Common trends in time series of exercise testing (WAnT) Bruno, Paula Marta Technical University of Lisbon, Faculty of Human Movement, CIPER Estrada da Costa, Cruz Quebrada 1495-688 Cruz Quebrada – Dafundo, Portugal E-mail: [email protected] Pereira, Fernando Duarte Technical University of Lisbon, Faculty of Human Movement, CIPER Estrada da Costa, Cruz Quebrada 1495-688 Cruz Quebrada – Dafundo, Portugal E-mail: [email protected] Fernandes, Renato Instituto Politécnico de Santarém, Escola Superior de Desporto de Rio Maior Rua José Pedro Inês Canadas, lote 1 – r/c 2040-324 Rio Maior E-mail: [email protected] In this paper we analyze time series related to pediatric exercise tests. A group of young athletes with two years of training practice were assessed in exercise laboratories in order to measure their anaerobic power output. The 30-s all-out Wingate Anaerobic Test (WAnT) was applied to assess performance of peak muscle power and local muscle endurance for both upper and lower limbs. Our objective in WAnT time series is detecting possible common trends. Dynamic factor analysis (DFA) is a technique especially designed for time series data, which enables underlying common patterns to be identified. This methodology allows to model short and nonstationary series, like 30-s all-out WAnT. The classical approach in the physiological field compare performance parameters (Peak Power, Average Power, Power Drop) between independent groups. This paper applies for the first time DFA to exercise tests results. 1. Wingate Anaerobic Test The most popular performance-based anaerobic test and widely used is the 30-s all-out WAnT. The test is usually performed on a mechanical braked cycle ergometer. The commonly available Monark ergometer has often been used. The subject pedals at maximal possible rate against a resistance and pedal revolutions are counting. The load or resistance to be applied is based on individual body weight (Maud, Berning and Foster, 2006). The 30-s all-out WAnT was developed in the mid-1970s at the Wingate Institute in Israel. It has been used extensively with able-bodied children and adults to assess performance of their lower (cycling) or upper (arm-cranking) limbs. This kind of laboratory-based exercise tests in pediatrics are used to aid in a diagnostic procedure or to assess the child’s physical fitness. In various pediatric disorders, peak muscle power and local muscle endurance are deficient. Knowledge of changes with time in these functions can help in evaluating the progression of a disease or the effects of a rehabilitation program (Bar-Or, 1993). Considerable research has evaluated the feasibility, reliability, validity, and sensitivity of the WAnT (Bar-Or, 1987). 2. Database A group of twenty young athletes engaged in competitive swimming and football for two years of practice were selected. This group of children was homogeneous in age, weight and body mass index. Each child was assessed in a cycle Monark ergometer for both arms and legs, and the power output given in rotations per minute (rpm) was registered along the time (Pereira, Fernandes and Mendonça, 2005). Figure 1 shows sequence plots of the rpm versus time. Both plots suggest some variability between children, and there seems to be different strategies in the execution of the exercise. Time plots are also important in what concerns identification of possible outlier. In this context, a time series with very low and constant values means that the children doesn’t pedal at maximal possible rate, and so doesn’t perform test correctly. In that sense, two subjects were leave out (initially there are twenty two child). We also detected in first visual approaches that the last times in many series were probably miss registered. The values showed some inconsistency and we suppose that it had no concern with the performance of the exercise. To outperform this problem we remove observations after 27-s in all time series. Figure 1 presents cleaned series analyzed in this paper. 3. Dynamic factor model Let y t be a n − dimensional vector of observed time series, a model for y t with m common trends is given by y t = c + Zα t + εt , where c is the m − dimensional vector of constants, Z is the n × m matrix of factor loadings, α t is the m − dimensional vector of common trends, ε t is the noise component, and it is assumed that εt ~ N ( 0, Σε ) . The trends represent the underlying common patterns over time and they can be modeled as α t = α t −1 + ft , where ft ~ N ( 0, Σf ) , Σf is a diagonal error covariance matrix, and ft is independent of εt . For further details consult for example (Tsay, 2005). Estimated models presented in this paper were obtained with the software package Brodgar (www.brodgar.com). The software enables estimating a model under a chosen m . It also allows to decide between a diagonal or a symmetric positive-definite covariance matrix Σε . After a standardization of series required to better interpretation of factor loadings and common trends, sixteen models were applied, eight for each kind of limbs. The m value is varying between one and four, and for each m was tried a diagonal and a symmetric covariance matrix. In order to achieve a parsimony model, the aim is to set the number of common trends as small as possible, but still have a reasonable model fit. In time series analysis, the most widely function used to measure parsimony is the Akaike’s information criterion (AIC), see (Akaike, 1974). The AIC values for each tried model are presented in Table 1. A lot of similarity is seen when we compare both arms and legs models. In each case a model with a symmetric positive-definite covariance matrix Σε is preferable. Models with only one common trend are much less explicative than the others, and the AIC values for the models with four trends reveal that great number of parameters is no justification. So we need to decide between models with two or three trends. Only based on AIC we choose models with three trends, but physiological interpretation is also important. We think that third trend in these models is not important in terms of explanation to the underlying patterns for rpm series, then we considerer the model containing two common trends with a positive-definite covariance matrix as the “best” model for both upper and lower limbs. Results of the two models are reported below. 4. Results First we analyze results for upper limbs. In Figure 2 are presented the estimated common trends. One of the common trend show a gradual rise until peak power is achieved near 10-s, peak values is kept for during 5-s, and after that a drop power is observed (this trend will be called as late peak power and drop). The other common trend begins with the peak power, followed by a sharply decrease until near 15-s, and then a slight decrease almost constant (this trend will be designated early peak power and drop). We present factor loadings (after rotation) in Table 2. The factor loadings show that children from football are essentially associated with late peak power and drop trend, while early peak power and drop is important to children from swimming. In Figure 3 are presented the estimated common trends for the lower limbs and we can see a high similarity in shape of the underlying structure when compared with the arms. Factor loadings, in table 3, show once more that early peak power and drop trend is associated with swimming and late peak power and drop is related to football. It must be noted that, in general, children follow similar pattern for both arms and legs. For example, children 2 and 3 from swimming aren’t associated with early power peak trend like other swimming’s children, but they have the same pattern in both limbs. 5. Discussion DFA applied to power output time series enables to identify two common structures for each upper and lower limb. In the execution of the exercise, with arms are performed similar shape strategies as with legs. Also a very important detail is to know when peak power happens. Neither of these conclusions was possible before with classical approach. An improvement when dealing with this approach is it capability to find that each trend is associated with a different sport. In agreement with literature, peak power reflects muscle explosiveness and mean power reflects muscle endurance. Thus both peak power and mean power can be considerer components of child’s fitness. Both peak power and percentage fatigue are correlated with the preponderance of fast-twitch fibers, mainly in swimming athletes. In short it can be said that the anaerobic test reflects the changes due to training, the specificity of the muscle group (arms or legs) on one hand and the individual differences in muscle function on the other, due probably to development (maturation, growth) factors. Despite this is a preliminary draft, the authors are very confident with the results, they are assured that DFA could be an important tool in the study of the performance based on anaerobic functions. Figure 1 125 power output (rpm) power output (rpm) 125 100 100 75 50 75 50 25 25 0 5 10 15 time (sec) 20 25 0 5 10 15 time (sec) 20 25 Upper limbs rpm series in the left graph and lower limbs rpm series in the right graph Table 1 AIC values obtained from DFA for the time series in study common trends 1 2 3 4 upper limbs diagonal symmetric matrix matrix 929.842 -425.524 -113.800 -463.282 -302.693 -478.629 -376.260 -479.164 lower limbs diagonal symmetric matrix matrix 898.827 -266.537 137.168 -300.753 72.329 -320.020 -175.344 -328.512 Figure 2 8 8 4 4 0 0 -4 -4 -8 0 5 10 15 20 time (sec) 25 -8 0 5 10 15 20 time (sec) 25 Common trends for rpm upper limbs series Table 2 Estimated factor loadings for rpm upper limbs series (label F for football and S for swimming, number associated represent children, and in bold are factor loadings above the cutoff of 0.2) serie trend 1 trend 2 serie trend 1 trend 2 factor loadings factor loadings F1 F2 F3 F4 0.29 0.24 0.11 0.22 0.07 0.15 0.26 0.14 S1 S2 S3 S4 0.11 0.29 0.29 -0.03 0.27 -0.04 -0.09 0.29 F5 0.15 0.26 S5 0.03 0.29 F6 F7 0.26 0.09 0.03 0.28 S6 S7 0.03 -0.01 0.29 0.23 F8 0.24 0.17 S8 0.02 0.29 F9 F10 F11 0.29 0.31 0.27 0.08 0.05 0.12 S9 0.06 0.26 Figure 3 8 8 4 4 0 0 -4 -4 -8 0 5 10 15 20 time (sec) 25 -8 0 5 10 15 20 time (sec) 25 Common trends for rpm lower limbs series Table 3 Estimated factor loadings for rpm lower limbs series (label F for football and S for swimming, number associated represent children, and in bold are factor loadings above the cutoff of 0.2) factor loadings factor loadings serie trend 1 trend 2 serie trend 1 trend 2 F1 F2 0.17 0.22 0.22 0.18 S1 S2 0.10 0.31 0.26 -0.04 F3 0.13 0.26 S3 0.27 0.08 F4 F5 0.26 0.14 0.10 0.24 S4 S5 0.06 -0.04 0.28 0.29 F6 0.29 0.05 S6 0.06 0.29 F7 F8 0.26 0.24 0.11 -0.18 S7 S8 0.03 0.03 0.30 0.31 F9 F10 F11 0.25 0.29 0.22 0.08 0.03 0.18 S9 0.00 0.30 REFERENCES Akaike, H. (1974). A new look at the statistical model identification. IEEE Tranactions on Automatic Control AC19, 716-723. Bar-Or, O. (1987). The Wingate Anaerobic Test: An update on methodology, reliability and validity. Sports Medicine 4, 381-394. Bar-Or, O. (1993). Pediatric laboratory exercise testing clinical guidelines: Champaign, ILL.: Human Kinetics. Maud, P. J., Berning, J. M., and Foster, C. (2006). Physiologival assessement of human fitness, 2nd edition: Champaign, ILL.: Human Kinetics. Pereira, F. D., Fernandes, R., and Mendonça, G. (2005). Physiological impact of swimming and football on prepubertal young athletes. AIESEP World Congress. Tsay, R. S. (2005). Analysis of Financial Time Series, 2nd edition: Wiley.