2014年美赛数模B题-Finalist

Team Control Number

For office use only For office use only T1 ________________ F1 ________________ T2 ________________ F2 ________________ T3 ________________ F3 ________________

Problem Chosen T4 ________________ F4 ________________ 24270 B

2014 Mathematical Contest in Modeling (MCM) Summary Sheet

Summary

In order to estimate the excellence of different sports coaches and to give a ranking result, two distinct models are developed. The first model is a comprehensive evaluation method. And the second model is a ranking algorithm analogous to the Journal Influence Algorithm. In the first model, we take into account a variety of metrics, and divide them into two

categories: Objective Metrics and Subjective Metrics. In the Objective Metrics, we consider four factors, the number of wins, winning percentage, champions and final fours. All these factors have contributions to the excellence of a coach. We deem that the total number of

games in a year could affect the number of wins, and the unevenness of team quality could

affect the winning percentage. By employing statistical regression method to process collected data, we establish two functions of influence coefficient to eliminate the

discrepancy caused by the two kinds of effect. In the Subjective Metrics: we consider two factors, media popularity and tenure. We employ Fuzzy Analysis Method to quantify these two subjective factors. We further incorporate Analytic Hierarchy Process (AHP) and Gray Relational Analysis Grade Method (GRAP) to determine the weight allocation to different metrics. The final ranking gives a comprehensive result by weighing results returned by these two methods. Using data from Sports Reference and other websites, the rankings in basketball, football and baseball accord with previous media commentaries.

In the second model, we deem that the excellence of a certain coach can be reflected from the media impact over the span of history and that the interactions between two coaches can reflect the disparity of skill level between them. We use search results returned by Google to quantify the impact of one coach on another. Based on the search results, we build a

cross-reference matrix to represent relationships between coaches. In view that the different time periods that two coaches were in may largely affect the interaction between them, and the personal reputation may influence the number of search results, we develop a weight function of two variables to compensate the influence of time and to rule out the redundant information.

In consideration of the similarity between personal influence and journal influence, we refer to the Journal Influence Algorithm introduced by Eigenfactor and establish a new ranking algorithm. The basic idea of the algorithm is subtle: using weight function to modify the cross-reference matrix, and taking into consideration of individual influence, the algorithm gives an evaluation vector to rank different coaches. To test the validity of this algorithm, we apply the algorithm into basketball, football and baseball. The algorithm gives a result that is similar to the result obtained in the first model. The ranking also agrees with

previous media commentaries. Furthermore, by slightly adjusting the coefficients, we can apply the algorithm into various sports.

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“Dream Team” of College Coaches

# Team 24270

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Team # 24270 Page 2 of 26

Contents

1. Introduction ...................................................................................................................... 3

1.1. Restatement of the Problem..................................................................................... 3 1.2. Model Overview ...................................................................................................... 3 2. Assumptions ...................................................................................................................... 3 3. Model Ⅰ ........................................................................................................................... 4

3.1. Additional assumptions ........................................................................................... 4 3.2. Notations ................................................................................................................. 4 3.3. Evaluation System ................................................................................................... 5 3.3.1. The influence of time on the total number of wins ........................................................ 6 3.3.2. The influence of time on the winning-percentage .......................................................... 6 3.3.3. Fuzzy Analysis............................................................................................................... 7 3.3.4. Nondimensionalization process ..................................................................................... 8

3.3.5. Final result ..................................................................................................................... 8

3.4. Solutions to ModelⅠ ............................................................................................ 10 3.4.1. Basketball..................................................................................................................... 10 3.4.2. Football ........................................................................................................................ 11

3.4.3. Baseball........................................................................................................................ 12

4. Model Ⅱ ......................................................................................................................... 13

4.1. Additional assumptions ......................................................................................... 14 4.2. Notations ............................................................................................................... 14 4.3. The Individual Influence Vector............................................................................ 14

3.4.4. Sensitivity analysis....................................................................................................... 13

4.3.1. Original data................................................................................................................. 15 4.3.2. The influence coefficient of time ................................................................................. 15 4.3.3. The influence coefficient of reputation ........................................................................ 16 4.3.3. The individual influence vector ................................................................................... 16 4.4. The Cross-Reference Matrix ................................................................................. 16 4.4.1. The weight function ..................................................................................................... 17 4.4.2. The final cross-reference matrix .................................................................................. 17 4.5. The Evaluation Vector........................................................................................... 18 4.6. Solutions to Model II............................................................................................. 18 4.6.1. Basketball..................................................................................................................... 18 4.6.2. Football ........................................................................................................................ 19 4.6.3. Baseball........................................................................................................................ 19 4.6.4. Sensitivity analysis....................................................................................................... 19

5. Applicability.................................................................................................................... 20 6. Strengths and Limitations ............................................................................................. 21

6.1. ModelⅠ................................................................................................................. 21 6.2. Model II ................................................................................................................. 21 7. Conclusions ..................................................................................................................... 21 8. The Article for Sports Illustrated ................................................................................... 22 References .............................................................................................................................. 23 Appendix ................................................................................................................................ 24

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Team # 24270 Page 3 of 26 1. Introduction

1.1. Restatement of the Problem

Sports, by definition, is all forms of usually competitive physical activity which aim to use

physical ability while providing entertainment to participants and spectators[1]. No wonder the word “sports” gives us a first impression of fierce competition, agitated spectators, sweating on the running track, combined with a joy of victory. It is the uncertainty that makes the sports game so intriguing. However, where there is competition, there will always be victory,

defeat, and ranking. Loyal sport fans could debate day and night over the question who is the best player or coach. These debates have called forth a need for certain criterion of sports coaches and players. The criterion has to be: (1) all-encompassing to take into consideration a variety of factors; (2) applicable to various sports; (3) robust enough to remain unaffected by

fluctuation.

1.2. Model Overview

? ModelⅠ

The evaluation method in ModelⅠis based on a comprehensive method sophistically combining Analytic Hierarchy Process(AHP) and Gray Relational Analysis Grade Method(GRAP). In the evaluation process, we take into consideration the influence of time horizon, and incorporate Fuzzy Analysis Method, which make it feasible to compare diverse factors on the same level. The ranking results in three different sports accord with previous media report, which attest the validity of this method. ? Model Ⅱ In model II, we assume that the excellence of a certain coach can be reflected from the media impact over the span of history and thus can be gauged by the impact on another coach within or without the same period of time. We use Google search results to quantify the impact of one coach on another. The relationship between coaches can be established as a cross-reference matrix. By further taking into account the influence of time, influence of reputation, and a modification to rule out the redundant information, we obtain a final evaluation vector. The final ranking result is roughly approximate to the result in model I. To sum up, we only need the search results returned by Google search engine to estimate the excellence of certain coach with high accuracy.

2. Assumptions

? We assume that the competition rules of each sport do not change.

Although sports are developing, we do not take into account of time in the competition rules in order to compare the coaches of different years more fairly.

? We neglect tied competitions since they have the same effect on the two compared

teams.

? We only take the Division I into consideration.

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Team # 24270 Page 4 of 26

Competitions are divided into three parts: Division I, II and III according to the level of sport strengths of different colleges. Since Division I always concludes top coaches, we only take Division I into consideration. ? The selected data are valid.

? Additional assumptions are made to simplify analysis for individual sections. These

assumptions will be discussed at the appropriate sections.

3. Model Ⅰ 3.1. Additional assumptions

? The evaluation system includes two parts: Objective Metrics(OM) and Subjective

Metrics (SM).

? We assume that OM include four specific indexes: the total number of wins, the

winning-percentage, the number of final fours and the number of champions.

? Tenure and media popularity are considered in SM.

In the subjective metrics of ranking coaches, some factors are hard to investigate

qualitatively and quantitatively due to lacking data, such as, his or her influence to players, range of knowledge, studying ability, team spirits, searching talents, acting in competitions, salary and so on. Therefore, we neglect these indexes in SM. ? Time only makes a difference in the total number of wins, and the winning

percentage.

In fact, the numbers of final fours and champions have no effect on the other two in OM, since the number of teams which are able to enter into final fours and even achieve champions is fixed. And we neglect the influence of time on media popularity in order to simplify the model.

3.2. Notations

Table 1: Notations and Descriptions

Notations Descriptions

Si Evaluation object xj Evaluation index

n The number of evaluation objects m The number of evaluation indexes Evaluation index matrix x, x , x’’, x*

t Time pi , qi Influence coefficients of time W(t) The total number of competitions in t

s(t) The standard deviation of all winning-percentage in t Mj Maximum of xij mj Minimum of xij

’

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Team # 24270 Page 5 of 26 Notations Descriptions

f(x) Subordinate function A Pairwise comparison matrix

λ The largest eigenvalue

w Weight vector CI Consistency index RI Random consistency index CR Consistency ratio B Evaluation vector of AHP

?0??, Grey relational coefficient Absolute difference

??? ???

Δmin Minimum difference

Δmax Maximum difference r Relation degree vector

C Evaluation vector of Grey Relation Degree α , β Partial coefficient U Ultimate evaluation vector

3.3. Evaluation System

We define n as the number of evaluation objects, and S1, S2,…, Sn (n>1) are the evaluation objects. m is the number of evaluation indexes, and x1, x2,…, xm are the evaluation indexes. Evaluation index vector is

T

?????????????1 2The total evaluation indexes include OM: the total number of wins, the winning-

percentage(pct.), the number of final fours and the number of champions and SM: tenure and media popularity. So m?? 6 ,

T

, x? x ??????? 1 x , , mx . m???

Where:

x x x x x ????????????, , , , ,?x x 1 2 3 4 5 6? x1 — the total number of wins vector. ? x2 — the winning-percentage vector. ? x3 — the number of final fours vector. ? x4 — the number of champions vector. ? x5 — tenure vector.

? x6 — media popularity vector.

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Team # 24270 Page 6 of 26

Figure 1: Flow chart of model I

Undoubtedly, time plays an important role in evaluating top coaches. According to the assumptions, time only makes a difference in the total number of wins, the winning- percentage.

3.3.1. The influence of time on the total number of wins

With the development of sports, the competition is getting relatively fiercer than ever, which means the disparity between teams become wider. The total number of games also increases with time going on. Therefore, when evaluating coaches in the previous century, the later certain coach begin his coaching career, the more likely he will get more wins. So we should put less weight on the coaches active in a later time period. And we can get a fairer evaluation of coaches within different time periods.

In order to compensate the influence of t, we establish Influence Coefficients of Time (ICT)

pi??i?? 1,2,?, n? . We assume that the total number of competitions in t is W??t?? . W??t?? can be obtained by statistical regression and simulating and curve fitting of selected data. So we define:

i

p???

1 W??tmi???'

where tmi is the middle year of tenure of Si . And then x1i?? x1i?? pi??i?? 1,2,?, n? .

3.3.2. The influence of time on the winning-percentage

As for the winning-percentage, sports were underdeveloped at an earlier time, and the quality disparity between teams is comparatively narrow. Therefore, the standard deviation of winning-percentage of each coach is closer to zero. Thus we should put less weight on the coaches active in a “mediocre” time period. We define ICT here as qi (i=1,2,…,n), we assume that the standard deviation of all winning-percentage in t is s??t?? . s??t?? can be obtained by statistical regression and simulating and curve fitting of selected data. So we define:

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and x2i?? x2i?? qi??i?? 1,2,?, n? . '

Team # 24270 Page 7 of 26 i

q???

1 s??tmi???

3.3.3. Fuzzy Analysis

As for SM indexes, we assume that they can be divided into five levels: “ Excellent, Very Good, Good, Not Good, Bad”. And we correspond the five levels into 5,4,3,2,1 successively For continuous quantification, we assume:

? As for “Excellent”, we suppose f??5??? 1. ? As for “Very Good”, f??3??? 0.7 .

? As for “Bad”, f??1??? 0.1 . We employ partial large Cauchy distribution and the logarithmic function as the subordinate function[2]:

f (x)?????

??c ln x?? d , 3?? x?? 5

where a, b, c, d stands for undetermined constants. We use the initial conditions above to define their values. And solution of the subordinate function( Figure 2) is:

???? 121 a x b???????????????,1 3x????????????????????

??????? 121 2.8049 0.4417x?????????????????????????????,1 f ( x)???? 3 x ???????????????????????????? (1) ??0.5873ln x?? 0.0548, 3?? x?? 5

Figure 2: Trend of f(x)

Media popularity is measured by the number of search results via Google. The impact of duplication of names can be neglected by means of adding search keywords in order to rule out the redundant information.

We map xj ( j=5,6) into interval [1,5], through function (1),we can obtain:

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Team # 24270 Page 8 of 26 ????4 ji jx m??????????? M 1x f??????12 5,6i n j??? (2)??????????????

j?? m j???where M j?? maxxij , m j?? minxij1????1???i? n

????? j?? 5,6? . As for x3 and x4, we define that x3’= x3, x4’= x4. we use x'j?? j?? 1, 2,?,6? to proceed the following calculation. 3.3.4. Nondimensionalization process

We employ extreme difference method to nondimensionalize the different indexes so that we can compare them[2]on the same level. The method is as follows:

x'ji?? m j

?M j?? m j

T

and ' ' ' ' '1 2 3, , , , j j j jnx x x x????????????????????x?????????????????????1, 2, ,6j??????where M j?? max1???? x1ij???i? , n

m j?? min?xij???? j?? 1, 2,?,6? . and then we obtain the final evaluation index matrix:

T

, , , , ,????????????????????* '' '' '' '' '' ''1 2 3 4 5 6x x x x x x x 3.3.5. Final result

By using AHP as the subjective evaluation method and GRAP as the objective method, the final represents a comprehensive evaluation combined the merits of these two methods.

? Analytic Hierarchy Process[3] (AHP)

By comparing the effect of two indexes x'j ,the weights of the two method w? x'j???(j=1,2,…,m)are given. Then we construct the pairwise comparison matrix A .

1??1 5 ??3 7 2???11 ??

? ?? 1 1 3 5???????5 3???A???3 ?? 3 1 1 6???????

5 3?????????????????????

5??

??5 5 7 1 21 1 1 3???????

?? 3 3 5???????3 1????We can obtain the largest eigenvalue of A:λ=6.0496 and its weight vector :

??????????????????????????0.1248,0.1469,0.4593,0.8125,0.0775,0.2928?w T

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(4)

Team # 24270 Page 9 of 26 After that, we must check the consistency of matrix A. The consistency index is calculated as follows:

CI???????n???? 9.92??10?3 n??1

From Table 2, the random consistency index RI=1.24

Table 2: The Quantitative Values of RI[2]

n 1 2 3 4 5 6 7 8 9 10 11 RI 0 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 1.51 Then, we can obtain consistency ratio:

CR???CI RI

0.008 0.1???????Therefore, we can safely draw the conclusion that the inconsistent degree of matrix A is in

a tolerable range, and we can take its eigenvector as weight vector w[3].

We define B as the evaluation vector of AHP, and B can be calculated as follows:

B?? x'?? w (5)

In evaluation vector, the greater Bi is, the higher ranking Si is. ? Gray Relational Analysis Grade Method[4] (GRAP)

We use integral grey relational degree to analyze the metrics data. And we take the total number of wins as the reference sequence:

0 0

????????????1,2, ,x i n??????x???????????????and then we can obtain the gray relational coefficient[4]:

???ji ?? ir x x, ???? ?? m m??? ?? i?????????????????? 1, 2,?, n, j?? 1,2,?,6 ???0 maxji?????????Where:

???0??????0j????ji i x ??? ix????

—absolute difference.

????? min minj

minj i i?????????? —minimum difference of all indexes data. ?????j

max max maxj ii ???????????? —maximum difference of all indexes data. ??? —resolution ration.

For every coach Si , we determine its weight as wi , which should satisfy the requirements:

n

0?? wi?? 1, 1 w ??i i??1

??

After determining the weight, we can obtain the relational degree[4]:

??????,0 0

jr r x xn

??? j ?i i iw ??? , r x x?????j i??1

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(6)

Team # 24270 Page 10 of 26 r r r r And then we construct the relation degree vector????????????1 2 3 4 5 6r , , , , , rr = , where 1 1r?? . T

We define C as the evaluation vector of AHP, and C can be calculated as follows::

C?? x'?? w (7) In evaluation vector, the greater Ci is, the higher ranking Si is.

? Combination of AHP and GRAP

At first, we employ extreme difference method to nondimensionalize the two evaluation vector B and C. And then, we construct an ultimate evaluation vector:

U???? B???? C (8)

where?? ,?? respectively stands for the weight of AHP and GRAP, which should satisfy the requirements of α + β =1.

Finally, we sort the value of Ui (i=1,2,…,n), and Si that corresponds to the top 5 of Ui are top five coaches.

3.4. Solutions to ModelⅠ

We choose three sports to verify our model and get the results, which include basketball, football and baseball.

3.4.1. Basketball

? Searching and selecting data

We search and select data through the Internet[5][6][7]. For example, first, we search 100 coaches and their evaluation index data. Secondly, we rank them by comprehensively considering the total number of wins and the winning-percentage, so we can get top 40

coaches. And then, we consider other metrics and rank top 20 coaches. Finally, the evaluation system is based on the selected 20 data. Table A1 in Appendix show the selected coaches and their evaluation index data.

? Determining the final evaluation index matrix

At first, we determine vector x’. For x1, via the data in Table A2, we use W(ti)=Num2, where Num represents the total number of teams in ti.

We utilize software MATLAB to plot the graph of W(1)(t) by simulating and curve fitting of data (Figure 3). So we can get pi for each Si, and then we obtain the vector x1’.

Figure 3: Trend of W(1)(x) Figure 4: Trend of s(1)(x)

For x2, via the data in Table A3, we also plot the graph of s(1)(t) by simulating and curve fitting of data (Figure 4). So we can get qi for each Si, and then we obtain the vector x2’.

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Team # 24270 Page 11 of 26 For x5 and x4, from (1)(2), we can obtain x5’ and x6’.

Secondly, from (3), we can obtain x''j?? j?? 1,2,?,6? . Finally, from (4), we can obtain x* . We list the quantitative value of x* in Table A4.

? Obtaining the result via ultimate evaluation vector

At first, from (5), we use AHP and get B . Secondly, we use GRAP and define that ρ=0.3 and wi=0.05(i=1,2,…,n). From (6), we can get the relation degree vector r. And then, from (7), we can obtain C. Finally, from (8), by defining???? 0.6,???? 0.4 , we can obtain the ultimate evaluation vector:

U????0.293,0.288,0.468,0.241,0.422,0.160,0.521,0.868,0.713,0.311,0.481,0.836,0.168,0.998, ?0.130 0.318 0.243 0, , , .482,1.761,0.138 By sorting the value of Ui (i=1,2,…,n), we can obtain the ranking result of Si. And the ranking vector is:

T

????????????????????????????????19,14,8,12,9,7,18,11,3,5,16,10,1,2,17,4,13,6,20,15?1Rank

Table 3: Top 5 Coaches of Basketball

T

Therefore, we list top five coaches of basketball in the previous century in Table 3:

No.1 No.2 No.3 No.4 No.5 S19 S14 S8 S12 S9

John Wooden Dean Smith Mike Krzyzewski Adolph Rupp Bob Knight

This result is largely agreement with the widely accepted result[8][9].

3.4.2. Football

? Searching and selecting data

Like what we do in basketball, we search and select data through the Internet[5][10][11]. However, we calculate that the number of final fours is the sum number of times that teams can enter into Super Bowl.

? Determining the final evaluation index matrix

At first, we determine vector x’. For x1, we use W(ti)=Num2, where Num represents the total number of teams in ti.

We can obtain W(2)(t) by simulating and curve fitting of data. So we can get pi for each Si, and then we obtain the vector x1’.

For x2, we also obtain s(1)(t) by simulating and curve fitting of data. So we can get qi and the vector x2’.

Finally, from (4), we can obtain x* . We list the quantitative value of x* .

? Obtaining the result via ultimate evaluation vector

Like what we do in Basketball, we can obtain the ultimate evaluation vector:

U????0.234,0.879,0.738,0.106,0.359,0.296,0.383,0.193,0.291,0.453,0.494,0.248,0.180,0.615, T

?1.000 0.316 0.151 0, , , .047, 0.392,0.052 By sorting the value of Ui (i=1,2,…,n), we can obtain the ranking result of Si. And the ranking vector is:

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Team # 24270 Page 12 of 26 ?????????????????????????????????,2,3, ,11,10, ,7,5,16, ,9,12,1,8,1315 14 1 ,17,49 6 ,20,18?2Rank Table 4: Top 5 Coaches of Football

T

Therefore, we list top five coaches of basketball in the previous century in Table 4:

No.1 No.2 No.3 No.4 No.5

S15 S2 S3 S14 S11

Joe Paterno Bobby Bowden Bear Bryant Tom Osborne Don James

This result is largely agreement with the widely accepted result[12].

3.4.3. Baseball

? Searching and selecting data

Like what we do in basketball, we search and select data through the Internet[13][14]. But in this sport, we assume that the number of final fours is the number of champions of NCAA

competitions that teams can achieve. And we assume that the number of champions is the number of champions of National competitions that teams can get.

? Determining the final evaluation index matrix

At first, we determine x’. For x1, due to scarcity of the data, we can only search a little

information of several years[9]. We use W(ti)=Num2, where Num represents the total number of competitions of champion in ti.

We can obtain W(3)(t) by simulating and curve fitting of data. So we can get pi for each Si , and then we obtain the vector x1’.

For x2, due to lacking the standard difference of winning-percentage in every ten year, we choose another approach to get x2’. Considering the influence of time, first, we employ extreme difference method to nondimensionalize tm into tm’, where tmi is the middle year of tenure of Si. Then, we define

''mt???1 5m t???

'

and then we define x2' i?? x2i?? tmi'' . So from (3), we obtain x''2 .

Finally, from (4), we can obtain x*. We list the quantitative value of x*.

? Obtaining the result via ultimate evaluation vector

Like what we do in Basketball, we can obtain the ultimate evaluation vector:

U????0.353,0.278,0.587,0.223,0.205,0.302,0.208,0.189,0.314,0.494,0.203,1.000,0.214,0, T

?0.022,0.044,0.232,0.225, 0.138,0.118 By sorting the value of Ui (i=1,2,…,n), we can obtain the ranking result of Si. And the

ranking vector is:

????????????????????,3,10,1,9,6, 2,17,18, 4,13,7,5,11,8,19, 20,161 ,12 ,15 4?3Rank Table 5: Top 5 Coaches of Baseball

T

Therefore, we list top five coaches of basketball in the previous century in Table 5:

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Team # 24270 Page 13 of 26 No.1 No.2 No.3 No.4 No.5 S15 S2 S3 S14 S11

John Barry Mike Martin Rod Dedeaux Augie Garrido Jim Morris This result is largely agreement with the widely accepted result[15].

3.4.4. Sensitivity analysis

By changing the weight of AHP and GRAP in equation (8), we analyze the changing result of basketball. For example, we define???? 0.5,???? 0.5 , and the result is listed in Table 6. The coaches who rank top 5 do not change:

Table 6: Top 5 Coaches of Basketball

No.1 No.2 No.3 No.4 No.5 S19 S12 S14 S8 S9

John Wooden Adolph Rupp Dean Smith Mike Krzyzewski Bob Knight

When defined???? 0.4,???? 0.6 , the result changes, which is listed in Table 7. The coaches who rank top five change:

Table 7: Top 5 Coaches of Basketball

No.1 No.2 No.3 No.4 No.5 S19 S12 S14 S7 S8

John Wooden Adolph Rupp Dean Smith Hank Iba Mike Krzyzewski When defined???? 0.7,???? 0.3 , the result is listed in Table 8. The coaches who rank top

five do not change:

Table 8: Top 5 Coaches of Basketball

No.1 No.2 No.3 No.4 No.5 S19 S14 S12 S8 S9

John Wooden Dean Smith Adolph Rupp Mike Krzyzewski Bob Knight

As can be seen from above, when there is a slight change of weights, the result do not

change. But with a relatively greater change, weights have an effect on the result.

4. Model Ⅱ How could one’s reputation affect another’s? One way is to follow the implication in the saying: “You wouldn’t mention A and B in the same breath.” It means if the difference between two people is too wide, it would be unlikely for most of individuals to mention them in a same talk. The same holds true for the sports coaches. That means, if two coaches are absolutely not on the same level, more likely than not, there will be few reports on these two coaches. On the other hand, if two of them are top coaches, there will be a plethora of reports: such as “The Greatest Coaches Ever” “Basketball Hall of Fame”, on the two coaches. Informed by this natural law, we may find an innovative approach to estimate a coach’s level of excellence and popularity. The working flow is shown as follows:

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Team # 24270 Page 14 of 26

Figure 5: Flow chart of model II

4.1. Additional assumptions

? The excellence of a coach and association between two coaches can be accurately

reflected by the mass media.

? The attention that the mass media have on certain coach is related to the search results on

Google, in terms of number of pages, report orientation and report time.

? The media attention is related to time and the excellence of certain coach. The influence

of time and excellence on the media attention remains unchanged to different kinds of people.

4.2. Notations

Table 9: Notations and Descriptions

Notations Descriptions

?? The number of search results of coach i

k, b Coefficient of the function through linear regression ?? Characteristic year of coach i

u The number of search results about sports career ICT Influence coefficient of time ICR Influence coefficient of reputation l Individual influence vector Z Original cross-reference matrix

WF Weight function

W Weighted cross-reference matrix

α, β Partial coefficient

4.3. The Individual Influence Vector

By our hypotheses, the excellence of a coach can be accurately reflected by the mass media. There are several ways to evaluate the media attention on a celebrity. One of the most simple and direct way is to record the number of search results on Google. However, the search results can be influenced by a variety of factors, such as time periods, tenure, etc. By simulating and curve fitting of sorted data, we evaluate the impact of such factors separately. Finally, we obtain a normalized individual influence vector.

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Team # 24270 Page 15 of 26

4.3.1. Original data

Here we define ti as the characteristic year, the average of the year that the coach i start coaching and the year of his or her retirement. (If the coach i is still active, then t is the average of the year that the coach i start coaching and this year, that is, 2014)

The search results vector a is the original data we use to estimate the individual influence, where ai is the number of search results of coach i. Particularly, the coaches here are sorted by

characteristic year in a descend order. This can be a great convenience to our later discussion about time factor.

4.3.2. The influence coefficient of time

According to the growth law of web information[16], the information aiming at a certain field is similar to an exponent increase. To test this hypothesis and better apply it to sports, we entered the Google website. Using “1910 basketball”, “1920 basketball” ， and “1930 basketball” as the “exact keywords”[17] respectively. The numbers of search results are shown

in Table 10:

Table 10: The Numbers of Search Results

Year 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Results 2750 5160 7440 11700 16200 26400 40200 25800 27000 67700 326000

Assuming that this is an exponential function: y1?? c?? edt . We use the least squared method

to obtain the unknown numbers in the function. See Figure 6.

Figure 6: Trend of exponential function y1 Figure 7: Trend of linear function y2

The result gives a satisfying simulation to the numbers of search results. However, the distinction between 2000s and 1900s is too large. In our observation, the search results of coaches at different period of time is almost of the same magnitude of as each other. So, here we use the natural logarithm of the search results. Again we obtain a linear function y2 as showed in Figure 7.

The difference between maximum and minimum is about half of the minimum value. This is a modest value that we can safely put into use to estimate ICT. Common sense told us that the greater number of total reports is, the more “valuable” the search result is, the greater weight the search result will get. So, we define ICT as

(influence coefficient of time)=iICT 1

1, 2, ,i n??????????????kti?? b

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Team # 24270 Page 16 of 26 where k and b are the unknown variables related to searching data.

4.3.3. The influence coefficient of reputation

As mentioned in the overview, by searching data with different methods and using diverse keywords, we observe that on the track of fame, the media will turn at first to the

achievements of sports one has then to the other aspects in his or her life. Therefore, the media attention can be interpreted and quantified by using the overall search results of one coach and his or her triumph and achievement.

To extract the information about news reports on the triumph of a certain coach from the search engine, first, we use Wordnet[18] as our tool to obtain a host of synonyms of the word “winning”, the result is:

“booming; flourishing; palmy; prospering; prosperous; roaring; thriving; in; made; no-hit; productive; self-made; sure-fire; triple-crown; victorious; successful”

Using these words as our “alternative keywords”, we can obtain the numbers of winning search results ui for coach i.

The result turns out to be an indication that the ratio between winning search results and overall search results is negatively correlated with the degree of reputation. That is, the higher reputation one coach gets, the more likely the mass media will concentrate on the other

aspects of life of this coach. According to this rule, we establish a function of ICR:

2

ia????

ICRi (influence coefficient of reputation)=1????????i?? 1,2,?, n??? ui???

4.3.3. The individual influence vector

The individual influence vector l can be interpreted as the overall search results modified by ICT and ICR.

i

l???

i i in j??1

a ICT ICR??????j j j a ICT CRI????????????????1, 2, ,i n?????(9)

The individual influence vector gives an accurate estimation of the media attention certain

coach got. It is a normalized vector, so that we can conveniently put it into use in the later section.

4.4. The Cross-Reference Matrix

With the help of the Google search engine, the degree of correlation of two coaches can be measured by the number of search results using two names as “Citation Keywords”[17] simultaneously. And we define the original cross-reference matrix Z. The entries of the matrix is:

Zij =number of search results number of coach i and coach j

Since exchange of the two names does not affect the result, the matrix Z is symmetrical. And we set all of the diagonal elements of this matrix to be 0.

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Team # 24270 Page 17 of 26

4.4.1. The weight function

The elements of cross-reference matrix are influenced by the distinct period of time and reputation. However, things get a bit more complicated here: if two coaches exist in the same period of time, then there could be more reports on competitions they engaged. The

competition reports are the redundant information that we want to avoid. What we are aiming to do is to evaluate a certain coach’s impact over the span of the sports history and to rule out

the redundant information. To do that, we assign a lower weight to the matrix element in which the two coaches is in the same period of time, and a higher weight to those in a different period of time.

First, let ti and tj to be the characteristic year of two coaches, we want to construct a weight function related to ti and tj. Because we previously set the diagonal elements to be 0, the weight on the diagonal can be zeros. Assuming that the average term of office is 2σ, according to the 2σ principle[3], there is little likelihood that the two coaches can encounter each other in the sport games. Then the corresponding weight can be approximate to be 1 outside the interval 2σ. By carefully weighing the pros and cons of diverse types of function, we establish the original weight function as

1 2????(1 e??

??2( )i jt 2? 2

)

Additionally, to take into consideration of ICT, we simply transform the one variable function IIT to two variable function multiplied by the original weight function. The final weight function WF is:

1 1 ??=WF

1 ??1 kt 2????i jkt 2 2

(1 e??

??2( )i jt t??2? 2

?????????) , 1, 2, ,i j n?????(10)

From (10), the values of the function in intervals (1910, 2010), (1910, 2010) are drawn in

3D graph as showed in Figure 8 and Figure 9.

Figure 8: Contour plot of weight function Figure 9: 3D graphic of weight function

As can be seen from the graphs above, as the characteristic year increases, the weight decreases linearly. And with the gap between two characteristic years increase, the weight increases somehow similar to a normal distribution curve. 4.4.2. The final cross-reference matrix

The weighted cross-reference matrix is defined as the original cross-reference matrix multiplied by the function value of weight function, via:

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Team # 24270 Page 18 of 26 Wij?? Zij??WF (ti , t j )

Normalizing the weighted cross-reference matrix by column sums, we obtain the final cross-reference matrix N.

N???

ijW ij

n

(11)

ijk??1

W??4.5. The Evaluation Vector

So far we have obtained the individual influence vector and the cross-reference matrix. Each of them partly reflect the impact that a certain coach has over the course of sports history. Furthermore, if one is related to a very influential coach, then this relationship can be

more “valuable” to him. This law is far better than merely counting the correlated search

our data.

Following the algorithm described in Eigenfactor Article Influence Algorithm, a new matrix is calculated as:

results. Following this law, we sophistically combine the Article Influence Algorithm[19] with

S???? N???? l?? eT (12)

Where α andβ are partial coefficients, α + β =1, and ? ? is a row vector of 1’s. We find out the greatest eigenvalue of matrix S, and the corresponding eigenvector is defined as the evaluation vector v. Through the interpretation given by Eigenfactor, [19]the values of each element of the leading vector represent an average fraction of time spent on each article[19]. However, the situation here is a little different in three respects: (1) the matrix N is a symmetrical matrix; (2) The individual influence vector here is the corresponding individual search results divided by the sum of search results; (3) the relative magnitude of α and β cannot be determined by previous data.

Finally, the evaluation vector is sorted in descending order. The location of element is the ranking of corresponding coach i.

4.6. Solutions to Model II

4.6.1. Basketball

? Searching and selecting data

First, we choose 100 coaches and their evaluation index data as our candidate pool. Secondly, we rank them by comprehensively considering the total number of wins and the winning-percentage. In this way, we get top 20 coaches from the candidates. The evaluation system is based on the selected 20 data.

Through Google search results, we obtain the original search result vector a(the third column in Table A5) and the original cross-reference matrix Z(Table A6).

? The influence coefficient of time

We use linear regression method, and the result turns out to be: k?? 0.0368, b????62.2719 .

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Team # 24270 Page 19 of 26 Therefore , ICT???

1

0.0368ti?? 62.2719

. ? The individual influence vector

Using software Microsoft Excel, we can easily get the individual influence vector from the data shown in Table A5. ? Weight function

Using the linear regression result in the previous section, from (10), the weight function can be written as:

1 1 ??=WF

1 1 2???12.5 0.0368 0.0368i jt t b??????????????????2 2

? The final cross-reference matrix

2 12.5(1 )e??????2( )i jt t??2 By following several steps of simple Matlab matrix operation, the final cross-reference matrix is listed in Table A7. ? Evaluation vector

In the equation S???? N???? l?? eT , by following the recommendation from Eigenfactor, we first set the variable?? and?? to be 0.8 and 0.2 separately. The elements of the leading vector are all negative. Then we change?? and?? to be 0.85 and 0.15, the result turns out to be satisfying: the greatest eigenvalue and elements in the leading vector are all positive. After that, we alter the value of?? and?? , the result changed in a manageable range. The top five coaches are:

1. John Wooden 2. Dean Smith 3. Mike Krzyzewski 4. John Thompson 5. Rick Pitino. Three of the top five coaches are also in the top five list of Model I. 4.6.2. Football

The data selection and processing method are completely in consistent with the method used in 4.6.1 section.?? and?? remains to be 0.85 and 0.15. The top five coaches are: 1. Frank Thomas 2. Joe Paterno 3. Urban Meyer 4. Don James 5. Bear Bryantlisted Three of the top five coaches are also in the top five list of Model I. 4.6.3. Baseball

Like how we process the data in basketball, the final top five coaches are: 1.John Barry 2. Jim Morris 3. Gary Ward 4. Gene Stephenson 5. Rod Dedeaux Again, three of the top five coaches are also in the top five list of Model I. 4.6.4. Sensitivity analysis

In the process of calculating evaluation vector, the?? and?? are defined to be 0.85 and 0.15. But it is not often the case: in various sports, the partial coefficient can change slightly. We only take basketball for example here. First, we set the partial coefficient?? and???to be 0.8 and 0.2, the amount of change(in percentage) in the returned evaluation vector is showed in Table 14:

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Team # 24270 Page 20 of 26

Table 14: The Amount of Change

Number 1 2 3 4 5 6 7 8 9 10

Change (in %) -3.73 -3.06 -1.31 -7.50 -5.61 -1.66 -2.13 -4.97 -1.48 -5.80 Number 11 12 13 14 15 16 17 18 19 20 Change (in %) -6.65 12.05 -7.09 -6.45 -5.73 15.23 -1.69 -6.25 14.74 -3.35

The ranking list of top five is:

1. John Wooden 2.Dean Smith 3.Mike Krzyzewski 4.Bob Knight 5.Rick Pitino

This list is closer to the list obtained in Model I(Table 3), in which four of the coaches are the same.

Secondly, we set the partial coefficient?? and?? to be 0.9 and 0.1, the returned evaluation vector changed in a moderate range. See Table 16.

Number 1 2 3 4 5 6 7 8 9 10

Change (in %) 3.03 2.12 0.29 7.23 5.21 0.5 1.3 4.38 0.99 4.99 Number 11 12 13 14 15 16 17 18 19 20 Change (in %) 6.3 -12.74 6.77 6.03 5.4 -16.02 1.45 5.64 -16.05 2.29 The ranking list of top 5 in this case is:

1. John Wooden 2.Mike Krzyzewski 3.Dean Smith 4.John Thompson 5.Rick Pitino Three of the coaches are listed in Model I.

As can be seen from above, with a slight change in the partial coefficient, the result

changed in a controllable range and remain close to the result in Model 1. The lager?? is, the more weight it is attributed to individual influence and vice versa.

Table 16: The Amount of Change

5. Applicability

? Genders

As for gender factors, we can employ two models to evaluate coaches with different genders due to the same evaluation indexes. Through the data collected by Internet[20], we use and modify model I to obtain the women basketball result. See Table 16.

Table 16: Top 5 Coaches of Women Basketball

No.1 No.2 No.3 No.4 No.5

Pat Summitt Tara VanDerveer Barbara Stevens C. Vivian Stringer Geno Auriemma

The ranking list of top coaches is agreement with the data from Internet. ? Other Sports

As for other sports，like Track and Field Events or Swimming Events, these evaluation systems of two models can be applied very well. For model I, we only need to change evaluation indexes and modify the data, while the analytical method is the same. For model II, because we do not need to consider other evaluation indexes but the data of search result, we also evaluate and rank top coaches of other sports conveniently.

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Team # 24270 Page 21 of 26 6. Strengths and Limitations

6.1. ModelⅠ

? Strengths

─ High accuracy. We combine objective and subjective indexes in modelⅠ, and

meanwhile, we employ subjective and objective methods to evaluate the rank of coaches in the previous century. Therefore, this evaluation system has a high accuracy and is accordance with reality.

─ Extendibility. With so many sports and fields, this model extracts the common

features of different sports so that it can be adapted to a large range of sports and fields via comparing the same metrics.

─ Easy to understand. This model is succinct and clear, which can be easily

understood.

? Limitations

─ The lack of data. When a game is not hot, the data are difficult to find, which may

result in the lack of data.

─ Difficult to determine weights. In order to obtain an accurate result, this model

needs to determine weights carefully. However, it is difficult to choose appropriate values of weights.

6.2. Model II

? Strengths

─ Efficient. Once the model has available data, the final result can be obtained

efficiently, which means that it do not rely largely on the work of human. ─ The lack of data do not occur. Since the data of model is based on the searching

number of Google, the needed data is always sufficient. ─ Simple. Information of coaches do not need to be specific. ? Limitations

─ The influence of media. Since this model is based on the degree of media attention,

it do not take comprehensive factors in consideration.

7. Conclusions

In model I, we have defined evaluation indexes and determined the weights of their importance. By searching and selecting needed data, we obtained the final amount of evaluation of each coach in a certain sport. Finally, we compared the evaluation amounts and listed top five coaches, which is accordant with widely accepted result. And in the sensitivity analysis, the result is satisfying and of robustness. Further work of model I should choose more evaluation indexes to rank coaches in a more fairly way.

In model II, the algorithm offers us a convenient way to evaluate the excellence of certain coach, without any need for detailed information. That is, by a series of repetitive search on the Internet, one can get a grasp of rankings in any given sports. Because of the simplicity of the search job, a program can be developed to perform the rote tasks. Given that the algorithm

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Team # 24270 Page 22 of 26

does not need any details, the method can be applied to any fields that need a ranking system. However, the partial coefficient in this algorithm must be determined on a mass of trail and

error adjustments, which we cannot adequately explain here.

8. The Article for Sports Illustrated

“Dream Team” of College Coaches

Where there is a competition, there will always be victory, defeat, and ranking. Loyal sport fans could debate day and night over the question who is the best coach. However, people are born to be biased to their own preference. Without any objective analysis, the ranking is

dubious at best. Nevertheless, after a reference to various data about college coaches from Sports Reference[5], we can safely put forward our unbiased evaluation. Here are 5 greatest college coaches over the course of sports history in basketball, football and baseball, as shown below:

Sport No.1 No.2 No.3 No.4 No.5 Basketball John Wooden Dean Smith Mike Krzyzewski Adolph Rupp Bob Knight Football Joe Paterno Bobby Bowden Bear Bryant Tom Osborne Don James Baseball John Barry Mike Martin Rod Dedeaux Augie Garrido Jim Morris

There is a dream team of college coaches in every sport fan’s heart. People tend to have different criteria for “excellence”. Some people would agree that trophies are the touchstones for a great coach, while others may prefer winning percentage. In view that there are a host of criteria, we consider objective and subjective factors together with different weight degrees to obtain our final result. We have incorporate total wins, the winning percentage, champions, final fours, tenure and their media influence as our basic criterion. With this method, we select five greatest college basketball coaches including John Wooden, Dean Smith, Mike Krzyzewski, Adolph Rupp, and Bob Knight. Amazingly, the results obtained from our comprehensive analysis models are quite similar to ideal ones hold by the majority[9]. John Wooden is thought to be one of the best college basketball coaches, or even the best coach ever. During 12 tenure years at UCLA, he won 10 national championships. The win-loss record is 664-162 and winning-percentage is 0.804, the highest in NCAA history[21]. The 27 tenure years are considered to be a glorious history of UCLA basketball team, which put the name John Wooden into spotlights. Take various factors into consideration, there is no doubt that Wooden was the greatest college basketball coach. We appreciate the honors of Wooden, furthermore, we praise his personality and contributions he had made to

basketball[20]. We can tell that, a great coach can not only help his own team become better, but also promote the development of basketball.

Whenever people speak of great coaches, they often compare them with other great coaches. So we can abandon time limits to compare coaches with previous coaches, and in the same era, we can also compare any two of coaches. The relation between two great coaches can make up the network of relationships. Here we employ the data of search results from media to consider this network of relationships when two coaches were mentioned together. We also consider the effect of the tenure periods to evaluate the historical status of coaches.

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Team # 24270 Page 23 of 26

We find that through using two methods to find top five coaches, we get quite similar results.

Similarly, we can employ these methods into college football and baseball coaches，though their data to evaluate coaches are not the same as basketball. For instance, their games have ties while basketball only has wins and loses. But our methods can also be used into these two sports and get satisfying results.

Maybe you want to know whether women coaches can be evaluated by our methods, we can confirm that they also fit the methods. The result can tell everything. By our methods, the ranking of top five women basketball coaches is: Pat Summitt, Tara VanDerveer, Barbara Stevens, C. Vivian Stringer, Geno Auriemma[20]. We find that the result is quite similar with public ideal ones. So the gender factor dose not make a difference in our methods. College sports are various, can we put these methods into use in other sports such as hockey, track and field? We give out these two methods and they fit almost overall sports by collecting the data.

In brief, these two methods to evaluate college coaches have a high applicability towards different sports and genders. Whether you have enough data or not, you will find ways to construct your “Dream Team” of college coaches.

References

[1] Sport. Wikipedia.[2014-2-7].http://en.wikipedia.org/wiki/Sports.

[2] Baidu Library.The evaluation model of mathematical modeling method[EB/OL]. [2014-2- 7].

http://wenku.http://www.wodefanwen.com//link?url=jt3jrHz6x47Q0TGQsbx3Cb0BHnSTXGDWyxXPPCCZ7G wHSZC6ZbTHgTrSO_Ee4uGoItw1h8EPNKD1pEiYewEXE0u6Vi0mWIYQSn-gVt6ERdm. [3] Qiyuan Jiang, Jinxing Xie, Jun Ye.Mathematical Modeling[M].4th ed.Higher Education P ress, 2011 :249-257.

[4] Baidu Library. Grey Relational Degree[EB/OL].[2014-2-7]. http://wenku.http://www.wodefanwen.com//view/ aba402ccba0d4a7302763a83.html. [5] Sports Reference.[2014-2-7]. http://www.sports-reference.com/. [6] Sports Reference. College Basketball. Coaches.[2014-2-7]. http://www.sports-reference.com/ cbb/coaches/.

[7] Sports Reference. College Basketball. Seasons.[2014-2-7]. http://www.sports-reference.com/cbb/seasones/. [8] Men’s College Basketball.[2014-2-7].

http://insider.espn.go.com/ncb/insider/columns/story?columnist=bilas_jay&id=2423815. [9] 10 Greatest Coaches in NCAA Basketball History.[2014-2-7].

http://bleacherreport.com/articles/1341064-10-greatest-coaches-in-ncaa-basketball-history/pa ge/6.

[10] Sports Reference. College Football. Coaches. [2014-2-7]. http://www.sports-reference.com/cfb/coaches/.

[11] Sports Reference. College Football. Seasons. [2014-2-7]. http://www.sports-reference.com/cfb/years/.

[12] Top Ten College Football Coaches of All-Time. [2014-2-7].

http://www.barrystickets.com/blog/top-ten-college-football-coaches-of-all-time/.

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Team # 24270 Page 24 of 26

[13] NCAA. College Baseball. Championship History. [2014-2-7]. http://www.ncaa.com/history/baseball/d1.

[14] NCAA. College Baseball. Records. [2014-2-7]. http://fs.ncaa.org/Docs/stats/SB_Records/2011/coaches.pdf. [15]10 Best College Baseball Coaches Ever. [2014-2-7].

http://www.mademan.com/mm/10-best-college-baseball-coaches-ever.html.

[16] Jingsong Yan.Study on the Growth Law of Web Information[D].Lanzhou university, 200 6.

[17]Machang Zhao.Secret of Google-Interpretation of PageRank.[2014-2-8].

http://www. itlearner.com/ good/pagerank_cn.htm.

[18] Princeton. Wordnet Search.[2014-2-8]. http://wordnetweb.princeton.edu/ perl/webwn. [19] Eigenfactor. Eigenfactor Method Overview.[2014-2-8]. http://www.eigenfactor.org/ methods.pdf.

[20] List of college women's basketball coaches with 600 wins.[2014-2-8].

http://en.wikipedia.org/wiki/List_of_college_womens_basketball_coaches_with_600_wins. [21] John Wooden. [2014-2-8].

http://www.sports-reference.com/cbb/coaches/john-wooden-1.html.

Appendix

Table A1: Selected 20 Basketball Coaches and Their Metrics Data Name win-lose years pct. champions final fours season Google

S1 Phog Allen 719-259 48 0.735 1 3 1905 182000 S2 Jim Boeheim 942-314 38 0.75 1 4 1976 918000 S3 Jim Calhoun 877-382 40 0.697 3 4 1972 2660000 S4 John Calipari 585-171 22 0.774 1 4 1988 2470000 S5 Denny Crum 675-295 30 0.696 2 6 1971 426000 S6 Don Haskins 719-353 38 0.671 1 1 1961 1540000 S7 Hank Iba 752-333 40 0.693 2 4 1929 5020000 S8 Mike Krzyzewski 975-302 39 0.764 4 11 1975 1430000 S9 Bob Knight 899-374 42 0.706 3 5 1965 34000000 S10 Lute Olson 776-285 34 0.731 1 5 1973 298000 S11 Rick Pitino 681-239 28 0.74 2 7 1978 1880000 S12 Adolph Rupp 876-190 41 0.822 4 6 1930 375000 S13 Nolan Richardson 509-207 22 0.711 1 3 1980 1020000 S14 Dean Smith 879-254 36 0.776 2 11 1961 60100000 S15 Bill Self 524-169 21 0.756 1 2 1993 2390000 S16 Jerry Tarkanian 761-202 30 0.79 1 4 1969 1150000 S17 John Thompson 596-239 27 0.714 1 3 1972 4210000 S18 Roy Williams 715-187 26 0.793 2 7 1988 2140000 S19 John Wooden 664-162 29 0.804 10 12 1946 39700000 S20 Gary Williams 668-380 33 0.637 1 2 1978 2130000

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Team # 24270 Page 25 of 26

Table A2: The Number of Teams in Every Ten Year

Year 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 Num 136 174 171 158 152 173 203 264 295 318 345

Table A3: The Standard Difference of Winning-percentage in Every Ten Year

Year 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 s 0.069 0.063 0.068 0.050 0.055 0.034 0.075 0.040 0.075 0.043 0.068

Table A4: The final evaluation index matrix x* of Basketball

x1 x2 x3 x4 x5 x6

S1 0.592 0.284 0.182 0.000 1.000 0.055 S2 0.157 0.389 0.273 0.000 0.794 0.083 S3 0.149 0.246 0.273 0.222 0.841 0.145 S4 0.030 0.415 0.273 0.000 0.136 0.138 S5 0.103 0.331 0.455 0.111 0.552 0.064 S6 0.165 0.322 0.000 0.000 0.794 0.106 S7 0.854 0.476 0.273 0.110 0.841 0.219 S8 0.170 0.445 0.909 0.333 0.818 0.102 S9 0.187 0.368 0.364 0.222 0.885 0.748 S10 0.122 0.397 0.364 0.000 0.685 0.059 S11 0.082 0.399 0.545 0.111 0.473 0.118 S12 1.000 1.000 0.455 0.333 0.864 0.062 S13 0.026 0.310 0.182 0.000 0.136 0.087 S14 0.251 0.736 0.909 0.111 0.742 1.000 S15 0.000 0.300 0.091 0.000 0.055 0.136 S16 0.147 0.710 0.273 0.000 0.552 0.092 S17 0.076 0.405 0.182 0.000 0.428 0.195 S18 0.063 0.453 0.545 0.111 0.380 0.127 S19 0.520 0.995 1.000 1.000 0.514 0.813 S20 0.069 0.000 0.091 0.000 0.655 0.127

Table A5: Overall search results and winning search results of basketball Year Name Overall Winning 2003.5 Bill Self 2390000 256000 2001 Roy Williams 2140000 883000 1999 John Calipari 2470000 578000 1995 Jim Boeheim 918000 254000 1994.5 Mike Krzyzewski 1430000 476000 1994.5 Gary Williams 2130000 1470000 1992 Jim Calhoun 2660000 136,000 1992 Rick Pitino 1880000 671,000 1991 Nolan Richardson 1020000 31,500

'' '' '' '' '' ''

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Team # 24270 Page 26 of 26

1990 Lute Olson 298000 275,000 1986 Denny Crum 426000 31,500 1986 Bob Knight 34000000 215,000 1985.5 John Thompson 4210000 1,160,000 1984 Jerry Tarkanian 1150000 59,000 1980 Don Haskins 1540000 313,000

1979 Dean Smith 60100000 397,000 1960.5 John Wooden 39700000 1,010,000 1950.5 Adolph Rupp 375000 94,000 1949 Hank Iba 5020000 31,500 1929 Phog Allen 182000 32,200

Table A6: Parts of Original Cross-Reference Matrix of Basketball

Bill Self Roy Williams John Calipari Jim Boeheim Mike Krzyzewski

Bill Self 0 92900 97100 282000 271000 Roy Williams 92900 0 111000 86300 111000 John Calipari 97100 111000 0 81400 69700 Jim Boeheim 282000 86300 81400 0 140000

Mike Krzyzewski 271000 111000 69700 140000 0

Table A7: Parts of Final Cross-Reference Matrix of Basketball

Bill Self Roy Williams John Calipari Jim Boeheim Mike Krzyzewski

Bill Self 0 0.002142 0.00986 0.096381 0.0426 Roy Williams 0.001882 0 0.002294 0.015612 0.009703 John Calipari 0.006253 0.001656 0 0.006777 0.003033 Jim Boeheim 0.060131 0.011086 0.006667 0 7.82E-05 Mike Krzyzewski 0.063972 0.016584 0.007182 0.000188 0

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