The purpose of this study was to define a new index the Robust Exponential Decreasing Index (REDI), which is capable of an improved analysis of the cumulative workload. This allows for precise control of the decreasing influence of load over time. Additionally, REDI is robust to missing data that are frequently present in sport.

200 cumulative workloads were simulated in two ways (Gaussian and uniform distributions) to test the robustness and flexibility of the REDI, as compared with classical methods (acute:chronic workload ratio and exponentially weighted moving average). Theoretical properties have been highlighted especially around the decreasing parameter.

The REDI allows practitioners to consistently monitor load with missing data as it remains consistent even when a significant portion of the dataset is absent. Adjusting the decreasing parameter allows practitioners to choose the weight given to each daily workload.

Computation of cumulative workload is not easy due to many factors (weekends, international training sessions, national selections and injuries). Several practical and theoretical drawbacks of the existing indices are discussed in the paper, especially in the context of missing data; the REDI aims to settle some of them. The decreasing parameter may be modified according to the studied sport. Further research should focus on methodology around setting this parameter.

The robust and adaptable nature of the REDI is a credible alternative for computing a cumulative workload with decreasing weight over time.

It is sometimes difficult to obtain regular long-term monitoring data. Training camps, injuries and time-off periods may prevent close follow-up.

When longitudinal sports data are missing, a new method improves existing indices for monitoring load or fatigue.

The Robust Exponential Decreasing Index takes into account the decreasing effect of the past training; it remains meaningful even in the presence of many missing data points.

It is flexible for all situations by adjusting a specific coefficient for each discipline.

One of the fundamental reasons that athletes are monitored is to measure their progress in response to their training.

Banister

Foster proposed the rating of perceived exertion (RPE) session assessment, an index that can be adapted to a large number of activities.

More recently, Gabbett

The ACWR is based on specific sport and injury data.

Menaspà's paper, although presented in an editorial format (ie, the lowest level of evidence), puts forward another limit of the ACWR.

On top of the aforementioned limitations, Lolli

More recently Gabbett

ACWR is a useful method to analyse training load. However, like all tools, it has its limitations, which we have tried to address through our theoretical study.

Williams

The EWMA

where

where

A first limitation of the EWMA lies in the complexity of this recursive equation, which may complicate the interpretation, implementation and computation of the coefficients. Moreover, the way each workload is weighted only depends on the number (N) of days considered in the calculation.

In terms of a long-term follow-up, the EWMA weight coefficients of loads tend to be equivalent to the ACWR and very small (eg, with 100 days, the most recent load accounts for just 2/101 of the total average). Thus, in this context, the EWMA value merely approximates the unweighted average load over N days and decreases the importance of recent workloads in favour of historical cumulated ones.

Therefore, EWMA is more consistent and accurate than the ACWR with a small value for N and a rolling average. However, in this context, both the EWMA and ACWR become very sensitive to missing data and need a period of initialisation that cannot be computed.

On the other hand, the impact of acute load differs greatly according to various sports disciplines. Currently, none of the presented methods provide a parameter able to adjust the decreasing influence of load, depending on sport context.

Using what has already been proposed in the literature and taking into account the different limitations outlined earlier, the purpose of this study was to propose a new way to compute cumulative training loads.

We designed the REDI, a new measurement of cumulated workload, adapted to each sport’s specificities, and allows us to address the issue of missing data. It is defined as

where:

N is the total number of previous days before in our dataset.

λ is a parameter that can be adjusted in order to decrease the weighting.

The main concept of this index is to introduce an explicit exponential weight that multiplies each workload. All of the weighted workloads are added together and subsequently divided by the sums of the weights in order to normalise the index. In this sense, the index is defined as a weighted mean of the workloads.

In our model, the values of the weights decrease as one moves away from the current day (as the value of
i
increases). Moreover, this weight equals 0 when the workload value is missing. This allows consideration of all days instead of ignoring some. However, even if the value of

One advantage of this index is that the coefficients are completely explicit and easily computable for each past workload, through the definition of

The ACWR, EWMA and REDI were studied and compared in different situations with simulated datasets. These simulated data were monitored, throughout a full season, using the same methods that would be implemented for real data, with features that assess workload over time.

In the first situation, 200 consecutive workload values were drawn from a Gaussian distribution N(500 100). This represents rather stable loads with slight perturbations. The second dataset is composed of 200 consecutive workload values, drawn from a uniform distribution U([0,1000]). This situation depicts unstable loads that potentially change sharply over time. These datasets were used to enlighten several properties and behaviours of the different indices according to the context.

In both cases, a portion of the load values (5%, 30% and 50%) were randomly removed in order to simulate the issue of missing data. In this context, errors exist that differentiate the true values—values that result from the indices computed on complete datasets—from those that result from incomplete ones. In order to compare these errors, numerically as well as graphically, the percentage of relative difference was defined as such:

The three methods (ACWR, EWMA and REDI) were applied on the eight series (0%, 5%, 30%, and 50% missing data for both the N(500 100) and U([0,1000] datasets), providing a fictional follow-up of 200 time instants in the different situations described previously.

EWMA and ACWR could not be computed with 30% missing data, as it was impossible to build the indices due to the requirement of 28 consecutive days of load to compute an unbiased index (

The value of the EWMA (on the left) and ACWR (on the right) over 200 instant time period on a dataset with 5% missing load. ACWR, acute:chronic workload ratio; EWMA, exponentially weighted moving average.

Comparison of REDI on a series of 200 consecutive workload values (simulated data according to a Gaussian distribution) with 0%, 5%, 30% and 50% missing data (left). Same comparison on a series of uniformly distributed loads (right). REDI, Robust Exponential Decreasing Index.

In order to test the consistency of the results in

Mean relative errors between REDI values from datasets with missing values, as compared with the complete one

% of missing data | 5% | 15% | 30% | 40% | 50% |

Gaussian data, mean error±SD | 0.65%±0.19% | 1.48%±0.28% | 2.39%±0.43% | 3.16%±0.60% | 3.76%±0.69% |

Uniform data, mean error±SD | 2.13%±0.6% | 4.69%±2.35% | 7.48%±2.09% | 9.65%±2.67% | 11.57%±2.88% |

The mean error and SD come from a REDI computation on 100 simulations of complete and incomplete datasets for each of the six configurations.

REDI, Robust Exponential Decreasing Index.

For both distributions, the REDI curve precisely follows the trend of the EWMA curve (with a 0.65% mean error). In addition, REDI starts earlier than EWMA since it does not need the initial follow-up period for the index to be built (

Comparison between REDI (λ=0.1) and EWMA on a series of 200 consecutive workload values of simulated Gaussian data (left). Comparison between REDI and EWMA on a series of 200 uniformly distributed loads (right). EWMA, exponentially weighted moving average; REDI, Robust Exponential Decreasing Index.

These simulated data were defined with a mean value of 500. In this case, the REDI better reflects the nature of the data than the EWMA; it dependably produces values around 500, while the EWMA sits slightly lower at about 425 on average. Due to its definition, EWMA consistently underestimates the cumulative load. The REDI behaves similarly to the EWMA, only for the particular case when λ=0.1.

Different values of the λ coefficient in the function of the decreasing impact of chronic workload by time windows

λ | 3 days | 7 days | 14 days | 21 days | 28 days |

5 | 0 | 0 | 0 | 0 | 0 |

1 | 0.05 | 0 | 0 | 0 | 0 |

0.5 | 0.22 | 0.03 | 0 | 0 | 0 |

0.3 | 0.41 | 0.12 | 0.01 | 0 | 0 |

0.1 | 0.74 | 0.5 | 0.25 | 0.12 | 0.06 |

0.07 | 0.81 | 0.61 | 0.38 | 0.23 | 0.14 |

0.05 | 0.86 | 0.7 | 0.5 | 0.35 | 0.25 |

0.03 | 0.91 | 0.81 | 0.66 | 0.53 | 0.43 |

0.01 | 0.97 | 0.93 | 0.87 | 0.81 | 0.76 |

0.001 | 1 | 0.99 | 0.99 | 0.98 | 0.97 |

A λ of 0.1 is used as reference. It is close to EWMA behaviour. One can adjust the value of λ according to the table.

This study provides a new training load index including three improvements. First, the REDI is more robust to missing data than either the ACWR or EWMA as it can handle datasets with missing data, preserving the global tendency of the cumulated workload. Second, the REDI allows for better control of the decreasing influence of load over time, through a coefficient that can be computed according to the sport, the disciplines or the event. Third, the REDI can be calculated starting as soon as the second time point within a consecutive series.

From a sporting point of view, the contributions of this work allow for some flexibility when it comes to the collection of data, which is a well-known difficulty in the monitoring of athletes, especially for those at high level. The REDI can facilitate load monitoring throughout the year, whereas the EWMA and ACWRs (which are usually computed over 28 days) lead to biases when missing workload values are not considered. The REDI is capable of considering periods of limited data, such as career duties, international breaks or the lack of monitoring during the off-season,

The use of the REDI avoids the necessity to simulate or ignore the data that are missing in a monitoring context. This adaptive index stabilises the acute and chronic load variables. It is more robust and usable for contextualising the performance, injuries and readiness of athletes. To determine what percentage of missing data are acceptable, REDI users can refer to

The choice of an exponential is explained by its flexible nature. Moreover, the exponential reflects the laws of the body. In physiology, the phenomena of fatigue and overcompensation after an effort are exponentially expressed. One can also control the decay intensity of the exponential through a single parameter, λ.

As an example choice for practitioners, we proposed 10 values of λ for five time windows (3, 7, 14, 21 and 28 days) to illustrate the influence of λ on the decreasing weights assigned to each load. This choice was inspired by what was previously documented in the literature,

A λ value close to 0 provides a very slow decay to the weight of the past training load. In other words, the closer the coefficient λ is to 0, the greater the former load impact. The oldest training loads have as much weight as the most recent ones. Conversely, with an increasing λ value, the weight of the oldest loads is neglected. In other words, recent workload have a much higher impact.

The λ values in this work are only proposed as an example. At this point, one cannot state with certainty which λ should be used for each situation since its optimal value probably depends on discipline. Deciding on the appropriate value must take into account the characteristics of the discipline, as well as the specific influence of the load. We emphasise that this parameter should be tuned using experts’ knowledge and/or statistical optimisation from data to suit the context at best.

In order to choose an appropriate and meaningful λ, several objective methods could be used. For example, λ could be optimised according to a criterion (eg, likelihood maximisation) or via cross-validation. It could also be optimised in order to connect workload to several features (injury

The REDI offers the possibility to manage the λ coefficient for a specific decreasing weight load according to sport, even by position or athlete physiology.

The present paper offers a theoretical study of a new index that can be implemented with simplicity and adaptability. Although REDI’s properties seem promising, the index must now prove its ability to link variables of interest, such as fatigue or injury, like the ACWR. Moreover, while the λ coefficient provides flexibility by decreasing the importance of the load over time and we are able to provide insights on its influence, it is a new parameter that must be justified with experimental data. Future studies, with real-world data, are needed to test the full potential of REDI and to prove its efficacy in practice.

The REDI is a cumulative load analysis tool. It is both robust to missing data and flexible according to the discipline. The robustness of REDI has been demonstrated by its ability to analyse datasets with missing data while preserving the main trend. It is therefore consistent when periods without data (injury, international duties and off-season) are numerous. Moreover, its single control parameter, λ, allows the practitioner to control for the decreasing impact of past training loads. Finally, the REDI can be computed after only 2 days of monitoring and is adaptable by sport, position or level of performance.

We deeply thank Stephanie Duncombe for her proofreading and advice in English.

IM, AL, GS and AS were involved in the design. AL, JS and IM simulated the data. AL refined the model. IM, AL, JS and GS interpreted the results. IM, AL, JS, JFT and AS contributed to the writing and revision of the manuscript. JFT and AS managed the conduction of the research. All authors read and approved the final manuscript.

The authors have not declared a specific grant for this research from any funding agency in the public, commercial or not-for-profit sectors.

None declared.

Not commissioned; externally peer reviewed.

All data relevant to the study are included in the article or uploaded as supplementary information.

Not required.