Switched Latent Force Models for Movement Segmentation

Transcription

1 Switche Latent Force Moels for Movement Segmentation Mauricio A. Álvarez, Jan Peters, Bernhar Schölkopf, Neil D. Lawrence 3,4 School of Computer Science, University of Manchester, Manchester, UK M3 9PL Max Planck Institute for Biological Cybernetics, Tübingen, Germany School of Computer Science, University of Sheffiel, Sheffiel, UK S 4DP 4 The Sheffiel Institute for Translational Neuroscience, Sheffiel, UK S HQ Abstract Latent force moels encoe the interaction between multiple relate ynamical systems in the form of a kernel or covariance function. Each variable to be moele is represente as the output of a ifferential equation an each ifferential equation is riven by a weighte sum of latent functions with uncertainty given by a Gaussian process prior. In this paper we consier employing the latent force moel framework for the problem of etermining robot motor primitives. To eal with iscontinuities in the ynamical systems or the latent riving force we introuce an extension of the basic latent force moel, that switches between ifferent latent functions an potentially ifferent ynamical systems. This creates a versatile representation for robot movements that can capture iscrete changes an non-linearities in the ynamics. We give illustrative examples on both synthetic ata an for striking movements recore using a Barrett WAM robot as haptic input evice. Our inspiration is robot motor primitives, but we expect our moel to have wie application for ynamical systems incluing moels for human motion capture ata an systems biology. Introuction Latent force moels [] are a new approach for moeling ata that allows combining imensionality reuction with systems of ifferential equations. The basic iea is to assume an observe set of D correlate functions to arise from an unobserve set of R forcing functions. The assumption is that the R forcing functions rive the D observe functions through a set of ifferential equation moels. Each ifferential equation is riven by a weighte mix of latent forcing functions. Sets of couple ifferential equations arise in many physics an engineering problems particularly when the temporal evolution of a system nees to be escribe. Learning such ifferential equations has important applications, e.g., in the stuy of human motor control an in robotics [6]. A latent force moel iffers from classical approaches as it places a probabilistic process prior over the latent functions an hence can make statements about the uncertainty in the system. A joint Gaussian process moel over the latent forcing functions an the observe ata functions can be recovere using a Gaussian process prior in conjunction with linear ifferential equations []. The resulting latent force moeling framework allows the combination of the knowlege of the systems ynamics with a ata riven moel. Such generative moels can be use to goo effect, for example in ranke target preiction for transcription factors [5]. If a single Gaussian process prior is use to represent each latent function then the moels we consier are limite to smooth riving functions. However, iscontinuities an segmente latent forces are omnipresent in real-worl ata. For example, impact forces ue to contacts in a mechanical ynamical system (when grasping an object or when the feet touch the groun) or a switch in an electrical circuit result in iscontinuous latent forces. Similarly, most non-rhythmic natural mo-

2 tor skills consist of a sequence of segmente, iscrete movements. If these segments are separate time-series, they shoul be treate as such an not be moele by the same Gaussian process moel. In this paper, we extract a sequence of ynamical systems motor primitives moele by secon orer linear ifferential equations in conjunction with forcing functions (as in [, 6]) from human movement to be use as emonstrations of elementary movements for an anthropomorphic robot. As human trajectories have a large variability: both ue to planne uncertainty of the human s movement policy, as well as ue to motor execution errors [7], a probabilistic moel is neee to capture the unerlying motor primitives. A set of secon orer ifferential equations is employe as mechanical systems are of the same type an a temporal Gaussian process prior is use to allow probabilistic moeling []. To be able to obtain a sequence of ynamical systems, we augment the latent force moel to inclue iscontinuities in the latent function an change ynamics. We introuce iscontinuities by switching between ifferent Gaussian process moels (superficially similar to a mixture of Gaussian processes; however, the switching times are moele as parameters so that at any instant a single Gaussian process is riving the system). Continuity of the observe functions is then ensure by constraining the relevant state variables (for example in a secon orer ifferential equation velocity an isplacement) to be continuous across the switching points. This allows us to moel highly non stationary multivariate time series. We emonstrate our approach on synthetic ata an real worl movement ata. Review of Latent force moels (LFM) Latent force moels [] are hybri moels that combine mechanistic principles an Gaussian processes as a flexible way to introuce prior knowlege for ata moeling. A set of D functions {y (t)} D = is moele as the set of output functions of a series of couple ifferential equations, whose common input is a linear combination of R latent functions, {u r (t)} R r=. Here we focus on a secon orer orinary ifferential equation (ODE). We assume the output y (t) is escribe by A y (t) t + C y (t) t + κ y (t) = R r= S,ru r (t), where, for a mass-spring-amper system, A woul represent the mass, C the amper an κ, the spring constant associate to the output. We refer to the variables S,r as the sensitivity parameters. They are use to represent the relative strength that the latent force r exerts over the output. For simplicity we now focus on the case where R =, although our erivations apply more generally. Note that moels that learn a forcing function to rive a linear system have proven to be well-suite for imitation learning for robot systems [6]. The solution of the secon orer ODE follows y (t) = y ()c (t) + ẏ ()e (t) + f (t, u), () where y () an ẏ () are the output an the velocity at time t =, respectively, known as the initial conitions (IC). The angular frequency is given by ω = (4A κ C )/(4A ) an the remaining variables are given [ by c (t) = e α t cos(ω t) + α sin(ω t) ], e (t) = e α t sin(ω t), ω S f (t, u) = A ω t S G (t τ)u(τ)τ = A ω t ω e α (t τ) sin[(t τ)ω ]u(τ)τ, with α = C /(A ). Note that f (t, u) has an implicit epenence on the latent function u(t). The uncertainty in the moel of Eq. () is ue to the fact that the latent force u(t) an the initial conitions y () an ẏ () are not known. We will assume that the latent function u(t) is sample from a zero mean Gaussian process prior, u(t) GP(, k u,u (t, t )), with covariance function k u,u (t, t ). If the initial conitions, y IC = [y (), y (),..., y D (), v (), v (),..., v D ()], are inepenent of u(t) an istribute as a zero mean Gaussian with covariance K IC the covariance function between any two output functions, an at any two times, t an t, k y,y (t, t ) is given by c (t)c (t )σ y,y + c (t)e (t )σ y,v + e (t)c (t )σ v,y + e (t)e (t )σ v,v + k f,f (t, t ), where σ y,y, σ y,v, σ v,y an σ v,v are entries of the covariance matrix K IC an k f,f (t, t ) = K t G (t τ) t G (t τ )k u,u (t, t )τ τ, ()

3 where K = S S /(A A ω ω ). So the covariance function k f,f (t, t ) epens on the covariance function of the latent force u(t). If we assume the latent function has a raial basis function (RBF) covariance, k u,u (t, t ) = exp[ (t t ) /l ], then k f,f (t, t ) can be compute analytically [] (see also supplementary material). The latent force moel inuces a joint Gaussian process moel across all the outputs. The parameters of the covariance function are given by the parameters of the ifferential equations an the length scale of the latent force. Given a multivariate time series ata set these parameters may be etermine by maximum likelihoo. The moel can be thought of as a set of mass-spring-ampers being riven by a function sample from a Gaussian process. In this paper we look to exten the framework to the case where there can be iscontinuities in the latent functions. We o this through switching between ifferent Gaussian process moels to rive the system. 3 Switching ynamical latent force moels (SDLFM) We now consier switching the system between ifferent latent forces. This allows us to change the ynamical system an the riving force for each segment. By constraining the isplacement an velocity at each switching time to be the same, the output functions remain continuous. 3. Definition of the moel We assume that the input space is ivie in a series of non-overlapping intervals [t q, t q ] Q q=. During each interval, only one force u q (t) out of Q forces is active, that is, there are {u q (t)} Q q= forces. The force u q (t) is activate after time t q (switche on) an eactivate (switche off) after time t q. We can use the basic moel in equation () to escribe the contribution to the output ue to the sequential activation of these forces. A particular output z (t) at a particular time instant t, in the interval (t q, t q ), is expresse as z (t) = y q (t t q ) = c q (t t q )y q (t q ) + e q (t t q )ẏ q (t q ) + f q (t t q, u q ). This equation is assumme to be vali for escribing the output only insie the interval (t q, t q ). Here we highlighte this iea by incluing the superscript q in y q (t t q ) to represent the interval q for which the equation hols, although later we will omit it to keep the notation uncluttere. Note that for Q = an t =, we recover the original latent force moel given in equation (). We also efine the velocity ż (t) at each time interval (t q, t q ) as ż (t) = ẏ q (t t q ) = g q (t t q )y q (t q ) + h q (t t q )ẏ q (t q ) + m q (t t q, u q ), where g (t) = e αt sin(ω t)(α ω + ω ) an [ ] h (t) = e α α t sin(ω t) cos(ω t), m (t) = S ( t ω A ω t ) G (t τ)u(τ)τ. Given the parameters θ = {{A, C, κ, S } D =, {l q } Q q= }, the uncertainty in the outputs is inuce by the prior over the initial conitions y q (t q ), ẏ q (t q ) for all values of t q an the prior over latent force u q (t) that is active uring (t q, t q ). We place inepenent Gaussian process priors over each of these latent forces u q (t), assuming inepenence between them. For initial conitions y q (t q ), ẏ q (t q ), we coul assume that they are either parameters to be estimate or ranom variables with uncertainty governe by inepenent Gaussian istributions with covariance matrices K q IC as escribe in the last section. However, for the class of applications we will consier: mechanical systems, the outputs shoul be continuous across the switching points. We therefore assume that the uncertainty about the initial conitions for the interval q, y q (t q ), ẏ q (t q ) are proscribe by the Gaussian process that escribes the outputs z (t) an velocities ż (t) in the previous interval q. In particular, we assume y q (t q ), ẏ q (t q ) are Gaussian-istribute with mean values given by y q (t q t q ) an ẏ q (t q t q ) an covariances k z,z (t q, t q ) = cov[y q (t q t q ), y q (t q t q )] an kż,ż (t q, t q ) = cov[ẏ q (t q t q ), ẏ q (t q t q )]. We also consier covariances between z (t q ) an ż (t q ), this is, between positions an velocities for ifferent values of q an. Example. Let us assume we have one output (D = ) an three switching intervals (Q = 3) with switching points t, t an t. At t, we assume that y IC follows a Gaussian istribution with 3

4 mean zero an covariance K IC. From t to t, the output z(t) is escribe by z(t) = y (t t ) = c (t t )y (t ) + e (t t )ẏ (t ) + f (t t, u ). The initial conition for the position in the interval (t, t ) is given by the last equation evaluate a t, this is, z(t ) = y (t ) = y (t t ). A similar analysis is use to obtain the initial conition associate to the velocity, ż(t ) = ẏ (t ) = ẏ (t t ). Then, from t to t, the output z(t) is z(t) = y (t t ) = c (t t )y (t ) + e (t t )ẏ (t ) + f (t t, u ), = c (t t )y (t t ) + e (t t )ẏ (t t ) + f (t t, u ). Following the same train of thought, the output z(t) from t is given as z(t) = y 3 (t t ) = c 3 (t t )y 3 (t ) + e 3 (t t )ẏ 3 (t ) + f 3 (t t, u ), where y 3 (t ) = y (t t ) an ẏ 3 (t ) = ẏ (t t ). Figure shows an example of the switching ynamical latent force moel scenario. To ensure the continuity of the outputs, the initial conition is force to be equal to the output of the last interval evaluate at the switching point. 3. The covariance function z(t) The erivation of the covariance function for the switching moel is rather involve. For continuous y (t t ) output signals, we must take into account constraints y (t t ) at each switching time. This causes initial conitions for each interval y (t ) y (t t ) y (t ) y 3 (t ) to be epenent on final conitions for the previous y (t t ) y 3 (t t ) interval an inuces correlations t t t across the inter- Figure : Representation of an output constructe through a switching ynam- vals. This effort is worthwhile ical latent force moel with Q = 3. The initial conitions y q (t q ) for each though as the result- interval are matche to the value of the output in the last interval, evaluate at ing moel is very flexible the switching point t q, this is, y q (t q ) = y q (t q t q ). an can take avantage of the switching ynamics to represent a range of signals. As a taster, Figure shows samples from a covariance function of a switching ynamical latent force moel with D = an Q = 3. Note that while the latent forces (a an c) are iscrete, the outputs (b an ) are continuous an have matching graients at the switching points. The outputs are highly nonstationary. The switching times turn out to be parameters of the covariance function. They can be optimize along with the ynamical system parameters to match the location of the nonstationarities. We now give an overview of the covariance function erivation. Details are provie in the supplementary material. (a) System. Samples from the latent force. 4 (b) System. from the output. Samples (c) System. Samples from the latent force. 3 () System. from the output. 6 Samples Figure : Joint samples of a switching ynamical LFM moel with one output, D =, an three intervals, Q = 3, for two ifferent systems. Dashe lines inicate the presence of switching points. While system respons instantaneously to the input force, system elays its reaction ue to larger inertia. 4

5 In general, we nee to compute the covariance k z,z (t, t ) = cov[z (t), z (t )] for z (t) in time interval (t q, t q ) an z (t ) in time interval (t q, t q ). By efinition, this covariance follows cov[z (t), z (t )] = cov [ y q (t t q ), y q (t t q )) ]. We assumme inepenence between the latent forces u q (t) an inepenence between the initial conitions y IC an the latent forces u q (t). With these conitions, it can be shown that the covariance function 3 for q = q is given as c q (t t q )c q (t t q )k z,z (t q, t q ) + c q (t t q )e q (t t q )k z,ż (t q, t q ) +e q (t t q )c q (t t q )kż,z (t q, t q ) + e q (t t q )e q (t t q )kż,ż (t q, t q ) +k q f,f (t, t ), (3) where k z,z (t q, t q ) = cov[y q (t q )y q (t q )], k z,ż (t q, t q ) = cov[y q (t q )ẏ q (t q )], kż,z (t q, t q ) = cov[ẏ q (t q )y q (t q )], kż,ż (t q, t q ) = cov[ẏ q (t q )ẏ q (t q )]. k q f,f (t, t ) = cov[f q (t t q )f q (t t q )]. In expression (3), k z,z (t q, t q ) = cov[y q (t q t q ), y q (t q t q )] an values for k z,ż (t q, t q ), kż,z (t q, t q ) an kż,ż (t q, t q ) can be obtaine by similar expressions. The covariance k q f,f (t, t ) follows a similar expression that the one for k f,f (t, t ) in equation (), now epening on the covariance k uq,u q (t, t ). We will assume that the covariances for the latent forces follow the RBF form, with length-scale l q. When q > q, we have to take into account the correlation between the initial conitions y q (t q ), ẏ q (t q ) an the latent force u q (t ). This correlation appears because of the contribution of u q (t ) to the generation of the initial conitions, y q (t q ), ẏ q (t q ). It can be shown 4 that the covariance function cov[z (t), z (t )] for q > q follows c q (t t q )c q (t t q )k z,z (t q, t q ) + c q (t t q )e q (t t q )k z,ż (t q, t q ) +e q (t t q )c q (t t q )kż,z (t q, t q ) + e q (t t q )e q (t t q )kż,ż (t q, t q ) +c q (t t q )X k q f,f (t q, t ) + c q (t t q )X k q m,f (t q, t ) +e q (t t q )X 3 k q f,f (t q, t ) + e q (t t q )X 4 k q m,f (t q, t ), (4) where k z,z (t q, t q ) = cov[y q (t q )y q (t q )], k z,ż (t q, t q ) = cov[y q (t q )ẏ q (t q )], kż,z (t q, t q ) = cov[ẏ q (t q )y q (t q )], kż,ż (t q, t q ) = cov[ẏ q (t q )ẏ q (t q )], k q m,f (t, t ) = cov[m q (t t q )f q (t t q )], q q n= i= xq i+ an X, X, X 3 an X 4 are functions of the form q q (t q i+ t q i ), with x q i+ being equal to c q i+, e q i+, g q i+ or h q i+, epening on the values of q an q. A similar expression to (4) can be obtaine for q > q. Examples of these functions for specific values of q an q an more etails are also given in the supplementary material. 4 Relate work There has been a recent interest in employing Gaussian processes for etection of change points in time series analysis, an area of stuy that relates to some extent to our moel. Some machine learning relate papers inclue [3, 4, 9]. [3, 4] eals specifically with how to construct covariance functions Derivations of these equations are rather involve. In the supplementary material, section, we inclue a etaile escription of how to obtain the equations (3) an (4) See supplementary material, section... 3 We will write f q (t tq, uq ) as f q (t tq ) for notational simplicity. 4 See supplementary material, section.. 5

6 in the presence of change points (see [3], section 4). The authors propose ifferent alternatives accoring to the type of change point. From these alternatives, the closest ones to our work appear in subsections 4., 4.3 an 4.4. In subsection 4., a mechanism to keep continuity in a covariance function when there are two regimes escribe by ifferent GPs, is propose. The authors call this covariance continuous conitionally inepenent covariance function. In our switche latent force moel, a more natural option is to use the initial conitions as the way to transit smoothly between ifferent regimes. In subsections 4.3 an 4.4, the authors propose covariances that account for a suen change in the input scale an a suen change in the output scale. Both type of changes are automatically inclue in our moel ue to the latent force moel construction: the changes in the input scale are accounte by the ifferent length-scales of the latent force GP process an the changes in the output scale are accounte by the ifferent sensitivity parameters. Importantly, we also concerne about multiple output systems. On the other han, [9] proposes an efficient inference proceure for Bayesian Online Change Point Detection (BOCPD) in which the unerlying preictive moel (UPM) is a GP. This reference is less concerne about the particular type of change that is represente by the moel: in our application scenario, the continuity of the covariance function between two regimes must be assure beforehan. 5 Implementation In this section, we escribe aitional etails on the implementation, i.e., covariance function, hyperparameters, sparse approximations. Aitional covariance functions. The covariance functions kż,z (t, t ), k z,ż (t, t ) an kż,ż (t, t ) are obtaine by taking erivatives of k z,z (t, t ) with respect to t an t []. Estimation of hyperparameters. Given the number of outputs D an the number of intervals Q, we estimate the parameters θ by maximizing the marginal-likelihoo of the joint Gaussian process {z (t)} D = using graient-escent methos. With a set of input points, t = {t n} N n=, the marginal-likelihoo is given as p(z θ) = N (z, K z,z + Σ), where z = [z,..., z D ], with z = [z (t ),..., z (t N )], K z,z is a D D block-partitione matrix with blocks K z,z. The entries in each of these blocks are evaluate using k z,z (t, t ). Furthermore, k z,z (t, t ) is compute using the expressions (3), an (4), accoring to the relative values of q an q. Efficient approximations Optimizing the marginal likelihoo involves the inversion of the matrix K z,z, inversion that grows with complexity O(D 3 N 3 ). We use a sparse approximation base on variational methos presente in [] as a generalization of [] for multiple output Gaussian processes. The approximations establish a lower boun on the marginal likelihoo an reuce computational complexity to O(DNK ), being K a reuce number of points use to represent u(t). 6 Experimental results We now show results with artificial ata an ata recore from a robot performing a basic set of actions appearing in table tennis. 6. Toy example Using the moel, we generate samples from the GP with covariance function as explaine before. In the first experiment, we sample from a moel with D =, R = an Q = 3, with switching points t =, t = 5 an t =. For the outputs, we have A = A =., C =.4, C =, κ =, κ = 3. We restrict the latent forces to have the same length-scale value l = l = l = e 3, but change the values of the sensitivity parameters as S, =, S, =, S, =, S, = 5, S,3 = an S,3 =, where the first subinex refers to the output an the secon subinex refers to the force in the interval q. In this first experiment, we wante to show the ability of the moel to etect changes in the sensitivities of the forces, while keeping the length scales equal along the intervals. We sample 5 times from the moel with each output having 5 ata points an a some noise with variance equal to ten percent of the variance of each sample output. In each of the five repetitions, we took N = ata points for training an the remaining 3 for testing. 6

7 Q = Q = Q = 3 Q = 4 Q = 5 SMSE 76.7± ±.74.3±..3±.3.7±.56 MSLL.98±.46.79±.6.9±.3.87±.4.55±.4 SMSE 7.7±6.88.8±.5.±.5.6±.5.±.9 MSLL.79±.8.6±..5±..7±.3.6±.6 Table : Stanarize mean square error (SMSE) an mean stanarize log loss (MSLL) using ifferent values of Q for both toy examples. The figures for the SMSE must be multiplie by. See the text for etails. (a) Latent force toy example (b) Output toy example (c) Output toy example () Latent force toy example. 5 5 (e) Output toy example (f) Output 3 toy example Figure 4: Mean an two stanar eviations for the preictions over the latent force an two of the three outputs in the test set. Dashe lines inicate the final value of the swithcing points after optimization. Dots inicate training ata. Optimization of the hyperparameters (incluing t an t ) is one by maximization of the marginal likelihoo through scale conjugate graient. We train moels for Q =,, 3, 4 an 5 an measure the mean stanarize log loss (MSLL) an the mean stanarize mean square error (SMSE) [8] over the test set for each value of Q. Table, first two rows, show the corresponing average results over the 5 repetitions together with one stanar eviation. Notice that for Q = 3, the moel gets by the first time the best performance, performance that repeats again for Q = 4. The SMSE performance remains approximately equal for values of Q greater than 3. Figures 4(a), 4(b) an 4(c) shows the kin of preictions mae by the moel for Q = 3. Figure 3: Data collection was performe using a Barrett WAM robot as haptic input evice. We generate also a ifferent toy example, in which the length-scales of the intervals are ifferent. For the secon toy experiment, we assume D = 3, Q = an switching points t = an t = 8. The parameters of the outputs are A = A = A 3 =., C =, C = 3, C 3 =.5, κ =.4, κ =, κ 3 = an length scales l = e 3 an l =. Sensitivities in this case are S, =, S, = 5, S 3, =, S, = 5, S, = an S 3, =. We follow the same evaluation setup as in toy example. Table, last two rows, show the performance again in terms of MLSS an SMSE. We see that for values of Q >, the MLSS an SMSE remain similar. In figures 4(), 4(e) an 4(f), the inferre latent force an the preictions mae for two of the three outputs. 6. Segmentation of human movement ata for robot imitation learning In this section, we evaluate the feasibility of the moel for motion segmentation with possible applications in the analysis of human movement ata an imitation learning. To o so, we ha a human teacher take the robot by the han an have him emonstrate striking movements in a cooperative game of table tennis with another human being as shown in Figure 3. We recore joint positions, 7

8 Value of the log likelihoo (a) Log-Likelihoo Try Number of intervals Latent Force (b) Latent force Try. 5 5 Time HR.5 (c) HR Output Try Time 5 Value of the log likelihoo () Log-Likelihoo Try Number of intervals Latent Force 4 3 (e) Latent force Try. 5 5 Time SFE (f) SFE Output Try. 5 5 Time Figure 5: Employing the switching ynamical LFM moel on the human movement ata collecte as in Fig.3 leas to plausible segmentations of the emonstrate trajectories. The first row correspons to the loglikelihoo, latent force an one of four outputs for trial one. Secon row shows the same quantities for trial two. Crosses in the bottom of the figure refer to the number of points use for the approximation of the Gaussian process, in this case K = 5. angular velocities, an angular acceleration of the robot for two inepenent trials of the same table tennis exercise. For each trial, we selecte four output positions an train several moels for ifferent values of Q, incluing the latent force moel without switches (Q = ). We evaluate the quality of the segmentation in terms of the log-likelihoo. Figure 5 shows the log-likelihoo, the inferre latent force an one output for trial one (first row) an the corresponing quantities for trial two (secon row). Figures 5(a) an 5() show peaks for the log-likelihoo at Q = 9 for trial one an Q = for trial two. As the movement has few gaps an the ata has several output imensions, it is har even for a human being to etect the transitions between movements (unless it is visualize as in a movie). Nevertheless, the moel foun a maximum for the log-likelihoo at the correct instances in time where the human transits between two movements. At these instances the human usually reacts ue to an external stimulus with a large jerk causing a jump in the forces. As a result, we obtaine not only a segmentation of the movement but also a generative moel for table tennis striking movements. 7 Conclusion We have introuce a new probabilistic moel that evelops the latent force moeling framework with switche Gaussian processes. This allows for iscontinuities in the latent space of forces. We have shown the application of the moel in toy examples an on a real worl robot problem, in which we were intereste in fining an representing striking movements. Other applications of the switching latent force moel that we envisage inclue moeling human motion capture ata using the secon orer ODE an a first orer ODE for moeling of complex circuits in biological networks. To fin the orer of the moel, this is, the number of intervals, we have use cross-valiation. Future work inclues proposing a less expensive moel selection criteria. Acknowlegments MA an NL are very grateful for support from a Google Research Awar Mechanistically Inspire Convolution Processes for Learning an the EPSRC Grant No EP/F5687/ Gaussian Processes for Systems Ientification with Applications in Systems Biology. MA also thanks PASCAL Internal Visiting Programme. We also thank to three anonymous reviewers for their helpful comments. 8

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