lago - the department of statistics and actuarial science

Editorrsquos Note This article will be presented at the Technometrics invited session at the 2006 Joint ResearchConference in Knoxville TN June 2006

LAGO A Computationally Efficient Approachfor Statistical Detection

Mu ZHU and Wanhua SU

Department of Statistics and Actuarial ScienceUniversity of Waterloo

Waterloo ON N2L 3G1 Canada

Hugh A CHIPMAN

Department of Mathematics and StatisticsAcadia University

Wolfville NS B4P 2R6 Canada

We study a general class of statistical detection problems where the underlying objective is to detect itemsbelonging to a rare class from a very large database We propose a computationally efficient method toachieve this goal Our method consists of two steps In the first step we estimate the density functionof the rare class alone with an adaptive bandwidth kernel density estimator The adaptive choice of thebandwidth is inspired by the ancient Chinese board game known today as Go In the second step we adjustthis density locally depending on the density of the background class nearby We show that the amountof adjustment needed in the second step is approximately equal to the adaptive bandwidth from the firststep which gives us additional computational savings We name the resulting method LAGO for ldquolocallyadjusted Go-kernel density estimatorrdquo We then apply LAGO to a real drug discovery dataset and compareits performance with a number of existing and popular methods

KEY WORDS Adaptive bandwidth kernel density estimator Average precision Drug discovery Near-est neighbor Order statistic Radial basis function network Support vector machine

1 THE STATISTICAL DETECTION PROBLEM

Suppose that we have a large collection of items C of whichonly a fraction π (π 1) is relevant to us We are interestedin computational tools to help us identify and single out theseitems The reason why an item is considered relevant dependson the context of a specific problem For example for fraud de-tection (eg Bolton and Hand 2002) the relevant items are in-dividual transactions that are fraudulent for drug discovery therelevant items are chemical compounds that are active against aspecific biological target [such as the human immunodeficiencyvirus (HIV)] and so on

Typically we have a training dataset ( yixi)ni=1 where

xi isin Rd is a vector of predictors yi = 1 if observation i is rel-

evant and yi = 0 otherwise Supervised learning methods (egclassification trees neural networks) are used to build a predic-tive model using the training data The model is then used toscreen a large number of new cases it often produces a rele-vance score or an estimated probability for each of these newcases The top-ranked cases can then be passed onto furtherstages of investigation A summary of this process is providedin Figure 1

If it is decided that the top 50 cases should be investigatedfurther then we say that these 50 cases are the ones ldquodetectedrdquoby the model or the algorithm although strictly speaking thealgorithm really does not detect these cases per se it merelyranks them as being more likely than others to be what we want

The data structure and the types of supervised learning meth-ods often used here are similar to those encountered in a stan-dard two-class classification problem However the underlyingobjective is very different In particular automatic classificationis of little interest This is because further investigation of thetop-ranked candidates is almost always necessary For examplein the drug discovery application one never starts to manufac-ture a drug without further testing a compound in the fraud de-tection application one seldom terminates a credit card account

without confirming the suspected fraud Therefore we are mostinterested in producing an effective ranking of all of the candi-dates so that any further investigation (often very expensive) isleast likely to be carried out in vain

11 Performance Evaluation

Bolton and Hand (2002) emphasized that because relevantcases are quite rare for these detection problems ldquosimple mis-classification rate cannot be used as a performance measurerdquofor evaluating different methods For example if only 1 ofthe transactions are fraudulent then a model that simply clas-sifies everything as nonfraudulent will have a misclassificationrate of only 001 but it clearly is not a useful model

Suppose that an algorithm detects (in the sense made explicitearlier) a fraction t isin [01] from C and h(t) isin [0 t] is later con-firmed to be relevant these are often called ldquohitsrdquo (as opposedto ldquomissesrdquo) Figure 2 provides a schematic illustration Table 1is the corresponding confusion matrix

The function h(t) is called a ldquohit curverdquo (eg Wang 2005chap 2) Figure 3 shows some typical hit curves for a hypo-thetical case where only 5 of all cases are of interest Thedotted curve at the top hP(t) is an ideal curve produced by aperfect algorithm every item detected is an actual hit until allpotential hits (5 in total) are exhausted The diagonal dottedcurve hR(t) is the expected hits under random selection Thesolid and dashed curves hA(t) and hB(t) are those of two typi-cal detection algorithms

The hit curve h(t) tells us a lot about the performance of amodel or an algorithm For example if hA(t) gt hB(t) for every tthen algorithm A is unambiguously superior to algorithm B If

copy 2006 American Statistical Association andthe American Society for Quality

TECHNOMETRICS MAY 2006 VOL 48 NO 2DOI 101198004017005000000643

193

194 MU ZHU WANHUA SU AND HUGH A CHIPMAN

Figure 1 Illustration of the Typical Modeling and Prediction Process

an algorithm consistently produces hit curves that rise up veryquickly as t increases then it can often be considered a strongalgorithm In particular a perfect algorithm would have a hitcurve that rises with a maximal slope of 1 until t = π that iseverything detected is a hit until all possible hits are exhaustedAfterward the curve necessarily stays flat [see the curve hP(t)in Fig 3]

Table 1 also makes it clear that the hit curve h(t) is related tothe well-known receiver operating characteristic (ROC) curve(eg Cantor and Kattan 2000) but the two are not the sameAlthough both have h(t) on their vertical axes the horizontalaxis for the ROC curve is t minus h(t) instead of t In medicine thearea under the ROC curve is often used to measure the effec-tiveness of diagnostic tests

12 Average Precision

Whereas the hit curve provides an effective visual perfor-mance assessment modeling algorithms also require quanti-tative performance measures Often an algorithm will have anumber of tuning or regularization parameters that must be cho-sen empirically with a semiautomated procedure such as cross-validation For example for the K nearest-neighbor algorithm

Figure 2 Illustration of a Typical Detection Operation A small frac-tion π of the entire collection C is relevant An algorithm detects a frac-tion t from C and h(t) is later confirmed to be relevant

Table 1 The Confusion Matrix

Relevant Irrelevant Total

Detected h(t) t minus h(t) tUndetected π minus h(t) (1 minus t) minus (π minus h(t)) 1 minus t

Total π 1 minus π 1

we must select the appropriate number of neighbors K Thisis done by calculating a cross-validated version of the perfor-mance criterion for a number of different Krsquos and selecting theone that gives the best cross-validated performance For thispurpose we clearly prefer a numeric performance measure notan entire curve

In this article we use a numeric performance measure widelyused in the information retrieval community known as aver-age precision (AP) (see eg Deerwester Dumais LandauerFurnas and Harshman 1990 Dumais 1991 Peng Schuurmansand Wang 2003) Appendix A gives more details on how theaverage precision is defined and calculated As far as we knowthe precise connection between the AP and the area under theROC curve is not known In fact trying to establish this connec-tion is part of our ongoing research program For most readerswhose primary interest is in the modeling aspect of our workit suffices to know that the AP is a numeric summary of the hitcurve h(t) and that if method A has a higher AP than method Bthen method A can be considered the better method

2 A GENERAL PARADIGM FOREFFICIENT MODELING

We first propose a general paradigm that is efficient forconstructing computational models for the statistical detectionproblem Recall that we are most interested in the ranking ofall potential candidates Given a vector of predictors x the pos-terior probability is arguably a good ranking function that is

Figure 3 Illustration of Some Hit Curves for a Situation With 5 In-teresting Cases The curve hP (t) is an ideal curve produced by a per-fect algorithm hR(t) corresponds to the case of random detection andhA(t) and hB(t) are hit curves produced by typical detection algorithms

TECHNOMETRICS MAY 2006 VOL 48 NO 2

LOCALLY ADJUSTED GOndashKERNEL DENSITY ESTIMATOR 195

items with a high probability of being relevant should be rankedfirst By Bayesrsquo theorem the posterior probability can be ex-pressed as

g(x) equiv P(y = 1|x) = π1p1(x)

π1p1(x) + π0p0(x) (1)

Here π0 and π1 are prior probabilities and p0 and p1 are con-ditional density functions of x given y = 0 and y = 1 To rankitems from a new dataset xi i = 12 N it is clear that avery accurate estimate of g(xi) is not necessary as long as theestimated function ranks these observations in the correct orderAs far as ranking is concerned all monotonic transformationsof g are clearly equivalent Therefore it suffices to concentrateon the ratio function

f (x) = p1(x)

p0(x) (2)

because the function g is of the form

g(x) = af (x)

af (x) + 1

for some constant a gt 0 not depending on x which is clearly amonotonic transformation of f

A key feature for the statistical detection problems is thatmost observations are from the background class (class 0 C0)and only a small number of observations are from the class ofinterest (class 1 C1) This important feature allows us to makethe following assumptions

A1 For all practical purposes the density function p1(x)

can be assumed to have bounded local support possi-bly over a number of disconnected regions Sγ sub R

d γ = 12 in which case the support of p1 can bewritten as

S =⋃

γ=1

Sγ sub Rd

A2 For every observation xi isin C1 there are at least a certainnumber of observations say m from C0 in its immedi-ate local neighborhood moreover the density functionp0(x) in that neighborhood can be assumed to be rela-tively flat in comparison with p1(x) We will be morespecific about what we mean by the ldquoimmediate localneighborhoodrdquo in Section 32

Regions outside S are clearly not of interest because f (x)

would be 0 there and thus can be completely ignored see (2)Assumptions A1 and A2 then imply that to estimate the func-tion f we can simply estimate p1 and adjust it locally accordingto the density p0 nearby Note that this strategy would give us asignificant computational advantage because the number of ob-servations belonging to class 1 is typically very moderate evenfor very large datasets In what follows we exploit this generalprinciple to construct a computationally efficient method

3 LAGO UNIVARIATE MODEL CONSTRUCTION

To apply the general principles outlined in the previous sec-tion we first assume that the predictor x isin R is just a scalarWe generalize this to R

d for d gt 1 in Section 4

31 Step 1 Estimating p1

The first step is to estimate p1 To do so we use an adaptivebandwidth kernel estimator

p1(x) = 1

n1

sum

yi=1

K(x xi ri) (3)

where n1 is the number of observations belonging to class 1 andK(x xi ri) denotes a kernel function centered at xi with band-width ri For each xi isin C1 we choose ri adaptively to be the av-erage distance between xi and its K nearest neighbors from C0that is

ri = 1

K

sum

wjisinN(xiK)

|xi minus wj| (4)

where the notation N(xiK) is used to refer to the set that con-tains the K nearest class-0 neighbors of xi The number K hereis a tuning parameter that must be selected empirically (egthrough cross-validation)

The original inspiration of this particular bandwidth choicecomes from the ancient Chinese game of Go (Fig 4) In thisgame two players take turns placing black or white stones ontoa (19 times 19)-square board and try to conquer as many territoriesas possible At any given stage during the game any seriousplayer must evaluate the relative strength of his or her positionon the board as well as the strategic value of a number of pos-sible moves Ignoring the various rules of the game which areirrelevant for this article the basic principles behind such anevaluation can be roughly summarized as follows Any givenstone on the board exerts a certain amount of influence overits immediate neighborhood region The amount of influence isinversely related to how close the opponentrsquos stones are lyingaround it

For kernel density estimation the situation is analogousEach observation xi isin C1 exerts a certain amount of influenceover its immediate neighborhood region the extent of this in-fluence is controlled by the bandwidth parameter Following the

Figure 4 The Ancient Chinese Game of Go Is a Game in WhichEach Player Tries to Claim as Many Territories as Possible on the BoardImage taken from httpgoaradroIntroducerehtml



basic evaluation principle from the game of Go we allow eachxi to have a different bandwidth parameter ri depending on thedistance between xi and its neighboring observations from C0We refer to this special kernel density estimator as the ldquoGo-kernel density estimatorrdquo

32 Step 2 Local Adjustment of p1

The next step is to adjust the density estimate p1 locally ac-cording to the density of class 0 nearby Here we view (3) as amixture and adjust each mixture component (centered at xi) ac-cordingly Based on (2) we should estimate p0 locally aroundevery xi isin C1 say p0(x xi) and divide it into K(x xi ri)Assumption A2 implies that we can simply estimate p0(x xi)

locally as a constant say ci This means that our estimate of fshould be of the form

f (x) = 1

n1

sum

yi=1

K(x xi ri)

ci(5)

for appropriately estimated constants ci i = 12 n1 Herewe present an argument that makes the estimation of ci unnec-essary in practice which gives us an additional computationaladvantage

Intuitively if p0(x xi) asymp ci is roughly a constant nearbythen ci should on average be inversely proportional to the adap-tive bandwidth ri given in (4) that is if the density of class 0nearby is low then we will expect the average distance be-tween xi and its K nearest neighbors from C0 to be large andvice versa

Theorem 1 (see App B for a proof ) makes this intui-tion more precise in an idealized situation where insteadof saying p0(x xi) asymp ci we explicitly assume that for everyxi isin C1 there exist iid observations w1w2 wm from C0that can be taken to be uniformly distributed on the interval[xi minus 12ci xi + 12ci] Such an idealization does not make asignificant difference in practice but will allow us to provide aproof that is accessible to a much wider audience

Theorem 1 Let x0 be a fixed observation from class 1 Sup-pose that w1w2 wm are iid observations from class 0that are uniformly distributed around x0 say on the interval[x0 minus 12c0 x0 + 12c0] If r0 is the average distance be-tween x0 and its K nearest neighbors from class 0 (K lt m)then we have

E(r0) = K + 1

4(m + 1)c0

For the idealized situation the statement in Theorem 1 is ex-act In practice we can assume (see assumption A2) that thereare at least m observations from C0 distributed approximatelyuniformly around every xi isin C1 For K lt m then Theorem 1allows us to conclude that ri will be approximately proportionalto 1ci Therefore having already computed ri in the previousstep we can avoid the need to estimate ci separately for each xiand simply express our estimate of f as

f (x) = 1

n1

sum

yi=1

ri K(x xi ri) (6)

We use the acronym LAGO to refer to the resulting estimator fas the ldquolocally adjusted Go-kernel density estimatorrdquo

33 Discussion

It is important to realize that although assumptions A1and A2 are sufficient conditions for LAGO to work they areby no means necessary conditions Approximating functions lo-cally as a constant is not uncommon For example in regressiontrees (CART) (Breiman Friedman Olshen and Stone 1984)one approximates the regression function f (x) with a piecewiseconstant surface even if one knows a priori that the function isnot truly piecewise constant Therefore assumption A2 merelyprovides us with the motivation for estimating p0 locally as aconstant in step 2 (Sec 32) but one may proceed with such anapproximation regardless of whether one believes in assump-tion A2 or not

4 LAGO THE GENERAL MULTIVARIATE MODEL

We now extend the methodology presented in Section 3 to Rd

for d gt 1 For multivariate predictor x isin Rd we simply ap-

ply the naiumlve Bayes principle (see eg Hastie Tibshirani andFriedman 2001 sec 663) that is we model each dimensionindependently and multiply the marginal models together

1 For every training observation in class 1 xi isin C1 findits K nearest class-0 neighbors in R

d and compute a spe-cific bandwidth vector ri = (ri1 ri2 rid)

T where rij isthe average distance in the jth dimension between xi andits K nearest class-0 neighbors in R

d 2 For every new observation x = (x1 x2 xd)

T where aprediction is required score and rank x according to

f (x) = 1

n1

sum

yi=1

dprod

j=1

rijK(xj xij rij)

(7)

Generally speaking using the naiumlve Bayes principle is thesame as using ellipsoidal (eg Gaussian) or rectangular (eguniform) kernel functions whose principal axes are aligned withthe coordinate system If initial exploratory analysis indicatesthe existence of fairly strong correlations among the predic-tors making the naiumlve Bayes principle inappropriate then onecan transform the data with a method such as principal com-ponent analysis and apply LAGO in the transformed space asis often done in density estimation (see eg Silverman 1986pp 77ndash78)

41 Choice of Kernel

In practice we must choose a kernel function K(u) Com-mon choices include the uniform kernel the triangular kerneland the Gaussian kernel (see Fig 5) Our experience has indi-cated that if the uniform kernel is used then many observationscan be tied in terms of their rank score (7) this does not pro-duce an effective ranking Therefore significant improvementscan be obtained by choosing K(u) to be Gaussian or triangularIn addition there is usually no significant difference betweenthe Gaussian and the triangular kernel (see Sec 6) making thetriangular kernel the most attractive choice due to its compactlocal support



(a) (b) (c)

Figure 5 Some Commonly Used Kernel Functions (a) Gaussian (b) Triangular and (c) Uniform

42 A Radial Basis Function Network

The LAGO estimator can be viewed as a radial basis func-tion (RBF) network An RBF network (eg Hastie et al 2001sec 67) is a model for functional estimation that estimates anarbitrary function using a number of kernel functions as basisfunctions In general an RBF network has the form

f (x) =nsum

i=1

βiK(xmicroi ri) (8)

where K(xmicro r) is a kernel function centered at location micro

with radius (or bandwidth) vector r = (r1 r2 rd)T Clearly

to construct an RBF network one must make specific decisionsabout the centers microi and the radii ri for i = 12 n or specifyan algorithm to pick them For d-dimensional problems (d gt 1)it is not uncommon to specify the kernel function K as a prod-uct of d univariate kernel functions as in (7) (eg Hastie et al2001)

LAGO can be viewed as a highly specialized RBF networkIn particular we have specifically chosen to put basis functionsat and only at all of the class-1 observations in the trainingdata each with a rather specific radius vector defined separatelyin each dimension by (4) Once the centers and the radii arespecified the coefficients βi (i = 12 n) are often esti-mated empirically In constructing the LAGO model we have

instead specified these coefficients implicitly in the model aswell that is

βi = 1

n1

dprod

j=1

rij

43 A More General Parameterization

To the extent that LAGO can be viewed as an RBF networkit is possible to consider a more direct and slightly more gen-eral way of parameterizing such a model Start with the adap-tive bandwidth kernel density estimator for p1 described inSection 31 Figure 6 illustrates the effect of the density p0 onthe ranking function f that is of interest In this simplified andhypothetical illustration the density p1 has two componentsIf the background class has a flat density function p0 then theranking function f is clearly equivalent to p1 Now suppose thatthe background class has density function pprime

0 instead Figure 6shows that this will in general have two effects on the rankingfunction f For any given kernel component of f if pprime

0 is rela-tively low nearby then we should stretch its bandwidth (or ra-dius) and lift its height On the other hand if pprime

0 is relativelyhigh nearby then we should dampen its bandwidth (or radius)and lower its height We call the effect on the bandwidth (orradius) the α-effect and call the one on the height the β-effect

(a) (b)

Figure 6 Density Functions p0 and p1 (a) and the Ratio Function f(x) (b)



We can then parameterize these two effects explicitly Toconstruct an RBF network suppose that each component is akernel function belonging to a location-scale family that is

1

riK

(x minus xi

ri

)

If ri is chosen adaptively as in (4) and hence can be taken tobe on average inversely proportional to the density p0 in a lo-cal neighborhood around xi (Sec 32) then we can explicitlyparameterize the α- and β-effects as

rβ primei

1

αriK

(x minus xi

αri

)prop rβ primeminus1

i K(

x minus xi

αri

)equiv rβ

i K(

x minus xi

αri

)

where α and β are tuning parameters to be selected empiricallyIn Section 32 we in effect argued that the parameter β should

in theory be set to 0 (or β prime = 1) The parameter α on the otherhand can be beneficial in practice Theorem 1 makes it clearthat ri will generally increase with K For relatively large K wehave found that we can usually obtain a model with very similarperformance by setting α gt 1 and using a much smaller K thisis also quite intuitive Therefore by retaining the parameter α inthe model we can restrict ourselves to a much narrower rangewhen selecting K by cross-validation which is computationallyattractive Hence the final LAGO model that we fit is of the form

f (x) = 1

n1

sum

yi=1

dprod

j=1

rij K(xj xij αrij)

(9)

with two tuning parameters K and α

5 SUPPORT VECTOR MACHINES

The support vector machine (SVM) (eg Cristianini andShawe-Taylor 2000) has in recent years become a very pop-ular and successful supervised learning technique Schoumllkopfet al (1997) argued that SVM can be viewed as an automaticway of constructing an RBF network In this section we give avery brief highly condensed overview of what the SVM is howit can be used for the statistical detection problem and why itcan be viewed as an RBF network and hence becomes directlyrelevant for us

51 Overview of Support Vector Machines

A hyperplane in Rd is characterized by

f (x) = βTx + β0 = 0

Given xi isin Rd and yi isin minus1+1 (two classes) a hyperplane is

called a separating hyperplane if there exists c gt 0 such that

yi(βTxi + β0) ge c forall i (10)

Clearly a hyperplane can be reparameterized by scaling forexample

βTx + β0 = 0 is the same as s(βTx + β0) = 0

for any scalar s A separating hyperplane satisfying the condi-tion

yi(βTxi + β0) ge 1 forall i (11)

Figure 7 Two Separating Hyperplanes One With a Larger MarginThan the Other

that is scaled so that c = 1 in (10) is sometimes called acanonical separating hyperplane (Cristianini and Shawe-Taylor2000)

As illustrated in Figure 7 there exist an infinite number ofseparating hyperplanes if two classes are perfectly separableGiven a training dataset any separating hyperplane can be char-acterized by its margin a geometric concept illustrated in Fig-ure 7 Strictly speaking the margin is equal to 2 minyidiwhere di is the signed distance between observation xi and thehyperplane in particular di is equal to (see eg Cristianini andShawe-Taylor 2000 Hastie et al 2001 sec 45)

di = 1

β (βTxi + β0) (12)

It is then easy to see from (11) that a canonical separating hy-perplane has margin equal to 2 minyidi = 2β

The SVM is based on the notion that the ldquobestrdquo canonicalseparating hyperplane to separate two classes is the one withthe largest margin Therefore we are interested in solving thefollowing convex optimization problem

min1

2β2 + γ

nsum

i=1

ξi (13)

subject to ξi ge 0 and

yi(βTxi + β0) ge 1 minus ξi forall i (14)

Note that in general we must introduce the extra variables ξi torelax the separability condition (10) so that some observationsare allowed to cross over to the other side of the hyperplanebecause we cannot assume that the two classes are always per-fectly separable The term γ

sumξi acts as a penalty to control

the degree of such relaxationThe foregoing optimization problem has a quadratic objec-

tive function and linear inequality constraints making it a stan-dard quadratic programming problem We omit the details here(see Cristianini and Shawe-Taylor 2000) and simply state whatthe solution looks like The solution for β is characterized by

β =sum

iisinSV

αiyixi



where αi ge 0 (i = 12 n) are solutions to the dual optimiza-tion problem and SV the set of ldquosupport vectorsrdquo with αi gt 0strictly positive This means the resulting hyperplane can bewritten as

f (x) = βTx + β0 =sum

iisinSV

αiyixTi x + β0 = 0

Up to this point the SVM appears to be a strictly linear clas-sifier However it is easy to replace the inner product xT

i x witha kernel function K(xxi) to get a nonlinear decision boundary

f (x) =sum

iisinSV

αiyiK(xxi) + β0 = 0 (15)

The boundary is linear in the space of h(x) where h(middot) is suchthat K(uv) = 〈h(u)h(v)〉 is the inner product in the spaceof h(x) Note that the mapping h(middot) is not explicitly required

To fit the SVM two tuning parameters are required thepenalty parameter γ and a bandwidth parameter σ for thekernel K(xxi) We use the function svm from a librarycalled e1071 (Dimitriadou Hornik Leisch Meyer andWeingessel 2005) in R (R Development Core Team 2004)

52 Support Vector Machines for Detection Problems

To classify a new observation x all that we need to knowis whether f (x) is positive or negative For detection problemshowever we must assign every new observation a rank scoreA natural choice is to use the signed distance between x and theestimated decision boundary

d(x) = 1

β (βTx + β0)

For ranking purposes all monotonic transformations of d(x) areequivalent so it suffices to use just the inner product

βTx =sum

iisinSV

αiyixTi x

If the inner product xTi x is replaced with a kernel func-

tion K(xxi) then the final ranking function becomes

Fsvm(x) =sum

iisinSV

αiyiK(xxi) (16)

53 Support Vector Machines as an RBF Network

Notice the similarity between (16) and (8) The ranking func-tion produced by the SVM is a linear combination of a numberof kernel functions centered at all of the support points xi isin SV Hence SVM can be viewed as an automatic way of constructingan RBF network because the SVrsquos and the coefficients αi areautomatically determined by the algorithm Because LAGO canbe viewed as a specially constructed RBF network (Sec 42) anautomatically constructed RBF network provides a good bench-mark for us In addition it is also instructive to compare thecenters chosen by the automatic SVM algorithm (the supportvectors) with the ones that we choose (all of the class-1 obser-vations) see Section 65

54 Support Vector Machines With AsymmetricClass Weights

It is also possible to weigh observations from the two classesdifferently in SVM something that could potentially bene-fit these highly asymmetric detection problems Suppose thatw0 and w1 are weights for C0 and C1 The convex optimizationproblem (13) then becomes

min1

2β2 + γ1

sum

yi=1

ξi + γ0

sum

yi=0

ξi (17)

where γ1 = γ w1 γ0 = γ w0 and the constraints remain thesame as (14) Solving (17) is the same as solving (13) exceptthe penalty term involving γ is split into two separate terms onewith γ1 for yi = 1 and another with γ0 for yi = 0 We call thisthe asymmetric support vector machine (ASVM) which can befitted using the same function svm in R but with four tuningparametersmdashσ γ w0 and w1 Without loss of generality wecan fix w0 = 1 and tune the remaining three parameters σ γ and w1 simultaneously

6 AN APPLICATION DRUG DISCOVERY

In this section we illustrate the use of LAGO with a real drugdiscovery dataset First we give some background on the drugdiscovery problem

61 Drug Discovery

Recent technological advances in high-throughput screening(HTS) have allowed chemists to screen large numbers of chem-ical compounds against specific biological targets such as theHIV virus As a result of HTS each compound can be clas-sified as being active ( y = 1) or inactive ( y = 0) against thebiological target under investigation Here being active usuallymeans that the compound is able to inhibit a certain protein andhence protect the cells against the target such as a virus Activecompounds are then passed onto further stages of drug devel-opment

A pharmaceutical company usually has a large chemical li-brary a collection of different chemical compounds To reducecost we can screen only part of the chemical library and use theresults to build a computational model to screen the remainingcompounds and identify the ones that would have tested activeIn fact these models can even be used to virtually screen com-pounds not yet in the companyrsquos chemical library To build sucha computational model we need to have a number of predictorsto characterize each compound by say its molecular structureThese are often computed with various algorithms from compu-tational chemistry A commonly used set of predictors is BCUTnumbers (Burden 1989 Lam Welch and Young 2002) Herewe omit the details of exactly what the BCUT numbers are be-cause these details are not directly relevant to us

Figure 8 summarizes the process described earlier For anygiven biological target active compounds are almost alwaysquite rare Therefore building a computational model to iden-tify these active compounds for drug discovery is a statisticaldetection problem Figure 8 also illustrates an important featureof statistical detection problems the inexpensive availability ofpredictor variables for observations in which y has not yet beenmeasured



Figure 8 Illustration of the High Throughput Screening Process (Based on an illustration from Welch 2002)

62 Data

The dataset that we use is the AIDS antiviral database fromthe National Cancer Institute (NCI) The October 1999 releaseof the NCI data available at httpdtpncinihgovdocsaidsaids_datahtml were used in this analysis The publicly avail-able dataset identifies each compound by name and providesthe response value y There are 29812 chemical compoundsof which only 608 are active against the HIV virus Each com-pound is described by d = 6 BCUT numbers which were gen-erated and kindly provided to us by computational chemists atGlaxoSmithKline Inc The BCUT numbers can be thought ofas mapping compounds into a six-dimensional space in whichactivity occurs in local regions

63 Methods

We apply LAGO with three different kernel functions(uniform triangular and Gaussian) and use the following al-gorithms as benchmarks the SVM and ASVM for reasons dis-cussed earlier (Sec 5) and the K nearest-neighbor classifier(KNN) For both SVM and ASVM the Gaussian kernel is usedKNN is chosen here as a benchmark for two reasons First theway in which the adaptive bandwidths are computed in LAGOhas a similar flavor to the KNN algorithm Second an earlierstudy (Wang 2005) using the same dataset concluded that KNNis one of the best methods from among a number of techniquescompared including trees neural networks MARS (Friedman1991) generalized additive models and logistic regression Inthat study four experiments were conducted to evaluate the per-formance of different methods In each experiment the datasetwas randomly split using stratified sampling into a trainingset and a test set each with n = 14906 compounds of which304 are active compounds Performance was then assessed bythe AP on the test set Over the four replications a paired t-testconcluded that KNN significantly outperformed all other meth-ods at a 5 significance level

Here we also conduct four experiments using the same foursplits of the dataset The four experiments are also referred to asldquosplit 1rdquo ldquosplit 4rdquo in what follows In each experiment alltuning parameters are selected using five fold cross-validationon the training set whereas all performance measures are com-puted on the test set For KNN there is only one tuning parame-ter K For SVM there are two tuning parameters γ and σ (see

Sec 51) for ASVM there is an additional tuning parameterfor the class weights (see Sec 54) For LAGO there are twotuning parameters K and α (see Sec 43) The actual tuningparameters selected for LAGO are given in Table 2

Remark In deciding to use 50 of the data for training weattempt to represent a large-scale drug discovery problem witha smaller public-domain dataset We seek to balance the fol-lowing two factors

1 The NCI dataset consists of nearly 30000 compoundswhereas real pharmaceutical libraries can contain a mil-lion compounds

2 As such a training set selected from a real pharmaceu-tical library often would contain far fewer than 50 ofall compoundsmdashin fact it might well contain roughly thesame number of compounds as our training set (approxi-mately 15000)

Thus the size of our training set matches real problems butour test set is smaller than usual However despite this smallertest set the results in Section 64 are still statistically signif-icant In addition the proportion of active compounds in ourdataset (about 2) is also typical of real problems

64 Results

Figure 9 plots the average precision and Figure 10 showsthe hit curves when different methods are applied to differenttest data Because we measure performance four times usingslightly different training and test data it is also possible toperform an analysis of variance (ANOVA) comparison of thesedifferent methods Let microKmicroSmicroAmicroUmicroT and microG denote theaverage performance (measured by average precision) of KNNSVM ASVM and LAGO using the uniform triangular and

Table 2 Tuning Parameters Selected for LAGO Using Different Kernels

Uniform Triangular Gaussian

K α K α K α

Split 1 7 1 4 3 3 3Split 2 7 1 2 5 2 4Split 3 7 1 4 3 5 2Split 4 7 1 4 3 4 2



Figure 9 The Average Precision of All Algorithms Evaluated on theTest Data ( Gaussian triangular uniform KNN SVM

ASVM) The terms ldquoGaussianrdquo ldquotriangularrdquo and ldquouniformrdquo refer to ourLAGO model using the corresponding kernel functions

Gaussian kernels we construct five (nonorthogonal) contrasts(Table 3) The ANOVA result is given in Table 4 The mainconclusion at either a 1 or a 5 significance level is

(triangle LAGO sim Gaussian LAGO)

ASVM SVM (uniform LAGO sim KNN)

where ldquosimrdquo means ldquonot significantly different fromrdquo and ldquordquomeans ldquosignificantly better thanrdquo

Note that although the hit curve for ASVM can sometimesbe higher than that of LAGO for example for n gt 300 in splits2 and 4 the hit curve for LAGO is always higher initially Thismeans the top-ranked compounds identified by LAGO are morelikely to be active than the top-ranked compounds identified byASVM LAGO has consistently higher average precision be-cause this measure puts greater weight on early detection

65 Comparison With Support Vector Machines andAsymmetric Support Vector Machines

As mentioned in Section 5 because both LAGO and SVMcan be viewed as RBF networks (constructed using entirely dif-ferent approaches) it is instructive to examine the structure ofthese two RBF network models in more detail Recall that themain feature of our RBF network LAGO is that we choose toput basis functions at and only at all of the class-1 observationsin the training set which is very efficient because class 1 is therare class Recall also that the SVM will put basis functions atall of the support points

Table 5 lists the number of training observations from bothclasses chosen as support vectors by SVM and ASVM Inall cases almost every observation from class 1 (active com-pounds) is chosen as a support vector This suggests using allclass-1 observations in the training data is a good idea On theother hand both SVM and ASVM also use a fairly large num-ber of observations from class 0 (inactive compounds) as sup-

port vectors Given that the overall performance of SVM andASVM is inferior to that of LAGO these extra support vectorsseem to be quite wasteful

We must also stress here that fitting the ASVM is an ex-tremely expensive computational exercise Over 26 days ofCPU time were needed to produce the results for all four splitsThis was made feasible by a powerful parallel computing clus-ter with 38 nodes

66 Empirical Evidence for β = 0

Finally we present some empirical evidence to support thetheoretical claim that the extra parameter β in the more gen-eral parameterization of LAGO (Sec 43) should be set to 0 Todo so we treat β as an extra tuning parameter With K and αwe now have altogether three tuning parameters making cross-validation slightly more difficult and tedious To simplify thecomputation we conduct an experiment on the NCI AIDS an-tiviral data by fixing the parameter K and using cross-validationto select α and β simultaneously We use the Gaussian kerneland simply fix K = 5 which is a reasonable choice based on re-sults from Table 2 Figure 11 shows that cross-validation selectsβ = 0 completely in agreement with our theoretical choice

7 SUMMARY

Here we summarize the main contributions of this articleFirst we outlined a general paradigm for constructing efficientcomputational models for the statistical detection problem Thisparadigm can be summarized as follows Estimate the densityfor the rare but important class alone and adjust locally depend-ing on the density of the background class nearby The two stepsare estimation and adjustment Significant computational sav-ings can be achieved because we need to model only the rareclass

Second based on the general paradigm we developed thenew LAGO model A powerful feature of LAGO is that the twosteps (estimation and adjustment) are computationally com-bined into one step In particular we estimate p1 with an adap-tive bandwidth kernel density estimator and can show that theadjustment needed in the second step is proportional to theadaptive bandwidth already calculated in the first step Hencewe obtain additional computational savings because the adjust-ment step is automatic and requires no extra computation

Finally we showed that LAGO could be viewed as an RBFnetwork Based on this point of view we presented a more gen-eral parameterization for LAGO and illustrated its usefulnesswith a real dataset from drug discovery We showed that LAGOwas competitive and that when the Gaussian or the triangularkernel was used it outperformed both SVM and KNN

ACKNOWLEDGMENTS

This work is supported in part by the Natural Science andEngineering Research Council (NSERC) of Canada as wellas the Mathematics of Information Technology and ComplexSystems (MITACS) network Infrastructure provided by theCanada Foundation for Innovation and operated by the Aca-



Figure 10 The Hit Curve on Test Data ( Gaussian triangular uniform KNN SVM ASVM) Only the initial part of the curves(up to n = 500) are shown The terms ldquoGaussianrdquo ldquotriangularrdquo and ldquouniformrdquo refer to our LAGO model using the corresponding kernel functions

Table 3 Estimated Contrasts

Contrast Expression Estimate

Cntr1 microT minus microG 0027Cntr2 microG minus microA 0339Cntr3 microA minus microS 0230Cntr4 microS minus (microK + microU)2 0157Cntr5 microU minus microK 0014

Table 4 ANOVA Analysis of Differences Among Methods

Source SS (times10minus4) df MS (times10minus4) F0 p value

Methods 233504 5 46701 64307 lt0001Cntr1 140 1 140 193 6664Cntr2 22916 1 22916 31556 lt0001Cntr3 10534 1 10534 14505 0017Cntr4 6531 1 6531 8994 0090Cntr5 036 1 036 050 8258

Splits 18877 3 6292 8664 0014Error 10893 15 726

Total 263274 23

dia Centre for Mathematical Modelling and Computation wasused in this research The authors would like to thank S Youngfor providing the NCI AIDS Antiviral Data Set Duane Curriefor assistance with parallel computing and William J WelchR Wayne Oldford Jerry F Lawless and Mary E Thompson forvarious suggestions and comments The first author especiallythanks William Welch for providing the opportunity to spendtime in the statistics department at the University of BritishColumbia while working on this article

Table 5 Number of Support Vectors From Classes C0 and C1

SVM ASVM

C0 C1 C0 C1

Split 1 11531 294 5927 291Split 2 11419 303 3472 284Split 3 11556 293 11706 290Split 4 1863 293 6755 281

Total possible 14602 304 14602 304



Figure 11 Choosing α and β (while fixing K = 5) Using Five Fold Cross-Validation on the Training Data The contours are calculated on a 9 times 9grid for α = 1 25 5 1 15 2 3 4 5 and β = minus1 minus5 minus1 minus01 0 01 1 5 1

APPENDIX A THE AVERAGE PRECISION

Here we give details about how the average precision is de-fined and calculated Referring to Figure 2 let h(t) be the hitcurve let

r(t) = h(t)

πand p(t) = h(t)

t (A1)

The average precision is defined as

AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)

It is most instructive to examine a few concrete examples

Example A1 (Random selection) Suppose that h(t) = π tthat is the proportion of relevant items among those detectedso far stays constant at π This is the case of random selectionIn this case r(t) = h(t)π = t Therefore

AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π

Example A2 (Perfect detection) Suppose that

h(t) =

t t isin [0π]π t isin (π1]

that is everything detected is relevant until all relevant itemsare exhausted at t = π This is the case of ideal detection onecannot possibly do any better than this This implies that

r(t) = t

π t isin [0π]

1 t isin (π1]Therefore

AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1



Table A1 Two Algorithms for a Toy Dataset With ThreeRelevant Cases

Algorithm A Algorithm B

Item (i) Hit p(ti) ∆r(ti) Hit p(ti) ∆r(ti)

1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0

Example A3 (Practical calculation) In practice h(t) takesvalue only at a finite number of points ti = in i = 12 nHence the integral (A2) is replaced with a finite sum

int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)

where r(ti) = r(ti)minus r(timinus1) We illustrate this with a concreteexample Table A1 summarizes the performance of two hypo-thetical methods A and B

In this case we have

AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63

which agrees with our intuition that algorithm A performs bet-ter here because it detects the three hits earlier on

APPENDIX B PROOF OF THEOREM 1

Suppose w1w2 wm are iid observations uniformly dis-tributed on the interval [x0 minus 12c0 x0 + 12c0] Lettingzj = |wj minus x0| be the distance between wj and x0 we first claimthat

E(z(k)

) = k

2(m + 1)c0 (B1)

where z(k) is the kth order statistic of zj j = 12 m We canthen write the average distance between x0 and its K nearestneighbors from wj j = 12 m as

r0 = 1

K

Ksum

k=1

z(k) (B2)

Equation (B1) then implies that

E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)

which completes the proofTo prove (B1) first note that we can assume that x0 = 0

without loss of generality If w1w2 wm are uniformlydistributed on [minus12c012c0] then it is easy to show thatthe distribution for zj = |wj| p(z) is uniform on the interval[012c0] that is p(z) = 2c0 if 0 le z le 12c0

Now if z(k) is the kth order statistic of z1 z2 zm thenusing standard results on order statistics (eg Ross 2000 p 58ex 237) we can obtain the density for z(k)

fz(k) (z) = m(k minus 1)(m minus k) (2c0)(2c0z)kminus1(1 minus 2c0z)mminusk

(B4)

Using (B4) we can calculate E(z(k)) as

E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0

m(k minus 1)(m minus k) (2c0z)k(1 minus 2c0z)mminusk dz

Letting y = 2c0z this integral becomesint 1

0

m(k minus 1)(m minus k) ( y)k(1 minus y)mminusk 1

2c0dy

=int 1

0

(k

m + 1

)(m + 1)

k(m minus k) ( y)k(1 minus y)mminusk 1

2c0dy

= k

2(m + 1)c0

where the last step is due to the fact thatint 1

0

(m + 1)k(m minus k) ( y)k(1 minus y)mminusk dy = 1

because the foregoing integrand can be recognized as aBeta(k + 1m minus k + 1) density function

[Received February 2005 Revised September 2005]

REFERENCES

Bolton R J and Hand D J (2002) ldquoStatistical Fraud Detection A ReviewrdquoStatistical Science 17 235ndash255

Breiman L Friedman J H Olshen R A and Stone C J (1984) Classifi-cation and Regression Trees Belmont CA Wadsworth

Burden F R (1989) ldquoMolecular Identification Number for SubstructureSearchesrdquo Journal of Chemical Information and Computer Sciences 29225ndash227

Cantor S B and Kattan M W (2000) ldquoDetermining the Area Under the ROCCurve for a Binary Diagnostic Testrdquo Medical Decision Making 20 468ndash470

Cristianini N and Shawe-Taylor J (2000) An Introduction to Support Vec-tor Machines and Other Kernel-Based Learning Methods Cambridge UKCambridge University Press

Deerwester S Dumais S T Landauer T K Furnas G W andHarshman R A (1990) ldquoIndexing by Latent Semantic Analysisrdquo Journalof the Society for Information Science 41 391ndash407



Dimitriadou E Hornik K Leisch F Meyer D and Weingessel A (2005)ldquoe1071 Misc Functions of the Department of Statistics (e1071)rdquo R PackageVersion 15-11 TU Wien

Dumais S T (1991) ldquoImproving the Retrieval of Information From Exter-nal Sourcesrdquo Behavior Research Methods Instruments and Computers 23229ndash236

Friedman J H (1991) ldquoMultivariate Adaptive Regression Splinesrdquo The An-nals of Statistics 19 1ndash67

Hastie T J Tibshirani R J and Friedman J H (2001) The Elements ofStatistical Learning Data-Mining Inference and Prediction New YorkSpringer-Verlag

Lam R L H Welch W J and Young S S (2002) ldquoUniform CoverageDesigns for Molecule Selectionrdquo Technometrics 44 99ndash109

Peng F Schuurmans D and Wang S (2003) ldquoAugmenting Naive BayesClassifiers With Statistical Language Modelsrdquo Information Retrieval 7317ndash345

R Development Core Team (2004) R A Language and Environment for Statis-tical Computing Vienna R Foundation for Statistical Computing availableat httpwwwR-projectorg

Ross S M (2000) Introduction to Probability Models (7th ed) New YorkAcademic Press

Schoumllkopf B Sung K K Burges C J C Girosi F Niyogi P Poggio Tand Vapnik V (1997) ldquoComparing Support Vector Machines With GaussianKernels to Radial Basis Function Classifiersrdquo IEEE Transactions on SignalProcessing 45 2758ndash2765

Silverman B (1986) Density Estimation for Statistics and Data AnalysisNew York Chapman amp Hall

Wang Y (2005) ldquoStatistical Methods for High-Throughput Screening DrugDiscovery Datardquo unpublished doctoral dissertation University of WaterlooDept of Statistics and Actuarial Science

Welch W (2002) ldquoComputational Exploration of Datardquo course notes Univer-sity of Waterloo



Figure 1 Illustration of the Typical Modeling and Prediction Process

an algorithm consistently produces hit curves that rise up veryquickly as t increases then it can often be considered a strongalgorithm In particular a perfect algorithm would have a hitcurve that rises with a maximal slope of 1 until t = π that iseverything detected is a hit until all possible hits are exhaustedAfterward the curve necessarily stays flat [see the curve hP(t)in Fig 3]

Table 1 also makes it clear that the hit curve h(t) is related tothe well-known receiver operating characteristic (ROC) curve(eg Cantor and Kattan 2000) but the two are not the sameAlthough both have h(t) on their vertical axes the horizontalaxis for the ROC curve is t minus h(t) instead of t In medicine thearea under the ROC curve is often used to measure the effec-tiveness of diagnostic tests

12 Average Precision

Whereas the hit curve provides an effective visual perfor-mance assessment modeling algorithms also require quanti-tative performance measures Often an algorithm will have anumber of tuning or regularization parameters that must be cho-sen empirically with a semiautomated procedure such as cross-validation For example for the K nearest-neighbor algorithm

Figure 2 Illustration of a Typical Detection Operation A small frac-tion π of the entire collection C is relevant An algorithm detects a frac-tion t from C and h(t) is later confirmed to be relevant

Table 1 The Confusion Matrix

Relevant Irrelevant Total

Detected h(t) t minus h(t) tUndetected π minus h(t) (1 minus t) minus (π minus h(t)) 1 minus t

Total π 1 minus π 1

we must select the appropriate number of neighbors K Thisis done by calculating a cross-validated version of the perfor-mance criterion for a number of different Krsquos and selecting theone that gives the best cross-validated performance For thispurpose we clearly prefer a numeric performance measure notan entire curve

In this article we use a numeric performance measure widelyused in the information retrieval community known as aver-age precision (AP) (see eg Deerwester Dumais LandauerFurnas and Harshman 1990 Dumais 1991 Peng Schuurmansand Wang 2003) Appendix A gives more details on how theaverage precision is defined and calculated As far as we knowthe precise connection between the AP and the area under theROC curve is not known In fact trying to establish this connec-tion is part of our ongoing research program For most readerswhose primary interest is in the modeling aspect of our workit suffices to know that the AP is a numeric summary of the hitcurve h(t) and that if method A has a higher AP than method Bthen method A can be considered the better method

2 A GENERAL PARADIGM FOREFFICIENT MODELING

We first propose a general paradigm that is efficient forconstructing computational models for the statistical detectionproblem Recall that we are most interested in the ranking ofall potential candidates Given a vector of predictors x the pos-terior probability is arguably a good ranking function that is

Figure 3 Illustration of Some Hit Curves for a Situation With 5 In-teresting Cases The curve hP (t) is an ideal curve produced by a per-fect algorithm hR(t) corresponds to the case of random detection andhA(t) and hB(t) are hit curves produced by typical detection algorithms





π1p1(x) + π0p0(x) (1)


f (x) = p1(x)

p0(x) (2)


g(x) = af (x)

af (x) + 1






S =⋃

γ=1

Sγ sub Rd









p1(x) = 1

n1

sum

yi=1

K(x xi ri) (3)


ri = 1

K

sum

wjisinN(xiK)

|xi minus wj| (4)











f (x) = 1

n1

sum

yi=1

K(x xi ri)

ci(5)





E(r0) = K + 1

4(m + 1)c0


f (x) = 1

n1

sum

yi=1

ri K(x xi ri) (6)


33 Discussion











f (x) = 1

n1

sum

yi=1

dprod

j=1

rijK(xj xij rij)

(7)


41 Choice of Kernel




(a) (b) (c)




f (x) =nsum

i=1







βi = 1

n1

dprod

j=1

rij






(a) (b)





1

riK

(x minus xi

ri

)


rβ primei

1

αriK

(x minus xi

αri


i K(

x minus xi

αri

)equiv rβ

i K(

x minus xi

αri

)



f (x) = 1

n1

sum

yi=1

dprod

j=1

rij K(xj xij αrij)

(9)






f (x) = βTx + β0 = 0











di = 1

β (βTxi + β0) (12)



min1

2β2 + γ

nsum

i=1

ξi (13)







β =sum

iisinSV

αiyixi





iisinSV




f (x) =sum

iisinSV

αiyiK(xxi) + β0 = 0 (15)





d(x) = 1

β (βTx + β0)


βTx =sum

iisinSV

αiyixTi x



Fsvm(x) =sum

iisinSV

αiyiK(xxi) (16)





min1

2β2 + γ1

sum

yi=1

ξi + γ0

sum

yi=0

ξi (17)




61 Drug Discovery







62 Data


63 Methods








64 Results




K α K α K α


















7 SUMMARY




ACKNOWLEDGMENTS











Splits 18877 3 6292 8664 0014Error 10893 15 726

Total 263274 23



SVM ASVM

C0 C1 C0 C1


Total possible 14602 304 14602 304






r(t) = h(t)

πand p(t) = h(t)

t (A1)


AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)



AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π


h(t) =



r(t) = t

π t isin [0π]


AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1






1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES

























π1p1(x) + π0p0(x) (1)


f (x) = p1(x)

p0(x) (2)


g(x) = af (x)

af (x) + 1






S =⋃

γ=1

Sγ sub Rd









p1(x) = 1

n1

sum

yi=1

K(x xi ri) (3)


ri = 1

K

sum

wjisinN(xiK)

|xi minus wj| (4)











f (x) = 1

n1

sum

yi=1

K(x xi ri)

ci(5)





E(r0) = K + 1

4(m + 1)c0


f (x) = 1

n1

sum

yi=1

ri K(x xi ri) (6)


33 Discussion











f (x) = 1

n1

sum

yi=1

dprod

j=1

rijK(xj xij rij)

(7)


41 Choice of Kernel




(a) (b) (c)




f (x) =nsum

i=1







βi = 1

n1

dprod

j=1

rij






(a) (b)





1

riK

(x minus xi

ri

)


rβ primei

1

αriK

(x minus xi

αri


i K(

x minus xi

αri

)equiv rβ

i K(

x minus xi

αri

)



f (x) = 1

n1

sum

yi=1

dprod

j=1

rij K(xj xij αrij)

(9)






f (x) = βTx + β0 = 0











di = 1

β (βTxi + β0) (12)



min1

2β2 + γ

nsum

i=1

ξi (13)







β =sum

iisinSV

αiyixi





iisinSV




f (x) =sum

iisinSV

αiyiK(xxi) + β0 = 0 (15)





d(x) = 1

β (βTx + β0)


βTx =sum

iisinSV

αiyixTi x



Fsvm(x) =sum

iisinSV

αiyiK(xxi) (16)





min1

2β2 + γ1

sum

yi=1

ξi + γ0

sum

yi=0

ξi (17)




61 Drug Discovery







62 Data


63 Methods








64 Results




K α K α K α


















7 SUMMARY




ACKNOWLEDGMENTS











Splits 18877 3 6292 8664 0014Error 10893 15 726

Total 263274 23



SVM ASVM

C0 C1 C0 C1


Total possible 14602 304 14602 304






r(t) = h(t)

πand p(t) = h(t)

t (A1)


AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)



AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π


h(t) =



r(t) = t

π t isin [0π]


AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1






1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES



























f (x) = 1

n1

sum

yi=1

K(x xi ri)

ci(5)





E(r0) = K + 1

4(m + 1)c0


f (x) = 1

n1

sum

yi=1

ri K(x xi ri) (6)


33 Discussion











f (x) = 1

n1

sum

yi=1

dprod

j=1

rijK(xj xij rij)

(7)


41 Choice of Kernel




(a) (b) (c)




f (x) =nsum

i=1







βi = 1

n1

dprod

j=1

rij






(a) (b)





1

riK

(x minus xi

ri

)


rβ primei

1

αriK

(x minus xi

αri


i K(

x minus xi

αri

)equiv rβ

i K(

x minus xi

αri

)



f (x) = 1

n1

sum

yi=1

dprod

j=1

rij K(xj xij αrij)

(9)






f (x) = βTx + β0 = 0











di = 1

β (βTxi + β0) (12)



min1

2β2 + γ

nsum

i=1

ξi (13)







β =sum

iisinSV

αiyixi





iisinSV




f (x) =sum

iisinSV

αiyiK(xxi) + β0 = 0 (15)





d(x) = 1

β (βTx + β0)


βTx =sum

iisinSV

αiyixTi x



Fsvm(x) =sum

iisinSV

αiyiK(xxi) (16)





min1

2β2 + γ1

sum

yi=1

ξi + γ0

sum

yi=0

ξi (17)




61 Drug Discovery







62 Data


63 Methods








64 Results




K α K α K α


















7 SUMMARY




ACKNOWLEDGMENTS











Splits 18877 3 6292 8664 0014Error 10893 15 726

Total 263274 23



SVM ASVM

C0 C1 C0 C1


Total possible 14602 304 14602 304






r(t) = h(t)

πand p(t) = h(t)

t (A1)


AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)



AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π


h(t) =



r(t) = t

π t isin [0π]


AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1






1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES























(a) (b) (c)




f (x) =nsum

i=1







βi = 1

n1

dprod

j=1

rij






(a) (b)





1

riK

(x minus xi

ri

)


rβ primei

1

αriK

(x minus xi

αri


i K(

x minus xi

αri

)equiv rβ

i K(

x minus xi

αri

)



f (x) = 1

n1

sum

yi=1

dprod

j=1

rij K(xj xij αrij)

(9)






f (x) = βTx + β0 = 0











di = 1

β (βTxi + β0) (12)



min1

2β2 + γ

nsum

i=1

ξi (13)







β =sum

iisinSV

αiyixi





iisinSV




f (x) =sum

iisinSV

αiyiK(xxi) + β0 = 0 (15)





d(x) = 1

β (βTx + β0)


βTx =sum

iisinSV

αiyixTi x



Fsvm(x) =sum

iisinSV

αiyiK(xxi) (16)





min1

2β2 + γ1

sum

yi=1

ξi + γ0

sum

yi=0

ξi (17)




61 Drug Discovery







62 Data


63 Methods








64 Results




K α K α K α


















7 SUMMARY




ACKNOWLEDGMENTS











Splits 18877 3 6292 8664 0014Error 10893 15 726

Total 263274 23



SVM ASVM

C0 C1 C0 C1


Total possible 14602 304 14602 304






r(t) = h(t)

πand p(t) = h(t)

t (A1)


AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)



AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π


h(t) =



r(t) = t

π t isin [0π]


AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1






1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES
























1

riK

(x minus xi

ri

)


rβ primei

1

αriK

(x minus xi

αri


i K(

x minus xi

αri

)equiv rβ

i K(

x minus xi

αri

)



f (x) = 1

n1

sum

yi=1

dprod

j=1

rij K(xj xij αrij)

(9)






f (x) = βTx + β0 = 0











di = 1

β (βTxi + β0) (12)



min1

2β2 + γ

nsum

i=1

ξi (13)







β =sum

iisinSV

αiyixi





iisinSV




f (x) =sum

iisinSV

αiyiK(xxi) + β0 = 0 (15)





d(x) = 1

β (βTx + β0)


βTx =sum

iisinSV

αiyixTi x



Fsvm(x) =sum

iisinSV

αiyiK(xxi) (16)





min1

2β2 + γ1

sum

yi=1

ξi + γ0

sum

yi=0

ξi (17)




61 Drug Discovery







62 Data


63 Methods








64 Results




K α K α K α


















7 SUMMARY




ACKNOWLEDGMENTS











Splits 18877 3 6292 8664 0014Error 10893 15 726

Total 263274 23



SVM ASVM

C0 C1 C0 C1


Total possible 14602 304 14602 304






r(t) = h(t)

πand p(t) = h(t)

t (A1)


AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)



AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π


h(t) =



r(t) = t

π t isin [0π]


AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1






1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES

























iisinSV




f (x) =sum

iisinSV

αiyiK(xxi) + β0 = 0 (15)





d(x) = 1

β (βTx + β0)


βTx =sum

iisinSV

αiyixTi x



Fsvm(x) =sum

iisinSV

αiyiK(xxi) (16)





min1

2β2 + γ1

sum

yi=1

ξi + γ0

sum

yi=0

ξi (17)




61 Drug Discovery







62 Data


63 Methods








64 Results




K α K α K α


















7 SUMMARY




ACKNOWLEDGMENTS











Splits 18877 3 6292 8664 0014Error 10893 15 726

Total 263274 23



SVM ASVM

C0 C1 C0 C1


Total possible 14602 304 14602 304






r(t) = h(t)

πand p(t) = h(t)

t (A1)


AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)



AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π


h(t) =



r(t) = t

π t isin [0π]


AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1






1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES
























62 Data


63 Methods








64 Results




K α K α K α


















7 SUMMARY




ACKNOWLEDGMENTS











Splits 18877 3 6292 8664 0014Error 10893 15 726

Total 263274 23



SVM ASVM

C0 C1 C0 C1


Total possible 14602 304 14602 304






r(t) = h(t)

πand p(t) = h(t)

t (A1)


AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)



AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π


h(t) =



r(t) = t

π t isin [0π]


AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1






1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES





































7 SUMMARY




ACKNOWLEDGMENTS











Splits 18877 3 6292 8664 0014Error 10893 15 726

Total 263274 23



SVM ASVM

C0 C1 C0 C1


Total possible 14602 304 14602 304






r(t) = h(t)

πand p(t) = h(t)

t (A1)


AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)



AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π


h(t) =



r(t) = t

π t isin [0π]


AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1






1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES






























Splits 18877 3 6292 8664 0014Error 10893 15 726

Total 263274 23



SVM ASVM

C0 C1 C0 C1


Total possible 14602 304 14602 304






r(t) = h(t)

πand p(t) = h(t)

t (A1)


AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)



AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π


h(t) =



r(t) = t

π t isin [0π]


AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1






1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES


























r(t) = h(t)

πand p(t) = h(t)

t (A1)


AP =int 1

0p(r)dr =

int 1

0p(t)dr(t) =

int 1

0

πr(t)dr(t)

t (A2)



AP(Random) =int 1

0

πr(t)dr(t)

t=

int 1

0

π t

tdt = π


h(t) =



r(t) = t

π t isin [0π]


AP(Perfect) =int 1

0

πr(t)dr(t)

t

=int π

0

(π times t

πtimes 1

π

t

)dt +

int 1

π

(π times 1 times 0

t

)dt

= 1






1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES


























1 1 11

13 1 1

113

2 1 22

13 0 1

2 0

3 0 23 0 0 1

3 0

4 1 34

13 1 2

413

5 0 35 0 0 2

5 0

6 0 36 0 0 2

6 0

7 0 37 0 0 2

7 0

8 0 38 0 1 3

813

9 0 39 0 0 3

9 0

10 0 310 0 0 3

10 0


int 1

0p(t)dr(t) =

nsum

i=1

p(ti)r(ti) (A3)



AP(A) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

2+ 3

4

)times 1

3asymp 92

and

AP(B) =10sum

i=1

p(ti)r(ti) =(

1

1+ 2

4+ 3

8

)times 1

3asymp 63




E(z(k)

) = k

2(m + 1)c0 (B1)


r0 = 1

K

Ksum

k=1

z(k) (B2)


E(r0) = 1

K

Ksum

k=1

E(z(k)

) = 1

K

Ksum

k=1

k

2(m + 1)c0

= K + 1

4(m + 1)c0 (B3)





(B4)


E(z(k)

) =int 12c0

0z fz(k) (z)dz

=int 12c0

0



0


2c0dy

=int 1

0

(k

m + 1

)(m + 1)


2c0dy

= k

2(m + 1)c0


0




REFERENCES






















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