Speed conversion

[cosmicway]

Horses do not always run the same distance.
If you are looking into a 7 furlong racecard and you want to study the form of a particular horse, you will probably find that it's last six races were something like 5f, 5f, 8f, 7f, 6f, 7f.
So what do you do ? You look at the speed of the 4th and the 6th ? You can do that but it's archaic. The form of the horse is more likely in the last two, 5f and 5f.
To work out speed figures the experts work independent of distance but other experts are all the time looking for more exact methods.
Suppose the horse we are looking at run the last race at 57.00 secs (it was a 5 furlong race). Then the 7 furlongs equivalent is -according to my conversion formula- 87.56 secs or 1.27.56. Forget about track variants, weight, track to track conversions, or suppose we 've done those.

The problem I am posing to you is what about the sectional times ? Early speed and finish. A typical early speed figure for a horse running 57.00 over 5 furlongs is 23.00 sec / 2 furlongs from the gate and a typical figure for the late fraction is 25.50 sec / the 2 last furlongs. But those may vary.
Many people think that if two horses run 5 furlongs with the same speed, 57.00 sec wire to wire, then the less speedy type (at the gate) has the edge if they run 7 furlongs. Not always of course, but in the majority of circumstances.
Is this true ?
Has anyone worked out such formulas ?

[cj]

If you ever figure it out, don't tell a soul, quit your job, and start raking in the dough.

[Overlay]

To work out speed figures the experts work independent of distance but other experts are all the time looking for more exact methods.
That's true, but eventually the point of diminishing returns is reached, as far as whether all that extra effort is compensated for by the marginal benefit derived from it.

I personally don't think that it's necessary to get down in the weeds like that on distance. Supplementing a basic speed measure with other metrics that shed light on condition and pace can work just as well, and can also provide a means of judging not only which horse is most likely to finish first in any given race, but also whether that horse (or any other horse in the race) is worth a bet at its posted odds. To me, that is really the question to be addressed, as long as making money (as opposed to picking winners) is the ultimate objective.

[Overlay]

As an added thought: Another way to achieve the same objective would be by rating the effectiveness of a figure in predicting the outcomes of races in different distance categories (such as sprints versus routes, or whatever other scheme you might choose), rather than trying to incorporate distance into the formulation of the figure itself.

[gm10]

Horses do not always run the same distance.
If you are looking into a 7 furlong racecard and you want to study the form of a particular horse, you will probably find that it's last six races were something like 5f, 5f, 8f, 7f, 6f, 7f.
So what do you do ? You look at the speed of the 4th and the 6th ? You can do that but it's archaic. The form of the horse is more likely in the last two, 5f and 5f.
To work out speed figures the experts work independent of distance but other experts are all the time looking for more exact methods.
Suppose the horse we are looking at run the last race at 57.00 secs (it was a 5 furlong race). Then the 7 furlongs equivalent is -according to my conversion formula- 87.56 secs or 1.27.56. Forget about track variants, weight, track to track conversions, or suppose we 've done those.

The problem I am posing to you is what about the sectional times ? Early speed and finish. A typical early speed figure for a horse running 57.00 over 5 furlongs is 23.00 sec / 2 furlongs from the gate and a typical figure for the late fraction is 25.50 sec / the 2 last furlongs. But those may vary.

That is one speedy middle furlong.


Many people think that if two horses run 5 furlongs with the same speed, 57.00 sec wire to wire, then the less speedy type (at the gate) has the edge if they run 7 furlongs. Not always of course, but in the majority of circumstances.
Is this true ?
Has anyone worked out such formulas ?

I calculate early/middle/late for each horse, and then project those onto today's circumstances. That projection takes into account things like course & distance profile. Say we're looking at a 6.5F @ Turf Paradise, and while keeping it simple, I would calculate a (simplified)

Expected figure = 0.40 * early (in previous races) + 0.35 * middle + 0.25 * late.

The parameters (0.4, 0.35, 0.25) are determined by linear regression based on past races.

The highest Expected Figure wins about 34-35% of races, which is better than using last time out speed figures (they find around 30% winners, see Beyer's book).

[cosmicway]

I calculate early/middle/late for each horse, and then project those onto today's circumstances. That projection takes into account things like course & distance profile. Say we're looking at a 6.5F @ Turf Paradise, and while keeping it simple, I would calculate a (simplified)

Expected figure = 0.40 * early (in previous races) + 0.35 * middle + 0.25 * late.

The parameters (0.4, 0.35, 0.25) are determined by linear regression based on past races.

The highest Expected Figure wins about 34-35% of races, which is better than using last time out speed figures (they find around 30% winners, see Beyer's book).


So that's 5f from 6 1/2 f, is it ?
It's a probable candidate but the real question is, is that better than
< Expected figure = some constant x total > ?
Have both been checked ?
Also I don't use linear regression unless I have to.

There is something I somewhat don't like. "Late" depends on "early" does n't it ? After all when we see the clock going mad in the early stages we use to say "the stayers are having it", don't we ?

[cosmicway]

If you ever figure it out, don't tell a soul, quit your job, and start raking in the dough.

The old cynic philosopher in you ...
So take down the forums as well or talk about broads only from now on.
Then the secrets -if any- stay buried !
Your comment is not without a certain logic though.
It always scares me what will happen if I ever find a pirate treasure in the Carribean. How do I raise a crew ? What to do to keep inland revenue away from it ?

[Light]

If you are looking into a 7 furlong racecard and you want to study the form of a particular horse, you will probably find that it's last six races were something like 5f, 5f, 8f, 7f, 6f, 7f.
So what do you do ? You look at the speed of the 4th and the 6th ? You can do that but it's archaic. The form of the horse is more likely in the last two, 5f and 5f.


I disagree about what PL the form of a horse is in because form can disappear or re appear in any given race. In cases of sprints and routes, I generally use a sprint or route PL according to today's distance. If they are embedded races like your example,you will get nice enough prices to offset the use of more recent races.

In this specific case you presented,(the 2f difference of 5f and 7f is very significant) if the horse has run those two 7f races in the last 3 or even 4 months,and they are better figs than his 5f races,I'd go with the 7f races. The horse is telling you he prefers that distance. If the opposite is true,that the 7f races are significantly worse than the 5f races,I'd use the 5f races with reservation.

[cosmicway]

I disagree about what PL the form of a horse is in because form can disappear or re appear in any given race. In cases of sprints and routes, I generally use a sprint or route PL according to today's distance. If they are embedded races like your example,you will get nice enough prices to offset the use of more recent races.

In this specific case you presented,(the 2f difference of 5f and 7f is very significant) if the horse has run those two 7f races in the last 3 or even 4 months,and they are better figs than his 5f races,I'd go with the 7f races. The horse is telling you he prefers that distance. If the opposite is true,that the 7f races are significantly worse than the 5f races,I'd use the 5f races with reservation.

The problem, when looked upon in it's entirety, is quite complex indeed.
We are approaching it piecewise, so that's why I talk of the distance-to-distance conversion formula.

Ultimately, when after a 7f race, the 5f races prior to it have to be downgraded in significance, quite indeed. Use a "relevence coefficient" or something like that, if you are following a math model. To compute such it is advisable to use a stochastic criterion rather than least squares, because with least squares you eat dust.

[gm10]

So that's 5f from 6 1/2 f, is it ?
It's a probable candidate but the real question is, is that better than
< Expected figure = some constant x total > ?
Have both been checked ?
Also I don't use linear regression unless I have to.

There is something I somewhat don't like. "Late" depends on "early" does n't it ? After all when we see the clock going mad in the early stages we use to say "the stayers are having it", don't we ?

I guess they are dependent, but I don't see that as a problem.

Yes I have checked ... results follow

Period: January -> September 2011
Surface: Dirt
Going: Fast
Days since last runs (each horse): <= 45
Number of Lifetime runs (each horse): >= 1
Edge that the best rated horse has: >= 1 length

Best 'composite' Expected Figure: strike rate 34.1%
Best 'final time' Expected Figure: strike rate 32.5%
Best figure last time out: strike rate 31.1%

The method described earlier finds 34% winners. Using the same averaging of past races, but using only final times finds 32.5%. The basic "last time out" approach finds 31% winners (which is nicely above the 'Beyer benchmark' of 30%).

[Light]

The problem, when looked upon in it's entirety, is quite complex indeed.
We are approaching it piecewise, so that's why I talk of the distance-to-distance conversion formula.

Ultimately, when after a 7f race, the 5f races prior to it have to be downgraded in significance, quite indeed. Use a "relevence coefficient" or something like that, if you are following a math model. To compute such it is advisable to use a stochastic criterion rather than least squares, because with least squares you eat dust.

The problem with your theory is horses are not mathematical equations.

[cosmicway]

The problem with your theory is horses are not mathematical equations.

Everything is a mathematical equation.

[dansan]

if you see a horse or horses that have won at that distance 7furlongs bet him or them that simple

[Light]

Everything is a mathematical equation.

The reason mathematics will not work for a 5f race to a 7f race (in a lot of cases) is:

since physiological distribution of energy requirements are different for the two distances, some horses are just not comfortable with the physiological demands of one distance or another even though their mathematical formula says they are. Can you put a mathematical formula on the psychological and physiological comfort a horse feels in a race?

That being said, Beyer has a distance to distance equivalency chart in the back of his book "Beyer on Speed".

If you want sectional times based on class pars you can look at any Bris files for any given race.

If you want a personal formula for fractional conversions,try P eming "Trifecta Mike". The guru of horse racing mathematics on this board might be able to help you. But I have never understood his math.

[gm10]

The reason mathematics will not work for a 5f race to a 7f race (in a lot of cases) is:

since physiological distribution of energy requirements are different for the two distances, some horses are just not comfortable with the physiological demands of one distance or another even though their mathematical formula says they are. Can you put a mathematical formula on the psychological and physiological comfort a horse feels in a race?

That being said, Beyer has a distance to distance equivalency chart in the back of his book "Beyer on Speed".

If you want sectional times based on class pars you can look at any Bris files for any given race.

If you want a personal formula for fractional conversions,try P eming "Trifecta Mike". The guru of horse racing mathematics on this board might be able to help you. But I have never understood his math.

Actually, there is at least one very powerful mathematical model available for the physiological side (Bob Wilkinson). The lack of data can be a problem, however.

[Light]

Can you present this "model" for us to look at ?

[cosmicway]

You can never tell what will happen if a horse is upped in distance.
Most of the celebrated sprinters are disgraced when tried at 15f. Some of the stayers prove capable of that far but I don't remember a single sprinter.
You can't be a sprinter and have stamina as well. It's got to be a rare exception, the one who makes it in the long routes.
But most animals are good for up to the mile say. If your horse is good at 5-6f and can't run the mile, it's a problem horse really and you got to complain from where you bought it !

All the formulas are approximate, and they are better than none at all.
I 'll try the one suggested before by gm10 and see what happens. I don't remember having done this before. It requires 3x3 determinants to work out the coefficients and I don't remember having computed determinants before.

[thaskalos]

You can never tell what will happen if a horse is upped in distance.

You lost me here...

You made the above quote (which I agree with wholeheartedly), but you disagreed with the poster Light earlier...when he suggested that you would have a difficult time "equalizing" the 5 furlong and the 7 furlong distances by using a math model. When Light stated that "horses are not mathematical equations"...you replied that "everything is a mathematical equation".

If you really think that horses are "mathematical equations"...why do you now state that "You can never tell what will happen if a horse is upped in distance"?

In my opinion...it is the highest folly to rate a horse off of a 5 furlong race, when today's race is 7 furlongs (or 6 furlongs for that matter).

It is very common for a sprinter to register a much higher speed figure in a 5 furlong race than in a 6 or 7 furlong race.

Trust the 5 furlong speed ratings at your own risk...

[Red Knave]

You can never tell what will happen if a horse is upped in distance.

Actually you can in the example you gave. This horse has two 5F races and two 7F races. You have what you need for this horse. How does he convert from 5 to 7F? If you have enough of these examples for this track/class/sex you really have something.

All the formulas are approximate, and they are better than none at all.
Bingo.

[cosmicway]

Thaskalos means teacher. One might be tempted to make it "daskalos", in an attempt to anglicize it. But the "th" in the beginning of English words is not usually a delta, but a theta, as in the "pith of the Danaides". Examples of theta pronounciation are, theophany, thaw, theocracy, thanksgiving, etc. Examples of delta pronounciation are only the pronouns this-that-those-them etc. So I think it should be daskalos.

Anyway I just hope gm10's method does n't turn out to give some negative coefficients when I feed the data to the computer. If it does I have to declare it r.i.p..

But of course I don't know what happens when horses are upped in distance.
I told you a regular 6-7-8 furlonger going up to 15f (the longest we have in Greece) is a flop until proven otherwise.
A hose from 5f-6f going up to 7f is likely to perform, or his chest has a problem.

When I compute a likely value for 7f from 5f I give it a lower coefficient of relevence, in the histogram of the horse's history.
But I got to have a conversion formula to begin with.
I have such formulas but I 'm reorganising them now.

Those methods are of course, by the nature of things, applying to one race course. To analyse UK tracks, French tracks, USA tracks someone who has the data has to sit down and do the job.
But you should also know that communication with the various racing authorities sometimes proves difficult. Also the products commercially available in the market tend to be expensive.

[thaskalos]

Thaskalos means teacher. One might be tempted to make it "daskalos", in an attempt to anglicize it. But the "th" in the beginning of English words is not usually a delta, but a theta, as in the "pith of the Danaides". Examples of theta pronounciation are, theophany, thaw, theocracy, thanksgiving, etc. Examples of delta pronounciation are only the pronouns this-that-those-them etc. So I think it should be daskalos.

My friend, you have misinterpreted my intentions here...

By your petty criticism of the spelling of my username, you show me that you thought that I was mocking you with my post.

I was not.

I was just trying to point out an inconsistency in your reasoning...which you chose not to comment on.

Instead...you chose to to focus on the spelling of "thaskalos"...as a way of trying to prove that I am unfit to call myself a "teacher" because you think that I lack your grammatical expertise...

I admire your sharp wit. I hope you are putting it to good use...:ThmbUp:

[cosmicway]

I did n't say you are not a real teacher or that you broke grammar rules (I made up the rule in this case).

Also I tried to go over the irregularities you spotted as best as I could. Those irregularities exist because the approach one makes to such problems is an iterative one. There is no closed solution. Arguably there is no solution at all. One is only trying to improve the catch rates a bit.

[gm10]

Can you present this "model" for us to look at ?

Ah no. It's way too complicated. I recommend you read the book, it's really good.

http://www.highstakes.co.uk/shop/product.php/12888/0/

'It describes a scientific study of competitive running and develops a mathematical model which balances the energy supply from both anaerobic and aerobic sources with the energy required to accelerate the body, sustain running, and overcome air resistance.

When applied to horse racing it allows the relationships between distance, time, weight carried, going, and other factors, to be evaluated. The model is applied to racing on turf in Britain, but it is easily adapted to racing on other surfaces and tracks.

The result of the model is a Power Equation, which can be used to assess performance in a race in terms of a power rating. Two methods of assessing performance are examined in detail, based on race time, or on collateral form.. Examples are given of the calculation of time ratings (speed ratings) and form ratings.

This book is not about "how to pick winners" or racing "systems". It is about the link between equine exercise physiology and racehorse ratings. A basic under-standing of mathematics is required to follow the development of the model.'

[cosmicway]

That is one speedy middle furlong.



I calculate early/middle/late for each horse, and then project those onto today's circumstances. That projection takes into account things like course & distance profile. Say we're looking at a 6.5F @ Turf Paradise, and while keeping it simple, I would calculate a (simplified)

Expected figure = 0.40 * early (in previous races) + 0.35 * middle + 0.25 * late.

The parameters (0.4, 0.35, 0.25) are determined by linear regression based on past races.

The highest Expected Figure wins about 34-35% of races, which is better than using last time out speed figures (they find around 30% winners, see Beyer's book).


There is a paradox in this.
You say formula = A x early + B x middle + C x late
where A,B,C are constants to be detremined by least squares fitting.

But middle = total - early - late, so what you really say is:

formula = A' x total + B' x early + C' x late

if [ distance from > distance to ] then we should have B' > 0, C' < 0

and

if [ distance from < distance to ] then we should have B' < 0, C' > 0

for obvious reasons.

But the least squares won't always oblige.
Is there a fly in the ointment ?

[Handiman]

Didn't Carrol come up with a way to create distance adjusted speed figures in his book on speed?


Handi:)

[gm10]

There is a paradox in this.
You say formula = A x early + B x middle + C x late
where A,B,C are constants to be detremined by least squares fitting.

But middle = total - early - late, so what you really say is:

formula = A' x total + B' x early + C' x late

if [ distance from > distance to ] then we should have B' > 0, C' < 0

and

if [ distance from < distance to ] then we should have B' < 0, C' > 0

for obvious reasons.

But the least squares won't always oblige.
Is there a fly in the ointment ?

I'm not really following to be honest. I can't reveal every detail of my personal approach, but to keep it practical and intuitive, you need to set up your regression so that A, B, C are positive. You may also have to normalize so that they sum up to 1. The idea is to break down a horse's ability and see how compatible it is with today's circumstances.

[TrifectaMike]

I'm not really following to be honest. I can't reveal every detail of my personal approach, but to keep it practical and intuitive, you need to set up your regression so that A, B, C are positive. You may also have to normalize so that they sum up to 1. The idea is to break down a horse's ability and see how compatible it is with today's circumstances.
gm10,

I believe he is saying that the independent variables for the regression are collinear. (I think).

Mike (Dr Beav)

[gm10]

gm10,

I believe he is saying that the independent variables for the regression are collinear. (I think).

Mike (Dr Beav)

Yes, but I'm not too concerned about it. The objective is to find values of a, b, c that are more predictive than simply setting a = b = c= 1/3. You can also just iterate over different (a,b,c) combinations, for example 0.34, 0.32, 0.33 -> 0.35, 0.31, 0.33 -> 0.36, 0.30, 0.33 etc etc, and see which combination worked best at a certain course, surface, distance, going. I just use linear regression as a shortcut, I guess. In the past I used a multinomial logit model, but at some point I switched, I can't even remember why!

[cosmicway]

Yes, but I'm not too concerned about it. The objective is to find values of a, b, c that are more predictive than simply setting a = b = c= 1/3. You can also just iterate over different (a,b,c) combinations, for example 0.34, 0.32, 0.33 -> 0.35, 0.31, 0.33 -> 0.36, 0.30, 0.33 etc etc, and see which combination worked best at a certain course, surface, distance, going. I just use linear regression as a shortcut, I guess. In the past I used a multinomial logit model, but at some point I switched, I can't even remember why!

Your model is linear, hence can be solved by calculus.
With trial and error also, it's the same thing - only slower.
To go from 5f to 7f I worked out that A = 1.4273, B = 1.7598, C = 1.5453, using my data (quite a lot, some thousands).
In this case A < B checks, but C is < B.

Suppose we have early = 22, middle = 10, late = 24, the call points being at the 2 furlong pole, 3 furlong pole respectively. The total is 56.
The equivalent 7f speed is:

1.4273 x 22 + 1.7598 x 10 + 1.5453 x 24 = 86.09 = 1.26.09 (reasonable)

If I make that into 22, 11, 23 it gives me 56 again and 86.30 (also reasonable at first look).
That is, the horse was refreshed a little and finished better than before in the 5f, but the equivalent 7f time is bigger. I 'm not 100% sure that was well handled.

* We gotta tackle this I believe. Otherwise why cry for sectionals. Only to check the extreme cases, like the slow starters ?

[pondman]

The problem I am posing to you is what about the sectional times ?

Do it the old fashion way and convert it to feet per second. And then work on theories of change in velocity, between the half mile and the finish. You can still make money on middle to late runnings in S. California with this, because everyone and their brothers, mothers, and aunts, are hooked on beyers number. I guess that's what you pace guys must do.

[cosmicway]

Do it the old fashion way and convert it to feet per second. And then work on theories of change in velocity, between the half mile and the finish. You can still make money on middle to late runnings in S. California with this, because everyone and their brothers, mothers, and aunts, are hooked on beyers number. I guess that's what you pace guys must do.

I 'm investigating something now, along the previous lines.
If I do convert to velocity units what does change ?
It's something like U2 = U1 - f1 x t1 say. This works out the deceleration f1 in the early part.
Then U3 = U2 - f2 x t2. This works out the deceleration or acceleration in the middle part.
Then what ?
I remember dimly somewhere a talk about velocity units.

Anyway that's only the first part of the investigation. You gotta make histogram of past performences and calibrate it to give maximum probability. Then you can talk of Beyer-like figures (or losing to Beyer as the case may be).

[pondman]

. This works out the deceleration or acceleration in the middle part.
Then what ?
I remember dimly somewhere a talk about velocity units.


Break you fraction in the fps. Try and get as many results as possible for the track you are playing, and come up with your own theory as to front, middle, or final velocity. The change in velocity is going to be the difference between the fractions. I'm sure there are many people on here doing this. They probably use a slope. This still is an okay method for a number of S. California Races. It helps for 3/4 mile at GG. You'll find most horses are decelerating. But ever few month you'd find a horse which could run a 60 fps in it's final and the crowd would completely miss it, because they are using beyer's. You can set a floor for you final fraction as a filter. At one time it was 50 feet per second (final Fraction) on most 3/4 mile claimers at GG. But it's been years since I've paid attention to it. At that time 56.3 was the half filter in lower claimers. You could explore the idea of a slope and see what you could come up with.

[dansan]

or just keep it simple and have a firm grip on the obvious

[pondman]

or just keep it simple and have a firm grip on the obvious

Agreed.

Except sometimes stretch runs are illusions. People might be screaming and the jockeys might be beating, but the mules aren't running very fast and have slowed to a crawl.

[cosmicway]

I can work out mean velocities, start to finish that is.
I did it once. Turned out the mean velocities were equal more or less for all distances, but 5f was considerably faster.
Some independent confirmation about this came last year whan 5f races were stopped altogether, for being "too fast" and because "the track has some problems" (*).

(*) There are ancient statues buried under the Athens racetrack, the arcaeologists concur. If you dig then the entire racecourse is gone ! The law permits to leave things as they are, but prohibits any engineering work that might cause some damage to the archaeological objects underneath (such as making a really good pitch).

Maybe ...

[cosmicway]

To the extent that least squares can be trusted, it now seems that a good solution has the following form:

dist(1) < dist(2): new total = A x total / (1 + B x early + C x middle)

dist(2) < dist(1): new total = A x total / (1 + B x middle + C x late)

with A,B,C > 0.

The application of this formula is likely to be universal, while of course the values of the coefficients A,B,C are unique for the various tracks.

That's kinematics sensitive too, although the kinematics model mentioned earlier still escapes me.

It's a little dirty because the values A,B,C cannot be extracted analytically and you have to use slow numerical method.

[cosmicway]

That defeats gm10's formula.
In 41 samplings out of 42.
Only when I convert 7f to 8f, gm10 is slightly better.

Both formulas (gm10, mine) register an improvement compared to when we don't use sectionals.

[gm10]

That defeats gm10's formula.
In 41 samplings out of 42.
Only when I convert 7f to 8f, gm10 is slightly better.

Both formulas (gm10, mine) register an improvement compared to when we don't use sectionals.

Sounds promising ... do you mind expanding? What are typical A, B, C values, strike rates?
What exactly are you looking at? Is EARLY, the early number from only the last race, or some sort of average of all previous races?

[cosmicway]

Sounds promising ... do you mind expanding? What are typical A, B, C values, strike rates?
What exactly are you looking at? Is EARLY, the early number from only the last race, or some sort of average of all previous races?

Typical value:

6f to 7f A = 1.4937, B = 0.168, C = 0.120
The track variant corrected speed at 7f is predicted in this way from previous runs at 6f.
speed 7f + published track variant 7f = f(speed 6 + published track variant) + weight adjustment, where f is the function as described above.
That's a raw average. To make it into a speed line you have to make a histogram with weights for the last performences and you may also try to fit a pdf (probability distribution function).

The method of least squares is however fooling you sometimes.
It' s tempting to try speed = A . speed + B to convert from distance to other distance. That is without use of sectionals. This type of conversion has seen the light in periodicals too, in the form of graphs to assist the reader.
It gives a tearful line, but if you make it simpler, like speed = A . B (again without use of sectionals) the line becomes reasonable. I don't know why.

With your formula for 6f to 7f I get A=1.2474,B=1.4064, C=0.9322 (the values that make the sum of the square deviations minimum).
In this example we are close. The standard error (= sqr(2/pi) x std) is 1.36 for you, 1.35 for me.

The number of samples was 4500 horse histories.

[gm10]

Typical value:

6f to 7f A = 1.4937, B = 0.168, C = 0.120
The track variant corrected speed at 7f is predicted in this way from previous runs at 6f.
speed 7f + published track variant 7f = f(speed 6 + published track variant) + weight adjustment, where f is the function as described above.
That's a raw average. To make it into a speed line you have to make a histogram with weights for the last performences and you may also try to fit a pdf (probability distribution function).

The method of least squares is however fooling you sometimes.
It' s tempting to try speed = A . speed + B to convert from distance to other distance. That is without use of sectionals. This type of conversion has seen the light in periodicals too, in the form of graphs to assist the reader.
It gives a tearful line, but if you make it simpler, like speed = A . B (again without use of sectionals) the line becomes reasonable. I don't know why.

With your formula for 6f to 7f I get A=1.2474,B=1.4064, C=0.9322 (the values that make the sum of the square deviations minimum).
In this example we are close. The standard error (= sqr(2/pi) x std) is 1.36 for you, 1.35 for me.

The number of samples was 4500 horse histories.

Some good stuff in here. Made me think.

[Robert Goren]

Everything is a mathematical equation.That might be true, but every horse is a different mathematical equation and on top of that a lot of the variables are unknown.

[Tom]

And horses are living creatures, not constants.

[cosmicway]

It works - on the basis of least squares theory.
More practical results are awaited.
One mathematical equation for every horses is more difficult to achieve and nobody has one mathematical equation for each horse. Those that stay in racing up to the eage of 8-9-10 race about 100 times. It's a small number, also the performence drops with age.
It's useful to know peculiar properties of some horses but what we are trying to do here is use collective properties to predict the behaviour of the individual.

[cosmicway]

I 'm reviving this thread to report to you, like I promised.

My findings now are as follows:

First the above theories seem to be correct, when you try to predict speed.
The unfortunate thing is that if you rely upon reggression or even robust reggression, it falls foul of probability considerations (the fear I expressed in previous posts here came true).

The object under investigation is the races not the horses.
To make stochastic calibration the computing time is longer, and introduces instabilities - because of the increased number of variables.

Using a model without splits I managed to do some newer stochastic calibration and produce a reasonable speed line.
I 've done the same before but I have big problem now because I lost all my data from 2003-2009 in a disk crash and I don't know if they can be recovered.

I expect anyone trying to do such calculations here will come across the same situations.

[cosmicway]

Now I managed to include the splits also.
It's tricky because you need correction factors and linear reggression theory is a lying monster.