Biophysical Journal, Volume 98 Supporting Material Analysis of video-based Microscopic Particle Trajectories Using Kalman Filtering Pei-Hsun Wu,

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1 Biophysical Journal, Volume 98 Supporting Material Analysis of video-based Microscopic Particle Trajectories Using Kalman Filtering Pei-Hsun Wu, Ashutosh Agarwal, Henry Hess, Pramod P Khargonekar, and Yiider Tseng 1

2 SUPPLIMENTAL MATERIAL A The Kalman filter applied to Video-based Particle Tracking The Kalman filter uses a recursive procedure to estimate the true state of a linear process It preserves the intrinsic fluctuations of the measured object while effectively removing the external noise Here we briefly discuss how to use this method to estimate the true trajectory from a measured trajectory obtained from the video-based particle tracking experiment In the Cartesian coordinate system, the x- and y-components of a particle trajectory moving in the x-y plane are theoretically independent from one another; therefore, the use of the Kalman filter on a typical video-based particle tracking experiment can be simplified to a one-dimensional model of particle motion in each direction Individual time steps of the Kalman filter process employ two distinct processing phases: predict and update The elapsed time after onset is represented by k t, where k represents the observation time step Any given variable with a subscript, k l, is at time step, k, and in phase, l When l = k 1 the variable is in its prediction phase, and when l = k the variable is in the update phase Using this format, the position at the current time step, x ˆ k k 1, is first predicted by the sum of the previous position, x ˆ k 1 k 1, and the displacement, u 0, from either the freely diffusive or directed motion of the particle: x ˆ k k 1 = x ˆ k 1 k 1 + u 0 (S1) Once the position, x ˆ k k 1, is predicted, it is updated to a more accurate current position, x ˆ k k, in the update phase This is achieved by adding a correction term that uses an adjustable factor, K k, to weigh the difference between the measured position, z k, and x ˆ k k 1 : x ˆ k k = x ˆ k k 1 + K k (z k ) (S) The factor K k is at its optimal value and referred to as the Kalman gain when the x ˆ k k reaches the minimal error covariance, P kk ( E(( x ˆ k k ) )) At this minimal value, ( ) 1, (S3) K k = P kk 1 P kk 1 + R k ( ) and R k ( E((z k ) )) are the magnitudes of the error covariance for ˆ where P kk 1 E(( x ˆ k k 1 ) ) x k k 1 and z k, respectively (S1) The thermal fluctuation, Q k, propagates during this recursive process; therefore, P k+1 k must also be updated at each time step: P k +1 k = P k k + Q k In addition, P kk is concurrently updated with x kk in subsequent time steps: P k k = P k k 1 K k P k k 1 Substituting Eq S5 into Eq S4, we obtain the discrete algebraic Riccati equation: P k+1 k = P k k 1 K k P k k 1 + Q k (S4) (S5) (S6) SUPPORTING REFERENCE S1 Kalman RE (1960) J Basic Eng 8, 35-45

3 B The value of Q/R determines the best estimation of particle trajectories using the Kalman filter In the one-dimensional particle tracking case, Q k and R k are considered to be independent of k, so that the number of variables is significantly reduced Therefore, P k k 1, P k k and K k can reach their steady state values quickly In a steady state, P k k 1 and P kk are independent of k, and equal to each other, which can be denoted as P Together, Eq S3 can be simplified as K = K k = P (P + R) -1 and be further substituted into the simplified form of Eq S6, which is expressed as KP = Q Therefore, we get P QP QR = 0 (S7) Since P is the error covariance, it possesses possess a positive value Thus, the solution of P in Eq S7 is which leads to P = Q/+ Q + 4QR /, (S8) K = (Q/R) + (Q/R) + 4(Q/R) + (Q/R) + (Q/R) + 4(Q/R) (S9) Thus, K is solely determined by the value of Q/R When the Kalman filter is at its best performance, ie, the correct Q/R value has been applied, the MSD value estimated by filtered position can be expressed as: MS ˆ D = E[( x ˆ k k x ˆ k 1 k 1 ) ] (S10) The expression of x ˆ k k and x ˆ k 1 k 1 in Eq S1 and Eq S, respectively, is subsequently substituted into Eq S10 and we obtain: MS ˆ D = E[(K(z k ) + u 0 ) ] = E[(K((z k ) + (x k )) + u 0 ) ] (S11) From Eq 3 of the main text, we know that z k = v k We further use x kk 1 to represent ( x k x ˆ kk 1 ) and obtain MS ˆ D = E[(K(v k ) + u 0 ) ] = K E[(v k ) ]+ u 0 KE[(v k )] + u 0 (S1) Here, v k and x kk 1 are both random variables with E( v k )= R and E( x kk 1 )= P kk 1 = P Further, v k is not correlated with x kk 1, so E( v k x kk )= 0 Therefore, the MS ˆ D can be further simplified to: MS ˆ D = K ( R + P)+ u 0 = Q + u 0, (S13) where the relation K ( R + P)= Q can be obtained from Eqs S3 and S7 Analytically, the true MSD at the shortest time lag can be calculated based on the true positions state, x k, and Eq 1 in the main text: MSD = E[(x k 1 ) ] = E[(w k + u 0 ) ] = Q + u 0 Therefore, the MSD calculated from a filtered trajectory using the correct values of Q and R in Eq S13 is equal to the true MSD in Eq S14 (S14) 3

4 C Simulations illustrate the Applicability of the Kalman filter to Particle Tracking This result can be demonstrated by the simulation of a one-dimensional particle tracking experiment A model trajectory with an assigned Q (= Q T ) and R (R T = 100 Q T ) was simulated The Kalman algorithm was applied to this model trajectory using 500 random sets of (Q, R), with both variables ranging between 10-4 and 10 4 This yielded 500 RMSE and their corresponding normalized MSD values (or MSD/MSD T values, where the subscript T is denoted the true value of MSD) The simulated trajectory with the assigned Q T but without any positioning error, R, is the true trajectory, and the MSD T is calculated based on this trajectory Two scatter plots were generated by this simulation: RMSE versus normalized Q/R (Fig S1 A), and normalized MSD versus normalized Q/R (Fig S1 B) The fact that these two scatter plots form a continuous curve is in agreement with the analytical derivation that the Kalman filter performance is solely dependent on the assigned Q/R value In addition, the minimal value of RMSE localizes at a normalized Q/R = 1 in the RMSE curve, while the normalized MSD = 1 also occurs at normalized Q/R = 1 in the normalized MSD curve Since the estimated MSD from filtered positions ( x ˆ k k ) is equivalent to the one calculated from true positions ( x k ), these simulation results demonstrate how the Kalman filter algorithm with the appropriate choice of Q and R can effectively restore the true MSD value Figure S1 The best estimation of the trajectory via Kalman filter is determined by the Q/R values (A) The relation between RMSE and normalized Q/R suggest that the best trajectory restoration by Kalman filter is obtained when the Q/R = (Q/R) T This also support that the Kalman filter is an appropriate tool to correct the noisy trajectory (B) The relationship between the normalized MSD and normalized Q/R value suggested that the normalized Q/R value determines the outcome of the MSD of the estimated trajectory When normalized Q/R = 1, the application of the Kalman filter to the trajectory can obtain an estimated trajectory that possesses the same MSD value compared to the true trajectory In (A) an (B), each curve is constructed by 500 Kalman filtering results derived from random sets of Q and R inputs, ranging from 10-4 to 10 4 The reference trajectory for the calculation of RMSE and MSD R is the simulated trajectory without R 4

5 D The estimation of the correct initial steps of the trajectory using the Kalman filter The current position in the Kalman filter process is estimated from the updated position in the preceding time step As earlier positions are updated to be more accurate, the later positions in the trajectory can be better estimated However, the real positions of the first several positions of the trajectory cannot be reasonably estimated since there are no or only a few preceding steps available When the MSD is calculated, these initial positions should be omitted since these initial steps are less accurate than the later ones The reliability of the Kalman filter in correcting the first position was evaluated to further assess this claim The RMSE values of the filtered positions were calculated from 300 independent simulations with an assigned Q/R = 001 (Fig S A) and the results indicate that the first ten positions contain larger errors than later positions These results are also in agreement with the error covariance, P k, which indicates the performance of Kalman algorithms (Fig S B) The error covariance was calculated as described in reference (S1) To identify the minimum number of steps (N) that is required for the Kalman filter to achieve the best performance, we define that N is equal to the minimal k, where (P k-1 /P k ) becomes less than 1005 The N as function of (Q/R) T plot (Fig S C) suggests that when the R-value increases relative to the Q-value, more initial steps should be omitted Figure S Error of filtered step depends on the position of the step in the trajectory (A) RMSE of each step is estimated from 300 independent simulations at Q/R = 001 RMSE values decrease with the proceeding steps to a plateau (B) The performance of Kalman filter accessed by the error covariance (P k ) is in agreement with the RMSE values Error covariance is plotted against the step number as SNR = 01 and it reduces with propagating steps to a plateau value A smaller P k represents a better prediction of position These results suggest the Kalman filter yields the optimum prediction after a certain number of steps from beginning (C) The N-Q T /R T plot shows N decrease with increments of (Q/R) T, where N is the first step from onset that (P k- 1/P k ) becomes less than

ENGR352 Problem Set 02

engr352/engr352p02 September 13, 2018) ENGR352 Problem Set 02 Transfer function of an estimator 1. Using Eq. (1.1.4-27) from the text, find the correct value of r ss (the result given in the text is incorrect).