A regression method for the power–duration relationship when both variables are subject to error

Abstract

Purpose

The power–duration relationship has been variously modelled, although duration must be acknowledged as the dependent variable and is supposed to represent the only source of experimental error. However, there are certain situations, namely extremely high power outputs or outdoor field conditions, in which the error in power output measurement may not remain negligible. The geometric mean (GM) regression method deals with the assumption that also the independent variable is subject to a certain amount of experimental error, but has never been utilized in this context.

Methods

We applied the GM regression method for the two- and three-parameter critical power models and tested it against the usual weighted least square (WLS) procedure with our previous published data.

Results

There were no significant differences between parameter estimates of WLS and GM. Bias and limit of agreements between the two methods were low, while correlation coefficients were high (0.85–1.00).

Conclusions

GM provided equivalent results with respect to WLS in fitting the critical power model to experimental data and for its conceptual characteristics must be preferred wherever concerns on the precision of P measurement are present, such as for in-field power meters.

Appendix

The developments detailed in this Appendix describe how to apply the GM method to the hyperbolic function expressed in Eq. (1). The interested reader can refer to Ludbrook (2012) for practical considerations on linear GM regression and to Ebert and Russell (1994) for an example of adaptation of GM to nonlinear functions, as in the present study.

The assumption behind GM regression is that the ratio between the magnitude of the absolute error in y and in x is similar to the absolute slope of the fitted curve (Brace 1977). In the case of Eq. (1) we set y = T_lim and x = P, thus the slope is equal to:

$$\left|\frac{d{T}_{\mathrm{l}\mathrm{i}\mathrm{m}}}{dP}\right|=\left|-\frac{{W}^{^{\prime}}}{{\left(P-\mathrm{C}\mathrm{P}\right)}^{2}}\right|\sim \frac{{W}^{^{\prime}}}{{P}^{2}}$$

(2)

Is this compatible with the ratio between the magnitude of the absolute error in T_lim and P (ε_Tlim/ε_P)? Yes, both for biological and technological reasons. We know in fact that ε_Tlim and ε_P are T_lim and in P are proportional to T_lim and P, respectively thus:

$$\frac{{\varepsilon }_{{T}_{\mathrm{l}\mathrm{i}\mathrm{m}}}}{{\varepsilon }_{P}}\sim \frac{{T}_{\mathrm{l}\mathrm{i}\mathrm{m}}}{P}$$

(3)

Inserting (1) into (3):

$$\frac{{\varepsilon }_{{T}_{\mathrm{l}\mathrm{i}\mathrm{m}}}}{{\varepsilon }_{P}}\sim \frac{{W}^{^{\prime}}}{P\left(P-\mathrm{C}\mathrm{P}\right)}+\frac{k}{P}\sim \frac{{W}^{^{\prime}}}{{P}^{2}}$$

(4)

Therefore,

$$\frac{{\varepsilon }_{{T}_{\mathrm{l}\mathrm{i}\mathrm{m}}}}{{\varepsilon }_{P}}\sim \left|\frac{d{T}_{\mathrm{l}\mathrm{i}\mathrm{m}}}{dP}\right|$$

(5)

Thus, the assumption required for GM regression is met. Since GM is independent of the choice of the dependent variable, the demonstration is still valid when inverting x and y.

We, therefore, define the loss function as the grey area of Fig. 1, that is absolute difference between the rectangles formed by the thin lines and the integral of the function in the same P interval (white area between the curve and the P-axis). For a generic data point with coordinates (P_i, T_i)

$${T}_{i}\left(\frac{{W}^{{^{\prime}}}}{{T}_{i}-k}+\mathrm{C}\mathrm{P}-{P}_{i}\right)-{\int }_{{P}_{i}}^{\frac{{W}^{{^{\prime}}}}{{T}_{i}-k}+\mathrm{C}\mathrm{P}}\left(\frac{{W}^{{^{\prime}}}}{P-\mathrm{C}\mathrm{P}}+k\right)dP$$

(6)

It is noteworthy that Eq. (6) in this formulation has always a positive value thus there is not the need of a modulo operator. Solving and simplifying yields to the following loss function:

$$\left({T}_{i}-k\right)\left({P}_{i}-\text{CP}\right)-{W}^{{^{\prime}}}\left\{1+\text{ln}\left[\left({T}_{i}-k\right)\left({P}_{i}-\text{CP}\right)\right]-\text{ln}{W}^{{^{\prime}}}\right\}$$

(7)

By setting k = 0 we obtain the 2-p version of the loss function, while $k=-\frac{{W}^{^{\prime}}}{{P}_{0}-\mathrm{C}\mathrm{P}}$ leads to the alternative parameterization of Eq. (1) (Morton 1996). The commands for SPSS are derived from this demonstration are reported Table

Table 2 SPSS commands for GM regression

2, although it must be remembered that equivalent results are granted if starting the demonstration assuming P instead of T_lim as the dependent variable.