To answer the existence of optimal swimmer learning/teaching strategies, this work introduces a two-level clustering in order to analyze temporal dynamics of motor learning in breaststroke swimming. Each level have been performed through Sparse Fisher-EM, a unsupervised framework which can be applied e ciently on large and correlated datasets. The induced sparsity selects key points of the coordination phase without any prior knowledge.
The development of Dynamical Systems Theory [
From a theoretical point of view, motor learning is viewed as a process
involving active exploration of a so-called perceptual-motor workspace which is
learner dependent and de nes all the motor possibilities available to him. Few
studies have already highlighted this exploratory behavior during learning a ski
simulator task [
The article will describe at rst the context of the research insisting on the
way data have been collected, what are the long-term expectations in sport
science eld and what are the short term locks in machine learning eld. Then
we will give a brief view of the Fisher-EM algorithm [
In breaststroke swimming, achieving high performance requires a particular
management of both arm and leg movements, in order to maximize propulsive
effectiveness and optimize the glide and recovery times [
Based on this context, the breaststroke swimming task was deemed as suitable in investigating the dynamics of learning, mainly as it implies at a macroscopic scale the acquisition of an expert arm-leg coordination that can be easily assessed. however, the investigation of potential di erences in learning strategies required a continuous movement assessment. In that sense, the use of motion sensors allowed a fast, accurate and cycle per cycle movement assessment.
Previously, two analysis methods were used in the cycle per cycle study of
motor learning. A previous study [
Legspropulsion
Streamlinedglide
by each participant during learning to kick a ball. The authors therefore linked
these di erent behaviors to a search strategy. However, the cluster analysis was
performed individually and there was no comparison done between the learners
(e.g. did they use identical behaviors?), it implied only few participants (i.e.
four learners), it was performed only with 120 kicks per learner (i.e. 10 kicks per
session during 12 sessions) and like the previous study of [
40 60 Percentage of cycle duration (%) 80 100 For this study, 26 novices were involved in 16 lessons of breaststroke swimming, with two sessions per week for a total duration of two months. The general goal of learning for all the 26 swimmers was to increase the distance per stroke, while maintaining the speed stable. Then the 26 learners were divided into four different groups, each group receiving a di erent instruction during the learning process: 1) Control group (N=7): This group received only the general goal of learning, increase the distance per stroke 2) Analogy group (N=7): In addition to the general goal of learning, this group received a single additional instruction: "glide two seconds with your arms outstretched" 3) Pacer group (N=6): In addition to the general goal of learning, this group had to follow an auditory metronome trying to perform one cycle every single auditory signal. The frequency of the metronome was decreased every two sessions, in order to promote a decrease in the stroke frequency of the learners that should lead to an increase in the distance per stroke 4) Prescription group (N=6): In addition to the general goal of learning, this group received multiple additional instructions: "keep your arms outstretched forward when you extend your legs; then glide with your arms and legs outstretched; then keep your legs outstretched when you ex your arms; recover both arms and legs together". These di erent instructions were supposed to have a speci c impact on the learning strategies of the learners.
Each learner performed 10 trials of 25-m swim during each session, with 1 x
25-m consisting approximatively in 8 recorded cycles (one cycle correspond to
the period between two successive maximal knee exion). During every learning
session, all learners were equipped with small motion sensors on both arms and
legs (3-D gyroscopes, 3-D magnetometers, 3-D accelerometers) including a data
logger and recording elbow and knee angles at a frequency of 200 Hz. Following
the literature in coordination dynamics [
To sum-up, we have recorded 4160 trials (26 swimmers 16 sessions 10 trials) and there is an average of 8 cycles per trials. Thus, the dataset is composed by 33280 cycles, each cycle is represented by 100 continuous relative phase samples. 2.3
From a sport sciences point of view, the speci c aims of the study were twofold: { Assessing the dynamics of learning: In other words, the aim was to assess not only the di erent behaviors used during learning but also the transitions between these behaviors, that is the potential search strategy exhibited by learners (e.g. they used preferably behavior no 1 then no 4, then no 3 . . . ). { Assessing the impact of di erent learning conditions on the dynamics of learning: In other words, the aim was to investigate the possible existence of di erent behaviors exhibited by the learners regarding their learning condition, as well as the possible existence of di erent search strategy exhibited by the di erent groups.
A last point in this experiment was the possibility to transfer the results of
the analysis towards practical application or guidelines for teachers. From a
pedagogical point of view, it appeared di cult to teach novice swimmers by giving
instruction on the arm-leg coordination during all the cycle and the de nition
of key points within the entire cycle re ects a paramount aspect for teaching.
Indeed, a strong literature in sports pedagogy highlights the role played by
attentional focalization during motor learning, as a focalization on a key point of
the swimming cycle may be highly bene cial in seeking to reorganize the entire
arm-leg coordination [
From a machine learning point of view, there are two locks to tackle: 1) Each cycle is described by 100 features which are highly correlated due to the fact that they are samples of the relative phase which is a continuous time signal. Nevertheless, we don't want to bias the study by preprocessing the data, a transformation like lters, wavelet transform or sample selection that will embedded our a priori knowledge. 2) The number of cycles are not equal on all the trials, that is why a trial can not be directly described by a xed number of features.
Those two problems were address by 1) using a clustering by Fisher-EM [
A clustering can be derived from a mixture of Gaussians generative model. A Gaussian, which is parameterized by a covariance matrix and a mean in the observation space, represents a cluster. An observation is labeled according to its ownership (likelihood ratio) to each Gaussian. Knowing the number of clusters, the mixture and Gaussian parameters are learned from the observation data trough an Expectation-Maximization (EM) algorithm.
The Fisher-EM algorithm [
We consider that the n observations y1; y2; : : : ; yn are realizations of a random vector Y 2 Rp. We want to cluster these observations into K groups. For each observation yi, a variable zi 2 Z = f1; : : : ; Kg indicates which cluster its belong to. This clustering will be decided upon a generative model, namely a mixture of K Gaussians which lies in a discriminative latent space X 2 Rd where d K 1.
This latent space is linked to the observation space through a linear transformation,
Y = U X + ; (1) where U 2 Rp d and U tU = Id(d) where Id(d) is the identity matrix of size d, i.e. U is an orthogonal matrix and non-discriminative noise.
Let be W = [U; V ] 2 Rp p such that W tW = Id(p). V is the orthogonal complement of U . Thus, a projection U ty of an observation y from space Y of dimension p, lies on the latent discriminative subspace X of dimension d and the projection V tyi lies on the non-discriminative complement subspace of dimension p d.
Conditionally to Z = k, random variables X and Y are assumed to be Gaussian, XjZ=k N ( k; k) ; and YjZ=k N (mk; Sk) ; where k 2 Rd, k 2 Rd d, mk 2 Rp and Sk 2 Rp p.
With the help of equation 1, we can deduce parameters of the distribution YjZ=k in the observation space from the parameters of the distribution XjZ=k in the latent space, mk = U k and Sk = U kU t + ; where 2 Rp p is the covariance matrix of which is assumed to follow a 0-centered Gaussian distribution. To ensure that represents non-discriminative noise, we will impose that the covariance of , , projected into the discriminative space is null, i.e. U U t = 0(d), and that projected into the non-discriminative subspace is diagonal, i.e. V V t = Id(p d). Thus,
W tSkW =
The iterative Expectation-Maximization (EM) algorithm can be extended by a
Fisher Step (F-Step) in-between the E-Step and the M-Step where the latent
discriminative subspace is computed [
E-Step In this step, for each observation i, its posterior probability to each cluster k is computed by oik
k (yi; ^k) PK l=1 l (yi; ^l) ; where ^k = fU; ; k; kg. From these probabilities, each observation can be given to a cluster by zi = arg max oik.
k F-Step The projection matrix U is computed such that Fisher's criterion is maximized in the latent space,
U arg max trace (U tSU ) 1 U tSBU
U w:r:t: U tU = Id(d) ; where S is the variance of the whole dataset and SB = n1 PK k=1 nk(mk y)t where nk = Pi oik and y the mean of the dataset. y)(mk M-Step Knowing the posterior probabilities oik and the projection matrix U , we compute the new Gaussian parameters by maximizing the likelihood of the observations, ^k nk n ; ^k 1 n
X oikU tyi; ^k nk i=1
U tCkU; ^k where uj is the j-th column of U and Ck = n1k Pn i=1 oik(yi empirical covariance matrix of the cluster k. trace(Ck)
Pd
j=1 utj Ckuj p d
; mk)(yi mk)t the 3.3
Yet, the use of latent space introduces dimension reduction and computation e
ciency. Nevertheless the back-projection from the latent space to the observation
space can involve all the original features. To do feature selection, the projection
matrix U has to be sparse. [
The clustering is done in two steps: 1) A clustering on cycle data. Here an observation is just one swimming cycle. This clustering has two purposes, a) give a label to each cycle b) select which phase samples over the 100 are informative through sparsity. 2) A clustering on trials. Each trial can be described now by a sequence of cycle labels learned at the rst step. Features for this clustering consist in the transition matrix of the sequence with its diagonal put to zero. The number of cluster is chosen by analysis of the Bayesian information criterion (BIC).
For the rst clustering level, analysis of the BIC (Tab. 1) highlights the existence of 11 clusters within the whole set of data. The mean coordination of these clusters are represented at Figure 2a. This result advocates for qualitative reorganizations of motor behavior during motor learning, as each learner visited between 9 and 11 di erent clusters during their sessions. For instance, the mean and standard deviation of one cluster (no8) is presented in Figure 2b.
In order to di erentiate the e ect of the di erent instructions on the learning process, Table 2 shows the distribution of each emerging cluster across the di erent learning conditions. Interestingly, the use of di erent additional instructions led to the exhibition of di erent preferred patterns of coordination. For instance, the group who received an analogy exhibited preferably clusters 3, 7, 8 and 9, whereas clusters 2, 4 and 10 were inhibited. In the meantime, the use of the prescriptive instruction preferably led to the use of cluster 5 and inhibited the use of clusters 2, 6 and 10. This result is a key point of the experiment, validating the possibility of guiding the exploration during learning and by extension the result of the learning process with using di erent types of instructions during the practice.
On Figure 2c, we have superimposed a typical coordination curve and, in gray bars, the back-projection of latent space into observation space to see induced sparsity from the rst level. The height of a bar at a feature i 2 [1 : : : p] is proportional to Pd
j=1 jUij j. A null value shows that the corresponding feature is not involved in the projection to the latent space, i.e. it is not selected by the FStep or it is squeezed by the sparsity; therefore it can be considered not relevant to build the clusters. Interestingly, only key points of the movement have high values, thus the Fisher-Em algorithm is able to select key points without any prior knowledge.
The second level of cluster analysis, based on the transition matrix during each trial showed the existence of six di erent clusters. More speci cally, Figure 3 highlights the preferred transitions exhibited by each emerging cluster. Interestingly, the group who showed the highest number of preferred transition (i.e. cluster 6) was associated with the learning group that did not receive any instruction. In that sense, this second level of cluster analysis allowed to highlight the use of temporary additional information during learning in order to modify the learning search strategy, namely by impacting the preferred transitions. 5
These preliminary experiments show that we can apply e ciently the FisherEM clustering on highly correlated features. Interestingly, the induced sparsity corresponds to key points of the coordination phase. Now, a qualitative work needs to be undertaken to qualify clusters of trials in term of learning condition and learning dynamics. a) Mean patterns of coordination for each b) Mean pattern for cluster 8 (black line), cluster standard deviation (dotted line)