Introduction

               Plotting the results of the Head of the River Race on a chart with 3 axes for the percentage of the Course Time spent in each of the 3 main sections - Start to Barnes, Barnes to Hammersmith, Hammersmith to Finish - shows that there is always a small cluster of the best crews at one spot.  This spot is defined as the "Hot Spot" and its location varies from year to year.    The location of every other crew indicates how well they performed relative to the best.  Thus the chart is referred to as a Performance Chart.

               In some years the basic figures do not conform to the same pattern, and it is shown that the departure is time related - probably a function of the wind - and that its effect can be calculated for each crew.  When the basic figures are adjusted by this amount, all the figures for the 5 years considered follow the same basic pattern.  In fact, there is an adjustment for every year, but for some years it is small.  Where there are significant adjustments the resultant figures may not be quite as precise as when they are small,  but there is a clear pattern indicative of performance relative to the best, and in which section(s) of the course any particular crew may have underperformed.

General Description - 2000

       .  An example of a Performance Chart for the Head of the River Race is the First 25 for 2000.   Across the top is the percentage for Start to Barnes - down the left side is that for Barnes to Hammersmith - and down the right is the Hammersmith to Finish scale.  Note that Barnes to Hammersmith increases towards the bottom of the chart, Hammersmith to Finish increases towards the top of the chart.  Each place is plotted where it comes from the crews figures.  Where two ( or more ) places would overlap the first is printed as white in a black rectangle, and the overlap noted at the bottom of the chart.   Thus in First 25 for 2000 1 is in this format, with a note at the bottom that 3 is the overlaping crew.   Thus this chart shows that 1, 3 and 4 were nearly coincident.  It is found that there is always a point in the First 25 where there is a cluster of the most successful crews, and this is clearly a good definition of the best combination of percentages for the particular year, and is called The Hot Spot.  Where there are places outside the range of the chart, the fact is noted at the bottom - in this case 10 and 12 were off the top of the chart, and 11 off to one side.

          First 25 for 2000 has another point of great interest - the crew placed second is well away from the Hot Spot - on the right hand side just beside the 39.1 figure for the Hammersmith to Finish scale.  The crew second in 2000 was Leander - the winner was Queens Tower.  The location of the 2 says that Leander were about 0.3% faster from Start to Barnes than the other 3 in the first 4.  17 mins is a little over 1000 secs, so 0.1% is about 1 sec for the fastest crews - about 1.5 secs for those further down the order.  Thus Leander were about 3 secs faster than the other 3 for the first leg.  For Barnes to Hammersmith, they were about 0.7% ( 7 secs ) slower, and then about 0.3% ( 3 secs ) faster from Hammersmith to the Finish.  The other results on this chart indicate that these were unusual figures, and hence it would seem that Leander probably lost the first place rather than Queens Tower winning it !!

Detailed Consideration - 2001

            The First 25 for 2001 is fairly similar to that for 2000, but with The Hot Spot selection only defined by the First Crew, although there are a number of other places fairly close which make this selection really the only possible.  The Hot Spot is manually selected - if any reader would prefer a different selection there is no reason he shouldn't make the relevant alterations - having the Hot Spot defined makes comparisons with charts for other groups of crews simple. 

            The remaining places are grouped as 26 to 99, 100 to 199, 200 to 299, 300 to the end.  This grouping is chosen to avoid too much clutter, and for the same reason the hundreds digit is usually omitted ( places in the 400's are the exception ) - the omitted digit is flagged up in the title area at the top of the chart.  All the Performance Charts have a red, and a green line on them.  The green line is the fit for all the crews in the race - the red one is for the crews on this chart.  The lines generally run roughly along a nearly constant Start to Barnes line, and hence the centre of the points for both cases has been calculated, and indicated by a short bar across the main line.  Generally the green line is not far off the red one, and this is true of the bars as well, indicating that this chart is a fairly representative sample of the whole the whole set of results.

             This discussion will concentrate on the 100 to 199 group, but the others are similar, and can be accessed if desired.  Looking at 100 to 199 for 2001, it is fairly typical for any of these groupings, with the bulk of places just below the Hot Spot, and extending down around about 1% on the Hammersmith to Finish scale.  Places off the top, centre and bottom have been discussed above, as have the black square indicating an overlap.  It is worth noting at this point the indications of a crew "blowing up" - it will have a larger than normal Hammersmith to Finish percentage, with the other 2 legs smaller - which means that it will be in the top right corner of the chart.  Clearly there is some sort of boundary to this area - it is not known at present whether it is sloping, or precipitous, or what is its exact location.  On the whole there are not many crews with results above the Hot Spot and even less in this particular corner, but any in this area should be considered from this point of view, although they may well be below this ( as yet undefined ) boundary.

The next simple year - 2002

              The First 25 for 2002 is next for consideration.  In this case the Hot Spot has been placed between nos 1, 2 and 3, with another 5 of the first 25 quite close - a better definition than 2001.

              Considering next the 100 to 199 for 2002, there is not a great deal to be said about it other than it is fairly similar to the 2001 chart. In general the fact that all these charts are fairly similar, and that the fit lines indicate there is nothing very special about any of them, strongly suggest that there is only one set of percentages for the whole race.  Clearly this varies from year to year.   These charts can be produced for Status groupings, or Trophy groupings, as well as the numerical grouping of finish places, but all are fairly similar in principle.   The Senior 4 for 2002 is produced here as a sample.   The actual places for the later crews do become slightly more scattered, but there is not a lot more to say about them.  This further adds to the conclusion above that all the charts for a particular year are fairly similar, and indicates that there are fairly narrow bands of percentages for each leg, and the Hot Spot is roughly central to the Start to Barnes range, and at the point where the minimum Barnes to Hammersmith percentage for most crews cuts this Start to Barnes figure,  This endorses the view that many people hold that the Barnes to Hammersmith leg is the crucial one.

The External Influence - 2005

             Passing on, the next item to consider is First 25 for 2005.  This has a Hot Spot with 1 and 2 virtually coincident, and 3,4,8 and 23 quite close.  There is nothing else that justifies comment.

             The next chart to consider is 100 to 199 for 2005,  and here there is a big departure from the previous pattern - the bulk of the data has moved down the chart by about 0.6%, and a good deal of it is off the chart at the bottom.  In order to understand this, and how to adjust for it, consider Initial Barnes to Hammersmith fit for 2005.  This is an example of a different type of chart - across the top is a scale of percentages for a particular leg, in this case Barnes to Hammersmith, and down the side is a scale of the Start time.  The individual points are plotted as a crew number - note crew number here, and place on Performance Charts - and it is immediately apparant that there are significant changes to the percentage and that they are generally time related.  There is a good deal of scatter, but the general trend can easily be found by fitting a curve to the points by least squares, as has been done on this chart.

             There can be little doubt that there is a consistent ( but varying ) effect due to some external influence, of which the most likely is changes in the wind.  If the Performance Chart for 2005 is to be comparable with those for 2001 and 2002, there clearly needs to be an adjustment to allow for this external influence.  The line which has been fitted goes through the center of the data, and it is clear that, below about 25 mins, for any time there is a good range of data, which must represent different standards of crew from good to not so good, and the line can be taken as representative of an average crew.  However, above, say, 20 mins there is not the same spread of crews, which is to be expected since the crews are mostly good, and thus the top part of the line can be said to be an average for good crews.

              Whilst this situation of an average line for good crews becoming a line for average crews is not satisfactory from the point of view of quantified the effect, it is worth noting that, for the part of the line referring to good crews, if the percentage for the start time of crew 1 given by the line is subtracted from the similar percentage for some other crew, the resultant difference can be considered a good enumeration of the relative effect of this change on the two crews.  This can be subtracted ( this in a mathematical sense - the correction may be negative resulting in an addition ) from the second crews percentage, which will adjust it to remove the effect of this change of conditions relative to the first crew.

               In order to produce a satisfactory position for the average crews reference should be made to Barnes to Hammersmith fit for 2005.  The crews with a result in the first 90 are plotted in red, and the red line has been fitted to these points.  The black line is the fit for all the rest.  The black line is now a reasonable definition of the effect on average crews, and the red line is its counterpart for the good crews.  Clearly adjustments can be calculated for good crews from the red line as outlined above, but using the black line for average crews in the same way needs a further adjustment for the difference between good crews ( including crew 1 ) and average crews.

               The red line is only meaningful for the good crews at the top of the chart, and becomes meaningless further down the chart where there are only a few, or no, points, and, similarily, the black line is without meaning at the top of the chart.  There is, however, a zone where both lines are meaningful, the two lines initially converging, and then diverging.  The percentage indicated by the minimum difference is taken as the adjustment necessary to convert from an average crew to a good one.  The time for this point is referred to as the "Swap Time".  Thus crews starting before the swap time have their adjustment relative to crew 1 calculated from the red line, and those starting after the swap time are calculated from the black line, with an additional adjustment for those after the swap time of the difference at the swap time.  It is freely admitted that there are several assumptions in this procedure which do not have any very good theoretical basis, but reference to Adjusted 100 to 199 for 2005 does show that this chart is now very similar to the charts for 2001 and 2002.  An additioal piece of information on the "fit" charts is that the Hot Spot is shown as a magenta line.

               Reference to Hammersmith to Finish fit for 2005 shows it is similar in principle to the Barnes to Hammersmith one discussed above, and the swap time is approximately the same.  The Start to Barnes fit for 2005 requires little comment - whilst there is a small variation, it is nothing like as big as for the other two legs. By inspection, the Swap Time chosen from the other 2 legs fits reasonably, and it is clearly possible to deduce alterations for this leg..

Adjusted Performance Curves - 2005

               Using the principles discussed above, a set of adjustments for 2005 can be generated.   The Adjusted Top 25 for 2005 shows that the general effect is to tighten the grouping around the Hot Spot.  As mentioned above the Adjusted 100 to 199 for 2005 is similar to its counterpart for 2001 and 2002.  The number of points above the Hot Spot on the low Start to Barnes side may be slightly more than might be expected, but with this proviso the resultant chart is quite believable.   These adjusted charts should be treated with some care as they are dependant on the assumptions made - they are clearly much better than without the adjustment, but it could be that they are slightly high on Hammersmith to Finish, low on Barnes to Hammersmith figures.  The total percentages for the 3 legs do sum to very nearly 100% on all the various steps in the procedure, and a minor correction is included to make the summations exactly 100%.

               The weaknesses of these adjustments are that the choice of 90 crews as fast, and the difference between fast and average being the minimum separation of the two lines, are arbitary, although they are reasonable  However they do not have any theoretical basis.  Thus any deductions from the Adjusted charts with regard to the performance of a particular crew relative to the best, whilst clearly of the right order, may be slightly wrong in detail.  The indications are that the adjustments may place crews slightly nearer the top of the charts than they would be if a rigorous analysis were possible.  The adjusted figures are clearly most reasonable for crews away from the Hot Spot, but their validity is more open to question nearer that ideal.

Reconsideration - 2001 and 2002

                In the light of the discussion for 2005, the charts for 2001 can be reconsidered.   In order to get the curves giving a satisfactory point for the swap time, it has been necessary to extend the definition of good crews from 90 to 150 for this particular year.   The Barnes to Hammersmith fit for 2001 shows that the variations are quite small, and this is borne out by the Hammersmith to Finish fit for 2001.  Likewise the Start to Barnes fit for 2001 is not very interesting.  The Adjusted Top 25 for 2001 shows, as might be expected, only small variations from the unadjusted version and the same is true of the Adjusted 100 to 199 for 2001.

                 The Barnes to Hammersmith fit for 2002 shows a little bit more variation than the 2001 version, and this is borne out in the Hammersmith to Finish fit for 2002.  Again Start to Barnes for 2002 shows little of interest.  The Adjusted Top 25 for 2002 shows a bit more variation than for 2001, but no major changes.   Comparing the 100 to 199 for 2002 with its counterpart for 2001 does show the main block of results slightly further down the Hammersmith to Finish scale, and this is corrected in the Adjusted 100 to 199 for 2002.  The Adjusted Senior 4 for 2002 shows the same slight differences as the unadjusted version.

A more complex case - 2000

                  So far there has been only a trivial mention of 2000 in the opening paragraph, and none of 2003, although data is available.  It has already been mentioned that the First 25 for 2000 is reasonably typical, with a Hot Spot defined by 1, 3 and 4, and 2, Leander, in another meadow.  Looking at 100 to 199 for 2000, this is not too dissimilar to its counterparts for 2001 and 2002 ( 100 to 199 for 2001, 100 to 199 for 2002 ).  However 300 to End for 2000 is very different to 300 to End for 2002, in that the approximate rectangle of Places has dropped well down the chart.  300 to End for 2001 is in agreement with that for 2002.

                  Before the data for 2005 became available this was somewhat puzzling, and there was some doubt about what should be expected.  In the light of the 2005 data, it becomes clear that this behaviour is caused by the variation of external conditions discussed above for 2005.    As for 2001, it has proved necessary to extend the definition of good crews from 90 to 150 for this year to get a satisfactory swap time.  The Barnes to Hammersmith fit for 2000 immediately confirms not only that the variation of external conditions is present, but it applies primarily to the second part of the race.  The Hammersmith to Finish fit for 2000 confirms this, and Start to Barnes fit for 2000 again shows little of interest.  In the light of this, the Adjusted 300 to End for 2000 can be produced, and the anomaly can be seen to have been removed.  The Adjusted 300 to End for 2001 and the Adjusted 300 to End for 2002 can be compared to verify this conclusion.  Adjusted First 25 for 2000 and Adjusted 100 to 199 for 2000 show, as might be expected, only minor variations from the unadjusted versions.

The most complex year - 2003

                  2003 is a rather more complicated - the Top 25 for 2003 does not show the same pattern as for the other 4 years, and this, as for the 2000 data, was somewhat puzzling before the 2005 data became available.  Indeed with 2001 and 2002 presenting a consistent pattern, and 2000 and 2003 not agreeing, there was considerable doubt about exactly what should be expected, which problem was clarified by the 2005 data.

                  Inspection of the Start to Barnes fit for 2003 shows that it has rather more variation than its counterparts for the other 4 years, and the Barnes to Hammersmith fit for 2003 and the Hammersmith to Finish fit for 2003 show a definate trend in the first 40 minutes in the reverse direction to the others.  In the light of the experience of the 2005 data, where an increasing Barnes to Hammersmith percentage with time caused the points on the performance charts to move downwards away from the hot spot, the higher numbered crews in the Top 25 for 2003 can be seen to have moved towards the top of the chart.  The 100 to 199 for 2003 shows this trend to a greater extent.  The Adjusted  Top 25 for 2003 shows that the adjustment process has more or less corrected this tendancy, and the Adjusted 100 to 199 for 2003 is very near to what has come to be expected.

                 In the 2003 race, No 300 - Mortlake Anglian and Alpha II, a S.3 crew, started with  No 301 - St Paul's School II, also a S.3 crew, overlapping, and the two crews finished overlapping, having maintained approximately the same relative positions over the whole course.  Mortlake Anglian and Alpha II finished 95, and St Pauls School II finished 93, and both crews can be found fairly near the Hot Spot.  St Peter's School York, a S.4 crew, overtook 7 crews to finish 99, and they also can be found near the Hot Spot.   See Adjusted 26 to 99 for 2003 for these results.

Further Comments

                  However this is not all the story with 2003.  The fits have been deliberately done using low order polynomials in order to pick up the overall trends, without getting too involved in the detail.  The program is capable of fitting two different polynomials to a set of data.   The Special Hammersmith to Finish fit for 2002 shows fits for 2 different polynomials - the 4th and 5th - for those unfamiliar with polynomial fitting this simply means that the 5th order has extra freedom to fit curves in the data.  The 4th order is in black and red as explained above.  In the 5th the red for the good crews is replaced by green, and the black for average crews by the blue line.  It is clear there is virtually no difference between the 2 sets of lines, and hence the 4th is an adequate representation of the data.

                 Looking at the Special Barnes to Hammersmith fit for 2003, there is clearly a difference between the black and blue, and between the red and green, and the fact that the blue and green have more curves says that there is greater variation in the external influence.  Reference to the Barnes to Hammersmith fit for 2003 shows that it is the 5th order polynomial which has been used to establish the adjustments.  It seems probable that the external influence is the wind, and the conclusion is that in 2003 it was more variable.  This suggests that the charts for 2003 may contain more external effects on individual crews than for other years.   With this proviso, the 2003 charts generally follow the same general conclusions as the others discussed above.

                 This is a suitable place to say that the 4th order polynomial has proved satisfactory for 2000, 2002 and 2005.  The 5th order has only proved necessary for 2001 and 2003.  Just for the record the swap times are defined by a particular crew number as follows:

                 The Special Hammersmith to Finish fit for 2002 shows another aspect of information which can be obtained by fitting curves to the data in various ways.  In this case the data has been broken into 2 sections - crews above and below 44 - and there is clearly a definate pattern to the good crews in the first 44.  Similar effects can be found in other sets of data for the various years.  As previously mentioned, the fitting has been confined to the lower order polynomials in order to identify the overall effects, and adjust the data to identify and remove, as far as possible, the external effects which have influenced the overall patterns.  There are clearly other patterns in the data, particularily for the fast crews in the first part of the race, which can be identified.  Identifying such patterns further down in the order is likely to be more difficult because of the mix of faster and more average crews.