k-means and Kaboom!

Little boys are thrilled by the idea of big boats brandishing big guns.  Some big boys and girls are thrilled by the idea of using big tools on big data.  Let me try to put those two joys together!

The Story

In February 1940, the German Navy launched the Bismarck, one of the most famous battleships in history.  Immediately recognized as one of the most powerful battleships in the world, she went on to sink the British flagship HMS Hood and to single-handedly fight a British fleet.  The Bismarck’s sea battles launched a flotilla of booksmovies, and documentaries.

However, just two years prior to Bismarck’s launch, the German navy had launched a pair of twin sister warships, the Scharnhorst and the Gneisenau (pronounced in English something like “nice an now,” but with the hard “g” sound at the beginning.)  The proper categorization of these two warships has been long argued by naval architects and historians, and will continue to be argued for eternity by armchair admirals playing warship simulation games.  Today, we will look at the data and see how an important machine learning tool used for categorization addresses the question.

The unresolved question has been whether these two capital ships should be properly categorized as battleships or battlecruisers.  Battleships were large, heavily-armored big-gun ships capable of operating with a fleet and slugging it out with other similarly armed and armored warships.  They were designed to be the most powerful warships in existence, with significant focus on gun size.  Battlecruisers were a short-lived class of large specialty warships.  Though very similar in external appearance to battleships, they sacrificed some battleship capabilities in order to focus on speed.  In the days before radio, one of the primary purposes of battlecruisers was to independently seek out the enemy fleet, get close enough to determine the enemy ships’ identify, speed, direction, and disposition, and then rush back to report to the home team’s main fleet.  With this mission of rushing towards large enemy fleets with their guns blazing, battlecruisers were considered dashing ships with prestigious crews.

The crux of these two ships’ identity argument relates to these ship’s main gun size.  In an era when battleships were being built with 16-inch guns and the armor to survive 15-inch hits, these ships were fitted with 11-inch guns.  No battleship had been launched with 11-inch guns in 25 years.  It has been argued that these were special high-velocity, faster loading, more accurate 11-inch guns.  However, gun range is roughly related to projectile size.  Projectile impact kinetic energy is very related to projectile size.  Contemporary opponent battleships would have been expected to hit these ships at long range, and these two ships would not have been expected to be able to penetrate opponent battleships’ armor until they got within a suicidal close range.

In 1940 and since, the German position was that the Scharnhorst and the Gneisenau were battleships.  This argument focused on intentions.  With the advent of radio, radar, and aircraft, nobody in the 1930s intended to build any battlecruisers as fleet scouts.  The Germans did acknowledge that, primarily due to treaty restrictions, they had been forced to build “small” battleships with small guns.  This argument uses “appeal to authority” logic.  If the ships’ builders and owners categorized their ships as battleships, then who could question this?

The British position was that the Scharnhorst and the Gneisenau were actually battlecruisers.  This argument touched on intentions, but focused on results.   Since battleships were expected to slug it out with other contemporary battleships, there was no such thing as a “small battleship” with small guns.  (In appearance if not in function, the heavy cruiser class were the ships that usually resembled scaled-down battleships.)  Deploying a “small” battleship against a fleet of contemporary battleships with bigger guns would be like an NFL football team deploying a high-school player on the offensive line.  While a talented high-school player in the backfield might be effective, a “small center” deployed on the line of scrimmage would shortly be in the hospital.   The British pointed out that the stated purpose of these ships was commerce raiding and independently hunting down enemy cruisers.  These were actually among the secondary purposes of traditional battlecruisers.  The British pointed out that the design of these ships appeared to be based on the Mackensen, the last class of German true battlecruisers, built some 20 years before.  To me, the next point was the most interesting.  While acknowledging that building battlecruisers may have never been intended, if German battlecruiser design had continued and had followed the previous design “trajectory,” the results would have been ships similar to what was built in Scharnhorst and Gneisenau.  In other words, it doesn’t matter that you set out to build a goose.  If what you end up with looks like a swan, swims like a swan, and flies like a swan, then it is a swan.

The Data

Battleship design focused on achieving a balance between three main characteristics.  I have narrowed my analysis to these core measurements.  The three core aspects were hitting power (measured in gun diameter in inches,) protection (measured in armor thickness in inches) and speed (measured in nautical miles per hour.)

With balanced characteristics, battleships were expected to be able to slug it out with other battleships of similar class.  They were expected to be able to do damage to contemporary opponents, to absorb hits from their contemporary opponents, and have sufficient speed to be able bring their contemporary opponents to battle.

Battlecruisers were externally similar to their peer battleships, but their design characteristics were different.  The traditional British and Japanese battlecruiser design philosophy was to sacrifice armor for speed.  The traditional German battlecruiser design philosophy was to sacrifice gun size (and the significant weight of all the attendant gun turret mechanisms) for speed.

While there were several different armored structures on warships, for simplicity I have selected to focus on the size of the main armor feature, known as “belt armor.”  For simplicity, my analysis ignores all the other warship dimensional data, such as length, beam, draft, and number of guns.

I have collected, charted, and analyzed the data in an Excel workbook.  The workbook’s contents and format are deeply described in another posting.  The workbook is available for your inspection on on both GitHub and Google Drive.

The Ships

The battleship HMS Dreadnought (launched in 1906) revolutionized warship design and instantly obsoleted all other battleships then afloat.  I have excluded from analysis the “pre-dreadnought” class battleships built before 1906, as well as a couple of pre-dreadnought classes that it took France and Austria-Hungary a few years after 1906 to complete.  Warships were generally built in small numbers of identical “sister” ships.  For example, the US tended to build battleships in pairs, with one ship each being built in shipyards on each US coast.  Eventually, US battleships were built in fours.  The group of sister ships were called a “class,” usually named after the first sister completed.  My analysis is performed by ship class.

There is occasionally some small disagreement over the names of certain classes, most especially the class of sister ships under consideration here:  the Scharnhorst and the Gneisenau.  I am following the lead of my reference book in calling this the Gneisenau class because that ship was laid down first and commissioned first.

I have excluded from the analysis a small number of battleships from other countries.  The British and the Americans built a small number of battleships for regional powers, such as Greece, Spain, Argentina, and Brazil.  These tended to be based on older design patterns, or made design compromises that the producers would not entertain for their own ships.

I have excluded a number of specialty warships that were considered by their builders (and many historians) to be unusually large cruisers.  I have also excluded a small number of warships that completed construction as aircraft carriers.

The Visualizations

Here are the visualizations I prepared to chart the key points about these ships’ identity.  In all these charts, the gold data point represents the Gneisenau class as built.  The red points are not historical data but rather “alternative history.”  In some charts, the red point indicates data for the related German battlecruiser Mackensen, which was launched but never commissioned.  In other charts, the red point indicates alternate data for the Gneisenau class if the two ships had been delayed and built with the originally planned 15-inch guns.  In some charts, blue and purple points indicate data for other historical ships of particular interest and comparison.

New Battleship Gun Size

My first chart provides a timeline of main gun size for new battleship classes.  Each data point represents the first commissioning of a new ship in a new class.  The darker data points represent multiple new battleship classes being commissioned in the same year with the same main gun size.

BB Gun Capture 2


This timeline highlights the main issue with the Gneisenau class.  Their 11-inch guns make them the most visible data outlier for ships designated as battleships.  No new battleship classes with 11-inch guns have been commissioned in over 30 years.  Other navies have been producing 16-inch gun battleships for over 15 years.  These facts are the key contributors to the argument that this class did not represent a true battleship for its time.

The red point above shows what would have been the Gneisenau class data point if the ships could have been completed according to original plan with 15-inch guns by waiting one year.  Interestingly, waiting a year or two would have put the resulting data point very close to the global trend line for new battleship gun diameter.

German Battlecruiser Speed Timeline

My next chart provides a timeline of maximum speed for new German battlecruiser classes.

BC Speed Capture

This timeline highlights one dimension of the most interesting British claim about the Gneisenau class.  Key characteristics of the Gneisenau class closely align with an extrapolation over time of those same characteristics for the previous generations of German battlecruisers.  In other words:  whether intentional or not, in 1939 the German naval architects had built ships with characteristics that their own fathers would likely have contemplated for a future generation of  battlecruisers.  In this chart, that key characteristic is the most important capability of a battlecruiser:  speed.

The maximum speed of the Gneisenau class is consistent with trends in the maximum speed of German battlecruiser classes of the previous generations.  The pale blue trend line is mathematically calculated, but includes future data in calculating a trend in past data.  I have visually added the “gray black” trend line based purely upon the earlier historical battlecruiser data.

The red point above shows what would have been the maximum speed data point of the German battlecruiser Mackensen class if the ship could have been commissioned when planned.

German Battlecruiser Armor Timeline

My next chart provides a timeline of maximum belt armor thickness for new German battlecruiser classes.

BC Armor

This timeline highlights another dimension of that claim that the Gneisenau class had key characteristics that align with an extrapolation over time from the previous generations of German battlecruisers.  In this chart, that key characteristic is the one that historically most distinguished German battlecruisers from British ones:  armor.

The belt armor thickness of the Gneisenau class is roughly consistent with trends in the belt armor thickness of German battlecruiser classes of the previous generations.

The red point above shows what would have been the belt armor thickness data point of the Mackensen class if the ship could have been commissioned when planned.

German Battlecruiser Gun Size Timeline

My next chart provides a timeline of main gun size for new German battlecruiser classes. BC Gun Capture The gun size of the Gneisenau class is disruptive to both arguments about the correct classification of these ships.  Gun size complicates the argument that these are a natural evolution of previous German battlecruisers, because even 20 years previously German battlecruisers were built with significantly larger guns.  However, this chart reinforces the point that no capital ships of this era would be expected to carry such a small main gun.

The red point above shows the main gun size data point of the Mackensen class if the ship could have been commissioned when planned.

Contemporary Battleship Characteristics

The next group of three charts together provide a kind of orthographic view of the key characteristics of the contemporaries of the Gneisenau class.  These are the capital ships commissioned about a dozen years before or after Gneisenau (all designated as battleships by their builders.)

BB Contemp Capture

The gold points are the Gneisenau class.  The purple dots are the Japanese monster Yamato class.  The blue dots are the French Dunkerque class, of which there has been a similar but quieter discussion about proper classification.  The red dots represent what would have been the Gneisenau class data point if the ships had been completed later with the 15-inch guns.

To me, these visualizations tend to show most battleships of that era bunched together, with the Gneisenau class and the Dunkerque class as a “light” outlier and the Yamato class as a “heavy” outlier.  The exception is the Speed versus Belt Armor chart, which shows the Gneisenau class as a strong member of that central group and only shows the Dunkerque class as the “light” outlier.


Cluster Analysis

The favored method for grouping multi-dimensional data points into categories is k-means clustering.  This method enables the analyst to choose a number indicating into how many categories the objects should be grouped.  The mathematical method then “sorts” the objects into that many clusters, based upon the objects’ “similarity” as determined by their characteristic data.

Historians have tended to group the big-gun capital ships into some half-dozen categories, primarily based upon development history.  Using external tools, I performed k-means clustering operations on the ships’ gun size, armor thickness, and maximum speed.  For this clustering analysis, I used only real historical data for ships actually commissioned.  (No “alternate history” data was included.)

Prior to performing the clustering analysis, I prepared this three-chart orthographic view chart of the key characteristics of all the historical ships in order to help visualize any obvious clustering.

All Ships Capture

For my first analysis set, I made use of a k-means tool using the standard Lloyd’s algorithm.  I ran separate calculations using cluster counts between five and twelve.  The results were pretty clear and consistent:

  • In all cases, for all cluster counts between five and twelve, the analysis placed twelve of the last sixteen classes in one group.  (I have nicknamed this group of the most recent ships the “Modern” group.)  In all cases, the Gneisenau class is included in this group of battleships.  In all cases, the Hood and the Nagato class are included as the earliest ships in this group.
  • In all cases, the battlecruisers are clearly segregated from the battleships.  With five clusters, the British, Japanese, and German battlecruisers are all in one group.  It takes analysis with twelve clusters to separate the German battlecruisers from the British and Japanese battlecruisers (which prefer the big gun / lighter armor philosophy.) Interestingly, at twelve clusters, the two earliest British battlecruiser classes separate from the Japanese and later British battlecruisers.
  • Interestingly, in all cases, the French Dunkerque class is included in the battlecruiser group(s.)  There has been a similar but much quieter discussion about whether the ships of this class should be considered battleships or battlecruisers.


The Scharnhorst and Gneisenau both had illustrious service careers and engaged in famous sea actions.  The ships frequently sortied together, but found their ends separately.  The circumstances of their separate destruction highlighted the same identity question as at their construction.

In 1943, the decision was made to convert Gneisenau to the kind of battleship originally planned.  She had all her 11-inch guns and their turrets removed in preparation for installation of 15-inch guns in new turrets.  She endured much aerial bombing during the transformation, and the land war forced abandonment of the the conversion.  After an ignoble scuttling, she was scrapped.  One of the original turrets with the 11-inch guns survived, and can be seen today in a fort in Norway.

In December 1943, the British battleship HMS Duke of York and 13 escort ships ambushed the Scharnhorst off the coast of Norway.  The two capital ships exchanged heavy fire, mostly to the Scharnhorst’s detriment.  However, the ship was primarily sunk by torpedoes, one of the weapons that made all battleships obsolete.  The ship and the remains of her crew still rest on the bottom of the Barents Sea.

I welcome any thoughts or suggestions you might have on the data, the use of the Excel tool, the analysis, and the results.  Feel free to contact me directly.

(Image courtesy of United States Navy.)


One thought on “k-means and Kaboom!

  1. Pingback: Guns, Armor, Speed, and Steel | Carl Gusler

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