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Hypernasal speech detection by acoustic analysis of unvoiced plosive consonants
φ ( x ) : ℜn1 → ℜn 2 is generally a nonlinear function which maps
vector x into what is called a feature space of higher dimensionality (possibly infinite) where classes are linearly separable. The vector w defines the separating hyper-plane in such a space and w0 represents a possible bias (Webb, 2002).
2.5. Rademacher complexity model Rademacher complexity is a measure proposed in (Koltchinskii, 2001) which attempts to balance the complexity of the model with its fit to the data by minimizing the sum of the training error and a penalty term. Let {Xi,Yi}ni=1 be a set of training instances, where Xi is the pattern or example associated with features {Fj}qj=1, and Yi is the label of the example Xi. Let h(xi) be the class obtained by the classifier h, trained using {Xi,Yi}ni=1. Then, the training error is defined as eˆ ( h ) =
1 n where, ∑I n i =1 {h( x1 )≠ y1}
I{h( x1 )≠ y1} =
h( X ) ≠ Y {0,1, when when h ( X ) = Y i
i
i
i
Let {σi}ni=1 be a sequence of Rademacher random variables (i.i.d.) independent of the data {Xi}ni=1 and each variable takes values +1 and -1 with probability 1/2. According to this, computation of the Rademacher complexity involves the following steps (Delgado et al., 2007): – Generate {σi}ni=1 – Get a new set of labels, doing zi= σi yi . – Train the classifier hR using {Xi,Zi}ni=1 .
– Compute the Rademacher penalty, given by
Rn =
1 n ∑ σ i I{hR≠Yi } n i =1
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