mathematical formulas for daniels paces?

I don't know what those paces mean. If you give a source, I can try to explain the math.

check out p.54 and 55 on this link for a source

check out p.54 and 55 on this link for a source

http://books.google.com/books?id=mv4gURVxf84C&pg=PA169&lpg=PA169&dq=jack+daniels+r+pace++books&source=bl&ots=Ksv84DYUX7&sig=ESN63bcziPVIId5K5mZQYLQv3nA&hl=en&ei=kcGOTNq_KsGqlAeHwuHrAw&sa=X&oi=book_result&ct=result&resnum=10&ved=0CDwQ6AEwCQ#v=onepage&q&f=false

Check out

http://www.attackpoint.org/trainingpaces.jsp

.

I don't think they provide the formula, but (assuming their calculations are correct) you can extract the data for your spreadsheet. Or you could ask them for their formula.

And a german description:

But the formulas are Haskell code and pretty straightforward to understand.

There is also that huge spreadsheet from

It already has the Daniels paces calculated. But I find Excel formulas horrible to reverse-engineer.

Not sure about his formulas and how they apply. He uses a 10% reduction in speed (multiply by 2.2 to go from distance X to distance 2 X where this is applied to 800 and 1600 meters, for instance.

Here is a long post I made in a thread about drop off in speed by event. Here I was making (in the second table) an adjustment for the time lost getting up to speed.

As for the numbers, there is a dropoff in speed as you go from anaerobic to aerobic distance and so you need a larger factor early (e.g., 400 to 800, 800 to 1600) when doubling the distance and then it declines until you reach an equilibrium. Now, these values are 'elasticities' "%change in pace"/"%change in distance". If you want to discuss more, see the original thread first and then PM me there.

[http://mb.trackandfieldnews.com/discussion/viewtopic.php?f=1&t=41279&p=668920#p668920]

400 to 800 -24.5% -25.5% -26.5% -27.5% -28.6% -29.7%

800 to 1600-14.2% -14.6% -15.1% -15.5% -15.9% -16.3%

1500 to 3000 -10.2% -10.4% -10.6% -10.8% -11.0% -11.2%

3000 to 5000 -6.40% -6.49% -6.59% -6.68% -6.78% -6.88%

The percentage change basis used here can give a slightly partial view of things. Thus, being an economist I will bring in a couple of things, Acceleration Lag (getting up to speed, including reaction time), because the underlying item of interest is in the speed that the runners can carry through for the distance, and if you timed the athletes from 20m to 120m you would be getting a somewhat different picture at the short end. Also, I used Elasticities; %ChangePace/%ChangeDistance. Finally, I calculating Elasticities from midpoints rather than end points for the percentage change of the distance interval calculations.

The first table has the Dist of the end point (e.g., 200 means 100 to 200 changes) and the last column has an adjustment for starting Lag

Dist Elas Pace Elas Adj (1.5 seconds)

200.0 -0.16% -8.65%

400.0 -11.1% -15.1%

800.0 -14.5% -16.2%

1500.0 -8.06% -8.76%

3000.0 -6.45% -6.81%

5000.0 -3.10% -3.24%

10000 -3.99% -4.09%

21097 -4.80% -4.85%

42195 -5.97% -5.99%

Next I generalize the adjustment for different starting lags; we can see that taking into account the time it takes to get up to speed significantly alters the picture at the short end. For a long enough Lag Factor, there is not the initial flat portion. Interestingly, as you get to the long distances (off the track) you get a slow down. Part of this is reduced peak efforts in the population over fast courses, all of which are slower than good tracks. Also, it indicates that the switch in energy systems seems to be an increasing limitation.

0.5000 1.0000 1.5000 2.0000 2.5000 3.0000

-4.31% -8.74% -13.6% -18.8% -24.6% -31.0%

-19.9% -22.1% -24.5% -27.0% -29.6% -32.2%

-24.5% -25.5% -26.5% -27.5% -28.6% -29.7%

-14.2% -14.6% -15.1% -15.5% -15.9% -16.3%

-10.2% -10.4% -10.6% -10.8% -11.0% -11.2%

-6.40% -6.49% -6.59% -6.68% -6.78% -6.88%

-6.16% -6.21% -6.27% -6.32% -6.37% -6.42%

-6.92% -6.94% -6.97% -6.99% -7.01% -7.04%

-9.24% -9.25% -9.26% -9.27% -9.28% -9.29%

I am missing something here. If you have the tables already, why not just enter the data into the cells? You do not need the formulae for that.

You can scan the tables and then import into Excel if you know how to do it. Do not ask me to explain it as the explanation would probably take me an hour to write and then I would still screw it up. But if you scan it to a .pdf or .html you can do a search on how to import.

luv2run's answer might be the shortest for you, just retype the table by hand.

I have some ideas though, how I would approach the problem.

First, it helps to know how Daniels/Gilbert calculates VDOTs (or a pseudo-"vo2max") from a performance:

percent_max = 0.8 + 0.1894393 * e^(-0.012778 * time) + 0.2989558 * e^(-0.1932605 * time)

vo2 = -4.60 + 0.182258 * velocity + 0.000104 * velocity^2

vo2max = vo2 / percent_max

where time is in minutes and velocity is in meters per minute.

I don't know, but I suspect then that Daniel's E/T/R/I paces are some nice constant percentage of his calculated "vo2max", e.g. 70%/90%/105%/110% (I don't know exactly).

Once you have the "vo2max", and the desired percentage, you can solve the above equations for velocity and/or time, and then figure out the desired pace.

But I won't do the math now to see if my hunch is right.

Daniels arrives at his paces as a percentage of velocity of VO2 max. If I recall his I pace is 98%, and his T pace is 86 to 88%. I don't remember the R pace? The problem with arriving at a formula to plug in based on 5k time is that this time for a fixed distance. Basing it on velocity of VO2 max is using a fixed effort. It is the velocity that a person could run at for 11 minutes. In this instance, the pace changes, but it's for a fixed amount of time, so the proper intensity can be easily calculated.

You could fudge the formula based on 5k time, but it won't be exactly the same for everyone.

His R pace is roughly 24 seconds per mile faster than I pace, which is roughlyu 24 seconds per mile faster than T pace. Youi could just use 5k pace, plus 24 seconds per mile for T pace, or minus 24 seconds per mile for R pace. The farther your 5k time is from 11 minutes, the slower all these paces would be.

A mathematical analysis of Daniels tables -- do simple formulas exist?

A close look at the tables suggests a couple of things:

1) How big are rounding errors in the table? For example, I calculate, using Daniel's exact equations, the mile time for a VDOT of 30 as 9:10.37, or 9:10, but the table indicates 9:11.

2) Have the values been manually adjusted later? Creating formulas and comparing them highlights some ranges are not always "smooth". This site

http://www.attackpoint.org/trainingpaces.jsp

notes that "Note that the new E/L paces are slightly slower than recommended in previous editions."

Nevertheless, here are some formulas that come reasonably close.

Daniel's/Gilbert's formulas for determining VDOT: The equations for converting a race performance to a VDOT is well documented in Daniels/Gilbert's Oxygen Power. Given a race time T, and a race distance DIST:

PERCENT_MAX = 0.8 + 0.1894393 * EXP(-0.012778 * T*1440) + 0.2989558 * EXP(-0.1932605 * T*1440)

VO2 = -4.6 + 0.182258 * (DIST/T/1440) + 0.000104 * (DIST/T/1440)^2

vDOT = VO2 / PERCENT_MAX

Note that multiplying and dividing times by 1440 allow to enter and display times in Excel's "hh:mm" format.

Assuming some target paces are at a fixed percentage of VDOT, we can substitute PERCENT_MAX, and solve the equations for T, yielding a general Excel formula, based on DIST, VDOT, and PERCENT_MAX:

TARGET =(DIST*2*0.000104)/(-0.182258+SQRT(0.182258^2-4*0.000104*(-4.6-PERCENT_MAX*VDOT)))/1440

Note that dividing times by 1440 allows us to display the times with a more user-friendly "hh:mm" time format.

E-Pace: PERCENT_MAX = 67.0%

TARGET =(DIST*2*0.000104)/(-0.182258+SQRT(0.182258^2-4*0.000104*(-4.6-0.67*VDOT)))/1440

Note this seems to fit well for VDOTS 36-85, but for VDOTS 30-35 the table times are a couple seconds faster, up to 67.5% for a VDOT of 30.

M-Pace: PERCENT_MAX = ~82%

M Pace is harder to calculate, in a single formula, as the PERCENT_MAX is not fixed across all VDOTS. While 82% yields reasonably close values, in fact, they range from 80.74%, for a VDOT of 30, to 84.27%, for a VDOT of 85. Re-arranging the equations above, and solving for T, doesn't seem straightforward. It's easier to write a script, or a program which implements an iterative search for the marathon pace.

T-Pace: PERCENT_MAX = 88.0%

TARGET =(DIST*2*0.000104)/(-0.182258+SQRT(0.182258^2-4*0.000104*(-4.6-0.88*VDOT)))/1440

I-Pace: PERCENT_MAX = 97.5%

TARGET =(DIST*2*0.000104)/(-0.182258+SQRT(0.182258^2-4*0.000104*(-4.6-0.975*VDOT)))/1440

R-Pace: PERCENT_MAX = ~120%

Like M-Pace, R-Pace is not fixed to a single PERCENT_MAX, but more likely, something like "mile" pace, requiring another iterative solution. In fact here:

http://www.coacheseducation.com/endur/jack-daniels-aug-00.htm

he says that R-PACE should be specific to your event. That is a VDOT 60 miler, and a VDOT 60 marathoner should have different R-Pace targets.

Assuming that 3K pace is the "fastest" aerobic pace, I did a simple curve fitting exercise, which produces reasonably close times.

200m target: 92.95% of the time to run 200m at 3K race pace

400m target: 94.50% of the time to run 400m at 3K race pace

800m target: 95.28% of the time to run 800m at 3K race pace

Note the values fit the table reasonably well, but some table ranges seem to me not to be smooth. For example my formula for 800m for VDOT 72 is off by 2 seconds. Maybe it's my formula, but the table progression doesn't seem quite right to me.

More notes:

Given that the tables can be off by at least 1 second (see first point at top), and in most cases 1 or 2 seconds doesn't matter much anyway, as targets may have to be customized for individual athletes anyway, these formulas seem reasonably close.

You can use Excel formatting for times: "[m]:ss", or "[s]", to "mimic" the table. In this case, times can be converted back and forth by multiplying and dividing by 1440.

I-Pace is limited to 5:00 or below

R-Pace is limited to 2:30 or below

I think one of the best ways to look at training paces is as a percentage of velocity at VO2 Max. This is consistent with what Daniels does, although we have slight differences on how we breakout our pacing groups. I have 3 more pace groups than he formally uses I think.

I use the following standard for determing velocity at VO2 Max (or what Daniels terms vVO2) = the pace which you can hold for 11 minutes in an all out effort. I believe this is consistent with what Daniels does, but you may want to refer to one of his books or e-mail him to confirm.

Based on this vVO2 Max pace I utilize the following training paces:

Recovery Pace: 65-70% of vVO2 Max

Easy Pace: 70-75.5% of vVO2 Max

Steady State: 75.5% - 81.5% (core range: 77-80%)

Aerobic Threshold: 81.5% - 87.5& (core: 84-86%)

Lactate Threshold: 87.5% - 92.5% (core: 89-91%)

Groove Pace: 92.5% - 97.5% (core: 94-96%)

VO2 Max Pace: 97.5% - 101.5% (core: 99-100%)

Fast Pace: 101.5% - 110% (core 104-108%)

Sprint Pace: 110% - 120% (core 112-118%)

If you don't know your current vVO2, it can be closely estimated based on your most recent race times.

More on my pace charts and calculating your vVO2 based on race times can found here:

http://www.mprunning.com/pace_guide.php

Daniels also talked about a simpler 2.2 + 6 seconds rule.

That is, for fairly fit runners, if, for any reason, you don't have, or don't use the tables, the following approximations are good enough:

- Doubling the distance takes about 2.2x the time.

- The difference between I-pace and R-pace is about 6 seconds, per 400m.

This is a rather funny comparison but these "rules of thumb" that many runners use to judge training paces, because they don't understand the math or science behind it, are similar to the many "rules of thumb" that many Investment Bankers use because they don't understand the exact math and finance behind the deals.

What I've noticed about "Rules of Thumb" is that they are often misremembered or pass along incorrectly and thus implemented improperly, at best they only get you in the right ball park, and they don't account for exception or special circumstances.

If you don't understand the phsiology behind proper training paces, then use charts from someone who does, rather than trying to remember some rule of thumb that may or may not apply or even be passed along properly.

A Daniels' formula spreadsheet already exists. It does pretty much everything. It even has a section that calculates change in projected race time due to temperature changes or changes in weight, VDOT, age, gender, etc. I downloaded it a couple of years ago. It's an excel spreadsheet. With some searching, you should be able to track it down online. The title within the spreadsheet is Daniels' Training Tables Rev2.9f.00

The title of the download is SPREADSHEET.xls, I think.

I see your point about messages getting passed along incorrectly. But I hope you don't imply that I'm passing the message wrongly. It comes straight from his book, and what I said is basically correct, except that I left out one thing:- Not only R-Pace and I-pace, but the difference between T pace and R pace is also about 6 seconds per 400m.It's only for fairly well trained runners though. He says if your mile time is slower than 5:30, better to use his charts.I also have to disagree with your ultimate advice. In spite of my own inclination towards match and science, I don't think runners should need to understand the math or science or physiology. Ideally, they should be able to dial into the right pace by knowing how it feels, and what is required, without the need for a watch or a track. This feeling also factors in individual variability, strengths and weaknesses.

precise numbers wrote:

This is a rather funny comparison but these "rules of thumb" that many runners use to judge training paces, because they don't understand the math or science behind it, are similar to the many "rules of thumb" that many Investment Bankers use because they don't understand the exact math and finance behind the deals.

What I've noticed about "Rules of Thumb" is that they are often misremembered or pass along incorrectly and thus implemented improperly, at best they only get you in the right ball park, and they don't account for exception or special circumstances.

If you don't understand the phsiology behind proper training paces, then use charts from someone who does, rather than trying to remember some rule of thumb that may or may not apply or even be passed along properly.

I was NOT directly referring to you, the information you gave was accurate and directly from Daniels and useful. Sorry for the confusion on that.

I also agree that feel is the ultimate goal. But if you are going to use a pre-determined guide for pace, than I am suggesting using a guide from someone like Daniels, McMillan or MPR who understands the math and science behind it.

Apologies for digging up an old post, but I've gone through the same effort trying to crack the actual formulas behind the training paces and this is what I've found.

As mentioned above, E, T and I paces are relative to one's VO2max. M pace is not. It is (perhaps quite logically) derived from a person's predicted marathon race performance. Use the race prediction formulas as explained by Larry Simpson (http://www.simpsonassociatesinc.com/runningmath1.htm) to calculate one's marathon race performance, then divide by 42.195 (or 26.219 if miles is your thing). Voila, M pace, which aligns with the values in Jack Daniels' tables.

R pace is calculated by subtracting 6 seconds per 400m from I pace. Easy as that.

Anyone have a formula or rule of thumb to adjust Daniels training paces for training in heat and humidity (maybe by heat index)?

I need a formula to calculate my Daniel's paces specific for my height and weight. It should also take into account my hydration levels and the performance indicator on my Garmin heart rate monitor, along with the weight of my shoes and the terrain I am running on. It's probably embarrasing to ask this, but obviously should also include adjustments for barometric pressure.