### Abstract

I always had the feeling that with a good algorithm, it's possible to have a good confidence about knowing who to bet on in sport. Sure, it won't be an unrisked strategy, or if I would have find such an algorithm, you wouldn't be reading about it by now ! But if you find such, feel free to contact me ;)
Even if I do have in mind the fact that "if they do that, it's because they win at the end", I may found an algorithm that will lower my risk by giving me good prediction given the history and a confidence interval, preventing me from remembering all the history of each player, and maybe it will make me little money ! For several reason I think it may be possible.
The first one is that I think that they and I don't have the same output to predict. If I were to design an algorithm for a bet institution, I wouldn't predict who will win the match, but what will be the distribution of the bets for each team. Let's give an example with the last football world cup. I leave in France. In quarter-final there were France vs Germany. Around me everybody bet France, and I amuse that in Germany everybody bet Germany. So if as a bet institution I only predict who will win to compute my odds, and predicted Germany as a winner in France, if France win I will lose money because most people would bet France. On the other hand, if I know the distribution of the bets, I can choose my odds in order to be winning regardless the winner of the match.
The second reason is that I can use as an input for my model the odds computed by the bet institution. And their scores reflects a non linear and non trivial informations about the game.
The third reason is that I can construct a strategy of bets knowing the global precision of my model. Let's take a simplified example and suppose I have a good algorithm : if I'm wrong I give them y and if I'm right they give me (1+x)*y. Moreover, let's suppose that I tested my algorithm, and I know I'm right 4 times over 5. So I have to bet on matchs such that the money I will get the 4 rights times is greater than the money I will lose the fifth time :
4*(1+x)*y-4*y > y. So if x>0.25, I'm winning ! Of course, the precision of my model will be lower in tight match, and when my precision will be good x will be closer to 0.01 I think. So I think a strategy exists, if and only if I have a really really good model for tight matchs.

#### But why tennis ?

First, I like tennis, so it's easier for me to think about good variables that would impact the result. Second, I have to limit the biais, and I think the biais is lower in tennis than in other sports like football for instance. To me it's a question of convergence and personal motivation. The biais is smaller in indivual sport because during a match the pressure is always on you, and to keep winning you must be always at your top. Regarding the convergence, scoring in tennis is much more frequent than in football where just one mistake could cost you a goal and the match whereas in tennis, you could always come back.

I get all the data from here. I take the match from 2006 to 2014. They have data since 2000 but for 2000 they don't have the bet and for 2005 they don't have the number of points of a match, so I started in 2006, which gives me 23 942 matchs. Then I suppressed the line where one of the player's rank was greater than 100, because too little observations are available for such player and a top 10 against a top 100 needs no modelisation to predict with a good confidence rate who will win... I ended up with 15 338 matchs.

## Model construction

In order to construct an effective model, I first tried to list what are the key point that will lead to a player to win the match. For each key point I will then compute variables and features. My distinction between variable and feature is that a variable is the result of a frequency count (for instance the winning rate) and a feature is a combination/transformation of variables. Each key point impact will be validated graphically and mathematically. Given those key features, I will finally be able to develop the model and make the prediction.
The key point I think about are :
• The relative strength of the players
• The relative dynamic of the players (winning trend)
• Who has the psychological ascendant ?
• The tiredness
• The context advantage (who is the best on this specific surface, indoor/outdoor, tournament, ...)

A visualisation of those key point for each match in my dataset will be available in the viz part here.

### Strength In order to measure the relative strength of the players, one needs their ATP points, their global and per surface winning rates and their trophies (Grand Slam + Master Cup, Master 1000, ATP 500, ATP 250). For winning rate variables I computed a ratio between player 1 and player 2, so a variable of 1 correspond to a same strength. For the trophies, I compute the difference of trophies divided by the max number so this gives me a percentage of more (or less) trophies player 1 has compared to player 2. For the ATP points, I computed the difference divided by the minimum of the two ATP points, in order to have a higher score when the worst ranked player has little ATP points. Thus, a difference of 200 points between high ranked players will give lower difference than between two low ranked players who have difficulties to win 200 points. I prefer the ATP points to the rank because the rank gives the same difference between player 3 and 4 then between player 43 and 44 (which is much more thin).
This gives me for each match two strengths vectors (player 1 relative to player 2 and player 2 relative to player 1) s of dimension 9. From this dataset, I performed dimensionality reduction by applying an EM algorithm with K clusters in order to take into account non-linearity effects. In order to choose the number K of clusters, I performed an EM algorithm for K from 3 to 15 and plotted the winning rate by cluster for each K. My will was to have a least one cluster with a high winning rate, one with a low and one around 50%, no matter if several cluster would have a similar winning rate, maybe they will help to differentiate once crossed with the other keys. See here for the plot for each key point. I end up with for each match two vectors s' of dimension 11 where the k-th value correspond the probability of this relative strength to belong to cluster k. To choose the value of K, I performed several K-means for K from 3 to 20 and then use the elbow method to determine the region of optimal K (around 11-12) and choose the best among this region graphically.

### Dynamic The dynamic of a player is the conjunction of two things : his current winning rate compared to his global winning rate and his number of match played rate by day during the last period. For the first one I computed for each player the ratio between his winning rate of the last two weeks, one month, three months and his global past winning rate. For the second one I computed the number of match and sets played during the same three periods. This gives me 9 variables for each player, and their relative dynamic comes from their ratio. I then performed the same dimensionality reduction through EM algorithm with 3 to 10 clusters following the same process as for strength (results). Here the elbow method gives a K region around 7-8. I chose K=9.

### Psychological Ascendant The psychological ascendant depends on (excluding strength and dynamic) the direct confrontations, the number of confrontations, the reputation of the player on the surface (winning rate on this surface), the experience (number of matchs played before and number of time the player reach such a round of a similar competition), and the winning rate at this round of a competition. Computing experiences ratio we end up with a 5 dimensions vector, turned into a 8 dimensions vectors as usual. Here the elbow method gives a K region around 8-9.

### Tiredness The tiredness is somehow related to the dynamic. In order to have a tiredness variable, instead of computing the number of match/sets played by day in the past, I computed the ratio between this rate in the last period over this rate over the last year. Thus, if a player has played an important number of match in the last period but also during the whole year, we can assume that he is used to it and has a good physical condition thus he is not tired versus some player who has played much more match by day than usual. We then have a 6 dimensions variables ... turned into a 7 as usual. The elbow method gives a K region around 7.

### Context I lack a lot of context information... The most significant context I lack of his the weather. But I also lack of a gold information that sportsman not communicate about, that is their physical health. Sometimes we can know about it just looking at him and seeing many straps around the knees, or the help of a kine during the previous match, or knowing that Nadal has an appendicitis ... but those are information I haven't in my database. But what I do have is the odds of the match which I hope has been computed taking those information into account. This winning odd ratio will be my first context variable. The other variables will be a description of the match : Tournament (ATP250/ATP500/Master1000/GS/Masters/Others), Surface. Those description gives me binary vector of dimension 10, so a 11 dimension vector for the context variable and I didn't make any further transformation for this one. The first reason why is that I can't get a better clustering of the surface variable than the binary vector with only one 1. So a clustering won't give me any more informations besides the fact that a given tournament is played on a specific surface. The second reason is that for each tournament and each surface taken has a standalone, exactly half player have won and half player have lost, so there is nothing to gain. So thus binary variables will help later when combined with the others key points. I will nonetheless provide the winning rate histogram given the relative bet variable to check it's effectiveness.

#### EM algorithm clustering

For the first four graphs, the abscissa correspond to the number of cluster and the ordered to the winning rate of a cluster. Hover the balloon to see the winning rate and the number of player by cluster. The size of the balloon is proportional to the number of players in it. The bet graph give the winning rate in function of the bet ratio. Thus when the bet ratio is close to 1, to the player have the same level and the winning rate is close to 0.5. When the winning rate is close to 0 (low score / high score), the winning rate is high for the one who has a high score.

### Features dimensionality reduction

In order to take non linear information in the features I chose to perform non linear dimensionality reduction on two possible dataset.
The first one is obtain by pasting every key points in columns so this gives me for each match a vector of length 46 = 11(strength)+9(dynamic)+8 (psychological ascendant)+7(tiredness)+11+(context)
The second one is obtain by computing the outer product between all my key points so this gives me for each match a vector of length 95 040 = (11+1)(strength)*(9+1)(dynamic)*(8+1) (psychological ascendant)*(7+1)(tiredness)*(11+1)*(context). The +1 coming from the fact that I had a constant vector of 1 for each key point to keep in my variable each variable as a standalone. This dataset gave a lot a trouble to manage that many columns !
Then for each dataset, in order to reduce the dimension, I implemented the Laplacian Eigenmap method, which had been presented by Mikhail Belkin and Partha Niyogi. Finally I tried a projection into a subspace of several dimension from 3 to 19, and performed for each an SVM classifier with different kernel (linear, polynomial and gaussian).

## Results

For each dataset and dimension chose for the dimensionality reduction I performed 5 cross validation learning on a random subset of 80% of the data and testing on the 20% left. Here is the graph of the precision (not the precision-recall, see the remark why) for each dimension. (The precision is the ratio between the true positive and the positive prediction, and the recall is the ratio between the true positive and the total of real positive value). I also did the same for the first dataset without dimensionality reduction to see the impact of the reduction. I can't do it for the second one because the number of variable is far to big for an SVM on my computer !
Remark : There can be only one winner ! So there is no decision value to consider. I predict for each player of a match his winning probability. Then I can't say "if this probability is greater than x then he won" because I could get two winners or two losers. So if his winning probability is greater than is opponent's then he won. Thus, if I'm right for one of the two players, I'm automatically right for both, and I predict exactly one positive value and one negative value for each match. Consequently the number of positive value is equal to the number of positive prediction and the precision equal the recall equal the F-score.
Remark 2 : The precision is always greater or equal than 0.5. Indeed, if the precision is lower than 0.5, I choose the opposite strategy which gives me a precision greater than 0.5. When this happen, the reversed parameter of the frame while be set to true.

Dataset - - - - -
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Prediction
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Reality
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TP = ?FN = ?
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FP = ?TN = ?
The results being quite good, this confirm the interest of such a methodology. It seems that computing the outer product of the variables has globally a negative impact on the prediction for this model. I'm very surprised by the result with the first dataset in dimension 3. I ran the model several times to check it and it gives that good a prediction each time. I'm then going to re-run the model on more data and make a real prediction for the future with this parametrization in an other article and see what happen...