Measuring Breakaway Gaps

[rokopt]

Regarding E' = T2' - T1' = 1/V2 - 1/V1 = (V1 - V2) / (V1 * V2)"

While sensible, not completely true. It makes sense to measure the rate of change of the time gap, but it isn't such a simple equation, because it doesn't only depend on V1 and V2, but also on d (distance between group A and B). If V1 and V2 are constants, we can say d(t)=t*(V1-V2) and input that variable into E'=f(V1,V2,d(t)). Moreover, V1 and V2 are also depending on time and on our prediction, so even more complex.

Ah, I'm not differentiating with respect to time. In the notation I used, E was a function of _location_ (A through D were functions of time) -- specifically, E is the stopwatch gap at a given location, given by E = T2 - T1. So E' is the derivative of E not with respect to time, but with respect to location. I skipped some steps in the above, but here they are fleshed out:

E = T2 - T1 (definition of E)
E' = (T2 - T1)' (differentiate both sides)
= T2' - T1' (differentiation is linear)
= (L2inv)' - (L1inv)' (definition of T as inverse of L)
= (1/(L2' o L2inv)) - (1/(L1' o L1inv)) (inverse function rule for derivatives)
= (1/(S2 o L2inv)) - (1/(S1 o L1inv)) (definition of S as L')
= (1/(S2 o T2)) - (1/(S1 o T1)) (definition of T as inverse of L)
= 1/V2 - 1/V1 (definition of V as S o T).

This completes the proof. The reason it is simple is that this one is a function of location. Indeed, that's probably the most compact summary I can think of of my entire analysis: the question "What is the time gap at a particular location?" has one simple, easy, accurate, unequivocal answer; it's the stopwatch gap at the given location. But the question "What is the time gap at a particular _time_?" is difficult -- there is no single accurate answer to that one; all we have is an infinite variety of ways of estimating the answer. One simple way of estimating it is method A -- the stopwatch gap at the chasers' location -- and that often works well, but it's also sometimes massively misleading, fundamentally because a method-A gap of N minutes takes no account of what the leaders have been doing for the past N minutes. If they've been going at a good, steady clip (with respect to the terrain they're on and the other conditions they're facing), then the method-A gap is probably a good representation of reality. But if the leaders have been going like mad for the past N minutes, the method-A gap is probably an underestimate, and if the leaders have been soft-pedaling for the past N minutes, then the method-A gap is probably an overestimate.

As for the rest of your comments, I actually agree with you. I am not seriously proposing that anyone devise a variety of possible "C" algorithms and do some kind of regression analysis over the course of a huge number of races comparing them to "D". (I certainly would not be capable of such an analysis myself; I'm no statistician.) I just mentioned the theoretical possibility to illustrate the point that there are aspects of the difficult question "What is the time gap at a particular time?" that we can answer more accurately after the race than during it. And I didn't think you were seriously proposing it either -- I just wanted to make sure to give you credit for having illustrated an understanding of the issues by raising the theoretical possibility.

There are some extremely limited (and therefore relatively predictable) conditions under which some forms of gap prediction do get used, of course -- such as Chapatte's Law, which does at least get used by Paul Sherwen.

 

[Ildabaoth]

Well in nortehrn classics a surface parameter could be included, too, in order to get real speeds. But really these are very few races were we have that problem.
A wind factor could be include, too. But normaly the wind in a break and the peleton shouldn't be to much different. After all usually there are just a few kilomters between break and peleton. But in theory, a wind parameter could be easily included too. The motorcycles would just need a wind measurement advice.

Hi, Bavarian. It is good you are interested in the problem and you try to solve it. I praise you for that. The problem is, parameters aren't so easily included as you think. Multivariable prediction systems are very complex and difficult. Lets start with a simple characterisation problem: we need to characterise a simple 2D circular shape, so it is a 2-dimension system. We could say we need at least 10 points to roughly define the shape. Now extrapolate that problem to a 3-dimension one, and use points to define a simple sphere. You probably need 100 or more points to have a very broad approximation. According to this oversimplification (shapes in n-dimension aren't so simple and convex and 100 points aren't clearly enough), we can say that to describe a n-dimension problem we need a number of samples roughly related to k^n (actually, this isn't exactly the case, real n-dimension systems are even more complex, but we aren't using any deep math and our intention is just to show the difficulty of n-dimension math).

How is this related to our prediction problem? We need a lot of variables in order do create a better predictor. Just for the sake of debate, because a lot of these variables can't be estimated or known, lets say we have access to all the data. So we throw in speeds, distances, temperatures, wind speeds and every variable we consider may be useful to build a better predictor. Granted, regularly having a higher number of variables is useful to get a better prediction, but that only happens if there is an adequate correlation between the variable and the output; otherwise it actually decreases the performance (unless each variable is treated as a weak classifier in a boosted system, but we won't discuss that here). Even with this extreme simplification, we are hitting a wall now. As stated, we need an exponentially higher number of samples in order to generate the predictor, so now we are in a dylemma: do we use a relatively low number of variables so our samples are enough to define the problem, knowing that we are discarding important information? Or either do we throw in more variables, realising that the number of samples isn't clearly enough to train the classifier, so there won't be a proper training? And even if we neglect the reduced sampling, we have other real limits to the number of variables, such as memory and processing power. Not as in: well, not now, but in a couple of years. We are talking about k^n, so if n is, lets say, just 40, the problem is already incredibly huge. So huge that no number of races in the whole history would be enough to get just a tiny fraction of the required batch.

According to what we have shown, multivariable prediction is hard. So much that real multivariable systems normally don't use more than 10, maybe 20 variables. And even in that case they normally approach the problem using each variable as a single predictor or a group of them instead of a more powerful all-variable approach and then combine the results (so they aren't really multivariable, but single variable meta predictors, actually). However, while this simplify the problem, the accuracy is a lot lower.

This means, there is no such thing as "it is just a matter of including more variables"; the problem is more complex than that. Besides, reality is even harder. Our problem is non linear (as in output doesn't change in a linear way with changes in the input) and chaotic too (as in identical known conditions might result in very different outputs) and the data is more likely non-convex (and non convexities in n-dimension math are a nightmare), which not only increases the difficulty, but makes it impossible to build a reliable predictor.

This explanation lacks of some deep analysis (feel free to throw me a PM if interested), but I tried to make it clear and understable.

 

[rokopt]

Hi, Bavarian. It is good you are interested in the problem and you try to solve it. I praise you for that. The problem is, parameters aren't so easily included as you think. Multivariable prediction systems are very complex and difficult. Lets start with a simple characterisation problem: we need to characterise a simple 2D circular shape, so it is a 2-dimension system. We could say we need at least 10 points to roughly define the shape. Now extrapolate that problem to a 3-dimension one, and use points to define a simple sphere. You probably need 100 or more points to have a very broad approximation. According to this oversimplification (shapes in n-dimension aren't so simple and convex and 100 points aren't clearly enough), we can say that to describe a n-dimension problem we need a number of samples roughly related to k^n (actually, this isn't exactly the case, real n-dimension systems are even more complex, but we aren't using any deep math and our intention is just to show the difficulty of n-dimension math).

How is this related to our prediction problem? We need a lot of variables in order do create a better predictor. Just for the sake of debate, because a lot of these variables can't be estimated or known, lets say we have access to all the data. So we throw in speeds, distances, temperatures, wind speeds and every variable we consider may be useful to build a better predictor. Granted, regularly having a higher number of variables is useful to get a better prediction, but that only happens if there is an adequate correlation between the variable and the output; otherwise it actually decreases the performance (unless each variable is treated as a weak classifier in a boosted system, but we won't discuss that here). Even with this extreme simplification, we are hitting a wall now. As stated, we need an exponentially higher number of samples in order to generate the predictor, so now we are in a dylemma: do we use a relatively low number of variables so our samples are enough to define the problem, knowing that we are discarding important information? Or either do we throw in more variables, realising that the number of samples isn't clearly enough to train the classifier, so there won't be a proper training? And even if we neglect the reduced sampling, we have other real limits to the number of variables, such as memory and processing power. Not as in: well, not now, but in a couple of years. We are talking about k^n, so if n is, lets say, just 40, the problem is already incredibly huge. So huge that no number of races in the whole history would be enough to get just a tiny fraction of the required batch.

According to what we have shown, multivariable prediction is hard. So much that real multivariable systems normally don't use more than 10, maybe 20 variables. And even in that case they normally approach the problem using each variable as a single predictor or a group of them instead of a more powerful all-variable approach and then combine the results (so they aren't really multivariable, but single variable meta predictors, actually). However, while this simplify the problem, the accuracy is a lot lower.

This means, there is no such thing as "it is just a matter of including more variables"; the problem is more complex than that. Besides, reality is even harder. Our problem is non linear (as in output doesn't change in a linear way with changes in the input) and chaotic too (as in identical known conditions might result in very different outputs) and the data is more likely non-convex (and non convexities in n-dimension math are a nightmare), which not only increases the difficulty, but makes it impossible to build a reliable predictor.

This explanation lacks of some deep analysis (feel free to throw me a PM if interested), but I tried to make it clear and understable.

Nice explanation; I learned from that (I think). The bold parts in particular are new and interesting information to me.

 

[Buffalo Soldier]

Didn't read a lot of it, but it seems to me that you should also show the standard deviation of the predicted variables?

 

[Morbius]

I'm looking forward to seeing all these variables put on the blackboard on a moto

The cyclists will all learn to handle the maths, just like darts players calculate finishes

 

[Magnus]

breakaway 1KM ahead of peloton, both doing 20KPH, the time gap is 3 minutes.

Then both breakaway and peloton increase speed to 40kph. The time measured gap is halved, which should mean the breakaway is now in a much worse position. Except they clearly aren't.

The breakaway is in a worse position if they are 1 km ahead of a peloton that's doing 40 km/h than if they are 1 km ahead of a peloton doing 20 km/h which is clearly reflected in the time gap between the two.

In the understanding it might help to think about the effort it takes to build a 1 km gap over a peloton doing 20 and a peloton doing 40 km/h.

Or think about it this way: If you have a 250m gap and have to change a wheel you'll be caught by a peloton doing 60 km/h but not by a peloton doing 15 km/h.
So if you have a 250m gap at a peloton doing 15 km/h and they increase the speed to 60 km/h you really are in a much worse position. Irrespective of you're own speed.

Of course you'd be right in stating that time-gap in no way reflects the breakaways actual speed. Nor does it say anything about whether or not the breakaway will be caught before the finish line. It reflects in a very beautiful way the speed of the peloton and the breakaway up until the point where you measure the time gap.

 

[DenisMenchov]

I don't get it why is this even being considered. The whole point of racing is in finishing the race in fastest time. Hence they measure the gap between breakaway and peloton in time. After all they are always going to cover same distance.
In addition to that, it is much more easier to calculate how faster you must go to catch a breakaway if you measure in time, because you know both the distance left, and the time gap you need to make. If you have only distance you have two unknowns both time and speed you need.

 

[Magnus]

In addition to that, it is much more easier to calculate how faster you must go to catch a breakaway if you measure in time, because you know both the distance left, and the time gap you need to make. If you have only distance you have two unknowns both time and speed you need.

No matter what you don't know how fast you'll have to go to catch the breakaway because you don't know how fast the breakaway is going to complete the remaining part of the course.

 

[rokopt]

I don't get it why is this even being considered. The whole point of racing is in finishing the race in fastest time. Hence they measure the gap between breakaway and peloton in time.

Yes, I would say that most of the thread reached the conclusion long ago that the gap ought to be measured in time. That is not, by and large (there have been exceptions), what has been being considered since then. The question being examined since then is what the value of the gap, in units of time, should be considered to be at a particular moment during the race.

 

[DenisMenchov]

No matter what you don't know how fast you'll have to go to catch the breakaway because you don't know how fast the breakaway is going to complete the remaining part of the course.

OFC, but you know how fast they are probably going to go.

 

[TeoSheva]

Nearly. Say the breakaway is 10km ahead on dead flat ground, for argument's sake, let's have the break at 100km in and the peloton at 90km in, and they're both going 20km/h.

So the break completes the 10km between 90km and 100km at 20km/h, so it takes them 30 minutes. Now suppose that as the break go over the 100km point (and the peloton 90km to go), they both accelerate to 40km/h. Then the peloton completes that 10km in just 15 minutes.

So, your reasoning is almost correct, but they are actually in a worse position than they were - the reason the time gap decreases is because the peloton has raced a particular section faster than they have. That is being genuinely worse off.

just this is right

the istantaneous speed does not count anything, what counts is the average speed.

 

[rokopt]

OFC, but you know how fast they are probably going to go.

It is precisely that which Ildabaoth's last post explains the enormous difficulty of.

 

[Caruut]

Yes, I would say that most of the thread reached the conclusion long ago that the gap ought to be measured in time. That is not, by and large (there have been exceptions), what has been being considered since then. The question being examined since then is what the value of the gap, in units of time, should be considered to be at a particular moment during the race.

I personally can't see past the stopwatch method. That is, the gap is as though the finish line were on the front wheel of the first rider in the chasing group. For sheer simplicity of calculation and understanding from all concerned, I think it wins.

 

[rokopt]

I personally can't see past the stopwatch method. That is, the gap is as though the finish line were on the front wheel of the first rider in the chasing group. For sheer simplicity of calculation and understanding from all concerned, I think it wins.

I won't argue against using it. I will, however, make note that there are cases in which it will differ radically from what I think most people would consider a sensible gap value to be. Probably the most extreme case is the following: imagine the peloton has been pursuing some lone leader with both the leader and the peloton working steadily and the gap having held at five minutes for a long time (if there's an extended steady state, then all gap method calculations will eventually come to agree, so we can say that at some point, the gap is five minutes according to all methods). Then, the leader crashes, takes four minutes to get back on the bike, and then resumes riding at the same work rate as before the crash. The peloton, during that same period, simply continues riding at the same steady work rate as always. What happens to the gap?

I think that most people would agree that the way a sensible gap method would work is that the gap would drop by one second for each second during the four minutes that the leader was off the bike and the peloton was steadily gaining, gradually declining to one minute and then holding there once the leader resumed riding. But that's not at all what happens to the gap you describe (the stopwatch gap at the chaser's position, AKA the method-A gap). The method-A gap holds steady at five minutes up until the peloton reaches the point where the leader crashed, and then instantly drops to one minute when the peloton reaches that point. The reason is that the method-A gap doesn't know anything about what the leader has been doing since passing the point where the peloton is now. The method-A gap therefore doesn't know about the crash -- it suddenly "finds out" at the moment when the peloton reaches the location of the crash.

 

[Magnus]

I think that most people would agree that the way a sensible gap method would work is that the gap would drop by one second for each second during the four minutes that the leader was off the bike and the peloton was steadily gaining, gradually declining to one minute and then holding there once the leader resumed riding. But that's not at all what happens to the gap you describe (the stopwatch gap at the chaser's position, AKA the method-A gap). The method-A gap holds steady at five minutes up until the peloton reaches the point where the leader crashed, and then instantly drops to one minute when the peloton reaches that point. The reason is that the method-A gap doesn't know anything about what the leader has been doing since passing the point where the peloton is now. The method-A gap therefore doesn't know about the crash -- it suddenly "finds out" at the moment when the peloton reaches the location of the crash.

So you're suggesting that at a given time point you look at where the peloton is and then measure how much time it took for the leader to ride from there to his present position?

If so you have the same 'problem' you just described if the peloton crashes.

 

[cineteq]

I think that most people would agree that the way a sensible gap method would work is that the gap would drop by one second for each second during the four minutes that the leader was off the bike and the peloton was steadily gaining, gradually declining to one minute and then holding there once the leader resumed riding. But that's not at all what happens to the gap you describe (the stopwatch gap at the chaser's position, AKA the method-A gap). The method-A gap holds steady at five minutes up until the peloton reaches the point where the leader crashed, and then instantly drops to one minute when the peloton reaches that point. The reason is that the method-A gap doesn't know anything about what the leader has been doing since passing the point where the peloton is now. The method-A gap therefore doesn't know about the crash -- it suddenly "finds out" at the moment when the peloton reaches the location of the crash.

How are you going to use to process that information? I mean what do we know? Gap (time) and both speeds (km/h)?

 

[Caruut]

I won't argue against using it. I will, however, make note that there are cases in which it will differ radically from what I think most people would consider a sensible gap value to be. Probably the most extreme case is the following: imagine the peloton has been pursuing some lone leader with both the leader and the peloton working steadily and the gap having held at five minutes for a long time (if there's an extended steady state, then all gap method calculations will eventually come to agree, so we can say that at some point, the gap is five minutes according to all methods). Then, the leader crashes, takes four minutes to get back on the bike, and then resumes riding at the same work rate as before the crash. The peloton, during that same period, simply continues riding at the same steady work rate as always. What happens to the gap?

I think that most people would agree that the way a sensible gap method would work is that the gap would drop by one second for each second during the four minutes that the leader was off the bike and the peloton was steadily gaining, gradually declining to one minute and then holding there once the leader resumed riding. But that's not at all what happens to the gap you describe (the stopwatch gap at the chaser's position, AKA the method-A gap). The method-A gap holds steady at five minutes up until the peloton reaches the point where the leader crashed, and then instantly drops to one minute when the peloton reaches that point. The reason is that the method-A gap doesn't know anything about what the leader has been doing since passing the point where the peloton is now. The method-A gap therefore doesn't know about the crash -- it suddenly "finds out" at the moment when the peloton reaches the location of the crash.

I think these overly-complicated gap-measuring methods are based on the assumption that this one number has to tell us everything about the gap. If the leader has crashed, I think people are smart enough to realise that he isn't still holding the time gap. All that it would take is for the commentators to occasionally say "And remember, the time gap is still holding because it's based on the time of the leader at the point where the chasing group are now, not what he's done since then", and everyone would be in the know.

 

[rokopt]

So you're suggesting that at a given time point you look at where the peloton is and then measure how much time it took for the leader to ride from there to his present position?

If so you have the same 'problem' you just described if the peloton crashes.

The quantity you describe is the exact same quantity. "How much time it took for the leader to ride from the peloton's current position to the leader's current position" is just another way of saying "how long it has been since the leader passed the peloton's current position", AKA the stopwatch gap at the peloton's current position, AKA the method-A gap.

 

[rokopt]

I think these overly-complicated gap-measuring methods are based on the assumption that this one number has to tell us everything about the gap. If the leader has crashed, I think people are smart enough to realise that he isn't still holding the time gap. All that it would take is for the commentators to occasionally say "And remember, the time gap is still holding because it's based on the time of the leader at the point where the chasing group are now, not what he's done since then", and everyone would be in the know.

Correct. I think this would be a perfectly sensible answer to cineteq's question in the post before yours, too. A similar situation is when a breakaway group of non-climbers hits the slopes of a massive final climb with the peloton still on the flat in pursuit. At that point, the announcers might say, "We've been getting a [method-A] reading of five minutes for a while, but the breakaway is already suffering on these slopes, so that gap is going to start tumbling as soon as the leaders hit the climb". (Indeed, I'm sure I have heard announcers say that many times.)

 

[Magnus]

The quantity you describe is the exact same quantity. "How much time it took for the leader to ride from the peloton's current position to the leader's current position" is just another way of saying "how long it has been since the leader passed the peloton's current position", AKA the stopwatch gap at the peloton's current position, AKA the method-A gap.

No it isn't.

Lets say everybody's riding at 60 km/h and the leader has a 5 min gap on the peloton, which is then a 5 km gap if measured in distance.

Assume the leader crashes at km 100 of the race. Then at the time of the crash the peloton is at km 95 of the race.

The method-A gap is then 5 min at km 100.

Assuming the rider takes 4 min to get back on the bike and then continues to ride at 60 km/h the method-A gap will be 1 min at every point of the route from km 99.99 onwards.

Using the method I described would measure the gap to be 5 min when the crash takes place. And then for every second the rider lies on the ground it declines one second.
For instance after one minute the peloton will be at km 96 of the race. Since it took the leader 4 min to get from km 96 to km 100 the gap will be 4 minutes by the method I described.

 

[Magnus]

Correct. I think this would be a perfectly sensible answer to cineteq's question in the post before yours, too. A similar situation is when a breakaway group of non-climbers hits the slopes of a massive final climb with the peloton still on the flat in pursuit. At that point, the announcers might say, "We've been getting a [method-A] reading of five minutes for a while, but the breakaway is already suffering on these slopes, so that gap is going to start tumbling as soon as the leaders hit the climb". (Indeed, I'm sure I have heard announcers say that many times.)

If the gap drops in this situation it's because the chasers are climbing faster than the breakaway.

If you take two groups who ride at the same speed on any given point of the route the time gap will remain the same at all times.

 

[rokopt]

No it isn't.

Lets say everybody's riding at 60 km/h and the leader has a 5 min gap on the peloton, which is then a 5 km gap if measured in distance.

Assume the leader crashes at km 100 of the race. Then at the time of the crash the peloton is at km 95 of the race.

The method-A gap is then 5 min at km 100.

Assuming the rider takes 4 min to get back on the bike and then continues to ride at 60 km/h the method-A gap will be 1 min at every point of the route from km 99.99 onwards.

Using the method I described would measure the gap to be 5 min when the crash takes place. And then for every second the rider lies on the ground it declines one second.
For instance after one minute the peloton will be at km 96 of the race. Since it took the leader 4 min to get from km 96 to km 100 the gap will be 4 minutes by the method I described.

Oh, I see what you mean. Yes, you're right. This illustrates that I violated my earlier simplifying assumptions by using the illustration of a crash, where one group's speed drops to zero altogether. If speeds remain non-zero, the two quantities, "How much time it took for the leader to ride from the peloton's current position to the leader's current position" and "how long it has been since the leader passed the peloton's current position" are the same. If the leader's speed drops to zero, then "How much time it took for the leader to ride from the peloton's current position to the leader's current position" becomes, in effect, a predictive gap measurement which estimates the gap based on the assumption that the peloton will ride the intervening distance at the same speed the leader did. That sounds like a sensible substitute for while the leader is crashed.

 

[rokopt]

If the gap drops in this situation it's because the chasers are climbing faster than the breakaway.

If you take two groups who ride at the same speed on any given point of the route the time gap will remain the same at all times.

In the situation I was describing, the chasers hadn't started climbing yet. They were on the flat leading up to the climb, and the method-A gap was holding steady. However, the announcers were anticipating (almost certainly correctly) that the (method-A) gap would start falling once the chasers hit the climb, because the breakaways were climbing it so slowly that they could foresee that the chase group (which includes the real climbers) would climb it faster. The method-A gap has not yet started to reflect what we, watching on television, can already perceive -- that the breakaway is "losing time with every pedalstroke".

Regarding your second statement, yes, that's true, and when it really does apply to the entire route, and all the riders start at the same spot, then they'll all be at the same spot throughout. Sounds like a Cav stage.

 

[Magnus]

In the situation I was describing, the chasers hadn't started climbing yet. They were on the flat leading up to the climb, and the method-A gap was holding steady. However, the announcers were anticipating (almost certainly correctly) that the (method-A) gap would start falling once the chasers hit the climb, because the breakaways were climbing it so slowly that they could foresee that the chase group (which includes the real climbers) would climb it faster. The method-A gap has not yet started to reflect what we, watching on television, can already perceive -- that the breakaway is "losing time with every pedalstroke".

Fair enough.

It's just that often when people talk about that situation they make it sound like the breakaway is losing time because the peloton is going faster at that point in time and not because they will be going faster at that point of the course (rereading your post I see that you're quite clear about this point).

 

[Master50]

Take a very simple and easy method of measuring gaps and make it so complicated that to explain it to the general public we need a math degree and finite element analysis. Try explaining any of this to the TV audience and people just change the channel. Your standing at the edge of the road watching the racers go by. You have a watch on your wrist and now we have to go out on the road to measure distance between gaps? Or you watch the ;eager go by and time the gaps. At that point on the road everyone has exactly the same climbing, decending and distance covered. How do you get their speed to calculate their distance apart? Ask? He andy how fast are you going? Time is the only objective method of knowing the gap. If the break and the chase maintain the same time gap the distance between them will change according to their speed and that is often changing by terrain.

BTW a timed crit is still a distance race unless the time ends exactly at the line. The winner is the fist guy to travel the next full lap and all riders are judged at a specific distance.

This thread while interesting has been an absurd exercise and I challenge anyone to make distance work as an easy tool to show TV audiences the gap. On a climb he is 200 meters ahead and on the decent he is 1000 meters with exactly the same time gap. IT is time we use and it is time we will always use. Distance is the length of the course.