Using multi-class classification methods to predict baseball pitch types

1.1Literature review

Previous research has mostly focused on a binary prediction, commonly a fastball vs. non-fastball split. This prediction has resulted in accuracy around 70% (Ganeshapillai and Guttag, 2012), with varying degrees of success for individual pitchers and different methods. This binary classification using support vector machines was expanded upon using dynamic feature selection by Hoang (2015), improving the results by approximately 8 percent. Because many classification methods were originally designed for binary classification, this prediction method makes sense, but we wish to expand it further into predicting multiple pitch types.

Very limited attempts have been made at multi-class prediction prior to our work. Bock (2015) used a support vector machine approach with a linear kernel function, focusing mostly on finding a measure of predictability for a pitcher and comparing that predictability against long-term pitcher performance. The authors only examined four pitch types, and had a very limited out-of-sample testing set, looking at the methods accuracy on 14 pitchers during a single World Series, with an average accuracy of just under 61%. Woodward (2014) used decision trees to again only predict up to four different types of pitches, and only shows results for two individual pitchers. We expand on their work by considering up to seven different pitch types and apply machine learning to many more pitchers over a longer amount of time. Along with a much larger test set, we also investigate which methods perform the best, if our results share any correlation with standard pitching statistics, what variables are most important for prediction, and finally our ability to predict the next pitch type in a live game situation.

Fig.1

League-wide Batting Average and ERA over the past decade.

2.1Data

The introduction of PITCHf/x was a revolutionary development in baseball data collection. Cameras, installed in all 30 MLB stadiums, record the speed and location of each pitch at 60 Hz, and data is made available to the public through a variety of online resources. MLB Advanced Media uses a neural-network based algorithm to classify those pitches, giving a confidence in the classification along with the type of pitch (Fast, 2010). This information is added to the PITCHf/x database, along with the measure characteristics of the pitch and details about the game situation. Using the pitch data provided at gd2.mlb.com, we were able to retrieve every pitch from the 2013, 2014, and 2015 seasons. Using 22 features from each pitch, we created data sets for every individual pitcher, adding up to 81 additional features to each data set, depending on how many types of pitches the individual threw. Here we consider seven pitch categories (with given integer values), fastball (FF, 1), cutter (CT, 2), sinker (SI, 3), slider (SL, 4), curveball (CU, 5), changeup (CH, 6), and knuckleball (KN, 7), and those that had a type confidence (the MLBAM algorithm’s confidence that its classification is correct) greater than 80%.

We restricted our data set only to pitchers who threw at least 500 pitches in both the 2014 and 2015 seasons, which left us with 287 total unique pitchers, 150 starters and 137 relievers as designated by ESPN. The average size of the data set for each pitcher was 4,682 pitches, with the largest 10,343 pitches and the smallest 1,108 pitches. Because each pitcher threw a different number of unique pitch types, not all the datasets are the same size. At the most, a pitcher could have 103 features associated with each pitch and at the minimum he could have 63. The average pitcher had 81 features.

Table 1 gives a list of the features used, both those that can be taken from the immediate game situation and the features we generated using the historical data on for both pitcher and batter. Because of the size of the feature set, similar features are grouped together in the table, i.e. group 16 contains the previous pitch’s type, result, break angle, break length, break height, and the zone where it crossed the plate. Groups 19–26 have variable sizes due to the number of types of pitches each pitcher throws. Groups 27–29 are unique for each batter, containing the percent of each type of pitch he puts in play, has a strike on, or takes a ball on.

2.2Model development

We decided to employ three different classification-based methods to compare and contrast results from all 287 pitchers in our data set. First, we used multi-class Linear Discriminant Analysis because its speed and efficiency. Then, to compare results to Bock (2015) and Woodward (2014) we used support vector machines and classification trees. In an effort to reduce model variance between different models and increase accuracy, we employed a committee method, using ten of each model type and taking the majority vote as output.

2.3Linear Discriminant Analysis

Linear Discriminant Analysis (LDA) is a method descended from R.A. Fisher’s linear discriminant first introduced by Fisher (1936). Assuming two classes of observations have respective mean covariance pairs (μ→0,Σ0) and (μ→1,Σ1) , then the linear combinations w→·x→ have the mean covariance pairs (w→·μ→i,w→TΣiw→) for i = 0, 1. Fisher determined the separation between the two distributions (and therefore classes) as the ratio of the variance between the two classes to the variance within each class, i.e.

where the maximum separation between the classes is found when w→∝(Σ0+Σ1)-1(μ→1-μ→0) .

To go from the linear discriminant to LDA, we use the assumption that the class covariances are the same, i.e. Σ₀ = Σ₁ = Σ, then the proportional equation leads to w→·x→>c , where

and so the decision of which class x→ belongs to depends on whether the linear combination satisfies the inequality. In order to extend LDA to multi-class classification, the same assumption is made that each of N classes has a unique mean μ→i but the same covariance Σ. The between class covariance is found by

where μ→ is the mean of the means, and then class separation is determined by

We use a regularization parameter to shrink the covariance matrix closer to the average eigenvalue of Σ. In the MATLAB implementation of LDA, N (N - 1)/2 separations are made, making it comparable to the one-vs-one (OvO) technique for support vector machines. The final class decision is made by minimizing the sum of the misclassification error (MathWorks, 2016a).

2.4Multi-class Support Vector Machines

Support vector machines (SVM) were first designed as a method of binary classification. The extension of binary SVMs to a multi-class method has led to two common approaches to multi-class classification. Unlike Unlike Bock (2015), however, we employed a C-SVM formulation with the radial basis kernel function. For a problem with N distinct classes, the one-vs-all (OVA) method creates N SVMs, and places the unknown value in whatever class has the largest decision function value.

For our model, we used the one-vs-one (OVO) method, which creates N (N - 1)/2 SVMs for classification. For a set of p training data, (x₁, y₁) , …, (x_p, y_p) where x_l ∈ Rⁿ, l = 1, …, p and y_l ∈ {1, …, N} is the class label for x_l, then the SVM trained on the i^th and j^th classes solves

where φ (x) = e^{γ||x_i-x_j||} is the radial basis function used to map the training data x_i,j to a higher dimensional space and C is the penalty parameter. After all comparisons have been done, the unknown value is classified according to whichever class has the most votes from the assembled SVMs.

Figure 2 shows a basic representation of the differences between the OVA and OVO methods. The middle triangle formed by the OVA method leaves a gap where the classification algorithm can fail to place an unknown value, but since the OVO method does not have any blind spots, we used it for our classification. We used a modified grid search from Finkel (2003) to optimize for both parameters C and γ, using the five-fold cross-validation accuracy of the training set to find the optimal parameters.

2.5Classification trees

In order to compare our results to Woodward (2014), we also implemented classification trees, using random forests. We used the MATLABCART (Classification And Regression Trees) implementation that creates aggregate random forests of trees, known as TreeBagger.

Fig.2

A visual representation of the one-vs-all method (thin lines) compared to the one-vs-one method (thick line) from (Aisen, 2006).

Classification Trees are a specific type of binary decision tree that give a categorical classification as an output. The input set is used in the nodes of the tree, determining which feature to split at each node and what criterion to base that decision on. To determine node impurity, MATLAB classification trees use the Gini diversity index (gdi), given by I=1-∑i=1Np2(i) , where N is the number of classes and p (i) is the fraction of each class i that reaches the node. The gdi is a measure of the expected error rate at the node if the class is randomly selected according to the distribution of the classes at that node.

For each node, the tree first computes the impurity at the node, then sorts the observation to determine what features can be used as splitting criteria or cut points. For all splitting points, the tree seeks to maximize the decrease in impurity, ΔI, by splitting the observations at the node into two child nodes, then measuring ΔI for each node with different cut points. Once a feature is chosen as the best splitting candidate, then the feature is split using the best cut point, and the process is repeated until the total impurity is minimized and the end leaf nodes are found. To combat overfitting, however, classification trees are pruned by merging leaves that have the most common class per leaf (MathWorks, 2016b).

3.1Overall prediction accuracy

To establish a value for comparison, we found the best “naive” guess accuracy, similar to that used by Ganeshapillai and Guttag (2012). We define this naive best guess to be the percent of time each pitcher throws his preferred pitch from the training set in the testing set. Consider some pitcher who throws pitch types 1, 2, 4, and 5 with distribution P = {p₁, p₂, p₄, p₅} where ∑p_i = 1, and his preferred training set pitch is max(P_train) = p₂, then the naive guess for the pitcher is P_test (p₂). For example, since Jake Arrieta threw his sinker the most in the training set (26.31%), we would take the naive guess as the percentage of the time he threw a sinker in the testing set, which gives a naive guess of 34.03%. With the random forest method, we predicted 48.33% of his pitches correctly, so we beat the naive guess by 14.30% in his case. For all 287 pitchers, the naive guess was 54.38%.

Table 2 shows a breakdown of each type of pitch predicted for Odrisamer Despaigne by the random forest method, as well as the accuracy for each specific pitch, showing that for each pitch type, the accuracy of the model prediction improves the naive guess. We take this style of comparison from Woodward (2014), who gives an outline of a decision tree based prediction model, but does not go into detail or use more than a handful of examples, so we cannot fully compare to his results.

Table 3 shows the prediction results from each individual method. On average accuracy alone, Classification Trees had the best prediction accuracy of 66.62%. The number of pitchers we predicted better than naive is given, as well as the percentage of the 287 total pitchers that number represents. The average prediction accuracy is shown in Table 4, given along with the overall average improvement over the naive guess, denoted PI¯ , the average improvement for those pitchers who did beat the naive guess, denoted as PB¯ , and the average amount the pitchers who did not beat the naive guess failed by, denoted by PW¯ . Given the number of pitchers N with respective prediction value P_i and naive guess G_i, the number who did better than the naive guess, N_B, the number who did worse than the naive guess N_W, we find

We also give the average range of accuracy between the most and least accurate members of each committee as well as the average time for each pitcher’s model to be trained and tested. As shown in Table 3, the random forests of classification trees outperformed both LDA and SVM by a wide margin. Basing the judgement solely on how many pitchers were predicted better, the random forests were near-perfect, leading the average prediction accuracy and improvement to also be higher. LDA outperforms the random forests only when we examine the average improvement for those pitchers who we are able to beat the naive guess for, but conversely also has much worse performance for the pitchers we do not beat the naive guess for. At this stage, we undertook further comparative analysis to determine if the random forests were the best method overall.

3.2Prediction by count

A common analysis in any pitch prediction is breaking down prediction success rate by each pitch count. There are twelve possible counts for any at-bat, three where the batter and pitcher are even, three where the pitcher is ahead in the count (more strikes than balls), and six where the batter is ahead (more balls than strikes). Similar to other works, On the very batter-favored count of 3-0, we were able to predict 104 pitchers (36.24% of 287 pitchers) totally correct, i.e. for every pitch they threw on a 3-0 count, we predicted them all exactly. The total counts and average success rates for the random forest classification tree method are given in Table 4. Pitcher ahead counts are bolded, batter-favored counts are italicized.

The high success rate on counts in which the batter is ahead is not surprising, given that a pitcher is more likely to throw a controllable pitch in order to even the count or to avoid a walk. Batter-behind counts give the pitcher much more freedom, which explains the lower average predictability.

3.3Comparison with standard statistics

In an effort to determine if the prediction success correlated with any standard measure of pitcher success, we ran a linear regression analysis to find the correlations between the random forest model prediction accuracy and the improvement over the naive guess to the pitchers’ wins-above-replacement (WAR) and fielding-independent-pitching (FIP) statistics. FIP is an extension of a pitchers’ earned run average (ERA) that examines only outcomes over which the pitcher had control. To find a measure of the diversity a pitcher’s pitch selection, we also compared the prediction accuracy, improvement, WAR, and FIP to the Herfindahl-Hirschman Index (HHI). HHI is inversely proportional to the diversity of pitch selection, and can be expressed as

Overall, given the very low R² values for most pairs of variables we examined, the only correlation that we can draw any conclusions from is between the HHI and overall model accuracy, shown in Fig. 3. The higher the HHI (and therefore the less diversity in pitch selection), the more accurate the model is, most likely due to the fact that the pitcher is throwing a single pitch type a lot. While the regression slopes of the other pairings might suggest interesting correlations, due to the excessively small R² values there is little to no mathematical strength to any conclusions we could draw.

Fig.3

Linear regression fit line between the Herfindahl-Hirschman Index and the random forest model prediction accuracy, with intercept 0.332 and correlation 0.746.