Fire Detection Algorithms Using Multimodal ... - Bilkent University

CHAPTER 6. WILDFIRE DETECTION 96Solution can be obtained by using Lagrange multipliers:L = ∑ i(w i (n) − w i ) 2 + λ(D T (x, n)w − y(x, n)) (6.23)Taking partial derivatives with respect to w i :and setting the result to zero:∂L∂w i= 2(w i (n) − w i ) + λD i (x, n), i = 1, ..., M (6.24)a set of M equations is obtained:2(w i (n) − w i ) + λD i (x, n) = 0, i = 1, ..., M (6.25)w(n + 1) = w(n) + λ D(x, n) (6.26)2The Lagrange multiplier, λ, can be obtained from the condition equation:as follows:D T (x, n)w − y(x, n) = 0 (6.27)y(x, n) − ŷ(x, n) e(x, n)λ = 2 = 2(6.28)||D(x, n)|| 2 ||D(x, n)|| 2where the error, e(x, n), is defined as e(x, n) = y(x, n) − ŷ(x, n) and ŷ(x, n) =D T (x, n)w(n). Plugging this into Eq. 6.26w(n + 1) = w(n) +is obtained. Note that this is identical to Eq. 6.30The projection vector w(n + 1) is calculated as follows:w(n + 1) = w(n) +e(x, n)D(x, n) (6.29)||D(x, n)||2e(x, n)D(x, n) (6.30)||D(x, n)||2This equation is the same as the NLMS equation, Eq. 6.19, with µ = 1.Whenever a new input arrives, another hyperplane based on the new decisionvalues D(x, n) of sub-algorithms, is defined in R My(x, n + 1) = D T (x, n + 1)w (6.31)