1. MYOELECTRIC PROSTHESES : NOVEL METHODOLOGIES FOR ENHANCING
USABILITY AND CONTROL
A Dissertation Submitted
to the Graduate School
University of Arkansas at Little Rock
in partial fulfillment of requirements
for the degree of
DOCTOR OF PHILOSOPHY
in Engineering Science and Systems
in the Department of Systems Engineering
of the Donaghey College of Engineering and Information Technology
May 2014
Ghulam Rasool
B.E., National University of Sciences and Technology, Pakistan, 1999
M.Sc., Center for Advanced Studies in Engineering, Pakistan, 2010
3. This dissertation, ``Myoelectric Prostheses : Novel Methodologies for Enhancing Usability and
Control,'' by Ghulam Rasool, is approved by:
Dissertation Advisor: Kamran Iqbal
Professor of Systems Engineering
Dissertation Committee: Andrew Wright
Associate Professor of Systems Engineering
Cang Ye
Associate Professor of Systems Engineering
Keith Bush
Assistant Professor of Computer Science
Gannon White
Assistant Professor of Health, Human Performance, and Sport Management
Nidhal Bouaynaya
Assistant Professor of Electrical Engineering, Rowan University
Program Coordinator: Yupo Chan
Professor of Systems Engineering
Interim Graduate Dean: Paula J. Casey
Professor of Law
4. Fair Use
This dissertation is protected by the Copyright Laws of the United States (Public Law 94-553,
revised in 1976). Consistent with fair use as defined in the Copyright Laws, brief quotations from
this material are allowed with proper acknowledgement. Use of this material for financial gain
without the author's express written permission is not allowed.
Duplication
I authorize the Head of Interlibrary Loan or the Head of Archives at the Ottenheimer Library at
the University of Arkansas at Little Rock to arrange for duplication of this dissertation for
educational or scholarly purposes when so requested by a library user. The duplication will be at
the user's expense.
5. MYOELECTRIC PROSTHESES : NOVEL METHODOLOGIES FOR ENHANCING
USABILITY AND CONTROL by Ghulam Rasool, May 2014
ABSTRACT
Control of powered prostheses using the information from leftover muscles in amputees is a
challenging problem. I investigated the problem with a view to find clinically reliable and robust
schemes that can interpret human intent and control a powered prosthesis in real-time. Recently,
pattern classification systems have been suggested to map muscle activation patterns to the
human intent (or the movement). However, due to the underlying generic structure,
system-specific information cannot be exclusively incorporated into pattern classification
algorithms. I have proposed two new methodologies in my research. The first one extended the
pattern classification with a new feature set comprised of autoregressive-autoregressive
generalized conditional heteroscedastic (AR-GARCH) coefficients, that outperformed the
conventional feature set (p < .01). The second methodology addressed the problem by suggesting
a novel physiologically-relevant mathematical model. The hypothesis of muscle synergies was
employed to formulate a state-space representation of the neural signals. The proposed model is
based on the assumption that the neural drive originating from the central nervous system (CNS)
and terminating at the peripheral muscles contains task-specific information. Therefore, the
neural drive was estimated accurately using a recursive Bayesian technique. Later, the estimated
neural drive and task-specific muscle synergies were used to identify the intended/performed task.
Detailed performance analysis was performed in off-line as well as real-time using a virtual
prosthesis. The proposed scheme outperformed the pattern classification systems in off-line
accuracy (p < .001) and real-time controllability (p < .01).
6. ACKNOWLEDGEMENTS
I would never have been able to finish my dissertation without the guidance of my committee
members, help from friends, and support from my family and wife.
I would like to express my deepest gratitude to my advisor, Dr. Kamran Iqbal, for his excellent
guidance, caring, patience, and providing me with an excellent atmosphere for doing research.
It is with immense gratitude that I acknowledge the support and help of Dr. Nidhal Bouaynaya
in all matters related to research and beyond, especially correcting my writings patiently and
financially supporting my research. I would like thank Dr. Gannon White and the Chair of
department of Health, Human Performance, and Sport Management, Dr. Donna Quimby, for
letting me use the Human Performance lab for my research and data collections. Special thanks
goes to Dr. Andrew Wright, who was willing to participate in my final defense committee at the
last moment.
I would like to thank the Dean of College of Engineering and Information Technology, Dr.
Eric Sandgren for providing a handsome amount to purchase equipment for my research. I would
also like to thank the Associate Dean College of Engineering and Information Technology, Dr.
Abhijit Bhattacharyya for valuable advices and financial support.
I am indebted to my parents and family for assisting me in my all endeavors. Special thank
goes to my father, mother, brothers, sisters and especially my wife for patience, help and support.
7. PROTOCOL APPROVAL STATEMENT
Name of project : Human Movement Analysis for Myoelectric Controlled Prostheses.
Committee Name : Ghulam Rasool and Kamran Iqbal
Approval date : 12 February, 2013
Protocol number : 13-144
Approved By : Dr. Elisabeth Sherwin, IRB Chair, UALR
Name of project : Human Movement Analysis for Myoelectric Controlled Prostheses.
Committee Name : Ghulam Rasool and Gannon White
Approval date : 14 January, 2014
Protocol number : 13-144 C1
Approved By : Dr. Elisabeth Sherwin, IRB Chair, UALR
8. MEMORANDUM
TO: Ghulam Rasool, Dr. Kamran Iqbal (EIT)
CC: Rhiannon Morgan, Research Compliance Officer
FROM: Dr. Elisabeth Sherwin, IRB Chair
UALR Institutional Review Board
DATE: February 12, 2013
RE: IRB Request for Exemption
Thank you for your recent Institutional Review Board Request for Exemption
(Protocol # 13-144) titled “Human Movement Analysis for Myoelectric Controlled
Prostheses.” We have reviewed this request and find that it meets the IRB’s
criteria for protection of human participants. Your project has IRB approval from
today until 2/11/14 and you are free to proceed with data collection. After this
date, all interaction with human subjects and data collection for this project must
cease.
Continuations
If you would like this study to continue unchanged for more than one year, you
will need to submit a Request for Continuing Review 4 weeks prior to the above
expiration date. Expired protocols will not be granted a continuation or extension
by the IRB. Please note that if your original protocol was approved by review at
a convened meeting, then your request for continuation or annual review must
follow the same process.
Modifications to Approved Research
It is the responsibility of the PI (Principal Investigator) to notify the IRB if
substantive changes to the research are desired. This would include, but is not
limited to: modifications to recruitment strategies, the addition of new personnel
9. to the study, changes in target population, or data collection methods. Any and
all changes to the study must be reviewed and approved by the IRB prior to
implementation. Please submit a Request for Modification/Amendment form if
you wish to modify your research. Changes implemented without IRB approval
are in violation of 45 CFR 46 and UALR institutional policy.
Adverse Event Reporting
Principle Investigators are required to report unanticipated problems or adverse
events involving human subjects to the IRB promptly. If you have an event to
report, please document the incident and call the UALR Research Compliance
Officer, Rhiannon Morgan, at (501) 569-8657. For regulatory guidance and
definitions on adverse events and reporting timelines, please see the link below.
http://www.hhs.gov/ohrp/policy/advevntguid.html
10. MEMORANDUM
TO: Ghulam Rasool/Dr. Gannon White
CC: Rhiannon Gschwend, Research Compliance Officer
FROM: Dr. Elisabeth Sherwin, IRB Chair
UALR Institutional Review Board
DATE: January 14, 2014
RE: IRB Request for Continuation
Thank you for your recent Institutional Review Board Request for Continuation
(Protocol # 13-144 C1) titled “Human Movement Analysis for Myoelectric
Controlled Prostheses.” We have reviewed this request and find that it meets the
IRB’s criteria for protection of human participants. Your project has IRB approval
from today until 1/13/15 and you are free to proceed with data collection. After
this date, all interaction with human subjects and data collection for this project
must cease.
Continuations
If you would like this study to continue unchanged for more than one year, you
will need to submit a Request for Continuing Review 4 weeks prior to the above
expiration date. Expired protocols will not be granted a continuation or extension
by the IRB. Please note that if your original protocol was approved by review at
a convened meeting, then your request for continuation or annual review must
follow the same process.
Modifications to Approved Research
It is the responsibility of the PI (Principal Investigator) to notify the IRB if
substantive changes to the research are desired. This would include, but is not
limited to: modifications to recruitment strategies, the addition of new personnel
11. to the study, changes in target population, or data collection methods. Any and
all changes to the study must be reviewed and approved by the IRB prior to
implementation. Please submit a Request for Modification/Amendment form if
you wish to modify your research. Changes implemented without IRB approval
are in violation of 45 CFR 46 and UALR institutional policy.
Data Storage and Retention
Federal regulations require researchers to store data from human subject
research in a secure manner for a period of three years after completion. All
data, whether in electronic or paper format, must be maintained on the UALR
campus and made accessible to the IRB, Compliance Office or other auditors as
needed. The requirement applies to all consent and assent forms, receipts of
incentives paid to subjects, and related documents. You are also required to
maintain documentation of your IRB approval and all applicable ethics training.
For questions about data storage and retention, please contact the Research
Compliance Office.
Adverse Event Reporting
Principle Investigators are required to report unanticipated problems or adverse
events involving human subjects to the IRB promptly. If you have an event to
report, please document the incident and call the UALR Research Compliance
Officer, Rhiannon Morgan, at (501) 569-8657. For regulatory guidance and
definitions on adverse events and reporting timelines, please see the link below.
http://www.hhs.gov/ohrp/policy/advevntguid.html
19. List of Tables
2.1 Mathematical definitions of TDAR features. . . . . . . . . . . . . . . . . 21
3.1 Heteroscedasticity in the myoelectric signal. . . . . . . . . . . . . . . . . 52
3.2 A summary of statistical tests to model the myoelectric signal as an AR-
GARCH process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Feature sets selected for the pattern classification algorithms. . . . . . . . 57
3.4 Sensitivity and specificity of the LDA algorithm. . . . . . . . . . . . . . 59
4.1 TAC test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
xix
20. 1
Chapter 1
Introduction
1.1 Motivation
The ease and accuracy of volitional activities, like reaching and grasping that we perform
routinely without much effort, hide the complexity of underlying neuromusculoskeletal processes
that take place to perform these. In addition, changes in the environmental conditions, sensitivity
of the task to be performed in terms of accurate vs. coarse movements and changes in actuation
mechanisms, i.e., the musculature, are readily incorporated in the movement plan or learned as a
routine. The complexity of the movement generation system may be revealed in some cases, e.g.,
when a part of body (e.g. a limb) is impaired or missing as a result of a traumatic incident or
congenital deficit. In upper-extremity limb impairment, functional deficiencies are more
pronounced as compared to lower-extremity. The situation may be even worst for bilateral and
proximal amputations. In any case, the loss of a limb has multiple distressing facets. These may
include a reduced functionality, social and psychosocial problems, and the worst may be the
acceptance of own amputation condition. Prostheses are a suitable alternate and may mitigate the
21. 2
distressing conditions. Cosmetic prostheses help in public appearance, while body-powered
prostheses increase functionality in activities of daily living (ADLs). In contrast, the externally
powered prostheses controlled by the neural signals, such as surface myoelectric signal, provide
enhanced dexterity, low cognitive burden and improved functionality. Myoelectric prostheses
(prostheses driven by the control information extracted from the myoelectric signals) fall under
the umbrella of a larger area known as the body-machine interface (BMI) [1].
1.2 Background
The idea of controlling externally powered prostheses for upper-extremity using myoelectric or
electromyogram (EMG) signals is not a new one [2], [3], [4], [5], [6], [7]. The first effort to use
myoelectric signals as a control source for externally powered prosthetic device was done as early
as 1945 by Reinhold Reiter, a physics student at Munich University [7]. The prosthetic arm was
designed for an amputee factory worker. Since that time, different research groups from all over
the world have been involved in advancing the field of myoelectric prostheses [2], [7].
In myoelectric control, the electrical activity in the muscles representing their activation is
recorded using surface electrodes. The recorded signal is processed through statistical and signal
processing techniques to identify the movement intent of the user. A naive approach to estimate
the movement intent is to measure the intensity of the myoelectric signal [8]. Based on the
measured intensity, pre-defined control commands are issued. An artificial limb consisting of
mechanical links and motors is driven through external power using the control commands. Such
a control paradigm enables basic functionality in the form of an on/off control. Even in a
rudimentary form, myoelectric prostheses have found an important place in clinical setups. In
22. 3
upper-limb amputees, myoelectric prostheses have much less rejection rate than passive and body
powered prostheses (39%, 53% and 50% respectively) [9]. However, despite a lower rejection
rate, the lack of dexterity and the absence of sensory feedback have been reported as the design
issues requiring the most attention [9], [10], [11], [12].
The primary challenge in the myoelectric prosthesis design is the myoelectric signal itself. The
signal represents a complex biological process, i.e., the electrical activity in a muscle and can be
considered an indirect representative of the force [13]. The signal can be detected with sensors
placed on the surface of the skin or with the needle/wire sensors introduced into the muscle
tissue [14], [15]. The surface EMG being the least invasive is commonly employed in prostheses
applications, albeit intramuscular and intra-neural signal acquisition and processing are also
active research areas [16].
1.3 Research Objectives
The literature review for the research included a vast array of topics including but not limited to
body/brain-machine interfaces (BMIs), myoelectric signal processing, myoelectric prostheses,
pattern classification for myoelectric prostheses, human motor control, hypothesis of muscle
synergies and related topics in neuroscience, physiology and anatomy. Based on an extensive
literature review, following research objectives were identified.
1. Develop a basic understanding of the BMIs with a focus on prosthetic devices for
upper-extremity disabilities. With enhanced breadth of knowledge in BMIs, develop
in-depth understanding of neural signals especially the myoelectric signal. Understand
23. 4
generation mechanism of the signal in the spinal cord and its manifestation as muscle
activations.
2. Develop in depth understanding of the state-of-the-art in the field of myoelectric prostheses,
i.e., the pattern classification systems. Critically analyze the functional principles of pattern
classification systems, especially the process of information extraction from the
myoelectric signals in the form of representative feature sets.
3. Analyze a commonly employed feature set, i.e., the autoregressive (AR) model coefficients
in myoelectric prosthesis. Special emphasis is to be laid on the heteroscedasticity of the
myoelectric signal, i.e., the time-varying variance of the AR modeling residuals. In the light
of new findings, suggest an improved feature set focusing on the heteroscedastic modeling
of the myoelectric signal.
4. Investigate myoelectric prostheses problem to understand the dynamics of the CNS in
planning, execution and coordination of voluntary movements. Emphasis is to be laid on
the role of spinal cord in all three phases. The focus of the research will be the hypothesis
of muscle synergies.
5. By employing hypothesis of muscle synergies, develop a mathematical model for the neural
drive estimation. Further, explore novel methods to discriminate between different tasks /
movements based on the estimated neural drive.
6. Propose a comprehensive experimental protocol for myoelectric data collection to validate
the theoretical results. Experimental protocols should be designed in a way that the derived
24. 5
results can be generalize to all upper-extremity movements. However, the designed
protocols should be specific enough to verify proposed methodologies also.
7. Explore various methodologies for virtual testing and performance evaluation of
myoelectric prostheses. Newly proposed algorithm should be evaluated in the virtual
environment with respect to the controllability of the system as a whole.
1.4 Research Contributions
Keeping the research objectives in view, a multi-pronged investigation was undertaken. Based on
preliminary findings, two specific research paths were identified. The first approach consisted of
surface myoelectric signal classification using the Autoregressive-Generalized Autoregressive
Conditional Heteroscedastic (AR-GARCH) model and is discussed in detail in Chapter 3. The
other approach explored muscle synergies for task discrimination and is presented in Chapter 4.
Original research contributions for both adopted approaches are enumerated below.
1.4.1 Myoelectric Signal Classification Using the AR-GARCH Model
1. AR modeling of the myoelectric signal was analyzed using the Ljung-Box Q-test. Later, the
modeling residuals were tested for heteroscedasticity, i.e., the time-varying variance, using
the Engle's test. The myoelectric signal was, in general, found to exhibit heteroscedasticity.
2. The AR-GARCH process was used to model the myoelectric signal, where the GARCH
process was used to capture the heteroscedasticity in the AR modeling residuals. Various
25. 6
statistical tests including the goodness of fit confirmed the validity of the proposed
AR-GARCH model.
3. A new feature set based on the proposed modeling scheme, i.e., the AR-GARCH process,
was used in the pattern classification system. The newly proposed feature set significantly
outperformed (p < .01) the conventional AR feature set in the classification accuracy.
4. Experimental data was collected from human forearm muscles (six participants) to verified
the increased classification accuracy.
1.4.2 Task Discrimination Using the Hypothesis of Muscle Synergies
1. The problem of myoelectric prostheses was investigated in its true domain, i.e., the human
motor control domain. The hypothesis of muscle synergies was explored in detail to
understand the fixed spatial inter-muscle coordination. The focus of the investigation was
to find a physiologically-relevant mathematical model for task discrimination.
2. Based on the hypothesis of muscle synergies, a physiologically-relevant mathematical
model of the neural drive was proposed. The proposed model captures the time evolution of
the task-specific neural drive.
3. A modified form of the Kalman filter, i.e., the state-constrained Kalman filter was
employed to solve the state-space model and estimate the neural drive, i.e., the latent state
in the proposed state-space model.
4. The estimated neural drive was used to discriminate between a set of selected tasks using
different similarity measures.
26. 7
5. The discrimination decision by the proposed model was further processed to increase
classification accuracy using a novel post-processing routine.
6. Experimental data was collected from twelve participants to ascertain the efficacy of the
proposed discrimination scheme. In addition to off-line performance analysis, a virtual
prosthesis was used to quantitatively measure the controllability of the algorithm. The
proposed scheme outperformed the conventional classification algorithms in off-line
accuracy (p < .001) and as well as real-time controllability (p < .01).
27. 8
Chapter 2
Background and Literature Review
The chapter provides a detailed description of the related work in the field of myoelectric
prostheses. We start with the hypothesis of muscle synergies and later provide a detailed review
of state-of-the-art in the field of myoelectric prosthesis.
2.1 Human Motor Control and Myoelectric Prostheses
How the central nervous system (CNS) masters the many degrees-of-freedom (DOFs) of the
musculoskeletal system to control and coordinate a goal-directed movement is a long standing
question. Myriad of hypotheses exist to explain the control strategies adopted by the CNS [17].
While planning and executing a voluntary movement, the CNS is confronted with the problem (or
bliss [18]) of redundancy (or abundance [18]) at both the neuromuscular and the biomechanical
levels. For example, to perform a reaching movement, an infinite number of trajectories can be
28. 9
planned; and for every trajectory there are more DOFs available than actually required. Once a
trajectory is selected, an inverse dynamics model has to be solved to find joint torques (and
ultimately neural commands) for moving the limb on the desired trajectory to complete the
intended movement. However, skeletal muscles can only contract and generate force in one
direction. Therefore, muscles across a joint function as an agonist/antagonist pair to generate
torque by their collective action. Due to such a pairing, an infinite levels of individual muscle
forces can be combined to produce the required torque at each joint. Furthermore, bi-articular
muscles (muscle crossing multiple joints), if present around the joint of interest, complicate the
task of calculating a desired torque at that joint. Finally, many different combinations of motor
units within a muscle can fire to achieve the same force level [19], [20], [21]. Therefore, the
above mentioned inverse dynamics problem, i.e., calculation of neural activation signals required
to perform a planned movement is termed as an ill-posed one [19].
Similarly, due to the tight coupling between force, position, and velocity of the end effector
(hand / end of the limb), a forward model of the actuator has to be incorporated by the
CNS [22], [23]. Also, the CNS has to incorporate afferent (feedback) information into the
movement planning and execution. Similarly, during the execution of the movement, any changes
in the forward model and/or environmental conditions have to be taken into account by the
CNS [17], [19]. In the following section we will briefly describe different theories that have been
proposed to answer the above mentioned questions.
29. 10
2.1.1 Background
Earlier in the twentieth century, some researches proposed that the feedback of stretch reflex from
muscle spindles (MS) may cause initiation of the movement [24]. Similarly, later in 1970s it was
hypothesized by Merton that the central commands initiate movement via the gamma (γ) system
(muscle spindles), rather than activating muscles directly through the alpha (α) motor
neurons [25]. However, due to some obvious limitations in this feedback-only approach,
feed-forward control solution was proposed. The feed-forward scheme states that the CNS may
be carrying out inverse kinematic and dynamic computations while moving a limb in a purposeful
way [26]. However, for a multi-joint system the inverse dynamic computations can be enormous.
A few research efforts were undertaken to expound the underlying mechanics and it was proposed
that a fixed set of solutions in the form of a lookup table may already be stored by the CNS,
whereby a value of toque has been pre-calculated and stored for each possible value of position,
velocity and acceleration [27], [28]. However, an apparent limitation was that for a complete arm
model with seven joint coordinates, the entries in the table exponentially grow to 104
. Some
efforts were done to reduce the number of entries in the look-up table but there were other
problems also, e.g., interaction with the environment which is not accounted for by the
feed-forward only solution.
2.1.1.1 Equilibrium Point Hypothesis
Another effort by the name of equilibrium point hypothesis was proposed as an alternative to the
problem of inverse dynamic computations; the hypothesis specifies physiological variables
controlled by the CNS [29], [30]. The hypothesis states that centrally generated motor commands
30. 11
modulate the stiffness and rest length of the muscles that act as flexors and extensors about the
joint; and as a result, the elastic behavior of the muscles, like that of an opposing spring, defines a
single equilibrium position of the forearm [17]. A sequence of equilibrium positions produced
during movement by all the muscular activations has been called the virtual trajectory [31].
Generalizations of the equilibrium point hypothesis have also been proposed and further work is
still underway [32], [33], [34], [35], [36].
2.1.1.2 Uncontrolled Manifold (UCM)
The idea of uncontrolled manifold (UCM) was proposed with an emphasis on the variability of a
movement in multiple trials [37], [38], [39]. The roots of the UCM hypothesis can be traced back
to the experiments of Bernstein [40], where the end effector paths were less variable than the joint
level trajectories. Under the UCM hypothesis, to make a comparison of both variances, i.e.,
overall trajectory variance and the joint space variance, a joint space as an embedding space is
used, in which all variance is measured and the structure of the variance is interpreted relative to
spatial task variables. This is done by defining a subset, the UCM that contains all those
combinations of joint angles that are consistent with one particular end-effector position. There is
such an UCM for any position of the end effector. According to the UCM hypothesis, at any
given point during the movement, joint configurations vary primarily within that subset rather
than outside of it [34], [35], [36]. Recently, Neilson and Neilson extended the UCM to a
computational theory, named as the adaptive model theory (AMT) [41].
31. 12
2.1.2 Muscle Synergies
It is hypothesized that the activity of motor system may result from combination of discrete
elements or primitive building blocks, called the muscle
synergies [42], [17], [43], [44], [45], [46], [47] or muscle modes (M-modes) [48], [49], [38]. By
definition, muscle synergies are fixed relative levels of activation of different muscles [50], and the
recruitment of a large number of muscles may be performed by a reduced number of independent
control signals (i.e., the synergy activation coefficients). The activation signals are translated into
individual muscular activation patterns by a transformation matrix, called the synergy matrix [51].
This definition of muscle synergies is different from the one by Mark L. Latash, i.e., ``synergy is
a neural organization of a multi-element system that, 1) organizes sharing of task among a set of
elemental variables; and 2) ensures co-activation among elemental variables with the purpose to
stabilize performance variables" [38]. A closer look at both definitions reveal that the former
definition of muscle synergies is a subset of the later. However, we will use the former definition
of muscle synergies in our research, being mathematically and computationally more tractable.
The fundamental movement primitives or muscle synergies are assumed to serve two
purposes. First, they constrain the neuromusculoskeletal system, thus making the problem of
controlling and coordinating movement a well-posed one (which was previously an ill-posed).
Second, they increase the efficiency of the CNS, i.e., using fewer commands to control a large
space. Experimental evidence for the hypothesis of muscle synergies in the spinal motor system
comes from a number of experiments on spinalized frogs [52], [53], [54], freely moving
frogs [55], [56], rats [57], primates [58], human arm and hand
movements [50], [51], [59], [60], [61], [62], human postural task [63], [45], and locomotion [64].
32. 13
A B C D
A B C D
ActivationStrengths
Observed muscle activations
𝑦𝑦𝑖𝑖
A B C D
𝑦𝑦𝑖𝑖 = � 𝑥𝑥𝑖𝑖 𝑤𝑤𝑖𝑖
𝑛𝑛
𝑗𝑗=1
𝑤𝑤2
𝑤𝑤1
Muscles
0.5
0.5
�
(a)
Muscle Synergies
EMG Data
M1
M2
M3
Coefficients
Time
Time
𝒚𝒚𝟏𝟏
𝒘𝒘𝟏𝟏 𝒘𝒘𝟐𝟐
𝒙𝒙𝟏𝟏
𝒙𝒙𝟐𝟐
𝒚𝒚𝟐𝟐
𝒚𝒚𝟑𝟑
(b)
Figure 2.1: Two simple cases of muscle synergies, where activation levels of muscles are relatively
fixed within a muscle synergy. a) Each of the two muscle synergy consists of four muscles. A
discrete coefficient set is being specified by the CNS to combine the muscle synergies [65]. b) A
simple muscle synergy scheme for three muscles and two muscle synergies. A continuous stream
of parameters is being provided by the CNS to activate three muscles [66].
Muscle synergies come in two different flavors, i.e., synchronous muscle synergies and
asynchronous or time-varying muscle synergies [56]. In the former case, no temporal delay is
allowed in the activation of different muscles. In contrast, a time-varying synergy is a time
sequence of vectors representing a collection of muscle activation waveforms [56]. For activating
each synergy, the number of variables to be specified by the CNS in the case of asynchronous
muscle synergies are two as compared to one for the synchronous case [55]. Fig. 2.1 presents two
simple cases of synchronous muscle synergies. In Fig. 2.1(a), we present two muscle synergies
each consisting of four muscles. The CNS specifies two discrete parameters to combine the two
muscle synergies for activating four muscles. In Fig. 2.1(b), we present two muscle synergies
each consisting of three muscles. In this case, the CNS specifies two continuous streams of
parameters to activate three muscles continuously.
While the muscle synergies focus on relatively fixed activation levels of muscles, some
researchers favor the idea of individual motions and emphasize the uncoupled control of
33. 14
individual joints and muscles to generate necessary kinetics and kinematics of the movement [67].
Whether the muscle synergies have a neural origin or not is still an open question [65], [68], [69].
However, various aspects of muscle synergies are being vigorously
investigated [70], [71], [72], [73], [74], [75], [76], [77], [78], [79].
2.1.3 Extraction of Muscle Synergies
Various matrix factorization algorithms have been suggested for the decomposition of muscle
activity patterns, recorded through the myoelectric signal, into muscle synergies [80]. These
algorithms include the PCA [81], ICA [82], [83], [84], probabilistic ICA (pICA) [85],
NMF [50], [51], [46], [58], [52], [86] and factor analysis (FA) [87].
Considering the case of synchronous muscle synergies, we have
yi(k) =
n
j=1
wijxj(k), {yi, wij, xj} ≥ 0, ∀ i = 1, . . . , m; j = 1, . . . , n, (2.1)
where n is the number of muscle synergies, m is the number of muscles, yi represents activation
level of the ith
muscle, wij is the gain of jth
neural command for ith
muscle, and xj is the applied
neural command. We can write (2.1) in the vector form
y(k) = Wx(k), (2.2)
where y(k) = [y1(k), . . . , ym(k)]T
specifies activation levels of m muscles, W is a m × n matrix
whose columns are the muscle synergies and x(k) = [x1(k), . . . , xn(k)]T
is the neural command
vector (i.e., muscle synergy activation coefficients or the neural drive) at time k. The relations
34. 15
[(2.1) and (2.2)] can be extended for multiple trials, i.e., k = 1, . . . , K.
Ym×K = Wm×nXn×K. (2.3)
The subscripts in (2.3) indicate the dimensions of the matrices with Y a known quantity and W
and X being unknowns. This problem is known as a blind source separation (BSS) problem in
estimation theory. Various unsupervised learning algorithms can be employed to estimate the
parameters of the mixing system W and true physical sources X. Such algorithms exploit a priori
knowledge about the true nature or structure of the hidden variables such as non-negativity,
statistical independence, sparseness, spatio-temporal de-correlation, smoothness or lowest
possible complexity [88]. Due to the physiological constraints, all elements in W and X matrices
must be non-negative, i.e., the muscle activations cannot be a negative quantity [47].
2.1.4 Muscle Synergies for Myoelectric Prosthesis
There has been no explicit effort on using the hypothesis of muscle synergies to discriminate
between various tasks for the myoelectric prosthesis application. Recently, Ajiboye and Weir used
a subset of static hand postures of the American Sign Language (ASL) to estimate the muscle
synergies and then predicted other remaining ASL postures [59]. The authors reported that as few
as 11 basic postures were sufficient to predict all 33 postures of the ASL with an accuracy of
90%. However, the authors did not extend their work to tackle myoelectric prosthesis problem.
35. 16
Feature ExtractionData Segmentation Pattern Classification Control Scheme
Data Segmentation Change Point Detection Control Scheme
Data Segmentation Finite State MachineFeature Extraction
(a) Signal amplitude based control scheme.
Feature ExtractionData Segmentation Pattern Classification Control Scheme
Data Segmentation Change Point Detection Control Scheme
Data Segmentation Finite State MachineFeature Extraction
(b) Event driven finite state based control scheme.
Feature ExtractionData Segmentation Pattern Classification Control Scheme
Data Segmentation Change Point Detection Control Scheme
Data Segmentation Finite State MachineFeature Extraction
(c) Pattern classification based control scheme.
Figure 2.2: Three most common control paradigms for myoelectric prostheses [16].
2.2 Control Paradigms for Myoelectric Prostheses
Various paradigms have been proposed for controlling the hand prosthesis using the myoelectric
signals [16]. Fig. 2.2 presents three most common control schemes found in the myoelectric
prostheses literature. Fig. 2.2(a) presents a simple on/off control scheme. After myoelectric data
segmentation, a change point is detected using statistical techniques and the control algorithm
performs already established movements. As evident, the applicability of such a framework is
limited to one, or at the most two degrees-of-freedom (DOFs) devices. Fig. 2.2(b) presents an
event driven finite state (EDFS) control scheme. Dalley et al. proposed the EDFS approach for a
multi-grasp prosthetic hand with an average grasp completion rate of 99.2% [89]. Fig. 2.2(c)
presents the state-of-the-art control paradigm, i.e., the pattern classification system. In such an
arrangement, a representative feature set from the myoelectric signal is extracted and a
classification algorithm is trained. Later, the classifier is used to perform classification [90], [16].
A detailed description of pattern classification systems is presented in Section 2.3.
36. 17
Feature
Extraction
Data
Segmentation
Pattern
Classification
Data
Preprocessing
Post-
processing
Movement
Class Labels
Myoelectric
Signal
Figure 2.3: Schematic layout of the pattern classification system for extraction of control
information from the EMG signal [90].
2.3 Pattern Classification in Myoelectric Prostheses
The pattern classification for myoelectric control is based on the assumption that there exist
distinguishable and repeatable signal patterns among different types of muscular activations [91].
A schematic layout of pattern classification schemes used in myoelectric prostheses is given in
Fig. 2.3 [92]. Myoelectric signals are recorded from multiple muscle sites, followed by the data
segmentation process to form analysis windows. In some cases, data preprocessing may precede
the data segmentation. A representative set of features is extracted from the analysis windows and
a classification algorithm is trained. Depending upon the type of the classification algorithm, it
may require supervised or semi-unsupervised learning before going into the operational mode.
The output of the classification algorithm is a single class label. The post-processing routines may
also be included in the overall scheme to further improve the classification results. Most of the
classification schemes reported in the literature are capable of performing classification of four or
more distinct movements. Pattern classification systems have produced promising results, i.e.,
classification accuracy in the range of 90% or above [90], [93], [16], [94], [95].
2.3.1 Performance Parameters for Pattern Classification Systems
A general layout of the pattern classification systems is shown in Fig. 2.3. However, exact
configuration may vary considerably due to the type of selected features, pre-/post-processing
37. 18
algorithms, classification algorithms, experimental protocols used for the EMG data collection,
and number and type of movements. A brief discussion on all related issues is presented in the
following.
2.3.2 Preprocessing
A few preprocessing techniques have been investigated in the literature to improve classification
accuracy. One recently proposed scheme consists of pre-whitening the myoelectric signal. The
pre-whitening process de-correlates the myoelectric signal and may improved classification
accuracy by approximately 5% for small analysis windows (< 100 ms) [96]. The principal
component analysis (PCA) algorithm has also been proposed for dimensionality reduction of the
myoelectric signal before the feature extraction, which results in an improved classification
accuracy as well as reduced computational time for the feature extraction and classification [97].
2.3.3 Analysis Windows
The stochasticity and the variability inherent in the myoelectric signal limits its direct use in the
classification, although such efforts were attempted with limited success [98]. Therefore, a
segmentation operation on the EMG signal is performed to form analysis windows. Owing to the
real-time operational requirement, the size of analysis window is critical [99], [100]. A large
analysis window results in a better classification accuracy but may introduce delay in intent
interpretation (decision of the movement class by the classifier). On the other hand, a small
analysis window presents reduced amount of information to the feature extraction routine, and
resultantly increases the variance of the classifier, thereby increasing the classification errors.
38. 19
D1
𝑇𝑇𝑎𝑎 𝑇𝑇𝑑𝑑
𝑇𝑇𝑎𝑎
D2
𝑇𝑇𝑑𝑑
𝑇𝑇𝑑𝑑 : Processing time 𝑇𝑇𝑖𝑖 𝑖𝑖𝑖𝑖 : Window increment time𝑇𝑇𝑎𝑎 : Analysis window size
Decision Di = 𝑇𝑇𝑎𝑎 + 𝑇𝑇𝑑𝑑
(a) Non-overlapping analysis windows.
D1
𝑇𝑇𝑎𝑎 𝑇𝑇𝑑𝑑
𝑇𝑇𝑖𝑖 𝑖𝑖𝑖𝑖
𝑇𝑇𝑎𝑎
D2
𝑇𝑇𝑑𝑑
𝑇𝑇𝑑𝑑 : Processing time 𝑇𝑇𝑖𝑖 𝑖𝑖𝑖𝑖 : Window increment time
𝑇𝑇𝑖𝑖 𝑖𝑖𝑖𝑖
𝑇𝑇𝑎𝑎 : Analysis window size
First decision D1 = 𝑇𝑇𝑎𝑎 + 𝑇𝑇𝑑𝑑
Subsequent decisions Dk = 𝑇𝑇𝑖𝑖 𝑖𝑖𝑖𝑖
(b) Overlapping analysis windows.
Figure 2.4: Two different types of segmentation operations for the myoelectric signal are shown,
non-overlapping and overlapping analysis windows. Symbols D1 and D2 represent decisions from
the classification algorithm, Ta represents the analysis windows size, Td the processing time and
Tinc the increment time for the overlapping analysis windows. Modified from [92].
Additionally, in most cases, a sliding (overlapping) analysis window is introduced to provide a
rich stream of input data to the classifier. Effect of the size of analysis window on the
classification accuracy has been investigated in detail and optimal values have been found for
different feature sets and classification algorithms [92], [100], [99]. An analysis window of 150
ms to 250 ms may be a reasonable choice based on the feature set and the classification
algorithm [100]. In Fig. 2.4, we present non-overlapping and overlapping analysis windows.
2.3.4 Feature Sets
After data segmentation, representative features are extracted from analysis windows. A large
variety of features, individually and in groups, can be extracted to increase information density.
Broadly, four categories of features have been identified [16].
1. Time Domain (TD) features include mean absolute value (MAV), integrated absolute
value (IAV), variance (VAR), mean absolute value slope (MAVS), Willison amplitude
(WA), zero crossing (ZC), slope sign change (SSC), mean square value (MSV), and
39. 20
waveform length
(WL) [101], [102], [103], [104], [105], [94,106], [107], [108], [109], [110].
2. Time serial features include the AR model
coefficients [111], [106], [112], [94], [109], [113], [114], [115], [116], [117], and cepstral
coefficients [118], [112], [94].
3. Spectral domain and time-scale features include power spectrum, mean and median
frequency, frequency ratio, wavelet transform (WT), wavelet packet transform (WPT) and
short-time Fourier transform (STFT) [119], [120], [121], [94], [122], [117], [114].
4. Advanced spectral features, such as multifractal singularity spectrum [123].
It has been shown that for a slowly varying myoelectric signal, a combination of the AR model
coefficients and the TD features (MAV, MAVS, ZC, SSC, WL) gives good performance with
respect to computational time and classification accuracy [124], [90], [116]. Mathematical
definitions of the TD and AR features (usually referred to as TDAR in the literature) are given in
Table 2.1.
2.3.4.1 Feature set dimensionality reduction
In case, a large number of features are extracted from the analysis windows, the final feature set
consisting of multiple myoelectric channels may become large. Such practice may result in an
increased computational time for classifier training and/or degraded classification accuracy. To
circumvent, dimensionality reduction methodologies are often used [125], [90], [16]. Some
adopted techniques include the PCA [97], [126], independent component analysis (ICA) [127],
nonlinear projection [128], [129] or the fuzzy discriminant analysis [125].
40. 21
Table 2.1: Mathematical definitions of TDAR features. xi(k) is the kth
signal sample of the ith
segment, N is the number of samples in the segment i, xth is a predefined threshold [16], [92].
Feature Mathematical Definition
Mean absolute value MAVi = 1
N
N
k=1
|xi(k)|
Integrated absolute value IAVi =
N
k=1
|xi(k)|
Variance V ARi = 1
N
N
k=1
(xi − ¯xi)2
, where ¯xi = 1
N
N
i=1
xi
Mean absolute value slope MAV Si = MAVi+1 − MAVi
Willison amplitude WAMPi
N
k=1
f(|xi(k) − xi(k + 1)|)
with f(x) =
1 if x > xth
0 otherwise
Zero crossing ZCi =
N
k=1
f(k)
with f(k) = 1 if xi(k) ∗ xi(k + 1) < 0 and
|xi(k) − xi(k + 1)| > xth
Slope sign change SSCi =
N−1
k=1
f[{xi(k) − xi(k + 1)}{xi(k) + xi(k + 1)}]
with f(x) =
1 if x > xth
0 otherwise
Meas square value MSVi = 1
N
N
k=1
[xi(k)]2
Waveform length WLi =
N
k=1
(|xi(k) − xi(k + 1|)
Autoregressive coefficients xi(k) =
n
j=1
ajxi(k − j), an nth
order AR model with aj features
2.3.4.2 Feature set stability
Under dynamic contractions and noisy signal acquisition conditions, the performance of the
selected feature set may degrade. Tkach et al. investigated this phenomenon for different feature
sets under common conditions, such as, the electrode location shift, variation in the muscle
contraction effort, and muscle fatigue [130]. The authors reported that the muscle fatigue had the
smallest effect on the selected feature set, while electrode location shift and varying effort level
significantly reduced the classification accuracy for most of the feature sets.
41. 22
2.3.5 Classification Algorithms
Considerable efforts have been invested in exploring various classification algorithms for
myoelectric prostheses [131]. A classification algorithm has to take into account the stochastic
nature of the myoelectric signal as well as the noise added during the recording process. On the
other hand, the constraint of real-time decision making limits the computational time for the
classification algorithm. The classification algorithms reported in the literature include but are not
limited to the list presented here. Please refer to recent reviews and references therein for detailed
description of the classification algorithms [16], [90], [93], [132], [95]
1. Linear discriminant analysis (LDA) [101], [111], [119], [122].
2. Gaussian mixture model (GMM) [116], [133], [134] and log-linear GMM (LLGMM) [135].
3. Support vector machine (SVM) [120], [113], [110], [136].
4. Parallel multiple binary classification (MBC) and uncorrelated linear discriminant analysis
(ULDA) [137].
5. K-nearest neighbor (K-NN) with lazy learning [138] and genetic algorithms [139].
6. Parzen classifier, Fisher discriminant analysis and Quadratic discriminant analysis
(QDA) [131].
7. Hidden Markov models (HMM) [115].
8. Various types of artificial neural networks (ANN) in different configurations, i.e.,
multilayer perceptron (MLP), Radial Basis Functions (RBF), time-delayed ANN
42. 23
(TDANN), Elman neural network
(ENN) [102], [109], [140], [128], [106], [141], [121], [142], [143], [144], [98], [145].
9. Fuzzy and neuro-fuzzy based classifiers [114], [146], [112], [147], [107], [105], [148].
It is evident that a myriad of classification algorithms have been used in the myoelectric control
literature. However, it is known that once sufficient EMG channels are available (in the case of
forearm, a symmetrical arrangement of 4 to 6 EMG channels [124], [149]) and appropriate
features are selected, most of the modern classification algorithms perform well [90], [124], [131].
The analyses performed in various studies using the same set of EMG data revealed no statistical
differences between various classification algorithms [124], [131]. It is also known that with an
appropriate feature representation the myoelectric classification reduces to a linear problem [90].
2.3.6 Post-processing
The post-processing of the classification decision may improve overall pattern classification
system accuracy [92], [150], [151]. However, the post-processing may have additional
computational cost. Generally, the decision to include a post-processing routine may depend on
the type of classifier, number and type of features, analysis window size and available
computational power.
Majority voting (MV) is a common post-processing technique [92]. For a given decision point,
the MV decision includes previous m samples (of the decision stream) and next m samples. The
final decision by the post-processing algorithm will be the class with the greatest number of
occurrences in the 2m + 1 decision stream. The number of samples used in the MV (the
parameter m) is determined by the available processing time and the acceptable delay. However,
43. 24
it has been reported that despite the fact that MV improves classification accuracy, it does not
improve system usability and controllability [151].
Recently Simon et al. proposed a velocity ramp based post-processing scheme [151]. The
authors propose to keep the initial velocity of the movement low and increase it once an error-free
decision steam is received to the post-processing algorithm.
2.3.7 Sensor Orientation and Site Configurations
In the daily usage of a prosthesis, certain assumptions of the controlled laboratory experiments
may not hold. One such assumption is related to myoelectric sensors. In controlled laboratory
environment, the EMG electrodes are placed over precisely selected muscle sites parallel to
muscle fibers for optimal signal acquisition. On the other hand in actual prosthesis, EMG sensors
are permanently attached inside the socket of the prosthesis. Such an arrangement of the EMG
sensors makes it impossible for a user to adjust sensors exactly at the same spot each time he/she
dons the device. The resulting phenomenon is referred to as the socket misalignment. To quantify
the effects of socket misalignment, Young et al. investigated the pattern classification system
under various conditions of socket misalignment [152], [111]. Specifically the authors
investigated effects of socket misalignment on electrode orientation (parallel or perpendicular to
muscle fibers), inter-electrode distance (IED; which is fixed at the time of installation of the EMG
electrodes inside the socket), the size of electrode detection-surface (fixed once the electrode is
manufactured), and number of electrodes (EMG channels). The authors used two common feature
sets, i.e., the TD and AR. Some relevant conclusions by the authors are presented:
44. 25
1. Maximum classification accuracy was achieved while electrodes were oriented parallel to
the muscle fibers (p < .01) under both shift and no-shift conditions.
2. Electrode shifts perpendicular to muscle fibers increased classification error more than the
electrode shifts parallel to muscle fibers (p < .01).
3. An increase in IED from 2 cm (which is a considered a standard in myoelectric literature) to
4 cm improved system performance (p < .01).
4. An increase in electrode detection surface increased the classifier accuracy but not the
prosthesis usability.
5. Four to six channels performed well under shift/no-shift conditions.
6. The LDA classifier with an AR feature set outperform the TD feature set under electrode
shift condition (p < .01).
7. A combination of longitudinal and transverse electrodes yielded highest system usability
under electrode shift condition.
2.3.8 EMG Signal Sources : Surface vs. Intramuscular EMG
Given the fact that surface myoelectric signal exhibits some undesirable characteristics (e.g., the
muscle crosstalk, amplitude cancellation and limited access to deep musculature), feasibility of
using intramuscular EMG in myoelectric prosthesis has been investigated. Farrell and Weir
explored the effect of surface vs. intramuscular myoelectric signals on classification
accuracy [149]. The authors studied targeted and un-targeted conditions for both surface and
intramuscular EMG. In summary, the authors investigated four distinct cases, i.e., targeted
45. 26
surface, targeted intramuscular, un-targeted surface and un-targeted intramuscular. After a
detailed statistical analysis authors concluded that there were no statistical differences in the
classification accuracy of any of the four conditions. Similar results were earlier reported by
Hargrove et al. [124]. Both studies concluded that 4 to 6 equally spaced EMG electrodes around
the forearm provided more than 90% classification accuracy with TDAR feature set and the LDA
classifier.
2.3.9 Training Regimes
In myoelectric prostheses, the classification accuracy of the system also depends on the training
dataset as well as training conditions. In the following we discuss some related issues.
2.3.9.1 Transient vs. steady-state part of the myoelectric signal
In experimental studies, any voluntary movement (e.g., the wrist flexion) can be roughly divided
into three phases: the initial transient phase, middle steady-state phase, and the final transient
phase. The same three phases are also reflected in the myoelectric signal. The steady-state phase
is also called the isometric contraction as the muscle length do not change in this phase. The
other two phases are referred to as dynamic contraction phases. Fig. 2.5 presents all three phases
of a myoelectric signal recorded from the forearm. For extraction of features from the myoelectric
signal, few initial studies used only the transient phase [119], or the steady-state phase of the
signal [121]. However, later investigations revealed that both the steady-state and transient phases
of the signal provided better usability of the prosthesis, although the accuracy of the classifier
may be reduced [153], [154]. Another way to handle the transient information is by specifying the
46. 27
Initial
transient
phase
Terminal
transient
phase
Steady-state phase
cTp = 50%
cTp = 100%
Figure 2.5: Transient and steady-state phases in the myoelectric signal recorded from the forearm
muscle during a reaching movement.
contraction time percentage (cTp) [155]. A cTp value of 50% will remove all transient
information (from both sides of the signal) while a value of 100% will retain all information.
2.3.9.2 Variations in initial limb position
Variations in the limb position associated with normal use of the prosthesis and the position at
which the classifier was trained could have a substantial impact on the accuracy of the pattern
classification system [156], [157]. An increase in average classification error from 3.8% to 18%
has been reported in the literature when classifier was trained at one initial limb position and used
for classification with different initial position. Fougner et al. proposed to use accelerometers to
measure position of the arm and train the classifier in different limb positions. The authors
reported a reduction in the classification error from 18% to 5% using the proposed solution [156].
47. 28
2.3.9.3 Training using the kinematic and kinetic Data
Jiang et al. proposed to use the kinetic data (force and torques) in the training phase of the
classification algorithms [158]. The authors proposed a nonnegative matrix factorization (NMF)
algorithm based semi-unsupervised training regime. The same approach was used by Nielsen et
al., however, the kinetic data was recorded from the contra-lateral limb and an ANN based
classifier was trained to estimate force [159]. Later, Muceli et al., and Jiang et al. extended the
same concept to kinematic data where joint angles were mapped to muscle activations using a
MLP classifier [160], [109].
Instead of training the classifier on some preselected movements, user-selected training
regimes have also been explored where a user is at liberty to specify and make any movements for
the classifier training [107]. Momen et al. allowed users to select movements and induce muscle
activations for classifier training. The authors reported a classification accuracy of 92.7% ± 3.2%
for four different movements with a training time of 2 minutes [107].
Bunderson and Kuiken studied the effect of already acquired experience on the training
regime [161]. In the reported study, the participants included novice and experienced users of the
pattern classification system. The authors concluded that after a brief exposure to the training
scheme, the classification errors in novice users were reduced, although not to the level of
experienced ones. While the level of intraclass variability in novice users was similar to that of
the experienced users, they did not achieve the same level of interclass distance.
48. 29
2.3.10 Weight and Inertia of the Prosthesis
Unlike native human limb, a prosthesis is an alien object which an amputee has to lift and carry
by using force from his remaining muscles. Therefore, while wearing a prosthesis, some relevant
muscles of the forearm remain in a pre-activated state. In such a condition, any classifier trained
in relaxed muscle states will exhibit increased error rate [162].
2.4 Challenges in Surface EMG Processing
The extraction of a desired information from the surface myoelectric signal has its own
challenges. The contributing factors may include biological stochasticity, intrinsic
non-stationarity, and the heteroscedasticity of the signal. Additionally, the recording procedures,
instrument noise and skin condition also affect the quality of the acquired signal. Some relevant
challenges in the myoelectric signal processing with regards to the extraction of control
information are discussed here.
2.4.1 Muscle Crosstalk
A common problem typical to the surface myoelectric signal is the muscle crosstalk. Formally
defined as ``a signal detected over a muscle, however, generated by another muscle close to the
first one'' [163]. The phenomenon of crosstalk is referred as one of the most important sources of
error in the surface myoelectric signal interpretation [163], [164]. The effect of crosstalk worsens
in surface myoelectric recordings where muscles are relatively smaller in size and located inside a
compact area. For example, muscles responsible for hand postures (extrinsic muscle of the hand)
and wrist movements are densely packed in layers in the human forearm [165]. While the
49. 30
myoelectric signal is being recorded using multiple surface electrodes, the crosstalk adds a
correlation between independent sources of information, thus making some of the information
redundant.
2.4.2 Amplitude Cancellation
As the myoelectric signal represents a temporal and spacial sum of electrical spikes in multiple,
may be thousand or more, muscle fibers, some relevant information may be lost in the summation
process. The phenomenon is referred to as ``amplitude cancellation'' [166], [167], [168], [169].
The phenomenon of amplitude cancellation may cause loss of important control information in
the myoelectric signal.
2.4.3 Absence of Physiologically Relevant Musculature
A challenging difficulty in myoelectric signal recording (and in turn processing) may be the
absence of the physiologically relevant musculature [170]. Congenitally or after acquired
amputation, there may not exist relevant muscles for an access to myoelectric information. The
problem is intensified for individuals with proximal amputations, e.g., intrinsic and extrinsic hand
muscles found in the forearm are altogether absent in a shoulder dis-articulation amputee.
Recently proposed novel surgical procedure, called the targeted muscle reinnervation (TMR),
circumvents the problem by remapping the leftover nerves to different and biomechanically less
active sites [171]. After the TMR procedure, surface myoelectric signals can be recorded from
new sites using conventional procedures.
50. 31
2.4.4 Limited Access to Deep Muscles
The surface EMG recording procedure has inherently limited access to the deep musculature. The
phenomenon is more evident for surface myoelectric recordings from the forearm, where intrinsic
hand muscles are deep inside under a layer of superficial muscles.
2.5 Classification Accuracy vs. Usability of Myoelectric
Prostheses
The issue of accuracy of the pattern classification system vs. the usability of the myoelectric
prosthesis was raised by Lock [172]. The author designed a virtual environment where a
prosthetic arm was used to perform a clothespin test [173], [174]. Specifically, in an experimental
setup after classifier training, participants were asked to remove a clothespin from a horizontal
bar and attach it to the vertical bar. The time to complete a clothespin task, i.e., removing the
clothespin from a horizontal bar and placing it over the vertical bar, was the only metric used. The
author used three feature sets, i.e., the AR, TD, and a combination of both (TDAR) with three
classifiers, i.e., the LDA, ANN and GMM. The author reported that the correlation between the
classification accuracy of the selected pattern classification system and the usability of the
myoelectric prosthesis (time for clothespin test) was altogether absent or weak at best
(R2
= 0.238) [172].
51. 32
2.5.1 Performance Evaluation Metrics
Given the fact that the accuracy of the classification system does not translate into prostheses
usability, it was imperative to define new performance measures. These measures/metrics must
focus on the clinical usability of the device in addition to classification accuracy. Some relevant
measures/metrics proposed in the literature are discussed here.
2.5.1.1 The Motion Test
The Motion test is used to evaluate the real-time classification accuracy of the pattern
classification system [171]. The performance is evaluated using three metrics, i.e., the motion
selection time, motion completion time, and motion completion rate. The motion selection time is
the time taken to correctly select a target motion and is defined as the time from movement onset
to the first correct classification. The metric quantity measures how quickly motor commands can
be translated into correct motion predictions. The motion completion time is defined as the time
from movement onset to a preset number of correct classifications. The motion completion rate is
the percentage of successfully completed motions out of the total attempted.
2.5.1.2 Target Achievement Control (TAC) Test
The target achievement control (TAC) test is used to evaluate the resulting controllability of the
prosthetic systems as a whole [175]. The TAC test is relatively a challenging test for prosthesis
performance evaluation. The users are required to activate specific DOFs to achieve a target
posture. The TAC test performance metrics include the completion time, completion rate, and the
path efficiency. The completion time is the time from trial start to the successful achievement of
the target posture, not including the dwell time at the target posture. The completion rate is the
52. 33
percentage of successfully completed postures in a set of trials. Finally, the path efficiency is the
shortest path to the target divided by the total distance traveled.
2.5.2 Virtual Environments for Prosthesis Performance Evaluation
A virtual environment with a prosthetic limb (arm and hand) can be used for performance
evaluation of the myoelectric prosthesis [171], [155], [176], [177], [178]. The virtual prosthetic
arm mimics the hand/arm movements of the user in real-time. First, the EMG data is collected
from the participant and a classification algorithm is trained. Later, the trained algorithm
classifies the EMG signal is real-time and the virtual prosthetic limb performs the selected
movement in the virtual environment. A virtual environment provides a reasonable alternative to
actual prosthetic device.
53. 34
Chapter 3
Myoelectric Signal Classification Using the
AR-GARCH Model
In the pattern classification paradigm for controlling myoelectric prosthesis, the autoregressive
(AR) model coefficients are generally considered an efficient and robust feature set. However, no
formal statistical methodologies or tests are reported in the literature to analyze and model the
myoelectric signal as an AR process. We analyzed the myoelectric signal as a stochastic
time-series and found that the signal is heteroscedastic, i.e., the AR modeling residuals exhibit a
time-varying variance. Heteroscedasticity is a major concern in statistical modeling because it
can invalidate statistical tests of significance which may assume that the modeling errors are
uncorrelated and that the error variances do not vary with the effects being modeled. We
subsequently proposed to model the myoelectric signal as an Autoregressive-Generalized
Autoregressive Conditional Heteroscedastic (AR-GARCH) process and used the model
54. 35
parameters as a feature set for signal classification. Multiple statistical tests including the
Ljung-Box Q-test, Engle's test for heteroscedasticity, Kolmogorov-Smirnov (K-S) test and the
goodness of fit test were performed to show the validity of the proposed model. Our experimental
results show that the proposed AR-GARCH model coefficients, when used as a feature set in two
different classification schemes, significantly outperformed (p < .01) the conventional AR model
coefficients.
3.1 Background
In the myoelectric control, the choice of an appropriate feature set is crucial for the pattern
classification. Parker et al. have shown that the classification accuracy in a pattern pattern
classification scheme is more affected by the choice of the feature set than by the classification
algorithm [179]. A common and promising feature set considered in the myoelectric control
literature consists of AR coefficients [16], [95], [90]. Tkach et al. investigated different features
under conditions of change in muscle effort level, muscle fatigue and electrode shift, and found
the AR coefficients to be one of the most robust features [130]. Similarly, Young et al. found that,
under conditions of socket misalignment, the AR features performed better than the time-domain
features [111]. However, in the myoelectric control literature, there has not been an exclusive
effort to model the myoelectric signal as an AR process using formal statistical tools. A generally
adopted procedure in the literature is to adjust (increase or decrease) the AR model order based on
the required classification accuracy and available computational resources. Such practice usually
results in an AR model order in the range of 4 to 6 [16], [95]. We believe that a detailed statistical
study is essential to, 1) identify the parsimonious AR model order for the myoelectric signal, 2)
55. 36
measure the the goodness of fit of the identified model, and 3) evaluate various characteristics of
the modeling residuals, such as the sample autocorrelation and/or the heteroscedasticity.
In this chapter, we show that modeling the myoelectric signal as an AR process results into the
residual errors that exhibit heteroscedasticity, i.e., ``variability" in the variance. [180]. In order to
illustrate the concept of heteroscedasticity, let us consider a simple linear regression model
between two variables x and y,
y = α + βx + , (3.1)
where represents the regression error. We call homoscedasticity the assumption that the
expected size of the error is constant. We call heteroscedasticity the assumption that the expected
size of the error term is not constant [181]. The assumption of homoscedasticity is standard in
regression theory because of its mathematical convenience. However, in many applications, this
assumption may be unreasonable. A classic example of heteroscedasticity is that of income
versus expenditure. Those with higher income will display a greater variability in consumption
than lower-income individuals who tend to spend a rather constant amount. The concept of
heteroscedasticity generalizes to many other linear and non-linear models including the AR
model. The presence of heteroscedasticity in a time-series can invalidate statistical tests that
assume that the model residual variances are uncorrelated. It is possible to specify a stochastic
process for the residual errors and predict the average size of the error terms [181] using the
Autoregressive Conditional Heteroscedasticity (ARCH) [182] or the Generalized ARCH
(GARCH) models [183]. The GARCH process is a generalization of the ARCH process, and can
model the heteroscedasticity more parsimoniously.
56. 37
In engineering applications, the GARCH models have been employed to extract features from
the electroencephalogram (EEG) signals within the framework of wavelet transforms [184]. The
GARCH models have also been used within a state-space framework and the Kalman filter for
modeling non-stationary variance in the EEG signal [185], or modeling covariance for generation
of the EEG signal [186]. On the other hand, GARCH models have been successfully used to
model and enhance speech signals in the time-frequency domain [187]. Furthermore, voice
activity detection (VAD) is another area where GARCH models have been successfully used to
distinguish between the voice and the noise in a speech signal [188], [189]. In the case of
myoelectric signals, the GARCH process has been used for noise suppression with wavelet
coefficients [190]. However, to the best of our knowledge, no exclusive effort is reported in the
literature to model the myoelectric signal as an AR-GARCH process and use its parameters as a
basis for statistical inference and biophysical interpretation. In Fig 3.1, we contrast the proposed
approach with the conventional AR feature approach. The proposed AR-GARCH model extracts
more information from the myoelectric signal as compared to the convectional scheme as the AR
residuals further modeled as a GARCH process.
3.2 Heteroscedastic Processes and Myoelectric Signal
Modeling
Heteroscedastic processes are characterized by a volatile nature, and are often encountered in
econometrics and finance, as for instance in stock prices, which exhibit periods of large inter-day
price variability followed by periods of relative stability. The ARCH and GARCH processes are
58. 39
3.2.1 The GARCH Process
Let Zt be a sequence of i.i.d. random variables with zero mean and unit variance from some
specified probability density function. The process Yt is called GARCH(p, q) if
Yt = σtZt, t ∈ Z, (3.2a)
σ2
t = α0 +
q
i=1
αiY 2
t−i +
p
j=1
βjσ2
t−j, (3.2b)
where α0 > 0, αi ≥ 0 and βj ≥ 0, ∀ i, j; σt ≥ 0 and Zt ∼ N(0, 1). If the parameter p = 0 in the
second summation in (3.2b), the GARCH(p, q) process reduces to an ARCH(q) process [182]. In
an ARCH(q) process, the conditional variance is a linear function of past sample variances only,
whereas the GARCH(p, q) process takes into account lagged conditional variances as well [183].
If both p = q = 0, the process Yt is a pure white noise with variance specified by the parameter
α0. Some relevant properties of the GARCH process are discussed below. The expected value
(E[ . ]) of the GARCH (p, q) process is given by
E[Yt] = E[σtZt] = E[σt]E[Zt] = 0. (3.3)
Similarly, the variance (V [ . ]) of the GARCH (p, q) process is given by
V [Yt] = E[Y 2
t ] − [E(Yt)]2
= E[Y 2
t ] = E[σ2
t ],
= α0 +
q
i=1
αiE[Y 2
t ] +
p
j=1
βjE[σ2
t ],
=
α0
1 −
q
i=1
αi −
p
j=1
βj
. (3.4)
59. 40
From Eqs. (3.3) and (3.4), it is clear that the GARCH(p, q) is a zero mean process with a
constant variance, which is specified by the parameters {αi}q
i=0 and {βj}p
j=1. The GARCH
process is also statistically uncorrelated, i.e.,
E[YtYs] = E[σtZtσsZs] = E[σtZtσs]E[Zs] = 0. (3.5)
We show next that the conditional variance of the GARCH process is time-varying, reflecting the
volatility of the stochastic signal. Let Ft = {Ys : s ≤ t} represent the history of the process up to
time t. We have
E[Yt|Ft−1] = E[σtZt|Ft−1]
= σtE[Zt|Ft−1] = σtE[Zt] = 0 (3.6)
V [Yt|Ft−1] = E[σ2
t Z2
t |Ft−1]
= σ2
t V [Zt] = σ2
t (3.7)
From (3.6) and (3.7), we see that the conditional expected value of the GARCH process is also
zero but the conditional variance is equal to σ2
t , which is a time-varying quantity. In summary, we
have shown that the GARCH(p, q) process is a white noise process with time-varying conditional
variance.
To understand the difference between a white noise process and the GARCH process, in Fig.
3.2, we generated a Gaussian white noise, wt with zero-mean and unit variance, and a GARCH
process, Yt, using Gaussian innovations. A total of 10,000 realizations for each process were
generated. The sample autocorrelations of the two precesses, wt, Yt, and the squared processes,
60. 41
0 5000 10000
-5
0
5
Samples
Amplitude
Gaussian White Noise
0 5000 10000
-100
-50
0
50
100
Samples
Amplitude
GARCH(1,1) Process
0 5 10 15 20
-0.5
0
0.5
1
Lag
SampleAutocorrelation
White Noise
0 5 10 15 20
-0.5
0
0.5
1
Lag
SampleAutocorrelation
White Noise (Squared Samples)
0 5 10 15 20
-0.5
0
0.5
1
Lag
SampleAutocorrelation
GARCH(1,1) Process
0 5 10 15 20
-0.5
0
0.5
1
Lag
SampleAutocorrelation
GARCH(1,1) Process (Squared Samples)
Figure 3.2: A comparison of the Gaussian white noise and the GARCH(1,1) process. The first
column presents Gaussian white noise and the second column presents GARCH(1,1) process. Top
row shows both processes, mid row shows the samples autocorrelation values for 20 lags, and the
last row presents the sample autocorrelations of the squared processes. We can see that the GARCH
process is a white noise just like the Gaussian white noise, but with a unique characteristic, i.e.,
there is a significant autocorrelation in its squared sample values as is evident in the lower right
figure. The Gaussian noise was generated using standard normal distribution, and the GARCH
process was generated using Zt ∼ N(0, 1), α0 = 2, α1 = 0.09 and β1 = 0.9.
61. 42
w2
t , Y 2
t , were computed for 20 lag values. It is evident from Fig. 3.2 that the squared sample
autocorrelation of the GARCH process is much higher than the defined significance level.
Evidently, the GARCH process is a white noise with statistically significant autocorrelation in its
squared sample values.
3.2.2 The AR-GARCH Process
AR processes are driven by white noise [192] and we have shown that the GARCH process is
white noise. We can, therefore, generalize an AR process by considering a white noise of the form
specified by a GARCH process. The resulting process is called the AR-GARCH, where the AR
residuals t are given by σtZt and σt follows a GARCH volatility specifications in terms of
historical values of t. This formulation is a generalization of the family of AR and GARCH
models to exploit features of both models [193]. For an observed signal Xt, generated by an
AR(m) and a GARCH(p, q) process,
Xt =
m
k=1
φkXt−k + t, (3.8a)
t = σtZt, (3.8b)
σ2
t = α0 +
q
i=1
αi
2
t−i +
p
j=1
βjσ2
t−j, (3.8c)
where αi, βj and Zt are defined in Eq.(3.2).
62. 43
3.2.3 Modeling Heteroscedastic Processes
The first step towards modeling a heteroscedastic time-series is to confirm the presence of
heteroscedasticity. Once confirmed, different statistical tests can be performed to model and
quantitatively measure the validity of the modeling process. We discuss four statistical tests to
establish the heteroscedastic nature of the myoelectric signal and later model the same as an
AR-GARCH process. These tests include the Ljung-Box Q-test, Engle's test of heteroscedasticity,
the K-S test, and the goodness of fit testing and are discussed in the following.
3.2.3.1 The Ljung-Box Q-test
The Ljung-Box Q-test is a type of portmanteau statistical test of whether any of a group of
autocorrelations for a time-series are different from zero [194]. The outcome of the Ljung-Box
Q-test is a parsimonious model for which the sample autocorrelation of the residuals falls below
the critical value (determined by the significance level), i.e., the modeling residuals become
approximately white noise. The Ljung-Box Q-test assesses the null hypothesis that the series of
residuals exhibits no autocorrelation for a fixed number of lags m versus the alternative
hypothesis that the autocorrelation sequence is non-zero. Under the null hypothesis the
asymptotic distribution of the Q-statistic is χ2
m with m degrees of freedom. The modified
Q-statistic is defined as,
Q(r) = n(n + 2)
m
k=1
r2
k
n − k
, (3.9)
rk =
n
t=k+1
t t−k
n
t=1
2
t
,
63. 44
where rk is the autocorrelation estimate of the residuals at lag k, m is an appropriate number of
lags of the sample autocorrelation, n is the sample size, and represents the modeling residuals.
We employ the Ljung-Box Q-test to find a statistically correct AR model order for the
myoelectric signal.
3.2.3.2 Engle's test of heteroscedasticity
Robert Engle, in his seminal paper, proposed a test to establish the presence of heteroscedasticity
in a given time-series [182]. According to this test, in an ARCH model setting, under the null
hypothesis all ARCH parameters will be zero and under the alternate hypothesis, at least one
parameter will be nonzero. Formally, we have
H0 : α1 = . . . = αp = 0,
H1 : αi = 0, for some i = 1, · · · , p. (3.10)
The procedure is to run ordinary least squares regression and find the residuals. The next step is
to regress the squared residuals on a constant and p lags to test for Lagrange Multiplier statistic
TR2
as chi-square (χ2
p), where T is the sample size and R2
is the coefficient of determination of
regression. Under the null hypothesis, the asymptotic distribution of the test statistic is χ2
p with p
degrees of freedom. We employ the Engle's test to confirm that the AR residuals are
heteroscedastic.
64. 45
3.2.3.3 The Kolmogorov-Smirnov test
The two sample K-S test is a nonparametric test for the equality of continuous, one-dimensional
probability distributions that can be used to compare two samples. The K-S statistic quantifies a
distance between the empirical distribution functions of two samples. The null distribution of this
statistic is calculated under the null hypothesis that the samples are drawn from the same
distribution [195]. For the two sample test, the K-S statistic is given by:
Dm,n = sup
x
|F1,n(x) − F2,m(x)|, (3.11)
where F1,n and F2,m are the empirical cumulative distribution functions of the first and second
sample respectively. The empirical cumulative distribution function is given by
Fk(t) =
1
k
k
i=1
1{xi < t}, (3.12)
where 1{.} is the indicator function and is equal to 1 for xi < t and 0 otherwise. The null
hypothesis is rejected at level α if
n + m
nm
Dn,m > Kα, (3.13)
where Kα is the critical value calculated from the defined significance level α [196]. We use the
K-S test to quantitatively measure how close the AR residual distribution is to that of a GARCH
process.
65. 46
3.2.3.4 Goodness of fit
The goodness of fit of a statistical model describes how well the model fits the observations. In
our setting, we need to quantitatively measure how well the AR-GARCH model fits the
myoelectric signal. There are various tests available in the literature which can be used to find the
goodness of fit of a proposed statistical model [197]. We used the Akaike information criteria
(AIC) [198] and the Bayesian information criteria (BIC) [199]. Lower AIC/BIC values
correspond to a better fit of the model to the observed data.
3.3 Material and Methods
The study received approval from the Institutional Review Board at the University of Arkansas at
Little Rock. All participants provided written informed consent before start of the data collection.
Experimental protocol for data collection was designed as per the recommendations of Surface
Electromyography for the Non-Invasive Assessment of Muscles (SENIAM) project [200] and
standard physiology and anatomy text [201].
3.3.1 Participants
It has been shown that there is a significant positive correlation between myoelectric data
collected from able-bodied participants and amputees [109]. Therefore a set of healthy
participants can form a reasonable basis for testing newly proposed feature set. A total of six
able-bodied participants, male and female volunteered for the study. All selected participants
were right hand dominant with no neuromuscular disorder history.
66. 47
Posterior Anterior
Medial epicondyle
of the humerus
Styloid process of the ulna
1/3
(a) Electrode arrangement. (b) Wireless myoelectric sensors. (c) Snapshot of the GUI.
Figure 3.3: (a) Symmetrical placement of electrodes around the forearm on anterior and posterior
sides. (b) Arrangement of wireless myoelectric sensors on the forearm of a participant. (c) A
snapshot of the graphical user interface (GUI) for data collection. The GUI provided both visual
and auditory cues for movement initiation and termination. Duration of movement, number of
repetitions of a movement in a single trial and inter-movement break duration can be specified
separately for each trial by the experimenter.
3.3.2 Movements
We identified eleven movements for our data collection, i.e., five hand movements and six wrist
movements. Hand movements included lateral grasp (LG), cylindrical grasp (CG), tripod grasp
(TG), hand open (HO) and index finger point (IF) [95]. The wrist movements included pronation
(PR) and supination (SP), ulnar (UD) and radial deviation (RD), and flexion (FX) and extension
(XT). For the classification purpose a ``no movement (NM)'' class was also added, thus making
twelve classes in total.
3.3.3 Myoelectric Data Recording
It has been suggested that four to six surface sensors (myoelectric channels) capture adequate
muscle activation information required for good classification accuracies [202], [90]. Any further
increase in the number of channels may not necessarily increase the classification accuracy [149].
The same fact was observed in the preliminary data collection with as much as eight myoelectric
67. 48
channels. After analysis of initial results, five channels were considered adequate for assessing
the performance of the proposed feature set. We used wireless sensors to record and transmit
myoelectric data at a sampling rate of 1500 Hz with a Noraxon TeleMyo Direct Transmission
System (DTS) (Noraxon U.S.A. Inc, Scottsdale, Arizona). We used disposable, self-adhesive
silver/silver chloride (Ag/AgCl) snap electrodes with two circular conductive areas of 1 cm each
and an inter-electrode distance of 2 cm. It is known that a comparable accuracy can be achieved
without targeting specific muscle sites [149]. Therefore, we placed all five electrodes around the
circumference of the forearm symmetrically with the first electrode placed beneath the medial
epicondyle of the humerus. Electrodes were placed proximally around the forearm at a location
that is 1/3 of the distance between medial epicondyle of the humerus and styloid process of the
ulna. Electrode placement for the data collection is shown in Figs. 3.3(a) and 3.3(b).
3.3.4 Data Collection Protocol
Before start of the data collection experiment, each participant was sitting comfortably in a chair
with adjustable height armrests. Height of the right armrest was adjusted for each individual
keeping in view his/her comfort and ease. Initial/rest position of the arm was defined as: the
dorsopalmar axis pointing inside parallel to coronal plane (palm facing to the medial side of the
body with forearm parallel to the ground), elbow flexed at 90o
and arm abducted 10o
. Forearm
was supported at two places, i.e., under the styloid process of the ulna and the distal end of the
humerus (the elbow joint). A graphical user interface (GUI) based software was developed to
provide visual and auditory cues to participants for guiding through the data collection process
[Fig. 3.3(c)]. The GUI provided the flexibility to control different experimental parameters
68. 49
including the movement duration, the number of movement repetitions in a trial and the duration
of the inter-movement break. For this data collection, a single trial consisted of one repetition of
each movement appearing in front of the participant in a random order. There was a short break of
4 seconds between consecutive movements. Once indicated by the software through visual and
auditory cues, the user performed a movement in a comfortable way for a duration of 5 seconds.
Between every two trials, there was an additional break, the duration of which was left at the
discretion of the participant. A total of ten trials were recorded for each participant. Participants
were instructed to start the movement from defined initial (rest) position, stay in the prescribed
posture (shown in the GUI) and then return back to initial position. Participants were instructed to
maintain comfortable and repeatable force levels for all movements. Furthermore, participants
were asked to report any signs of pain and/or fatigue immediately to the experimenter. Before
start of the data collection, all participants were given approximately ten minutes of training to
familiarize them with the experimental protocol and movements to be performed. Myoelectric
data was collected using the Noraxon software (MyoResearch XP) and stored in the hard disk for
later processing. All data processing was performed in Matlab (The MathWorks, Natick, MA).
3.3.5 Myoelectric Signal Processing
We developed a custom software in Matlab for processing of the myoelectric data, i.e., data
segmentation, feature extraction and classification. Myoelectric data from a 3-seconds period was
extracted from the raw myoelectric data (originally recorded for 5 seconds) from all five channels.
The data was segmented to form non-overlapping analysis windows of 200 ms.
69. 50
3.4 Results
We present results in two separate parts. In the first part, we analyzed the myoelectric signal for
heteroscedasticity and modeled it as an AR-GARCH process. Later, we used the AR-GARCH
model parameters as a new feature set and performed classification of the myoelectric signal
using two different classification algorithms.
3.4.1 Myoelectric Signal Modeling and Analysis
We modeled the myoelectric signal as an AR process with the condition that the AR modeling
residuals were white noise. Later, we analyzed the AR modeling residuals for heteroscedasticity
and modeled the residuals as a GARCH process. Finally, we performed a goodness of fit analysis
of the proposed model.
3.4.1.1 Myoelectric Signal as an AR process
In order to find a statistically correct AR model, we increased the AR model order till the
modeling residuals became white noise. Specifically, for each AR model, the residuals formed an
input to the Ljung-Box Q-test which analyzed residuals. In addition, we also used squared values
of the residuals as an input to the Ljung-Box Q-test. The later input was used to highlight the
heteroscedastic nature of the myoelectric signal. We used 95% significance level for the test, with
the AR model order ranging from 5 to 15. An analysis window of 200 ms was used in this test. In
Fig. 3.4, it is evident that for the AR(9) model, the residuals are approximately white noise, while
the Q-statistic for the squared AR residuals does not decrease with increasing model order, which
70. 51
5 6 7 8 9 10 11 12 13 14 15
0
50
100
150
200
250
300
350
400
450
500
AR Model Order
Q-Statistic
AR Residuals
Squared AR Residuals
Critical Value
Figure 3.4: Ljung-Box Q-test results for the myoelectric signal. The AR modeling residuals are
approximately white noise for AR(9), while for the squared residuals, the Q-statistic does not
decrease and stays above the critical value for all AR models.
is a typical characteristic of a heteroscedastic time-series (in accordance with the lower right
figure in Fig. 3.2).
3.4.1.2 Heteroscedasticity in the Myoelectric Signal
Once a statistically correct AR model was found (i.e., the modeling residuals were white noise),
we used the Engle's test to establish whether the residuals exhibited heteroscedasticity or
otherwise. For this test, we used analysis windows of various sizes and two different AR models,
i.e., AR(5) and AR(9). In Table 3.1, we present results of the heteroscedastic testing. We varied
the size of analysis windows from 50 ms (75 samples) to 300 ms (450 samples). For both AR
models and all analysis windows of 100 ms or above, the null hypothesis was rejected, i.e., the
myoelectric signal was found to exhibit heteroscedasticity.
Further, we used a two sample K-S test to quantify whether the AR modeling residuals follow
the same distribution as that of a GARCH process. For this purpose, one sample for the K-S test
71. 52
Table 3.1: Heteroscedasticity in the myoelectric signal for different analysis windows and AR
models. Myoelectric data was recorded at a sampling frequency of 1500 Hz. Critical value for all
tests was 18.71.
Window size Samples Statistic Value Result
AR(5)
50 ms 75 11.55 homoscedastic
100 ms 150 29.44 heteroscedastic
150 ms 225 48.37 heteroscedastic
200 ms 300 44.86 heteroscedastic
250 ms 375 46.27 heteroscedastic
300 ms 450 48.84 heteroscedastic
AR(9)
50 ms 75 10.24 homoscedastic
100 ms 150 33.08 heteroscedastic
150 ms 225 55.53 heteroscedastic
200 ms 300 40.79 heteroscedastic
250 ms 375 50.03 heteroscedastic
300 ms 450 54.66 heteroscedastic
consisted of the AR residuals and the other sample was the GARCH process generated with
parameters estimated using the same AR residuals. The test was performed on various
myoelectric signals selected at random from experimental data and modeled with an AR(5) or
AR(9) process. Invariably, in all cases, the test was unable to reject the null hypothesis indicating
that the GARCH process and the residual samples were following the same probability
distribution function. In Fig. 3.5, we present the cumulative distribution function (CDF) and
histogram plots for the AR(5) residuals and the GARCH(1,1) process generated using α0, α1 and
β1 parameters estimated from the same AR modeling residuals and Gaussian innovations for Zt
(Eq. 3.2).
3.4.1.3 Goodness of fit
We perform goodness of fit testing using the AIC and BIC to establish whether the proposed
AR-GARCH modeling scheme actually fits the data well. We used different number of models,
