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Additional info for General Training Air conditioning - Module 3 Refrigerant Piping

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172) subject to the constraints M yi αi = 0, i=1 In other words, as C approaches inﬁnity, the solution of the L1 support vector machine approaches that of the associated L2 support vector machine. In the following, we discuss this in more detail. 154), but as C approaches inﬁnity, from our previous discussions, the weight vector converges to w1 . Namely, the following theorem holds. 8. For C in [max(Cmax , Cmax ), ∞], the sets of support vectors Smax and Smax are the same, and for L1 and L2 support vector machines, the weight vectors in the feature space converges to vector w1 as C approaches inﬁnity.

M, i = s}}. Then the rank of HL1 is Ng [3, pp. 311–12]. 138) Ng ≤ l is satisﬁed. Therefore, if M > (l + 1), HL1 is positive semideﬁnite. For the linear kernel, l = m, where m is the number of input variables, for the polynomial kernel with degree d, l = m+d Cd [109, pp. 38–41], and for the RBF kernel, l = ∞. The Hessian matrix HL2 in which one variable is eliminated, for the L2 support vector machine, is expressed by HL2 = HL1 + yi yj + δij C . 139) The matrix HL1 is positive semideﬁnite, and the matrix {δij /C} is positive deﬁnite.

4. 153) where yB = (· · · ys yi · · ·)T (i ∈ B) and 1B is a |B|-dimensional vector with all elements equal to 1. Therefore, if B = φ, namely, 0 < αi < C for all support vectors, αU = HU−1 (1U − yU ). 5. 159), and w1T w2 = 0. 155) 44 2 Two-Class Support Vector Machines Proof. 158) i∈U ti yi (g(xi ) − g(xs )) + w2 = i∈U yi (g(xi ) − g(xs )). 157)). 160) Now consider changing the margin parameter C. Let [Ck , Ck+1 ] be the interval of C, in which the set of support vectors does not change. Here, we consider that if unbounded support vectors change to bounded support vectors, the set is changed.