Mathematical modeling of the power duration relationship while cycling, Part 1 - A primer on modeling.

1. The New Power-Duration Model in WKO4
November 5, 2013 | Andrew R. Coggan, Ph.D.

2. The power-duration relationship
Linear time scale
Log time scale
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
2

3. The new power-duration model in WKO4
• A heart-warming tale in which the mean maximal
power chart and the power profiling table meet, fall
in love, get married, and have a child that turns out
to be somewhat of a child prodigy.
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
3

4. The new power-duration model in WKO4
• The most advanced mathematical model of the
exercise intensity-duration relationship ever
developed.
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
4

5. Webinar series topics
•
•
•
•
Part I: A primer on modeling
Part II: Limitations of current models
Part III: Introduction of a superior approach
Part IV: Present and possible future applications
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
5

6. The $64,000 question
Why?
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
6

7. The power-duration relationship
Linear time scale
Log time scale
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
7

8. The $64,000 answer
Because having a robust mathematical description
of an individual’s power-duration relationship provides
quantitative insight re. their unique abilities and paves
the way for other analyses.
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
8

9. Importance of new power-duration model
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
9

10. Evolution of rider phenotyping
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
10

11. Uses for the power-duration model
1. Auto FTP
2. Pmax
3. FRC
4. dFRC
5. Phenotyping
6. Individual adaptations scores
7. Auto match finding
8. Identification of outliers
9. Optimal pacing strategy
10. Application to other sports
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
11

12. The new power-duration model in WKO4
Part I: A primer on modeling
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
12

13. Some mathematical models
Coggan AR, Ruby BC, Zderic TW. FASEB J 1999; 13: A1055
Herrero P, et al. J Nucl Med 2007; 48: 955-964.
Herrero P, et al. J Nucl Med 2007; 48: 2046-2055.
Coggan AR, et al. J Nucl Cardiol 2009; 16: 562-570.
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
13

14. Caveats re. mathematical modeling
• Mathematical models are theoretical constructs that
may, or may not, have a basis in reality
• All models are wrong, but some are useful
• The usefulness of any model depends in part upon
its domain of validity
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
14

15. Example of a limited domain of validity
Model structure
Model validation
Coggan AR, et al. J Nucl Cardiol 2009; 16: 562-570.
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
15

16. How do you evaluate mathematical models?
1. Statistical criteria
a. F test
b. Runs test
c. AIC
2. Residuals
3. Parameter estimates
a. Number
b. Independence
c. Precision
4. External validation
a. Distribution
b. Magnitude
c. Bias
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
16

17. Most important take-home message?
R2 is not an adequate indicator of the goodness-of-fit of
a multi-parameter, non-linear model!
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
17

18. R2 as a measure of goodness-of-fit
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
18

19. R2 as a measure of goodness-of-fit
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
19

20. R2 as a measure of goodness-of-fit
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
20

21. How do you evaluate mathematical models?
1. Statistical criteria
a. F test
b. Runs test
c. AIC
2. Residuals
3. Parameter estimates
a. Number
b. Independence
c. Precision
4. External validation
a. Magnitude
b. Distribution
c. Bias
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
21

22. What is a “normalized residual”?
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
22

23. Residuals of linear fit
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
23

24. How do you evaluate mathematical models?
1. Statistical criteria
a. F test
b. Runs test
c. AIC
2. Residuals
3. Parameter estimates
a. Number
b. Independence
c. Precision
4. External validation
a. Magnitude
b. Distribution
c. Bias
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
24

25. The critical power model
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
25

26. Distribution of normalized residuals
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
26

27. Normalized residuals vs. duration
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
27

28. Independence of parameter estimates
Correlation matrix for parameters of critical power model
W’
W’
CP
1.00
0.07
CP
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
1.00
28

29. External validation
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
29

30. Model vs. reality
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
30

31. Distribution of normalized residuals
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
31

32. Normalized residuals vs. duration
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
32

33. Limitations of the critical power model
“The critical power concept is not without its limitations...in
particular, it tends to greatly overestimate the maximal power
that can be generated for only a few seconds, and it predicts that
there should be a power output below which fatigue will never
occur. In addition, the exact values obtained for W' and critical
power depend in part on the testing protocol, e.g., the exact
combination of powers and durations used to define the curve...”
- Yours truly, post to the wattage list on 2/11/2002
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
33

34. Summary
• Modeling the power-duration relationship provides
many potential benefits
• Such benefits, however, will only be realizable with a
sufficiently robust model
• Standard criteria and approaches (which do not
include R2) exist for evaluating mathematical models
• The classic critical power model fails to adequately
describe the entire power-duration relationship
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
34

35. Stay Updated
Receive the latest updates on WKO4 and the science of
training.
Get Updates >
Andrew R. Coggan, Ph.D., Peaks Coaching Group, and/or TrainingPeaks are the owner(s) and/or licensee(s) of all intellectual property rights to this
presentation and the material it contains. These works are protected by copyright laws and treaties around the world. All such rights are reserved.
35