The following series of posting to the derby list were first done in 1998.    The posting were done

to show if it is possible to model the bounce in horse racing.  I have not done any further work in this

area since 1998.  Larry Wellman


Subject: Modeling the Bounce

I've come across a few articles from the Journal of Applied 
Physiology that might help in explaining the reason for the bounce or 
reduction in performance.  The articles are relative to modeling 
performance based on the quantity of training (impulse) and the 
duration of the training.  The models have two components and one is 
relative to fatigue.  I've gone through the articles and plan on 
trying to apply the method using past performance data of race horses.
  One point that might be of interest is a term that relates the time 
needed to recover from the introduction of a training stimulus before 
the effects of fatigue are dissipated.  This time constant varies 
based on the existing conditioning level of the individual.  For 
example if two horses had the same reduction in time from one race 
too the next then the horse that was the fast would take longer to 
recover from the race.  We have two horse.  Horse A runs 1:15 in race 
number one and returns to run 1:14 in his next start.  He reduce his 
race time by one second out of 75 total sec from race one.  If the 
faster Horse B ran in 1:12 and then returned in 1:11 in the next race.
His recover time would be longer then Horse B.   Another point of 
interest is that when a new training program is started it takes 
almost 50 days of training at a constant work load before performance 
improvements exceed the initial performance condition.   However if 
the training is stopped after about two weeks the horse performance 
level will exceed the initial condition within the next two weeks.  
The performance curves show that the peak occurs around 16 days after 
continous training is halted.  This time interval is similar to what 
some trainers do once a horse is racing fit they do very little 
training between races.
I'll post more about theses article as I start to fully understand 
them and how to apply them to the race horse.

Subject: Modeling The Bounce:Part II

Hey guys.  I did not mean to start any flame wars with my earlier 
post about modeling the bounce.  I came across a few article that 
have implications about how performance changes based on work 
load/training.  I have gone over the articles a few more times and 
still plan on a few more times to completely understand how to model 
using past performance data.
Here's some more info.  In one article two of the researchers 
subjected themselves to a 28 day training program to test the 
algorithm for modeling there performance based on the training loads. 
After the 28 days they stop all training except for test runs at a 
standard distance (called criterion performance test).  One of the 
researcher was a runner while the other was not in any training 
before the experiment.  Each runners perfomance showed the same shape.
  During the 28 days of training the actual criterion runs dropped  
in value.  Similar to a lower Beyer.  After the training stopped they 
continued to run crtierion runs with the maximum performance 
occurring around day 50-60 into the test.  Almost 2 weeks after 
stopping the training.  The range of performance varied about plus or 
minus 15 percent about the initial condition.   Relative to a horse:  
Lets say a horse has an initial Beyer of 80 (Claimer 10-14K), he 
would drop to a low of 68 within 2-3 weeks and then return to 80 in 
another 2-3 weeks and then peak at 92 in another two weeks if the 
training was stopped after 30 days.  If training was continued his 
value might be in the mid 80's range.  I'm just giving an example of 
the trends and these do not represent an individual horse.
I'll post more later.



Modeling The Bounce: Part III (Technical)


Here's some more information relative to modeling

performance.  Performance is modeled based on two

components:  fitness and fatigue.  Both components

have an exponental form and are based on the amount of

training undertaken before a performance event.  In

horse racing the event would be a workout or a race.

We can also use a workout or a race as additional training

input for projecting the performance level for the next event.


The performance algorithm is of the following form:


p(t)= k1*g(t)-k2*h(t)  were g(t) and h(t) are fitness and

fatigue as a function of time and are based on previous

training. t-is time (days)


g(t) = g(t-i)*exp(-i/t1) + w(t)

h(t) = h(t-i)*exp(-i/t2) + w(t)

were w(t) is the amount of training undertaken, while i is the

intervening period between training days


The terms k1, k2, t1, and t2 are weighting factors and

time constants in days.  Experimental studies have

determined t1 ranging from 30 to 60 days, while values for t2

fall in the range of 2 to 15 days.  The four terms determine

the following factor called tn.  This term represents the time

from onset of training to the day of relative poorest performance.

I posted the number tn=16 days in one of my previous posts. 

The tn value has a range from 9 to 23 days.  The tn values appears

to be higher for the elite performer.


So how does this relate to horse racing?  Each horse just like each

individuals has different constants for his fitness and fatigue

equations.  These terms can change based on the current fitness level

of the horse.  Based on the values of k1,k2,t1, and t2 a horse can

benefit or lose fitness based on the spacing of the training events

relative to a performance event (race).  Using tn of 16 days means that

any work or race within 16 days of the next race actual hurts the horses

performance in his next start.  Races going back up to 40 days or more

actual help the the performance.  The Triple Crown series represents

an interesting test for a trainer because of the spacing of the races

relative to the 16 days.



Modeling the Bounce:  Part IV Example


Boxcar ask for a real world example on the algorithm

that I posted in my last post.  I've been working

on an example that I will share.  I will use Beyer numbers

in the example.  I found a horse who's past performance

record showed continous running with no large breaks

in his record and his races were all within a half furlong

of each other.  The horse I used is Marie's Topgun from the

DRF issue of 31 January 1998 who ran at Oaklawn.  I use

four races to predict the next start using the four constant

and the days between starts.  The first three starts were

at Lone Star with the remaining at LaD.  All races on a fast dry



BSFPro= R1*(K1*exp(tp-t1)/tau1-k2*exp(tp-t1)/tau2)



       +R4*(....)  last two are the same as the first two


BSFPro is the projected Beyer Speed Figure

R1 represents the last race Beyer, R2 the next to last and so on

tp represents todays date, t1 thru t4 represent the dates of the

last four starts ( converted to a number)


k1,k2, tau1, tau2 are the constants for the horse.  I adjusted the

constants using a least squared error for all races that I projected

a BSFPro.  The constants fell within the ranges given for humans.

I did not calculate the tn values, however tn falls around 11-16 days.


1997   Race    Race

Date   Dist    BSF    BSFPro

9 Nov   51/2   89     81.4

1 Nov   6      90     90.9

17 Oct  61/2   77     95.5 <In this race the horse was one length

27 Sept 51/2   75     79.3  off at the stretch call running 1:10.4

18 Sept 6      81     88.2  for 6f while finishing 3 behind the winner

31 Aug  6      77     75.0

21 Aug  6      68     Horse was claimed out of this race

24 July 6      81           

12 July 6      87

4 July  61/2   51


I did not include any workouts as additional training input or

daily gallops.  Plus I did not adjust the performance criterion

for changes in distance.  To do this correctly each training

input (race) would be normalized to a standard which would include

distance and relative level of effort (pace).  The Beyer number is a relative

standard, however distance or pace are not included.  A horse that

runs 6f would get a lower score when compared to a horse that ran

9f.  The level of effort would also be included.  Actually I think

the projection is pretty good.  Remember the BSF has an error of

around plus/minus 3-4 points or more.


Is this what you wanted to see Boxcar??


Modleing the Bounce: Part V



Boxcar presented his case about the example I posted as the iron

horse (Marie's Topgun).  I selected this example because all the

races were run near the same distance.  If you look back at the

example you will see that the horse  had a drop in performance on

9/27 and the projected Beyer also showed the drop.  I have three more

example I will present.  All are from the same DRF. 

Remember I use the first four races to begin the projection.  I have

also shown the tn value and a new term called tg.  Tn is the number

of days after a race that contribute to fatigue.  The tg value is the

number of days before an event where the race will maximize

performance.  If you pull out the DRF from that day you will notice

that the better the horse the higher the tn and tg values.  I am

using the BFS as the criterion perfomance for these example with no

changes for distance or pace.  In Quiet Hunts races he ran near the

front when routing before finishing up the track.  If I had used only

6f rating I think we would see better agreement.



Marie's Topgun:  tn = 8.9 days, tg= 28.7 days


Quiet Hunt (same race as Marie's)  tn=5.2 days tg=22.2 days

Date     Dist       BSF     BSFPro

11/9     1'70       57       67.8

10/25    6           70       61.7

10/2      6           53       45.2

9/26      6           67       49    <- New trainer

9/6        7           43       49.2

8/22     11/16     34       45.2   <- sloppy track

8/10     11/16     48

7/17      7           61

6/13     11/16     46

5/3       11/16     66


King Roller (Aqu)  tn= 10.6 days, tg= 32.2

Date      Dist       BSF     BSFPro

1/18/98  6          104     117

12/27     6            97      93.9  

12/14     6          104      103.2

11/27     6          107      104.5

11/8      7            89        91.5  <- sloppy track

10/24    61/2       96        93

 10/5     6           101

9/11      6           100

8/14      61/2      104

7/30      7             85


Laredo (Aqu), tg= 9.6 days, tg= 30.6 days

Date      Dist       BSF      BSFPro

12/27     6          111       119.7   

12/6       6          104       107.3

11/8       6          102         97.1

10/10     6           101        84.1

9/13       6            68

8/3         6            76

7/11       6            96

6/15       6            88