Fire Detection Algorithms Using Multimodal ... - Bilkent University

CHAPTER 6. WILDFIRE DETECTION 956.3.1 Set Theoretic Analysis of the Weight Update AlgorithmThe weight update algorithm summarized in Fig. 6.5 can be also analyzed in avector space framework without using stochastic signal processing concepts.Ideally, weighted decision values of sub-algorithms should be equal to thedecision value of y(x, n) the oracle:y(x, n) = D T (x, n)w (6.20)which represents a hyperplane in the M-dimensional space, w ∈ R M . A hyperplaneis a close and convex set in R M . At time instant n, D T (x, n)w(n) may notbe equal to y(x, n). The next set of weights are determined by projecting thecurrent weight vector w(n) onto the hyperplane represented by Eq. 6.20. Thisprocess is geometrically depicted in Fig. 6.4. The orthogonal projection w(n + 1)of the vector of weights w(n) ∈ R M onto the hyperplane y(x, n) = D T (x, n)w isthe closest vector on the hyperplane to the vector w(n) (cf. Fig 6.3).Figure 6.3: Orthogonal Projection: Find the vector w(n + 1) on the hyperplaney(x, n) = D T (x, n)w minimizing the distance between w(n) and the hyperplane.Let us formulate the problem as a minimization problem:w(n + 1) = arg minw||w − w(n)|| (6.21)s.t. D T (x, n)w = y(x, n) (6.22)