DTIC ADA249623: Maximum Likelihood Estimation of Fractional Brownian Motion and Markov Noise Parameters

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Maximum Likelihood Estimation of Fractional Brownian
Motion and Markov Noise Parameters
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Matthew E. Skeen, 2d Lt
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NSN 7540-01-280-5500 Stardard Porm 298 (Rev 2-89)
Title: Maximum Likelihood Estimation of Fractional Brownian Motion and
Markov Noise Parameters
Author: Matthew Edward Skeen
Military Rank: Second Lieutenant
Service Branch: Air Force
Date: 1991
Number of Pages: 138
Degree Awarded: Master of Science in Aeronautics and Astronautics
Institution: Massachusettes Institute of Technology
ABSTRACT
Maximum likelihood estimation and power spectral density analysis are
developed as tools for the analysis of stochastic processes. Some useful results from
the theory of Markov stochastic processes are then presented followed by the
introduction of fractional Brownian motion and fractional Gaussian noise as non-
Markov models for systems with power spectral density proportional to f0, where
-3 < P < -1 and -1 < P < 1 over all frequencies.
Maximum likelihood system identification is applied to estimating the
unknown parameters in a Markov model which approximates fractional Brownian
motion. The algorithm runs a Kalman filter on states and a maximum likelihood
estimator on parameters. Results are presented from estimating trend, white noise,
random walk, and exponentially correlated noise parameters from fits to simulated
and real test data.
Maximum likelihood estimation is applied to the batch estimation of
parameters in the non-Markov fractional Brownian motion model. New in this
thesis is the use of partial derivatives to minimize the resulting likelihood function,
and the capability to estimate the unknown parameters of additional trend and
Markov noise processes. Results are presented from fits to computer simulated
sample paths.
PRIMARY REFERENCES
Lundahl, Torbjijrn, William J. Ohley, Steven M. Kay and Robert Siffert, "Fractional
Brownian Motion: A Maximum Likelihood Estimator and Its Application to
Image Texture," IEEE Transactions on Medical Imaging, Vol. MI-5, No. 3,
September 1986, pp. 152-161.
Mandelbrot, Benoit B. and John W. Van Ness, "Fractional Brownian Motions,
Fractional Noises and Applications," SIAM Review, Vol. 10, 1969, pp. 422-437.
Sandel, Nils R. Jr. and Khaled I. Yared, "Maximum Likelihood Identification of State
Space Models for Linear Dynamic Systems," M.I.T. Electronic Systems Laboratory
92-11977
Report ESL-R-814, Cambridge, MA, April 1978.
91r 5 01 010
MAXIMUM LIKELIHOOD ESTIMATION OF FRACTIONAL BROWNIAN
MOTION AND MARKOV NOISE PARAMETERS
by 4'f:
MATTHEW E. SKEEN
B.S., Astronautical Engineering, United States Air Force Academy
Colorado Springs, Colorado (1990)
Ace w34o& For
Submitted to the Department of Aeronautics and Astronautics NT IS Qgi"
in Partial Fulfillment of the Requirements for the Degree of 971 tA-
MASTER OF SCIENCE in AERONAUTICS AND ASTRONAUTICS ttat
at the
__
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 1992 -VB±1 ,
Diet ropoojal
© Matthew E. Skeen, 1991. All Rights Reserved
The author hereby grants to M.I.T. permission to reproduce and
to distribute copies of this thesis document in whole or in part.
Signature of Author
Department of Aeronautics and Astronautics
20 December 1991
Approved by
Dr. Michael E. Ash
Principal Member Technical Staff, Charles Stark Draper Laboratory
Thesis Supervisor
Certified by
Professor Wallace E. Vander Velde
Department of Aeronautics and Astronautics
Thesis Advisor
Accepted by
Professor Harold Y. Wachman
Chairman, Department Graduate Committee
£
2
MAxM ULIKEuHOOD ESTIMATION OF FRACTIONAL
BROWNIAN MOTION AND MARKOV NOISE PARAMETERS
MATTHEW EDWARD SKEEN
Submitted to the Department of Aeronautics and Astronautics
on December 20, 1991
in partial fulfillment of the requirements for the degree of
Master of Science in Aeronautics and Astronautics
ABSTRACT
Maximum likelihood estimation and power spectral density analysis
are developed as tools for the analysis of stochastic processes. Some useful
results from the theory of Markov stochastic processes are then presented
followed by the introduction of fractional Brownian motion and fractional
Gaussian noise as non-Markov models for systems with power spectral
density proportional to f0, where -3 < 3 <-1 and -1 < 3 < 1 over all
frequencies.
Maximum likelihood system identification is applied to estimating the
unknown parameters in a Markov model which approximates fractional
Brownian motion. The algorithm runs a Kalman filter on states and a
maximum likelihood estimator on parameters. Results are presented from
estimating trend, white noise, random walk, and exponentially correlated
noise parameters from fits to simulated and real test data.
Maximum likelihood estimation is applied to the batch estimation of
parameters in the non-Markov fractional Brownian motion model. New in
this thesis is the use of partial derivatives to minimize the resulting
likelihood function, and the capability to estimate the unknown parameters
of additional trend and Markov noise processes. Results are presented from
fits to computer simulated sample paths.
Thesis Supervisor: Dr. Michael E. Ash
Principal Member Technical Staff, C. S. Draper
Laboratory
Thesis Advisor: Professor Wallace E. Vander Velde
Department of Aeronautics and Astronautics
3
4
Acknowledgements
My sincerest gratitude is extended to my thesis supervisor, Dr. Michael
E. Ash, not only for his enthusiastic assistance in my research, but also for his
interest in my overall education. I would like to thank my thesis advisor,
Prof. Wallace E. Vander Velde, for his guidance in the preparation of this
document and for the care with which he reviewed the drafts.
I thank my friends at Draper Laboratory for all the help that they have
given me and for making my time at MIT enjoyable. I give special thanks to
Tanya for her love, support, and perseverance over the last 17 months.
5
This thesis was researched and written at the Charles Stark Draper
Laboratory under Corporate Sponsored Research Project C68.
Publication of this thesis does not constitute approval by the laboratory
of the findings or conclusions contained herein. It is published for the
exchange and stimulation of ideas.
I hereby assign my copyright of this thesis to the Charles Stark Draper
Laboratory, Inc., of Cambridge, Massachusetts.
Matthew E. Skeen
Permission is hereby granted by the Charles Stark Draper Laboratory,
Inc. to the Massachusetts Institute of Technology to reproduce and to
distribute copies of this thesis document in whole or in part.
6
Table of Contents
Ab stract ..................................................................................................................... 3
Acknowledgements .............................................................................................. 5
Table of Contents .................................................................................................. 7
List of Illustrations ................................................................................................. 13
List of T ables ........................................................................................................... . 13
N otation ................................................................................................................... 15
Chapter 1 Introduction and Summary ............................................................. 19
1.1 Modeling Noise Processes and Estimating Noise Parameters.. 19
1.2 Background Material .......................................................................... 21
1.2.1 Maximum Likelihood Estimation .................... 21
1.2.2 Power Spectral Density Analysis ............................................. 22
1.2.3 Markov Noise Processes ............................................................ 22
1.2.4 Fractional Brownian Motion ................................................... 24
1.3 Summary of Maximum Likelihood Fits to Real and
Simulated Data ......................................................................... 24
1.3.1 Maximum Likelihood System Identification for Markov
Pr ocesses .......................................2.4.............................................
1.3.2 Maximum Likelihood Estimation of Fractional
Brownian and Other Parameters .................... 25
7
Chapter 2 Maximum Likelihood Estimation ................................................. 27
2.1 Parameter Estimation ........................................................................ 27
2.2 Likelihood Function and Negative Log-Likelihood Function 27
2.3 Fisher Information Matrix ................................................................ 28
2.4 Cramer-Rao Lower Bound ................................................................ 29
2.5 Properties of Maximum Likelihood Estimates .............. 31
2.6 Iterative Determination of Maximum Likelihood Estimates... 32
2.7 Relation to Least Squares Estimates ................................................ 34
2.8 Relation to Other Estimators ............................................................ 36
Chapter 3 Power Spectral Density Analysis ..................................................... 39
3.1 Stochastic Processes ............................................................................ 39
3.2 Autocorrelation Function of a Stochastic Process ....................... 40
3.3 Estimation of the Autocorrelation Function ................ 41
3.4 Power Spectral Density of a Stationary Stochastic Process ......... 43
3.5 Estimation of the Power Spectral Density .................. 43
3.5.1 Continuous Data ........................................................................ 43
3.5.2 Interpretation of the PSD as a Power Spectrum ................... 45
3.5.3 D iscrete D ata ................................................................................ . 47
3.6 Frequency Averaging ........................................................................ 49
Chapter 4 Markov Stochastic Processes ........................................................... 53
4.1 Martingale and Markov Processes .................................................. 53
4.2 Weiner Brownian Motion Process ................................................. 54
4.3 White Noise PSD Slope ..................................................................... 56
4.4 PSD Slope of Random Walk, Trend, and Quantization
N oise .......................................................................................... . 57
4.5 Ito Stochastic Integral ........................................................................ 61
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