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13.10. Behavior of characteristic roots of a nonlinear threshold AR model 391

13.11. Time history of a nonlinear threshold model that has charac teristic

roots that behave as Fig. 13.10 392

TABLES

Page

2.1 . Aliasing frequencies 22

2.2. Effect of spectral windows 43

2.3. Statistical effects of the change of M 45

2.4. Coefficients otj for various windows 46

3.1. Summary of seaways and particulars of measurement

and analysis for test runs 80

5.1. Results of estimation by model fitting â€” [ I ] 158

5.2. Results of estimation by model fitting â€” [II] 180

Al.l. 600 observations from a pure random process X, = e, 252

A1.2. 600 observations from an AR(1) process X, - 0.5 X t _ { =â‚¬, 252

A1.3. 600 observations from an AR(2) process

X,-0.5X t _i+0.8X,_ 2 = e, 254

A1.4. 600 observations from an ARMA(2.1) process (1)

X,-0.3 X t _ x + 0.4X,_ 2 = e, - 0.7e,_j 254

A1.5. 600 observations from an ARMA(2.1) process (2)

X,-0.3 X t _ x + 0AX t _ 2 = â‚¬, + 0.7e,_, 256

A1.6. 600 observations from an MA(2) process

X, = e, + 0.2e,_, + 0.8e,_ 2 256

A1.7. 600 observations from an MA(1) process (1)

X t = e, + 0.7e,_ j 258

A1.8. 600 observations from an MA(1) process (2)

X, = e, - 0.7e,_ , 258

A 1.9. 600 observations from an ARMA(2.2) process

X,-0.5 X,_j + 0.8 X,_ 2 = e, + 0.2e,_! + 0.8e ; _ 2 260

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FOREWORD

For several reasons, the publication of this report was postponed for several years.

Its contents were summarized at the time of this author's oral presentation at DTRC in

July 1985, reflecting the state-of-the-art up to 1984. The manuscript of the written

version was completed in August 1987 and is now to be published five years later in

1992. During these years the state-of-the-art has made considerable progress, and this

author finds the "review" to be insufficient, especially in Part III because of the recent

works of several researchers. In order to update the report this author has added a

"Supplement of References," (Refs. 101-124) listing publications that have appeared

since 1984, together with some other publications that he had failed to refer to in the

original manuscript.

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PREFACE

The David W. Taylor Lectures were initiated as a living memorial to our founder in

recognition of his many contributions to naval architecture and naval hydrodynamics.

Admiral Taylor was a pioneer in the use of hydrodynamic theory and mathematics for the

solution of naval problems. The system of mathematical lines developed by Taylor was

used to develop many ship designs for the Navy long before the computer was invented.

He founded and directed the Experimental Model Basin; perhaps most important of all.

he established a tradition of applied scientific research at the "Model Basin'* which has

been carefully nurtured through the decades and which we treasure and protect today. In

the spirit of this tradition, we invite an eminent scientist in a field closely related to the

Centers work to spend a few weeks with us, to consult with and advise our working staff,

and to give a series of lectures on subjects of current interest.

Our tenth lecturer in this series is Dr. Yasufumi Yamanouchi who is currently

Technical Advisor to Mitsui Engineering and Shipbuilding Co.. Ltd. Dr. Yamanouchi

graduated from the Tokyo University in 1943 and received his Dr. Eng. from there in

1962. For many years (1946-73), he served with the Governmental Research Institute on

Ship Technology, that changed its name from Railway Technical Research Institute

(1946-50), then to Transportation Technical Research Institute (1950-63), and finally to

Ship Research Institute, Ministry of Transport. He was Head of the Ship Performance

Division (1962-63) and Head of the Ship Dynamics Division (1963-69), then Deputy

General Director (1969-72) and finally Director-General of the Ship Research Institute

(1972-73), Ministry of Transport in Tokyo, Japan. He joined Mitsui Engineering and

Shipbuilding Co., Ltd. as Senior Deputy Director, advancing to Technical Director upon

his retirement in 1983. He contributed to the construction and organization of a new

research laboratory, and was the first General Manager of the Akishima Laboratory of

Mitsui from 1975 to his retirement in 1983. He was Professor at the College of Science

and Technology (1982-89) of Nihon University in Tokyo, Japan.

Dr. Yamanouchi has had a long and distinguished scientific career, with pioneering

publications on ship dynamics, ocean waves and stochastic processes. He is no stranger

to the United States, having done research at the Davidson Laboratory of Stevens Insti-

tute of Technology from 1957-60 and having served as Visiting Professor at the U.S.

Naval Academy during 1984-85. We are most honored that he agreed to be a David W.

Taylor Lecturer.

ABSTRACT

SDectnt^l, 1 ' ** f neraJ L pr0Cedures of ^ conventional method of correlation or

specmim analysts of a random process (nonparametric method) are reviewed stressing

Â« â„¢ ^TuiT r ts - A f r suggestions for ~* ssss

The mode, mnâ„¢ f . CharaCtenstlcs of AR, MA, and ARMA models are discussed.

I He model-fitting techmque supported by AIC criteria is introduced, with the examnles

m^Sl t0 ^ SeakeepiDg ^ * Pan m ' Â«* S * tistical treatmen s ofno"r e Ses

CHAPTER 1

INTRODUCTION

1.1 PROLOGUE

In the summer of 1951, a group of research naval architects who were scheduled to

be on board the cargo ship Nissei-Maru cruising the Pacific from Japan to the United

States was discussing the design of a system for measuring, recording, and analyzing ac-

tual ship performance data. Such a system was needed for the first large-scale

cooperative test, 1 organized by the Japan Towing Tank Committee (J'l'l'C) of the Society

of Naval Architects of Japan (SNAJ) and scheduled to start at the end of the year, to pro-

vide a basis for the post-World War II study of hydrodynamic ship performance in all the

laboratories and universities in Japan.

To fmd the relationships between ship's propeller revolutions, shaft horsepower,

modes of motion, rudder angle, and so on, we decided to record, simultaneously and as

accurately as possible, as many elements of ship performance as possible. Such measure-

ments would provide an overview of the response of the ship. Many practical limitations,

such as budget constraints, had to be considered, however. In addition we didn't know the

real limits of the accuracy of these kinds of measurements, nor did we know how to

choose the proper duration of observation times. After lengthy discussion, it was decided

to record each type of measurement for three minutes.

Very few naval architects at that time had a good knowledge of probability or of

statistics. We did not know how to analyze the data taken at sea, and we did not know

how to estimate the ensemble characteristics of performance from a single sample obser-

vation. The simultaneous records of the averages of the 3-minute responses gave us

valuable knowledge of actual sea performance, but we realized that simple averages

sometimes cover or conceal important information. For ship oscillations, of course, mean

values are not significant, but this author could not be satisfied with merely noting the

average frequency and mean amplitudes.

During the test, the author 2 was responsible for measuring the ship's relative speed

and developed a new type of ship speed meter, based on an idea of Dr. Shiha. This new

meter measured the frequency of Karman vortices produced behind a triangular cylinder

towed by the ship and gave instant indications of speed variations. Although the towing

point was selected to minimize the effects of pitching, heaving, and rolling, these motions

did affect the speed of the towed body. The objective was to eliminate the effects of all

motions except surging and to derive from these records, by some method, the real varia-

tions in the ship's speed relative to the water. The author later found that an advanced

technique of multiple input analysis would have been useful.

After the test at sea, we continued to look for methods for analyzing the data and

found a stochastic process analysis used in weather forecasting to predict temperature,

humidity, precipitation, and so on. The author started at once to study time series analysis

and tried first several kinds of periodogram analysis and then correlogram and spectrum

analysis. 3 Because of the poor communications in the engineering and scientific fields in

Japan at that time (even five years after the war) we had very little information about the

outside world until 1954 and were unaware of the pioneering work of Dr. M. St. Denis

and Prof. W. J. Pierson Jr. 4 in 1953. We had to start on our own independent of the

achievements in the USA.

1 -2 PURSUING THE IMPROVEMENT OF COHERENCY FUNCTIONS

After the bases for the mathematical and statistical theories were established by N

Wiener, 5 -* J. L. Doob, 7 C. E. Shannon, 8 S. O. Rice, 9 and others, mostly in the 1940's the

stochastic analysis techniques were applied in many scientific and engineering fields

besides communications and control engineering. These techniques were adopted rather

early in the analysis of ocean waves and of a ship's response at sea. This method has been

pretty well formulated by the efforts of oceanographers like G. Neumann 10 W J Pierson

Jr., M. S. Longuet-Higgins,^ D. E. Cartwright and M. S. Longuet-Higgins, 13 and by

the pioneering work of M. St. Denis and W J. Pierson, Jr., 4 succeeded by E V Lewis "

and J. F. Dalzell and Y. Yamanouchi, 15 and is now rather popular with us. We now know

that, m practical applications and when applying certain theories, a few statistical consid-

erations are necessary in the numerical computations in order to get reliable results.

In Part I, the author tried to show the problems encountered in sample computa-

tions, in the so-called correlation method that are also closely related to the basis for

model fitting techniques (that is, the parametric method treated in Part H) and reviews the

conventional nonparametric method. The author did not intend to go into detail about the

analysis technique.

The coherency function, if properly calculated, is a good index of the extent to

which spectrum analysis, as a linear process, is valid for application to a stochastic pro-

cess. A few results of this author's efforts in this field, presented later in Part I are related

to improvement of the techniques for obtaining a good estimate of coherencies.

1 .3 TIME DOMAIN CHARACTERISTICS

Time series analysis is sometimes called spectrum analysis, thus showing that esti-

mation of rehable spectral functions is very important in the analysis of time series.

a Â«J!*T* faactiam A m surel y PÂ° werful ^tions which provide good information on

a stochastic process and also resolve the tangled relations of convoluted types in the time

domain. However, because of this fact, and also partly because (auto) correlation did not

appear m an important way, in St. Denis and Pierson's pioneering paper, 4 rather little at-

tention has been paid by naval architects to the time domain relations over the past few

decades. From his first involvement in this study, this author has thought that tine do-

mam functions deserved more attention and has made some efforts along this line.

Of course spectral functions and correlation functions are actually the same function

expressed in different ways. Sometimes, however, in sample computations, applying the

ume domain expression helps us understand the characteristics of the stochastic process

better because we are accustomed to expressing the physical process by differential equa-

tions that are expressions in the time domain. The time domain expression of stochastic

processes helps us in the analysis to use already known characteristics of the process

Moreover, the correlation window operation in the sample computation is just a multiply-

ing operation, whereas the spectral window operation is now an entangled convolution

operation that is more complex in real sample computations. In sample computations

consideration of the windows is very important and also a troublesome problem to get

reliable results.

Parametric estimation of the spectral function, explained in Part n, is a method of

fitting a certain statistical model to the process to be analyzed and then estimating the

parameters of that model. This method is based on time domain characteristics, and the

model fitted is closely related to the equations of motion of the process itself. The method

represents a different approach to spectrum analysis and, this author believes, that is a

promising one, supplementing the conventional nonparametric approach, treated in Part I.

So, in Part II, several types of discrete parameter models will be introduced in detail, and

then the results of application of the model fitting method to the analysis of seakeeping

data will be shown. By this method, the actual response system, in which the output is fed

back to the input to some extent, can also be tackled. This kind of system has been hard to

be analyzed by the conventional method.

1 .4 TREATMENT OF NONLINEARITIES

One of the reasons for low coherencies in linear spectral analysis is the existence of

nonlinearities in the response or in the input. So in Part HI, the treatment of nonlinearities

in process analysis is reviewed. One of the greatest achievements in this field for the anal-

ysis of seakeeping data is due to John F. Dalzell 16 * 17 in the application of polyspectra.

This application will not be described in detail, except the basic idea, of this treatment but

another review of the treatment of nonlinearities will be given, to make clear the mutual

relationships of these several different approaches to the problem. In addition, the treat-

ment of nonlinearity in response characteristics during one trial has been introduced in

Section 3.4 in Part I, as an example of multi-input single-output analysis.

1.5 SCOPE OF STUDIES

In statistical studies on seakeeping, there are roughly two kinds of applications. One

is based on the invariant characteristics of a ship itself, and its behavior is studied statisti-

cally, assuming the excitement from the environment is also stationary. The other

involves the macroscopic probabilistic distributions of seakeeping behavior, assuming a

variety of changes in environmental conditions. The former is sometimes called short-

term statistics, and the latter, long-term statistics. They are, of course, closely related, and

the short-term statistics are usually used as the basis for studying the long-term statistics.

Here mostly short-term statistics will be treated.

PARTI

A REVIEW OF SPECTRAL ANALYSIS THROUGH PERIODOGRAM

(NONPARAMETRIC SPECTRAL ANALYSIS)

To clarify the problems and difficulties encountered in sample computations and

make them the basis for further discussions, the rough scheme of the techniques of spec-

trum analysis (through periodograms, the popular nonparametric method) will be

reviewed first in Chapter 2. Then some ideas proposed by this author for solving these

difficulties will be summarized in Chapter 3, Parti. Many text books}^ especially the

comprehensive one by Priestley, 23 were used as references in Chapter 2.

CHAPTER 2

BASIC PROCEDURE OF THE SPECTRAL ANALYSIS AND

THE PROBLEM OF SAMPLE COMPUTATIONS

2.1 RANDOM PROCESS AND ITS CHARACTERISTICS

and thfnrn^HTnT COntinUOUS P rocess on ' is ex P r ^ by X(t), its realization as x(t),

and toe probability density distribution function related to this process as Pt (x) The gen-

tln^T Pr0CeSS ^ \ " S reallZati0D iS *' "* Pr0babmt y distribution density function

is pt(x). Then, as expected values,

00

mean [X(t)} = E [X(t)] = I x(t)p t (x)dx = fi{t)

(2.1)

00

var. [X(t)] = E [{(Â») - M of] = J [x(t) - M f)] 2 p t (x)dx = o\t). (2.2)

â€” 00

Usually, the probability density distribution function p t (x) is a function of t ; accord-

ingly, the meaner) and the variance a\t) are also functions of time.

Joint probability density distribution functions, p^ _ tn (x u x 2 ,x 3 , . ..*â€ž) exist

for all n,n = 1,2,. . . â€ž,(â€ž - Â») and all p tl ( Xl ),p, 2 (x 2 ), ...â– p,^^), . . . . .

Pt,i 2h (xuX2,x 3 ), . . . ; .... are defined as their marginal functions.

Theoretically, if all the joint distribution functions of all orders

Pt,(.Xi),p t2 (x 2 ), . â– .

Pt,t 2 (xi,x 2 ),p h t 3 (X2,X3), ...-

Pv 2 t 3 (.X h X 2 ,X3), ....

Ptfoti â– â– â– tn(X\,X2,Xs, . . . Xâ€ž),

are known, the probability structure of X, is completely specified.

2.1.1 Completely Stationary

If

Pt,t 2 â– â– -/ B (*l,*2, â€¢ â€¢ â€¢ xâ€ž)=p h+ k i t 2 +k l - â– â– t n +k i Axi,x 2 , .... x n ), (2.3)

for any t\,t2, . . . . tâ€ž and any k, this process is completely stationary.

2.1.2 Stationary Up to Order m

In this case, the joint moment up to order m should be the same.

E[{X(r 1 )} m '{X(r 2 )} m2 [X(t n )} m ']

= E[{X(t l + k)} m >{X(t2 + k)} m > {X(t n + k)} m ^, (2.4)

for all positive integers m\, mi, . . . m,

where

mi + m 2 + . . . . m n < m.

2.13 Stationary Up to Order 2

Especially when the order is m = 2, the process is called weakly stationary. When

the probability density distribution functions are Gaussian, they are completely deter-

mined by the means, variances, and covariances of two variables, and accordingly they

are completely stationary.

2.1.4 Ergodicity

When the ensemble of the averages across the processes converge to the corre-

sponding time averages along the process over period N (when N tends toward infinity

and the mean square is consistent), the process is called ergodic. The ergodicity is a more

strict condition than the stationality as is shown in Fig. 2.1.

EVOLUTIONARY

STATIONARY UP TO 1ST ORDER

STATIONARY UP TO 2ND ORDER

ERGODIC UP TO 2ND ORDER

COMPLETELY STATIONARY

AND ERGODIC

For example,

Fig. 2.1. Stationarity and ergodicity.

R(r) = E [X(t)-X(t + r)]

x(t)x(t + x)p{x t , x t+z )dx4x t +r

J -a J -co

1 f 7

= lim-

t-* 00 1 Jo

x(t)x(t + x)dt.

(2.5)

2.15 Summary of Gaussian (Normal) Probability Distribution Functions

For convenience, the Gaussian probability distribution functions for various num-

bers of variables are summarized here. For a single variable,

P(x) =

1

Qm&* eXP

Normalized by z = (x -fi x )/o x .

1 (x-fi x f

2 a\

(2.6)

1 _Zi

P(Z)= l2^ e ' 2 -

(2.7)

For two variables X and Y

2nra;ay(l-e-)

i I (x-^) 2 2e0c-fixy-pj (y-n>) 2

2 (i-e 2 )o? (l-eV^ (i-eVr

^A^ CXP

-lOr./O 2 - ^7^(x-/fcKy-/Â«r) + "T Cv^

^^ CXP

- i {^(x-^) 2 - 2p xY (x^fi x )(y-My) + y3yyO'-y"y) 2 )

(2.8)

Here,

A =

= o 2 o 2 ,-Q 2 o 2 qj

= oWy{l-Q 2 ),

- â€ž2

yxx = E[X X] = ai , yxy = E[X â– Y] = Q ojj y

- â€ž2

y^ = E[Y- Y] = a* , y yx = E[Y-X] = QOyO x .

(2.10)

w -

>xx Hxy

lyx Pyy

1

ty]" 1

l

7yy >V

'Yxy Yxx

OxOfo-Q 2 )

Oy -QOjOy

-QOjOy o\

(2.11)

For n variables X\,X^, Xâ€ž,

p(xi,x 2 , Xâ€ž) =

Qjtf^b}! 2

exp

1 n n

Â»=1^1

, (2.12)

where

A = |x| =det(2

x= fa

(2.13)

and a tJ is an element of inverse matrix, so that

P(xi,x 2 , x n ) =

(2ji) n / 2 & 1 / 2

exp

-\[{x-n)'?rHx-n))

(2.14)

2.2 PROPERTIES REQUIRED FOR ESTIMATOR

The following properties are required for the estimator of some statistical value.

22.1 Unbiasedness

As the number of samples n tends toward infinity, for the estimator 6 of real 0,

bias(0) = Â£(0) -0â€”0

(2.15)

222 High Relative Efficiency

If we suppose that both estimators 0â€ž0 2 are unbiased, then the higher the

rel. effic. = var. (0\)/var. (0 2 ) (2.16)

is, the better estimator 0, is than 8 2 .

22 J Small Mean Square Error

Mean square error is defined as

M 2 (0) = Â£[(0 - 0) 2 ]

B {0 - E(0)] + JÂ£(0) - 0}

{0 - Â£(0)j 2 + {Â£(0) - 0J 2 + 2JÂ£(0) - 0jÂ£[0 - Â£(0)]

= var. (0) + b\6), (2.17)

which should be small. WhenM 2 (0i) < M 2 (6 2 ), we adopt 6\ as better than0 2 â€¢

22.4 Consistency

As the number of the samples n tends toward infinity, var. (9) -* 0, and bias

b 2 {6) -Â» must be satisfied. Then the mean square error M 2 (6) -â€” also stands and is

called the "mean square consistency."

225 Sufficiency

The estimator 6 must contain all the information Xi,X 2 , . . . . Xâ€ž in the sample,

relevant to the estimator of 6

6=6 (X lt X 2 ,

X n ).

2.3 AUTOCOVARIANCE FUNCTION AND ITS ESTIMATES

When the process X(t) is stationary up to an order of 2, the covariance function

cov. \X(t) X(t + r)] = E [{X(t)-fi}[X(t + T)-fi}] = R(t) (2.18)

is a function oft only. Then,e(r) = R(r)/R(0) is called an autocovariance coefficient,

and

R(0) = E [{X(t)-fi} 2 ] = var. [X(t)] = a 2 . (2.19)

When the process is real valued, R(-t) = R(r) as in Fig. 2.2, and when the process is

complex valued, R(-r) = R*(t). This function R(r) is a measure of similarity in a sense,

and is also a measure of the memory of the process. R(r) plays a big role later, in the

parametric analysis in Part II, in identifying the statistical model that will fit the process.

Mean/*, variance a 2 , R(r), and g(r) are constant by t.

Fig. 2.2. Autocovariance function.

10

23.1 Estimates of R(r)

For the discrete parameter process*, ,

R(r) =E [(X r -fi)(X t+r -[i t+r )]

= E[X r X t+r ], (2.20)

when [i, = pi t+r = E[X t ] = 0.

23.2 Unbiased Estimate

Instead of averaging the ensemble, take the time average as

, N-\r\

RÂ°(r) =â€” - 2 (X,-X) (X t+r -X) (2.21)

^ ''' *=l

where r = 0, Â± 1, Â± 2 Â± (N-l).

If we ignore the effect of estimating^ by X, then RÂ°(r) is an unbiased estimate of

R(r), because

, N-M

E [RÂ°(r)] = -^-j- X SCCfr-^CXi+r-At)]

N-\r\

1 â„¢ iv-lrl

= tt-z; I *(r) = tt-t m = *<r).

tf-lrl Â£f N-\r\

(2.22)

2 J. 5 Biased Estimate

On the contrary,

, N-Irl

*< r ) = 77 X (^-^)(^ + w- / ") (2.23)

is a biased estimate, because

1 AMH 1 / I I \

E &W = # X *W = ^ CW-IH) *(r) = il-j-\ R (r)

\r\

= R(r)-â€”R(r), (2.24)

11

biased by (\r\/N) R(r). WhenN -Â» oo , E[R(r)] â€” R(r), and is therefore asymptotically

unbiased.

By statistical mathematics, we can get

var. [RÂ°(r)] -* 0(l/(N-\r\)) (2.25)

var. [R(r)] -* 0(1 /N). (2-26)

1. When r is small relative to N, the difference between RÂ°(r) and R(r) is

small, and the bias of [R(r)] is also small.

2. When r becomes large relative to N and approaches N-l,

bias [R(r)] -* R(r). However, when r â€” Â» , R(r) -* 0. Therefore, when

N is very large, the bias remains small at all r.

3. When r â€” (N-l),

var. [RÂ°(r)] - 0(1) (2.27)

var. [R(r)] - 0(1/AO- ( 2 - 28 )

Therefore, the tail of correlation RÂ°(r) shows a wild and erratic behavior. Also, from

statistical mathematics, cov. [i?(r) R(r + s)] was computed and fairly high correlations

between neighboring points were found when s is small. When r tends toward infinity,

/?(r) -* 0. It can be concluded here that R(r) will be less damped than R(r) and will not

decay as quickly asi?(r).

2.4 SPECTRUM ANALYSIS

The spectrum function decomposes a time varying quantity into a sum (or integral)

of sine and cosine functions.

2.4.1 Spectrum for Various Processes

a. For Deterministic Periodic Functions â€” Fourier Series

For periodic functions with period T,

12

X(t) = - a + X

(jtnt\ . [nnt

a n cos^â€” j + bâ€ž sm^â€”

where

r i

1 f _. . Tint

a " = t I (r) cos T~ *

2

x

2

t 1 f _,. . ;rrc? ,

t ] (f) sm T~

(2.29)

(2.30)

The aâ€ž and 6â€ž functions are called Euler-Fourier coefficients. Equation 2.30 is

based on the orthogonality of cos â€” sin â€”

T ' T "

For the existence of a n , b n , and for the convergence of the series, the conditions

J \X{t)\dt < oo , J \X(t)\ 2 dt < 00

are sufficient.

Replacing a n ,b n by c = a o , c n =

the total energy [- J/2 to J/2] as

â– ^(al + b})

, gives

(2.31)

2 i-

= r ZÂ«3

(2.32)

and the total power [-T/2 to J/2] as

13

1 d,

(2.33)

ci is the contribution of the Â«th term to the total power, as in Fig. 2.3.

POWER

3 2 10 1 2 3

- T~TT TTt

Fig. 2.3. Power spectrum of periodic function.

b. For Nonperiodic Deterministic Functions â€” Fourier Integral

For nonperiodic functions with finite energy as in Fig. 2.4,

-\

X{t) = X(co) e tmt do),

(2.34)

X(fi>)

2jt J

X(t) e- l( Â°'dco.

(2.35)

Fig. 2.4. Nonperiodic function with finite energy.

14

For the existence and convergence of this expansion, conditions

13.11. Time history of a nonlinear threshold model that has charac teristic

roots that behave as Fig. 13.10 392

TABLES

Page

2.1 . Aliasing frequencies 22

2.2. Effect of spectral windows 43

2.3. Statistical effects of the change of M 45

2.4. Coefficients otj for various windows 46

3.1. Summary of seaways and particulars of measurement

and analysis for test runs 80

5.1. Results of estimation by model fitting â€” [ I ] 158

5.2. Results of estimation by model fitting â€” [II] 180

Al.l. 600 observations from a pure random process X, = e, 252

A1.2. 600 observations from an AR(1) process X, - 0.5 X t _ { =â‚¬, 252

A1.3. 600 observations from an AR(2) process

X,-0.5X t _i+0.8X,_ 2 = e, 254

A1.4. 600 observations from an ARMA(2.1) process (1)

X,-0.3 X t _ x + 0.4X,_ 2 = e, - 0.7e,_j 254

A1.5. 600 observations from an ARMA(2.1) process (2)

X,-0.3 X t _ x + 0AX t _ 2 = â‚¬, + 0.7e,_, 256

A1.6. 600 observations from an MA(2) process

X, = e, + 0.2e,_, + 0.8e,_ 2 256

A1.7. 600 observations from an MA(1) process (1)

X t = e, + 0.7e,_ j 258

A1.8. 600 observations from an MA(1) process (2)

X, = e, - 0.7e,_ , 258

A 1.9. 600 observations from an ARMA(2.2) process

X,-0.5 X,_j + 0.8 X,_ 2 = e, + 0.2e,_! + 0.8e ; _ 2 260

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FOREWORD

For several reasons, the publication of this report was postponed for several years.

Its contents were summarized at the time of this author's oral presentation at DTRC in

July 1985, reflecting the state-of-the-art up to 1984. The manuscript of the written

version was completed in August 1987 and is now to be published five years later in

1992. During these years the state-of-the-art has made considerable progress, and this

author finds the "review" to be insufficient, especially in Part III because of the recent

works of several researchers. In order to update the report this author has added a

"Supplement of References," (Refs. 101-124) listing publications that have appeared

since 1984, together with some other publications that he had failed to refer to in the

original manuscript.

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PREFACE

The David W. Taylor Lectures were initiated as a living memorial to our founder in

recognition of his many contributions to naval architecture and naval hydrodynamics.

Admiral Taylor was a pioneer in the use of hydrodynamic theory and mathematics for the

solution of naval problems. The system of mathematical lines developed by Taylor was

used to develop many ship designs for the Navy long before the computer was invented.

He founded and directed the Experimental Model Basin; perhaps most important of all.

he established a tradition of applied scientific research at the "Model Basin'* which has

been carefully nurtured through the decades and which we treasure and protect today. In

the spirit of this tradition, we invite an eminent scientist in a field closely related to the

Centers work to spend a few weeks with us, to consult with and advise our working staff,

and to give a series of lectures on subjects of current interest.

Our tenth lecturer in this series is Dr. Yasufumi Yamanouchi who is currently

Technical Advisor to Mitsui Engineering and Shipbuilding Co.. Ltd. Dr. Yamanouchi

graduated from the Tokyo University in 1943 and received his Dr. Eng. from there in

1962. For many years (1946-73), he served with the Governmental Research Institute on

Ship Technology, that changed its name from Railway Technical Research Institute

(1946-50), then to Transportation Technical Research Institute (1950-63), and finally to

Ship Research Institute, Ministry of Transport. He was Head of the Ship Performance

Division (1962-63) and Head of the Ship Dynamics Division (1963-69), then Deputy

General Director (1969-72) and finally Director-General of the Ship Research Institute

(1972-73), Ministry of Transport in Tokyo, Japan. He joined Mitsui Engineering and

Shipbuilding Co., Ltd. as Senior Deputy Director, advancing to Technical Director upon

his retirement in 1983. He contributed to the construction and organization of a new

research laboratory, and was the first General Manager of the Akishima Laboratory of

Mitsui from 1975 to his retirement in 1983. He was Professor at the College of Science

and Technology (1982-89) of Nihon University in Tokyo, Japan.

Dr. Yamanouchi has had a long and distinguished scientific career, with pioneering

publications on ship dynamics, ocean waves and stochastic processes. He is no stranger

to the United States, having done research at the Davidson Laboratory of Stevens Insti-

tute of Technology from 1957-60 and having served as Visiting Professor at the U.S.

Naval Academy during 1984-85. We are most honored that he agreed to be a David W.

Taylor Lecturer.

ABSTRACT

SDectnt^l, 1 ' ** f neraJ L pr0Cedures of ^ conventional method of correlation or

specmim analysts of a random process (nonparametric method) are reviewed stressing

Â« â„¢ ^TuiT r ts - A f r suggestions for ~* ssss

The mode, mnâ„¢ f . CharaCtenstlcs of AR, MA, and ARMA models are discussed.

I He model-fitting techmque supported by AIC criteria is introduced, with the examnles

m^Sl t0 ^ SeakeepiDg ^ * Pan m ' Â«* S * tistical treatmen s ofno"r e Ses

CHAPTER 1

INTRODUCTION

1.1 PROLOGUE

In the summer of 1951, a group of research naval architects who were scheduled to

be on board the cargo ship Nissei-Maru cruising the Pacific from Japan to the United

States was discussing the design of a system for measuring, recording, and analyzing ac-

tual ship performance data. Such a system was needed for the first large-scale

cooperative test, 1 organized by the Japan Towing Tank Committee (J'l'l'C) of the Society

of Naval Architects of Japan (SNAJ) and scheduled to start at the end of the year, to pro-

vide a basis for the post-World War II study of hydrodynamic ship performance in all the

laboratories and universities in Japan.

To fmd the relationships between ship's propeller revolutions, shaft horsepower,

modes of motion, rudder angle, and so on, we decided to record, simultaneously and as

accurately as possible, as many elements of ship performance as possible. Such measure-

ments would provide an overview of the response of the ship. Many practical limitations,

such as budget constraints, had to be considered, however. In addition we didn't know the

real limits of the accuracy of these kinds of measurements, nor did we know how to

choose the proper duration of observation times. After lengthy discussion, it was decided

to record each type of measurement for three minutes.

Very few naval architects at that time had a good knowledge of probability or of

statistics. We did not know how to analyze the data taken at sea, and we did not know

how to estimate the ensemble characteristics of performance from a single sample obser-

vation. The simultaneous records of the averages of the 3-minute responses gave us

valuable knowledge of actual sea performance, but we realized that simple averages

sometimes cover or conceal important information. For ship oscillations, of course, mean

values are not significant, but this author could not be satisfied with merely noting the

average frequency and mean amplitudes.

During the test, the author 2 was responsible for measuring the ship's relative speed

and developed a new type of ship speed meter, based on an idea of Dr. Shiha. This new

meter measured the frequency of Karman vortices produced behind a triangular cylinder

towed by the ship and gave instant indications of speed variations. Although the towing

point was selected to minimize the effects of pitching, heaving, and rolling, these motions

did affect the speed of the towed body. The objective was to eliminate the effects of all

motions except surging and to derive from these records, by some method, the real varia-

tions in the ship's speed relative to the water. The author later found that an advanced

technique of multiple input analysis would have been useful.

After the test at sea, we continued to look for methods for analyzing the data and

found a stochastic process analysis used in weather forecasting to predict temperature,

humidity, precipitation, and so on. The author started at once to study time series analysis

and tried first several kinds of periodogram analysis and then correlogram and spectrum

analysis. 3 Because of the poor communications in the engineering and scientific fields in

Japan at that time (even five years after the war) we had very little information about the

outside world until 1954 and were unaware of the pioneering work of Dr. M. St. Denis

and Prof. W. J. Pierson Jr. 4 in 1953. We had to start on our own independent of the

achievements in the USA.

1 -2 PURSUING THE IMPROVEMENT OF COHERENCY FUNCTIONS

After the bases for the mathematical and statistical theories were established by N

Wiener, 5 -* J. L. Doob, 7 C. E. Shannon, 8 S. O. Rice, 9 and others, mostly in the 1940's the

stochastic analysis techniques were applied in many scientific and engineering fields

besides communications and control engineering. These techniques were adopted rather

early in the analysis of ocean waves and of a ship's response at sea. This method has been

pretty well formulated by the efforts of oceanographers like G. Neumann 10 W J Pierson

Jr., M. S. Longuet-Higgins,^ D. E. Cartwright and M. S. Longuet-Higgins, 13 and by

the pioneering work of M. St. Denis and W J. Pierson, Jr., 4 succeeded by E V Lewis "

and J. F. Dalzell and Y. Yamanouchi, 15 and is now rather popular with us. We now know

that, m practical applications and when applying certain theories, a few statistical consid-

erations are necessary in the numerical computations in order to get reliable results.

In Part I, the author tried to show the problems encountered in sample computa-

tions, in the so-called correlation method that are also closely related to the basis for

model fitting techniques (that is, the parametric method treated in Part H) and reviews the

conventional nonparametric method. The author did not intend to go into detail about the

analysis technique.

The coherency function, if properly calculated, is a good index of the extent to

which spectrum analysis, as a linear process, is valid for application to a stochastic pro-

cess. A few results of this author's efforts in this field, presented later in Part I are related

to improvement of the techniques for obtaining a good estimate of coherencies.

1 .3 TIME DOMAIN CHARACTERISTICS

Time series analysis is sometimes called spectrum analysis, thus showing that esti-

mation of rehable spectral functions is very important in the analysis of time series.

a Â«J!*T* faactiam A m surel y PÂ° werful ^tions which provide good information on

a stochastic process and also resolve the tangled relations of convoluted types in the time

domain. However, because of this fact, and also partly because (auto) correlation did not

appear m an important way, in St. Denis and Pierson's pioneering paper, 4 rather little at-

tention has been paid by naval architects to the time domain relations over the past few

decades. From his first involvement in this study, this author has thought that tine do-

mam functions deserved more attention and has made some efforts along this line.

Of course spectral functions and correlation functions are actually the same function

expressed in different ways. Sometimes, however, in sample computations, applying the

ume domain expression helps us understand the characteristics of the stochastic process

better because we are accustomed to expressing the physical process by differential equa-

tions that are expressions in the time domain. The time domain expression of stochastic

processes helps us in the analysis to use already known characteristics of the process

Moreover, the correlation window operation in the sample computation is just a multiply-

ing operation, whereas the spectral window operation is now an entangled convolution

operation that is more complex in real sample computations. In sample computations

consideration of the windows is very important and also a troublesome problem to get

reliable results.

Parametric estimation of the spectral function, explained in Part n, is a method of

fitting a certain statistical model to the process to be analyzed and then estimating the

parameters of that model. This method is based on time domain characteristics, and the

model fitted is closely related to the equations of motion of the process itself. The method

represents a different approach to spectrum analysis and, this author believes, that is a

promising one, supplementing the conventional nonparametric approach, treated in Part I.

So, in Part II, several types of discrete parameter models will be introduced in detail, and

then the results of application of the model fitting method to the analysis of seakeeping

data will be shown. By this method, the actual response system, in which the output is fed

back to the input to some extent, can also be tackled. This kind of system has been hard to

be analyzed by the conventional method.

1 .4 TREATMENT OF NONLINEARITIES

One of the reasons for low coherencies in linear spectral analysis is the existence of

nonlinearities in the response or in the input. So in Part HI, the treatment of nonlinearities

in process analysis is reviewed. One of the greatest achievements in this field for the anal-

ysis of seakeeping data is due to John F. Dalzell 16 * 17 in the application of polyspectra.

This application will not be described in detail, except the basic idea, of this treatment but

another review of the treatment of nonlinearities will be given, to make clear the mutual

relationships of these several different approaches to the problem. In addition, the treat-

ment of nonlinearity in response characteristics during one trial has been introduced in

Section 3.4 in Part I, as an example of multi-input single-output analysis.

1.5 SCOPE OF STUDIES

In statistical studies on seakeeping, there are roughly two kinds of applications. One

is based on the invariant characteristics of a ship itself, and its behavior is studied statisti-

cally, assuming the excitement from the environment is also stationary. The other

involves the macroscopic probabilistic distributions of seakeeping behavior, assuming a

variety of changes in environmental conditions. The former is sometimes called short-

term statistics, and the latter, long-term statistics. They are, of course, closely related, and

the short-term statistics are usually used as the basis for studying the long-term statistics.

Here mostly short-term statistics will be treated.

PARTI

A REVIEW OF SPECTRAL ANALYSIS THROUGH PERIODOGRAM

(NONPARAMETRIC SPECTRAL ANALYSIS)

To clarify the problems and difficulties encountered in sample computations and

make them the basis for further discussions, the rough scheme of the techniques of spec-

trum analysis (through periodograms, the popular nonparametric method) will be

reviewed first in Chapter 2. Then some ideas proposed by this author for solving these

difficulties will be summarized in Chapter 3, Parti. Many text books}^ especially the

comprehensive one by Priestley, 23 were used as references in Chapter 2.

CHAPTER 2

BASIC PROCEDURE OF THE SPECTRAL ANALYSIS AND

THE PROBLEM OF SAMPLE COMPUTATIONS

2.1 RANDOM PROCESS AND ITS CHARACTERISTICS

and thfnrn^HTnT COntinUOUS P rocess on ' is ex P r ^ by X(t), its realization as x(t),

and toe probability density distribution function related to this process as Pt (x) The gen-

tln^T Pr0CeSS ^ \ " S reallZati0D iS *' "* Pr0babmt y distribution density function

is pt(x). Then, as expected values,

00

mean [X(t)} = E [X(t)] = I x(t)p t (x)dx = fi{t)

(2.1)

00

var. [X(t)] = E [{(Â») - M of] = J [x(t) - M f)] 2 p t (x)dx = o\t). (2.2)

â€” 00

Usually, the probability density distribution function p t (x) is a function of t ; accord-

ingly, the meaner) and the variance a\t) are also functions of time.

Joint probability density distribution functions, p^ _ tn (x u x 2 ,x 3 , . ..*â€ž) exist

for all n,n = 1,2,. . . â€ž,(â€ž - Â») and all p tl ( Xl ),p, 2 (x 2 ), ...â– p,^^), . . . . .

Pt,i 2h (xuX2,x 3 ), . . . ; .... are defined as their marginal functions.

Theoretically, if all the joint distribution functions of all orders

Pt,(.Xi),p t2 (x 2 ), . â– .

Pt,t 2 (xi,x 2 ),p h t 3 (X2,X3), ...-

Pv 2 t 3 (.X h X 2 ,X3), ....

Ptfoti â– â– â– tn(X\,X2,Xs, . . . Xâ€ž),

are known, the probability structure of X, is completely specified.

2.1.1 Completely Stationary

If

Pt,t 2 â– â– -/ B (*l,*2, â€¢ â€¢ â€¢ xâ€ž)=p h+ k i t 2 +k l - â– â– t n +k i Axi,x 2 , .... x n ), (2.3)

for any t\,t2, . . . . tâ€ž and any k, this process is completely stationary.

2.1.2 Stationary Up to Order m

In this case, the joint moment up to order m should be the same.

E[{X(r 1 )} m '{X(r 2 )} m2 [X(t n )} m ']

= E[{X(t l + k)} m >{X(t2 + k)} m > {X(t n + k)} m ^, (2.4)

for all positive integers m\, mi, . . . m,

where

mi + m 2 + . . . . m n < m.

2.13 Stationary Up to Order 2

Especially when the order is m = 2, the process is called weakly stationary. When

the probability density distribution functions are Gaussian, they are completely deter-

mined by the means, variances, and covariances of two variables, and accordingly they

are completely stationary.

2.1.4 Ergodicity

When the ensemble of the averages across the processes converge to the corre-

sponding time averages along the process over period N (when N tends toward infinity

and the mean square is consistent), the process is called ergodic. The ergodicity is a more

strict condition than the stationality as is shown in Fig. 2.1.

EVOLUTIONARY

STATIONARY UP TO 1ST ORDER

STATIONARY UP TO 2ND ORDER

ERGODIC UP TO 2ND ORDER

COMPLETELY STATIONARY

AND ERGODIC

For example,

Fig. 2.1. Stationarity and ergodicity.

R(r) = E [X(t)-X(t + r)]

x(t)x(t + x)p{x t , x t+z )dx4x t +r

J -a J -co

1 f 7

= lim-

t-* 00 1 Jo

x(t)x(t + x)dt.

(2.5)

2.15 Summary of Gaussian (Normal) Probability Distribution Functions

For convenience, the Gaussian probability distribution functions for various num-

bers of variables are summarized here. For a single variable,

P(x) =

1

Qm&* eXP

Normalized by z = (x -fi x )/o x .

1 (x-fi x f

2 a\

(2.6)

1 _Zi

P(Z)= l2^ e ' 2 -

(2.7)

For two variables X and Y

2nra;ay(l-e-)

i I (x-^) 2 2e0c-fixy-pj (y-n>) 2

2 (i-e 2 )o? (l-eV^ (i-eVr

^A^ CXP

-lOr./O 2 - ^7^(x-/fcKy-/Â«r) + "T Cv^

^^ CXP

- i {^(x-^) 2 - 2p xY (x^fi x )(y-My) + y3yyO'-y"y) 2 )

(2.8)

Here,

A =

= o 2 o 2 ,-Q 2 o 2 qj

= oWy{l-Q 2 ),

- â€ž2

yxx = E[X X] = ai , yxy = E[X â– Y] = Q ojj y

- â€ž2

y^ = E[Y- Y] = a* , y yx = E[Y-X] = QOyO x .

(2.10)

w -

>xx Hxy

lyx Pyy

1

ty]" 1

l

7yy >V

'Yxy Yxx

OxOfo-Q 2 )

Oy -QOjOy

-QOjOy o\

(2.11)

For n variables X\,X^, Xâ€ž,

p(xi,x 2 , Xâ€ž) =

Qjtf^b}! 2

exp

1 n n

Â»=1^1

, (2.12)

where

A = |x| =det(2

x= fa

(2.13)

and a tJ is an element of inverse matrix, so that

P(xi,x 2 , x n ) =

(2ji) n / 2 & 1 / 2

exp

-\[{x-n)'?rHx-n))

(2.14)

2.2 PROPERTIES REQUIRED FOR ESTIMATOR

The following properties are required for the estimator of some statistical value.

22.1 Unbiasedness

As the number of samples n tends toward infinity, for the estimator 6 of real 0,

bias(0) = Â£(0) -0â€”0

(2.15)

222 High Relative Efficiency

If we suppose that both estimators 0â€ž0 2 are unbiased, then the higher the

rel. effic. = var. (0\)/var. (0 2 ) (2.16)

is, the better estimator 0, is than 8 2 .

22 J Small Mean Square Error

Mean square error is defined as

M 2 (0) = Â£[(0 - 0) 2 ]

B {0 - E(0)] + JÂ£(0) - 0}

{0 - Â£(0)j 2 + {Â£(0) - 0J 2 + 2JÂ£(0) - 0jÂ£[0 - Â£(0)]

= var. (0) + b\6), (2.17)

which should be small. WhenM 2 (0i) < M 2 (6 2 ), we adopt 6\ as better than0 2 â€¢

22.4 Consistency

As the number of the samples n tends toward infinity, var. (9) -* 0, and bias

b 2 {6) -Â» must be satisfied. Then the mean square error M 2 (6) -â€” also stands and is

called the "mean square consistency."

225 Sufficiency

The estimator 6 must contain all the information Xi,X 2 , . . . . Xâ€ž in the sample,

relevant to the estimator of 6

6=6 (X lt X 2 ,

X n ).

2.3 AUTOCOVARIANCE FUNCTION AND ITS ESTIMATES

When the process X(t) is stationary up to an order of 2, the covariance function

cov. \X(t) X(t + r)] = E [{X(t)-fi}[X(t + T)-fi}] = R(t) (2.18)

is a function oft only. Then,e(r) = R(r)/R(0) is called an autocovariance coefficient,

and

R(0) = E [{X(t)-fi} 2 ] = var. [X(t)] = a 2 . (2.19)

When the process is real valued, R(-t) = R(r) as in Fig. 2.2, and when the process is

complex valued, R(-r) = R*(t). This function R(r) is a measure of similarity in a sense,

and is also a measure of the memory of the process. R(r) plays a big role later, in the

parametric analysis in Part II, in identifying the statistical model that will fit the process.

Mean/*, variance a 2 , R(r), and g(r) are constant by t.

Fig. 2.2. Autocovariance function.

10

23.1 Estimates of R(r)

For the discrete parameter process*, ,

R(r) =E [(X r -fi)(X t+r -[i t+r )]

= E[X r X t+r ], (2.20)

when [i, = pi t+r = E[X t ] = 0.

23.2 Unbiased Estimate

Instead of averaging the ensemble, take the time average as

, N-\r\

RÂ°(r) =â€” - 2 (X,-X) (X t+r -X) (2.21)

^ ''' *=l

where r = 0, Â± 1, Â± 2 Â± (N-l).

If we ignore the effect of estimating^ by X, then RÂ°(r) is an unbiased estimate of

R(r), because

, N-M

E [RÂ°(r)] = -^-j- X SCCfr-^CXi+r-At)]

N-\r\

1 â„¢ iv-lrl

= tt-z; I *(r) = tt-t m = *<r).

tf-lrl Â£f N-\r\

(2.22)

2 J. 5 Biased Estimate

On the contrary,

, N-Irl

*< r ) = 77 X (^-^)(^ + w- / ") (2.23)

is a biased estimate, because

1 AMH 1 / I I \

E &W = # X *W = ^ CW-IH) *(r) = il-j-\ R (r)

\r\

= R(r)-â€”R(r), (2.24)

11

biased by (\r\/N) R(r). WhenN -Â» oo , E[R(r)] â€” R(r), and is therefore asymptotically

unbiased.

By statistical mathematics, we can get

var. [RÂ°(r)] -* 0(l/(N-\r\)) (2.25)

var. [R(r)] -* 0(1 /N). (2-26)

1. When r is small relative to N, the difference between RÂ°(r) and R(r) is

small, and the bias of [R(r)] is also small.

2. When r becomes large relative to N and approaches N-l,

bias [R(r)] -* R(r). However, when r â€” Â» , R(r) -* 0. Therefore, when

N is very large, the bias remains small at all r.

3. When r â€” (N-l),

var. [RÂ°(r)] - 0(1) (2.27)

var. [R(r)] - 0(1/AO- ( 2 - 28 )

Therefore, the tail of correlation RÂ°(r) shows a wild and erratic behavior. Also, from

statistical mathematics, cov. [i?(r) R(r + s)] was computed and fairly high correlations

between neighboring points were found when s is small. When r tends toward infinity,

/?(r) -* 0. It can be concluded here that R(r) will be less damped than R(r) and will not

decay as quickly asi?(r).

2.4 SPECTRUM ANALYSIS

The spectrum function decomposes a time varying quantity into a sum (or integral)

of sine and cosine functions.

2.4.1 Spectrum for Various Processes

a. For Deterministic Periodic Functions â€” Fourier Series

For periodic functions with period T,

12

X(t) = - a + X

(jtnt\ . [nnt

a n cos^â€” j + bâ€ž sm^â€”

where

r i

1 f _. . Tint

a " = t I (r) cos T~ *

2

x

2

t 1 f _,. . ;rrc? ,

t ] (f) sm T~

(2.29)

(2.30)

The aâ€ž and 6â€ž functions are called Euler-Fourier coefficients. Equation 2.30 is

based on the orthogonality of cos â€” sin â€”

T ' T "

For the existence of a n , b n , and for the convergence of the series, the conditions

J \X{t)\dt < oo , J \X(t)\ 2 dt < 00

are sufficient.

Replacing a n ,b n by c = a o , c n =

the total energy [- J/2 to J/2] as

â– ^(al + b})

, gives

(2.31)

2 i-

= r ZÂ«3

(2.32)

and the total power [-T/2 to J/2] as

13

1 d,

(2.33)

ci is the contribution of the Â«th term to the total power, as in Fig. 2.3.

POWER

3 2 10 1 2 3

- T~TT TTt

Fig. 2.3. Power spectrum of periodic function.

b. For Nonperiodic Deterministic Functions â€” Fourier Integral

For nonperiodic functions with finite energy as in Fig. 2.4,

-\

X{t) = X(co) e tmt do),

(2.34)

X(fi>)

2jt J

X(t) e- l( Â°'dco.

(2.35)

Fig. 2.4. Nonperiodic function with finite energy.

14

For the existence and convergence of this expansion, conditions

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