A NEW STOCHASTIC ESTIMATOR FOR TREMOR FREQUENCY TRACKING
Alp Kucukelbir, Azadeh Kushki and Konstantinos N. Plataniotis
The Edward S. Rogers Sr. Dept. of Electrical & Computer Engineering
University of Toronto, Canada
falp.kucukelbir, azadeh.kushkig@utoronto.ca, kostas@comm.utoronto.ca
ABSTRACT
An important parameter in analysis of physiological tremor is the
diagnosis and study of neurological disorders. The instantaneous
tremor frequency (ITF) is an important parameter in tremor analy-
sis. This paper proposes a novel stochastic filter, the multiple ex-
tended Kalman filter (M-EKF), for tracking of ITF from neural mi-
croelectrode recordings. The M-EKF mitigates degradations in fil-
ter performance resulting from a mismatch between assumed initial
conditions and those of a particular realization of a stochastic sys-
tem. Specifically, the M-EKF is comprised of a bank of extended
Kalman filters (EKF), each initialized with different conditions, se-
lected according to the unscented transform. The final estimate is a
weighted average of the individual estimates provided by each EKF
where the weights reflect how closely the assumed EKF initial con-
ditions match those of the true system. The M-EKF is applied to a
synthetic tremor model to display its superior performance to that of
the EKF and the unscented Kalman filter.
Index Terms— Tremor frequency, state-space model, nonlinear
estimation, unscented transform, extended Kalman filtering.
1. INTRODUCTION
Tremor is an involuntary, quasi-periodic, oscillation of one or more
muscles of the body, often expressed as a symptom of various neu-
rological diseases [1]. Tremor activity is therefore a principal aspect
of the diagnosis and study of neurological movement disorders. This
paper addresses the problem of tracking or continuous estimation of
the instantaneous tremor frequency (ITF), an important parameter in
tremor analysis [2].
The work of [3] proposes the use of the stochastic filtering frame-
work for ITF tracking from binary spike trains observed in neural
microelectrode recordings (MER). In this approach, domain knowl-
edge, such as dynamics of the system and statistical characteristics
of observation noise, are incorporated into the estimation process
through a state-space description of the system. The following state-
space model is considered herein
x(k) = g(x(k 1)) + u(k); (1)
z(k) = h(x(k)) + v(k); (2)
where x(k) is a state vector containing all variables required to
describe system dynamics, z(k) is the observed measurement vector,
u(k)  N (0;Q(k)) and v(k)  N (0;R(k)) are white Gaussian
process and measurement noise, respectively. The functions g(
)
and h(
) are nonlinear and real valued, and k is the discrete time
This work is partially supported by the Natural Sciences and Engineering
Research Council (NSERC) of Canada.
index. The filter estimate at time step k given the measurement set
fz(1); : : : ; z(j)g is denoted as bx(kjj).
Given this state-space formulation, well-known stochastic fil-
ters can be used to estimate the unobservable state vector from noisy
measurements over time [4]. When the system and measurement
equations are both linear and noise distributions are Gaussian, the
Kalman filter provides the optimal estimate in the minimum mean
square error (MMSE) sense. In the case of nonlinear system or mea-
surement equations, the optimal estimate cannot be directly deter-
mined and the extended Kalman filter (EKF) [4], unscented Kalman
filter (UKF) [5], and the particle filter [6] are commonly used to pro-
vide suboptimal solutions. Since ITF is nonlinearly related to the
measurements obtained from neural recordings, the work of [3] uses
an extended Kalman smoother as the tracking solution.
The model of (1) describes the evolution of a stochastic sys-
tem in a recursive manner and therefore requires specification of ini-
tial conditions. Due to the stochastic nature of the system, resulting
from the presence of noise and modeling inaccuracies, the initial
state of the system is assumed to be random and generally modeled
as a Gaussian random variable with distributionN (bx(0j0);P(0j0)).
The mean value bx(0j0) is used as the initial state in the Kalman filter
and its variants. However, discrepancies between the assumed con-
ditions by the filter and the true realization of state lead to a degrada-
tion in accuracy of the estimates computed by the filter. To mitigate
such adverse effects, this paper proposes a novel stochastic filter, the
multiple-EKF (M-EKF) for nonlinear estimation problems such as
ITF tracking. As shown in Figure 1, the M-EKF is comprised of a
bank of EKFs running in parallel, initialized with a different set of
initial conditions. The estimates provided by these filters are then
fused using a weighted average where the weights are proportional
to how well the respective initial conditions represent the true con-
ditions.
The rest of this paper is organized as follows. Section 2 dis-
cusses the effect of initial conditions, Section 3 introduces the details
of the M-EKF algorithm, Section 4 applies the M-EKF to the prob-
lem of ITF tracking, and Section 5 concludes the paper and provides
directions for future work.
2. EFFECT OF INITIAL CONDITIONS ON THE EKF
To motivate the development of the M-EKF, this section examines
the effect of initial conditions on the EKF. The EKF [4] uses a first
order Taylor series expansion of the functions g(
) and h(
) in (1)
and (2) to obtain a linear approximation to the state-space descrip-
tion of the system. Define the Jacobians G(k) and H(k) as follows
G(k) =
@g
@x




bx(k1jk1)
and H(k) =
@h
@x




bx(kjk1)
(3)