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Major League Baseball has begun collecting a set of physics data under the name "Statcast", including such information as the exit velocity of notable home runs. I would like to know if this data is permanently archived somewhere, so that I can give stable references to it (e.g. in a paper).

All I have found so far is the "Statcast Leaderboard", which shows, e.g. the longest or fastest home runs of the season (or postseason) to date. But presumably the leaderboard data will eventually be replaced with next season's data. Even during a season, a particular home run might "fall off" the leaderboard. So if I refer to that particular home run in my paper, readers may eventually not be able to find the data to verify it, unless I can give a link to some permanent home for the data.

Is there such a "permanent home" for this data, provided either by MLB or someone else?

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The data are not generally publicly available, at least not yet. The stuff that is available can be found here:http://baseballsavant.com/pitchfx_search.php. I don't know how "permanent" it is. You can read a lot about Statcast at my own web site: http://baseball.physics.illinois.edu.

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Raw statcast data isn't available to the public, but one hack is to use the JSON files that the MLB AtBat App uses to populate its feed. These are accessible from: statsapi.mlb.com/api/v1/game/[GAME_PK]/feed/color, where GAME_PK can be found in the boxscore.xml of a standard gameday directory(i.e. gd2.mlb.com/components/game/mlb/year_2015/month_09/day_10/gid_2015_09_10_detmlb_clemlb_1/)

From this JSON file, you can use some simple regex to get exit velocities, batted ball distances and launch angles.

I've developed some simple scripts to parse and import statcast, pitchfx and gameday data into a database. You can find those here: https://github.com/brianhuey/mlbatbat_pgsql

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