[AusRace] Engineering Analysis of Thoroughbred Racing
Tony Moffat
tonymoffat at bigpond.com
Sat Feb 2 18:40:27 AEDT 2019
I removed some lines from this -they were advertising
I corrected some grammar anomalies - from US of A to strine.
The inputs can be modified to use 200,400 and 600 metre times, and the
overall (of course) although the author uses the expression 'furlongs' (a
metre longer, then use
201 as a furlong,401,601)
'e' cannot be explained nor defined nor extrapolated and an early equation
remains unsolved because of this - in addition to the overuse of nor.
Otherwise a couple of reads and it comes to you.
I presume I have permission to post it here - it is again freely available
There is an explanation of 'e' at the end, read on
Engineering Analysis of Thoroughbred Racing BY RUBIN BOXER, RETIRED ENGINEER
Engineering Analysis of Thoroughbred Racing
Note: This engineering report, long treasured by those who make a living
betting on the horses, has rarely been available to the general public. For
contained within this twelve-page report, written using the objective
language of mathematics by retired engineer Rubin Boxer, is a solid,
scientific foundation to the art of horse racing. Those who take the time to
learn how to apply the methodology as laid out in this report will have a
decided advantage over their competition, other horse handicappers.
ABSTRACT:
While the process of assessing the capability of horses is, to a large
extent, an art, some aspects of how a horse actually races can be analyzed.
We show that by considering the energy used by a horse, it is possible to
derive "Capability Constants" which are inherent descriptors of the racing
horse. In a further derivation we obtain the horse's internal "friction".
We develop a way to account for changes to the weight carried by a horse.
In essence, this paper provides an engineering dimension to describe and
compare thoroughbred horses.
I-INTRODUCTION
The art of analyzing how a horse races is a fairly involved process. Some of
the factors we have to consider are:
The horse's class which measures the quality of horses in its races.
The horse's breeding which determines whether a horse is better suited for
long races (routes), short races (sprints), dirt or turf surfaces, and
whether the horse can be expected to run well on a muddy track.
The horse's sex and age. Generally, male horses will defeat females and
horses reach a peak and then decline with age.
The horse's physical fitness. How long has it been since its last race; has
it been working out regularly; has it been improving?
The trainer's competence.
The jockey's competence.
The weight carried by the horse, or more important, the change in this
weight since a previous race.
The horse's post position. Post positions are numbered from the track rail
outward. A particular position can cause a horse, among other things, to run
wide around turns adding to its race distance, or, to be blocked by other
horses.
The type of race the horse runs, e.g., does the horse set the pace by being
a front runner, or does the horse normally come from behind to vie for the
lead?
The condition of the track. Track maintenance is done daily and between
races. Each race itself affects the track. Wind, which, for convenience, we
also associate with a track, can change the outcome of a race. Track related
factors such as these cause racing performance to vary from day to day and
race to race.
The differences among tracks. Some tracks are inherently faster or slower
than others.
The horse's performance in previous races.
It may be presumptuous to apply the term "Engineering Analysis" to something
as flesh-and- blood and non-mechanical as a thoroughbred race horse. The
analysis, however, leads to valid results and provides terminology that
allows us to think about, and describe, horse racing in useful ways.
II-ANALYSIS
A. Capability Constants and Race Time.
Of all the factors noted above, we can argue that the most important
indicator of a horse's potential capability is its performance in previous
races.
We assert that a horse has a certain store of energy at the start of a race,
which is used as the race progresses.
A simple analogy is useful. When a weight is dragged along the ground,
energy is used in overcoming the friction between the weight and the ground.
The greater the friction, the greater the energy use.
The case of a racing horse is, of course, much more complicated. The
"friction" is caused by track condition effects, the contact between the
horse's hooves and ground, internal interactions of muscle and bone, and
other biological processes. We assume here that all factors can be included
in what we term "equivalent friction".
We also assume that the rate at which horses can draw on this energy reserve
varies. There is a maximum rate, which translates into maximum speed, and
lower rates that yield lesser speeds. Whatever the rate of energy use, it is
not sustainable indefinitely.
Horses can differ in their rates of energy expenditure. Consider sprint and
route races. In sprints the horses usually race at full effort during the
entire race. Speed near the start is at its maximum and then decreases. In
routes there is often a need to save energy at the start for a greater
effort at the end. In some instances, especially in turf races, horses will
increase speed during part of the race.
The concept of "equivalent friction" is defined by equation 1:
fe = uW (eq. 1)
where:
fe is equivalent friction force
u is friction coefficient
W is weight that the horse carries
We structure the analysis in terms of the more usual case where speed
decreases, by assuming that the decrease is proportional to the energy used
to overcome friction. (For convenience we consider constants of
proportionality and units to be included in u.)
Let the distance along the track from the race start be x. Then:
v(t) = Vm-uWx(t) (eq. 2)
where:
v(t) is velocity, (speed)
Vm is starting velocity
x(t) is distance
t is time
With v(t) = dx/dt, equation 2 is a first order differential equation:
dx/dt = Vm-uWx(t) (eq. 3)
Solving equation 3:
x(t) = (Vm/uW)(1-e-uWt) (eq. 4)
which gives the position of the horse at any specified time.
The horse's speed is:
v(t) = Ve-uWt (eq. 5)
To obtain useful results we need numerical values for the constants in
equations 4 and 5. If we let k=uW for convenience, equation 4 becomes:
x(t) = (Vm/k)(1-e-ekt) (eq. 6)
Equation 6 contains an exponential term which we approximate, using a series
expansion, to make the calculation of the constants easier. This leads to:
x(t)=(Vm/k)(1-(1-kt+k2t2/2+..)) (eq. 7)
Using the first terms of the series:
x(t) = Vmt - Vmkt2/2 (eq. 8)
Equation 8 has two unknowns, Vm and k. We can solve for them by using data
from a previous race. We need the position of the horse, x(t) at two times.
One approach is to use the data
at the first or second calls and at the end of the race. (The first and
second calls are the times, from the start of the race, that the leading
horse arrives at standard intermediate distances of the race.) The data,
provided by a number of publications, also show how far the horse we are
considering was behind the leading horse at the time of those calls. This
gives us both the time and position of the horse.
The distance behind the leading horse is given in "lengths". Supposedly,
this originated from the length of a horse itself. The data for lengths
behind are subjective, provided by humans who observe the race. Values of
about 10 or 11 feet have typically been used for a length; in this paper we
use 10.
Using lengths behind as an actual distance is more accurate than equating a
length to 1/5th second of time, as is often done, because the time to run
one length varies at different points in the race.
In equation 8, let z = Vmk, then:
t12z-2t1Vm = -2x1 (eq. 9)
where x1 and t1 are the position and time of the horse at the first call. A
second equation would be similar for either the second call or finish of the
race.
However, greater accuracy may be obtained, if we used more data, for
example, three points. These can be the first and second calls and the race
finish. Using finish time and position data an equation similar to equation
9, with variables of time and distance of tf and xf would be a third
equation.
This is a case where the number of data points and equations that we have,
3, is larger than the number of unknowns, 2. A way to deal with this is to
obtain a least mean squares fit. Only the results of that standard procedure
are given here.
We define:
A = t12+t12+tf2
B = t13+t23+tf3
C = t14+t24+tf4
D = t1x1+t2x2+tfxf
E = t12x1+t22x2+tf2xf (eq. 10)
The results are:
Vm = (BE-CD)/(B2-AC)
z = (2AE-2BD)/(B2-AC) (eq. 11)
With Vmand using k = z/Vm we have all the unknowns needed. We call Vm and k
the horse's capability constants because they are useful in describing how
the horse raced. More discussion of this is in Section III.
Having the capability constantswe can obtain the horse's time at any
distance in the race from the quadratic form of equation 8.
(Vm/2)(kt2)-Vm+x = 0 (eq. 12)
The solution of this quadratic is:
t = 1/k - (1/Vmk)(Vm2-2Vmkx)1/2 (eq. 13)
In particularif we enter xfthe length of a racewe obtain tfthe horse's
finish time. Next we derive a way to adjust for any change in weight that
the horse carries.
In the introduction we noted that the performance of a horse will be
affected by the weight it carries. In the United Stateswith the exception of
a few tracksas opposed tosayJapanthe weight of the horse itself is not
usually known. What we are given is the weight that the horse must carry.
B-Weight Change Effects
Each race has certain weight conditions that must be met by each horse
entered in the race. One number is provided which represents the weight of
the jockey plus any weights added to meet those specified conditions. We are
limited in our analysis to using the weight carriedwithout knowledge of the
weight of the horse itself. This may not be too much of a limitationsince
the actual weight of a fit horse remains fairly constant and a horse's
baseline capability might be associated with that weight. The fact that our
analysis yields results that agree with "rules of thumb" of trainers and
other experts lends some credence to this conjecture.
>From equation 4 we can obtain an expression for t(without resorting to the
series approximation that was used to simplify obtaining the capability
constants):
t = ln(1-kx/Vm)/-k (eq.14)
Differentiating to obtain dt/dk:
dt/dk=(x/k)/(Vm-kx)+(1/k2)(ln((Vm-kx)/(Vm)) (eq. 15)
We can obtain values for dt/dkfor different race lengthsxfusing equation 15.
To do this we need values for Vm and k. Although these varywe can choose
typical values by assuming that the weight results are not strongly
sensitive to such variance. We choose
Vm = 0.0934
k = 0.0025
where the units of the basic equation terms are furlongs for distance and
furlongs per second for velocity or speed.
Using x = 8 for an 8 furlong race equation 15 yields:
dt/dk = 5042 (eq. 16)
In horse racingit is standard to deal in time units of 0.2 seconds(1/5th of
a second).
Using 0.2 seconds for dt in equation 16 yields 3.97 x 10-5 for dk which is
1.59% of the typical 0.0025 value for k. Since k = uW and u the coefficient
of friction is assumed constant then whatever percentage change we obtain
for k must be caused by the same percentage change in weight carried.
A typical value for weight carried is 116 pounds. A 1.59% change in this
weight is 1.84 pounds. So for 8 furlong (1 mile) races, every change in
weight of 1.84 pounds will change the horse's race time by 0.2 seconds.
The one fifth of a second standard time interval is called a "tick". It's
useful to give the time change in terms of the number, or fraction, of ticks
for each pound of weight change. Expressed this way, the above result is
that for each pound of weight change in an 8 furlong race, the time a horse
will take to run the race will change by 0.54 ticks.
In Table 1, we show the same results, for a number of common race distances.
TABLE 1
Time Change per One Pound Weight Change Normalized to 116 pounds RACE
DISTANCE
(FURLONGS) TIME CHANGE
(TICKS)
5 0.187
5.5 0.231
6 0.280
6.5 0.336
7 0.398
7.5 0.467
8 0.543
8.5 0.627
9 0.719
10 0.930
12 1.479
15 2.716
16 3.276
(Ainsley[Reference1.] refers to a weight formula used by racing experts: At
sprint distances, (say 6 furlongs), four pounds slows a horse by 1/5th
second. Three pounds doesit at a mile, (8 furlongs), two pounds for a mile
and one eighth, (9 furlongs), and one pound for a mile and a quarter, (10
furlongs). Using the pound per tick format of Table I, Ainsley's numbers are
equivalent to 0.25,0.33,0.5, and 1.0 pounds at these distances.
Agreement is close to our results at 6 and 10 furlongs, and "ballpark" at
the other distances.)Our example uses 116 pounds as a typical weight. In the
table we retain this weight as a standard because the basic calculation
involves a percentage change from a particular weight. Other weights would
lead to different values. In practice this is not a problem, as we explain
in part of the discussion that follows.
III-DISCUSSION
A. Capability Constants Discussion.
We derived Vm and k and called them the capability constants of the horse.
The interpretation of Vm is straight forward; it's the speed of the horse
just after the start of the race. In some races this turns out to be
important because it yields some idea of which horses "set the pace"i.e.are
out in front at the start.
What interpretation can we put on k? Recall that we defined k = uW. This is
also the equivalent friction of equation 1. From equations 4 and 5we see
that k can also be thought of as a degradation factor since it appears as a
negative power of the exponential terms in those equations.
The larger the value of k, the faster the horse uses energy, and the faster
is its drop in speed. If k were to equal zero it would mean that the horse
ran at constant speed, while a negative value of k would result from a horse
that sped up.
We must exercise care when interpreting k. Equivalent friction was described
as being made up of components which depend on the surface condition of the
track and the physical condition of the horse. If the track condition is
such that it is termed "slow"meaning that it causes horses to tire easily, k
will be large. Without regard to track condition however, if a horse is not
in fit condition when it races, it can tire easily, also causing a large
value for k.
To evaluate k for the horse itself we need to separate the part of race
performance due to track condition from the part due to horse capability.
Since both are contained in equivalent friction, the equations alone do not
allow us to separate the effects.
One way to deal with this is to use the concept of track variant.
The need to separate horse capability from track condition effects is not a
problem unique to the engineering analysis given here. It is classical to
handicapping. As a result there are many sources of information available
which describe the condition of tracks for particular days and races.
Numbers are provided to adjust race times for track condition.Such numbers
are called track variants.
Track variants are derived by experts who account for race class and the way
races are run. They describe track condition in terms of these variants. If
we use these variants at the calls and finish of a race to adjust the times
at those points, then k will be based more on horse capability than on track
condition.
B. Race Time Discussion.
We showed that we can take data from a previous race to derive the
capability constants which can be used in equation 13 to obtain race time.
Of course we already know the race time of a previous race. But we have
accomplished two things. First, we have a method that analyzes races and
gives us the constants that describe how particular horses raced.Second, we
have tools that can be used to try to predict the outcome of future races.
Factors that make prediction difficult were touched on in Section I. Here we
show how to deal with some of them.
To compare horses we need to proceed with a sequence of steps. These steps
have both judgmental and analytical aspects to them, i.e., some art and some
science. Typically we start by selecting a previous race.
Let's say we are going to examine today's races. Publications like the,
"Daily Racing Form", provide us with data about each horse's previous races.
(Some races have horses that never raced before, so no race data is
available for them. For such horses we obviously cannot apply our methods
but must rely on clues from a horse's breeding and it's workouts.)
Since previous race data is fundamental to our approach, selecting that race
is all important. Some things to consider:
It's preferable to use races that were at today's distance, surface, and
class. If a race at today's distance is not available choose one close to
it.
Choose a recent race over an earlier one as an indicator of a horse's
present fitness. (Even this simple statement has a caveat. If a horse hasn't
raced in a "reasonable" time, then it might be better to choose the earlier
race because the horse may have needed the layoff and rest period to return
it to an earlier fitness. Recent good workouts may be a sign that this is
so.)
After choosing a previous race the next step is to use its data to obtain
the capability constants and an estimate of finish time. (The description of
how to do this, which follows, may seem tedious, but bear in mind that it is
easily accomplished, accurately, with a computer.
We adjust horse position at the calls and finish where:
x1a = x1-FL1
x2a = x2-FL2
xfa = xf-FLf (eq. 17)
x1ax2aand x2a are the call and finish positions after adjustment.
L1L2and Lf are the lengths behind at the calls and finish.
F = 1/66furlongs per length.
Earlier we showed how track variant and weight carried affect race data sowe
must take them into account.
A useful way is to "normalize" previous races. We adjust the race results so
that they represent horses carrying 116 pounds racing on a normal track. A
normal track is one where the horses race at "par"i.e.the race outcomes are
what we expect from horses of their class and racing style. (One measure of
a par track is a variant of zero in a method used by the publication"Today's
Racing Digest". In their system, variants are given in number of "ticks".
Positive numbers mean that the track was fast so they are added to finish
time while negative numbers mean the track was slow and are subtracted.) If
the previous race was at a different track, then we add a refining
adjustment to account for their speed differences. Some sources provide
variants for each of the calls. In our examples,we assume we have track
variant for finish time only.
Since both weight carried and track variant affect race time and are
measured in "ticks"we can deal with them together to normalize call and
finish times:
t1a = t1+.2(x1a/xfa)(p+(wf)(116-Wp))
t2a = t2+.2(x2a/xfa)(p+(wf)(116-Wp))
tfa = tf+.2(p+(wf)(116-Wp) (Eq. 18)
wheret1at2aand tfa are adjusted call and finish times p is track
variantwf is the weight factor in Table Iand Wp is the weight carried in the
previous race.
Equations 18 show that weight and variant time adjustments are distributed
over the track in proportion to the call and finish distances. It's correct
to expect that this is proper for weight. For exampleat the half mile point
of a mile racethe time change due to weight should be half that at the
finish. For varianthoweverthis may not be the case. Track conditions can
vary during the racee.g. a headwind can change to a tailwind. But in the
absence of variant numbers at the internal race callsthe assumption of a
proportional distribution seems reasonable.
Equations 17 and 18 provide us with all we need to apply equations 10, 11,
and 13 to obtain the capability constants and finish time. (We use the
adjusted time and position values in equations 10, 11, and 13.) If we make
these calculations for each horse in a race, we obtain finish times for all
so we have a predicted order of finish. This prediction is based on
normalized results. (Recall normalized results are for horses carrying 116
pounds racing on a par track.) In today's race, horses may be carrying
different weights and the track will probably not be at par. We can argue
that we need not concern ourselves with the value of "today's" track variant
because unless there are special conditions present,each horse racing over
the same track will be affected in the same way. The predicted finishing
order should not change.
We make the time adjustment due to weight:
.2(116-Wt)(wf)
where Wt is the weight carried today. Adding this time adjustment to each
horse leads to a "denormalized" finish order prediction for "today's" race.
Now and again a few further refinements are possible. The data of previous
races include qualitative comments such as:
"Four-wide into turn" "Wide early" "Broke slowly" "Boxed in."
These comments can be very important. The first two are examples of where a
horse may actually have run a longer race the last two are examples of a
horse that might have been able to run its race in a faster time.
If an entire race were run without turns a horse that ran wide would not be
at a disadvantage. But almost every race has at least one turn so horses
that run wide around a turn have to cover more ground and the extra distance
can be appreciable.
To see this consider the distance around one turn i.e. half a circle:
C = πr (eq. 19)
where C is the distance around the turn and r is the radius of the circle
that the rail horse travels around the turn.
(π of course is the ratio of a circle's circumference to its diameter
approximately equal to 3.1416.)
If the separation of horses racing side by side is then the distance
traveled by those racing wide around the turn is:
Cn = π(r+ns) (eq. 20)
where n takes on the values 1, 2, 3, etc., representing wide horse positions
starting with the horse next to the rail horse.
The extra distance α around a turn traveled by a non-rail horse compared to
the rail horse is the difference between Cn and C:
α = πns (eq. 21)
This equation of extra distance does not depend on circle radius which means
that it is correct regardless of the length of the turn.
If a conservative value for horse separation, s, is 3 feet then the extra
lengths around a turn is shown in Table II.
TABLE II
Extra Distance Around a Turn
POSITION FROM RAIL HORSE (NUMBER) EXTRA DISTANCE (LENGTH)
1 0.94
2 1.88
3 2.83
4 3.77
5 4.71
As we can see, the extra lengths a wide horse must travel can make a
difference in race outcome. If the separation between horses is greater than
the assumed 3 ft, as it can very well be, the extra distance traveled is
greater still. Further, if a horse goes wide early and stays there
throughout a two turn race, the extra distance traveled doubles.
C. Additional Discussion.
Other important facts about a race are available from our equations. For
example, once the capability constants are obtained, equation 5 can be used
to obtain a horse's speed at different times, which in turn means that the
ratio of speeds at the calls and finish can be calculated. These internal
call and finish values can be useful in evaluating a horse's form.
We showed that the normalizing process gave us information on how the horses
would have raced on a par track. In other words the effects of track variant
were eliminated. This means the capability constants refer more to the horse
itself than to track condition. To the extent this is true, we can obtain an
estimate of a horse's "equivalent friction", independent of the track.
Since k = uW:
ueh = k/W (Eq. 22)
where ueh is the equivalent friction coefficient of the horse itself. A low
value of ueh means that the horse ran efficiently and implies that the horse
was fit. A negative value may also imply that a horse was able to race
without reaching its limit on rate of energy use. Further study of this
conjecture is needed.
V-CONCLUSION
This paper develops tools to analyze thoroughbred race horses.
We should bear in mind that the non-mechanical nature of horses, and the
occurrence of the unexpected during a thoroughbred horse race, caution us
that no analytical method is infallible. In a statistical sense, however, we
expect the better horses to win, and a method that helps us identify such
horses is valuable. The engineering approach discussed in this paper does
that.
Reference:
1. Ainsley's Encyclopedia of Thoroughbred Handicapping Ainsley, Tom,
William Morrow &
Company, New York, 1978; Page 253
E = Most tracks are dirt, grass or synthetic. Race times vary from track to
track depending on the surfaces. Horses wear horseshoes to protect their
feet and increase the "grippiness" on the track. Similarly to running
wearing spikes the shoes allow horses to turn with more power at high speeds
without slipping. We will examine the coefficient of fiction on dirt, grass,
and synthetic tracks. The coefficient of friction is the "measure of the
amount of resistance that a surface exerts on or substances moving over it,
equal to the ratio between the maximal frictional force that the surface
exerts and the force pushing the object toward the surface."[2] Because
racehorses are in motion we use kinetic coefficient of friction. The average
coefficient of friction for grass is .35, for synthetic tracks (rubberish)
.68, and for dirt .35.[3] [4] The safest track is a synthetic track and then
grass and dirt tracks. This is important to know because jockeys have to be
careful during turns, slowing down, but on a synthetic track they do not
need to slow down as much because it is grippier.
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