lactic and anaerobic alactic systems. In swimming, collection of gas exchanges to calculate the aerobic contribution is possible with the use of respiratory snorkel.
Towards Estimation of Frontcrawl Energy Expenditure Using the Wearable Aquatic Movement Analysis System (WAMAS) Farzin Dadashi, Kamiar Aminian
Florent Crettenand, Gregoire P. Millet
Laboratory of Movement Analysis and Measurement École Polytechnique Fédérale de Lausanne (EPFL) Lausanne, Switzerland {farzin.dadashi, kamiar.aminain}@epfl.ch
Institute of Sport Sciences University of Lausanne Lausanne, Switzerland {florent.crettenand, gregoire.millet}@unil.ch
Abstract— Inertial measurement unit (IMU) is a promising tool in the quantification of energy expenditure for human on-land activities, though has never been deployed before to calculate the aquatic activities energy expenditure. Investigating the factors that influence the required energy in aquatic locomotion can help the biomecanicians to better understand the biophysics of swimming. We used a set of three waterproofed IMUs worn on the forearms and sacrum of twelve swimmers to estimate the front-crawl energy expenditure. The swimmers performed three 300-m trials at 70%, 80% and 90% of their 400-m personal best time. At the end of each 300-m the reference value of energy expenditure was measured based on indirect calorimetry and blood lactate concentration. The three IMUs were used to extract the main spatio-temporal determinants of the front-crawl energy expenditure. Extraction of these parameters using IMU was previously validated. We used a combination of a linear estimator and kernel smoother on the residuals of the linear part to derive the mapping between the spatio-temporal inputs and reference energy expenditure. The algorithm validation on test data shows a strong association between the estimated and reference energy expenditure (Spearman’s rho = 0.97, p-value 0, as in sprint events). In order to estimate the beginning and the end of arm propulsion to calculate IdC, all three IMUs were used. We explained the extraction of these parameters with similar IMU configuration in [14]. D. Energy Expenditure Estimation Model To find the mapping between spatio-temporal parameters of front-crawl and EE of a 300-m trial the average IdC of trial, standard deviation of IdC, average velocity, IVV and CVV were selected as input feature space. However, it is well-known that the weight and height of the swimmer also affect the EE [13]. Therefore, these two parameters were also added to form the 7 input variables. ! Let !!"#$% = !! ! !!! ⊂ ℝ be the N-training input data -constructed from abovementioned features- and !! ! !!! presents the corresponding EE. We aimed to find a mapping in form of !! = ! !! , Θ + !! where Θ shows the set of model parameters and !! is an i.i.d Gaussian noise with zero mean and precision of !!! . We applied a linear combination of input variables using a least-squares regression to capture the main structure of the input-output mapping such that: !! = !! !! + ℎ + !! = !! !!,! + ⋯ + !! !!,! + ℎ + !!
(8)
where ! ∈ ℕ is the dimension of input space, !!,! shows the jth component of the !! , h is the estimated offset of the linear model and !! denotes the corresponding residual of
!! ∈!! !!
!! !! − !! . !!
!! ∈!! !!
!! !! − !!
(9)
where !! . denotes the distance of !! from the center of the kernel, !! , and B shows the bandwidth of the kernel. In (9), !! ∈ !! if and only if !! < !. The distance !! . is given by: !! !! − !! =
!! − !!
!
! !! − !!
(10)
where M is a diagonal matrix as in (11): ! = !"#$ !"# !! !!"#
%$ =
!!! ⋮ 0
⋯ ⋱ ⋯
0 ⋮ !!!
(11)
where !"# !! (. ) shows the inverse of covariance matrix. The set Θ = !, ℎ, !, ! determines the model parameters that were estimated on the training set. E. Input Selection and Statistical Evaluation of the Algorithm The final model in our algorithm has the form: ! ! ∗ , Θ(! ∗ ) = !! ! ∗ + ℎ + ! ! ∗
(12)
where ! ∗ represents the explanatory variables that are most relevant in determining the EE using the model in (12). We used the mean squared error criterion in the backward elimination arrangement [21] to find ! ∗ : Backward Elimination Algorithm 1: Let X be the full set of all inputs 2: while the modeling error is less than a constant: a)
for each !!!,! ∈ ! i.
set ! ! ← !\!,!
ii.
train the model ! ! ! , Θ with ! ! and calculate the mean squared error
b)
∗ ∗ set ! ← !\!!!,! where !!!,! is the input dimension changed the mean squared error the least in step 2a
c)
repeat the steps 2a and 2b with the new X
3: Return the best inputs as ! ∗
The leave one out cross validation was conducted to evaluate the algorithm performance on different test sets. Indeed, we trained the regression model using the data
from all subjects but one and then the data from the remaining subject were used to test the algorithm. This procedure was performed for all the swimmers one by one. The relative accuracy and precision (Mean and SD) and RMS error were calculated as the difference between the reference EE values, denoted by EEmeasure and the values estimated by our algorithm (EEmodel). The error indicators were provided for the full order model as well as reduced dimension models after backward elimination. Error analysis plots were also provided for the selected model. The concurrent validity of the proposed method was examined by correlation analysis and comparing the trend of EE in different velocity levels between the welltrained and recreational group. IV.
RESULTS
The correlation analysis showed a strong linear correlation between SF and the trial velocity (Pearson’s rho > 0.96). Consequently, we did not select SF as an input parameter of our model although in the literature it was considered as a determining factor of EE [4, 18]. Table II summarizes the relative difference between the EEmeasure and EEmodel. These values are presented for the model including all parameters and the result of backward elimination in two consecutive steps. The precision of the estimation after the third parameter elimination decreases to 20.1% that is comparable to the performance of the linear model including all parameters (26.3% in Table II). Fig. 3(a) demonstrates the quantile-quantile (Q-Q) plot of the sample quantiles of estimation error, !! , for the model including all seven input parameters versus theoretical quantiles from a normal distribution. If the distribution of !! is normal, the plot will be close to the normal line. In Fig. 3(b) values of !! were plotted against EEmeasure to examine the error dependency on the reference value range. High association between the EEmodel and EEmeasure was observed (Spearman’s rho = 0.97, p-value
