Multi-variate Quickest Detection of Significant Change Process

The aim of this consideration is to construct the mathematical model of a multivariate surveillance system. It is assumed that there is net N of p nodes. At each node the state is the signal at moment n ∈ N which is at least one coordinate of the vector -→ x n ∈ E ⊂ ℜ m . The distribution of the signal at each node has two forms and depends on a pure or a dirty environment of the node. The state of the system change dynamically. We consider the discrete time observed signal as m ≥ p dimensional process defined on the fixed probability space (Ω, F , P). The observed at each node process is Markovian with two different transition probabilities (see [17] for details). In the signal the visual consequence of the transition distribution changes at moment θ i , i ∈ N is a change of its character.

To avoid false alarm the confirmation from other nodes is needed. The family of subsets (coalitions) of nodes are defined in such a way that the decision of all member of some coalition is equivalent with the claim of the net that the disorder appeared. It is not sure that the disorder has had place. The aim is to define the rules of nodes and a construction of the net decision based on individual nodes claims. Various approaches can be found in the recent research for description or modeling of such systems (see e.g. [23], [16]). The problem is quite similar to a pattern recognition with multiple algorithm when the fusions of individual algorithms results are unified to a final decision. The proposed solution will be based on a simple game and the stopping game defined by a simple game on the observed signals. It gives a centralized, Bayesian version of the multivariate detection with a common fusion center that it has perfect information about observations and a priori knowledge of the statistics about the possible distribution changes at each node. Each sensor (player) will declare to stop when it detects disorder at his region. Based on the simple game the sensors’ decisions are aggregated to formulate the decision of the fusion center. The sensors’ strategies are constructed as an equilibrium strategy in a non-cooperative game with a logical function defined by a simple game (which aggregates their decision).

The general description of such multivariate stopping games has been formulated by Kurano, Yasuda and Nakagami in the case when the aggregation function is defined by the voting majority rule [9] or the monotone voting strategy [24] and the observed sequences of the random variables are independent, identically distributed. It was Ferguson [5] who substituted the voting aggregation rules by a simple game. The Markov sequences have been investigated by the author and Yasuda [21].

The model of detection the disorder at each sensor are presented in the next section. It allows to define the individual payoffs of the players (sensors). Section 3 introduces the aggregation method based on a simple game of the sensors. Section 4 contains derivation of the non-cooperative game and existence theorem for equilibrium strategy. The final decision based on the state of the sensors is given by the fusion center and it is described in Section 6. The natural direction of further research is formulated also in the same section. A conclusion and resume of an algorithm for rational construction of the surveillance system is included in Section 7.

Following the consideration of Section 1, let us suppose that the process { -→ X n , n ∈ N}, N = {0, 1, 2, . . .}, is observed sequentially in such a way that each sensor, e.g. r (gets its coordinates in the vector -→ X n at moment n). By assumption, it is a stochastic sequence that has the Markovian structure given random moment θ r , in such a way that the process after θ r starts from state -→ X n θr-1 . The objective is to detect these moments based on the observation of -→ X n • at each sensor separately. There are some results on the discrete time case of such disorder detection which generalize the basic problem stated by Shiryaev in [18] (see e.g. Brodsky and Darkhovsky [2], Bojdecki [1], Yoshida [25], Szajowski [20]) in various directions.

Application of the model for the detection of traffic anomalies in networks has been discussed by Tartakovsky et al. [22]. The version of the problem when the moment of disorder is detected with given precision will be used here (see [17]).

The observable random variables { -→ X n } n∈N are consistent with the filtration F n (or

. The random vectors -→ X n take values in (E, B), where E ⊂ ℜ m . On the same probability space there are defined unobservable (hence not measurable with respect to F n ) random variables {θ r } m r=1 which have the geometric distributions:

The sensor r follows the process which is based on switching between two, time homogeneous and independent, Markov processes {X i rn } n∈N , i = 0, 1, r ∈ N with the state space (E, B), both independent of {θ r } m r=1 . Moreover, it is assumed that the processes {X i rn }

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