By Ivan Bratko (auth.), Andrej Dobnikar, Uroš Lotrič, Branko à ter (eds.)

The two-volume set LNCS 6593 and 6594 constitutes the refereed lawsuits of the tenth foreign convention on Adaptive and ordinary Computing Algorithms, ICANNGA 2010, held in Ljubljana, Slovenia, in April 2010. The eighty three revised complete papers offered have been conscientiously reviewed and chosen from a complete of a hundred and forty four submissions. the 1st quantity comprises forty two papers and a plenary lecture and is geared up in topical sections on neural networks and evolutionary computation.

**Read Online or Download Adaptive and Natural Computing Algorithms: 10th International Conference, ICANNGA 2011, Ljubljana, Slovenia, April 14-16, 2011, Proceedings, Part I PDF**

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**Extra resources for Adaptive and Natural Computing Algorithms: 10th International Conference, ICANNGA 2011, Ljubljana, Slovenia, April 14-16, 2011, Proceedings, Part I**

**Example text**

By Theorem 5 (i), span GKγd (X) is dense in (C(X), . sup ) and by Proposition 1, Ez is continuous on (C(X), . sup). Thus by Proposition 2, fa+ is an argminimum of Ez over C(X). As the set of argminima is convex, the statement holds. Corollary 2 shows that in the space of continuous functions C(X), for each width a > 0 there is an argminimum of the empirical error formed by a linear combination of Gaussians with the width a. All these Gaussians have the same centers given by the input data. The set of all these argminima is linearly independent and all their convex combinations are also argminima.

Where step-response matrices are ⎡ k1 (k)s1,1 j ⎢ .. S j (k) = ⎣ . n ,1 kny (k)sj y ⎤ u . . k1 (k)s1,n j ⎥ .. ⎦ . ny ,nu . . kny (k)sj (10) 36 M. Lawry´ nczuk for j = 1, . . , N . Step-response coeﬃcients of the linear part of the model are denoted by sm,n for all j = 1, . . , N , n = 1, . . , nu , m = 1, . . , ny [11]. j Using (9), the output prediction vector can be expressed as a sum of a forced trajectory which depends only on the future (on future control moves u(k)) and a free trajectory y 0 (k), which depends only on the past where ˆ (k) = G(k) u(k) + y 0 (k) y (11) ⎤ ⎡ 0 ⎤ yˆ(k + 1|k) y (k + 1|k) ⎢ ⎥ ⎢ ⎥ ..

Then the set of argminima of the expected error m Ez over C(X) contains the set conv{fa+ | a > 0 }, where fa+ = i=1 cai Kγad u with i ca = Kγad [u]+ v. Proof. By Theorem 2(i), the argminimum of Ez over HKγa (X) has the form fa+= d m a + a a i=1 ciKγd ui and thus fa ∈ span GKγd (X). By Theorem 5 (i), span GKγd (X) is dense in (C(X), . sup ) and by Proposition 1, Ez is continuous on (C(X), . sup). Thus by Proposition 2, fa+ is an argminimum of Ez over C(X). As the set of argminima is convex, the statement holds.