Fractional Order Signal Processing Techniques, Applications and Urgency

1
Fractional Order Signal Processing Techniques,
Applications and Urgency

• YangQuan Chen
• Director, Center for Self-Organizing and
  Intelligent Systems
• Associate Professor, Dept. of Electrical
  Computer Engineering
• Utah State University, Logan Utah, USA
• E yqchen_at_ieee.org T 1(435)797-0148 F
  1(435)797-3054
• W http//fractionalcalculus.googlepages.com/
  

Wednesday, November 5, 2008 Third IFAC
International Workshop on Fractional Derivative
and Applications (IFAC FDA 2008), Nov. 5-7,
Ankara, Turkey
2
Outline

• Introduction Me and CSOIS Research Strength
• Fractional Order Signal Processing An Enticing
  Example.
• Fractional Order Signal Processing What Are The
  Techniques There?
• Applications Why Urgent?
• Concluding Remarks

3
Outline

• Introduction Me and CSOIS Research Strength
• Fractional Order Signal Processing An Enticing
  Example?
• Fractional Order Signal Processing What Are The
  Techniques There?
• Applications Why Urgent?
• Concluding Remarks

4
YangQuan Chen

• Ph.D. 1998 Nanyang Tech. Univ. Singapore
• Now Associate Prof. at USU with tenure (s08).
  Director of CSOIS (s04) Graduate Coordinator of
  ECE Dept. (s08).
• First heard of FC in 1999-2000 when I worked in
  Seagate as HDD servo product engineer (Luries
  TID patent, later Manabes control paper in 60s,
  then CRONE )
• Learned a lot from Blas and Igor, and many of you
  here
• Strong curiosity in the applied side of FC, see
  http//fractionalcalculus.googlepages.com/
• Still learning FC fraction by fraction

5
CSOIS - CSRA ResearchCenter for Self-Organizing
and Intelligent Systems

• CSOIS is a research center in USUs Department of
  Electrical and Computer Engineering that
  coordinates most CSRA (Control Systems, Robotics
  and Automation) research
• Officially Organized 1992 - Funded for 7 (seven)
  years by the State of Utahs Center of Excellence
  Program (COEP)
• Horizontally-Integrated (multi-disciplinary)
• Electrical and Computer Engineering (Home dept.)
• Mechanical Engineering
• Computer Science
• Vertically-integrated staff (20-40) of faculty,
  postdocs, engineers, grad students and undergrads
  
• Average over 2.0M in funding per year from
  1998-2004
• Three spin-off companies from 1994-2004.
• Directors 92-98, Bob Gunderson 98-04, Kevin
  Moore 04-now, YangQuan Chen

6
CSOIS research impacts (1998-2004)

• Educational
• 2 PhD graduated with 2 others expected this year
• 38 MS and ME students graduated
• Numerous ECE and MAE Senior Design Projects
• Scholarly
• Five faculty collaborating between three
  different departments
• Four books
• Over 100 refereed journal and conference
  publications
• 18 visiting research scholars from 7 countries (3
  month to 1 year visits)
• Economic
• 14 full-time staff employed (average of 7 FTE per
  year)
• 8 PhD students employed
• 64 MS and ME students employed
• 31 Undergraduate students employed
• 12 Other staff employed
• A payroll of over 5 million in salaries paid to
  students, faculty, and staff
• Purchases of over 1.5M in the U.S. economy

7
CSOIS Core Capabilitiesand Expertise

• Control System Engineering
• Algorithms (Intelligent Control)
• Actuators and Sensors
• Hardware and Software Implementation
• Intelligent Planning and Optimization
• Real-Time Programming
• Electronics Design and Implementation
• Mechanical Engineering Design and Implementation
• System Integration
• We make real systems that WORK!

8
CSRA/CSOIS Courses

• Undergraduate Courses
• MAE3340 (Instrumentation, Measurements)
  ECE3620/40 (Laplace, Fourier)
• MAE5310/ECE4310 Control I (classical, state
  space, continuous time)
• MAE5620 Manufacturing Automation
• ECE/MAE5320 Mechatronics (4cr, lab intensive)
• ECE/MAE5330 Mobile Robots (4cr, lab intensive)
• Basic Graduate Courses
• MAE/ECE6340 Spacecraft attitude control
• ECE/MAE6320 Linear multivariable control
• ECE/MAE6350 Robotics
• Advanced Graduate Courses
• ECE/MAE7330 Nonlinear and Adaptive control
• ECE/MAE7350 Intelligent Control Systems
• ECE/MAE7360 Robust and Optimal Control
  ltlt FOC
• ECE/MAE7750 Distributed Control Systems ltlt
  FOC

9
Selected CSOIS Research Strengths

• ODV (omni-directional vehicle) Robotics
• Iterative Learning Control Techniques
• MAS-net (mobile actuator and sensor networks) and
  Cyber-Physical Systems (CPS)
• Smart mechatronics, vision-based measurement and
  control, HIL simulation, networked control
  systems
• UAV (Unmanned Aerial Vehicles) based Cooperative
  Remote Sensing Engineered Swarms
• Fractional Dynamic Systems, Fractional Order
  Signal Processing and Fractional Order Control

10
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USU ODV Technology

• USU has worked on a mobility capability called
  the smart wheel
• Each smart wheel has two or three independent
  degrees of freedom
• Drive
• Steering (infinite rotation)
• Height
• Multiple smart wheels on a chassis creates a
  nearly-holonomic or omni-directional (ODV)
  vehicle

12
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13
ODIS On Duty in Baghdad
Putting Robots in Harms Way, So People Arent
14
Outline

• Introduction Me and CSOIS Research Strength
• Fractional Order Signal Processing An Enticing
  Example?
• Fractional Order Signal Processing What Are The
  Techniques There?
• Applications Why Urgent?
• Concluding Remarks

15
chemotaxis behavior quantification

• Soil remediation
• Genetically engineered bacterium
• Targeted chemotractant

16
Video microscopy
17
Sample trajectory of one bacterium
18
Sample variances
19
Good news NOT second order processes!

• Research opportunities related to FOSP
• Find the most sensitive index to quantify the
  chemotaxis behavior (H parameter?)
• Most robust to bacteria population size, temporal
  sample size
• Complex Hurst parameter for complex (2D) motion?
• 2-D LRD (Long range dependent) processes?
• On-going efforts.

20
Outline

• Introduction Me and CSOIS Research Strength
• Fractional Order Signal Processing An Enticing
  Example.
• Fractional Order Signal Processing What Are The
  Techniques There?
• Applications Why Urgent?
• Concluding Remarks

21

• Main references (101 references cited)
• YangQuan Chen and Rongtao Sun and Anhong Zhou.
  An Overview of Fractional Order Signal
  Processing (FOSP) Techniques. DETC2007-34228 in
  Proc. of the ASME Design Engineering Technical
  Conferences, Sept. 4-7, 2007 Las Vegas, NE, USA,
  3rd ASME Symposium on Fractional Derivatives and
  Their Applications (FDTA'07), part of the 6th
  ASME International Conference on Multibody
  Systems, Nonlinear Dynamics, and Control (MSNDC).
  18 pages.
• See also
• Ortigueira, M.D. An introduction to the
  fractional continuous-time linear systems the
  21st century systems IEEE Circuits and Systems
  Magazine, IEEEVolume 8, Issue 3, Third Quarter
  2008 Page(s)19 - 26

22
FOSP Techniques

• Fractional derivative and integral
• Fractional linear system
• Autoregressive fractional integral moving average
• 1/f noise
• Hurst parameter estimation
• Fractional Fourier Transform
• Fractional Cosine, Sine and Hartley transform
• Fractals
• Fractional Splines
• Fractional Lower Order Moments (FLOM) and
  Fractional Lower Order Statistics (FLOS)

23
Introduction

• The first reference to this area appeared during
  1695 in a letter from Leibnitz to LHospital.
• Only in the last three decades the application of
  FOSP deserved attention, motivated by the works
  of Mandelbrot on Fractals.
• Recently, many more fractional order signal
  processing techniques has appeared, such as
  fractional Brownian motion, fractional linear
  systems, ARFIMA (FIARMA) model, 1/f noise,
  Hurst parameter estimation, fractional Fourier
  transform, fractional linear transform,
  fractional splines and wavelets.
• It is very necessary to review them, and find out
  their relationships.

24
Fractional derivative and integral

• Fractional derivative and integral, also called
  Fractional calculus are the basic idea of FOSP.
• The first notation of was
  introduced by Leibniz. In 1695, Leibniz himself
  raised a question of generalizing it to
  fractional order.
• In last 300 years the developments culminated in
  two calculi which are based on the work of
  Riemann and Liouville (RL) and Grunwald and
  Letnikov (GL).

25
Some related functions

• Gamma function
• Beta function
• Mittag-Leffler function

26
Operator

• A generalization of fractional derivative and
  integral operator

27
Grunwald-Letnikov definition

• From integer order exponent
• where
• The GL definition is then given as
• where

28
Riemann-Liouville definition

• Similarly, the common formulation for the
  fractional integral can be derived directly form
  the repeated integration of a function
• Then Riemann-Liouville fractional integral can be
  written as
• The RL differointegral is thus defined as

29
Properties

• If f(t) is an analytic function of t, the
  derivative is an analytic function
  of t and .
• The operator gives the same result as the
  usual integer order n.
• The operator of order is the
  identity operator.
• Linearity
• The additive index law
• Differintegration of the product of two
  functions
  
• 
  

30
Laplace transform of

• From the GL definition
• From the RL definition

31
Fractional Linear System

• Consider fractional linear time-invariant (FLTI)
  systems described by a differential equation with
  the general format
• According to the Laplace transform of , the
  transfer function can be obtained as

32
Impulse response

• Start from the simple transfer function
• its impulse response is
• Proceed to a further step

33

• For transfer function like
• we can perform the inversion by the following
  steps
• 1,Transform from H(s) into H(z), by substitution
  of for z.
• 2,The denominator polynomial in H(z) is the
  indicial polynomial. Perform the expansion of
  H(z) in partial fractions.
• 3,Substitute back for z, to obtain the
  partial fractions in the form
• 4,Invert each partial fraction.
• 5,Add the different partial impulse responses.

34
Autoregressive fractional integral moving average
(ARFIMA)

• Using a fractional differencing operator which
  defined as an infinite binomial series expansion
  in powers of the backward-shift operator, we can
  generalize ARMA model to ARFIMA model.
• where L is the lag operator, are error
  terms which are generally assumed to be sampled
  from a normal distribution with zero mean

35
Properties of ARFIMA

• Fractionally differenced processes exhibit
  long-term memory (long rang dependence) or
  anti-persistence (short term memory)
• An ARFIMA (p, d, q) process may be differenced a
  finite integral number until d lies in the
  interval (-½, ½), and will then be stationary and
  invertible. This range is the most useful set of
  d.
• 1, d-½. The ARFIMA (p, -½, q) process is
  stationary but not invertible.
• 2, -½ltdlt0. The ARFIMA (p, d, q) process has a
  short memory, and decay monotonically and
  hyperbolically to zero.

36

• 3, d0. The ARFIMA (p, 0, q) process can be white
  noise.
• 4, 0ltdlt½. The ARFIMA (p, d, q) process is a
  stationary process with long memory, and is very
  useful in modeling long-range dependence (LRD).
  The autocorrelation of LRD time series decays
  slowly as a power law function.
• 5, d½. The spectral density of the process is
• as . Thus the ARFIMA (p, ½, q) process
  is a discrete-time 1/f noise.

37
1/f noise

• Models of 1/f noise were developed by Bernamont
  in 1937
• where C is a constant, S(f) is the power
  spectral density.
• 1/f noise is a typical process that has long
  memory, also known as pink noise and flicker
  noise.
• It appears in widely different systems such as
  radioactive decay, chemical systems, biology,
  fluid dynamics, astronomy, electronic devices,
  optical systems, network traffic and economics

38
1/f noise spectrum
39

• We may define 1/f noise as the output of a
  fractional system as discussed before. The input
  could be white noise.
• Also, we can consider 1/f noise as the output of
  a fractional integrator. The system can be
  defined by the transfer function
• with impulse
  response
• Therefore, the autocorrelation function of the
  output is

40
Fractional Gaussian Noise (FGN)

• FGN is a kind of 1/f noise.
• FGN can be seen as the unique Gaussian process
  that is the stationary increment of a
  self-similar process, called fractional Brownian
  motion (FBM).
• The FBM plays a fundamental role in modeling
  long-range dependence.
• The increments time series
• of the FBM process BH are called FGN.

41
Relationship between fractional order dynamic
systems, long range dependence and power law
42
Long-range dependence

• History The first model for long range
  dependence was introduced by Mandelbrot and Van
  Ness (1968)
• Value financial data
• communications networks data
• video traffic
• biocorrosion data

43
Long-range dependence

• Consider a second order stationary time series
• Y Y (k) with mean zero. The time series Y
  is said to be long-range dependent if

44
Hurst parameter

• The Hurst parameter H characterizes the degree of
  long-range dependence in stationary time series.
• A process is said to have long range dependence
  when
• Relationships
• 1,
• 2, d is the differencing parameter of ARFIMA

45
Models with Hurst phenomenon

• Fractional Gaussian noise (FGN) models
  (Mandelbrot, 1965 Mandelbrot and Wallis, 1969a,
  b, c)
• Fast fractional Gaussian noise models
  (Mandelbrot, 1971)
• Broken line models (Ditlevsen, 1971 Mejia et
  al., 1972)
• ARFIMA/FIARMA models (Hosking, 1981, 1984)
• Symmetric moving average models based on a
  generalised autocovariance structure
  (Koutsoyiannis, 2000)

46
Hurst parameter estimation methods

• R/S Analysis
• Aggregated Variance Method
• Dispersional Analysis Method
• Absolute Value Method
• Variance of Residuals Method
• Local Whittle Method
• Periodogram Method
• Wavelet-based
• Fractional Fourier Transform (FrFT) based

47
Comparison of some important Hurst parameter
estimation methods, tested with 100 FGN of known
Hurst parameters from 0.01 to 1.00
48
Fractional Fourier Transform (FrFT)

• Rotation concept of Fourier Transform
• where is the rotating
• angle

49
Rotation concept of FrFT

• The FrFT rotates over an arbitrary angle
  , when a1 it correspond to Fourier
  transform.
• From ,we can
  define FrFT for the angle by
• Any function f can be expanded in terms of these
  eigenfunctions
  
• , with
• where Hn(x) is an Hermite polynomial

50
Definition of FrFT

• By applying the operator , and use of
  Mehlers formula, as well as possible choice for
  the eigenfunctions of F we can get the definition
  for FrFT in linear integral form as

51
Properties of the Kernel Function of FrFT

• If is the kernel of the FrFT,
  then

52
Convolution of FrFT

• Concerns the convolution of two functions in the
  domain of the FrFT
• If
• Then its Fourier transform becomes
• Thus

53
FrFT of a Delta Function
54
FrFT of a Sine Function
55
Fractional Linear Transform

• Generalize the FrFT method to linear transform.
  Given linear
• transform T, the procedure to find its fractional
  transform is
• Find the eigenfunctions and
  enginvalues of T
• The kernel functionis defined by
• The fractional transform T is then given by

56
Fractional Hartley transform

• Hartley transform
• According to the linear fractional transform
  method, the fractional Hartley transform can be
  given by

57
Relations between FrHT and FrFT
where is the fractional Hartley
transform and is the fractional
Fourier transform.
58
Fractional Cosine and Sine transform

• Cosine and Sine transforms
• A.W. Lohmann, et al, in 1996, have derived the
  fractional Cosine/Sine transforms by taking the
  real/imaginary parts of the kernel of FrFT.

59
Fractals

• The term fractal was coined in 1975 by
  Mandelbrot, from the Latin fractus, meaning
  "broken" or "fractured.
• A fractal is a geometric shape which
• is self-similar and
• has fractional (fractal) dimension.
• Fractals can be classified
• according to their self-similarity.
• 
  
  Sierpinsky Triangle

Y. Chen, Fractional order signal processing in
biology/biomedical signal analysis," in
Fractional Order Calculus Day at Utah State
University, April 2005,http//mechatronics.ece.usu
.edu/foc/event/FOC Day_at_USU/.
60
Fractal dimension estimation

• In fractal geometry, the fractal dimension is a
  statistical quantity that gives an indication of
  how completely a fractal appears to fill space,
  as one zooms down to finer and finer scales.
• Box counting dimension
• Information dimension
• Correlation dimension
• Rényi dimensions

61

• Long-range dependent time series can also be
  described by a fractal dimension D which is
  related to the Hurst parameter through D
  2 - H . Here, the fractal dimension D can be
  interpreted as the number of dimensions the
  signal fills up.
• Besides, porous media model for the hydraulic
  system has fractal dimension. For example, the so
  called porous ball built by the French group
  CRONE has been used in car's hydraulic circuit.
• Also, the preparation of nanoparticles coated
  bio-electrodes is by polishing the surface with
  fractal shapes.
• In addition, the diffusion behavior of
  bioelectrochemical process will be fractional
  order dynamic, which is related with FD.

62
Fractional Splines

• The fractional splines are an extension of the
  polynomial splines for all fractional degrees a gt
  -1.
• The fractional splines with one-sided power
  function can be written as
• where xk are the knots of the spline.

63
Fractional B-Splines

• One constructs the corresponding fractional
  B-splines through a localization process similar
  to the classical one, replacing finite
  differences by fractional differences.
• are in L1 for all agt-1
• are in L2 for agt-1/2
• 
  Fractional
  B-Splines

http//bigwww.epfl.ch/index.html
64
Properties of Fractional Splines

• If a is an integer, fractional splines are
  equivalent to the classical polynomial splines.
• 2) The fractional splines are a-Hölder continuous
  for a gt 0.
• 3) The fractional B-splines satisfy the
  convolution property and a generalized fractional
  differentiation rule. Besides, they decay at
  least like xa-2.
• 4) The fractional splines have a fractional order
  of approximation a 1.
• 5) Fractional spline wavelets essentially behave
  like fractional derivative operators.

65
Fractional Lower Order Moments (FLOM) and
Fractional Lower Order Statistics (FLOS)

• The stable model can be used to characterize the
  non-Gaussian processes. including underwater
  acoustic signals, low frequency atmospheric noise
  and many man-made noises.
• It has been proven that using stable model and
  fractional lower-order statistics (FLOS),
  additional benefits can be gained using this type
  of fractional order signal processing technique.
• Note that, for stable distribution the density
  function has a heavier tail than Gaussian
  distribution.
• It has been noticed that there is a natural link
  between LRD and heavy tail or thick/fat/heavy
  processes characterized by FLM/FLOS. A special
  case is the so-called SaS (symmetrical a-stable)
  process, which finds wide applications in
  engineering and non-engineering domains.

66
Summary of FOSP Techniques

• Fractional derivative and integral
• Fractional linear system
• Autoregressive fractional integral moving average
• 1/f noise
• Hurst parameter estimation
• Fractional Fourier Transform
• Fractional Cosine, Sine and Hartley transform
• Fractals
• Fractional Splines
• Fractional Lower Order Moments (FLOM) and
  Fractional Lower Order Statistics (FLOS)

67
Outline

• Introduction Me and CSOIS Research Strength
• Fractional Order Signal Processing An Enticing
  Example.
• Fractional Order Signal Processing What Are The
  Techniques There?
• Applications Why Urgent?
• Concluding Remarks

68
Urgency-1 Weierstrass function

• Nowhere differentiable!

Fractional order derivative exists
differentiability order 0.5 or less
sprott.physics.wisc.edu/phys505/lect11.htm
Wen Chen. Soft matters. Slides presented at
2007 FOC_Day _at_ USU.
69
Urgentcy-2 Infinite variance

• Normal distribution N(0,1)

Sample Variance
70

• Uniformly distributed

Sample Variance
71
Sample trajectory of one bacterium
72
Sample variances
73
Fractional Lower Order Statistics (FLOS) or
Fractional Lower Order Moments (FLOM)
Shao, M., and Nikias, C. L., 1993. Signal
processing with fractional lower order moments
stable processes and their applications.
Proceedings of the IEEE, 81 (7) , pp. 986 1010.
74
Important Remarks

• In fact, for a non-Gaussian stable distribution
  with characteristic exponent a, only the moments
  of orders less than a are finite. Therefore,
  variance can no longer be used as a measure of
  dispersion and in turn, many standard signal
  processing techniques such as spectral analysis
  and all least squares (LS) based methods may give
  misleading results.

75
Urgency-3 GSL - Do you care about it?
76
Long-term water-surface elevation graphs of the
Great Salt Lake
77
Elevation Records of Great Salt Lake

• The Great Salt Lake, located in Utah, U.S.A, is
  the fourth largest terminal lake in the world
  with drainage area of 90,000 km2.
• The United States Geological Survey (USGS) has
  been collecting water-surface-elevation data from
  Great Salt Lake since 1875.
• The modern era record-breaking rise of GSL level
  between 1982 and 1986 resulted in severe economic
  impact. The lake levels rose to a new historic
  high level of 421185 ft in 1986, 12.2 ft of this
  increase occurring after 1982.
• The rise in the lake since 1982 had caused 285
  million U.S. dollars worth of damage to lakeside.
  
• According to the research in recent years,
  traditional time series analysis methods and
  models were found to be insufficient to describe
  adequately this dramatic rise and fall of GSL
  levels.
• This opened up the possibility of investigating
  whether there is long-range dependence in GSL
  water-surface-elevation data so that we can apply
  FOSP to it.

78
Two recent papers

• Rongtao Sun and YangQuan Chen and Qianru Li.
  Modeling and Prediction of Great Salt Lake
  Elevation Time Series Based on ARFIMA.
  DETC2007-34905 in Proc. of the ASME Design
  Engineering Technical Conferences, Sept. 4-7,
  2007 Las Vegas, NE, USA, 3rd ASME Symposium on
  Fractional Derivatives and Their Applications
  (FDTA'07), part of the 6th ASME International
  Conference on Multibody Systems, Nonlinear
  Dynamics, and Control (MSNDC). 11 pages.
• Qianru Li and Christophe Tricaud and YangQuan
  Chen. Great Salk Lake Level Forecasting Using
  FIGARCH Model DETC2007-34909 ibid. 10 pages
  (Fractional Integral
• Generalized Auto-Regressive Conditional
  Heteroskedasticity)

79
Urgency-4 In between time and frequency
WIGNER-VILLE DISTRIBUTION OF A SIMPLE CHIRP
SIGNAL
http//www.wavelet.org/tutorial/tf.htm
80
In between time and frequency
WVD OF TWO SIGNALS CONSISTING OF HARMONIC
COMPONENTS OF FINITE DURATION
http//www.wavelet.org/tutorial/tf.htm
81
Optimal filtering in fractional Fourier domain
Slide credit HALDUN M. OZAKTAS
82
Optimal filtering in fractional order Fourier
domain
Slide credit HALDUN M. OZAKTAS
83
More at

• IFAC FDA 2008 Plenary Lecture VII
• On Best Fractional Derivative to Be Applied in
  Fractional Modelling The Fractional Fourier
  Transform by J. J. Trujillo
• 1100-1140  Plenary Lecture VII- Conference Hall
  
• Thursday 6th, 2008 

84
HRV, LRD, FOSP, Fractal Physiology and Feedback
Control of Emotion

• YangQuan Chen
• Center for Self-Organizing and Intelligent
  Systems (CSOIS),
• Dept. of Electrical and Computer Engineering
• Utah State University
• E yqchen_at_ieee.org T 1(435)797-0148 F
  1(435)797-3054

Wednesday, March 26, 2008 1200-1230 PM SIG
FOC Weekly Meeting http//mechatronics.ece.usu.edu
/foc/yan.li/ See also http//fractionalcalculus.g
ooglepages.com/
85
Main references

• 1 Where Medicine Went Wrong Rediscovering the
  Path to Complexity Bruce J. West, World
  Scientific, 2006. ISBN 9812568832, 337 pages,
  US42.
• http//ieeexplore.ieee.org/iel5/51/4268305/0427229
  0.pdf?isnumber4268305prodJNLarnumber4272290a
  rSt10ared12arAuthorMagin2CR.L.
• 2 Phyllis K. Stein, Ph.D., Anand Reddy, M.D.
  Non-Linear Heart Rate Variability and Risk
  Stratification in Cardiovascular Disease Indian
  Pacing and Electrophysiology Journal (ISSN
  0972-6292), 5(3) 210-220 (2005)
• 3 Bradley M. Appelhans and Linda J. Luecken.
  Heart Rate Variability as an Index of Regulated
  Emotional Responding Review of General
  Psychology. 2006, Vol. 10, No. 3, 229240.

86
Main points 1

• West disease and pathology reflect a general
  loss of complexity. 1
• Wests book shows strong evidence for the
  presence of fractal dynamics underlying the
  complexity observed in physiological systems,
  dynamics measured by allometric order parameters
  derived from fractal time series analysis.
• Fractal physiology - heart rate, breathing rate
  and gait
• Multifractal cases

87
Main points 3

• Appelhans and Luecken 3 Heart rate variability
  (HRV) analysis is emerging as an objective
  measure of regulated emotional responding
  (generating emotional responses of appropriate
  timing and magnitude). The study of individual
  differences in emotional responding can provide
  considerable insight into interpersonal dynamics
  and the etiology of psychopathology.
• Conclusion - HRV is an accessible research tool
  that can increase the understanding of emotion in
  social and psychopathological processes.
• Note No mentioning of fractal or fractional

88
3
89
3
90
Log-log plot of the HRV power spectrum over 24
hours. The region between 0.01 and 0.0001 Hz is
used to calculate power law slope. (x-axis
frequency Hz)
2
91
short-term fractal scaling exponent (1995)
Examples of the power law slope in a) a patient
with cardiac disease. And b) a healthy person.
92
Whats next

• FOSP Instrument with fractal indicator
• Emotionometer??
• Dynamic data-driven computational psychology?
• Exploration of multifractal nature
• Feedback control of emotion?
• Motion picture classification?
• Elder care.

93
Outline

• Introduction Me and CSOIS Research Strength
• Fractional Order Signal Processing An Enticing
  Example.
• Fractional Order Signal Processing What Are The
  Techniques There?
• Applications Why Urgent?
• Concluding Remarks

94
Conclusions

• 7/13/1865 - Go west, young man. Go West and grow
  up with the country. Horace Greeley
  (1811-1872)
• Go Fractional. Its urgent! YangQuan Chen

http//upload.wikimedia.org/wikipedia/commons/1/12
/American_progress.JPG
95
Rule of thumb

• Self-similar
• Scale-free/Scale-invariant
• Power law
• Long range dependence (LRD)
• 1/f a noise

• Porous media
• Particulate
• Granular
• Lossy
• Anomaly
• Disorder
• Soil, tissue, electrodes, bio, nano, network,
  transport, diffusion, soft matters

96
Acknowledgments

• IFAC FDA08. In particular, Prof. Dr. Dumitru
  Baleanu.
• SDL Skunkworks Grant (2005-2006). FOSP for
  bioelectrochemical sensors. (PI YangQuan Chen,
  co-PI Anhong Zhou, 2005-2007)
• NIH R15. Whole Cell Biosensing Bacterial
  Adhesion/Chemotaxis. (PI Anhong Zhou, co-PIs
  YangQuan Chen et al, 2006-2009)
• Prof. Anhong Zhou, BIE Dept. of USU
• Prof. Blas Vinagre, Univ. of Extremadura, Spain
• Prof. Igor Podlubny, Tech. Univ. of Kosice,
  Slovakia
• Prof. Richard Magin, BioE Dept. of UIChicago.
• Prof. Wen Chen of Hohai University, China
• Dr. Bruce West, US Army Research Office.
• Prof. Hyo-Sung Ahn, Gwangju Institute of
  Technology, Korea
• Past graduate students Jinsong Liang (Msc. 05)
  Rongtao Sun (MSc.07) Tripti Bhaskaran (MSc. 07)
  Varsha Bhambhani (MSc. 08) Qianru Li (Ph.D. 08,
  Econ)
• Current graduate students Christophe Tricaud,
  Yiding Han, Shayok Mukhopadhyay, Hu Sheng, Cal
  Coopmans, Haiyang Chao, Ying Luo, Yongshun Jin,
  Di Long, Les Mounteer and Victoria Kmetzsch .