# Probabilistics Search for Tracking Targets: Theory and Modern Applications

# Probabilistics Search for Tracking Targets: Theory and Modern Applications

## Irad Ben-Gal, Eugene Kagan

Language: English

Pages: 348

ISBN: 2:00320018

Format: PDF / Kindle (mobi) / ePub

**Presents a probabilistic and information-theoretic framework for a search for static or moving targets in discrete time and space.**

*Probabilistic Search for Tracking Targets* uses an information-theoretic scheme to present a unified approach for known search methods to allow the development of new algorithms of search. The book addresses search methods under different constraints and assumptions, such as search uncertainty under incomplete information, probabilistic search scheme, observation errors, group testing, search games, distribution of search efforts, single and multiple targets and search agents, as well as online or offline search schemes. The proposed approach is associated with path planning techniques, optimal search algorithms, Markov decision models, decision trees, stochastic local search, artificial intelligence and heuristic information-seeking methods. Furthermore, this book presents novel methods of search for static and moving targets along with practical algorithms of partitioning and search and screening.

*Probabilistic Search for Tracking Targets* includes complete material for undergraduate and graduate courses in modern applications of probabilistic search, decision-making and group testing, and provides several directions for further research in the search theory.

*The authors:*

• Provide a generalized information-theoretic approach to the problem of real-time search for both static and moving targets over a discrete space.

• Present a theoretical framework, which covers known information-theoretic algorithms of search, and forms a basis for development and analysis of different algorithms of search over probabilistic space.

• Use numerous examples of group testing, search and path planning algorithms to illustrate direct implementation in the form of running routines.

• Consider a relation of the suggested approach with known search theories and methods such as search and screening theory, search games, Markov decision process models of search, data mining methods, coding theory and decision trees.

• Discuss relevant search applications, such as quality-control search for nonconforming units in a batch or a military search for a hidden target.

• Provide an accompanying website featuring the algorithms discussed throughout the book, along with practical implementations procedures.

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solution of the search problem can be obtained and there exist a number of algorithms, for example, Huffman search [26] or dynamic programming search [27, 28], that obtain the required solution. Nevertheless, these algorithms are mainly applicable to the search for a static target that corresponds to the search for a solution under the assumption that the uncertainty regarding the solution depends on the searcher’s actions and not on external factors. In the second solution type, the neighborhood

of this problem. For reviews of the search and screening theory and various military applications, see [5, 6, 33–36]. Recent results of the theory are also presented in [31, 37]. Further developments of the problem as formulated by Brown [32] are presented in [38]. In search and screening theory, it is assumed that the searcher has such available resources that the reaction of the target to the searcher’s actions may be omitted. In special cases, such a reaction is represented by the detection

density v t=100 . (f) Search density ut=100 ; available search effort K = 1000 A search density that corresponds to an optimal allocation as obtained by Algorithm 2.1. (a) Search density ut=0 = w ∗ ; available search effort K = 10. (b) Search density ut=0 = w ∗ ; available search effort K = 100 Dynamics of Markovian target location density and search density for optimal allocations provided by Algorithm 2.2. (a) Target location density v t=0 . (b) Search density ut=0 ; available search effort K =

x2 with search effort k22 = h2 /σ2 . Then, by similar considerations of the pointwise Lagrangian l[x2 , λ, k22 ] as above, this results in λ = σ3 p(x3 ) and c(x2 , k) = const, and we can specify c(x2 , k) = k22 . 40 PROBABILISTIC SEARCH FOR TRACKING TARGETS Let us apply these values to the Lagrangian l[x, λ, k] at point x1 . We need to deﬁne a search effort k such that the equivalence equation holds: d l[x1 , λ, k] = p(x1 )σ1 e−σ1 k − σ3 p(x3 ) = 0. dk Denote by k12 = h2 /σ1 the joint search

evading the searcher in PPEG with respect to the pair (d[ser] , d[tar] ) of the searcher’s and target’s policies. a Nash equilibrium of PPEG. A pair (d∗[ser] , d∗[tar] ) of the searcher’s and target’s policies is called the solution of the game. an expected reward that is the reward expected to be obtained − by the decision-maker while a policy d is chosen. A maximal expected reward over all possible policies is denoted by −∗ −∗ E(R| d ), where d is the optimal policy. an expected discounted