{"title": "Mixture of time-warped trajectory models for movement decoding", "book": "Advances in Neural Information Processing Systems", "page_first": 433, "page_last": 441, "abstract": "Applications of Brain-Machine-Interfaces typically estimate user intent based on biological signals that are under voluntary control. For example, we might want to estimate how a patient with a paralyzed arm wants to move based on residual muscle activity. To solve such problems it is necessary to integrate obtained information over time. To do so, state of the art approaches typically use a probabilistic model of how the state, e.g. position and velocity of the arm, evolves over time \u2013 a so-called trajectory model. We wanted to further develop this approach using two intuitive insights: (1) At any given point of time there may be a small set of likely movement targets, potentially identified by the location of objects in the workspace or by gaze information from the user. (2) The user may want to produce movements at varying speeds. We thus use a generative model with a trajectory model incorporating these insights. Approximate inference on that generative model is implemented using a mixture of extended Kalman filters. We find that the resulting algorithm allows us to decode arm movements dramatically better than when we use a trajectory model with linear dynamics.", "full_text": " \n\n \n\n \n \n \n\nMixture of time -w arped trajectory models for \n\nmovement decoding \n\nElaine A. Corbett, Eric J. Perreault and Konrad P. K\u00f6rding \n\n \n \n \n\nNorthwestern University \n\nChicago, IL 60611 \n\necorbett@u.northwestern.edu \n\n \n \n \n\nAbstract \n\nApplications of Brain-Machine-Interfaces typically estimate user intent \nbased on biological signals that are under voluntary control. For example, \nwe might want to estimate how a patient with a paralyzed arm wants to \nmove based on residual muscle activity. To solve such problems it is \nnecessary to integrate obtained information over time. To do so, state of the \nart approaches typically use a probabilistic model of how the state, e.g. \nposition and velocity of the arm, evolves over time \u2013 a so-called trajectory \nmodel. We wanted to further develop this approach using two intuitive \ninsights: (1) At any given point of time there may be a small set of likely \nmovement targets, potentially identified by the location of objects in the \nworkspace or by gaze information from the user. (2) The user may want to \nproduce movements at varying speeds. We thus use a generative model with \na trajectory model incorporating these insights. Approximate inference on \nthat generative model is implemented using a mixture of extended Kalman \nfilters. We find that the resulting algorithm allows us to decode arm \nmovements dramatically better than when we use a trajectory model with \nlinear dynamics. \n\n \n\n1 \n\nI n t ro d u c t i o n \n\nWhen patients have lost a limb or the ability to communicate with the outside world, brain \nmachine interfaces (BMIs) are often used to enable robotic prostheses or restore \ncommunication. To achieve this, the user's intended state of the device must be decoded \nfrom biological signals. In the context of Bayesian statistics, two aspects are important for \nthe design of an estimator of a temporally evolving state: the observation model, which \ndescribes how measured variables relate to the system\u2019s state and the trajectory model which \ndescribes how the state changes over time in a probabilistic manner. Following this logic \nmany recent BMI applications have relied on Bayesian estimation for a wide range of \nproblems including the decoding of intended human [1] and animal [2] movements. In the \ncontext of BMIs, Bayesian approaches offer a principled way of formalizing the uncertainty \nabout signals and thus often result in improvements over other signal processing techniques \n[1]-[3]. \n\nMost work on state estimation in dynamical systems has assumed linear dynamics and \nGaussian noise. Under these circumstances, efficient algorithms result from belief \npropagation. The most frequent application uses the Kalman filter (KF), which recursively \ncombines noisy state observations with the probabilistic evolution of state defined by the \ntrajectory model to estimate the marginal distribution over states [4]. Such approaches have \nbeen used widely for applications including upper [1] and lower [5] extremity prosthetic \n\n \n\n1 \n\n\fdevices, functional electric stimulation [6] and human computer interactions [7]. As these \nalgorithms are so commonly used, it seems promising to develop extensions to nonlinear \ntrajectory models that may better describe the probabilistic distribution of movements in \neveryday life. \n\nOne salient departure from the standard assumptions is that people tend to produce both slow \nand fast movements, depending on the situation. Models with linear dynamics only allow \nsuch deviation through the noise term, which makes these models poor at describing the \nnatural variation of movement speeds during real world tasks. Explicitly incorporating \nmovement speed into the trajectory model should lead to better movement estimates. \n\nKnowledge of the target position should also strongly affect trajectory models. After all , we \ntend to accelerate our arm early during movement and slow down later on. Target \ninformation can be linearly incorporated into the trajectory model, and this has greatly \nimproved predictions [8]-[12]. Alternatively, if there are a small number of potential targets \nthen a mixture of trajectory models approach [13] can be used. Here we are interested in the \ncase where available data provide a prior over potential t argets but where movement targets \nmay be anywhere. We want to incorporate target uncertainty and allow generalization to \nnovel targets. \n\nPrior information about potential targets could come from a number of sources but would \ngenerally be noisy. For example, activity in the dorsal premotor cortex provides information \nabout intended target location prior to movement and may be used where such recordings are \navailable [14]. Target information may also be found noninvasively by tracking eye \nmovements. However, such data will generally provide non-zero priors for a number of \npossible target locations as the subject saccades over the scene. While subjects almost \nalways look at a target before reaching for it [15], there may be a delay of up to a second \nbetween looking at the target and the reach \u2013 a time interval over which up to 3 saccades are \ntypically made. Each of these fixations could be the target. Hence, a probabilistic \ndistribution of targets is appropriate when using either neural recordings or eye tracking to \nestimate potential reach targets \n\nHere we present an algorithm that uses a mixture of extended Kalman Filters (EKFs) to \ncombine our insights related to the variation of movement speed and the availability of \nprobabilistic target knowledge. Each of the mixture component s allows the speed of the \nmovement to vary continuously over time. We tested how well we could use EMGs and eye \nmovements to decode hand position of humans performing a three -dimensional large \nworkspace reaching task. We find that using a trajectory model that allows for probabilistic \ntarget information and variation of speed leads to dramatic improvements in decoding \nquality. \n\n \n2 \n\nG e n e r a l D e c o d i n g S e t t i n g \n\nWe wanted to test how well different decoding algorithms can decode human movement, \nover a wide range of dynamics. While many recent studies have looked at more restrictive, \ntwo-dimensional movements, a system to restore arm function should produce a wide range \nof 3D trajectories. We recorded arm kinematics and EMGs of healthy subjects during \nunconstrained 3D reaches to targets over a large workspace. Two healthy subjects were \nasked to reach at slow, normal and fast speeds, as they would in everyday life. Subjects were \nseated as they reached towards 16 LEDs in blocks of 150s, which were located on two \nplanes positioned such that all targets were just reachable (Fig 1A). The target LED was lit \nfor one second prior to an auditory go cue, at which time the subject would reach to the \ntarget at the appropriate speed. Slow, normal and fast reaches were allotted 3 s, 1.5s and 1s \nrespectively; however, subjects determined the speed. An approximate total of 450 reaches \nwere performed per subject. The subjects provided informed consent, and the protocol was \napproved by the Northwestern University Institutional Review Board. EMG signals were \nmeasured from the pectoralis major, and the three deltoid muscles of the shoulder. This \nrepresents a small subset of the muscles involved in reaching, and approximates those \nmuscles retaining some voluntary control following mid -level cervical spinal cord injuries. \n\n \n\n2 \n\n\fThe EMG signals were band-pass filtered between 10 and 1,000 Hz, and subsequently anti -\naliased filtered. Hand, wrist, shoulder and head positions were tracked using an Optotrak \nmotion analysis system. We simultaneously recorded eye movements with an ASL \nEYETRAC-6 head mounted eye tracker. \n\nApproximately 25% of the reaches were assigned to the test set, and the rest were used for \ntraining. Reaches for which either the motion capture data was incomplete, or there was \nvisible motion artifact on the EMG were removed. As the state we used hand positions and \njoint angles (3 shoulder, 2 elbow, position, velocity and acceleration, 24 dimensions). Joint \nangles were calculated from the shoulder and wrist marker data using digitized bony \nlandmarks which defined a coordinate system for the upper limb as detailed by Wu et al. \n[16]. As the motion data were sampled at 60Hz, the mean absolute value o f the EMG in the \ncorresponding 16.7ms windows was used as an observation of the state at each time -step. \nAlgorithm accuracy was quantified by normalizing the root -mean-squared error by the \nstraight line distance between the first and final position of the endpoint for each reach. We \ncompared the algorithms statistically using repeated measures ANOVAs with Tukey post -hoc \ntests, treating reach and subject as random effects. \n\n In the rest of the paper we will ask how well these reaching movements can be decoded \nfrom EMG and eye-tracking data. \n\n \n\n \n\nFigure 1: A Experimental setup and B sample kinematics and processed EMGs for one reach \n\n \n3 \n\nK a l ma n Fi l t e r s w i t h Ta r g e t i n f o r ma t i o n \n\nAll models that we consider in this paper assume linear observations with Gaussian noise: \n\n \n\n \n\n \n\n(1) \n\nwhere x is the state, y is the observation and v is the measurement noise with p(v) ~ N(0,R), \nand R is the observation covariance matrix. The model fitted the measured EMGs with an \naverage r2 of 0.55. This highlights the need to integrate information over time. \n\nThe standard approach also assumes linear dynamics and Gaussian process noise: \n\n \n\n \nwhere, xt represents the hand and joint angle positions, w is the process noise with p(w) \n~ N(0,Q), and Q is the state covariance matrix. The Kalman filter does optimal inference for \nthis generative model. \n\n \n\n \n\n(2) \n\nThis model can effectively capture the dynamics of stereotypical reaches to a single target by \nappropriately tuning its parameters. However, when used to describe reaches to multiple \ntargets, the model cannot describe target dependent aspects of reaching but boils down to a \nrandom drift model. Fast velocities are underestimated as they are unlikely under the \ntrajectory model and there is excessive drift close to the target (Fig. 2A). \n\n \n\n3 \n\n\fIn many decoding applications we may know the subject\u2019s target. A range of recent studies have \naddressed the issue of incorporating this information into the trajectory model [8, 13], and we \nmight assume the effect of the target on the dynamics to be linear. This naturally suggests adding \nthe target to the state space, which works well in practice [9, 12]. By appending the target to the \nstate vector (KFT), the simple linear format of the KF may be retained: \n\n \n\n \n\n \n\n \n\n(3) \n\nwhere xTt is the vector of target positions, with dimensionality less than or equal to that of \nxt. This trajectory model thus allows describing both the rapid acceleration that characterizes the \nbeginning of a reach and the stabilization towards its end. \n\nWe compared the accuracy of the KF and the KFT to the Single Target Model (STM), a KF \ntrained only on reaches to the target being tested (Fig. 2). The STM represents the best possible \nprediction that could be obtained with a Kalman filter. Assuming the target is perfectly known, we \nimplemented the KFT by correctly initializing the target state xT at the beginning of the reach. We \nwill relax this assumption below. The initial hand and joint angle positions were also assumed to \nbe known. \n\n \n\nFigure 2: A Sample reach and predictions and B average accuracies with standard errors for KFT, \n\n \n\nKF and MTM. \n\n \n\nConsistent with the recent literature, both methods that incorporated target information produced \nhigher prediction accuracy than the standard KF (both p<0.0001). Interestingly, there was no \nsignificant difference between the KFT and the STM (p=0.9). It seems that when we have \nknowledge of the target, we do not lose much by training a single model over the whole \nworkspace rather than modeling the targets individually. This is encouraging, as we desire a BMI \nsystem that can generalize to any target within the workspace, not just specifically to those that are \navailable in the training data. \n\nClearly, adding the target to the state space allows the dynamics of typical movements to be \nmodeled effectively, resulting in dramatic increases in decoding performance. \n \n4 \n\nTi me Wa r p i n g \n\n4 . 1 \n\nI m p l e m e n t i n g a t i m e - w a r p e d t r a j e c t o r y m o d e l \n\nWhile the KFT above can capture the general reach trajectory profile, it does not allow for \nnatural variability in the speed of movements. Depending on our task objectives, which \nwould not directly be observed by a BMI, we might lazily reach toward a target or move a t \nmaximal speed. We aim to change the trajectory model to explicitly incorporate a warping \nfactor by which the average movement speed is scaled, allowing for such variability. As the \nmovement speed will be positive in all practical cases, we model the logarithm of this factor, \n\n \n\n4 \n\n\fand append it to the state vector: \n\n \n\n \n\n \n\n(4) \n\nWe create a time-warped trajectory model by noting that if the average rate of a trajectory is \nto be scaled by a factor S, the position at time t will equal that of the original trajectory at \ntime St. Differentiating, the velocity will be multiplied by S, and the acceleration by S2. For \nsimplicity, the trajectory noise is assumed to be additive and Gaussian, and the model is \nassumed to be stationary: \n\n \n\n \n\n \n \n \n \n \n \n \n\n \n \n\n \n\n \n\n \n \n \n \n\n \n\n \n \n \n \n \n \n \n\n \n\n \n\n \n\n \n\n(5) \n\n \n \n \n \n \n \n \n\n \n \n\n \n\n \n \n \n \n\n \n\nwhere Ip is the p-dimensional identity matrix and is a p p matrix of zeros. Only the \nterms used to predict the acceleration states need to be estimated to build the state transition \nmatrix, and they are scaled as a nonlinear function of xs. \n\nAfter adding the variable movement speed to the state space the system is no longer linear. \nTherefore we need a different solution strategy. Instead of the typical KFT we use the \nExtended Kalman Filter (EKFT) to implement a nonlinear trajectory model by linearizing \nthe dynamics around the best estimate at each time-step [17]. With this approach we add only \nsmall computational overhead to the KFT recursions. \n \n4 . 2 \n\nTr a i n i n g t h e t i m e w a r p i n g m o d e l \n\nThe filter parameters were trained using a variant of the Expectation Maximization (EM) \nalgorithm [18]. For extended Kalman filter learning the initialization for the variables may \nmatter. S was initialized with the ground truth average reach speeds for each movement relative to \nthe average speed across all movements. The state transition parameters were estimated using \nnonlinear least squares regression, while C, Q and R were estimated linearly for the new \nsystem, using the maximum likelihood solution [18] (M-step). For the E-step we used a \nstandard extended Kalman smoother. We thus found the expected values for t he states given \nthe current filter parameters. For this computation, and later when testing the algorithm, xs \nwas initialized to its average value across all reaches while the remaining states were \ninitialized to their true values. The smoothed estimate fo r xs was then used, along with the true \nvalues for the other states, to re-estimate the filter parameters in the M-step as before. We \nalternated between the E and M steps until the log likelihood converged (which it did in all cases). \nFollowing the training procedure, the diagonal of the state covariance matrix Q corresponding to \nxs was set to the variance of the smoothed xs over all reaches, according to how much this state \nshould be allowed to change during prediction. This allowed the estimate of xs to develop over the \ncourse of the reach due to the evidence provided by the observations, better capturing the \ndynamics of reaches at different speeds. \n \n4 . 3 \n\nP e r f o r m a n c e o f t h e t i m e - w a r p e d E K F T \n\nIncorporating time warping explicitly into the trajectory model pro duced a noticeable \nincrease in decoding performance over the KFT. As the speed state xs is estimated \nthroughout the course of the reach, based on the evidence provided by the observations, the \ntrajectory model has the flexibility to follow the dynamics of the reach more accurately (Fig. \n3). While at the normal self-selected speed the difference between the algorithms is small, \nfor the slow and fast speeds, where the dynamics deviate from average, there i s a clear \nadvantage to the time warping model. \n\n \n\n \n\n5 \n\n\f \n\nFigure 3: Hand positions and predictions of the KFT and EKFT for sample reaches at A slow, \n\nB normal and C fast speeds. Note the different time scales between reaches. \n\n \n\nThe models were first trained using data from all speeds (Fig. 4A). The EKFT was 1.8% \nmore accurate on average (p<0.01), and the effect was significant at the slow (1.9%, p<0.05) \nand the fast (2.8%, p<0.01), but not at the normal (p=0.3) speed. We also trained the models \nfrom data using only reaches at the self-selected normal speed, as we wanted to see if there \nwas enough variation to effectively train the EKFT (Fig. 4B). Interestingly, the performance \nof the EKFT was reduced by only 0.6%, and the KFT by 1.1%. The difference in \nperformance between the EKFT and KFT was even more pronounced on aver age (2.3%, \np<0.001), and for the slow and fast speeds (3.6 and 4.1%, both p< 0.0001). At the normal \nspeed, the algorithms again were not statistically different (p=0.6). This result demonstrates \nthat the EKFT is a practical option for a real BMI system, as it is not necessary to greatly \nvary the speeds while collecting training data for the model to be effective over a wide range \nof intended speeds. \n\nExplicitly incorporating speed information into the trajectory model helps decoding, by \nmodeling the natural variation in volitional speed. \n\n \n\n \n5 \n\nFigure 4: Mean and standard error of EKFT and KFT accuracy at the different subject-\nselected speeds. Models were trained on reaches at A all speeds and B just normal speed \nreaches. Asterisks indicate statistically significant differences between the algorithms. \n\n \n\nM i x t u re s o f Ta r g e t s \n\nSo far, we have assumed that the targets of our reaches are perfectly known. In a real-world \nsystem, there will be uncertainty about the intended target of the reach . However, in typical \napplications there are a small number of possible objectives. Here we address this situation. \nDrawing on the recent literature, we use a mixture model to consider each of the possible \ntargets [11, 13]. We condition the posterior probability for the state on the N possible targets, \nT: \n\n \n\n \n\n \n\n \n\n \n\n \n\n(6) \n\n \n\n6 \n\n\fUsing Bayes' Rule, this equation becomes: \n\n \n\n \n\n \n\n \n\n \n\n \n\n \n\n(7) \n\nAs we are dealing with a mixture model, we perform the Kalman filter recursion for each \npossible target, xT, and our solution is a weighted sum of the outputs. The weights are \nproportional to the prior for that target, , and the likelihood of the model given that \ntarget . is independent of the target and does not need to be calculated. \n\nWe tested mixtures of both algorithms, the mKFT and mEKFT, with real uncert ain priors \nobtained from eye-tracking in the one-second period preceding movement. As the targets \nwere situated on two planes, the three-dimensional location of the eye gaze was found by \nprojecting its direction onto those planes. The first, middle and last eye samples were \nselected, and all other samples were assigned to a group according to which of the three was \nclosest. The mean and variance of these three groups were used to initialize three Kalman \nfilters in the mixture model. The priors of the three groups were assigned proportional to the \nnumber of samples in them. If the subject looks at multiple positions prior to reaching, this \nmethod ensures with a high probability that the correct target was accounted for in one of the \nfilters in the mixture. \n\nWe also compared the MTM approach of Yu et al. [13], where a different KF model was \ngenerated for each target, and a mixture is performed over these models. This approach \nexplicitly captures the dynamics of stereotypical reaches to specific targets. Given perfect \ntarget information, it would reduce to the STM described above. Priors for the MTM were \nfound by assigning each valid eye sample to its closest two targets, and weighting the \nmodels proportional to the number of samples assigned to the corresponding target, divided \nby its distance from the mean of those samples. We tried other ways of assigning priors and \nthe one presented gave the best results. \n\nWe calculated the reduction in decoding quality when instead of perfect priors we provide \neye-movement based noisy priors (Fig. 5). The accuracies of the mEKFT, the mKFT and the \nMTM were only degraded by 0.8, 1.9 and 2.1% respectively, compared to the perfect prior \nsituation. The mEKFT was still close to 10% better than the KF. The mixture model \nframework is effective in accounting for uncertain priors. \n\n \n\n \n\nFigure 5: Mean and standard errors of accuracy for algorithms with perfect priors, and \nuncertain priors with full and partial training set. The asterisk indicates a statistically \n\nsignificant effects between the two training types, where real priors are used. \n\n \n\nHere, only reaches at normal speed were used to train the models, as this is a more realistic \ntraining set for a BMI application. This accounts for the degraded performance of the MTM \nwith perfect priors relative to the STM from above (Fig. 2). With even more stereotyped \ntraining data for each target, the MTM doesn't generalize as well to new speeds. \n\n \n\n7 \n\n\fWe also wanted to know if the algorithms could generalize to new targets. In a real \napplication, the available training data will generally not span the entire useable worksp ace. \nWe compared the algorithms where reaches to all targets except the one being tested had \nbeen used to train the models. The performance of the MTM was significantly de graded \nunsurprisingly, as it was designed for reaches to a set of known targets. Performance of the \nmKFT and mEKFT degraded by about 1%, but not significantly (both p>0.7), demonstrating \nthat the continuous approach to target information is preferable when the target could be \nanywhere in space, not just at locations for which training data is available. \n\n \n6 \n\nD i s c u s s i o n a n d c o n c l u s i o n s \n\nThe goal of this work was to design a trajectory model that would improve decoding for \nBMIs with an application to reaching. We incorporated two features that prominently \ninfluence the dynamics of natural reach: the movement speed and the target location. Our \napproach is appropriate where uncertain target information is available. The model \ngeneralizes well to new regions of the workspace for which there is no training data, and \nacross a broad range of reaching dynamics to widely spaced targets in three dimensions. \n\nThe advantages over linear models in decoding precision we report here could be equally \nobtained using mixtures over many targets and speeds. While mixture models [11, 13] could \nallow for slow versus fast movements and any number of potential targets, this strategy will \ngenerally require many mixture components. Such an approach would require a lot more \ntraining data, as we have shown that it does not generalize well. It would also be run-time \nintensive which is problematic for prosthetic devices that rely on low power controllers. In \ncontrast, the algorithm introduced here only takes a small amount of additional run-time in \ncomparison to the standard KF approach. The EKF is only marginally slower than the \nstandard KF and the algorithm will not generally need to consider more than 3 mixture \ncomponents assuming the subject fixates the target within the second pre ceding the reach. \n\nIn this paper we assumed that subjects always would fixate a reach target \u2013 along with other \nnon-targets. While this is close to the way humans usually coordinate eyes and reaches [15], \nthere might be cases where people do not fixate a reach target. Our approach could be easily \nextended to deal with such situations by adding a dummy mixture component that all ows the \ndescription of movements to any target. \n\nAs an alternative to mixture approaches, a system can explicitly estimate the target position \nin the state vector [9]. This approach, however, would not straightforwardly allow for the \nrich target information available; we look at the target but also at other locations, strongly \nsuggesting mixture distributions. A combination of the two approaches could further \nimprove decoding quality. We could both estimate speed and target position for the EKFT in \na continuous manner while retaining the mixture over target priors. \n\nWe believe that the issues that we have addressed here are almost universal. Virtually all \ntypes of movements are executed at varying speed. A probabilistic distribution for a small \nnumber of action candidates may also be expected in most BMI applications \u2013 after all there \nare usually only a small number of actions that make sense in a given environment. While \nthis work is presented in the context of decoding human reaching, it may be applied to a \nwide range of BMI applications including lower limb prosthetic devices and human \ncomputer interactions, as well as different signal sources such as electrode grid recordings \nand electroencephalograms. 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