Combining player statistics to predict outcomes of tennis matches

^** Email: tbarnett{at}swin.edu.au^*** Email: sclarke{at}swin.edu.au With the growth in sports betting, it is possible to bet, bothbefore and during a match, on a range of outcomes in tennis.This paper shows how the standard statistics published by theATP can be combined to predict the serving statistics to beobtained when two given players meet. These statistics are thenused in a spreadsheet model to predict further match outcomes,such as the length of the match and chance of either playerwinning. These calculations can be updated as the match progresses.The method is demonstrated by focusing on a very long men'ssingles match between Roddick and El Aynaoui played at the 2003Australian Open.     

Modelling a baseball game to optimise pitcher substitution strategies incorporating handedness of players

^** Email: hirotsu.nobuyoshi{at}jiss.naash.go.jp^*** Email: m.wright{at}lancaster.ac.uk This paper proposes a method for identifying the optimal strategyfor substituting players in a baseball game, taking into considerationthe handedness of players, which is one of the main factorsin terms of managerial decision-making for substitution. Usinga Markov chain model, we incorporate the effect of the handednessof players by introducing the concept of the defensive earnedrun average as a measure of the defensive ability of pitchersand calibrating the batting probabilities of players dependingon their handedness. We then develop a dynamic programming formulationincluding the effect of the handedness of players. This methodis illustrated using a match based on the real line-ups of theColorado Rockies and the San Francisco Giants in the NationalLeague of Major League Baseball, especially focusing on theintroduction of a relief pitcher in consideration with his handedness.     

Rating players in test match cricket

In general, the evaluation of player performance in test cricket is based on measures such as batting and bowling averages. These measures have a number of limitations, among which is that they fail to take into account the context in which runs are made or conceded and wickets are taken or given away. Furthermore, batting and bowling averages do not allow the comparison of performances in these two disciplines; this is because batting and bowling performances are measured using different metrics. With these issues in mind, we develop a new player rating system for test cricket. We use multinomial logistic regression to model match outcome probabilities session by session. We then use these probabilities to measure the overall contribution of players to the match outcome based on their individual batting, bowling and fielding contributions during each session. Our measure of contribution has the potential for rating players over time and for determining the ‘best’ player in a match, a series or a calendar year. We use results from 104 matches (2010–2012) to illustrate the method.     

Computation of non-smooth local centre manifolds

^** Email: msjolly{at}indiana.edu^*** Email: rrosa{at}ufrj.br An iterative Lyapunov–Perron algorithm for the computationof inertial manifolds is adapted for centre manifolds and appliedto two test problems. The first application is to compute aknown non-smooth manifold (once, but not twice differentiable),where a Taylor expansion is not possible. The second is to asmooth manifold arising in a porous medium problem, where rigorouserror estimates are compared to both the correction at eachiteration and the addition of each coefficient in a Taylor expansion.While in each case the manifold is 1D, the algorithm is well-suitedfor higher dimensional manifolds. In fact, the computationalcomplexity of the algorithm is independent of the dimension,as it computes individual points on the manifold independentlyby discretising the solution through them. Summations in thealgorithm are reformulated to be recursive. This accelerationapplies to the special case of inertial manifolds as well.     

Role and Results of statistical methods in protein fold class prediction

Statistical methods of discrimination and classification are used for the prediction of protein structure from amino acid sequence data. This provides information for the establishment of new paradigms of carcinogenesis modeling on the basis of gene expression. Feed forward neural networks and standard statistical classification procedures are used to classify proteins into fold classes. Logistic regression, additive models, and projection pursuit regression from the family of methods based on a posterior probabilities; linear, quadratic, and a flexible discriminant analysis from the class of methods based on class conditional probabilities, and the nearest-neighbors classification rule are applied to a data set of 268 sequences. From analyzing the prediction error obtained with a test sample (n = 125) and with a cross validation procedure, we conclude that the standard linear discriminant analysis and nearest-neighbor methods are at the same time statistically feasible and potent competitors to the more flexible tools of feed forward neural networks. Further research is needed to explore the gain obtainable from statistical methods by the application to larger sets of protein sequence data and to compare the results with those from biophysical approaches.     

Minimization of a Quadratic Function of Many Variables Subject only to Lower and Upper Bounds

Methods for solving this problem are considered with particularreference to achieving maximum efficiency. A streamlined versionof Fletcher's (1971) method for quadratic programming is consideredand also a new approach based on the use of partial LDL^T factorizations.Results on a wide variety of test problems indicate that theLDL^T method is superior in both efficiency and error control.This method can often be expected to solve the problem in atime comparable to that required for a Choleski factorization,and always in a small multiple of this time.     

Win–win match using a genetic algorithm

As the service industries grow, tasks are not directly assigned to the skills but the knowledge of the worker which is to be valued more in finding the best match. The problem becomes difficult mainly because the match has to be seen with the objectives of both sides. Assignment methods fail to respond to a multi-objective, multi-constraint problem with complicated match; whereas, metaheuristics is preferable based on computational simplicity. A conditional genetic algorithm is developed in this study to propose both global and composite match using different fitness functions. This algorithm kills the infeasibilities to achieve the maximum number of matches. The proposed algorithm is applied on an academic problem of multi-alternative candidates and multi-alternative tasks (m × n problem) in two stages. In the first stage, four different fitness functions are evaluated and in the second stage using one of the fitness functions global and composite matching have been compared. The achievements will contribute both to the academic and business world.     

Groups Quasi-Isometric to H2xR

The most powerful geometric tools are those of differentialgeometry, but to apply such techniques to finitely generatedgroups seems hopeless at first glance since the natural metricon a finitely generated group is discrete. However Gromov recognizedthat a group can metrically resemble a manifold in such a waythat geometric results about that manifold carry over to thegroup [18, 20]. This resemblance is formalized in the conceptof a ‘quasi-isometry’. This paper contributes toan ongoing program to understand which groups are quasi-isometricto which simply connected, homogeneous, Riemannian manifolds[15, 18, 20] by proving that any group quasi-isometric to H²xRis a finite extension of a cocompact lattice in Isom(H²xR) orIsom((2, R)).     

Linear system approximation via covariance equivalent realizations

A truncation technique for model reduction to simultaneously match a specified number of output covariance derivatives is described. These COVariance Equivalent Realizations which match q + 1 covariance derivatives are called “q-COVERs.” In general q-COVERs are not unique. The additional freedom is used herein so that the q-COVER obtained also matches q Markov parameters. The truncation technique uses a form of the observability matrices of the full-order system to determine a priori the order required of the reduced order model to match a specified number of output covariance derivatives and Markov parameters. The resulting realization is shown to be independent of the basis of the complete model to within a unitary transformation. Stability conditions for the reduced order model are also described, and the relationships are established between stochastically equivalent realizations and the q-COVERs.     

An analysis of strategy in the first three innings in test cricket: declaration and the follow-on

This paper analyses declaration and the follow-on decisions in test cricket. We model the match outcome given the end of first, second and third innings positions; data on 391 test matches, from the period 1997 to 2007, are used to fit the models. We then investigate how declaration strategy should vary from innings to innings, and how the nature and strength of the covariate effects vary. As the match progresses, the explanatory power of the covariates increases (from 44% at the end of the first innings to 80% at the end of the third). Home advantage and the effects of team strengths decrease. Overs-remaining, or equivalently overs used, and the number of runs by which the reference team lead their opponents remain important throughout. The follow-on decision problem is also briefly considered, and surprisingly, we find that the decision to enforce the follow-on or otherwise has no effect on match outcome.     

On a modified version of ILDM approach: asymptotic analysis based on integral manifolds

^** Email: vbykov{at}cs.bgu.ac.il^*** Email: goldfarb{at}cs.bgu.ac.il^**** Email: vladimir{at}bgumail.bgu.ac.il^***** Email: umaas{at}itt.mach.uni-karlsruhe.de Using the method of integral (invariant) manifolds, the intrinsiclow-dimensional manifolds (ILDM) method is analysed. This isa method for identifying invariant manifolds of a system's slowdynamics and has proven to be an efficient tool in modellingof laminar and turbulent combustion. It allows treating multi-scalesystems by revealing their hidden hierarchy and decomposingthe system dynamics into fast and slow motions. The performedanalysis shows that the original ILDM technique can be interpretedas one of the many possible realizations of the general framework,which is based on a special transformation of the original coordinatesin the state space. A modification of the ILDM is proposed basedon a new definition of the transformation matrix. The proposednumerical procedure is demonstrated on linear examples and highlynon-linear test problems of mathematical theory of combustionand demonstrates in some cases better performance with respectto the existing one.     

On defining OLAP formulations

^** Email: vutsinas{at}upatras.gr Recently there has been increasing interest in On Line AnalyticalProcessing (OLAP) to satisfy the organizational needs of high-levelinformation delivery and advanced data analysis. The actualapplication of OLAP tools involves the use of various functions,such as the common drilling down and slicing and dicing. Usuallyeach particular OLAP function is comprehensive and intuitive.However, sophisticated use of OLAP tools requires complicatedcombinations of different OLAP functions that are not straight-forwardfor end users or designers. In this paper we attempt to enumerateand formally define OLAP functions by defining a new OLAP modelthat provides a broader view of OLAP. We demonstrate the expressiveadequacy of the new OLAP model with application examples.     

Bayesian networks applied to credit scoring

Email: kchang{at}gmu.eduEmail: RobertFung{at}Fairlsaac.comEmail: alan.lucas{at}hotmail.com^¶Email: BobOliver{at}Fairlsaac.com^||Email: NShikaloff{at}Fairlsaac.com The objectives of this paper are to apply the theory and numericalalgorithms of Bayesian networks to risk scoring, and comparethe results with traditional methods for computing scores andposterior predictions of performance variables. Model identification,inference, and prediction of random variables using Bayesiannetworks have been successfully applied in a number of areas,including medical diagnosis, equipment failure, informationretrieval, rare-event prediction, and pattern recognition. Theability to graphically represent conditional dependencies andindependencies among random variables may also be useful incredit scoring. Although several papers have already appearedin the literature which use graphical models for model identification,as far as we know there have been no explicit experimental resultsthat compare a traditionally computed risk score with predictionsbased on Bayesian learning algorithms. In this paper, we examine a database of credit-card applicantsand attempt to ‘learn’ the graphical structure ofthe characteristics or variables that make up the database.We identify representative Bayesian networks in a developmentsample as well as the associated Markov blankets and cliquestructures within the Markov blanket. Once we obtain the structureof the underlying conditional independencies, we are able toestimate the probabilities of each node conditional on its directpredecessor node(s). We then calculate the posterior probabilitiesand scores of a performance variable for the development sample.Finally, we calculate the receiver operating characteristic(ROC) curves and relative profitability of scorecards basedon these identifications. The results of the different modelsand methods are compared with both development and validationsamples. Finally, we report on a statistical entropy calculationthat measures the degree to which cliques identified in theBayesian network are independent of one another.     

On the Structure of Minimal Left Ideals in the Largest Compactification of a Locally Compact Group

This paper is centred around a single question: can a minimalleft ideal L in G^LUC, the largest semi-group compactificationof a locally compact group G, be itself algebraically a group?Our answer is no (unless G is compact). In deriving this conclusion,we obtain for nearly all groups the stronger result that nomaximal subgroup in L can be closed. A feature of our work isthat completely different techniques are required for the connectedand totally disconnected cases. For the former, we can relyon the extensive structure theory of connected, non-compact,locally compact groups to derive the solution from the commutativecase, using some reduction lemmas. The latter directly involvestopological dynamics; we construct a compact space and an actionof G on it which has pathological properties. We obtain otherresults as tools towards our main goal or as consequences ofour methods. Thus we find an extension to earlier work on therelationship between minimal left ideals in G^LUC and H^LUC whenH is a closed subgroup of G with G/H compact. We show that thedistal compactification of G is finite if and only if the almostperiodic compactification of G is finite. Finally, we use ourmethods to show that there is no finite subset of G^LUC invariantunder the right action of G when G is an almost connected groupor an IN-group.     

The Elimination of All but Two from a Tournament

In a tournament with N participants, the smallest number Q(N)of decisive matches which may be necessary for the identificationof the runner-up has been observed by Schreier (1932) and Steinhaus(1950) to be the number M(N) defined in Section 1. Here we determinethe smallest number P(N) of such matches which may be necessaryfor the identification of the top pair, champion and runner-up,without necessarily distinguishing the champion. In particularwe show that P(N) = M(N)–1 if and only if N is of theform 2ⁿ+1, and that the optimal strategy has some unexpectedfeatures.     

Using dynamic programming to determine an optimal strategy in a contract bridge tournament

One form of contract bridge tournament comprises several rounds of matches in which players compete as pairs for ‘masterpoints’ awarded for each match won or drawn and for being highly placed at the end of the tournament. In the second and subsequent rounds, pairs are matched against other pairs that have been approximately equally successful so far. There has been vague discussion among bridge players of the possible advantages of ‘throwing’, that is, deliberately losing a match in round r so as to obtain a lower ranking and face an easier opponent in round r+1. This paper uses dynamic programming and data from one tournament to identify an optimal strategy (either ‘throw’ or ‘play to win’). The optimal strategy is a function of a pair's ability, its placing after each round, the relative rewards for winning a match and finishing high, the opponent's skill, and the round. For example, we find that a very weak pair should ‘throw’ and ‘play to win’ in alternate rounds.     

A mathematical analysis of badminton scoring systems

The International Badminton Federation recently introduced rule changes to make the game faster and more entertaining, by influencing how players score points and win games. We assess the fairness of both systems by applying combinatorics, probability theory and simulation to extrapolate known probabilities of winning individual rallies into probabilities of winning games and matches. We also measure how effective the rule changes are by comparing the numbers of rallies per game and the scoring patterns within each game, using data from the 2006 Commonwealth Games to demonstrate our results. We then develop subjective Bayesian methods for specifying the probabilities of winning. Finally, we describe how to propagate this information with observed data to determine posterior predictive distributions that enable us to predict match outcomes before and during play.     

Computing non-smooth minimizers with the mesh transformation method

^** Corresponding author. Email: lizp{at}math.pku.edu.cn A regularized mesh transformation method is applied to solvea variational problem allowing a non-smooth minimizer. Sincethe mesh lines can be made to match the discontinuity set ofthe minimizer, the method efficiently improves the approximatingproperty of the numerical solution. Error bounds dominated bythe error of the energy approximation have been derived, whichverify that the numerical solution obtained by the mesh transformationmethod is the optimal finite element solution in the sense thatthe corresponding error norm is minimized among all admissiblemesh distributions. Numerical experiments are given to showthe efficiency of the method.     

A New F Approximation for the Pillai—Bartlett Trace Under H 0

Abstract

Pillai suggested two approximations for the Pillai-Bartlett trace statistic in the null case. The first one matches one moment of a β₁ random variable, and corresponds to an F random variable, and the second matches four moments in the Pearson system. Although intuitively appealing and widely used in current statistical packages, the first lacks accuracy even with moderate sample size. The second matches two moment ratios in the Pearson system and provides much greater accuracy. Two new approximations match two moments of a β₁ random variable, and hence correspond to an F random variable, yet achieve most of the accuracy of Pillai's second approximation. The second of the two new approximations provides the best combination of logical properties and numerical accuracy.     

RIGHT FOCAL BOUNDARY VALUE PROBLEM WITH SINGULARITY

This article proves existence results for singular problem (-1)^n-px⁽ⁿ⁾(t)=f(t, x(t), ..., x^(n-1)(t)), for 0(i)(0)=0, i=1, 2, ..., p-1, x⁽ⁱ⁾(1)=0, i=p, p+1, ... , n-1. Here the positive Carathedory function f may be singular at the zero value of all its phase variables. The interesting point is that the degrees of some variables in the nonlinear term f(t, x₀, x₁, ..., x_n-1) are allowable to be greater than 1. Proofs are based on the Leray-Schauder degree theory and Vitali's convergence theorem. The emphasis in this article is that f depends on all higher-order derivatives. Examples are given to illustrate the main results of this article.