why the results are much alike is the structure of the network. The only variable, affecting to type probability variable (A) is variable B, which is a two valued variable. Thus all messages to B are two valued. Another reason is that due to the dynamic feature of the network, shapes of the Gaussian distributions in the D and C varies. The distributions in the varia-

ble D changes also because it is a linear combination of its continuous parent (Eq. 10). Therefore, if in the beginning observations are such that means of the distributions change, the effect is multiplied when time goes on. This phenomenon leads to the serious modelling problem. If the expert models a situation by dynamic hybrid Bayesian network and is not sure whether the model is correct or not, the results can be inconsistent. There are, however, some disciplines in order to avoid the problem described. The first is to model the structure of the hybrid Bayesian network so that the variable containing type probabilities has several children. In addition, if a child has multiple descendants the performance should be better if the interior structure of the variable is finer (that is, a variable has many values instead of just two). One should also consider carefully whether or not the continuous roots are really necessary.

Type 2 is correct

Type 1 is correct 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

type1 type2 type3 type4

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

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4

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8

10

12

14

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18

type1 type2 type3 type4

0

20

2

4

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8

1

0.9

0.9

0.8

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type1 type2 type3 type4

0.4

0.3

4

14

0.5

type1 type2 type3 type4

2

12

Type 4 is correct

Type 3 is correct 1

0

10

20

0 2

4

6

Figure 7. Results of the simulation.

8

10

12

14

16

18

20

8. CONCLUSIONS In this paper we have applied dynamic hybrid Bayesian networks to the identification task. The Bayesian networks are convenient method modelling attributes having different dimensions. In addition modelling with Bayesian networks is straightforward and easy, because all information needed is dependency relations between target and its attributes. There are some properties, which restricts the use of Bayesian networks. One of the these properties is that the Bayesian networks are very sensitive to modelling errors. In the simulation variable having two values dominated the updating of the type distributions. As the other value converged to one, there were no changes in the message transmitting procedure. This phenomenon is typical for dynamic Bayesian networks. In order to avoid the problem one can model state evaluation model (SEM) between the time instants. SEM is the answer to the question “if we know that at the moment t the type is friend, by which probability the target is friend at the next moment?”. In our scenario we assumed SEM to be an identity matrix, which emphasized the unpleasant features of the dynamic hybrid Bayesian network. Another problem with Bayesian networks is that they suffer from the computational burden. Bayesian networks are computationally complex and conditioning (in order to achieve single connected network and separate the discrete and continuous variables) slows down the reasoning even more. In this study we have briefly recapitulated the theory of a discrete and continuous Bayesian networks. In addition we have shown one method to build a hybrid Bayesian networks. Bayesian networks are very convenient in any application where one must combine observations of multiple features, because the reasoning is independent of the dimensions of the attributes. Moreover, the hybrid Bayesian networks can be used in situation where the problem is characterized by discrete and continuous features. Simulations showed that when applying the hybrid dynamic Bayesian networks, one should be very cautious and precise in the modelling.

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