### TABLE I VARIABLES USED IN KALMAN FILTER

2006

Cited by 1

### TABLE I VARIABLES USED IN KALMAN FILTER

2006

### Table I defines the variables used in the Kalman equations, which follow:

2006

Cited by 1

### Table I defines the variables used in the Kalman equations, which follow:

2006

### Table 2 Unscented Kalman fllter.

"... In PAGE 7: ... For comparison, the EKF guarantees only p = 1. The UKF is presented in Table2 . In the predictor phase, the sigma-points... In PAGE 20: ...tates (L gt; 1) is straightforward (see, e.g., [19,21]). Within the time interval [ti ti+1] we follow the notation of Table2 but avoid the use of indexes i and i + 1. Let x be a Gaussian random variable at the beginning of the time step, fea- turing mean ^ x and covariance P .... ..."

### Table 1 Extended Kalman fllter.

"... In PAGE 7: ...are assumed additive, uncorrelated white and Gaussian processes, with zero mean [9,17]; V and W are the relevant time-invariant covariance matrices. Starting at t0 from an a-priori estimation ^ x0 of the state vector, endowed with the relevant covariance matrix P 0, the EKF updates the current estimates of state and model parameters according to Table1 . Here E[2] denotes the expected value of the Gaussian random variable 2 and F i is the gradient, or Jacobian of the mapping (15).... In PAGE 17: ...Sensitivity analysis We report here the details of the computation of the gradient F i of the evo- lution equations necessary in the EKF approach (see Table1 ). To simplify the notation, unlike in Section 3 in what follows the current estimates are not denoted by the hat.... In PAGE 18: ... In fact, according to Eq. (15) and Table1 , # is updated by the EKF only at the end of the time step. In (A.... ..."

### Table 1: Reconstruction results for the linear and Kalman filter. The table also shows how the Kalman filter results vary with lag times (see text).

"... In PAGE 5: ... In the interest of simplicity, we consider a single optimal time lag for all the cells though evidence suggests that individual time lags may provide better results [15]. Using time lags of 0, 70, 140, 210 D1D7 we train the Kalman filter and perform reconstruction (see Table1 ). We report the accuracy of the reconstructions with a variety of error measures used in the literature including the correlation coefficient (D6) and the mean squared error (MSE) between the reconstructed and true trajectories.... In PAGE 5: ... We report the accuracy of the reconstructions with a variety of error measures used in the literature including the correlation coefficient (D6) and the mean squared error (MSE) between the reconstructed and true trajectories. From Table1 we see that optimal lag is around two time steps (or 140D1D7); this lag will be used in the remainder of the experiments and is similar to our previous findings [15] which suggested that the optimal lag was between 50-100D1D7. Decoding: At the beginning of the test trial we let the predicted initial condition equal the real initial condition.... In PAGE 5: ...eal initial condition. Then the update equations in Section 2 are applied. Some examples of the reconstructed trajectory are shown in Figure 2 while Figure 3 shows the reconstruction of each component of the state variable (position, velocity and acceleration in DC and DD). From Figure 3 and Table1 we note that the reconstruction in DD is more accurate than in... In PAGE 6: ... Compared with Figure 3, we see that the results are visually similar. Table1 , however, shows that the Kalman filter gives a more accurate reconstruction than the linear filter (higher correlation coefficient and lower mean-squared error). While linear filtering is extremely simple, it lacks many of the desirable properties of the Kalman filter.... In PAGE 6: ... In that case we showed that acceleration was redundant and could be removed from the state equation. The data used here is more natural , varied, and rapid and we find that modeling acceleration improves the prediction of the system state and the accuracy of the reconstruction; Table1 shows the decrease in... ..."

### Table 1: Reconstruction results for the xed linear and recursive Kalman lter. The table also shows how the Kalman lter results vary with lag times (see text).

2003

"... In PAGE 5: ... In the interest of simplicity, we consider a single optimal time lag for all the cells though evidence suggests that individual time lags may provide better results [15]. Using time lags of 0, 70, 140, 210 ms we train the Kalman lter and perform reconstruction (see Table1 ). We report the accuracy of the reconstructions with a variety of error measures used in the literature including the correlation coef cient (r) and the mean squared error (MSE) between the reconstructed and true trajectories.... In PAGE 5: ... We report the accuracy of the reconstructions with a variety of error measures used in the literature including the correlation coef cient (r) and the mean squared error (MSE) between the reconstructed and true trajectories. From Table1 we see that optimal lag is around two time steps (or 140ms); this lag will be used in the remainder of the experiments and is similar to our previous ndings [15] which suggested that the optimal lag was between 50-100ms. Decoding: At the beginning of the test trial we let the predicted initial condition equal the real initial condition.... In PAGE 5: ...eal initial condition. Then the update equations in Section 2 are applied. Some examples of the reconstructed trajectory are shown in Figure 2 while Figure 3 shows the reconstruction of each component of the state variable (position, velocity and acceleration in x and y). From Figure 3 and Table1 we note that the reconstruction in y is more accurate than in the x direction (the same is true for the xed linear lter described below); this requires further investigation. Note also that the ground truth velocity and acceleration curves are computed from the position data with simple differencing.... In PAGE 6: ... Compared with Figure 3, we see that the results are visually similar. Table1 , however, shows that the Kalman lter gives a more accurate reconstruction than the linear lter (higher correlation coef cient and lower mean-squared error). While xed linear ltering is extremely simple, it lacks many of the desirable properties of the Kalman lter.... In PAGE 6: ... In that case we showed that acceleration was redundant and could be removed from the state equation. The data used here is more natural , varied, and rapid and we nd that modeling acceleration improves the prediction of the system state and the accuracy of the reconstruction; Table1 shows the decrease in accuracy with only position and velocity in the system state (with 140ms lag). 4 Conclusions We have described a discrete linear Kalman lter that is appropriate for the neural control of 2D cursor motion.... ..."

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### Table 1: Reconstruction results for the fixed linear and recursive Kalman filter. The table also shows how the Kalman filter results vary with lag times (see text).

2003

"... In PAGE 5: ... In the interest of simplicity, we consider a single optimal time lag for all the cells though evidence suggests that individual time lags may provide better results [15]. Using time lags of 0, 70, 140, 210 a7a9a8 we train the Kalman filter and perform reconstruction (see Table1 ). We report the accuracy of the reconstructions with a variety of error measures used in the literature including the correlation coefficient (a7 ) and the mean squared error (MSE) between the reconstructed and true trajectories.... In PAGE 5: ... We report the accuracy of the reconstructions with a variety of error measures used in the literature including the correlation coefficient (a7 ) and the mean squared error (MSE) between the reconstructed and true trajectories. From Table1 we see that optimal lag is around two time steps (or 140a7 a8 ); this lag will be used in the remainder of the experiments and is similar to our previous findings [15] which suggested that the optimal lag was between 50-100a7a9a8 . Decoding: At the beginning of the test trial we let the predicted initial condition equal the real initial condition.... In PAGE 5: ...eal initial condition. Then the update equations in Section 2 are applied. Some examples of the reconstructed trajectory are shown in Figure 2 while Figure 3 shows the reconstruction of each component of the state variable (position, velocity and acceleration in a11 and a14 ). From Figure 3 and Table1 we note that the reconstruction in a14 is more accurate than in the a11 direction (the same is true for the fixed linear filter described below); this requires further investigation. Note also that the ground truth velocity and acceleration curves are computed from the position data with simple differencing.... In PAGE 6: ... Compared with Figure 3, we see that the results are visually similar. Table1 , however, shows that the Kalman filter gives a more accurate reconstruction than the linear filter (higher correlation coefficient and lower mean-squared error). While fixed linear filtering is extremely simple, it lacks many of the desirable properties of the Kalman filter.... In PAGE 6: ... In that case we showed that acceleration was redundant and could be removed from the state equation. The data used here is more natural , varied, and rapid and we find that modeling acceleration improves the prediction of the system state and the accuracy of the reconstruction; Table1 shows the decrease in accuracy with only position and velocity in the system state (with 140ms lag). 4 Conclusions We have described a discrete linear Kalman filter that is appropriate for the neural control of 2D cursor motion.... ..."

Cited by 17

### Table 1: Reconstruction results for the fixed linear and recursive Kalman filter. The table also shows how the Kalman filter results vary with lag times (see text).

"... In PAGE 5: ... In the interest of simplicity, we consider a single optimal time lag for all the cells though evidence suggests that individual time lags may provide better results [15]. Using time lags of 0, 70, 140, 210 D1D7 we train the Kalman filter and perform reconstruction (see Table1 ). We report the accuracy of the reconstructions with a variety of error measures used in the literature including the correlation coefficient (D6) and the mean squared error (MSE) between the reconstructed and true trajectories.... In PAGE 5: ... We report the accuracy of the reconstructions with a variety of error measures used in the literature including the correlation coefficient (D6) and the mean squared error (MSE) between the reconstructed and true trajectories. From Table1 we see that optimal lag is around two time steps (or 140D1D7); this lag will be used in the remainder of the experiments and is similar to our previous findings [15] which suggested that the optimal lag was between 50-100D1D7. Decoding: At the beginning of the test trial we let the predicted initial condition equal the real initial condition.... In PAGE 5: ...eal initial condition. Then the update equations in Section 2 are applied. Some examples of the reconstructed trajectory are shown in Figure 2 while Figure 3 shows the reconstruction of each component of the state variable (position, velocity and acceleration in DC and DD). From Figure 3 and Table1 we note that the reconstruction in DD is more accurate than in the DC direction (the same is true for the fixed linear filter described below); this requires further investigation. Note also that the ground truth velocity and acceleration curves are computed from the position data with simple differencing.... In PAGE 6: ... Compared with Figure 3, we see that the results are visually similar. Table1 , however, shows that the Kalman filter gives a more accurate reconstruction than the linear filter (higher correlation coefficient and lower mean-squared error). While fixed linear filtering is extremely simple, it lacks many of the desirable properties of the Kalman filter.... In PAGE 6: ... In that case we showed that acceleration was redundant and could be removed from the state equation. The data used here is more natural , varied, and rapid and we find that modeling acceleration improves the prediction of the system state and the accuracy of the reconstruction; Table1 shows the decrease in accuracy with only position and velocity in the system state (with 140ms lag). 4 Conclusions We have described a discrete linear Kalman filter that is appropriate for the neural control of 2D cursor motion.... ..."