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1、Team Control Number28015Problem ChosenBFor office use onlyT1 T2 T3 T4 For office use onlyF1 F2 F3 F4 2014 MathematicalContest in Ming(MCM) Summary SheetIn this paper, in order to eliminate the subjective impact, the coach evaluation system is mainly based on objective data. Using the authoritative d

2、ata on web, we dig up four representative new metrics, and make use of two original metrics. The final metrics we chosen are stability, win-loss percentage, number of champion, Efficiency values, improving ability and experience.First, we establish the Principal Component weighting method as an orig

3、inal m . After bal ng the eigenvalues and the cumulative contribution, we identified three Principalcomponents to calculate the compensive scores.Then, to avoid excessive objectivity, the Neural network is involved to modify the weightsand improve the original m.When considering the time line horizo

4、n, after creating two ms based on two period, wefind that time line horizon always make old times coaches be underestimated. So, we establishedan additive mof time series to eliminate the secular trends and calculate the final score.Finally, we make the Sensitivity Analysis based on the uncertainty

5、of metrics. The rankscalculated by the mwill not appear substal changes, which mean the sensitivity is nothigh. Our mhas strong applicability; it combines the subjectivity with objectivity as a way of judgment. And it is adapted well across gender and most possible sports.The metrics needed in our m

6、is relatively common. But for sports where theave morethan two competitors, we should find other useful metrics. The sports involved female is more unpopular than the tops. Since Principal component analysis requires a lot of data to support, sowe should increase the weight of Neural network to redu

7、ce the shortcoof lacking data.Team#28015Page 1 of 23ContentsIntroduction21.1 Overview21.2 Problem background31.3 Assumptions31.4 Terms and Notation412The evaluation m. 42.1 The metrics for assessment42.1.1 Data introduction42.1.2 Evaluation metrics Introduction52.2 Choose the appropriate method to c

8、alculate the weight62.3 Initial m. 62.3.1 Weighting method of principal component62.3.2 solving steps82.4 Improved m. 82.4.1The principal theory of neural network92.4.2The steps of the compensive method92.5 The solution and comparison102.5.1 The weight of the principal component102.5.2 The compariso

9、n with the principal and improved method12How time affect the coach's assessment123.1 Different times bring different coaches.133.2 Pick out the secular trend from series of coaches14About gender and other sports16The final result of three sports16Mtest and sensitivity analysis176.1 Mtest176.2 S

10、ensitivity analysis1734567Evaluation of our m. 187.1 Strengths187.2 Weaknesses18. 18The article19Reference20Appendix2089101110.1 The10.2 The10.3 The10.4 The10.5 Thecode of calculating improving capacity metric20code of calculating stability metric21code of calculating experience metric21code of PCA

11、method combined with BP network method21code of sectional fitting and eliminating time-series trends22Team#28015Page 2 of 231Introduction1.1 OverviewlSelecting the evaluation methods:In the evaluation activities of coach, the rationality of evaluation methods and types makes a direct impact on the c

12、onsolidated results of coach. From the survey results at home and abroad over the last decade, a variety of evaluation methods can be seen. Generally speaking, the evaluation methods can be divided into the following five parts.²Expert evaluation method. Select the evaluation object and analyze

13、 them by experts first. Then determine the value of each index and the consolidated results by using addition or multiplication scoring method.Questionnaire method.Subjective method and other mathematical method: such as AHP, fuzzy²²compensive evaluation.²Artificial neural network eva

14、luation method: gray comp method.ensive evaluation²Principal component evaluation method: Principal component evaluation method is an analysis method which based on the existing data to calculate the right weight of each component and the consolidated results.lConsidering the following principl

15、es when selecting evaluation methods:²²The evaluation method chosen must have a solid theoretical foundation;The method chosen must be simple and clear to minimize the complexity of the algorithm more and more;The method chosen must be able to reflect the actual situation of every evaluati

16、on object accurately;²Thus, combining the advantages and disadvantages of various evaluation methods, wedecide to choose the Principal Component Compensive Evaluation Method.lThe method of determining the weightsWhen we conducted a systematic assessment, each indicator of the evaluation index s

17、ystem has a different degree of impact on the description of the object, thus their weights are different.There are a variety of methods to determine the weight, which mainly be divided into two parts, subjective and objective weighting method. Subjective weighting method is mainly based on the expe

18、rience and the subjective judgment by the experts, such as the Analytic Hierarchy Process (AHP) method, Delphi method. However, the objectivity of thisTeam#28015Page 3 of 23method is poor. Another method is objective weighting method, which is based on thecomposition of the original data. It does no

19、t depend on the's subjective judgment. Sothese methods have a stronger objectivity, such as principal component weighting method, entropy method, etc.To empower each indicator accurately, objectively and reasonably, we should combine a variety of methods, and take the internal laws of the data e

20、valuation and past experience into consideration in order to transform this experience into knowledge. Along with the development of evaluation process, the method of determinate the weighting become more and more scientific and reasonable.Because there often exits a greater correlation between the

21、evaluation indexes, the duplicate information they provided will affect the authenticity and reliability of the analysis results. Principal component weighting method can converse related P indexes to P unrelated component by the way of coordinating rotation, and then determine the indexweight. This

22、 method just can overcome the shortco method is both objective and very reasonable.of other indicator systems. So this1.2 Problem backgroundWith the development of modern competitive sports and increasingly intense competition, the athletes who want to create excellent grades, not only depends on th

23、eirquality, but also the compensive quality of the coaches, so the requirements needed arehigher and higher. Quality of the coaches directly affects the quality of the sports trainingand the level of athletes. However, when comes to the matter how to reflect the overall quality of coaches in sports

24、training and competition and how to correctly evaluate coaches, we need a set of scientific and effective evaluation method.Based on the development of the modern sports, higher requirements are put forward on the coaches. We want to build the coaches evaluation system from the ability of quality, w

25、hich can not only make more scientific and reasonable evaluation for the coaches, but also play the role of guiding and motivating the coaches, and further improve their comp ensive quality. This paper are based on the above background to research on the problem of coaches overall quality evaluation

26、.1.3 AssumptionsIn order to make a simple mof the assessment, we make the following assumptionfor the whole paper. Others applied only for part of the paper will be necessary.clear whenlllllThe players results of a team change little because of an individual player. A teams performance can reflect t

27、he score of the coach.Suppose that a coach only teach in one team every year. All the indicators data of the coaches is accurate.The team scores of one coach were completely included in the data.Team#28015Page4of23llMost of the coaches have short shutdown time in their career. The data sorted by the

28、 coachs retired year is a time-series data.1.4 Terms and NotationAdditional terms and notation areto simplify analysis for individual sections.These assumptions will be discussed at the appropriate locations.2The evaluation m2.1 The metrics for assessment2.1.1 Data introductionIn view of the pros an

29、d cons of different evaluation ms in the above literaturereview, we decided to construct a mbased on the data itself and find the best weightsto avoid the influence of subjective factors. Therefore, we searched a lot of website and literature, and decided to use the data on "sports-reference&qu

30、ot; to ensure the authenticity and rationality of the data. Part of the original data is in the following table:Table1:1900-2013 annual competition information table (Football)Table2: the coach information table (Football)YRSCoach IDSchool IDWLPctCoachSchool1900124550.5Sam Boyleinson1900241550.5Fred

31、 BrownMassachusetts.563J.H. CallahanTri(CT)200.375TJ WeistConnecticutYYearsPWin-loss percentageGGamesWwinsLlossesSCompensive scoreliThe eigenvalues of the i-th principal componentX iThe i-th standard of evaluationwiThe weight of the i-th standardTeam#28015Page 5 of 232.1.2 Evaluation metrics Introdu

32、ctionThere are lots of metrics to represent a great coach. We can not only judged by the win-loss percent, years, games, but also the improving ability, performance, stability and soon. When it comes to choose an useful metrics, we take fullof the principle of easyavailability, usability, and compen

33、sive evaluation of other aspects. We abandon themetrics that can not be qufied, such as team u qufiable metrics.and coach moral, we just keep someTo make these metrics are applicable in every sport, we select metrics which are common to each sports , such as the win-loss, number of champion and so o

34、n. According tothe known data from websitewe use Microsoft SQL Server andsoftware make data conversion of raw data and other operations. step1.Standardizedstep2 Normalized. Ultimately we get the evaluation as shown below.：Stability，which means the stability of development of which team theX11)coach

35、in.Code is in the appendix.We assume that coach Tom taught in Harvard between 1948 and 1956, in Stanford between 1956 and1970. Standard deviation1 equals to Standard deviation of x2 when he was in Harvard.x1 = Standard deviation1+2) X 2 ：Performance, the percentage of win games and totally games.3)

36、X 3 ：The number of champion. It differs when it comes to different sport. Such as we use the bowl games to represent it.4) X 4 ：Improving capacity. If a coach promote the win-loss percentage of a teama%b%in n years. The improving capacity equalfromtoCoach IDCoachFromToYrsGWLTPctG(Bowl)1Sam Boyle9110

37、0.4502Fred Brown12800.601969TJ Weist200.3750Team#28015Page 6 of 23to (b% - a%) / n . Code is in the appendix.：Efficiency values. It is equal to Games/years, the number of games whichX 55)participate every year oage.X 6 :Experience: it is the transformation of the games. According to common6)sense, t

38、he marginal fuction is a Marginal decreasing function.So we assume that marginal fuction has the form like:f (x) = ae-bx + c(1)Based on the particular data we have. We use three equations to get the value of a, b and c.gamesò0Experience =f (x)dx(2)After searching and calculating, we find thousa

39、nds data of coaches of football, basketball and soccer as follows. You can find part of the data in the Appendix.Table 3：The coach number of each sport2.2 Choose the appropriate method to calculate the weight.When we conducted a systematic assessment, each indicator of the evaluation index system ha

40、s a different degree of impact on the description of the object, thus their weights are different.In order to calculate the appropriate weight of each metric accurately, objectively andreasonably, we have to find the best empowering way to construct compensiveevaluation indexes to judge whether the

41、coach is great or not. There are many ways to empower weight, but most of them are too subjective. Therefore, we used principal component weighting method to empower each indicator, and then principal component weighting method combined with neural network weighting method.2.3 Initial m2.3.1 Weighti

42、ng method of principal componentSince evaluation indexes have great correlation with each other, the repeatedsportscoach numberfootball1969basketball3405baseball351Team#28015Page 7 of 23information they provided will affect thes authenticity and reliability. Theprincipal component method can transfo

43、rm all the related indicators to irrelevndicatorsby rotation of coordinates. Then we can set the weight of each index and improve the authenticity and reliability of the results. The principal component method is as follows:P .So sample data can be expressedFor sample data, we set variables asas the

44、 following matrix:X12 .X1PX 22 .X 2 PX=（X .X ） (3)121PXn 2 .X nPAfter standardization, we transform initial data to X and we can find a new ensive variables instead of the initial variables by weighting method of principalcompcomponent, as follows:ìF1 = a11x1 + a21x2 + .ap1xpïF= a x + a x

45、+ .axï 212 122 2p 2 píï(4)ïF= ax + ax + .axîp1m 12m 2pm pa1i ,a2i ,.api (i = 1,2., m)means the eigenvalue of the Xs covariance matrix.Then we set:A = (aij ) pxm = (a1,a2 ,.am ) , Z × ai = liai(5)Z is the correlation coefficient matrix, li、ai means correlation coefficien

46、t matrixs eigenvalues and unit eigenvectors, l1 ³ l2 ³ . ³ lp ³ 0 .The above equations require precondition as follows:1. a2 + a2 +.a2 = 1(i = 1,m)(6)1i2ipi2. AT A = I ( A = (a )= (a ,a ,.a ) (7)mij pxm12mTeam#28015Page 8 of 23= ì0 i ¹ j3. Cov(F , F ) = ld ,dí(8)ij

47、i ijijî1 i = jF1was the first principal component ,it has the biggest variance and theuminformation. If the first principal component wasnt enough to represent the information of initial p index, we have to select F2 to reflect the original information effectively, The F3.FP can also be selecte

48、d as the same way.We can obtained the factor loading matrix B by SPSS, li、ai is its eigenvalues andunit eigenvectors. Then we determined the weight by characteristic root, asliw =(9)ipå lii=1We set B = (w ,.w ,.w )' , so synthesis evaluate function is:12pS = w1F1 + w2F2 + . + wp Fp(10)2.3.2

49、solving stepsStep1 ： This procedure has two dual steps, so all the indicators can bestandardization at first and normalization.Step2：Analysis the principal component and identify several main ingredients tostay.Step3：Calculate the principal components scores.Step4：Count the synthesized indicators ag

50、gregate.Although the principal component weighting method which is based on the data isobjective and practical, we also need to consider more realistic andto the real life.2.4 Improved mPrincipal component weighting method combines with neural network weighting method. Adoption of principal componen

51、t weighting method combined with neuralnetwork weighting method，inherent law of indexes and past evaluation experience can be，which rationalizes the determination method of weight.taken intoTeam#28015Page 9 of 232.4.1The principal theory of neural networkThe basic idea of principalneural network wei

52、ghting method are as below：firstaccording to the pre-existing index system, we work out the principal components by means of PCA( principal component analysis), then weight as the rate of contribution of each variance corresponding to the principal component, summarize and get the synthesis score. F

53、inally, we choose conversion of the index data and evaluation result as samples, combine experiment of experts with the inherent law of index data ， use the PCA-simplified parameter matrix as a input vector of neural network at the same time，and then simplify the network structure eventually.2.4.2Th

54、e steps of the compensive methodStep1: Determine the neural network training samples.,Yin ,Determine the scoring matrix Yi = Yi1,Yi 2 ,corresponded to each principalYi , Si component via PCA，as well as compas training sample.ensive scoring matrix Si , and choosingStep2: Set up a 3- layers BP neural network and determine training method and networkparameters of neural network.Below is a schematic diagram of three-layer BP neural network, where the input layer contains three input nodes, hidden layer includes four nodes and the output layer includes an