Measuring Breakaway Gaps

[Buffalo Soldier]

Not a hard choice for me

 

[Buffalo Soldier]

Method B should be funny in races with small hills
(And with funny, I mean useless)

 

[Bavarianrider]

Well a minute is not a minute. It depends on how fast the riders go.
Catching up a minute at 50km/h is much more difficult than with 20km/h on a climb. Therefore it's a bit misleading.
Thanks to modern technique, a more objective system would be better.
The "Realgapmeter"
Example:
Situation A.
The break is 4 minutes ahead of the peleton. the peleton rides at 50km/h the break at 46km/h. This makes an average speed of 48km/ for the field.

Realapmeters = 360 seconds/13,3(m/s)
Realgapmeters = 27

Situation B:
The break is 2 minutes ahead of the peleton. The peleton rides at 20km/h, the break at 18km/H

Average Fieldspeed: 19km/h
Realgapmeters = 180seconds/5,27(m/s)
Realgapmeters = 34

So you see the real gap in Situation B is bigger than in situation A, altough the time difference is twice as much in situation B. Hence the realgapmeter is a much better and objective way of measuring the real gap.

However, the problem of course is, that taking only the current speed in acount can be a bit misleading too. How big the gap really is heavily depends on what the speed will be like for the rest of the stage nad how many kilometers are left. Hence, the projected average speed for the rest of the stage as well a sthe distance left, are important, too.
So in order to geat a real objective measurement of gaps, we simply have to multiply our realgapmeter with the inverse quotient of the distance and the projected average speed for the rest of the stage:
So the Formula for the Realgapmeter looks like this in the end.

Realgapmeter = Time/Average Field Speed X (1/(Rest Distance/Projected rest average speed))

Examples:
Let's go back to our examples A and B

Let's say there are 40 flat km left and the projected average speed is 50km/H

Realgapmeter = 360 seconds/13,3(m/s)X (1/(40000m/13,8(m/s)))
Realgapmeter = 0,009

Situation B: Let say 20km with some mountains are left. so projected average speed is 30km/H

Realgapmeter = 180seconds/5,27(m/s) X (20000m/8,3(m/s))
Realgapmeter = 0.014

Of course those small numbers are a bit uncomfortable. Therefore, the result gets multiplied 1000 in the end. So that in our two examples we get the following realgapmeters
Situation A 9
Situation B 14

So you see that the real gap is bigger in situation B.

I really think this offers a good and objective measurements for gaps. Riders and spectators would only need to et to know the system and learn it. So that you immidately get a feeling what Realgapmeter is a big gap and what's a rather low one. But this learning affect would work pretty quickly. People would soon realise that A realgapmeter of like 15 offers the break an excellent chance of making it, while a 5 or so means that they most likely won't make it.
Rember all this calculation is irelevant for the riders and spectators as it's done by computers in real time. Thanks to modern technology with Gps and all that stuff it would not be aproblem.
Maybe smaller aces could not afford it or it would be to much work to install the systems needed. But for big races it would not be a problem at all.

 

[Potomac]

Maybe the gap should be in units of work.

Then all we need in addition to location and speed is wind velocity over each part of the course and the profile.

 

[Ildabaoth]

Maybe the gap should be in units of work.

Then all we need in addition to location and speed is wind velocity over each part of the course and the profile.

Too inaccurate. Throw in atmospheric preassure, temperature, humidity and drag coefficient and you might be up to something.

 

[richtea]

Thanks to modern technique, a more objective system would be better.
The "Realgapmeter"
Example:
Situation A.
The break is 4 minutes ahead of the peleton. the peleton rides at 50km/h the break at 46km/h. This makes an average speed of 48km/ for the field.

Realapmeters = 360 seconds/13,3(m/s)
Realgapmeters = 27

Situation B:
The break is 2 minutes ahead of the peleton. The peleton rides at 20km/h, the break at 18km/H

Average Fieldspeed: 19km/h
Realgapmeters = 180seconds/5,27(m/s)
Realgapmeters = 34

Wouldn't it be simpler just to present an estimate how long it will take for the peloton to catch the break at the current speed of the two groups - alongside existing measures of the time gap? (It could take a negative value where the leaders are travelling faster than the chasers, or it could be ignored).

 

[Ildabaoth]

Wouldn't it be simpler just to present an estimate how long it will take for the peloton to catch the break at the current speed of the two groups - alongside existing measures of the time gap? (It could take a negative value where the leaders are travelling faster than the chasers, or it could be ignored).

Not only that. Besides the inherent difficult to establish an accurate estimation of speeds (current and future), Bavarian's math is quite odd. For example, the units of his Realgapmeters are squared seconds over distance, which don't have a too clear physical abstraction. On the other hand, it seems he uses 90 seconds minutes, or at least that's what could be infered by his non clear equations:

The break is 4 minutes ahead of the peleton. the peleton rides at 50km/h the break at 46km/h. This makes an average speed of 48km/ for the field.

Realapmeters = 360 seconds/13,3(m/s)

Finally, his examples don't show Realgapmeters are a better indication of the relative gap between 2 groups. Quite the contrary, according to his values, in the first case the group would be caught in 46 minutes, while in the second one it would take just 17,85 minutes and in Realgapmeters the second value is higher than the first one. In any case, this is pointless ,because all those numbers would require that the speeds of the 2 groups are constant or, at least, known and predicted during the whole trajectory, which is impossible.

 

[Bavarianrider]

Not only that. Besides the inherent difficult to establish an accurate estimation of speeds (current and future), Bavarian's math is quite odd. For example, the units of his Realgapmeters are squared seconds over distance, which don't have a too clear physical abstraction. On the other hand, it seems he uses 90 seconds minutes, or at least that's what could be infered by his non clear equations:

Finally, his examples don't show Realgapmeters are a better indication of the relative gap between 2 groups. Quite the contrary, according to his values, in the first case the group would be caught in 46 minutes, while in the second one it would take just 17,85 minutes and in Realgapmeters the second value is higher than the first one. In any case, this is pointless ,because all those numbers would require that the speeds of the 2 groups are constant or, at least, known and predicted during the whole trajectory, which is impossible.

Ups i indeed used 90 seconds instead of 60 But i did thid mistake both in A as well as in B, so the result basically stays the same.
They would caught earlier cause the speed difference is bigger. 2km/h at 20km/h are a lot bigger than at 50km/h. It were just examples. But the general tendency that Bigger realgapmeter = bigger gap is correct.

 

[Ildabaoth]

Ups i indeed used 90 seconds instead of 60 But i did thid mistake both in A as well as in B, so the result basically stays the same.
They would caught earlier cause the speed difference is bigger. 2km/h at 20km/h are a lot bigger than at 50km/h. It were just examples. But the general tendency that Bigger realgapmeter = bigger gap is correct.

The problem is your units are weird, as I stated previously. Squared seconds over meters don't have a clear physical abstraction. And even if there is a general tendency that bigger realgapmeters means bigger gap, it is clear that estimated time to caught the breakaway not only has a tendency, but it would always accurately measure the relative gap, so it is superior to realgapmeters.

Another inherent problem in your calculation is the use of average speed. That means it would be exactly the same if the breakaway speed is 5 km/h and the chasers are rolling at 45 km/h or if the speeds are 20 km/h and 30 km/h, which is weird. Actually, it would be the same if both groups are rolling at 25 km/h. Or even weirder, it is the same if the speeds are the opposite and the breakaway is actually going on faster than the peloton.

With this I don't want to imply that a better metric can't be created. Of course it can, for example using relative speeds and distances. The whole matter, nevertheless, is that it shouldn't be used any metric that requires an estimation of instantaneous speed in the future, because it is a very very difficult non linear prediction and its chaotic nature make it impossible to have an adequatelly accurate estimation. Current time gaps, on the other hand, don't relly on impossible predictions: the time that you see is the time difference between the two groups as the chasers reach a point. Simple and effective.

 

[rokopt]

Waterloo Sunrise, you mentioned a few posts back that there were two alternative measures of "time gap" being suggested, called A (the "stopwatch gap", which in light of gerundium's comment could also be called the "motorcycle gap") and B ("Bavaria's quotient" -- a modest name; it's the one you used in your original post to this thread, so you could also have named it after yourself). There were also a couple of other methods described as well -- Master50 explained that some races do make estimates that take into account the chasers' speed, and Ildabaoth and Potomac have speculated about predictions that would take into account terrain, weather conditions, and other factors. It is even possible to imagine writing a program that would attempt to simulate riders' likely behavior as a function of race situation -- that is, to account for tactics as well. So there are far more than two alternative measures of "time gap". In this post, I'm going to try to summarize what has happened so far and then answer the following questions: "Why are there so many potential measures of 'time gap'?", "What are the relationships among them?", and "Is any of them 'right'?". I'll use (simple) mathematical symbols because I and some others will find them clearer and much more concise for computations, but I'll also try to express all the mathematical results in natural language so that anyone who's not interested in the math can just glean the conclusions.

First, to summarize what has already happened, I think that the thread started with some confusion and argument because of two separate differences of perspective. The first difference was actually the initial question in the initial post, which was (paraphrasing), "In what units should gaps be measured?" I believe that Libertine Seguros provided the definitive answer to that question, which is that cycling road races, and especially stage races, are scored by total time, and that, because we want to be able to answer questions like "Who is the virtual stage race leader at this moment?", the only unit in which a gap can be expressed that would allow us directly to answer that question is time. A method that produces an answer in some other unit, such as distance or work, may have value, but only as _part_ of a more complete method that combines the original method with a method for translating distance or work or whatever into time. Without that translation, the original method is incomplete, because it can't provide an answer, by itself, to questions such as "Who is the virtual stage race leader at this moment?".

So Libertine demonstrated unequivocally (in my opinion at least -- I don't think anyone has rebutted the point successfully) that a _complete_ gap-measuring method must provide an answer in units of time. That leads to the second difference of perspective, which is what method of obtaining a gap in units of time is "_the_ time gap". Waterloo's original post described the behavior of a method which ended up being dubbed "method B" or "Bavaria's quotient", and some other posters, such as richtea, also adopted this viewpoint, whereas answers from posters such as Buffalo Soldier and Caruut revealed that they considered "_the_ time gap" to be what ended up being dubbed "method A" or "the stopwatch gap". Some confusion ensued, but I think that confusion has now been cleared up by the observation that different groups of posters were talking about different methods for obtaining a time gap.

I believe we also have testimony in this thread that not all broadcasts use the same time gap computation method to produce the numbers they display, but I think there seems to be a general consensus that at least _most_ races use the stopwatch gap (method A) at least most of the time. (We also have testimony describing how errors can be injected into the measurements, but for the purposes of this post, I'm going to ignore that, and pretend that whatever methods we choose, we implement them accurately.) There are some specific instances in which the stopwatch gap is definitely being used: when the television broadcast starts a clock when a breakaway reaches the top of a climb, for example, and then stops it when the chase reaches the top of that climb, it is measuring the stopwatch gap between the break and the chase at the top of the climb. Most important, the final, official time gap between two riders on a given stage -- the one that actually gets used to score the GC of the stage race -- is the stopwatch gap at the finish line.

So the stopwatch gap is clearly important. The first questions I'll address, then, are these: if the stopwatch gap is so important, why are there so many other potential methods too? Is the stopwatch gap, or any other method, the single "right" measure of time gap?

This is where I'll start using (simple) symbolic math. Waterloo Sunrise has given us names A and B for the stopwatch gap and Bavaria's quotient, respectively. Both of those are functions from time to time -- you give them the current time, and they give you a measure of time gap. In all that follows, I'll assume that time and location are real numbers. I'll use "L" to mean a rider's location as a function of time. I'll assume that L is analytic and strictly increasing, meaning that riders always move in the correct direction (no one pulls an Abdel-Kader Zaaf) at a non-zero speed (no trackstands) without teleporting around (as interesting as that would be). For simplicity, I'll assume in most calculations that time and position both go off to infinity in both directions, and that L's image is the whole real line (an idealized infinitely long race in which everyone covers the whole course). I think that those assumptions, while not true in reality, are close enough for the purposes of this discussion. In most equations, I'll also focus specifically on two riders/groups -- a lead rider/group whose position at a given time is L1, and a chase rider/group whose position at a given time is L2. Since L1 is the leader, we have L1 > L2 (for all values of time -- an idealized infinitely long chase). Given our assumptions, L is invertible, so I shall call its inverse T, which is also an analytic, strictly increasing function. T is a function from locations to times -- you give it a location, and it gives you the time at which a rider reaches the given location. When we focus on two riders/groups with location functions L1 and L2, we shall also have their inverse functions T1 and T2, with T1 < T2 (meaning that, for any given location, the lead group gets there at an earlier time than the chase group).

Now we want to find an equation or equations describing candidate measurements of "time gap". The first observation I'll make in this post which I don't think has yet been made explicitly is that, just as there used to be confusion (until Libertine cleared it up) over which units should be the _output_ of a time gap function, we also have choices regarding which units should be the _input_ of a time gap function. For the purposes of analyzing a race situation live, and displaying some number as the "time gap" in the corner of the screen on the TV broadcast, we clearly want to be able to compute a time gap _at the current time_ -- so we really want a function whose _input_ is also in units of time. But the "stopwatch gap" gives us a gap at a particular _place_ -- it is a function whose input is in units of _location_. Its _output_ is in units of time -- it tells us how apart two groups are in time -- but it does so as a function of a particular _place_ (e.g. the top of a climb, or the finish line, or a point that a motorcycle pinged with a GPS).

This, I would say, is the reason that there are many candidates for "time gap" as a function of time -- there is a clear "one, true, right time gap" function, and it is the stopwatch gap, but it is the stopwatch gap _as a function of location_. The question "What is the time gap as a function of time?" has spawned much debate, but I think that the question "What is the time gap as a function of location?" has only one sensible answer, and it's the difference in the times between the chase group and the lead group at the given location: T2 - T1.

If there's one true time gap as a function of location, can we convert that to a time gap as a function of time, so that we know what to display on the TV screen at any given time? The difficulty is that, though we have a clear time gap as a function of location (T2 - T1), at any particular time during the race, we have _two different_ locations to think about -- L1, the location of the leader, and L2, the location of the chaser. Either one of those (or maybe neither!) might be the place that we should pass to (T2 - T1) to get the one true time gap as a function of time. If we give L2, the location of the chaser, to the (T2 - T1) function, then we get method A, the stopwatch gap at the chaser's position, but we could conceivably give it L1 instead, obtaining the stopwatch gap at the _leader_'s position.

(...continues in next post...)

 

[rokopt]

(...continued from previous post...)

So here, then, are some candidates for "one true time gap as a function of time". I'm going to start with the same A and B that Waterloo Sunrise did (keeping in mind that we have seen above that method A is the stopwatch gap at the chaser's position). I'll use "C" to represent any of the various predictive methods proposed by Bavarianrider, Ildabaoth, and Potomac -- I'm not going to examine their inner details, because they don't have precise equational definitions yet. Finally, I'll use "D" to represent the aforementioned stopwatch gap at the leader's position. In the following, I'm using the letter "o" to mean function composition (since it looks like the little circle that mathematicians usually use), and the prime symbol "'" to denote the derivative. Speed is the derivative of location with respect to time, so I'll write S for speed, so that we have S = L' (in particular, S1 = L1' and S2 = L2'). I'm also eventually going to use the suffix "inv" to mean "inverse of" (for invertible functions).

A = (T2 - T1) o L2
B = (L1 - L2) / S2
C = (... not precisely defined ...)
D = (T2 - T1) o L1

Since the quantity (T2 - T1) keeps appearing, I'm going to use a letter for that as well -- "E" is the next one available, so I'll write E = T2 - T1. Then we can summarize the above further:

A = E o L2
B = (L1 - L2) / S2
C = (... not precisely defined ...)
D = E o L1
E = T2 - T1

I have already argued that E is the one true time gap as a function of location. But is any of A through D the one true time gap as a function of time? We know that A is the one that usually shows up on our TV screens.

To try to determine what the one true time gap as a function of time is, I'll ask another question -- one which commentators sometimes ask aloud, particularly when there is a dangerous breakaway, or during time trials with the lead on the line. It's essentially Libetine's key question again: "If we stopped the stage right now, how much would the leader win this stage by?" (That answer would give us the answer to the further question, "If we stopped the stage right now, who would be the leader of the stage race as a whole, and by how much?", or, more concisely, "Who is the virtual GC leader?") To answer that question, we must realize that if we stopped _all_ the riders right now, we wouldn't be _able_ to answer the question, because, once again, of Libertine's point that the GC is scored by time. If we stopped all the riders, we would only have distance gaps, which don't translate to GC. So I assert that "If we stopped the stage right now" should, in a time-scored, fixed-distance race, really mean, "If we suddenly placed the finish line right where the leader is now". That would stop the race _for the leader(s)_, but the chasers would have to keep riding until they reached the magically-appearing finish line. A chaser's stopwatch gap from the leader at the finish line would be that chaser's time loss with respect to the leader for that stage -- which is to say, the amount of time the chaser takes to get to where the leader is right now, because where the leader is right now is where we have magically placed the finish line.

If that's what "if we stopped the race right now" means for the purposes of things like calculating the virtual GC leader on the road, and if "if we stopped the race right now" is the proper measure for the one true time gap as a function of time, then the one true time gap between a leader and a chaser as a function of time -- I'll call it G -- is the amount of time the chaser is _going to take_ to reach the point where the leader is now. In symbols, where I is the identity function (you give it a time, it gives you back the same time):

G = T2 o L1 - I

That is, G is the time at which the chaser reaches the leader's current position minus the current time. Doing some simple computations, we get:

G = T2 o L1 - I = T2 o L1 - T1 o L1 (since T1 and L are inverses) = (T2 - T1) o L1 (by the definition of composition) = E o L1 (by the definition of E) = D (by the definition of D).

So G = D -- they're the same thing. The one true time gap as a function of time, the one that tells us who the virtual GC leader is (among other things), is D, the stopwatch gap at the _leader_'s position. But the TV shows us A, the stopwatch gap at the _chaser_'s position, and B and C represent other proposed methods for what we should treat as the gap. Why do we have these alternatives, if D is the one true answer?

The reason is simple: calculating D requires us to see the future! We don't know what the stopwatch gap at the leader's position is going to turn out to be yet, because the chaser hasn't reached that spot. This, in turn, shows what the relationship is among A, B, all the possible C's, and D -- A, B, and all the possible C's are all different attempts to _predict_ what D, the one true time gap as a function of time, is going to turn out to be. This is another way of restating Ildabaoth's point that "The most accurate possible time measure system would require a real time speed predictor (as in it needs to predict future peloton speeds, not just estimate current ones)". A, B, and all the possible C's each have some conditions under which they work well as predictions of D, and other conditions under which they are a long way off from D. From this perspective, A is an estimate of D which assumes that the chaser is going to take the same time to cover the distance between the chaser's current point and the leader's current point as the leader did. B is an estimate of D which assumes that the chaser is going to maintain a constant speed between the chaser's current point and the leader's current point. All the possible C's are other estimates of D which take further factors into account, such as terrain, weather conditions, and who knows what else.

In that sense, A, B, and all the possible C's are peers -- they're estimates of D which are good (close to D) under some conditions and bad (far from D) under some conditions. The question "Is there one true time gap as a function of time?" is both settled and forever unsettled -- it's settled in that there is an answer, D, but it's unsettled in that we don't know what D _is_ during the race, so we can spend forever fiddling with measuring different factors and feeding them into different prediction functions to see if we can get better and better estimates of D.

However, our ignorance of the one true D does not last forever: once the race is over, we can actually go back and compute the exact value of D at any given time. This is another point that I do not think has made yet in this thread. Here's how you do it. First, I'm going to define a couple more auxiliary functions:

F = T2 o L1 (definition)
P = T1 o L2 (definition)

Here's what these functions mean. They're both functions from time to time. To calculate F, you give it the current time, and it feeds it to L1 -- the leader's current location -- and then it feeds the result of that to T2 -- the time at which the chaser reaches the leader's current location. Therefore, F is the function that, given the current time, tells you at what time in the future (hence "F") the chaser will reach the leader's position. Similarly, P is the function that, given the current time, tells you at what time in the past (hence "P") the leader was at the chaser's current position. For all times, we have P < I < F (the past is before the current time, and the future is after it -- brilliant, right?). P we can calculate during the heat of a race, but F requires us to know the future, so we can only figure it out retrospectively, after we have the race on tape. P and F are inverses: Finv = (T2 o L1)inv = L1inv o T2inv = T1 o L2 = P.

With those definitions made, here's a way of computing D:

D = E o L1 = E o I o L1 = E o L2 o L2inv o L1 = A o T2 o L1 = A o F.

This gives us a means of computing D in terms of A (the time gap we see on our screens): if we want to know the _true_ time gap at a given time, we wind the tape forward until the chaser reaches the place where the leader was at the original time (that's computing F), and then read what the screen says the time gap is at that future time (that's computing A o F).

How far forward will we have to wind the tape to compute D at a given time? Let's call that value "W" (for "wind"). We wound the tape forward to the time F, whereas the current time as a function of current time is the identity function I, so W = F - I. But the amount we've wound the tape forward is just enough for the chaser to reach the leader's current position -- which is the very D we're trying to calculate (in symbols, W = F - I = T2 o L1 - I = T2 o L1 - T1 o L1 = (T2 - T1) o L1 = E o L1 = D; note, for future reference, that we also have, symmetrically, A = E o L2 = (T2 - T1) o L2 = T2 o L2 - T1 o L2 = I - P, and therefore P = I - A -- that will be useful later). This suggests another perspective on method A, the gap we see on our screens. From the "estimate of D" perspective, A is just like B and all the C's, an estimate with some strengths and some weaknesses. But there's another perspective on A as well. If D = A o F, then we also have the following:

A = A o I = A o F o P = D o P = D o (I - A) (remember we computed earlier that P = I - A).

(...continues in next post...)

 

[rokopt]

(...continued from previous post...)

This equation, A = D o (I - A), is a special relationship between A and the one true gap D, which B and all the C's do _not_ have. What it means in words is that the time gap we see on our screens -- method A, also known as the stopwatch gap at the _chaser's_ position -- is exactly equal to what the one true gap D (the stopwatch gap at the _leader's_ position) _used to be_ at a time which is in the past by exactly the value of A itself. If we keep this in mind, it resolves potential problems with interpreting it. If we see on our screens in the heat of a race that the time gap is five minutes, what that really means is that the one true gap _was_ five minutes -- five minutes ago. The bigger the gap, the more out of date it is.

When viewed from that perspective, A isn't a prediction anymore. Instead, it's a measure of the most up-to-date information that we have for _certain_ -- it's the most recent value of the one true gap that we can calculate definitively, _without_ doing any prediction, and it's also exactly equal to the _age_ of that calculation (how out of date it is). The purpose of B and all the C's is to try to make some predictions, and here's the tradeoff in doing so: because B and the C's involve prediction, they become non-definitive and disputable, but they potentially provide more up-to-date information. A gives you something indisputable, but out of date. Because A is not only the value of the gap but also the value of how out of date it is, B and all the C's perhaps have more and more value as A gets bigger and bigger -- if our screens are telling us that the gap is half an hour, that really means that the One True Gap (method D) was half an hour half an hour ago, and a lot could have changed in the last half hour (the leader could have hit a big climb some time during the past half hour and completely blown up on it, for example, but method A wouldn't reflect that yet).

For those who want to do a little more math, I'll finish with an appendix about the instantaneous rates of changes of the main quantities I've examined: A, D, and E. I'll use one more definition: V (for "velocity", which in this model is pretty much a synonym for "speed") to mean a rider's speed as a function of that rider's location (whereas S is speed as a function of time). With that definition added to the existing ones, we have V = S o T. Note that, by some derivative laws, we can compute V = S o T = L' o Linv = 1 / (Linv)' = 1 / T'.

First, let's look at E' -- the rate of change of the stopwatch gap as a function of location (the one true gap):

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

This makes sense -- the gap E grows (E' > 0) at a particular location if the leader goes faster at that location than the chaser (V1 > V2), and shrinks (E' < 0) if the chaser goes faster than the leader (V2 > V1). That's why it also makes sense when we say things like "De Gendt lost two minutes to the GC favorites on the last climb" -- we can break down the final gap into the gaps at each location (the final gap is the integral of E' over the whole course).

What about the time we see on our screens -- what makes that change? A similar application of derivative laws (I'm going to start skipping all the intermediate steps now) gives us the following:

A' = ((S1 o P) - S2) / (S1 o P).

This tells us that the gap that shows up on our screens goes up when S1 o P > S2 -- that is, when the speed the leaders _were_ doing back when they were where the chasers are now is greater than the chasers' current speed. It goes down when the opposite is true.

But the one true gap changes in response to different inputs:

D' = (S1 - (S2 o F)) / (S2 o F).

D' > 0 when S1 > S2 o F -- the _true_ gap goes up when the leaders' speed is greater than what the chasers' speed _will_ be when they reach the spot where the leaders are now. I think that this corresponds to our intuition of the race -- when we see the break just start to fall apart on the last, gigantic climb, the time gap we see on our screens does not yet reflect the break's collapse (assuming the race is using method A, and assuming the measurements are being taken accurately), but we feel that the _true_ gap _is_ shrinking, because we know that when the GC contenders get to that same climb, they're not going to slow down by nearly as much as the daring but doomed breakaways (so they're going to be going faster than the breakaways are now). But note that we don't really know what D' is at a given point during the race, just as we don't know D.

Here, then (at last!) is a summary of my entire argument: at a given time during the race, D, the one true gap, and D', the rate of change of the one true gap, depend on quantities that we don't know yet. We don't know what the real gap is, and we don't know whether it's rising or falling. But we can make guesses, using various different functions such as B and the C's, and furthermore, after the race is over, we _can_ calculate D and D' precisely, so then we could even compare, over the course of many many races, how close various different candidates for C come to D, and thus, over the course of time, refine our methods for computing C (i.e. for estimating D during the race). Also, during the race, A gives us a continuous picture of what D really _was_ -- some time ago equal to A itself.

 

[CobbleStoner]

remind me again which way they have been doing for over 100 years? probably should stick with that one...
it is irrelevant which way is more accurate, whatever way the riders are used to is best for them when it comes to judging when to start riding

 

[cineteq]

All calculations are there

 

[quiqui]

Thanks rokopt, that was a good read and I fully agree with your analysis and conclusions.

 

[Ildabaoth]

Quite an impressive analysis, rokopt . A few opinions on the matter, tho.

Ildabaoth and Potomac have speculated about predictions that would take into account terrain, weather conditions, and other factors.

I didn't speculate about developing such predictors; I was joking. And unless my ironymeter is damaged, Potomac was joking too. I'd never trust a predictor when the system is so chaotic, multivariable and non linear as this one.

Given our assumptions, L is invertible, so I shall call its inverse T, which is also an analytic, strictly increasing function.

First we need to know the function L=f(t), which isn't possible for t>t0, where t0 is current time. Besides, even if calculable, it would be a complex function for t<t0 (and actually, we don't even need to know L for t<t0, so null point).

If there's one true time gap as a function of location, can we convert that to a time gap as a function of time.

Again the same problem, that function while existing, it is impossible to calculate or even to accurately predict.

First, let's look at E' -- the rate of change of the stopwatch gap as a function of location (the one true gap):

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.

over the course of many many races, how close various different candidates for C come to D, and thus, over the course of time, refine our methods for computing C (i.e. for estimating D during the race)

This is one of my main issues. Even if we consider many many many races, the estimation of D isn't by far a solvable problem. Especially because it would depend on a lot of variables, many of them non predictable even by the best system. Even if 2 samples are exactly the same (and it would be impossible to have 2 identical samples), the output might and will be different by even slight non predictable variables. No system may predict how willing is a peloton to chase a particular breakaway and how strong it will be to actually chase (do we input fatigue, lactate, will power, etc, as system variables?), for example, and that would change future speeds, which would change the predicted virtual gap. Granted, somebody can say that even if it is impossible to have a perfect predictor, a closely accurate one might be done. However, that isn't the case either. Not even meteorologic predictors are that accurate and in this case the prediction is even more multivariable and chaotic. So it isn't a matter of current predictors and metrics are obsolete, but better ones will come, but a matter of there are predictors that can't be created due to chaos.

Finally, just for the sake of discussion, I would hate if we could have a predictor such as the peloton will catch the breakaway in 12 minutes, 38 seconds. I enjoy the fact that we didn't actually know if Nibali would have been caught in LBL. Had we been able to predict speeds over time, including Nibali's and Iglinsky's ones, we could have skipped the whole finish of the race. Besides, didn't we agree that we hate this modern thing of breakaways being so controlled that the peloton knows where and how much to pull to catch them? Well, we are proposing something even more horrible!

 

[barmaher]

This is one of the most insane threads I have ever read on the internet.

Addressing a few of points that have been made:
1. The breakaway and the peloton both accelerate at the same time….this means that the time gap is artificially large. Say the breakaway have travelled 110k and the peloton has travelled 100k on a 200km stage. Both traveling at 40km/h. Time gap will be 15 minutes. That is if we do what has been done for 100 years, and measure the time gap at the latest available point on the course (as both groups cross the 100km point). If both instantaneously accelerate to 50km/h, that means the peloton will complete the course in 2 hours, and the breakaway in 1hr 48 minutes, so the argument seems to be that the 15 minute gap is artificially large. But how do we know that the peloton (or indeed the breakaway) will maintain this pace to the line? What if the speed of the bunch increased because of an attack and the speed of the break because they have decided to push on? The only possible way is to use time. The rest is just conjecture.

2. Using relative speeds and distance between the two groups would be better. This completely ignores the fact that racing takes place on different terrains, with vastly different speeds. Say the stage profile looks like this, flat with a 2km up section followed by a 2km descent. _____________________/\_______________. Speed on the flat bits is 40km/h. Speed on the uphill is 20km/h. Speed on the downhill is 60km/h. The gap is 10 minutes, the break hits the climb. Using any type of average speed, relative speed horsesh!t is going to make it look like the gap is artificially small as the break hits the ascent, artificially large as the break hits the descent, with similar but opposite effects as the peloton hits the climb / descent. Crazy. And even flat stages have gentle uphill drags and faster sections and corners and pee breaks and windier sections and cobbled sections and other things which affect both breaks and chasers, but which would generate way too much spurious volatility in the “gap” between the two groups.

3. Some type of multivariable model could solve for these difficulties. In theory, yes. But let’s look at it this way. When the racers cross the finish line, the time gap between each of them is used to decide relative merits. So having a checkpoint or numerous checkpoints is the purest way of estimating this. All of the factors above would dilute accuracy. You could then develop a multivariable model to account for these factors. But it would be almost impossible to do, and it would still (in my view) not get close to the accuracy of a simple stopwatch at a checkpoint.

And addressing some of the peculiar points raised on this thread.

“Personally I feel distance gives a more objective measure.”At various stages, TdG was 2.5km, then 1.5km, then 1.7km ahead of the pack and at the end (I think) 1.3km ahead of the pack. This means absolutely nothing to me.

“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.”
They clearly are. If those speeds are maintained to the line, the time gap at the line will be halved. It will be 1.5 minutes!!!!! And that 90 seconds will be used in GC, not the 500m!!!!!


“Wouldn't it be simpler just to present an estimate how long it will take for the peloton to catch the break at the current speed of the two groups - alongside existing measures of the time gap?”
No, as I explained above…if the peloton are on a slight incline, and the breakaway on a slight downhill, this will be horsesh!t.

“I think there should be a quotient of distance and current speed”
No there shouldn’t, again for the reasons outlined above.

 

[RownhamHill]

This is one of the most insane threads I have ever read on the internet.

I love this thread!

 

[Dekker_Tifosi]

 

[Bavarianrider]

With modern Gps technique it should not be a problem to no the actual graident of the road all the time. So in order to adjust speed differences caused by different gradients a mathematic formula could help:

0% gradient is 100%
1% climb = 105
2% climb is 110% and so on


1 % downhill = 95%
2% downhill = 90%....

/These are just odd example numbers. Of course an accurate calculation would be needed how gradient excactly influences speed.

So peleton is riding 40km%h on 1% climb: Real speed= 42km/h
Break is riding at 50km/h on a 1% downhill: Real speed is 47,5km/h

Those real speeds could than be used for formulas like my realgapmeter or stuff like that.

 

[Buffalo Soldier]

What about cobbles? Wind?

 

[Bavarianrider]

What with cobbles?

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.

 

[simoni]

“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.”
They clearly are. If those speeds are maintained to the line, the time gap at the line will be halved. It will be 1.5 minutes!!!!! And that 90 seconds will be used in GC, not the 500m!!!!!

Which makes sense as the breakaway has sat on its **** riding 20kph for that km whereas the bunch got its finger out and rode at 40kph. They'll cover that km twice as fast and gain 1.5minutes. What we'll see is a dropping of the time gap as the bunch hammers that km which makes perfect sense to anyone watching surely? And given that its impossible to predict when and where the relative speeds of break and bunch will change anything other than reporting the simplest variable to actually measure just doesn't make sense.

But the most important argument is surely that GC is decided on time so noting the difference in anything other than the latest measured time gap (whether you do it by GPS or a bloke on the side of the road) is just nuts!!

I'd love to have seen David Duffield explaining some of that algebra though...

 

[barmaher]

With modern Gps technique it should not be a problem to no the actual graident of the road all the time. So in order to adjust speed differences caused by different gradients a mathematic formula could help:

0% gradient is 100%
1% climb = 105
2% climb is 110% and so on


1 % downhill = 95%
2% downhill = 90%....

/These are just odd example numbers. Of course an accurate calculation would be needed how gradient excactly influences speed.

So peleton is riding 40km%h on 1% climb: Real speed= 42km/h
Break is riding at 50km/h on a 1% downhill: Real speed is 47,5km/h

Those real speeds could than be used for formulas like my realgapmeter or stuff like that.

Or just quote the time gaps.

 

[Ildabaoth]

This is one of the most insane threads I have ever read on the internet.

Completely agree.

2. Using relative speeds and distance between the two groups would be better.

In case you are addressing to:

With this I don't want to imply that a better metric can't be created. Of course it can, for example using relative speeds and distances. The whole matter, nevertheless, is that it shouldn't be used any metric that requires an estimation of instantaneous speed in the future, because it is a very very difficult non linear prediction and its chaotic nature make it impossible to have an adequatelly accurate estimation.

I think I made it clear that I wasn't talking about instantaneous relative speeds, but about predicted relative speeds along the course] if and only if such prediction could be done[/B]. That's the reason I pointed out that since it requires an estimation of speeds in the future, I wouldn't use it, due to the impossibility to create such predictor.