Category Archives: Machine Learning

Optimizing a Daily Fantasy Sports NBA lineup — Knapsack, NumPy, and Giannis

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Pic when I was sitting courtside on Oct 24th, 2018. If you zoom in a little, you can see Giannis about to make a screen, while Embiid watches from the other side of the court to help out if the screen is successful. Both players were in the optimal lineup that night.

Opener

In the data world, when looking for projects or an interesting problem, sports almost always gives you the opportunity. For a lot of what I write, I talk about getting the data because the sports leagues rarely if ever give the data away. That gets a little repetitive, so I wanted to change it up to something interesting that’s done after you get the data, like how to optimize a lineup for NBA Daily Fantasy Sports (DFS).

Before continuing, I’ll say that this isn’t about me making money from betting. In the past season I made lineups for some of the nights, but realized quickly that in order to win, you really need to know a ton about the sport. I love watching the NBA in the winter, love watching the Bucks, but don’t follow all other teams close to enough compared to others. Still, I found it worth it to keep getting the data during the regular season and found it most interesting to find out who would have been in the best lineup that night, and then look back at the highlights to see why a certain player did well.

Because of that, I took the code I used, refactored it some, and wrote this up to show what I did to get to the point where I can calculate the best lineup.

Knapsacking

This optimization is generally categorized as a Knapsack problem. The wiki page for the Knapsack Problem defines it as follows:

“””Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.”””

Or, self described – If you’re stealing items of known value, but only are able to carry a certain amount of weight, how do you figure out which items to take?

This DFS problem though is slightly different than the standard Knapsack problem, and makes it much more interesting.

FanDuel Rules

The DFS site I’m using for this is FanDuel, one of the two main Daily Fantasy Sports sites. Their rules for the NBA are that all players are assigned a position, Point Guard (PG), Shooting Guard (SG), Small Forward (SF), Power Forward (PF), and Center (C). A line up will have 2 PGs, 2 SGs, 2 SFs, 2 PFs, and 1 C. Each player is given a salary, and the combined salary of the players in your lineup must not be above $60,000. For reference, to give a sense of salary distribution, and what you’ll see in the final solution for best lineup of October 24th, 2018, MVP Giannis Antetokounmpo had a salary of $11,700, and ear blower and member of the Lakers Meme-Team, Lance Stephenson has a salary of $3,900. This data was given in csv files from FanDuel that we can download.

The amount of points a player gets depends on a bunch of stats for the night, positive points for things like actual points, rebounds, 3 point attempts made, assists, steals, and negative points for things like turnovers. This data comes from nba.com which I scraped and loaded into postgres.

Data

Below is a screenshot of what an example salary csv file that we can download looks like. Note that this is for a different date than the example day I’m using. I didn’t get the csv from FanDuel on that date, I had to scrape it from somewhere else, but it’s still important to give a look of what the csv file looks like. For our simple optimization, we only need the name, the position, and the salary of all the players.

Secondly, we need the stat lines which I then use to calculate the number of points a player got in a night. Below is a screenshot from stats.nba.com where it show’s how a player did that night. I have a script that scrapes that data the next day and puts that into the db.

If you look at the data csv files in the repo, all I have here is the name, position, salary, and points earned. This is a post about optimization, not about data gathering. If you’re wondering a little, here’s the query I used to get the data. I have players, positions, stat_lines, games, and some other tables. A lot of work goes into getting all this data synced up.

select p.id as pid, p.fd_name as name, sl.fd_positions as pos, sl.fd_salary as sal, sl.fd_points as pts from stat_lines sl join games g on sl.game_id=g.id join players p on p.id=sl.player_id where g.date='2018-10-24' and sl.fd_salary is not null order by sal desc

Code

Here’s the link to all the code on github. In it, you’ll find the csv files of data, three separate scripts to run the three different optimization methods, and three files for the Jupyter notebooks to look at a simplified example of the code.

Continuing

In the rest of the post, I’ll go through the three slightly different solutions for the problem. The first uses basic python elements and is pretty slow. The second brisk solution uses libraries like Pandas and NumPy to speed up the calculation quite a bit. The final fast solution goes beyond the second, ignoring most python structures, and uses matrices to improve the quickness an impressive amount.

In all cases, I made simple Jupyter files that go through how they each combine positions which hopefully give a little interactive example of the differences to try to show it more than words can do. In each case, when you go through them, you’ll see at the bottom they all return the same answer of what are the best players, what their combined salary is, and what their point totals are.

I talk about salaries, points, and indexes a lot. Salaries are the combined salaries of the players in a group, points are the combined points of the players in a group, and indexes are the the indexes from the csv file or the pandas dataframe which represent which players are in a group. Instead of indexes, we could use their names instead. Don’t get this confused when I talk about the indexes in the numpy arrays / matrixes that are needed to find which groupings are the best. To keep the index talk apart, I’ll refer to the indexes of the players as the player indexes. Also, I sometimes mix salary and cost, so if you see either of those words, they refer to the same thing.

If you have any questions, want clarification, or find mistakes, get in contact. Also I have twitter if you feel like looking at how little I tweet.

Basic solution

Time to talk about the solutions. There are effectively two parts to the problem at the start of basic. The first is combining the positions themselves together. From the FD rules, we need two PGs together. The goal of this is to return, for each salary as an input, the combination of players of the same position who have a combined salary less than the inputted salary with the most combined points.

Said a different way, for each salary of the test, we want to double loop through the same initial position array, and find the most successful combination where the combined salary is less than the salary we’re testing against.

The second part deals with combining each of those returned values together. Say we have the information about the best two PGs and the best two SGs. Again, for each salary as input, it returns the best combination of players below that salary. This is pretty much identical to what I said about the first part, with the only difference being that we didn’t start with two groups of the same players. Loop through the possible values of the salary possibilities, double loop through the arrays of positions, find the players who have the max points where the sum of their salaries is less than the salary value we’re testing.

There’s a lot of code in the solution, so I’ll post only a little, which was taken from the Jupyter file I created to further demonstrate. Click that, go through the lines of example code, and look at the double loops and see how the combinations are created. If you’re reading this, it’s worth it. To get a full look look here’s the link directly to the file on github.

#pgs, and sgs have the format of [(salary, points, [inds...])...]
#where salary is the combined cost of the players with inds in inds, points is the sum of points.

test_salary = 45000 #example test salary.
max_found_points = 0
for g1 in pgs:
    for g2 in sgs:
        if g1[0] + g2[0] > test_salary:
            break #assuming in sorted salary order, which they are
        points = g1[1] + g2[1]
        if points > max_found_points:
            max_found_points = points
            top_players = g1[2] + g2[2] #combining two lists
            top_points = points
            top_sal = g1[0] + g2[0]
return (top_sal, top_points, top_players)
#after the loop we have a new tuple of the same format (salary, points, [inds])
#where this is the best combo of players in pgs and sgs who don't have a total salary
#sum greater than the test salary

Here’s a slow gif of it running where you can see the time it takes to do the combinations. In the end, it prints out the names and info for the winners in the lineup. I also use cProfile and pstats to time the script, and also show where it’s being slow. This run took a tiny bit under 50 seconds to run (as you’ll see from the timing logs) so don’t think you’ll have to sit there and wait for minutes.

Brisk solution

After completing the first, most basic solution, it was time to move forward and write the solution which removes some of those loops by using numpy arrays.

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Using Clustering Algorithms to Analyze Golf Shots from the U.S. Open

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Cluster analysis can be considered one of the pillars of machine learning, and yet it’s one that’s difficult to talk about.

First off, it’s difficult to find specific use cases for clustering, other than pretty pictures. When looking through the wiki page on clustering, we’re told one of the uses is market research, where analysts use surveys to group together customers for market segmentation. That sounds great in theory, but the results don’t end with specific numbers telling the researchers what to do. Second, in so many cases, the hardest part of data science projects, or tutorials, is finding real world data that have the different results you want to show. In this case, I’m incredibly lucky.

I have a golf background, and on U.S. Open’s website, they have these interactive graphs that show where each ball was located after each stroke for every player. If you click around, you can see who hit what shot, how far the shot went and how far remains between the ball and the hole. For cluster analysis, we’re going to use the location. For you to check out how I got the data, look and read here.

Shinnecock Hills, the host of the 2018 U.S. Open last week, has a few parts of the course where balls roll to collection areas into groups, or, ya know, clusters. Here are the specific shots our clustered data is coming from.

Hole 10, Round 1, Off the Tee

The description that the USGA gives hole number 10 is

The player faces a decision from the tee: hit a shot of about 220 yards to a plateau, leaving a relatively level lie, or drive it over the hill. Distance control is critical on the approach shot, whether from 180 yards or so to a green on a similar plateau, or with a shorter club at the bottom of the hill or, more dauntingly, part of the way down the hill. The approach is typically downwind, to a green with a closely mown area behind it.

First I’ll say, always hit driver off the tee. Look at the cluster! If you get it down the hill you’ll be in the fairway! In the vast, vast majority of the time, it’s better to be closer to the hole. Golf tips aside, when I first saw this graph, it popped out as a great example to use as a clustering example.

Shift command 4 if you want selective screenshots

When looking at this picture, the dots represent where the players hit their tee shots on hole 10 in the first round, and the colors show how many strokes it took them to finish the hole in relation to par. For this, we’re ignoring the final score and only looking at the shots themselves.

Hole 10, Round 1, Approaching the green

One data set isn’t good enough to demonstrate the differences of the algorithms, and I wanted to find an example of a green with collection areas that would make approach shots group together. Little did I know, the 10th green, the same hole as the one above showing the drives, is the best example out there. If you’re short, it rolls back to you. If you’re long, it rolls away. You gotta be sure to hit the green. You can see that here.

So this will be a second example of data for all the algorithms.

Algorithms themselves

This time, in this blog post, I’m only looking for results, not going through the algorithms themselves. There are other tutorials online talking about them, but for now at least, we’re only getting little introductions to the algorithms and thoughts.

Instead, I use the Scikit-Learn implementations of the algorithms. Scikit-Learn offers plenty of clustering algorithms, which I could spend hours using and writing about, but for this post, the ones I chose are K Means, DBSCAN, Mean Shift, Agglomerative Clustering.

Other Notes

Before going in to the algorithms, here are a few notes on what to expect.

  • Elevation is key as to why there are clusters. If you look around the other holes, you won’t see close to as much distribution and clusters of shot results. Now, if we had elevation as a data point as well, then we could really do some great cluster analyses.
  • The X and Y values on the sides of the graphs represent yards from the hole, which is located at the (0,0) location. If you look at the first post, I show that if you measure the hypotenuse using those X and Y numbers, you’ll have the yardage to the pin.
  • This isn’t a vast data set. We have 156 points in the two data sets because that’s how many players there were in the tournament.
  • If you’re wondering which part took the longest, it was writing the matplotlib code to automatically create figures with multiple plots for different input variables, and have them all show up at once. Presentation is key, and that took tons of time.

Code

I put all the code and data on Github here, so if you want to see what’s going on behind the scenes and what it took to do the analysis, look there.

Questions, comments, concerns, notes, thoughts, etc: contact, twitter, and golf twitter if you’re interested in that too. Ok, algorithm time.

K Means

I’m starting with K Means because this was the clustering algorithm I was first introduced to, and one that I had to write myself during a machine learning class in college.

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U.S. Open Data — Gathering and Understanding the Data from 2018 Shinnecock

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After losing in a playoff to make it out of the local qualifying for the 2018 US Open at Shinnecock, I’m stuck at my apartment watching everyone struggle, wondering how much I’d be struggling if I was there myself.

Besides on TV, the US Open website offers some other way to follow what players are doing. As shown here, they very generously give us information on everyone’s shots on different holes. We’re able to see where people hit the ball, on which shot, and what their resulting score was on the hole. For example, why in the world did Tony Finau, the currently second ranked longest hitter on tour, hit it short off the first tee, leave himself 230 yards to the hole where he makes bogey?

Why didn’t Tony rip D?

One of the cool things these images show is the groupings of all the shots on a hole, like the tee shots here. And when I see very specific and interactive data like we have here, I know it comes from somewhere that I’m able to see myself. So I figured I should grab that data and do some cluster analysis on different holes to see if there are certain spots that players like to hit it.

Here, I’ll go through the data we have, what the values and the numbers mean, and also the code I wrote to eat up the data and display the graphs. Once I have this part going, I’ll be able to perform further analysis to most things that come to mind.

Any questions, comments, concerns, trash talking, get in touch: twitter, contact.

Current Posts

Using Clustering Algorithms to Analyze Golf Shots

Finding the data

First step was to search for where the data for the hole insights page was coming from. As always, open the dev tools, click on the network tab, and find what’s getting called with a pretty name.

Alert!

The file itself is quite dense and has all the information, which is really cool! It has IDs for all the players, all the shots they have on the hole, which include the starting distance from the flag and the ending distance to the flag.

First off, we’re given a list of Ps, meaning an array of player information, like this:

...
{u'FN': u'Justin', u'ID': u'33448', u'IsA': False, u'LN': u'Thomas', u'Nat': u'USA', u'SN': u'THOMAS'},
{u'FN': u'Dustin', u'ID': u'30925', u'IsA': False, u'LN': u'Johnson', u'Nat': u'USA',u'SN': u'JOHNSON D'}
{u'FN': u'Tiger', u'ID': u'08793', u'IsA': False, u'LN': u'Woods', u'Nat': u'USA', u'SN': u'WOODS'}
...

It looks like we have first name, player’s ID, whether or not they’re an amateur, last name, nationality, scoreboard name. The important part of this information is the ID, where we’ll be able to match players to shots.

Next, we’re given a few stats on the hole for the day:

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Classifying Amazon Reviews with Scikit-Learn — More Data is Better Turns Out

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Last time, I went through some basics of how naive Bayes algorithm works, and the logic behind it, and implemented the classifier myself, as well as using the NLTK. That’s great and all, and hopefully people reading it got a better understanding of what was going on, and possibly how to play along with classification for their own text documents.

But if you’re looking to train and actually deploy a model, say, a website where people can copy paste reviews from Amazon and see how our classifier performs, you’re going to want to use a library like Scikit-Learn. So with this post, I’ll walk through training a Scikit-Learn model, testing various classifiers and parameters, in order to see how we do, and also at the end, will have an initial, version 1, of a Amazon review classifier that we can use in a production setting.

Some notes before we get going:

  • For a lot of the testing, I only use 5 or 10 of the full 26 classes that are in the dataset.
  • Keep in mind, that what works here might not be the same for other data sets. We’re specifically looking at Amazon product reviews. For a different set of texts (you’ll also see the word corpus being thrown around), a different classifier, or parameter sets might be used.
  • The resulting classifier we come up with is, well, really really basic, and probably what we’d guess would perform the best if we guessed what would be the best at the onset. All the time and effort that goes into checking all the combinations
  • I’m going to mention here this good post that popped up when I was looking around for other people who wrote about this. It really nicely outlines going how to classify text with Scikit-learn. To reduce redundancy, something that we all should work towards, I’m going to point you to that article to get up to speed on Scikit-learn and how it can apply to text. In this article, I’m going to start at the end of that article, where we’re working with Scikit-learn pipelines.

As always, you can say hi on twitter, or yell at me there for messing up as well if you want.

How many grams?

First step to think about is how we want to represent the reviews in naive Bayes world, in this case, a bag of words / n-grams. In the other post, I simply used word counts since I wasn’t going into how to make the best model we could have. But besides word counts, we can also bump up the representations to include something called a bigram, which is a two word combos. The idea behind that is that there’s information in two word combos that we aren’t using with just single words. With Scikit-learn, this is very simple to do, and they take care of it for you. Oh, and besides bigrams, we can say we want trigrams, fourgrams, etc. Which we’ll do, to see if that improves performance. Take a look at the wikipedia article for n-grams here.

For example is if a review mentions “coconut oil cream”, as in some sort of face cream (yup, I actually saw this as a mis-classified review), simply using the words and we might get a classification of food since we just see “coconut” “oil” and “cream”. But if we use bigrams as well as the unigrams, we’re also using “coconut oil” and “oil cream” as information. Now this might not get us all the way to a classification of beauty, but it could tip us over the edge.

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Practical Naive Bayes — Classification of Amazon Reviews

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If you search around the internet looking for applying Naive Bayes classification on text, you’ll find a ton of articles that talk about the intuition behind the algorithm, maybe some slides from a lecture about the math and some notation behind it, and a bunch of articles I’m not going to link here that pretty much just paste some code and call it an explanation.

So I’m going to try to do a little more here, by hopefully writing and explaining enough, is let you yourself write a working Naive Bayes classifier.

There are three sections here. First is setup, and what format I’m expecting your text to be in for the classification. Second, I’ll talk about how to run naive Bayes on your own, using slow Python data structures. Finally, we’ll use Python’s NLTK and it’s classifier so you can see how to use that, since, let’s be honest, it’s gonna be quicker. Note that you wouldn’t want to use either of these in production, so look for a follow up post about how you might go about doing that.

As always, twitter, and check out the full code on github.

Setup

Data from this is going to be from this UCSD Amazon review data set. I swear one of the biggest issues with running these algorithms on your own is finding a data set big and varied enough to get interesting results. Otherwise you’ll spend most of your time scraping and cleaning data that by the time you get to the ML part of the project, you’re sufficiently annoyed. So big thanks that this data already exists.

You’ll notice that this set has millions of reviews for products across 24 different classes. In order to keep the complexity down here (this is a tutorial post after all), I’m sticking with two classes, and ones that are somewhat far enough different from each other to show that classification works, we’ll be classifying baby reviews against tools and home improvement reviews.

Preprocessing

First thing I want to do now, after unpacking the .gz file, is to get a train and test set that’s smaller than the 160,792 and 134,476 of baby and tool reviews respectively. For purposes here, I’m going to use 1000 of each, with 800 used for training, and 200 used for testing. The algorithms are able to support any number of training and test reviews, but for demonstration purposes, we’re making that number lower.

Check the github repo if you want to see the code, but I wrote a script that just takes the full file, picks 1000 random numbers, segments 800 into the training set, and 200 into the test set, and saves them to files with the names “train_CLASSNAME.json” and “test_CLASSNAME.json” where classname is either “baby” or “tool”.

Also, the files from that dataset are really nice, in that they’re already python objects. So to get them into a script, all you have to do is run “eval” on each line of the file if you want the dict object.

Features

There really wasn’t a good place to talk about this, so I’ll mention it here before getting into either of the self, and nltk running of the algorithm. The features we’re going to use are simply the lowercased version of all the words in the review. This means, in order to get a list of these words from the block of text, we remove punctuation, lowercase every word, split on spaces, and then remove words that are in the NLTK corpus of stopwords (basically boring words that don’t have any information about class).

from nltk.corpus import stopwords
STOP_WORDS = set(stopwords.words('english'))
STOP_WORDS.add('')
def clean_review(review):
  exclude = set(string.punctuation)
  review = ''.join(ch for ch in review if ch not in exclude)
  split_sentence = review.lower().split(" ")
  clean = [word for word in split_sentence if word not in STOP_WORDS]
  return clean

Realize here that there are tons of different ways to do this, and ways to get more sophisticated that hopefully can get you better results! Things like stemming, which takes words down to their root word (wikipedia gives the example of “stems”, “stemmer”, “stemming”, “stemmed” as based on “stem”). You might want to include n-grams, for an n larger than 1 in our case as well.

Basically, there’s tons of processing on the text that you could do here. But since this I’m just talking about how Naive Bayes works, I’m sticking with simplicity. Maybe in the future I can get fancy and see how well I can do in classifying these reviews.

Ok, on to the actual algorithm.

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Predicting PGA Tour Scoring Average from Statistics Using Linear Regression

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First off, I admit, that’s probably the most boring title for a blog post ever. It gets a negative value on the clickbait scale that is generally unseen in the modern, “every click equals dollars” era that we live in. On the other hand, it tells you exactly what this article is about — predicting scoring average using stats.

In this article, I’ll go through getting the data from the database, cleaning that data for use, and then running a linear regression in order to generate coefficients for each of the stats to generate scoring average predictions. Oh, and some analysis and commentary at the end!

Shameless shoutout to my other blog, Golf on the Mind. Check it out and subscribe to the newsletter / twitter / instagram if you’re into golf at all. Or ignore, and keep reading for some code!

Here's a pic of a golf course to get you in the mood.

Here’s a pic of a golf course to get you in the mood.

Getting the data

Last time if you remember, I spent all this effort taking the csv stat files, and putting the information into a database. Start there if you haven’t read that post yet. It’ll show how I grabbed the stats and formatted them.

Now that you’re back in the present we need to create a query that gets the stats for the players for a specific year. An example row in a CSV file of the data would be something like:

player_id, player_name, stat_1_value, stat_2_value, … , stat_n_value

for stats 1 to n where n (the number of stats), and the which stats themselves (driving distance, greens in regulation, etc.) vary depending on inputs.

Now let me say, I am not an expert in writing sql queries. And since people on the internet loooove to dole out hate in comments sections, I’m just going to say that there’s probably a better way of writing this query. Feel free to let me know and I can throw an edit in here, but this query works just fine.

select players.id,
  players.name,
  max(case when stat_lines.stat_id=330 then stat_lines.raw else null end) as putting_average,
  max(case when stat_lines.stat_id=157 then stat_lines.raw else null end) as driving_distance,
  max(case when stat_lines.stat_id=250 then stat_lines.raw else null end) as gir,
  max(case when stat_lines.stat_id=156 then stat_lines.raw else null end) as driving_accuracy,
  max(case when stat_lines.stat_id=382 then stat_lines.raw else null end) as scoring_average
from players
  join stat_lines on stat_lines.player_id = players.id
  join stats on stat_lines.stat_id=stats.id
where stat_lines.year=2012 and (stats.id=157 or stats.id=330 or stats.id=382 or stats.id=250 or stats.id=156) and stat_lines.raw is not null
group by players.name,players.id;

High level overview time! We’re selecting player id, and player name, along with their stats for putting average, driving distance, greens in regulation, driving accuracy and scoring average for the year 2012. In order to get the right stats, we need to know the stat id for the stats.

One more thing. This query is funky, and I probably could have designed the schema differently to make this prettier. For example, I could have just gone with one table, stat_lines, with fields for player_name and stat_name (along with all the current fields) and then the sql would be very simple. But there are other applications to keep in mind. What if you wanted to display all stats by a player? Or all of a players stats for a certain year? With the way I have the schema set up, those queries are simple and logical. For this specific case, I’ll deal with the complexity.

Loading the Data

That query above is great, but it’s not going to cut it if I have to specify what the year, and the stat ids in that string every time I run the script. Gotta be dynamic here.

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Defense Matters in the NBA, Apparently

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Interesting article here from Arxiv titled Finding Common Characteristics Among NBA Playoff Teams: A Machine Learning Approach.

Everyone should skim the paper because it’ll show you that these academic papers aren’t overwhelming, and a lot of times, they’re good at showing steps that go into tackling a machine learning problem. This paper, for example, goes over the basics of decision trees, pruning decision trees, as well as more high powered decision trees. Great progression.

As for what they found, apparently opponent’s stats are the most important determinants in whether a team makes the playoffs, with opponent field goal percentage and opponent points per game leading the way. Play good defense, and doesn’t matter how many points you score yourselves.

Only comment is that if you’re sitting out there thinking “well NBA teams don’t play defense anyway blah blah blah” you’re wrong. Defense is probably the most impressive aspect of a game to watch, especially in the playoffs currently going on.

Importance of variables is a really interesting topic in machine learning, sports especially. Knowing which variables matter can help someone in charge figure out who to draft (Moneyball style) or possibly what aspects to focus on during games or practice. Considering I have a bunch of PGA Tour data, maybe figuring out which stats are important for golfers should be something to focus on here…

Funny, they say that their “dataset was quite large”, so they only provided a sample of the data. 30 teams * 15 years * 44 variables = 19,800 data points. Too big to fit on a table in a paper sure, but don’t think I agree it’s quite large. More like big-ish. Fits in perfectly here.

Weekly Fantasy Golf Results and a System of Linear Equations

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I had an idea recently about how to use the “wisdom of the crowds” in forecasting performance for weekly fantasy golf. I see the benefits of crowdsourcing all the time working with prediction markets at Inkling Markets, so I figured I could leverage that to get an edge when choosing my lineups. I’ll get into the details of how I’m doing that in a later post since that’s worthy on it’s own, but I found a subproblem that I’d have to solve in order to make it work.

While some of the sites offer player salaries in an easily digestible csv format, they’re a little stingy when it comes to results. They only keep contest results for the last 30 days, and the only downloadable results they offer are the results from contests in the past week, which just show point totals for an entire lineup, not the point totals for individual golfers. And I need those player’s points for testing my forecasting algorithm, as well as using them in making the actual forecasts. One way to figure this out is to scrape the hole by hole data from pgatour.com and apply the site’s scoring algorithm to each of those holes, but I’ve found a better, simpler, and much more elegant solution.

I can model the results csv file as a system of linear equations, and by converting the different user’s lineups into a matrix and vector those equations, numpy can solve for the point totals that each golfer earned during the tournament.

Here’s an example row from that file, with the person’s username hidden.

"Rank","EntryId","EntryName","TimeRemaining","Points","Lineup"
263,85826104, "UNAME (1/100)", 0, 430.50, "(G) Ryan Palmer ,(G) Pat Perez ,(G) Kevin Kisner ,(G) Ben Martin ,(G) Shawn Stefani ,(G) John Peterson "

In order to solve, we need to create two data structures. The first is a num_lineups by num_players matrix of coefficients, where the value is a 1 if the player was used in that lineup, and 0 if he wasn’t. The second is an array of total points scored by that lineup.

The idea is that if we had an array of the points the players scored over the course of the tournament, we should be able to take the dot product of that and the corresponding row in the coefficient matrix to generate the point total of that lineup.

Here’s the code to generate the coefficient matrix and the point array:

points_label = "Points"
lineup_label = "Lineup"
player_coefficients = []
lineup_points = []
with open('outcome.csv', 'rb') as csvfile:
  rows = csv.reader(csvfile)
  headers = rows.next()
  points_index = headers.index(points_label)
  lineup_index = headers.index(lineup_label)
  for row in rows:
    points = float(row[points_index])
    lineup_points.append(points)
    names = [name.strip() for name in row[lineup_index].replace('(G)','').split(',')]
    lineup_players = [0] * player_count
    for name in names:
      lineup_players[player_list.index(name)] = 1
      player_coefficients.append(lineup_players)

From here, all we need to do is convert those arrays into numpy arrays, and run numpy’s linear algebra least squares algorithm to get the solution array!

coefficient_matrix = np.array(player_coefficients)
point_array = np.array(lineup_points).transpose()
solution = np.linalg.lstsq(coefficient_matrix, point_array)
player_points = list(solution[0])

Initially, I tried to use the numpy’s solve algorithm, but by looking at the docs realized that solve dealt with square coefficient matrices (something that I still remember from all those math classes). The lstsqrs method is used to get approximate results from rectangular matrices as is the case here.

Printing out the results yields the following results for the top and bottom 10:

Chris Kirk: 126.0
Jason Bohn: 105.5
Brandt Snedeker: 102.0
Jordan Spieth: 101.5
Kevin Kisner: 99.0
George McNeill: 96.5
Pat Perez: 95.0
Adam Hadwin: 92.0
Ian Poulter: 87.5
Brian Harman: 87.0
...
Kenny Perry: 18.0
Jason Kokrak: 17.5
Jonas Blixt: 16.5
Corey Pavin: 16.0
Bo Van Pelt: 15.0
David Toms: 14.0
Brian Davis: 11.5
Tom Watson: 4.61852778244e-13
Tom Purtzer: 3.87245790989e-13
Scott Stallings: 2.84217094304e-13

Chris Kirk won so him having the highest point total makes sense, and the three guys at the bottom with zero points all withdrew so they should be at 0 points. Unfortunate for the guys who didn’t take them out of their lineups, but they gotta pay a little more attention!

Only issue with the final result is that I only end up with point totals from 117 players, when 133 teed it up at the beginning of the tournament. That means that we’re missing point totals from some of the guys. That being said, I’m going to assume that those players weren’t picked because they weren’t likely to play well, so hopefully that 117 offer a good representation of point totals. Also, could easily be that draftkings in this case only listed 117 players to choose from. I’ll need to investigate this week.

In the end, it took about 3 hours to write the code and write this post. It’s fun little problems like this that really remind you that programming is fun and has value. Being able to create technical solution to something you’re interested in is probably the best part about being a programmer. Check out the entire script in this gist.

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