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... size, we can approximate this by the differential entropy, H(y)=−  p(y)log 2 p(y)dy = 1 2 log 2 (e(2 π ) d  )+ 1 2 log 2 det(C y ) (4.6) This is maximized by maximizing det(C y )=det(WCW  ) over ... we diagonalize Q rr ≡ A+A −1/2 BB  A −1/2 ≡U Q Λ Q U  Q , then the desired matrix of orthogonal column eigen- vectors is V mr ≡  A B   A −1/2 U Q Λ −1/2 Q (4.24) (so that K mm = V Λ Q V  ... Ψ = σ 2 1, that the d −d  smallest eigenvalues of the model covariance are the same and are equal to σ 2 , and that the sample covariance S is equal to the model covariance (so that σ 2 follows

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... Schonlau, 1998, Knorr et al., 2000, Knorr et al., 2001, Jin et al., 2001, Breunig et al., 2000, Williams et al., 2002, Hawkins et al., 2002, Bay and Schwabacher, 2003). Another class of outlier ... Han, 1994, Ramaswamy et al., 2000, Barbara and Chen, 2000, Shekhar and Chawla, 2002, Shekhar and Lu, 2001, Shekhar and Lu, 2002, Acuna and Rodriguez, 2004). Hu and Sung (2003) , whom proposed a method ... population (Schiffman et al., 1981,Ng and Han, 1994, Shekhar and Chawla, 2002, Shekhar and Lu, 2001, Shekhar and Lu, 2002, Lu et al., 2003). Some of the above-mentioned classes are further discussed

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... 9.3.12 Orthogonal (ORT) Criterion The ORT criterion was presented by Fayyad and Irani (1992). This binary criterion is defined as: ORT (a i ,dom 1 (a i ),dom 2 (a i ),S)=1 −cos θ (P y,1 ,P y,2 ) ... 2 }, the criterion is defined as: KS(a i ,dom 1 (a i ),dom 2 (a i ),S)=        σ a i ∈dom 1 (a i ) AND y=c 1 S     σ y=c 1 S   −   σ a i ∈dom 1 (a i )AND y=c 2 S     σ y=c 2 ...       σ a i ∈dom 1 (a i ) S    −    σ a i ∈dom 2 (a i )AND y=c i S       σ a i ∈dom 2 (a i ) S          2 When the target attribute is binary, the gini and twoing criteria

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... ∈dom(y)   σ y=c i S t   ·ln | S t | | σ y=c i S t | + | dom(y) | −1 2 ln | S t | 2 + ln π | dom(y) | 2 Γ ( | dom(y) | 2 ) where S t denotes the instances that have reached node t. The splitting ... res(a i ,dom 1 (a i ),dom 2 (a i ),a j ,dom 1 (a j ),dom 2 (a j ),S)=    σ a i ∈dom 1 (a i ) AND a j ∈dom 1 (a j ) S    | S | +    σ a i ∈dom 2 (a i ) AND a j ∈dom 2 (a j ) S    | S | ... described in (Crawford et al., 2002). Decision trees are useful for many application domains, such as: Manufacturing lr18,lr14, Security lr7,l10 and Medicine lr2,lr9, and for many data mining

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... 1 (2 π ) n/2 detR 1/2 io detR 1/2 in Γ ( ν in /2) Γ ( ν io /2) ( ν io σ 2 io /2) ν io /2 ( ν in σ 2 in /2) ν in /2 and the parameters are specified by the next updating rules: α i1n = ν io /2 +n/2 ... 4 11 θ 3 12 }{ θ 3 21 θ 4 22 }{ θ 1 311 θ 1 312 × θ 1 321 θ 0 322 × θ 2 331 θ 0 332 × θ 1 341 θ 1 342 }. The first two terms in the products are the contributions of nodes Y 1 and Y 2 to the likelihood, ... α i2 ) p( τ i )= 1 α α i1 i2 Γ ( α i1 ) τ α i1 −1 i e − τ i / α i2 where α i1 = ν io 2 , α i2 = 2 ν io σ 2 io . This is the traditional Gamma prior for τ i with hyper-parameters ν io and σ 2 io

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... r = ∑ i {log( σ 2 i0 / σ 2 i0 ) −(y i − μ i1 ) 2 / σ 2 i1 +(y i − μ i0 ) 2 / σ 2 i0 } where y i is the value of attribute i in the new sample to classify and the parameters σ 2 ik and μ ik are ... 1(t−l) ,y 2(t−1) , ,y 2(t−l) ,y 3(t−1) , ,y 3(t−l) is given by the product of the three factors: p(y 1(t) |h t )=p(y 1(t) |y 1(t−1) ,y 2(t−1) ) p(y 2(t) |h t )=p(y 2(t) |y 1(t−1) ,y 2(t−1) ) p(y ... (Sebastiani and Ramoni, 2000, Sebastiani and Ramoni, 2001B) to customer profiling (Sebastiani et al., 2000) and bioinformatics (Friedman, 2004,Sebastiani et al., 2004,2). Here we describe two

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... Framework 213 ¯ y|x =( β 0 − β 2 x a )+( β 1 + β 2 )x. (11.6) If β 2 is positive, for x ≥ a the line is more steep with a slope of ( β 1 + β 2 ), and lower intercept of ( β 0 − β 2 x a ).If β 2 is ... and Barto, 1999, Cristianini and Shawe-Taylor, 2000, Witten and Frank, 2000,Hand et al., 2001,Hastie et al., 2001,Breiman, 2001b,Dasu and Johnson, 2003), and associated with Data Mining are a variety ... often zero. 2 In a regression context, S 0 j is constructed from a function f (x) that replaces x with transformations of x. Then, we often require that f (x)= M ∑ m=1 β m h m (x), (11.2) 1 In much

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... two-dimensional classification example, the transformation is Φ : R 2 → R 3 , (x 1 ,x 2 ) → (z 1 ,z 2 ,z 3 ) ≡  x 2 1 , √ 2x 1 x 2 ,x 2 2  . The separating hyperplane is visible and the decision surface ... + b = +1} . x 2 x 1 Note: (w x 1 ) + b = +1 (w x 2 ) + b = −1 => (w (x 1 −x 2 )) =2 => (x 1 −x 2 ) = w ||w|| ( ) . . . . 2 ||w|| y i = −1 y i = +1 ❍ ❍ ❍ ❍ ❍ ◆ ◆ ◆ ◆ Fig. 12.1. A toy binary ... )+b) (12.12) where b is computed from Equation 12.9 and from the set of support vectors x i ,i ∈ I ≡ { i : α i = 0 } . b = 1 | I | ∑ i∈I  y i − n ∑ j=1 α j y j (x i ·x j )  (12.13) 12.2.2 The

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... classifier of Equations 12.17-12.18 with the following equations: min w,b,e 1 2  w  2 + γ 1 2 n ∑ i=1 e 2 i (12.29) Subject to y i ·((w · Φ (x i )) + b)=1 −e i , i = 1, ,n (12.30) Important differences ... (Equation 12.25) while replacing the quadratic function in Equation 12.26 with a linear function subject to constraints on the error of kernel expansion (Equation 12.25). Suykens et al. (2002) introduced ... in the literature (Platt, 1998, 12 Support Vector Machines 243 Joachims 1999, Smola et al. 2000, Lin 2001, Chang and Lin 2001, Chew et al. 2003, Chung et al. 2004). A fairly large selection of

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... ,x j )=(w 1   x i1 −x j1   g + w 2   x i2 −x j2   g + +w p   x ip −x jp   g ) 1/g where w i ∈ [0,∞) 14 A survey of Clustering Algorithms 271 14.2.2 Distance Measures for Binary Attributes ... Kamber, 2001): d(x i ,x j )=(   x i1 −x j1   g +   x i2 −x j2   g + +   x ip −x jp   g ) 1/g The commonly used Euclidean distance between two objects is achieved when g = 2. Given ... S. 2. d(x i ,x j )=0⇒ x i = x j ∀x i ,x j ∈ S. 14.2.1 Minkowski: Distance Measures for Numeric Attributes Given two p-dimensional instances, x i =(x i1 ,x i2 , ,x ip ) and x j =(x j1 ,x j2 ,

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... hierarchical agglomerative algorithms is O(m 2 ∗logm). 2. The space complexity of agglomerative algorithms is O(m 2 ). This is because a similarity matrix of size m 2 has to be stored. It is possible to ... data. 2: repeat 3: Associate a fitness value ∀structure ∈ population. 4: Regenerate a new generation of structures. 5: until some termination condition is satisfied Fig. 14.2. GA for Clustering. 286 ... centric object in the cluster, rather than by the implicit mean that may not belong to the cluster. 282 Lior Rokach The K-medoids method is more robust than the K-means algorithm in the pres- ence

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... illustrated in Figure 15.2. 15 Association Rules 303 {1, 2, 3} {1, 2} {1, 3} {2, 3} {1} {2} {3} { } 3 23 123 root 3 Fig. 15.1. Itemset lattice and tree of subsets of 1,2,3 C 1 := {{i} | i∈I }; ... section 15.2.1 states that for X 1 ⊂ X 2 ⊂ Z we have supp(X 1 ) ≥ supp(X 2 ) ≥ supp(Z), and therefore supp(Z)/supp(X 1 ) ≤ supp(Z)/supp(X 2 ) ≤ 1. Thus, the confidence value for rule X 2 → Z −X 2 will ... of plan failures (Zaki, 2001) • Web personalization (Mobasher et al., 2002) • Text data (Brin et al., 1997A,Delgado et al., 2002) • Publication databases (Lee et al., 2001) In the analysis of

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... {300,400} 2 50% {wine} {100,300} 2 50% {beer,chips} {100,200} 2 50% {beer,wine} {100} 1 25% {chips,pizza} {400} 1 25% {chips,wine} {100} 1 25% {pizza,wine} {300} 1 25% {beer,chips,wine} {100} 1 25% ... Frequent Set Mining 323 Table 16.2. Frequent sets, their cover, support, and frequency in D. Set Cover Support Frequency {} {100,200,300,400} 4 100% {beer} {100,200} 2 50% {chips} {100,200,400} 3 75% ... (eds.), Data Mining and Knowledge Discovery Handbook, 2nd ed., DOI 10.1007/978-0-387-09823-4_16, © Springer Science+Business Media, LLC 2010 322 Bart Goethals of the purchased products (Agrawal

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... al. 2000) yes yes (Emmanouilidis et al. 2002) yes yes (Guerra-Salcedo, Whitley 1998, 1999) yes (Ishibuchi & Nakashima 2000) yes yes yes (Llora & Garrell 2003) yes (Miller et al. 2003) ... algo- rithms proposed by (Krawiec 2002; Smith & Bull 2003; Firpi et al. 2005). Here we briefly discuss the former two, as examples of this approach. In (Krawiec 2002) each individual encodes a ... 2003) yes (Miller et al. 2003) yes (Moser & Murty 2000) yes yes (Ni & Liu 2004) yes (Pappa et al. 2002) yes yes (Rozsypal & Kubat 2003) yes yes yes (Terano & Ishino 1998) yes yes

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... Subsection 5.2, and there are also several GP algorithms for discovering classification rules (Freitas 2002a; Wong & Leung 2000) or for classification in gen- eral (Muni et al. 2004; Song et al. 2005; ... S (2004) Evolutionary multiobjective knowledge extraction for high- dimensional pattern classification problems. Proc. Parallel Problem Solving From Na- ture (PPSN-2004), LNCS 3242, 1123-1132, ... Sebag M (2004) Ensemble learning with evolutionary computation: application to feature ranking. Proc. Parallel Problem Solving from Nature VIII (PPSN- 2004), LNCS 3242, 1133-1142. Springer, 2004.

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... L (2003) Feature construction and selection using genetic programming and a genetic algorithm. Genetic Programming: Proc. EuroGP-2003, LNCS 2610, 229- 237. Springer. Smith MG and Bull L (2004) ... (eds.), Data Mining and Knowledge Discovery Handbook, 2nd ed., DOI 10.1007/978-0-387-09823-4_20, © Springer Science+Business Media, LLC 2010 402 Oded Maimon and Shahar Cohen obtained while following ... )V t (s t+1 ) (20.15) Rewriting Equation 20.14 with α t (s t ,a t )=1 results in: Q t+1 (s t ,a t )=r t + γ V t (s t+1 ) (20.16) The only difference between Equations 20.15 and 20.16 lies in the

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... ics (MEDINFO 2004), San Francisco, CA, September 2004, IOS Press, pp. 282–286. Bellman R. Dynamic Programming. Princeton University Press, 1957. 20 Reinforcement Learning 415 Fig. 20.3. The learning ... update rule in Equation 20.14 as follows. Q 159 (  4,2  ,  0,8  )=0.9 ·Q 158 (  4,2  ,  0,8  ) + 0.1 ·[r 158 + γ V 158 (0,8)] = 0.9 ·45 + 0.1 ·[29 + 0.9 ·25]=45.65 . (20.19) The consequence ... 60-26-1-4=29. The state for the next epoch is s 1 (159)=0 and s 2 (159)=8. The agent can calculate V 158 (s 1 (159),s 2 (159)) by maximizing the Q-values corresponding with s 1 (159) and s 2 (159),

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... neurons (Hinton, 1992). Sum Trans- form w 1 x 1 x 2 x 3 x d w 2 w 3 w d I n p u t Output Fig. 21.2. Information processing in a single neuron In Figure 21.1, let x =(x 1 ,x 2 , ,x d ) be ... three-layer neural network can be written as a nonlinear model of the form y = f 2 (w 2 f 1 (w 1 x)), (21.2) where f 1 and f 2 are the transfer functions for the hidden nodes and output nodes respectively. ... To understand how the network in Figure 21.1 works, we need first understand the way neurons in the hidden and output layers process information. Figure 21.2 provides the mechanism that shows how

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... 22.3, we have Table 22.4 and Table 22.5; they are isomorphic. Table 22.5 provides the topology of Table 22.4. Table 22.4 and 22.5 provide a better interpretation than that of Table 22.2 and 22.3. ... shown in Table 22.2. Table 22.2. Granular table: Construct granular table by naming each binary granulations and binary granules BALLs Granulation 1 Granulation 2 id 1 Having-RED W1 id 2 Having-RED ... Definition 6 A table (e.g. Table 22.2) whose attributes are equipped with binary relations (e.g. Table 22.3 for COLOR attribute) is called a topological relation. 22.6.3 New representations of topological

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