I really like generative art, or so-called algorithmic art. Basically, it means you take a pattern or a complex system of rules, and apply it to create something new following those patterns/rules.

When I finished my PhD, I got a beautiful poster of where the k-nearest neighbors algorithms was used to generate a set of connected points.

My first piece of generative art.

As we recently moved into our new house, I decided I wanted to have a brother for the knn-poster. So I did some research in algorithms I wanted to use to generate a painting. I found some very cool ones, of which I unforunately can’t recollect the artists anymore:

However, I preferred to make one myself. So we again turned to the work of the author that made the knn-poster: Marcus Volz.

He has written (in R) many other algorithms. And we found that one specifically nicely matched the knn-poster. His metropolis – or generative city:

However, I wanted to make one myself, so I download Marcus code, and tweaked it a bit. Most importantly, I made it start in the center, made it fill up the whole space, and I made it run more efficient so I could generate a couple dozen large cities quickly, and pick the one I liked most. Here’s the end result:

Ryan Holbrook made awesome animated GIFs in R of several classifiers learning a decision rule boundary between two classes. Basically, what you see is a machine learning model in action, learning how to distinguish data of two classes, say cats and dogs, using some X and Y variables.

These visuals can be great to understand these algorithms, the models, and their learning process a bit better.

Here’s the original tweet, with the logistic regression animation. If you follow it, you will find a whole thread of classifier GIFs. These I extracted, pasted, and explained below.

A thread of classifiers learning a decision rule. Dashed line is optimal boundary. Animations with #gganimate by @thomasp85 and @drob. #rstats

Logistic regression {stats::glm} with each class having normally distributed features. (1/n) pic.twitter.com/kKmqdO2zGy

Below is the GIF which I extracted using EZgif.com.

What you see is observations from two classes, say cats and dogs, each represented using colored dots. The dots are placed along X and Y axes, which represent variables about the observations. Their tail lengths and their hairyness, for instance.

Now there’s an optimal way to seperate these classes, which is the dashed line. That line best seperates the cats from the dogs based on these two variables X and Y. As this is an optimal boundary given this data, it is stable, it does not change.

However, there’s also a solid black line, which does change. This line represents the learned boundary by the machine learning model, in this case using logistic regression. As the model is shown more data, it learns, and the boundary is updated. This learned boundary represents the best line with which the model has learned to seperate cats from dogs.

Anything above the boundary is predicted to be class 1, a dog. Everything below predicted to be class 2, a cat. As logistic regression results in a linear model, the seperation boundary is very much linear/straight.

Logistic regression gif by Ryan Holbrook

These animations are great to get a sense of how the models come to their boundaries in the back-end.

For instance, other machine learning models are able to use non-linear boundaries to dinstinguish classes, such as this quadratic discriminant analysis (qda). This “learned” boundary is much closer to the optimal boundary:

Quadratic discriminant analysis gif by Ryan Holbrook

Multivariate adaptive regression splines gif by Ryan Holbrook

Next, we have the k-nearest neighbors algorithm, which predicts for each point (animal) the class (cat/dog) based on the “k” points closest to it. As you see, this results in a highly fluctuating, localized boundary.

K-nearest neighbors gif by Ryan Holbrook

Now, Ryan decided to push the challenge, and simulate new data for two classes with a more difficult decision boundary. The new data and optimal boundaries look like this:

On these data, Ryan put a whole range of non-linear models to work.

Like this support-vector machine, which tries to create optimal boundaries built of support vectors around all the cats and all the dohs (this is definitely not a technical, error-free explanation of what’s happening here).

Let’s jump into some tree-based algorithms and the resulting models. A decision tree classifies data based on multiple, sequential, binary splits. Here, Ryan trained a simple decision tree:

Decision tree gif by Ryan Holbrook

As well as it’s big brother, a random forest, which uses hundreds of trees in the back end and thus results in a more flexible boundary:

Random forest gif by Ryan Holbrook

Extreme gradient boosting is also a tree-based algorithm, which leverages many machine learning techniques to optimize the bias-variance tradeoff. Here’s an earlier blog on how to get started with Xgboost in Python or R:

I found this interesting blog by Guilherme Duarte Marmerola where he shows how the predictions of algorithmic models (such as gradient boosted machines, or random forests) can be calibrated by stacking a logistic regression model on top of it: by using the predicted leaves of the algorithmic model as features / inputs in a subsequent logistic model.

When working with ML models such as GBMs, RFs, SVMs or kNNs (any one that is not a logistic regression) we can observe a pattern that is intriguing: the probabilities that the model outputs do not correspond to the real fraction of positives we see in real life.

This is visible in the predictions of the light gradient boosted machine (LGBM) Guilherme trained: its predictions range only between ~ 0.45 and ~ 0.55. In contrast, the actual fraction of positive observations in those groups is much lower or higher (ranging from ~ 0.10 to ~0.85).

I highly recommend you look at Guilherme’s code to see for yourself what’s happening behind the scenes, but basically it’s this:

Train an algorithmic model (e.g., GBM) using your regular features (data)

Retrieve the probabilities GBM predicts

Retrieve the leaves (end-nodes) in which the GBM sorts the observations

Turn the array of leaves into a matrix of (one-hot-encoded) features, showing for each observation which leave it ended up in (1) and which not (many 0’s)

Basically, until now, you have used the GBM to reduce the original features to a new, one-hot-encoded matrix of binary features

Now you can use that matrix of new features as input for a logistic regression model predicting your target (Y) variable

Apparently, those logistic regression predictions will show a greater spread of probabilities with the same or better accuracy

Here’s a visual depiction from Guilherme’s blog, with the original GBM predictions on the X-axis, and the new logistic predictions on the Y-axis.

As you can see, you retain roughly the same ordering, but the logistic regression probabilities spread is much larger.

Now according to Guilherme and the Facebook paper he refers to, the accuracy of the logistic predictions should not be less than those of the original algorithmic method.

Much better. The calibration plot of lgbm+lr is much closer to the ideal. Now, when the model tells us that the probability of success is 60%, we can actually be much more confident that this is the true fraction of success! Let us now try this with the ET model.

In his blog, Guilherme shows the same process visually for an Extremely Randomized Trees model, so I highly recommend you read the original article. Also, you can find the complete code on his GitHub.

I came across this 1999-2003 e-book by Eric Raymond, on the Art of Unix Programming. It contains several relevant overviews of the basic principles behind the Unix philosophy, which are probably useful for anybody working in hardware, software, or other algoritmic design.

Brandon Rohrer — (former) data scientist at Microsoft, iRobot, and Facebook — asked his network on Twitter and LinkedIn to share their favorite resources on A/B testing. It produced a nice list, which I summarized below.

Hey Twitter, a contact just asked me about A/B testing. Do you have any posts or tutorials you would recommend for them?

Past days, I discovered this series of blogs on how to win the classic game of Battleships (gameplay explanation) using different algorithmic approaches. I thought they might amuse you as well : )

The story starts with this 2012 Datagenetics blog where Nick Berry constrasts four algorithms’ performance in the game of Battleships. The resulting levels of artificial intelligence (AI) seem to compare respectively to a distracted baby, two sensible adults, and a mathematical progidy.

The first, stupidest approach is to just take Random shots. The AI resulting from such an algorithm would just pick a random tile to shoot at each turn. Nick simulated 100 million games with this random apporach and computed that the algorithm would require 96 turns to win 50% of games, given that it would not be defeated before that time. At best, the expertise level of this AI would be comparable to that of a distracted baby. Basically, it would lose from the average toddler, given that the toddler would survive the boredom of playing such a stupid AI.

A first major improvement results in what is dubbed the Hunt algorithm. This improved algorithm includes an instruction to explore nearby spaces whenever a prior shot hit. Every human who has every played Battleships will do this intuitively. A great improvement indeed as Nick’s simulations demonstrated that this Hunt algorithm completes 50% of games within ~65 turns, as long as it is not defeated beforehand. Your little toddler nephew will certainly lose, and you might experience some difficulty as well from time to time.

A visual representation of the “Hunting” of the algorithm on a hit [via]

Another minor improvement comes from adding the so-called Parity principle to this Hunt algorithm (i.e., Nick’s Hunt + Parity algorithm). This principle instructs the algorithm to take into account that ships will always cover odd as well as even numbered tiles on the board. This information can be taken into account to provide for some more sensible shooting options. For instance, in the below visual, you should avoid shooting the upper left white tile when you have already shot its blue neighbors. You might have intuitively applied this tactic yourself in the past, shooting tiles in a “checkboard” formation. With the parity principle incorporated, the median completion rate of our algorithm improves to ~62 turns, Nick’s simulations showed.

Now, Nick’s final proposed algorithm is much more computationally intensive. It makes use of Probability Density Functions. At the start of every turn, it works out all possible locations that every remaining ship could fit in. As you can imagine, many different combinations are possible with five ships. These different combinations are all added up, and every tile on the board is thus assigned a probability that it includes a ship part, based on the tiles that are already uncovered.

Computing the probability that a tile contains a ship based on all possible board layouts [via]

At the start of the game, no tiles are uncovered, so all spaces will have about the same likelihood to contain a ship. However, as more and more shots are fired, some locations become less likely, some become impossible, and some become near certain to contain a ship. For instance, the below visual reflects seven misses by the X’s and the darker tiles which thus have a relatively high probability of containing a ship part.

An example distribution with seven misses on the grid. [via]

Nick simulated 100 million games of Battleship for this probabilistic apporach as well as the prior algorithms. The below graph summarizes the results, and highlight that this new probabilistic algorithm greatly outperforms the simpler approaches. It completes 50% of games within ~42 turns! This algorithm will have you crying at the boardgame table.

Relative performance of the algorithms in the Datagenetics blog, where “New Algorithm” refers to the probabilistic approach and “No Parity” refers to the original “Hunt” approach.

Reddit user /u/DataSnaek reworked this probablistic algorithm in Python and turned its inner calculations into a neat GIF. Below, on the left, you see the probability of each square containing a ship part. The brighter the color (white <- yellow <- red <- black), the more likely a ship resides at that location. It takes into account that ships occupy multiple consecutive spots. On the right, every turn the algorithm shoots the space with the highest probability. Blue is unknown, misses are in red, sunk ships in brownish, hit “unsunk” ships in light blue (sorry, I am terribly color blind).

The probability matrix as a heatmap for every square after each move in the game. [via]

This latter attempt by DataSnaek was inspired by Jonathan Landy‘s attempt to train a reinforcement learning (RL) algorithm to win at Battleships. Although the associated GitHub repository doesn’t go into much detail, the approach is elaborately explained in this blog. However, it seems that this specific code concerns the training of a neural network to perform well on a very small Battleships board, seemingly containing only a single ship of size 3 on a board with only a single row of 10 tiles.

Fortunately, Sue He wrote about her reinforcement learning approach to Battleships in 2017. Building on the open source phoenix-battleship project, she created a Battleship app on Heroku, and asked co-workers to play. This produced data on 83 real, two-person games, showing, for instance, that Sue’s coworkers often tried to hide their size 2 ships in the corners of the Battleships board.

Probability heatmaps of ship placement in Sue He’s reinforcement learning Battleships project [via]

Next, Sue scripted a reinforcement learning agent in PyTorch to train and learn where to shoot effectively on the 10 by 10 board. It became effective quite quickly, requiring only 52 turns (on average over the past 25 games) to win, after training for only a couple hundreds games.

The performance of the RL agent at Battleships during the training process [via]

However, as Sue herself notes in her blog, disappointly, this RL agent still does not outperform the probabilistic approach presented earlier in this current blog.

Reddit user /u/christawful faced similar issues. Christ (I presume he is called) trained a convolutional neural network (CNN) with the below architecture on a dataset of Battleships boards. Based on the current board state (10 tiles * 10 tiles * 3 options [miss/hit/unknown]) as input data, the intermediate convolutional layers result in a final output layer containing 100 values (10 * 10) depicting the probabilities for each tile to result in a hit. Again, the algorithm can simply shoot the tile with the highest probability.

Christ’s convolutional neural network architecture for Battleships [via]

Christ was nice enough to include GIFs of the process as well [via]. The first GIF shows the current state of the board as it is input in the CNN — purple represents unknown tiles, black a hit, and white a miss (i.e., sea). The next GIF represent the calculated probabilities for each tile to contain a ship part — the darker the color the more likely it contains a ship. Finally, the third picture reflects the actual board, with ship pieces in black and sea (i.e., miss) as white.

As cool as this novel approach was, Chris ran into the same issue as Sue, his approach did not perform better than the purely probablistic one. The below graph demonstrates that while Christ’s CNN (“My Algorithm”) performed quite well — finishing a simulated 9000 games in a median of 52 turns — it did not outperform the original probabilistic approach of Nick Berry — which came in at 42 turns. Nevertheless, Chris claims to have programmed this CNN in a couple of hours, so very well done still.

The performance of Christ’s Battleship CNN compared to Nick Berry’s original algorithms [via]

Interested by all the above, I searched the web quite a while for any potential improvement or other algorithmic approaches. Unfortunately, in vain, as I did not find a better attempt than that early 2012 Datagenics probability algorithm by Nick.

Surely, with today’s mass cloud computing power, someone must be able to train a deep reinforcement learner to become the Battleship master? It’s not all probability right, there must be some patterns in generic playing styles, like Sue found among her colleagues. Or maybe even the ability of an algorithm to adapt to the opponent’s playin style, as we see in Libratus, the poker AI. Maybe the guys at AlphaGo could give it a shot?

For starters, Christ’s provided some interesting improvements on his CNN approach. Moreover, while the probabilistic approach seems the best performing, it might not the most computationally efficient. All in all, I am curious to see whether this story will continue.

However, then I imagined that not everybody may be familiar with k-means, hence, I wrote the whole blog below.

Next thing I know, u/dashee87 on r/datascience points me to these two other blogs that had already done the same… but way better! These guys perfectly explain k-means, alongside many other clustering algorithms. Including interactive examples and what not!!

Seriously, do not waste your time on reading my blog below, but follow these links. If you want to play, go for Naftali’s apps. If you want to learn, go for David’s animated blog.

Naftali Harris built this fantastic interactive app where you can run the k-means algorithm step by step on many different datasets. You can even pick the clusters’ starting locations yourself! Next to this he provides a to-the-point explanation of the inner workings. On top of all this, he built the same app for the DBSCAN algorithm(wiki), so you can compare the algorithms’ performance… Insane!

David Sheehan (yes, that’s dashee) is more of a Python guy and walks you through the inner workings of six algorithmic clustering approaches in his blog here. Included are k-means, expectation maximization, hierarchical, mean shift, and affinity propagation clustering, and DBSCAN. David has made detailed step-wise GIF animations of all these algorithms. And he explains the technicalities in a simple and understandable way. On top of this, David shared his Jupyter notebook to generate the animations, along with a repository of the GIFs themselves. Very well done! Seriously one of the best blogs I’ve read in a while.

Let me walk you through what k-means is, why it is called k-means, and how the algorithm interally works, step by step.

Let me dissect that sentence for you, starting at the back.

Algorithm

Simply put, an algorithm (wiki) is a set of task instructions to be followed. Often algorithms are perform by computers, and used in calculations or problem-solving. Basically, an algorithm is thus not much more than a sequence of task instructions to be followed by a computer — like a cooking recipe.

In machine learning (wiki), algorithmic tasks are often divided in supervised or unsupervisedlearning (let’s skip over reinforcement learning for now).

The learning part reflects that algorithms try to learn a solution — learn how to solve a problem.

The supervision part reflects whether the algorithm receives answers or solutions to learn from. For supervised learning, examples are required. In the case of unsupervised learning, algorithms do not learn by example but have figure out a proper solution themselves.

Clustering

Clustering (wiki) is essentially a fancy word for grouping. Clustering algorithms seek to group things together, and try to do so in an optimal way.

Group things. As long as we can represent things in terms of data, clustering algorithms can group them. We can group cars by their weight and horsepower, people by their length and IQ, or frogs by their slimyness and greenness. These things we would like to group, we often call observations (depicted by the letter N or n).

Optimal way. Clustering is very much an unsupervised task. Clustering algorithms do not receive examples to learn from (that’s actually classification (wiki)). Algorithms are not told what “good” clustering or grouping looks like.

Yet, how do clustering algorithms determine the optimal way to group our observations then?

Well, clustering algorithms like k-means do so by optimizing a certain value. This value is reflected in an algorithm’s so-called objective function.

k-means’ objective function is displayed below. In simple language, k-means seeks to minimize (= optimize) the total distance of observations (= cases) to their group’s (= cluster) center (= centroid).

Fortunately, you can immediately forget this function. For now, all you need to know is that algorithms often repeat specific steps of their instructions in order for their objective function to produce a satisfactory value.

Hurray, we can now move to the k-means algorithm itself!

Why is it called k-means in the first place?

Let’s start at the front this time.

k. The k in k-means reflects the number of groups the algorithm is going to form. The algorithm depends on its user to specify what number of k should be. If the user picks k = 2 for instance, the k-means algorithm will identify 2 group by their 2 means.

As I said before, clustering algorithms like k-means are unsupervised. They do not know what good clustering looks like. In the case where the used specified k = 2, the algorithm will seek to optimize its objective function given that there are 2 groups. It will try to put our observations in 2 groups that minimizes the total distance of the observations to their group’s center. I will how that works visually later in this blog.

means. The means in k-means reflects that the algorithm considers the mean value of a cluster as its cluster center. Here, mean is a fancy word for average. Again, I will visualize how this works later.

You might have guessed it already: there are indeed variations of the k-means algorithm that do not use the cluster means, but, for instance, the cluster medians (wiki) or mediods (wiki). These algorithms are only slight modifications of the k-means algorithm, and simply called the k-medians (wiki) and k-medoids algorithms (wiki).

How does k-means work?

Finally, we get to the inner workings of k-means!

The k-means algorithm consists of five simple steps:

Obtain a predefined k.

Pick k random points as cluster centers.

Assign observations to their closest cluster center based on the Euclidean distance.

Update the center of each cluster based on the included observations.

Terminate if no observations changed cluster, otherwise go back to step 3.

Easy right?

OK, I can see how this may not be directly clear.

Let’s run through the steps one by one using an example dataset.

Example dataset: mtcars

Say we have the dataset below, containing information on 32 cars.

We can consider each car a separate observation. For each of these observations, we have its weight and its horsepower.

These characteristics — weight and horsepower — of our observations are the variables in our dataset. Our cars vary based on these variables.

Name

Weight

Horsepower

Mazda RX4

1188

110

Mazda RX4 Wag

1304

110

Datsun 710

1052

93

Hornet 4 Drive

1458

110

Hornet Sportabout

1560

175

Valiant

1569

105

Duster 360

1619

245

Merc 240D

1447

62

Merc 230

1429

95

Merc 280

1560

123

Merc 280C

1560

123

Merc 450SE

1846

180

Merc 450SL

1692

180

Merc 450SLC

1715

180

Cadillac Fleetwood

2381

205

Lincoln Continental

2460

215

Chrysler Imperial

2424

230

Fiat 128

998

66

Honda Civic

733

52

Toyota Corolla

832

65

Toyota Corona

1118

97

Dodge Challenger

1597

150

AMC Javelin

1558

150

Camaro Z28

1742

245

Pontiac Firebird

1744

175

Fiat X1-9

878

66

Porsche 914-2

971

91

Lotus Europa

686

113

Ford Pantera L

1438

264

Ferrari Dino

1256

175

Maserati Bora

1619

335

Volvo 142E

1261

109

Visually, we can represent this same dataset as follows:

Step 1: Define k

For whatever reason, we might want to group these 32 cars.

We know the cars’ weight and horsepower, so we can use these variables to group the cars. The underlying assumption being that cars that are more similar in terms of weight and/or horsepower would belong together in the same group.

Normally, we would have smart or valid reasons to expect a specific number of groups among our observations. For now, let’s simply say that we want to put these cars into, say, 2 groups.

Well, that’s step 1 already completed: we defined k = 2.

Step 2: Initialize cluster centers

In step 2, we need to initialize the algorithm.

Everyone needs to start somewhere, and the default k-means algorithm starts out super naively: it just picks random locations as our starting cluster centers.

For instance, the algorithm might initialize cluster 1 randomly at a weight of 1000 kilograms and a horsepower of 100.

We can use coordinate notation [x; y] or [weight; horsepower] to write this location in short. Hence, the initial random center of cluster 1 picked by the k-means algorithm is located at [1000; 100].

Randomly, k-means could initialize cluster 2 at a weight of 1500kg and a horsepower of 200. It’s intitial center is thus at [1500; 200].

Visually, we can display this initial situation like the below, with our 32 cars as grey dots in the background:

Now, step 2 is done, and our k-means algorithm has been fully initialized. We are now ready to enter the core loop of the algorithm. The next three steps — 3, 4, and 5 — will be repeated until the algorithm tells itself it is done.

Step 3: Assignment to closest cluster

Now, in step 3, the algorithm will assign every single car in our dataset to the cluster whose center is closest.

To do this, the algorithm looks at every car, one by one, and calculate its distance to every of our cluster centers.

So how would this distance calculating thing work?

The algorithm starts with the first car in our dataset, a Mazda RX4.

This Mazda RX4 weighs 1188kg and has 110 horsepower. Hence, it is located at [1188; 110].

As this is the first time our k-means algorithm reaches step 3, the cluster centers are still at the random locations the algorithm has picked in step 2.

The k-means algorithm now calculates the Euclidean distance (wiki) of this Mazda RX4 data point [1188; 110] to each of the cluster centers.

The Euclidean distance is calculated using the formula below. The first line shows you that the Euclidean distance is the square root of the squared distance between two observations. The second line — with the big Greek capital letter Sigma (Σ) — is a shorter way to demonstrate that the distance is calculated and summed up for each of the variables considered.

Again, please don’t mind the formula, the Euclidean distance is basically the length of a straight line between two data points.

So… back to our Mazda RX4. This Mazda is one of the two observations we need to input in the formula. The second observation would be a cluster center. We input both of the location of our Mazda [1188; 110] and that of a cluster center –, say cluster 1’s [1000; 100] — in the formula, and out comes the Euclidean distance between these two observations.

The Euclidean distance of our Mazda RX4 to the center of cluster 1 would thus be √((1118 – 1000)^{2} + (110 – 100)^{2}), which equals 192.2082 — or a rounded 192.

We need to repeat this, but now with the location of our second cluster center. The Euclidean distance from our Mazda RX4 to the center of cluster 2 would be √((1118 – 1500)^{2} + (110 – 150)^{2}), which equals 314.5537 — or a rounded 314.

Again, I visualized this situation below.

You can clearly see that our Mazda RX4 is closer to cluster center 1 (192) than to cluster center 2 (314). Hence, as the distance to cluster 1’s center is smaller, the k-means algorithm will now assign our Mazda RX4 to cluster 1.

Subsequently, the algorithm continues with the second car in our dataset.

Let this second car be the Mercedes 280C for now, weighing in at 1560 kg with a horsepower of 153.

Again, the k-means algorithm would calcalute the Euclidean distance from this Mercedes [1560; 153] to each of our cluster centers.

It would find that this Mercedes is located much closer to cluster 2’s center (560) than cluster 1’s (65).

Hence, the k-means algorithm will assign the Mercedes 280C to cluster 2, before continuing with the next car…

and the next car after that…

and the next car…

… until all cars are assigned to one of the clusters.

This would mean that step 3 is completed. Visually, the situation at the end of step 3 will look like this:

Step 4: Update the cluster centers

Now, in step 4, the k-means algorithm will update the cluster centers.

As a result of step 3, there are now actual observations assigned to the clusters. Hence, the k-means algorithm can let go of its naive initial random guesses and calculate the actual cluster centers.

Because we are dealing with the k-means algorithm, these centers will be based on the mean values of the observations in each group.

So for each cluster, the algorithm takes the observations assigned to it, and calculates the cluster’s mean value for every variable. In our case, the algorithm thus calculates 4 means: the average weight and the average horsepower, for each of our two clusters.

For cluster 1, the average weight of its cars is a rounded 939 kg. Its average horsepower is approximately 84. Hence the cluster center is updated to location [939; 84]. Cluster 2’s mean values come in at [1663; 171].

With the cluster centers updated, the k-means algorithm has finished step 4. Visually, the situation now looks as follows, with the old cluster centers in grey.

Step 5: Terminate or go back to step 3.

So that was actually all there is to the k-means algorithm. From now on, the algorithm either terminates or goes back to step 3.

So how does the k-means algorithm know when it is done?

Earlier in this blog post I already asked “how do clustering algorithms determine the optimal way to group our observations?”

Well, we already know that the k-means algorithm wants to optimize its objective function. It seeks to minimize the total distance of observations to their respective cluster centers. It does so by assigning observations to the cluster whose center is nearest according to the Euclidean distance.

Now, with the above in mind, the k-means algorithm determines that it has reached an optimal clustering solution if, in step 3, no single observation switches to a different cluster.

If that is the case, then every observation is assigned to the group whose center values best represent its underlying characteristics (weight and horsepower), and the k-means algorithm is thus satisfied with the solution given this number of groups (= k). These groups centers now best describe the characteristics of the individual observations at hand, given this k, as evidenced that each observations belongs to the cluster whose center values are closest to their own values.

Now, the k-means algorithm will have to check whether it is done somewhere in its instructions. It seems most logical to directly do this in step 3: quickly check whether every observations remained in its original cluster.

This is not so difficult if you have only 32 cars. However, what if we were clustering 100.000 cars? We would not want to check for 100.000 cars whether they remained in their respective cluster, right? That’s a heavy task, even for a computer.

Potentially there is an easier way to check this? Maybe we could look at our cluster centers? We update them in step 4. And if no observations have changed clusters, then the locations of our cluster centers will for sure also not have changed.

Even in our simple example it is less work to see check whether 2 cluster centers have remained the same, than comparing whether 32 cars have not changed clusters.

So basically, that’s what we will do in this step five. We check whether our cluster centers have moved.

In this case, they did. As can be seen in the visual at the end of step 4.

This is also to be expected, as it is very unlikely that the algorithm could have randomly picked the initial cluster centers in their optimal locations, right?

We conclude step 5 and, because the cluster center locations have changed in step 4, the algorithm is sent back to step 3.

Let’s see how the k-means algorithm continues in our example.

Step 3 (2nd time): Assignment to closest cluster

In step 3, the algorithm reassigns every car in our dataset to the cluster whose center is nearest.

To do this, the algorithm has to look at every car, one by one, and calculate its distance to every of our cluster centers.

Our cluster centers have been updated in the previous step 4. Hence, the distances between our observations and cluster centers may have changed respective to the first time the algorithm performed step 3.

Indeed two cars that were previously closer to the blue cluster 2 center are now actually closer to the red cluster 1 center.

In this step 3, again all observations are assigned to their closest clusters, and two observations thus change cluster.

Visually, the situation now looks like the below. I’ve marked the two cars that switched in yellow and with an exclamation mark.

The k-means algorithm now again reaches step 4.

Step 4 (2nd time): Update the cluster centers

In step 4, the algorithm updates the cluster centers. Because we are dealing with the k-means algorithm, these centers will be based on the mean values of the observations in each group.

For cluster 1, both the average weight and the average horsepower have increased due to the two new cars. The cluster center thus moves from approximately [939; 84] to [998; 94].

Cluster 2 lost two cars from its cluster, but both were on the lower end of its ranges. Hence its average weight and horsepower have also increased, moving the center from approximately [1663; 171] to [1701; 174].

With the cluster centers updated, step 4 is once again completed. Visually, the situation now looks as follows, with the old centers in grey.

Step 5 (2nd time): Terminate or go back to step 3.

In step 5, the k-means algorithm again checks whether it is done. It concludes that it is not, because the cluster centers have both moved once more. This indicates that at least some observations have changed cluster, and that a better solution may be possible. Hence, the algorithm needs to return to step 3 for a third time.

Are you still with me? We are nearly there, I hope…

Step 3 (3nd time): Assignment to closest cluster

In step 3, the algorithm assigns every car in our dataset to the cluster whose center is nearest. To do this, the algorithm looks at every car, one by one, and calculates its distance to each of our cluster centers. As our cluster centers have been updated in the previous step 4, so too will their distances to the cars.

After calculating the distances, step 3 is once again completed by assigning the observations to their closest clusters. Another car moved from the blue cluster 2 to the red cluster 1, highlighted in yellow with an exclamation mark.

Step 4 (3rd time): Update the cluster centers

In step 4, the algorithm updates the cluster centers. For each cluster, it looks at the values of the observations assigned to it, and calculates the mean for every variable.

For cluster 1, both the average weight and the average horsepower have again increased slightly due to the newly assigned car. The cluster center moves from approximately [998; 94] to [1023; 96].

Cluster 2 lost a car from its cluster, but it was on the lower end of its range. Hence, its average weight and horsepower have also increased, moving the cluster center from approximately [1701; 174] to [1721; 177].

With the renewed cluster centers, step 4 is once again completed. Visually, the situation looks as follows, with the old centers in grey.

Step 5 (3rd): Terminate or go back to step 3.

In step 5, the k-means algorithm will again conclude that it is not yet done. The cluster centers have both moved once more, due to one car changing from cluster 2 to cluster 1 in step 3 (3rd time). Hence, the algorithm returns to step 3 for a fourth, but fortunately final, time.

Step 3 (4th time): Assignment to closest cluster

In step 3, the k-means algorithm assigns every car in our dataset to the cluster whose center is nearest. Our cluster centers have only changed slightly in the previous step 4, and thus the distances are nearly similar to last time. Hence, this is the first time the algorithm completes step 3 without having to reassign observations to clusters. We thus know the algorithm will terminate the next time it checks for cluster changes.

Step 4 (4th time): Update the cluster centers

In step 4, the k-means algorithm tries to update the cluster centers. However, no observations moved clusters in step 3 so there is nothing to update.

Step 5 (4th): Terminate or go back to step 3.

As no changes occured to our cluster centers, the algorithm now concludes that it has reached the optimal clustering and terminates.

The clustering solution

With the clustering now completed, we can try to make some sense of the clusters the k-means algorithm produced.

For instance, we can examine how the observations included in each cluster vary on our variables. The density plot below is an example how we could go about exploring what the clusters represent.

We can clearly see what the clusters are made up of.

Cluster 1 holds most of the cars with low horsepower, and nearly all of those with low weights.

Cluster 2, in contrast, includes cars with horsepower ranging from low through high, and all cars are relatively heavy.

We could thus name our clusters respecitvively low horsepower and low weight cars, for cluster 1, and medium to high weight cars, for cluster 2.

Obviously, these clustering solutions will become more interesting as we add more clusters, and more variables to seperate our clusters and observations on.

A word of caution regarding k-means

Normalized input data

In our car clustering example, particularly the weight of cars seemed to be an important discriminating characteristic.

This is largely due to the fact that we didn’t normalize our data before running the k-means algorithm. I explicitly didn’t normalize (wiki) for simplicity sake and didactic purposes.

However, using k-means with raw data is actually a really easily-made but impactful mistake! Not normalizing data will cause k-means to put relative much importance on variables with a larger variance. Such as our variable weight. Cars’ weights ranged from a low 686 kg, to a high 2460 kg, thus spreading almost 1800 units. In contrast, horsepower ranged only from 52 hp to 335 hp, thus spreading less than 300 units.

If not normalized, this larger variation among weight values will cause the calculated Euclidean distance to be much more strongly affected by the weights of cars. Hence, these car weights will thus more strongly determine the final clustering solution simply because of their unit of measurement. In order to align the units of measurement for all variables, you should thus normalize your data before running k-means.

No categorical data

The need to normalize input data before running the k-means algorithm also touches on a second important characteristic of k-means: it does not handle categorical data.

Categorical data is discrete, and doesn’t have a natural origin. For instance, car color is a categorical variable. Cars can be blue, yellow, pink, or black. You can really calculate Euclidean distances for such data, at least not in a meaningful way. How far is blue from yellow, further or closer than it is from black?

Fortunately, there are variations of k-means that can handle categorical data, for instance, the k-modes algorithm.

No garuantees

The k-means algorithm does not guarantee to find the optimal solution. k-means is a fairly simple sequence of tasks and its clustering quality depends a lot on two factors.

First, the k specified by the user. In our example, we arbitrarily picked k = 2. This makes the algorithm seek specifically for solutions with two clusters, whereas maybe 3- or 4-cluster solutions would have made more sense.

Sometimes, users will have good theories to expect a certain number of clusters. At other times, they do not and are left to guess and experiment.

While there are methods to assess to some extent the statistically optimal number of clusters, often the decision for k will be somewhat subjective, though strongly affect the clustering solution..

Second, and more imporant maybe, is the influence of the random starting points for the initial cluster centers picked by the algorithm itself.

The clustering solutions that the algorithm produces are very sensitive to these initial conditions.

Due to its random starting points, it is also very likely that every time you run the k-means algorithm you will get different results, even on the same dataset.

To illustrate this, I ran the k-means visualization algorithm I wrote run a dozen of times.

Below is solution number 3 to which the k-means algorithm converged for our cars dataset. You can see that this is different from our earlier solution where the car at [1450; 270] belonged to the blue cluster 2, whereas here it is assigned to the red cluster 1.

Most k-means algorithms I ran on this dataset of cars resulted in approximately the same solution, like the one above and the one we saw before.

However, the k-means algorithm also produced some very different solutions. Like the one below, number 9. In this case cluster 2 was randomly initiated very muhc on the high end of the weight spectrum. As a result, the cluster 2 center remained all the way on the right side of the graph throughout the iterations of the algorithm.

Or this take this long-lasting iteration, where the k-means algorithm randomly located both cluster centers all the way in the bottom left corner, but fortunately recovered to the same solution we saw before.

One the one hand, the k-means algorithm sequences above illustrate the danger and downside of the k-means algorithm employing its random starting points. On the other hand, the mostly similar solutions produced even by the vanilla algorithm illustrate how the fantasticly simple algorithm is quite sturdy. Moreover, many smart improvements have fortunately been developed to avoid the random stupidity produced by the default algorithm — most notably the k-means++ algorithm (wiki). With these improvements, k-means continues to be one of the simplest though most popular and effective clustering algorithms out there!

Thanks for reading this blog!

While I intended to only share the link to the interactive visualization, I got carried away and ended up simulating the whole thing myself. Hopefully it wasn’t time wasted, and you learned a thing or two

Do reach out if you want to know more or are interested in the code to generate these simulations and visuals. Also, feel free to comment on, forward, or share any of the contents!

The cars dataset you can access in R by calling mtcars directly in your R console. Do explore it, as it contains many more variables on these 32 cars.

Some additional k-means resources

Here are some pages you can browse if you’re looking to learn more.