Facts and figures pertaining to the bivariate Rayleigh distribution

by

J.A. Battjes

Dept. of Civil Engineering Delft University of Technology

The Netherlands 1969

SUMMARY

There is some evidence that the heights and the periods squared of wind waves each are Rayleigh-distributed. If this is the case, it may be surmised that their joint distribution is also of the Rayleigh type. The bivariate Rayleigh distribution, which is known in the field of statistical communications theory, may thus be applicable to wind wave problems. However, 1n the field of coastal and ocean engineering, this distribution does not seem to be known. In this note attention is dra~m to its existence, and its ma1n properties are listed.

1. INTRODUCTION

2. THE BIVARIATE RAYLEIGH DISTRIBUTION ... .

2. 1. Backgr01.md

2.2. The probability density function ... ... .

2.3. The cumulative probability function ... ... .

2.4. The marginal probability density functions ... .

2.5. The moments ... ... . 2.6. The correlation coefficient ... ... .

2.7. Case of zero correlation

2.8. Case of 100% correlation • • • • • • • • • • • • • • 0' • • • • • • • • • • 2.9. Regression lines .. . . . .. .... . LIST OF REFERENCES paee 3 3 3 4 4 6 8 8 9 11

1. INTRODUCTION

In the fields of harbour-, coastal- and ocean-engineering, knowledge

of the statistical distributions of heights and periods of wind waves is

often of great importance. Many efforts, both theoretical and experimental, have been made to determine such distributions.

Empirical wave height distributions were first given by Putz

(1952).

At about the same time, Longuet-Higgins

(1952)

pointed out that theoretically

the Rayleigh distribution should be applicable to the heights of wind waves

with a narrow spectrum, provided the waves be not too steep. It has since

been established empirically that the Rayleigh distribution:-applies to wind

wave heights with a fair degree of accuracy, even if the waves are steep and

do not have a narrow spectrum. In the latter case, the wave height may be

defined as the difference of the maximum (crest) height and minimum (trough)

height between two successive zero up- or downcrossings.

The analytical determ~nation of the distribution of wave periods (intervals

between successive zero up- or down crossings) is difficult. So far only

approximate solutions have been obtained tonguet-Higgins,

1962).

Empirical

distribution -functions have been first proposed by Putz

(1952).

Bretschneider

(1959)

noted that the squares of the periods of wind waves had a Rayleigh

distribution with about the same ~egree of accuracy as the wave heights.

This conclusion seems to have also been reached by Russian authors (Bretschneider,

1965).

The distributions mentioned so far pertain to the heights (H) and periods

(T) separately. The problem of the joint distribution has not been solved by

theoretical means. Bretschneider

(1959)

approached this problem empirically.

. . . . . 2

Hav~ng observed that the marg~nal d~str~but~ons of Hand T are both of the

Rayleigh type, he assumed that this would also be the case for the joint

distribution;; However, he was unable to e,laborate . - this assum:r,tion because

the bivariate Rayleigh distribution was unknown to him. To quote:

"The joint distribution of wave heights and lengths (or wave heights and

pe-riods) in general is difficult to describe completely for all conditions of

correlation, ,' .. The fact that both marginal distributions .. are 9f the same type ~s

of some help. The bivariate asymptotic problem of joint distribution for the

It was unknovm to Bretschneider that this problem, which also ar1ses 1n the

field of communications theory, had already been solved by Uhlenbeck

(1943

)

and Rice

(1944, 1945).

The fact that the theoretical bivariate Rayleigh distribution is known

seems to have remained unnoticed in the fields of coastal- and ocean-engineering.

Apparently, there is a communications gap bet.Teen researchers dealing with wind waves and those working in the field of statistical communications theory.

It is the purpose of this note to contribute to bridging that gap .. One way

of doing this would be to: merely mention a few references. However, it 1S

deemed more useful to reproduce the main results, because not all of these are

available in any single pUblication known to the author. Also, some corrections

and additions will be

made~)

It is emphasized that this note in what follm'Ts deals solely with the

the0retical bivariate Rayleigh distribution. At this time no theoretical or

empirical evidence is offered to prove or to disprove its possible

applica-bility to wind waves.

This assumption cannot mathematically be justified. In fact, Frechet

(1951, 1956

)

has shown that corresponding to each given pair of marginal distribution functions, an infinite number of bivariate distribution functions can exist.

jY

Some IDlnor corrections appear in paragraph 2.5. The additions consist

2. THE BIVARIATE RAYLEIGH DISTRIBUTION

2. 1. Background

The envelope of a narro\~band random noise signal has a Rayleigh distri

-bution. Similarly, the values

X

(r) and

Y=

X

(t+r)

of such an envelope at

two different times separated by a certain time lag ~ have a joint Rayleigh

distribution. The derivation of this distribution seems to have been given

for the first time by Uhlenbeck

(1943)

and again, independently, by Rice

(1944

-

1945).

The method employed is essentially similar to the one used 1n

deriving the one-dimensional Rayleigh distribution. The derivation "Till

not be reproduced here.

2.2. The probability density function (p.d.f.)

The joint p. d. f.

1(Sc

,~

)

of two stochastic variables

X

and

Y

is

defined by

)( and )fare said to have a j0int Rayleigh distribution if their p.d.f. is

. given by

(2)

in which a bar denotes average values and

l'

_o is the modified Bessel

function of the first kind of order zero (Uhlenbeck,

1943;

Ri;e,

1944-'45;

Nakagami,

1964).

The parameter Afis defined by

in which C;;'(w) is the power spectral density of the narrow-band random

The case

£/

occurs when 7=0 , i.e. when

X::: Y ,

or when there is

. 100% correlation between

X

and

y.

For very large c, -(~ 0 , while

X

and

)I

approach stochastic independence and)therefore, become uncorrelated. Although in these special cases ~equals the coefficient of correlation between

X

and

y,

this is not generally the case. The general relation

-ship will be dealt with in section 2.6.

In the following, the definition of

~

as given by Eq.(3) will not be needed; ~ will merely function as a parameter which fulfills the inequality

K/~

For variables which have been normalized so as to make their average value equal, to one, the joint Rayleigh probability densitY'-becomes

(

4)

o

This is the form which will be used subsequently.

2.3. The cumulative probability function

The author has not succeeded in expressing the cumulative probability.

function, or the distribution function, in a finite number of known functions. It appears to be necessary to compute it numerically if and when the need arises.

2.4. The marginal probability density functions

Th~

marginal p.d.f. of

X

is defined by ao

IC~)

= /

-f{x,,~)

tiJ--Co

Substitution of Eq.

(4)

gives rise to an integral which may be evaluated by using the follow-ing series expression for

ID

(Abramowitz and Stegun,

1965) :

:zj /;

I).:t

and by integrating term~se. The result is, as expected, the one -dimensional Rayleigh p.d.f. = 0

lor

.::t:<o Similarly> = 0 2.5. The moments

The moment matrix U is defined by

r~?t

(8 )

Substituting Eqs.

(4)

and

(6)

and integrating term~se, it can be shown that

Apart from a constant factor, the infinite series ~ the right hand member is the same as the series expression for a bypergeometric function (Magnus

and Oberhettinger, 1943), so that Eq.(10) may be written as

(Middleton, 1960), or, after performing a Imovm transformation of the hypergeometric function:

(12)

(The equat ions 9.22 in Middleton (1960), corre sponding to Eqs. (10) an.d (11)

By definition resp. normali.ation, the zeroth- and first moments are equal to one. The second moments are given by

(

13)

and

The last expression is equivalent to

in vThich

!l(~)

and

E

(-I)

are the complete elliptic integrals of the first and second kind of modulus

1f.

Eq.

(15)

is due to Uhlenbeck

(1943)

.

It may be proved by using series expressions of the various transcendental functions in the right hand members of Eqs. (14) and (15), or of Eq'., (10 ) with 4 n

=

'11_ I

2.6.

The correlation coefficient

The coefficient of correlation between

X

and

Y

is defined by

Xy

xy

(16)

Substitution of

%.:0'

y=/

and of Eqs.

(13)

and

(15)

g1ves'

/ '-- .J!"

'f

A graph of

;0

~

.

~

is shown in figure 1 by the full line.

E

(f)

and

!La)

are even functions of

-:t.

It follows that

.p

1S also an

even function of

t.

Two special cases occur when -(

=

0 and

.f

=

1:

E(o)=

k(o)= :.

J --40

.;0=6

Y

-;(=0

IL

(-£)

- ? 00 Q4 - - ( -+ J)

--t,d {I-

-(~

kf-t)

~

0.> --<10

.F

=1

i

--'=1 . (19)

This confirms the remarks made in section 2.2 following the definition of

~

•

Utilizing series representations for ~ and }('in Eq. (17), or

for the gamma-functions in Eq. (10) with -?n- = /?t. = / , the following series

representation for

;0

may be obtained:

;0=

~ Cf~

+

?f

QO

/.

3

.

~

. . .

,

C

Y·

-.J)}

Z.

~

IZ'

V

J

L

~

rJ=:l.-

t;.

b'

g.

.

..

(2f) (20) or

p

+

-

--

--

J

in which CL_ 7T (22 )

It should be noted that

;0

is a non-negative function of ~.

Thus, two stochastic variables with a joint Rayleigh distribution cannot be negatively correlated. Bretschneider (1959) already arrived at a similar, though more restricted, result. He proved that two stochastic variables could not be jointly Rayleigh distributed if

? - / .

In order to obtain an explicit e.xpression for ;( as a tunction of

.? '

the series given by Eq. (21) has been inverted, y,ith the result

/

(

P).:<....

/

(

'p)3

iZ

a... - /Zg ~ ₍₂₃₎

The two series given by Eqs. (21) and (23) are rapidly converglng for

1..£ I

~ 1:,

I

f J ~ I • Graphs of

f

1Y.).

i

,

based on the first three terms

of the series·, are given in figure 1. Only for values o:f';o and -,( close

to 1 is there a visible de,nation from ,the exact relationship. The truncation

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2.7. Case of zero correlation

The stochastic variables

X

and Yare

coefficient of correlation

;0,

given by Eq.

Substitution of this value in Eq.

(4)

yields

i{

)

_

rrz

_![(x.:<-r(j<-]

::t.J/ -

fjX:f

-e

'<

= 0 o~

said to be uncorrelated if the

(16), is zero. If

?

=

0,

(24)

which is equal to the product of the marginal p.d. functions as given by

Eqs.

(7)

and

(8).

Thus, two' stochastic variables with a joint Rayleigh distri

-bution are stochasticallylltlependent if they are uncorrelated.

2.8. Case of 100% correlation

If 100% correlation occurs then the two variables

X

and Yare linearly .

dependent. Moreover,

X

and

'I

are identically distributed. Therefore,

X=

Y

if

f=

1. The two-dimensional p.d.f. actually degenerates to a one-dimensional one.

The joint probability density must then be zero for all

..c:/:t-

and be infinite

for .;c~o This may be shown formally by investigating the behavior of

./'rz,j-)

as

~~/. To (this end, the following asymptotic expression for

.z;,

(i)

(AbramOi-litz and

Stegun, 1965) is substituted in Eq. (4):

After some algebraic manipulations, the result can be written as follOvTs:

(~6)

in which

. .<.

u = _

Tr

The expression J.n the brackets is formally equal to a Gaussian p.d.f. wit'h

indep~

variable (x-t) .. zero mean and mean' square deviationcr-. As -;f!

approaches 1,0- approaches O. The Gaussian p.d.f. then a~proaches Dirac's unit

as a definition of

d

.

Thus,

(28)

which may also be v~itten as

or as

(30)

in which

~(x)

and

~(1)

are the marginal p.d. functions given by Eqs. ( 7 ) and (8).

2.9. Regression lines

The regression of )( on

)I

is given by

lCo

X

/(x

J fI )

cI~

fOOl

(~j/!)

dOK

The expression which results after substitution of Eq.

(4)

has been evaluated using equations 11.4.28 and 13.1.27 from Abramowitz and Stegun _...

(1965), with the result

(32 )

in which

11

is the confluent hypergeometric function, and Z is a coefficient given by

It appears that in general the regresswn of

X

on

Y

is non-linear.

Exception should be made for the two trivial cases of stochastic independence

and linear dependence of )( and

)I.

In the case of stochastic independence,

~

=

0, substitution of which

gives as expected.

X=/

;)

J

I

In the case of linear dependence of

X

and

y,

-I

= 1. As

-i

...

1 ,

~

-+

-

00. Using asymptotic expansions of the confluent hypergeometric

function, it may be shown that

Therefore,

as expected.

x_

l

if (36)

The regression of

Y

on

X

is obtained by interchanging

~

.

and

/f-

in

the preceding expressions, because the bivariate Rayleigh distribution is

symmetric with respect to ~ and ;J-.

References

Abramm·Titz, M. and Stegtul, LA. : "Handbook of Mathematical Functions",

Dover Publications, Inc., NevT York, 1046 pages, 1965.

Bretschneider, C.L.: "Wave Variability and Wave Spectra for Wind-generated

Gravity Waves", Techn. Mem. 118, B.E.B., Dept. of the Army, 19)2 pages, 1959.

Bretschneider, C.L.: "Generation of Waves by Wind; State of the Art", NESCO

Report SN- 134-6, 96 pages, 1965.

Lighthill, M.J.: "Fourier Analysis and Generalised Functions", Cambridge

Univ. Press, 79 pages, 1966.

LongueJi-Higgins, M.S. (1952); "On the Statistical Distribution of the Heights

of Sea Waves", Journal of Marine Research, vol. 11, No.3, p. 245 - .266.

Longuet-Higgins, M.S.: "The distribution of intervals betw·een zeros of a

stationary random ftulction", Phil. Trans. Roy. Soc., vol. 254, No. 1047,

p. 557-599, 1962.

Magnus, W. and Oberhettinger, F.: "Formeln tuld Satze fUr die speziellen Funktionen

der liathematischen Physik", Berlin, 172 p., 1943.

Middleton, D.: "Statistical Communication Theory", Mc. Graw-Hill Book Company,

Inc. , New York, 1140 pages., 1960 , ~ .. -~ _ ... - ~.

Nakagami, M.: "On the Intensity Distribution .• and its Application to Signal

Statistics", Radio Science, vol. 68D, No.9, p.995-1003, sept. 1964.

Putz, R.A.: "Statistical Distribution for Ocean Waves;' Trans. A.G.U., Vol. 33,

No.5, p. 685-692, 1952.

Rice, S.O.: "Mathematical Analysis of Random Noise", The Bell System Techn. Jour.,

Vol. 23, p. 282-332, 1944; Vol. 24, p. 46-156, 1945.

Uhlenbeck, G.E.: "Theory of Random Processes", MIT Radiation Lab. Rept. 454, 1943.

Frechet, M.: "Sur les tableaux de correlation dont les marges sont donnees",

Annales de l'Universite de Lyon, Section A, p. 53-77, 1951.

Frechet, M.: "Sur les tableaux de correlation dont les marges sont donnees",