Recent techniques such as Ultimate Zone Rating or the Plus-Minus system from The Fielding Bible (a must-read, by the way!) are based on the tabulation of both positive and negative fielding events. These statistics are more detailed and accurate measures of fielding ability. These techniques are also getting a bit more attention is the regular media, as evidenced by this recent Yahoo! Sports article. However, despite being obvious improvements on previous methods, both of these approaches are still based on dividing the baseball field into discrete zones and vectors, and tabulating events within each zone. Ideally, the baseball field could be treated as the continuous playing surface that it actually is, instead of a set of zones or vectors. Instead of tabulating fielding events within discrete zones, we fit continuous probability distributions to each fielder based on their past fielding events. The closest technique (at least in spirit) to our approach is the work by David Pinto.

1. Estimating starting locations for each position

Our BIS data does not provide a key piece of information for each g-bip: the location of each fielder before the ball was hit. We estimate the starting location for each fielder as the (x,y) location in the field where each position has the highest overall probability of making a successful play. For each grounder, we then convert the bip coordinates into the angle at which the grounder was hit off of the bat. An angle of 0 corresponds to the 3rd base line while an angle of 90 corresponds to the 1st base line.

2. Fitting smooth models for the average fielder at each position

We model the probability of a successful play on a grounder as a smooth function of the angle between fielder location and the BIP path. We model different functions for each velocity category, and also allow a different function for fielders moving to the left or the right. These models are calculated using the data from all infielders, and so represent the ability of an aggregate fielder at each position. In the figure below, we show the probability model at each position for successful fielding of grounders with an intermediate velocity.

We see that each position has a distinct probability model. Note that pitchers seem to have a much larger range than the other infield positions only because they are much closer to home plate and therefore do not have to travel as much distance to cover the same range of angles from home plate.

3. Fitting player-specific models and calculating differences

We calculate the same probability models using only the data for each individual fielder and allowing different parameters for each player. Since we have different models for each individual player, we can quantify the difference between players by comparing their individual probabilities of making an out relative to the aggregate probability of making an out. As an example, the figure below illustrates the comparison on grounders between the aggregate model for the SS position and the individual models for the best and worst shortstops.

4. Weighted sum of player-specific differences

For each possible angle, we can calculate the difference D between particular fielder's probability of success and the aggregate probability of success. A rough measure of fielder ability is the sum over all possible distances of the difference (individual player - aggregate) in probability of not making a successful play. This sum is carried out by simple numerical integration. However, since not all distances occur with equal frequency, our SAFE measures are actually calculated as a frequency-weighted sum, so that more frequent distances or angles are more important. In addition, our sum is also weighted by the average run consequence of each angle, which allows us to take into account the different consequences of grounders to different areas eg. a missed grounder down the first base line leads to more bases than a missed grounder to the shortstop. Thus, for an individual player, their SAFE statistic can be interpreted as their expected runs cost/saved relative to the average fielder. A good fielder will have a large positive SAFE, which means a high number of runs saved, whereas a bad fielder will have a large negative SAFE, which means a high number of runs cost.

1. Estimating starting locations for each position

Our BIS data does not provide a key piece of information for each a-bip: the location of each fielder before the ball was hit. We estimate the starting location for each fielder as the (x,y) location in the field where each position has the highest overall probability of making a successful play.

2. Fitting smooth models for the average fielder at each position

We model the probability of a successful play on a a-bip as a smooth function of the distance between the fielder starting location and the a-bip coordinates. We model different functions for each velocity category, and also allow a different function for fielders moving to the left or the right. These models are calculated using the data from all infielders, and so represent the ability of an aggregate fielder at each position.

3. Fitting player-specific models and calculating differences

We calculate the same probability models using only the data for each individual fielder and allowing different parameters for each player. Since we have different models for each individual player, we can quantify the difference between players by comparing their individual probabilities of making an out relative to the aggregate probability of making an out. As an example, the figure below illustrates the comparison on fly balls between the aggregate model for the CF position and the individual model for Darin Erstad in 2002

4. Weighted sum of player-specific differences

For each possible angle, we can calculate the difference D between particular fielder's probability of success and the aggregate probability of success. A rough measure of fielder ability is the sum over all possible (x,y) coordinates of the difference (individual player - aggregate) in probability of not making a successful play. This sum is carried out by simple numerical integration. However, since not all a-bip coordinates occur with equal frequency, our SAFE measures are actually calculated as a frequency-weighted sum, so that more frequent distances or angles are more important. In addition, our sum is also weighted by the average run consequence of each angle, which allows us to take into account the different consequences of a-bips to different areas eg. a missed a-bip into the outfield power alley has a higher consequence compared to a missed pop-up in shallow outfield. The figure below shows the major differences in consequences of a-bips to different areas of the field

Thus, for an individual player, their SAFE statistic can be interpreted as their expected runs cost/saved relative to the average fielder. A good fielder will have a large positive SAFE, which means a high number of runs saved, whereas a bad fielder will have a large negative SAFE, which means a high number of runs cost.

UPDATE TO VERSION 03 (2008-10-01): The results have been updated to include 95% posterior intervals in addition to the posterior mean for each player-year. The old version 02 results can still be found here

UPDATE TO VERSION 02 (2007-12-06): The results have been updated using better models for individual fielder curves, BIP frequency, and shared consequence. The old version 01 results can still be found here