Julien Gori

Olivier Rioul Ltci

Information-Theoretic Analysis of the Speed-Accuracy Tradeoff with Feedback

Human movements are inherently variable and involve some feedback mechanism. A study of the positional variance in a tapping task reveals that the variance profiles are unimodal in time. In the variance-decreasing phase, the aiming task can be modeled by a Shannon-like communication scheme where information is transmitted from a "source"-determined by the distance to the target at maximum variance-to a "destination"-the movement endpoint-over a "channel" with feedback perturbed by Gaussian noise. Thanks to the feedback link, we show that the variance decreases exponentially at a rate given by the channel capacity. This is confirmed on real data. The proposed information-theoretic model has promise to improve our understanding of human aimed movements.

I. INTRODUCTION

Variability is an important characteristic of human aimed movement. In fact, it has long been observed [START_REF] Woodworth | Accuracy of voluntary movement[END_REF] that a participant will produce disparate movements when asked to repeatedly perform the same task, while having to slow down to perform more accurate tasks-the so-called speed-accuracy tradeoff. Many attempts at modeling variability were proposed in the literature, e.g., [START_REF] Crossman | The speed and accuracy of simple hand movements[END_REF], [START_REF] Soukoreff | Towards a standard for pointing device evaluation, perspectives on 27 years of fitts' law research in hci[END_REF] which only characterize the variability of the movement's endpoints based on Fitts' paradigm [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF]. We focus instead on the variability of the position of the limb extremity used for aiming during the entire movement: How does the positional variance evolve over time?

As remarked in [START_REF] Todorov | Optimal feedback control as a theory of motor coordination[END_REF], trajectory variability has "surprisingly received little attention from researchers". The entropy of a set of trajectories over time was considered in [START_REF] Lai | Information entropy analysis of discrete aiming movements[END_REF], where unimodal profiles were observed. Positional variance was studied in [START_REF] Khan | Inferring online and offline processing of visual feedback in target-directed movements from kinematic data[END_REF], [START_REF] Van Der Meulen | Mechanisms underlying accuracy in fast goal-directed arm movements in man[END_REF] but only at specific kinematic markers (peak acceleration, velocity, deceleration, and movement time). Complete variance profiles for elbow flexion were reported in [START_REF] Darling | Changes in the variability of movement trajectories with practice[END_REF], [START_REF] Gutman | Basic functions of variability of simple pre-planned movements[END_REF], most resulting variance profiles being unimodal-yet human movements seem much more complex as they involve many joints.

The following simple reasoning suggests that positional variance profiles should be unimodal. Consider e.g., Fitts' tapping task [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF] where the participant aims at a target of a given width at a given distance:

1) all movements start at the same position: initial positional variance is null; 2) in the early stages of the movement, positional variance increases [START_REF] Darling | Changes in the variability of movement trajectories with practice[END_REF], [START_REF] Schmidt | Motor-output variability: A theory for the accuracy of rapid motor acts[END_REF]; 3) if time permits, the participant eventually reaches the target [START_REF] Darling | Changes in the variability of movement trajectories with practice[END_REF], [START_REF] Soechting | Invariant characteristics of a pointing movement in man[END_REF], [START_REF] Soechting | Effect of target size on spatial and temporal characteristics of a pointing movement in man[END_REF] and the movement ends, which implies that the positional variance vanishes asymptotically [START_REF] Wickelgren | Speed-accuracy tradeoff and information processing dynamics[END_REF]. We therefore expect a unimodal variance profile as illustrated in Fig. 1: A first variance-increasing phase for time ∈ [0,τ ],

where variance increases from 0 to σ 2 0 , is followed by a second variance-decreasing phase where the variance is decreased to arbitrarily low values and eventually vanishes.

Yet another important characteristic of human aiming is the use of feedback [15, p. 95] for generating movements. Feedback can be visual [START_REF] Elliott | Optimizing the use of vision in manual aiming: The role of practice[END_REF]- [START_REF] Chua | Visual regulation of manual aiming[END_REF] or kinesthetic [START_REF] Elliott | A century later: Woodworth's (1899) two-component model of goal-directed aiming[END_REF], [START_REF] Elliott | Goal-directed aiming: two components but multiple processes[END_REF] and affects movement as short as 100 ms [START_REF] Zelaznik | Rapid visual feedback processing in single-aiming movements[END_REF], [START_REF] Elliott | Goal-directed aiming: two components but multiple processes[END_REF], [START_REF] Carlton | Visual processing time and the control of movement[END_REF]. Feedback is often associated to control theory, which has been applied to human aiming movements but cannot easily handle trajectory variability nor noise [START_REF] Müller | Control theoretic models of pointing[END_REF]. Now, feedback is also frequently used in Shannon's information theory [START_REF] Shannon | A mathematical theory of communication, part i, part ii[END_REF], [START_REF] Cover | Elements of information theory[END_REF] to enhance communication in the presence of noise. Unfortunately, even though many previous studies have used information-theoretic tools to model human movements [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF], [START_REF] Mackenzie | A note on the information-theoretic basis for fitts' law[END_REF]- [START_REF] Gori | Speed-accuracy tradeoff: A formal information theoretic transmission scheme (FITTS)[END_REF], they did not account for feedback mechanisms. There wasn't even any explicit channel model described in Fitts' seminal paper [START_REF] Fitts | The information capacity of the human motor system in controlling the amplitude of movement[END_REF], and most recent attempts like [START_REF] Soukoreff | An informatic rationale for the speedaccuracy trade-off[END_REF], [START_REF] Gori | Speed-accuracy tradeoff: A formal information theoretic transmission scheme (FITTS)[END_REF] build on a feedforward channel without feedback. This may explain why information theory was said to be "no longer much of a factor" [START_REF] Luce | Whatever happened to information theory in psychology?[END_REF] for explaining human movements or more generally in experimental psychology.

In this paper, we present a complete information-theoretic model that uses feedback to predict that for t > τ (in the second variance-decreasing phase) the variability decreases exponentially at a rate given by Shannon's channel capacity. Our contributions are as follows. 1) Validate the unimodality hypothesis on empirical pointing data 2) Propose an information-theoretic model for the variance decreasing phase 3) Determine Shannon's capacity for this model, to provide an upper bound on the rate at which variance decreases over time 4) Validate the model and estimate the capacity on empirical pointing data for various participants.

II. UNIMODAL VARIANCE PROFILES IN REAL DATA

In this section, we evaluate the unimodality hypothesis on empirical data. Since there have been numerous published studies on pointing tasks, we have simply re-analyzed an existing dataset (hereafter named "G-dataset") from a study by Guiard et al. [START_REF] Guiard | Fitt's law as an explicit time/error trade-off[END_REF] which uses a discrete version of Fitts' tapping task [START_REF] Fitts | Information capacity of discrete motor responses[END_REF]. Using a stylus on a digital tablet, participants were instructed to reach a line located 150mm away from a starting point, under conditions ranging from (1) full speed emphasis to (5) full accuracy emphasis. The raw trajectory data was low-pass filtered and re-sampled closest to the average sampling frequency, and individual movements were extracted and synchronized so as to have a common time origin. 16 participants produced movements for five conditions, repeated five times, providing 80 different variance profiles 1 . As an illustration, Fig. 2 The unimodality hypothesis was verified by computing the positional variance profiles for each subset composed of all the trajectories acquired by a given participant under the same condition 3 . We checked unimodality by ensuring that there is only one sign change in the time derivative of the profile. We found that 64 out of 80 variance profiles in the dataset are strictly unimodal. Fig. 3 displays a typical empirical variance profile consistent with the ideal profile given in Fig. 1, as well as its derivative in orange, obtained from the same set as the one used for Fig. 2. There are a few profiles with multiple modes that are often due to outliers: Usually, the secondary modes are very small and the variance profile can still be reasonably well modeled by a unimodal profile.

For completeness, we also conducted the Hartigan & Hartigan DIP test for unimodality [START_REF] Hartigan | The dip test of unimodality[END_REF], where the null hypothesis is that the distribution is unimodal. Only 9 profiles (p < .05) were found to be significantly non-unimodal out of the total 80 profiles and the 1 No outliers were removed because removal procedures are based on arbitrary heuristics for which we found no satisfying method. As seen below, we nonetheless found compelling evidence in favor of the unimodality hypothesis.

2 Throughout the paper we will be using the example of participant 3. The study was carried out for every participant in the G-dataset and all participants behaved similarly. 3 As all traces end at different dates, the number of trajectories available upon which variance can be computed decreases with time; to ensure reliable estimations, we stopped computing variance when less than 10 trajectories were available. average p-value is just below 0.6. We also proceeded to a visual check and determined that 4 profiles that were labeled as multimodal had a massive outlier which distorted the variance profile.

There is thus substantial evidence to support the unimodality of variance profiles in tapping tasks.

III. A MODEL FOR THE VARIANCE-DECREASING PHASE A. Information-Theoretic Model Description

Whereas most existing models aim at predicting an average trajectory, the goal here is to predict how the variability of a set of trajectories evolves over time. Asymptotically, i.e., when the set is large enough, the position of the limb extremity (the 'limb' in short) is a continuous random variable. At the end of the first phase, we may assume that the position can be well approximated by a Gaussian distribution 4 , with some standard deviation σ 0 (see Fig. 1).

Thanks to the feedback, the limb position is known at the brain level. Due to eye-hand coordination and fast eye dynamics, the eye is usually pointing towards the target long before the end of the movement [START_REF] Elliott | Goal-directed aiming: two components but multiple processes[END_REF]. Hence the position of the target is also known at the brain level. The distance from limb to target can thus easily evaluated by the brain; in fact it can be readily estimated by the eye if the limb is close enough to the target.

Let A be the distance from limb to target; from the above assumptions, A is modeled as a Gaussian random variable with standard deviation σ 0 . Moreover, for rapid aiming, the first phase may equally well undershoot or overshoot the target [START_REF] Gori | A feedback information-theoretic transmission scheme for modelling aimed movements[END_REF], and we may assume zero mean: A ∼ N (0,σ 2 0 ). To successfully complete the movement, the brain has to send A to the limb; this is precisely the role of the second phase.

We consider the following scheme:

• From A the brain outputs a certain amplitude X 1 to be sent to the limb:

X 1 = f (A), (1)
where f is an unknown function performed by the brain. • To account for the variability of the human motor system [START_REF] Van Galen | Fitts' law as the outcome of a dynamic noise filtering model of motor control[END_REF], we consider a noisy transmission from brain to limb, where X 1 gets perturbed by additive white Gaussian noise (so-called AWGN channel). The output of the channel Y 1 is given by

Y 1 = X 1 +Z 1 where Z 1 ∼ N (0,N ). (2)
• Based on the channel output Y 1 , distance A 1 is actually covered, which is the result of some unknown function g applied by the motor organs to the received Y 1 .

• A 1 is returned to the brain via ideal (noiseless) feedback
where it is compared to A. From such a comparison a new amplitude X 2 is produced by the brain. The scheme then progresses iteratively for i = 1, 2, .... We assume that each step i, from the creation of X i to the reception of A i takes T seconds. Each such step is infinitesimal and thus, the whole process is an intermittent iterative correction model that becomes continuous at the limit.

At iteration i, the scheme is described by following equations (see Fig. 4):

1) The brain (the 'encoder') produces X i from A and all received feedback information A i-1 :

X i = f (A, A i-1
).

2) The motor organs (decoder) receive Y i contaminated by

Gaussian noise:

Y i = X i +Z i .
3) The covered distance A i is a function of all previous received amplitudes:

A i = g(Y i
). At this stage, f and g are still undetermined.

In Shannon's communication-theoretic terms, the aiming task in the second phase can thus be seen as the transmission of a real value from a "source" (distance from target at the end of the first phase) to a "destination" (limb extremity) over a noisy Gaussian channel with noiseless feedback. In human-centered terms, the second phase is the one which deals specifically with aiming-to make sure that the limb reliably reaches the target, once most of the distance has been covered. This second phase likely corresponds to the submovements of the stochastic optimized submovement model [START_REF] Meyer | Optimality in human motor performance: Ideal control of rapid aimed movements[END_REF], or to the second component in two-component models [START_REF] Woodworth | Accuracy of voluntary movement[END_REF], [START_REF] Elliott | A century later: Woodworth's (1899) two-component model of goal-directed aiming[END_REF], [START_REF] Elliott | Goal-directed aiming: two components but multiple processes[END_REF].

B. Bounds on Transmitted Information

We now leverage information-theoretic definitions.

• P i = E[X 2 i ]
, where E is the mathematical expectation, is the input's average power.

• The Shannon capacity C [START_REF] Shannon | A mathematical theory of communication, part i, part ii[END_REF], [START_REF] Cover | Elements of information theory[END_REF] of the AWGN Channel under power constraint P i ≤ P and noise power N is given by

C = 1 2 log 2 (1+P/N ). (3)
expressed in bits per channel use.

• D n = E[(A -A n ) 2 ]
is the quadratic distortion that represents the mean-squared error of the estimation of A by A n after n iterations (channel uses); the distortion essentially corresponds to the empirical variance. • I(A, A n ) is Shannon's mutual information [START_REF] Cover | Elements of information theory[END_REF] between A and A n .

Theorem 1. Consider the transmission scheme of Fig. 4 with an AWGN channel of Shannon capacity C and noiseless feedback. For a zero-mean Gaussian source A with variance σ 2 0 , we have that:

1 2 log σ 2 0 D n ≤ (a) I(A, A n ) ≤ (b) nC. ( 4
)
The proof is given in the Appendix. The Theorem expresses that enough information should be transmitted from the brain to the limb to reduce the positional variability from the initial variance (σ 2 0 ) to the variance at the end of the movement (D n ). However, it also expresses that the transmitted information can never exceed nC, where n is the number of iterations. Since the rate per iteration can never exceed C, being more accurate requires sending larger amounts of information, which in turn requires more iterations of the scheme.

IV. ACHIEVING CAPACITY For a given channel C and number of channel uses n, maximizing accuracy is equivalent to minimizing D n . Similarly, for a given accuracy (distortion), minimizing time is equivalent to minimizing n. Optimal aiming, which consists of achieving the best possible accuracy in the least amount of time is thus achieved when equality holds in (4):

1 2 log σ 2 0 D n = I(A, A) = nC. (5)
The goal of this section is to find the scheme that achieves optimality (Eq. ( 5)).

In all what follows we use the list notation i = ( 1 ,..., i ).

Lemma 1. Optimal aiming can be achieved if, and only if, we have the following conditions:

1) all considered random variables A, A i , A-A i , X i , Y i , Z i are Gaussian; 2) input powers P i = E[X 2
i ] are equal (to, say, P ); 3) endpoints A i are mutually independent; 4) channel outputs Y i are independent from the errors

A-A i ; 5) A i = g(Y i ) is a sufficient statistic of Y i for A.
The proof is given in the Appendix. Working with Gaussian variables considerably simplifies operations, as independence between Gaussian variables is equivalent to decorrelation, and the minimum mean-squared error (MMSE) estimator-which minimizes D n -reduces to a linear function in the Gaussian setting. As seen below, this implies that both f and g are linear-which is not too surprising as linear functions are known to preserve normality.

Notice that since g(Y i ) is a sufficient statistic of Y i for A, it does not matter if the feedback comes from the endpoints

A i A Brain f (A, A i-1 ) X i + Z i Y i A i-1 Sample Delay g(Y i ) Muscles A i
Fig. 4. Information-theoretic model for the aiming task, with initial distance A to the target at the end of the first phase; amplitude X i created by the brain at t = τ +iT ; noisy version Y i of X i ; distance A i actually covered by the limb at t = τ +iT , where T is the infinitesimal time between two iterations.

or from the outputs of the channel Y i . In this way our model can account for feedback information prior to the motor organs (e.g. kinesthetic feedback).

By working out the conditions of Lemma 1, we can derive the structure of the optimal scheme, namely the expression of f and g. We first obtain g by using the following result known as the orthogonality principle: if a certain quantity x is to be estimated from the observed data y by the unbiased estimator x(y), the orthogonality principle states that the following are equivalent:

• x(y) = E[x|y] = E[xy] t E[yy t ] -1 y is the unique MMSE estimator; • E[(x-x(y))y t ] = 0. Theorem 2. For the optimal transmission scheme, g(Y i ) is the MMSE estimator, g(Y i ) = E A|Y i . Proof. From condition 4 one has E[(A -g(Y i ))Y i ] = 0.
The result follows immediately from the orthogonality principle.

The optimal scheme thus yields an endpoint A i = g(Y i ) obtained as the best least-squares estimation of A from all the previous observations of channel outputs Y i = (Y 1 ,...,Y i ).

Theorem 3. For the optimal transmission scheme,

X i = f ( A i-1 ,A) = α i (A-A i-1 ) (6)
= α i (A-E A|Y i-1 ), (7)
where α i meets the power constraint E[X 2 i ] = P . The proof is given in the Appendix. The signal sent to the channel is thus simply the difference between the initial message A and its current estimate A i-1 , rescaled to meet the power constraint.

The previous two theorems formally define the encoding function f and decoding function g; both are linear functions. Incidentally, the motor system is known to be able to produce linear functions [START_REF] Rosenbaum | Human motor control[END_REF]. The functions f and g are mathematically simple and biologically feasible and the distance difference A-A i-1 can be readily estimated by the eye.

The next result shows that the procedure is incremental, yet optimal at each step, allowing optimal on-line control. Theorem 4. Let A i = X i /α i be the unscaled version of X i . We have

E[A|Y i-1 ] = i-1 j E[A|Y j ] = i-1 j E[A j |Y j ]. (8)
Again the proof is given in the Appendix. The theorem shows that the "decoding" process is recursive: At each step, a "message" A i that is independent from the previous ones

A i-1 = (A 1 ,...,A i-1
) is formed, and is then estimated optimally by least-square minimization. Finally, we check optimality in Eq. ( 5) by evaluating the distortion in the following Theorem 5. The quadratic distortion

D i = E[(A -A i ) 2 ]
decreases exponentially in i:

D i = σ 2 0 (1+P/N ) i . (9)
With this scheme, it is immediately checked that capacity C is exactly achieved and the distortion decreases geometrically (divided by (1 + P/N )) at each iteration. The scheme successfully makes the correspondence between the transmission of one real value A using feedback, with that of n independent channel uses (transmissions).

It is important to note that the obtained equations were already given in an information-theoretic context by Gallager and Nakiboglu [START_REF] Gallager | Variations on a theme by schalkwijk and kailath[END_REF] who discussed an older scheme by Elias [START_REF] Elias | Channel capacity without coding[END_REF]. To our knowledge, the constructive approach of the Elias scheme given here, as well as its application to model human aimed movements is novel.

V. EMPIRICAL EVIDENCE

Let t = nT be the total time spent during the second phase and notice that D n = σ 2 n is the variance of endpoints after n iterations. From Theorem 5, we have

log 2 σ n = log 2 σ 0 -C/T t = log 2 σ 0 -C t. ( 10
)
where C is the Shannon capacity (3) and C = C/T is the capacity in bits/s. This equation is illustrated on real data in

VI. CONCLUSION

this paper, we have established that for human aimed movements, the profiles of positional variance are unimodal in time, and that the variance-decreasing phase can naturally be modeled by an information-theoretic transmission scheme with feedback information from the current position. Positional variance decreases at best exponentially at a rate given by Shannon's capacity C. Increasing accuracy, i.e., decreasing the endpoint variance from σ 2 0 to σ 2 n , requires longer time t as shown in [START_REF] Gutman | Basic functions of variability of simple pre-planned movements[END_REF]. This is consistent with the well-known speed-accuracy tradeoff 5 . We believe that there is a lot of insight to be gained from information-theoretic considerations in the study of human aimed movements. An interesting perspective is to consider the effect of non-ideal feedback or multiplicative noise on the optimal rate, and their practical implications.

APPENDIX A

Proof of Theorem 1. The proof uses well known techniques and inequalities from information-theory [START_REF] Cover | Elements of information theory[END_REF]. For inequality (4a):

I(A; A n ) = H(A)-H(A| A n ) (11) = H(A)-H(A-A n | A n ) (12) ≥ H(A)-H(A-A n ) (13) ≥ H(A)- 1 2 log 2πeE[(A-A n ) 2 ] (14) = 1 2 log σ 2 0 D n (15)
Eq. ( 11) by definition of mutual information; Eq. ( 12) because of the conditioning by A n ; Eq. ( 13) because conditioning reduces entropy H; Eq. ( 14) because the Gaussian distribution maximizes entropy under power constraints; Eq. ( 15) by definition of the distortion and the entropy formula for a Gaussian distribution.

For inequality (4b):

I(A; A n ) ≤ I(A;Y n ) (16) = H(Y n )-H(Y n |A) (17) = i H(Y i |Y i-1 )-H(Y i |Y i-1 ,A) (18)
= i H(Y i |Y i-1 )-H(Y i |X i ) (19)
≤ i [H(Y i )-H(Z i )] (20)
≤ i 1 2 log(2πe(P i +N ))- 1 2 log(2πeN ) (21)
≤ i 1 2 log 1+ P i N ≤ nC (22)
Eq. ( 16) by the data processing inequality [START_REF] Cover | Elements of information theory[END_REF], where A -→ Y i -→ g(Y i ) = A i form a Markov chain; Eq. ( 17) by definition; Eq. ( 18) by applying the chain rule to both terms; Eq. ( 19) because of the feedback scheme; Eq. ( 20) because conditioning reduces entropy, Y i = Z i + X i and X i and Z i are independent; Eq. ( 21) because the Gaussian distribution maximizes entropy and X i and Z i are independent (where P i and N are the powers of respectively X i and Z i ); Eq. ( 22) by the concavity of the logarithm function.

Proof of Lemma 1. The proof consists of finding the conditions that turn the inequalities in the proof of Theorem 1 into equalities. Equality in Eqs. ( 13) and ( 16) directly imply condition 4. Equality in Eq. ( 14) implies that the A -A i are Gaussian. Equality in Eq. ( 16) implies that H(A|Y i ) = H(A|Y i ,g(Y i )) = H(A|g(Y i )), so that Y i -→ g(Y i ) -→ A should form a Markov chain, implying condition 5. Equality in Eq. ( 20) implies condition 3. Equality in Eq. [START_REF] Carlton | Visual processing time and the control of movement[END_REF] implies that the Y i 's are Gaussian. Eq. ( 22) implies condition 2 by concavity of the logarithm. Finally, X i is Gaussian, as the result of the sum of Y i and Z i , both Gaussian. Similarly, A i is Gaussian as both A and A-A i are Gaussian, which finally yields condition 1.

Proof of Theorem 3. On one hand, X i = E[X i |A, A i-1 ] is a linear function of A and A i-1 , because the conditional expectation is linear for Gaussian variables. On the other hand, condition 3 reads that E[

A i A i-1 ] = E[g(X i + Z i ) A i-1 ] = 0.
Because Z i is independent from X i and A i-1 , and g is linear, we get E[g(X i ) A i-1 ] = 0. Combining the two results, and because g is linear, we have E[α i (A-f ( A i-1 )) A i-1 ] = 0. The orthogonality principle clearly appears and f = E A|A i-1 is the MMSE estimator.

Proof of Theorem 4. The goal is to evaluate E[A|Y i-1 ]. We first use the operational formula for the conditional expectation E[A|Y i-1 ] = E(AY i-1 ) t E[Y i-1 (Y i-1 ) t ] -1 Y i-1 . We get, using the fact that the Y j are independent and have power P +N :

E[A|Y i-1 ] = i-1 j=1 (P +N ) -1 E[AY j ] Y j .
Second, it follows from previous computations that A-A i = E A|Y i-1 , which is a function of the observations Y i-1 and therefore independent of Y i . This leads to A-A i being independent of Y i and the following equality:

E[AY i ] = E[A i Y i ].
Combining both results, we thus get

E[A|Y i-1 ] = i-1 j=1 E[A|Y j ] = i-1 j=1 E[A j |Y j ].
Proof of Theorem 5. First notice that we can write

D i as E[(A i -E[A i |Y i ]) 2 ] as A -A i = A -A i-1 -E[A i |Y j ] = A i -[A i |Y j ]
. Next, we have that

D i = E(A 2 i )+E[E 2 (A i |Y i )]-2E(A i E(A i |Y i ))
From the proof above,

E[A i |Y i ] = E[A i Y i ](P +N ) -1 Y i = 1 α i P P +N Y i .
With some calculus and using E(A 2 i ) = D i-1 = P/α 2 i , we get that

D i = D i-1 1+P/N .
The proof is finished by applying this equation recursively.

Fig. 1 .

 1 Fig. 1. Ideal positional variance profile with maximum variance σ 2 0 at t = τ .

Fig. 2 .

 2 Fig. 2. The set of trajectories for participant 3 performing under condition 3 (balance speed versus accuracy) for the G-dataset.

1 Fig. 3 .

 13 Fig. 3. The unimodal empirical variance profile (in blue) of participant 3 performing in the balanced condition (3). The mode appears for τ 200 ms. The derivative of the variance profile (in orange) changes sign only once.

Fig. 5 .

 5 The figure shows the standard deviation profile from participant 3 performing under the balanced (3) instruction. For τ > 0.18 s, the logarithm log 2 σ(t) of the standard deviation decreases regularly and linearly with high goodness of fit (r 2 = 0.984, Student t-test for significance of the slope yields F (1,76) = 4693, p = 10 -16 ). Computing the same statistics on the other 11 participants, we found an average goodness of fit of r 2 = 0.93 and an average C = 4.52 bits/s. This provides strong empirical support for Theorem 5.

Fig. 5 .

 5 Fig. 5. Linear decrease of the logarithm of standard deviation of position over time for participant 3 of the G-dataset, performing in the balanced (3) condition.

This is an assumption that is often used[START_REF] Schmidt | Motor-output variability: A theory for the accuracy of rapid motor acts[END_REF],[START_REF] Meyer | Optimality in human motor performance: Ideal control of rapid aimed movements[END_REF], whose strict veracity depends on the conditions in which the movements were produced[START_REF] Hancock | The movement speed-accuracy relationship in space-time[END_REF]. Usually, this approximation is good enough as distributions are bell-shaped. In[START_REF] Lai | Information entropy analysis of discrete aiming movements[END_REF], the empirical entropy of the trajectory was compared with the theoretical entropy of a Gaussian distribution. The difference was never more than .3 bits, throughout the entire trajectory.

In[START_REF] Gori | A feedback information-theoretic transmission scheme for modelling aimed movements[END_REF], we show that our result is also consistent with the well known Fitts' law.

ACKNOWLEDGMENT

We thank Yves Guiard who kindly provided access to his dataset used throughout this paper.


Time-frequency learning machines

Paul Honeiné, Cédric Richard, Member, IEEE, Patrick Flandrin, Fellow, IEEE Abstract-Over the last decade, the theory of reproducing kernels has made a major breakthrough in the field of pattern recognition. It has led to new algorithms, with improved performance and lower computational cost, for non-linear analysis in high dimensional feature spaces. Our paper is a further contribution which extends the framework of the so-called kernel learning machines to time-frequency analysis, showing that some specific reproducing kernels allow these algorithms to operate in the time-frequency domain. This link offers new perspectives in the field of non-stationary signal analysis, which can benefit from the developments of pattern recognition and Statistical Learning Theory.

Index Terms-Time-frequency analysis, kernel machines, learning theory, support vector machines.

I. INTRODUCTION

T IME-frequency and time-scale distributions provide a powerful tool for non-stationary signal analysis. Unlike conventional spectral methods, they reveal the time-varying spectral content of one-dimensional signals by mapping them into a two-dimensional time-frequency domain. Substantial theoretical work has been carried out in this direction and has yielded many different classes of time-frequency distributions, parametric or otherwise, in which optimal solutions for a given signal or task can be selected. As an example, distributions dedicated to signal analysis are studied in [START_REF] Auger | Improving the readability of time-frequency and time-scale representations by the reassignment method[END_REF]- [START_REF] Gosme | Adaptive diffusion of timefrequency and time-scale representations: a review[END_REF] whereas optimal distributions for signal detection are considered in [START_REF] Flandrin | A time-frequency formulation of optimum detection[END_REF], [START_REF] Sayeed | Optimal detection using bilinear time-frequency and time-scale representations[END_REF].

Since the pioneering work of Aronszajin [START_REF] Aronszajn | Theory of reproducing kernels[END_REF], pattern recognition based on reproducing kernel Hilbert spaces (RKHS) has gained wide popularity. The most prominent recent developments include support vector machines (SVM) [START_REF] Boser | A training algorithm for optimal margin classifiers[END_REF], kernel principal component analysis (KPCA) [START_REF] Schölkopf | Nonlinear component analysis as a kernel eigenvalue problem[END_REF], kernel Fisher discriminant analysis (KFDA) [START_REF] Mika | Fisher discriminant analysis with kernels[END_REF], and its generalization to multiclass problems, kernel generalized discriminant analysis (KGDA) [START_REF] Baudat | Generalized discriminant analysis using a kernel approach[END_REF]. A key property behind such algorithms is that they can be expressed in terms of inner products only, involving pairs of input data. Replacing these inner products with a reproducing kernel provides an efficient way to implicitly map the data into a high, even infinite, dimensional RKHS and apply the original algorithm in this space. Because calculations are then carried out without making direct reference to the non-linear mapping of input vectors, this principle is commonly called the kernel trick. Kernel-based algorithms are computationally very efficient, and generally have their generalization performance guaranteed by Statistical Learning Theory [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], [START_REF] Cucker | On the mathematical foundations of learning[END_REF].

With the exception of [START_REF] Davy | Optimised support vector machines for nonstationary signal classification[END_REF], [START_REF] Rakotomamonjy | Non-parametric regression with wavelet kernels[END_REF], there are very few works combining kernel learning machines and time-frequency anal-P. Honeiné and C. Richard are with the laboratory ICD-M2S (FRE CNRS 2848), Université de technologie de Troyes, Troyes, France. P. Flandrin is with the Laboratoire de Physique (UMR CNRS 5672), ENS de Lyon, France ysis, although the interest in pattern recognition based on time-frequency representations remains strong. In [START_REF] Davy | Optimised support vector machines for nonstationary signal classification[END_REF], the authors solve a signal classification problem using a SVM and a reproducing kernel expressed in the time-frequency domain. Reproducing wavelet kernels are considered in [START_REF] Rakotomamonjy | Non-parametric regression with wavelet kernels[END_REF] for non-parametric regression. Clearly, time-frequency analysis still has not fully benefited from the rapid development of kernel learning machines. In this paper, we show that some specific reproducing kernels allow any kernel learning machine to operate in the time-frequency domain. For the sake of simplicity, we begin by describing our approach applied to the Wigner distribution. Next we apply it to other timefrequency distributions. But before, let us briefly review the basics concepts of kernel learning machines.

II. KERNEL LEARNING MACHINES: A BRIEF REVIEW OF BASIC CONCEPTS

Most kernel learning machines are statistical learning algorithms that take advantage of the geometric and regularizing properties of RKHS, which are established by the kernel trick and the representer theorem [START_REF] Kimeldorf | Some results on Tchebycheffian spline functions[END_REF], [START_REF] Schölkopf | A generalized representer theorem[END_REF]. In this section, we briefly introduce these concepts through an example.

A. Example of kernel-based method: the KPCA algorithm

Problems commonly encountered in machine learning start with a training set A n containing n instances x i ∈ X and, in a supervised context, their labels or desired outputs y i ∈ Y. The objective of the exercise is usually related to feature extraction, density estimation or classification. Linear methods have played a crucial role in the development of machine learning because of their inherent simplicity from conceptual and implementational points of view. However, in many fields of current interest such as biological engineering and communications, it is necessary to deal with non-linear complex phenomena. A possible way to extend the scope of linear learning machines is to map the input data from X into a feature space F via a non-linear mapping φ(•). The n instances φ(x i ) are then used as training samples. Clearly, this basic strategy may fail when F is a very high, or even infinite, dimensional space. As shown below with the KPCA algorithm, kernel learning machines overcome this limitation by using a powerful computational shortcut.

KPCA is a non-linear form of principal component analysis (PCA) which allows to extract features that are non-linearly related to the input variables. Consider a set of n data points φ(x i ) mapped into a feature space F and centered at the origin, that is, n i=1 φ(x i ) = 0. PCA is performed in F by solving the eigenvalue problem ΣΦ = µΦ with Φ = 0, where Σ = 1 n n i=1 φ(x i ) φ(x i ) is the covariance matrix. If F is infinite-dimensional, note that φ(x i ) φ(x i ) may be seen as the projection operator onto direction φ(x i ). The problem becomes n i=1 φ(x i ), Φ φ(x i ) = nµ Φ, where • , • is the inner product in F. Observe that any solution Φ k with µ k = 0 must lie in the span of φ(x 1 ) . . . , φ(x n ), and can then be expanded as follows

Φ k = n i=1 a i,k φ(x i ). (1)
As detailed in [START_REF] Schölkopf | Nonlinear component analysis as a kernel eigenvalue problem[END_REF], substituting this expression into the eigenvalue equation, and multiplying it from the right by φ(x j ), we obtain the following eigenvalue problem:

1 Ka k = λ k a k ,
where K is the n-by-n Gram matrix whose (i, j)-th entry is φ(x i ), φ(x j ) , and λ k = nµ k . The components of the k-th eigenvector a k of K are the a i,k 's defined in equation [START_REF] Auger | Improving the readability of time-frequency and time-scale representations by the reassignment method[END_REF]. Finally, the k-th principal component can be extracted from any φ(x) by projecting it onto Φ k , namely,

φ(x), Φ k = n i=1 a i,k φ(x), φ(x i ) . (2)
The most interesting characteristic of this algorithm is that the non-linear mapping φ(•) only appears in the form of inner products φ(x i ), φ(x j ) . Suppose we are given a kernel function κ : X × X → C with the property that there exists a map φ : X → F such that for all x i , x j ∈ X , we have κ(x i , x j ) = φ(x i ), φ(x j ) . For a survey, see e.g. [START_REF] Aronszajn | Theory of reproducing kernels[END_REF]. The socalled kernel trick consists of substituting each inner product φ(x i ), φ(x j ) by κ(x i , x j ). The power of this principle lies in that the inner products in the feature space F are computed without explicitly carrying out or even knowing the mapping φ(•), which results in computationally efficient algorithms. Classic examples of valid kernels are the q-th degree polynomial kernel κ(x i , x j ) = (1 + x i , x j ) q and the Gaussian kernel κ(x i , x j ) = exp(-x i -x j 2 /2σ 2 ), where σ is the kernel bandwidth. Note that the feature space corresponding to the latter is infinite dimensional.

B. The kernel trick, the representer theorem

The kernel trick can be used to transform any linear data processing technique into a non-linear one, on the condition that the algorithm can be expressed in terms of inner products only, involving pairs of the input data. This is achieved by substituting each inner product x i , x j by a kernel κ(x i , x j ), leaving the algorithm unchanged and incurring essentially the same computational cost. In conjunction with the kernel trick, the representer theorem [START_REF] Kimeldorf | Some results on Tchebycheffian spline functions[END_REF], [START_REF] Schölkopf | A generalized representer theorem[END_REF] is a solid foundation of kernel-based methods such as SVM, KFDA, KGDA and KPCA. Consider the learning machine L(x) = φ(x), Φ and the regularized risk functional

n i=1 V (L(x i ), y i ) + λ L 2 , (3)
with V (L(x), y) the cost of predicting L(x) when the desired output is y, and λ a positive parameter. The representer 1 The eigenvalue problem for non-centered data points φ(x i ) in feature space F is given by:

(K -1nK -K1n + 1nK1n)a k = λ k a k , with K(i, j) = φ(x i ), φ(x j ) and 1n(i, j) = 1 n [9].
theorem states that, under very general conditions on the loss function V , any Φ ∈ F minimizing the criterion (3) admits a representation of the form

Φ = n i=1 a i φ(x i ). (4)
This leads to the well-known kernel expansion L(x) = n i=1 a i κ(x, x i ). As an example, note that expressions (1) and ( 2) arise as a direct consequence of this theorem when it is applied to the problem of PCA in feature space.

III. TIME-FREQUENCY LEARNING MACHINES: GENERAL

PRINCIPLES

The reason for time-frequency analysis is to give a mathematical core to the intuitive concept of time-varying Fourier spectrum for non-stationary signals. For a survey, see [START_REF] Boashash | Time-frequency signal analysis and applications[END_REF], [START_REF] Flandrin | Time-Frequency/Time-Scale Analysis[END_REF] and references therein. Most of the parametric distributions of current interest belong to the Cohen class, which has proven useful in identifying non-stationarities in signals produced by a host of real-world applications. In this section, we investigate the use of kernel learning machines for pattern recognition in the time-frequency domain. To clarify the discussion, we will first focus on the Wigner distribution, which plays a central role in the Cohen class. This will be followed by an extension to other time-frequency distributions.

A. The Wigner distribution

Among the myriad of time-frequency representations that have been proposed, the Wigner distribution is considered fundamental in a number of ways. Its usefulness derives from the fact that it satisfies many desired mathematical properties such as the correct marginal conditions and the weak correct-support conditions [START_REF] Boashash | Time-frequency signal analysis and applications[END_REF], [START_REF] Flandrin | Time-Frequency/Time-Scale Analysis[END_REF]. This distribution is also a suitable candidate for time-frequency-based detection since it is covariant to time shifts and frequency shifts and it satisfies the unitarity condition [START_REF] Flandrin | A time-frequency formulation of optimum detection[END_REF]. Let X be a subspace of L 2 (C), the space of finite-energy complex signals, equipped with the usual inner product x i , x j = t x i (t) x * j (t) dt and its corresponding norm. The Wigner distribution of any signal x ∈ X is defined as

W x (t, f ) = x(t + τ /2) x * (t -τ /2) e -2jπf τ dτ. (5)
By applying conventional linear pattern recognition algorithms directly to time-frequency representations, we seek to determine a time-frequency pattern Φ(t, f ) so that machines eliminate this computational burden. It suffices to consider the following kernel

W x , Φ = W x (t, f ) Φ(t, f ) dt df (
κ W (x i , x j ) = W xi , W xj , (7)
and note that W xi and W xj do not need to be computed since, by the unitarity of the Wigner distribution, we have

κ W (x i , x j ) = | x i , x j | 2 . ( 8
)
Any kernel learning machine proposed in the literature can then be used with the kernel (8) to perform pattern recognition tasks in the time-frequency domain. The solution

L(x) = n i=1 a i | x, x i | 2
guaranteed by the representer theorem allows for a time-frequency distribution interpretation, L(x) = W x , Φ W , with

Φ W = n i=1 a i W xi . (9)
We should again emphasize that the coefficients a i are estimated without calculating any Wigner distribution. The timefrequency features are obtained from a decomposition into time-domain ones, which is an interesting consequence of the kernel trick and the representer theorem that is worth mentioning. The time-frequency pattern Φ W can be subsequently evaluated with [START_REF] Mika | Fisher discriminant analysis with kernels[END_REF], in an iterative manner, without suffering the drawback of storing and manipulating a large collection of Wigner distributions. The inherent sparsity of the coefficients a i produced by most of the kernel learning machines, a typical example of which is SVM, may speed-up the calculation of Φ W .

B. Example of time-frequency learning machine: the Wignerbased KPCA

We proceed now to illustrate the concept of time-frequency learning machines outlined above through an example involving KPCA, described in Section II-A, and the Wigner distribution. There have been so many contributions related to eigenvalue and singular value decompositions of timefrequency distributions that we can only mention one of the first papers [START_REF] Marinovitch | The singular value decomposition of the wigner distribution and its applications[END_REF] and recent applications [START_REF] Bernat | Decomposing ERP timefrequency energy using PCA[END_REF], [START_REF] Mamar | Probabilistic classifiers and time-scale representations: application to the monitoring of a tramway guiding system[END_REF]. Consider a collection of n signals x 1 , . . . , x n , each of length d. KPCA can be adapted to operate directly on their Wigner distributions by using the kernel κ W (x i , x j ). The central step of the algorithm is to perform eigendecomposition of the Gram matrix K W -

1 n K W -K W 1 n + 1 n K W 1 n , with K W (i, j) = κ W (x i , x j ) and 1 n (i, j) = 1
n . Let the eigenvectors and eigenvalues be denoted by a k and λ k , respectively, with λ 1 ≥ λ 2 ≥ . . . The k-th principal component can be extracted from any signal x as follows

L k (x) = W x , Φ k = n i=1 a i,k κ W (x, x i ), (10)
with Φ k = n i=1 a i,k W xi , and a i,k the i-th component of a k . As shown in [START_REF] Mika | Fisher discriminant analysis with kernels[END_REF], the normalization requirement Φ k = 1 leads to the condition λ k a k 2 = 1. We call Φ k the k-th eigendistribution. We call Φ k the k-th eigendistribution. Note, however, that it is not a valid time-frequency distribution in that no signal with time-frequency transform Φ k necessarily exists. This approach is summarized in Table I.

To show that KPCA is a potentially useful tool in timefrequency signal processing, a set of 1000 noisy signals of length 64 was generated. Each signal was made up of a linear chirp with normalized frequency increasing from 0.2 Hz to 0.45 Hz, a Gaussian pulse centered at time index 32 and normalized frequency 0.1 Hz, and an additive zero-mean white Gaussian noise with a standard deviation of 1. This resulted in a signal-to-noise ratio of -5.1 dB in the timefrequency domain. Wigner-based KPCA was performed to determine the eigendistributions Φ k . Their calculation was based on the discrete-time discrete-frequency Wigner distribution introduced in [START_REF] Chassande-Mottin | Discrete time and frequency Wigner-Ville distribution: Moyal's formula and aliasing[END_REF] since it satisfies most of properties of its continuous counterpart [START_REF] Sayeed | Optimal detection using bilinear time-frequency and time-scale representations[END_REF], in particular unitarity. The first eigendistribution Φ 1 represented in Figure 1(a) shows that significant information has been successfully extracted from noisy data. The two signal components can be clearly distinguished in it, as well as oscillating interferences that are characteristic of the Wigner distribution and often limit its expertise. This observation is corroborated by an increase in signal-to-noise ratio of 8.5 dB for Wigner distributions projected into the space spanned by the eigendistributions Φ 1 and Φ 2 .

Applying standard PCA directly to the set of Wigner distributions would lead to the same result. However, this approach usually suffers from the high computational cost of calculating the d 2 -by-d 2 covariance matrix of n time-frequency distributions, each of size d-by-d, and performing its eigendecomposi- I, whose computational complexities are O(dn 2 ) and O(n 3 ), respectively. Figure 2 shows the computation time of both methods plotted as a function of n, with d fixed to 64.

(i, j) = κ W (x i , x j ) O(dn 2 ) 3. Perform eigendecomposition of K W -1 n K W -K W 1 n + 1 n K W 1 n O(n 3
As expected, it is almost linear in n for PCA applied directly to Wigner distributions, and polynomial for Wigner-based KPCA. It can also be verified that the latter is computationally more efficient than PCA as long as n is less than d 2 , a condition often satisfied in practice. These conclusions remain valid for standard pattern recognition methods that require either inversion or eigendecomposition of covariance matrices, such as FDA and GDA.

C. Application to other time-frequency distributions

Obviously, the concept of time-frequency learning machine is not limited only to the Wigner distribution. In this subsection, we illustrate it with other popular time-frequency distributions, linear and quadratic.

The short-time Fourier transform is probably the most common example of linear time-frequency distribution. Denoting by w(t) an analysis window localized around the origin of the time-frequency domain, it is defined by

F x (t, f ) = x(τ ) w * (τ -t) e -2jπf τ dτ, (11)
or, in an equivalent way,

F x (t, f ) = x, w t,f with w t,f (τ ) = w(τ -t) e 2jπf τ . The kernel function κ F (x i , x j ) = F xi , F xj , namely, κ F (x i , x j ) = w 2 x i , x j . ( 12
)
can be used with any kernel learning machine proposed in the literature. The solution guaranteed by the representer theorem offers a time-frequency distribution interpretation:

L(x) = F x , Φ F with Φ F = n i=1 a i F xi .
The use of quadratic forms in non-stationary signal analysis is motivated by the need to collect information on the distribution of signal energy over time and frequency. Over the years, the Cohen class has received considerable attention because it contains all the distributions C x that are covariant with respect to time-frequency shifts applied to the signal. These are taking the form

C x (t, f ) = Π(t -t, f -f ) W x (t , f ) dt df ( 13
)
where Π is a weighting function. We can easily check that κ C (x i , x j ) = C xi , C xj is a valid kernel that can be used by any kernel learning machine. The solution can further be rewritten as

L(x) = Φ C , C x with Φ C = n i=1 a i C xi .
The advantage of κ C over κ W is that the correlative form [START_REF] Davy | Optimised support vector machines for nonstationary signal classification[END_REF] can be exploited to improve the readability of Φ C , that may be affected by the presence of troublesome oscillating interferences. Nevertheless, as can be seen on Figure 1(b) with an example of Choï-Williams-based KPCA, the same processing is simultaneously applied to interference terms and signal components, removing the former ones and spreading out the latter. The most popular quasi-interference-free distribution of the Cohen class is certainly the spectrogram. Formally defined as the squared magnitude of the short-time Fourier transform, the spectrogram is related to the kernel

κ S (x i , x j ) = | x i , w t,f x j , w t,f | 2 dt df .
Other examples of distributions include those that satisfy the unitary condition

C xi , C xj = | x i , x j | 2 ,
e.g., the Wigner distribution, the Page distribution and the Rihaczek distribution. Kernel learning machines L(x) = Φ C , C x based on unitary distributions share the same kernel [START_REF] Schölkopf | Nonlinear component analysis as a kernel eigenvalue problem[END_REF], and then have the same performance. They differ by their time-frequency pattern Φ C , which can be computed directly or using Φ W as follows:

Φ C (t, f ) = Π(t -t, f -f ) Φ W (t , f ) dt df . ( 14
)
In the general case of non-unitary distributions, the calculation of κ C (x i , x j ) can be a costs O(n 2 d 2 ). Definition [START_REF] Davy | Optimised support vector machines for nonstationary signal classification[END_REF] shows that extra computation cost of O(d 2 ) is also required to calculate time-frequency distributions other than the Wigner distribution, but the whole computation also takes time of order O(d 2 log d) per distribution [START_REF] Richard | Joint recursive implementation of timefrequency representations and their modified version by the reassignment method[END_REF]. Computation times of Wigner-based and Choï-Williams-based KPCA plotted in Figure 2 corroborate this analysis. It can also be observed that Choï-Williams-based KPCA is computationally more efficient than PCA as long as n is strictly less than d 2 , that is, as long as the size of Gram matrix is less than the size of covariance matrix.

In applications where computation time is a crucial factor, we suggest a simple heuristic procedure to derive rules of the form L(x) = C x , Φ C . It consists of training the kernel learning machine with κ W (x i , x j ) = | x i , x j | 2 . The time-frequency feature Φ C is then obtained from Φ W and equation ( 14). This strategy is clearly non-optimal when C x does not satisfy the unitarity condition. However, it greatly improves computational efficiency and also simplifies the use of signal-dependent methods of designing Π. Figure 1(c) provides an example of time-frequency-based PCA followed by the AOK algorithm [START_REF] Jones | An adaptive optimal-kernel time-frequency representation[END_REF]. This result demonstrates a significant visual improvement over those given in Figures 1(a) and 1(b). A complete analysis of this heuristic falls beyond the scope of this paper and will be addressed in the future.

D. Signal classification with time-frequency learning machines

The last ten years have seen an explosion of research in supervised [START_REF] Mika | Fisher discriminant analysis with kernels[END_REF], [START_REF] Baudat | Generalized discriminant analysis using a kernel approach[END_REF], [START_REF] Abdallah | An improved training algorithm for nonlinear kernel discriminants[END_REF] and unsupervised [START_REF] Schölkopf | Support vector method for novelty detection[END_REF] classification techniques based on kernels; see [START_REF] Vert | A primer on kernel methods[END_REF] for a recent survey. These include SVM, which map data into a high dimensional space where the classes of data are more readily separable, and maximize the distance -or margin -between the separating hyperplane and the closest points of each class [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF]. SVM basically involve formulating the margin maximization problem into dot product form in order to use kernels, and solving a quadratic programming problem to estimate the parameters a i of the test statistic

L(x) = n i=1 a i κ(x, x i ). (15)
The excellent performance of SVM has inspired countless works in discriminant analysis. In particular, KFDA is a powerful method of obtaining non-linear Fisher discriminants. It also uses the kernel trick and the representer theorem to design, via an eigenvalue problem, kernel-based classifiers of the form (15) that maximize the Rayleigh coefficient of the between and the within class scatter matrices [START_REF] Mika | Fisher discriminant analysis with kernels[END_REF], [START_REF] Abdallah | An improved training algorithm for nonlinear kernel discriminants[END_REF]. KGDA is an extension to this approach for handling multiclass problems [START_REF] Baudat | Generalized discriminant analysis using a kernel approach[END_REF]. We are now going to illustrate the concept of timefrequency learning machines through supervised classification problems involving SVM, KFDA and KGDA.

The first example deals with a binary classification task involving two classes of 1000 signals of length 64. Each signal was consisting either of a Gaussian pulse centered at time index 32 and normalized frequency 0.1 Hz, or of a frequency-modulated wave with a parabolic modulation between 0.1 Hz and 0.4 Hz, corrupted by an additive zeromean white Gaussian noise with a standard deviation of 2.2. Two time-frequency learning machines were evaluated, Wigner-based SVM and Wigner-based KFDA, obtained from SVM and KFDA with κ W (x i , x j ). Preliminary experiments were conducted on a cross-validation set of 1000 signals to select the best settings for each algorithm. The regularization parameter c of SVM was set to 1, and KFDA was regularized by adding 10 -3 to the diagonal elements of within class scatter matrix. Figure 3 provides, in both cases, the time-frequency pattern Φ that follows from the reformulation of (15) in terms of Wigner distributions. The signal components can be clearly distinguished, showing that discriminant information has been successfully extracted from training data. The positive orientation of the Gaussian pulse, and the negative orientation of the frequency-modulated wave and its interference terms, allow the test statistic [START_REF] Kimeldorf | Some results on Tchebycheffian spline functions[END_REF] to increase the separability of the two competing classes of signals. Note that Figure 3 KFDA estimated over a 10000-sample test set: 12.5% for the former and 13.9% for the latter. This illustrates the good generalization ability of SVM that makes them so popular. Our approach enables this technique to be adapted for use with time-frequency distributions. This experiment was extended to a wide variety of distributions of the Cohen class: smoothed pseudo-Wigner, Margenau-Hill, Choï-Williams, Born-Jordan, reduced-interference with Hanning window and spectrogram. Parameters of these distributions were set to the default values proposed by the Matlab toolbox TFTB2 . Figure 4 shows that the best performance was obtained with the Wigner distribution, which means that the filtering process (13) caused some loss of relevant information in the other distributions. Wigner-based learning machines were also computationally the most efficient.

Given a m-class classification problem, KGDA provides (m -1) test statistics of the form (15) that maximize the Rayleigh ratio of between and within class scatter matrices. The classification of new data is then achieved by comparing these test statistics to predetermined thresholds. As another application of time-frequency learning machines, we propose here to perform KGDA with kernel κ W (x i , x j ). This example deals with a classification problem involving three classes of 300 signals, each of length 64. Each signal was consisting either of a Gaussian pulse centered at time index 32 and normalized frequency 0.25 Hz, of a 0.1 Hz sine wave, or of a frequency-modulated wave with a parabolic modulation between 0.15 Hz and 0.45 Hz. These signals were corrupted by an additive zero-mean white Gaussian noise with a standard deviation of 0.45. Figures 5(a) and 5(b) represent the timefrequency patterns extracted by the Wigner-based KGDA algorithm, denoted by Φ a and Φ b , respectively. Each signal can be easily distinguished in these distributions, meaning that all the discriminant information has been successfully collected. In particular, note that the task of Φ a is to discriminate the parabolic modulation from the sine wave and the Gaussian pulse since the former is negatively oriented while the latter are positively oriented. The time-frequency pattern Φ b makes it possible to discriminate between the parabolic modulation, which is almost absent, the sine wave and the Gaussian pulse, which are respectively negatively and positively oriented. This analysis is confirmed in Figure 5(c), where each training data x i is represented in the Φ a Φ b plane by a point whose coordinates are W xi , Φ a and W xi , Φ b . The class of parabolic modulations is characterized by negative Φ acoordinates whereas the classes of sine waves and Gaussian pulses correspond to positive Φ a -coordinates. Along the Φ baxis, the coordinates of the sine waves, parabolic chirps and gaussian pulses are negative, close to zero and positive, respectively.

IV. CONCLUSION

The theory of reproducing kernels enabled the development of new learning algorithms for pattern recognition, whose formulation is independent of the representation space of data. Their success was largely influenced by the emerging field of Statistical Learning Theory, which simultaneously provided fundamental bounds on achievable performance. In this paper, we have focused our attention on the new concept of time-frequency learning machines. It takes advantage of this progress for implementing universal learning machines that extract time-frequency information from signals. We have illustrated the efficiency of these novel techniques for nonstationary signal analysis through unsupervised and supervised learning problems. Time-frequency learning machines can be used in many other applications, such as blind source separation [START_REF] Bach | Kernel independent component analysis[END_REF] and filtering [START_REF] Engel | Kernel recursive least squares[END_REF], where kernel-based methods have proved their efficiency. Their extension to higher order distributions also seems feasible.

In ongoing studies, we are investigating kernel-based methodologies that could be advantageously used to solve recurrent problems in the field of non-stationary signal analysis. For instance, we have recently proposed a method for selecting time-frequency distributions appropriate for given learning tasks [START_REF] Honeiné | Optimal selection of time-frequency representations for signal classification: a kerneltarget alignment approach[END_REF]. It is based on a criterion that has recently emerged from the machine learning literature: the kerneltarget alignment. Further work may contribute to strengthen these connections with the most recent methodological and theoretical developments of pattern recognition and Statistical Learning Theory, in order to offer new perspectives in the field of non-stationary signal analysis.

Fig. 1 .

 1 Fig. 1. First eigendistribution Φ 1 obtained using (a) Wigner-based KPCA, (b) Choï-Williams-based KPCA and (c) AOK algorithm applied to the Wignerbased KPCA result. The experiments were carried out using a collection of 1000 signals, each of length 64, consisting of a linear chirp and a Gaussian pulse in an additive white Gaussian noise with a standard deviation of 1.

) 4 .Fig. 2 .

 42 Fig. 2. Computation times of KPCA and PCA applied to time-frequency distributions, as a function of the number n of 64-sample signals. The Wigner distribution and the Choï-Williams distribution are considered. The former is a unitary distribution whereas the latter is not. All the codes were implemented in Matlab and run on a laptop PC with 1 GB RAM and Pentium M 1.60 GHz processor.

Fig. 3 .

 3 Fig. 3. Discriminant information extracted by (a) Wigner-based KFDA and (b) Wigner-based SVM. The experiments were carried out using two classes of 1000 signals, each of length 64, consisting of a frequency-modulated waves with a parabolic modulation and a Gaussian pulse, respectively, in an additive white Gaussian noise with a standard deviation of 2.2.

Fig. 4 .

 4 Fig. 4.Error rates of time-frequency-based SVM and KFDA using various distributions of the Cohen class: Wigner (W), smoothed pseudo-Wigner (SPW), Margenau-Hill (MH), Choï-Williams (CW), Born-Jordan (BJ), reduced-interference with Hanning window (RIDH) and spectrogram (SP). Same data as in Fig.3.

Fig. 5 .

 5 Fig. 5. The time-frequency patterns Φa and Φ b , in (a) and (b) respectively, were extracted by Wigner-based GDA in a three-class classification problem involving sine waves, frequency-modulated waves with a parabolic modulation and Gaussian pulses embedded in noise. In (c), the training data are represented in the ΦaΦ b plane.

  Compute the Wigner distribution of each one of the n signals O(d 2 log d) per signal [22] 2. Compute the Gram matrix K W

	TABLE I
	THE WIGNER-BASED KPCA ALGORITHM
	Instructions	Complexity
	1.

The Time-Frequency Toolbox (TFTB) is downloadable from http://tftb.nongnu.org/.