Bootstrapping

In their paper "Bootstrapping the Battle of Britain" which was published in The Journal of Military History (84(1), 151-186), Fagan, BT, Horwood, I, MacKay, NJ, Price, C, Richards, E & Wood, AJ discussed using the statistical technique of weighted bootstrapping to model a hypothesis about the battle. As I have my own data set I have explored using boostrap simulations to investigate aspects of the battle.

What is Boostrapping?

Bootstrapping uses sampling with replacement to generate an alternative data set based on the original. For example, if we have 30 data points then to create an equivalent bootstrap sample we would randomly select one of those data points 30 times, picking from the full data set each time meaning each original data point could appear more than once (or not at all) in the bootstrap sample. This is what's meant by "replacement", each time we pick a data point we place it back into the sample pool we are drawing from. By running this sampling process a large number of times the aggregate bootstrap data sets should be representative of the original data set.

Block boostrapping

The Battle of Britain is typically broken up into phases (e.g. Kanalcampf, Adlerangriff, the Blitz) so for bootstrapping the whole data set can be broken down into sub-sets that correspond to these phases. For example, if we define Adlerangriff as lasting from 12 August to 17 September then we can create bootstrap data sets from this block to represent a hypothetical Adlerangriff.

Re-weighting

Using blocks enables the creation of "what if?" scenarios where we model what would happen if Adlerangriff had been started earlier and run for longer by creating bootstrap data sets that are weighted differently to the original.

Modelling

For my modelling I have taken the approach of using a starting value (S0) and walking through the data set applying the value for each day (Lt) to create a simulated run of the battle.

The value on each day is thus: St = S(t-1) + Lt

Backtesting

In order to test the validity of the model we need to run it against the historical data to check it at least approximates the historical results, if it doesn't resemble the historical data then it suggests that there's something wrong with our model or assumptions.

For example if we look at the classic example of R.A.F. fighter pilot strength we can use the historical values for this during the battle given in Appendix 11 of The Narrow Margin:

Date Strength
1940-07-06 1,259
1940-07-13 1,341
1940-07-20 1,365
1940-07-27 1,377
1940-08-03 1,434
1940-08-10 1,396
1940-08-17 1,379
1940-08-24 1,377
1940-08-31 1,422

To run our model we start with the value for S0 of 1,341 on 13 July 1940 (as that's the first day that is in both our data set and the historical values) then for each day in our data set we subtract the number of pilots lost or wounded (Lt) and add on the number of replacement pilots per day that Fighter Command was receiving (r) this gives us a figure for the R.A.F.'s pilot strength on each day:

St = S(t-1) - Lt + r

Before creating our bootstrapped data sets we run this on our data set in chronological order. As r is not a figure for which I have a source, runs were done with different values to establish a best fit.

Bootstrappd R.A.F. pilot strength (simulated with replacement rate)

As the shape of these graphs matches the recorded pilot numbers it indicates that the model is valid and a value for r of six provides a reasonable fit for this data set.

This value of r is not to suggest that this was the actual pilot replacement value but merely a value that is required to make our model "work" - it's a fudge factor.

(There is a notable divergence for 31 August - notably one of the days where R.A.F. single day losses were very high - but, as that's the limit of my data set, I'm ignoring it for now. That is something that will need more data to investigate but my initial guess is that the value for r will need to change at this point if the model is to follow the historical data.)

Testing

Now we have some confidence in our model it's time to put the bootstrap to work in order to investigate some aspects of the battle.

Initially it's helpful to understand what the bootstrap simulation does. For the backtest we used the values of Lt in chronological order. For the boostrapped model we create new, randomly selected, sequences of days to run through the model and then then average (rounded mean) the values for St from each day of all the runs to give us a sequence of values. Unless otherwise stated, the simulations here are using 10,000 runs.

If we treat the period we're examining as a single phase the boostrapped result gives us a straight line between the starting strength and the ending strength. This intuitively makes sense as we're taking random samples of each day and averaging them out so we would expect it to produce a straight line average across all days.

Bootstrappd R.A.F. pilot strength (single phase)

I take two things away from this: The first is that viewing this data as a single phase isn't particularly helpful as the simulated result isn't a good match for the shape of historical data. The second is that picking the start and end point of phases is going to be very important as the result of that phase will be the average of the start and end points.

Phasing

The Battle of Britain is typically viewed as series of distinct phases with those established in 1943 by the official R.A.F. history of the battle being the most common definition.

Dates Phase
1940-07-13 - 1940-08-07 Phase I: Channel Battles
1940-08-08 - 1940-08-18 Phase II
1940-08-19 - 1940-08-23 Pause
1940-08-24 - 1940-09-06 Phase III
1940-09-07 - 1940-10-21 Phase IV

Running a bootstrap using these phases gives a good fit for the model.

Bootstrappd R.A.F. pilot strength (R.A.F. phase asignment)

Whilst the R.A.F. phases match up well with R.A.F. pilot numbers they don't match up with the phases of strategy employed by the Luftwaffe which are outlined as follows in "To Defeat the Few":

Dates Phase
1940-07-13 - 1940-08-10 Kanalcampf
1940-08-12 - 1940-08-16 Adlerangriff stage 1
1940-08-18 - 1940-08-22 Adlerangriff stage 2
1940-08-24 - 1940-09-06 Adlerangriff stage 3
1940-09-07 - 1940-09-17 Adlerangriff stage 4

This ends on 17 September as this was when Seelöwe was officially cancelled so, whilst fighting continued, it marks the end of the Luftwaffe's attempt to obtain air superiority for an invasion.

Notably the following dates are not assigned to a phase: 1940-08-11, 1940-08-17 & 1940-08-23. For the purposes of modelling these have been assigned to a "Pause" phase which is used for these three dates in the sequence.

Running a bootstrap for these phases doesn't give as good a fit as the R.A.F. phases - although it's still in the ballpark of the historically recorded figures.

Bootstrappd R.A.F. pilot strength (Luftwaffe phase asignment)

Digging deeper into this there is one day that stands out as having a disproportionate effect on the result: 11 August. This is not assigned to a phase but has a high number of pilot losses (Lt) and looking at the events of that day there was a large attack from Luftflotten 3 on Portland docks so, in terms of tactics employed, it's probably a better fit for Kanalcampf. Running a bootstrap for this revised phasing gives a better fit for the first and third stages of Adlerangriff.

Bootstrappd R.A.F. pilot strength (Revised Kanalcampf asignment)

There is still one outlier here and that's 18th August which, with attacks on satellite airfields, is more closely aligned to stage two of Adlerangriff. Assigning the two remaining "pause" phases to the nearest applicable stage gives us a phasing that is similar to the R.A.F. assignment.

Dates Phase
1940-07-13 - 1940-08-11 Kanalcampf
1940-08-12 - 1940-08-18 Adlerangriff stage 1
1940-08-19 - 1940-08-23 Adlerangriff stage 2
1940-08-24 - 1940-09-06 Adlerangriff stage 3
1940-09-07 - 1940-09-17 Adlerangriff stage 4

Running a bootstrap for this gives a good fit for the stages of Luftwaffe operations whilst matching their strategic phasing.

Bootstrappd R.A.F. pilot strength (Fitted Adlerangriff phase asignment)

This now gives us a reasonable basis for examining alternate scenarios of Luftwaffe strategy.

Radar plots appear

As the dawn mist burns off, radar picks up aircraft over Cap Griz Nez. (6+ over Cap Griz Nez and 3+ in the Straights.)

54 Sqn (Spitfires) are scrambled from Manston to deal with fighters and 610 Sqn (Spitfires) are scrambled from Hawkinge to deal with bombers.

The 6+ raid are Do 17 from KG 2 targeting Lympne.

15 Do 17 Bomb Lympne

15 Do 17s from I/KG 2 escorted by JG 54 come in low over Romney Marsh and rise up over the escarpment and drop 50kg bombs which explode in straight lines, the landing ground is cratered and a hanger is damaged.

Crews report dropping 90 110-lb bombs on hangars and buildings, although they also state that the full effect of their attack could not be gauged due to mist.

54 Sqn Spitfires are scrambled from Manston but are intercepted by JG 54's Bf 109s out of the sun - possibly over Dover - before they can reach the bombers. Two Spitfires force land at Lympne and are forced to swerve to avoid the craters.

All the Do 17s return home 45 minutes later, without radar Fighter Command are unable to find them.

This raid is intended to keep Lympne based interceptors on the ground and not interfere with the attacks on the radar stations.

OPERATIONS RECORD BOOK of (Unit or Formation) R.A.F. STATION, LYMPNE.

Place Date Time Summary of Events References to Appendices
LYMPNE. AUGUST
12TH.
Weather. Fair. 0800 HRS. STATION attacked by enemy aircraft. 141 Bombs dropped in a few seconds. Damage caused to hangers, offices and landing ground. NO CASUALTIES.

EprGr 210 attack four radar stations

Visibility is perfect with Dungeness being visible from Dover.

The Bf 110s of EprGr 210 lead by Swiss born Hptm Rübensdörffer come in at low level to attack the radar network. Lacking azimuth tracking and confusing range information by passing abeam the radar sites, the Filter Room at Fighter Command assigns the plot an 'X' code (doubtful origin).

Oberleutnant Otto Hintze leads 3 Staffel's eight Bf 109E-1/B Jabos - carrying one SC250 bomb each - attacking Dover. They approach the Dover radar at right angles to make them harder to detect but are picked up by the Rye Radar. The towers are slightly damaged and huts are smashed. Hintze reports seeing the towers clearly swaying.

Two minutes later Lieutenant Rössinger leads 2 Staffel's Bf 110s attacking Rye. Almost every building is hit, except the transmitting and receiving block. The Filter Room repeatedly tries to contact them eventually being told by a WAFF that "Your X raid is bombing us."

Hauptman Rübensdörffer leads the attack on Dunkirk. One 500kg bomb moves the concrete transmitter building several inches.

The final element of 1 Staffel's Bf 110s hit Pevensy with eight 500kg bombs, one of which cuts the power line taking the station off line.

A breach of 160 km has been opened in the radar coverage.

II/KG 2 bomb Canterbury airfield

54 Sqn and 56 Sqn Hurricanes are vectored towards the 24 Do 17s of II/KG 2, lead by Kommodore Oberst Johannes Fink, who it is assessed are heading for Manston after crossing the Channel from Calais to North Forland. The 30 Bf 109 escorts of I and III/JG 54 fail to see the camoflaged Hurricanes climbing to intercept the bombers.

The bombers throw out confetti, toilet paper and handgrenades to confuse the attacking fighters.

The ensuing combat between the fighters continues back across the Channel with 32 Sqn and 64 Sqn joining in.

Oberleutnant Albert Dressel of III/JG 54's Stabsschwarm belly lands his Bf 109 near Margate and is taken prisoner.

Luftwaffe records state that ‘Canterbury airfield’ and another reported as being west of Manston received a total of five 500kg, four 250kg and 439 50kg bombs. The bombs fall around Bekesbourne and Patrixbourne.

Bekesbourne was a WWI RAF airfield and had been used in May and June 1940 by Lysanders of 2 and 13 Sqns. supporting Operation Dynamo. They were widthdrawn on 8 June and the aifield obstructed by stakes to prevent it being used by invading forces.

Sorces: Airfields of Britain Conservation Trust

Ramsgate airfield bombed

20 He 111 (or 30+ Do 17) drop 13 tons of bombs onto Ramsgate's civilian airfield, a quater of the estimated 210 bombs fall on the airfield - the rest causing extensive damage to the town.

151 Sqn. (Hurricanes) intercept the Bf 109 escorts from III./JG 26 on the right flank over Manston and 610 Sqn. (Spirfires) engage the left flank escorts of III./(J)LG 2 over Ramsgate. The bombers are not engaged.

"To Defeat the Few" says it was He 111 but the "Battle of Britain Combat Archive" says Do 17s.

I./KG 3 bomb Sandgate auxiliary airfield

A '9+' group of 12 I./KG 3 Do 17s, lead by Obstlt Rudolf Gabelmann, covered by '20+' fighters of II./JG 51 pass east of Dover and turn inland south of Deal. 616 Sqn. (Spitfires), led by Sqn. Ldr. Marcus Robinson, are scrambled from Biggen Hill to intercept them but they turn south and dive towards the coast unloading 12.67 tons of bombs meant for the Sandagte auxillary airfield (now a parade ground) near Shorncliffe Redoubt but fail to hit anything of military significance. They cross back over the Channel before 616 Sqn. can catch them.

OKW releases ships allocated to invasion

The Armed Forces High Command
WFSt/Abt. L 1 Nr. 33294/40 g.Kdos.Chefs.
Berlin
19 Sep. 1940
Eight Copies
Top secret

Re: Operation "Seelöwe"

The Führer and Supreme Commander of the Armed Forces has made the following decisions:

  1. Concentration of the transport fleet is to be stopped, insofar as it has not already been completed.
  2. The ships assembled in the ports of departure are to be dispersed in such a way that losses from enemy air attacks will be reduced to a minimum. However, it should remain possible to reassemble the ships at the ports of departure on 10 days notice under favorable weather conditions.
  3. The 10 steamers from the Norway traffic earmarked for operation "Herbstreise"* should be returned to their normal tasks. Six other steamers earmarked for operation "Seelöwe" should be withdrawn over a period of time unnoticeably, and should be used until further notice to speed up supplies for group XXI.
  4. The Commander in Chief, Air is to increase the Air defenses in the invasion ports to the greatest possible extent.

The Chief of Staff, Armed Forces High Command

signed: Keitel

* Landing on the coast of Scotland planned as a diversionary operation in connection with "Seelöwe". Four empty ocean liners and 11 transport steamers heading towards Britain between Newcastle and Aberdeen whist the heavy cruiser Admiral Hipper attempted to break out into the Atlantic via Iceland and the Faroes.