Counting level crossings by a stochastic process by Loren Lutes

7/28/2019 Counting level crossings by a stochastic process by Loren Lutes

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Probabilistic Engineering Mechanics 22 (2007) 293300

www.elsevier.com/locate/probengmech

Counting level crossings by a stochastic process

Loren D. Lutes

Zachry Department of Civil Engineering, Texas A&M University, 101 Summit Edge Court, Glen Rose, TX 76043, USA

Received 14 September 2006; received in revised form 16 February 2007; accepted 27 February 2007

Available online 14 March 2007

Abstract

The number of times, N, that a stochastic process has up-crossings of a fixed level within a fixed time interval, T, is investigated. Existingintegral formulas for the moments of N for a stationary Gaussian process are shown also to apply to processes that are neither stationary nor

Gaussian, and explicit formulas are given for approximating the probability distribution of N from the moment formulas. Particular attention is

given to simplified results for the limiting situations of very small and very large values of T, and to the behavior of variance, skewness, and

kurtosis of N. For small T, the number N approaches the well-known Poisson distribution, but the results for large T are significantly different.

For many stationary processes it is shown that the variance of N tends to grow linearly with T when T is very large, but the large-T growth rate

is sometimes much smaller than that of the small-T Poisson process. More detailed results and some numerical examples are presented for the

special case of a stationary Gaussian process crossing its own mean value.c 2007 Elsevier Ltd. All rights reserved.

Keywords: Crossing rates; Multiple crossings; Number of crossings; Stochastic process

1. Introduction

Various models for estimating failure probabilities in

mechanical or structural systems use information regarding the

crossings of some level by some stochastic process. In addition

to the obvious situations involving crossings of a critical stress

or displacement level by the dynamic response of a system,

there are also situations involving the estimation of the largest

value of the response or of the occurrence of peaks and valleys

within a time history. This latter situation corresponds to zero-

crossings by the first derivative of the dynamic response.

This is surely not a new problem, dating back at least

to the work of Rice in 1944 and 1945 [1]. Rices work, in

particular, gives the mean number of crossings within a given

time interval, the mean rate of crossings of a given level, and,

in some instances, the conditional mean rate of crossings at

one time given the existence of a crossing at another time.

Various other investigators have made significant extensions

to this work, but most of these have restricted their attention

to the special case of stationary Gaussian processes (e.g. [2

5]). Particularly pertinent to the current work is the derivation

Tel.: +1 254 898 2631.E-mail address: [email protected].

by Cramer and Leadbetter [2] of formulas for the moments of

the number of crossings of a fixed level within a fixed timeinterval for this special case. Rigorous mathematical analyses

of various aspects of the crossing problem for stationary

Gaussian processes are available in the books by Cramer and

Leadbetter [6] and Leadbetter, Lindgren, and Rootzen [7].

Some of this work for stationary Gaussian processes can also

be applied to a category of stationary non-Gaussian processes.

In particular, crossings by a translation process [8], which can

be written as a monotonic nonlinear function of a Gaussian

process, are directly related to crossings by the underlying

Gaussian process.

One of the emphases of the current work is the development

of integral formulas relating to the crossings of a process

that is neither stationary nor Gaussian, before demonstrating

how these results simplify as one imposes conditions, first

of stationarity and finally of Gaussianity. The approach to

the nonstationary non-Gaussian development is more heuristic

than rigorous. For example, there is no consideration given to

measure theoretic demonstrations that uncertain quantities of

interest are actually random variables.

Another aspect of the current work is that particular attention

is given to counting crossings during either very short or very

0266-8920/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.probengmech.2007.02.003
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294 L.D. Lutes / Probabilistic Engineering Mechanics 22 (2007) 293300

long time intervals. Simplified formulas are derived for these

limiting situations.

The quantities studied here include the variance, skewness,

and kurtosis of the number of crossings during a given time

interval, and procedures are also given for approximating the

probability distribution of that number. The fact that the dual

crossing rate governing two crossings in an interval can beevaluated in closed form for a stationary Gaussian process

allows simple computation of the variance of the number of

crossings and the probability of two crossings within a small

time interval for this situation. Numerical results are presented

for selected examples.

Although the current work is restricted to the counting of up-

crossings of a given level, one can easily generalize the problem

presented here to investigate the up-crossing of multiple levels

or to consider down-crossings at some or all of the times of

interest.

2. Problem formulation

Let {X(t)} be a stochastic process with continuous time

histories, and let the random variable N be the number of

up-crossings of a fixed level b within a time interval [0, T].

Note that {X(t)} is not required to be either stationary or

Gaussian. One simple way to reach the number N in the limit is

to divide the time interval into n subintervals by introducing

points tn j = jn with n = T/n. Let Nn be the number

of subintervals containing exactly one up-crossing of b. The

{X(t)} process is now limited to be sufficiently smooth that the

probability of more than one crossing within any subinterval

tends to zero more rapidly than the probability of one crossing,

so that one can consider each subinterval to contain either 0 or

1 up-crossing for n sufficiently large. Then N is the limit of Nnas n tends to infinity. The indicator function for the event of one

up-crossing in the j th subinterval, for a given n value, will be

denoted by In j . That is, In j = 1 if there is an up-crossing in

subinterval j , and is zero otherwise. One can then say that

Nn =

nj =1

Inj . (1)

The focus here will be on calculating the moments of the

random variable N by taking the limits of the moments of the

random variable Nn . In particular,

E(Nkn ) =n

j1=1

n

jk=1

E(In j1In j2 In jk)

=

nj1=1

njk=1

P(In j1 = 1, . . . , In jk = 1) (2)

will tend to E(Nk) for n . Presume, now, that there exist

finite crossing rates of the form used by Rice [ 1]:

+k (t1, t2, . . . , tk) =

0

0

0

y1y2 yk

pX(b, y1, b, y2, . . . b, yk)

dy1dy2 dyk (3)

in which X = [X(t1), X(t1), X(t2), X(t2), . . . , X(tk), X(tk)]T.

Rices result for k = 1 is well known, and he also gave

some explicit consideration to the dual-crossing rate of(3) with

k = 2. One can now write the probabilities that are needed in

(2) using the crossing rate in (3), giving

P(In j1 = 1, . . . , In jk = 1)

=

tnj k+ntnj k

tnj1 +ntnj 1

+k (t1, . . . , tk)dt1 dtk

= +k (tn j1 , . . . , tn jk)kn + O(

k+1n ) (4)

provided that the (j1, . . . , jk) arguments in (4) are distinct. This

restriction of distinct time intervals is obviously required, since

the order of magnitude in (4) is not correct if some of the

subintervals are identical. For example, in the extreme situation

in which all the tn j are identical, one finds that the probability

on the left-hand side is O(n) rather than O(kn). Also, it may

be noted that the pX(b, y1, b, y2, . . . b, yk) probability densityfunction in (3) is singular along any surface in (t1, t2, . . . , tk)

space having any two or more of these arguments equal to

each other, which gives singularities on these surfaces for the

multiple crossing rate of(3).

The moments ofNn can now be evaluated from (2). Note that

this summation generally includes situations with identical tn jvalues, as well as those with distinct subintervals. Nonetheless,

(4) does give all the information necessary to compute the

moments when one counts the ways in which some tn j values

may be identical and uses the fact that Iln j = In j for any positive

value ofl. The results will be written out explicitly for the first

four moments:

E(Nn) =

j

E(In j ) = Mn1 +

j

O(2n)

= Mn1 + O(n1) (5)

E(N2n ) =

j

E(I2n j ) +

distinct (j1,j2)

E(In j1In j2 )

=

j

E(In j ) +

distinct (j1,j2)

(+2 (tn j1 , tn j2 )

2n

+ O(3n)) = Mn1 + Mn2 + O(n1) (6)

E(N3n ) = iE(I3ni ) + 3 distinct (j1,j2)

E(Ini1I2ni2

)

+

distinct (j1,j2,j3)

E(In j1In j2In j3 )

= Mn1 + 3Mn2 + Mn3 + O(n1) (7)

E(N4n ) =

i

E(I4ni ) + 3

distinct (i1,i2)

E(I2ni1I

2ni2

)

+ 4

distinct (i1,i2)

E(Ini1I

3ni2

)

+ 6

distinct (i1,i2,i3)

E(Ini1Ini2I

2ni3

)

+

distinct (j1,j2,j3,j4)

E(In j1In j2In j3In j4 )

= Mn1 + 7Mn2 + 6Mn3 + Mn4 + O(n1

) (8)


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L.D. Lutes / Probabilistic Engineering Mechanics 22 (2007) 29 3300 295

kurtosis(N) =E[(N M1)

4]

4N

=M4 + M3(6 4M1) + M2(7 12M1 + 6M

21 ) + M1 4M

21 + 6M

31 3M

41

(M2 + M1 M21 )

2.

Box I.

in which

Mnk = kn

distinct (i1,...,ik)

+nk(tni1 , . . . , tnik). (9)

Now one can take the limit as n , giving Mnk tending to

the integral

Mk

T0

T0

+k (t1, . . . , tk)dt1 dtk (10)

in which +k (t1, . . . , tk) is the continuous function that is equal

to the multiple crossing rate +k (t1, . . . , tk) whenever all the

time arguments are distinct. Since N is the limit of Nn , the

moments ofN are then given by (5)(8) with each Mnk replaced

by Mk.The terms Mk in (10) are certainly not new. They were used

by Cramer and Leadbetter [6] in the context of a stationary

Gaussian process {X(t)}. For this special case, Cramer and

Leadbetter gave a rigorous proof that

Mk = E[N(N 1) (N k + 1)] (11)

and called this a factorial moment of the number of crossings.

It is easily shown that the moments obtained as the limits of

(5)(9) with n are consistent with the factorial moments

of (11). The intent of the presentation here is not to introduce

any new idea, but to demonstrate that these factorial moments

and the usual E(Nk) moments can be applied to more generalprocesses that are nonstationary and/or non-Gaussian.

The variance of N is, of course, easily obtained by the usual

method from the limits of (5) and (6):

2N = M2 + M1 M21 . (12)

Similarly, the skewness is

skewness(N) =E[(N M1)

3]

3N

=M3 + 3M2(1 M1) + M1 3M

21 + 2M

31

(M2 + M1 M

2

1 )

3/2(13)

and the kurtosis is given by the equation given in Box I.

One may note that E(N) = M1 is of O(T) when T is

very small and the expressions for 2N, E[(N M1)3], and

E[(N M1)4] each contain a term M1 that will dominate in

this situation, since all the other terms are either powers of M1or contain an Mk representing an integral over a higher-order

cube of dimension T. This behavior is the same as that of the

well-known Poisson counting process, for which 2N = E(N),

E[(N M1)3] = E(N), and E[(N M1)

4] = 3[E(N)]2 +

E(N) for all values of time.

One can also use the limits of the moment expressions in

(6)(8) to learn more about the singularities in the crossing

rates, resulting from the singularity of the probability density

function in (4) when times converge. In particular, one can writea moment equation similar to (11) as

E(Nk) =

T0

T0

+k (t1, . . . , tk)dt1 dtk (14)

and the moment expressions in (6)(8) are consistent with this

if +1 (t) = +1 (t) and the higher-order crossing rates have

singularities as follows:

+2 (t1, t2) = +2 (t1, t2) +

+1 (t1) (t1 t2) (15)

+3 (t1, t2, t3) = +3 (t1, t2, t3)

+ +

2

(t1, t2) (t2 t3) + +

2

(t2, t3) (t1 t3)

+ +2 (t1, t3) (t1 t2)

+ +1 (t1) (t1 t2) (t1 t3) (16)

and

+4 (t1, t2, t3, t4) = +4 (t1, t2, t3, t4) +

+3 (t1, t3, t4) (t1 t2)

+ +3 (t1, t2, t4) (t1 t3)

+ +3 (t1, t2, t3) (t1 t4)

+ +3 (t1, t2, t4) (t2 t3)

+ +3 (t1, t2, t3) (t2 t4)

+ +3 (t1, t2, t3) (t3 t4)

+ +2 (t1, t3) (t1 t2) (t3 t4)

+ +2 (t1, t2) (t1 t3) (t2 t4)

+ +2 (t1, t2) (t1 t4) (t2 t3)

+ +2 (t1, t2) (t1 t3) (t1 t4)

+ +2 (t1, t3) (t1 t2) (t1 t4)

+ +2 (t1, t4) (t1 t2) (t1 t3)

+ +2 (t1, t2) (t2 t3) (t2 t4)

+ +1 (t1) (t1 t2) (t1 t3)(t1 t4) (17)

in which () denotes the Dirac delta function. Of course, the

integrals in (14) with these integrands containing Dirac deltafunctions are rigorously defined only in terms of generalized

functions, but the concept is well known.

The nature of the singularities can be illustrated by

considering +4 . Within the four-dimensional space spanned

by (t1, t2, t3, t4), there is a first-order singularity involving +3

whenever two of the times are equal and the other two are

different, a second-order singularity involving +2 whenever the

times are either two equal pairs or three the same and one

different, and a third-order singularity involving +1 along the

line with all four times being the same. Clearly, such an explicit

listing of the singularities would become tedious for larger

values ofk in +k . Nonetheless, the pattern is clear.


4/8


In some situations it is convenient to make use of conditional

rates of crossing given the existence of crossings at other

specified times. Using the finite subintervals introduced earlier,

one can define a conditional crossing rate as

+k (tn ji +1 , . . . , tn jk|upcr. at tn j1 , . . . , tn ji )

limn

1

kin

P(In j1

= 1, . . . , In jk

= 1)

P(In j1 = 1, . . . , In ji = 1)

=+k (tn ji +1 , . . . , tn jk)

+k (tnj i +1 , . . . , tn ji ). (18)

In particular, +1 (t1|upcr. at t2) = +2 (t1, t2)/

+1 (t2) is an

important special case that will be used here. This formula was

also used by Rice [1], and is easily shown to be consistent

with the Slepian model that applies for the special case of a

stationary Gaussian process (e.g. [7]).

One more type of information that can be obtained from the

moments relates to the discrete probability distribution of N. In

particular, let pj P(N = j ). The kth factorial moment from(11) is then

Mk =

j=0

(j [j 1] [j k+ 1])pj =

j=k

j !

(j k)!pj (19)

in which the first k terms in the summation have been dropped

in the final form because they are identically zero. The formula

in (19) is most useful when pj pj +1 for all j 0, as

would usually be the case when T is very small. In this situation

one can say that Mk = (k!)pk + O(pk+1) or pk Mk/(k!).

For T values that are not very small, one anticipates that the

largest pk value will occur when k is in the neighborhood of

M1 = +1 T, and that pk will be very small for k M1. Based

on this, one may choose to ignore the contribution to (19) of

all pk terms having k > , with M1, by truncating the

summation at j = . This finite set of simultaneous equations

with a triangular coefficient matrix is easily solved to give

pj 1

j !

ji =0

(1)iMj +i

i !. (20)

The generalization of the problem presented here to consider

the up-crossing of multiple levels (b1, b2, . . . , bk) at times

(t1, t2, . . . , tk) or to consider down-crossings at some or all of

the times is quite straightforward. The basic modifications are

in the crossing rates of (3). It should be noted, though, that

the singularities of the joint probability density function in (3)

and of the crossing rates, as illustrated in (15)(17), occur only

when at least one crossing event occurs more than once. That

is, when there are two or more up-crossings of the same level

and/or two or more down-crossings of the same level.

3. Limiting behavior for large time

Attention will now be limited to stochastic processes for

which the process and its derivatives at time t1 become

independent of these same quantities at time t2 when the

separation between t1 and t2 becomes large. For the special

case of a Gaussian process, of course, this occurs if the

autocovariance function of {X(t)} tends to zero for well-

separated time arguments, but the same behavior also occurs for

more general processes, including many of practical interest.

The variance behavior of N for large values of time in this

situation is better exhibited by using an alternate form given

by Cramer and Leadbetter [6] for the dual crossing rate. Inparticular, let

2(t1, t2) = +2 (t1, t2)

+1 (t1)

+1 (t2) (21)

so that (10) and (12) give

M2 = M21 + H2,

2N = M1 + H2 (22)

with H2 defined as

H2

T0

T0

2(t1, t2)dt1dt2. (23)

The imposed independence condition implies that 2(t1, t2)

tends to zero when |t2 t1| is large, and weak restrictions onthe decay of the function imply that H2 is O(T), rather than

O(T2). In particular, the contribution of2(t1, t2) is significant

only in the vicinity of the line t1 = t2. Thus, the variance from

(22) only grows as O(T).

Similarly, let

3(t1, t2, t3) = +3 (t1, t2, t3)

+1 (t1)

+2 (t2, t3)

+1 (t2)

+2 (t1, t3)

+1 (t3)+2 (t1, t2) + 2

+1 (t1)

+1 (t2)

+1 (t3) (24)

and

H3

T

0

T

0

T

03(t1, t2, t3)dt1dt2 dt3 (25)

so that (10) gives

M3 = H3 + 3M1M2 2M31 . (26)

One can divide the large-time domain of the arguments in (25)

into 5 subsets:

a: t1, t2, and t3 all well separated,b: t2 and t3 not well separated, but t1 well separated from them,c: t1 and t3 not well separated, but t2well separated from them,d: t1 and t2 not well separated, but t3 well separated from them,

ande: no separationthe vicinity of the line t1 = t2 = t3.

Using the independence argument employed with regard to

(21), one can show that 3(t1, t2, t3) tends to zero in all of these

subsets except e, so that weak restrictions give H3 as O(T) for

T .

One can now use (22) and (26) to rewrite the skewness

expression. In particular, the numerator in (13) becomes

E(N M1)3 = H3 + 3H2 + M1 (27)

which only grows as O(T) for T for appropriate +kfunctions. From (27) and (22)one can see that the skewness in

(13) tends to zero as O(T

1/

2) in this situation.


5/8


In the same way, one can show that the function

4(t1, t2, t3, t4) = +4 (t1, t2, t3, t4)

+1 (t1)

+3 (t2, t3, t4)

+1 (t2)

+3 (t1, t3, t4)

+1 (t3)

+3 (t1, t2, t4)

+1 (t4)

+3 (t1, t2, t3)

+2 (t1, t2)

+2 (t3, t4)

+2 (t1, t3)

+2 (t2, t4)

+2 (t1, t4)

+2 (t2, t3)

+ 2+1 (t1)+1 (t2)

+2 (t3, t4)

+ 2+1 (t1)+1 (t3)

+2 (t2, t4)

+ 2+1 (t1)+1 (t4)

+2 (t2, t3)

+ 2+1 (t2)+1 (t3)

+2 (t1, t4)

+ 2+1 (t2)+1 (t4)

+2 (t1, t3)

+ 2+1 (t3)+1 (t4)

+2 (t1, t2)

6+1 (t1)+1 (t2)

+1 (t3)

+1 (t4) (28)

tends to zero for large-time arguments for all situations except

the vicinity of the line t1 = t2 = t3 = t4, so that

H4

T0

T0

T0

T0

4(t1, t2, t3, t4)dt1dt2dt3dt4 (29)

can become O(T) for T . From (2) one now has

M4 = H4 + 4M1M3 + 3M22 12M

21M2 + 6M

41 (30)

which can be used along with (22) and (26) to rewrite theexpression for kurtosis. The numerator of Box I gives

E(N M1)4 = 3(M1 + H2)

2 + H4 + 6H3 + 7H2 + M1

= 34N + H4 + 6H3 + 7H2 + M1. (31)

The fact that the denominator in the kurtosis grows as 4N =

O(T2) then gives the kurtosis as tending to 3 + O(T1) if theHk terms all grow as O(T) for T . The skewness and

kurtosis results, of course, are consistent with N tending to a

Gaussian distribution as T tends to infinity. This can also bejustified by the central limit theorem under weak restrictions

on the {X(t)} process. In particular, it is sufficient to have

conditions that the autocovariance function of {X(t)} tendsto zero for well-separated time arguments plus a Lyapunov

condition that the nonstationarity of {X(t)} is not such that the

magnitude of N for T tending to infinity is dominated by thecontributions during a finite number of finite time intervals [9].

4. Simplification for a stationary process

For a stationary process some of the results can be simplified

somewhat by rewriting +1 (t) as a time-invariant crossing rate

+1 and

+k (t1, . . . , tk) as a function

+k (1, . . . , k1) ofk 1

arguments i = tk ti . Similarly, k(t1, . . . , tk) is rewritten

as k(1, . . . , k1). This allows various multiple integrationexpressions to be reduced by one order. For example, M1 =+1 T, and (10) and (23) become

M2 = 2

T0

(T ) +2 () d,

H2=

2T

0 (T ) 2() d

(32)

respectively. Presuming that 2() tends to zero faster than

2 for large values of , one can then use the approximation

H2 H20T H21 when T is very large, in which

H20 = 2

0

2()d, H21 = 2

0

2()d. (33)

The formulas in (22) then give

M2 (+1 T)

2 + H20T H21,

2N (+1 + H20)T H21

(34)

for large values ofT. Note that (34) gives an explicit expression

for the limiting linear growth of the variance. The effect of

the H20 term will be found to be quite significant in some

situations, so that the rate of growth of the variance may be

much smaller than for the corresponding Poisson process, for

which H2 = 0.

One can generalize (32) to show that

T0

T

0f(t1, . . . , tk)dt1 dtk

= k

T0

T0

[T max(1, . . . , k1)] f(1, . . . , k1)

d1 dk1 (35)

for any function f that has the properties exhibited by +k or kfor a stationary process. In particular, the f(t1, . . . , tk) function

is unchanged by any permutation of its k arguments and is a

function only of the k 1 time differences.

Ifk tends to zero faster than 2 for large values of any of

the arguments, then

Hk0 k

0

0k(1, . . . , k1) d1 dk1 (36)

and

Hk1 k

0

0

max(1, . . . , k1)k(1, . . . , k1)

d1 dk1 (37)

are finite and Hk Hk0T Hk1 as T tends to infinity. Using

(34), (27) and (31)now gives

skewness(N) (+1 + H30 + 3H20)T H31 3H21

[(+

1+ H

20)T H

21]3/2

(+1 + H30 + 3H20)

(+1 + H20)3/2T1/2

(38)

and

kurtosis(N) 3

+(+1 + H40 + 6H30 + 7H20)T H41 6H31 7H21

[(+1 + H20) T H21]2

3 ++1 + H40 + 6H30 + 7H20

(+1 + H20)2T

(39)

for this large-time situation.


6/8


5. Stationary Gaussian process with known autocorrelation

function

Some of the general expressions in the preceding sections

can be written explicitly in relatively simple forms in the special

case of a Gaussian {X(t)} process, and some of the integrals in

the general expressions can be evaluated in closed form in the

even more special case in which {X(t)} is also stationary and

the events of interest are crossings of its mean value. Note that

none of the mathematical results for this stationary Gaussian

case are fundamentally new, since it is the special case that

has received considerable study in the past (e.g. [27]). It is

introduced here, not so much because it is extensively used

in practical applications, as to provide a simple illustration of

the general relationships developed in this study. Formulas here

will be written for the common situation of zero-crossings of a

mean-zero Gaussian process, but they also apply to any other

situation with b = X.

The mean up-crossing rate of zero for a mean-zero stationary

Gaussian {X(t)} process is well known to be +1 = X/(2 X)

and the joint probability density functions needed for evaluation

of higher-order crossing rates from (4) are also well known. For

the dual-crossing rate the integration can be performed in closed

form and the result written as

+2 () =2X

2X(2 )2(1 20 )

1/2

A B (A2 B2) tan1

B

A

(40)

in which

A =

1 2 21

1 + 0

1/2,

B =

1 + 2

21

1 0

1/2 (41)

with

0 =cov[X(t1), X(t2)]

2X

, 1 =cov[X(t1), X(t2)]

XX,

2 =cov[ X(t1), X(t2)]

2X

.

(42)

Note that 0, 1, and 2 are all functions of t2 t1,

even though that has not been included explicitly in the

equations. If one knows the autocovariance function G()

cov[X(t), X(t+ )] for the process {X(t)}, then the correlation

coefficients can be written in terms of that function and its first

two derivatives.

The expression in (40) confirms the anticipated result that

+2 () tends to (+1 )

2 for very large , so that 2() tends to

zero. In particular, all the correlation coefficients go to zero

for large , so that A and B both tend to unity. This makes

the first term in (40) to be unity and the second one to be

zero, confirming the desired result. No closed-form solution has

been found for the integrals of 2() in (32). Thus, numerical

integration is required for evaluating M2, which appears in the

variance of N, even for this simplified Gaussian problem. The

integration is simple, though, since it is one-dimensional. Also,

no closed-form solution has been found for the integrals in (3)

to evaluate +k () for k 3.

In order to obtain more detailed information about the dual

crossing rate and associated properties of the counting processN for small values of time, one can consider a power-series

expansion for the autocovariance function G():

G() =

j=0

cj ||j . (43)

Obviously c0 = 2

X and it is also necessary for the present

situation to have c1 = 0. In particular, it is necessary that G()

be twice differentiable in order that finite crossings rates exist.

This gives 2X= 2c2 and

+1 () = (2c2/c0)

1/2/(2 ).

Three distinct conditions must be considered in evaluating

the dual crossing rate related to the autocovariance function in

(43). With no further restriction other than c1 = 0, the formulasin (40) and (41) give the limiting dual crossing rate for small

as

+2 (0) =c

1/20 c3(3

3/2 )

6 2(2c2)1/2. (44)

For small values ofT one can substitute (44) into (32) to obtain

M2 +2 (0) T

2. Clearly, this term coming from +2 () makes

a very small contribution to the variance of N, as given in

(12), when T is small. Also the probability p2 M2/2 of two

crossings within a small time interval is very small.

In some applications, one may choose to use a model with c3

also equal to zero. In particular, this is a necessary condition forany process with a finite value ofX. For c3 = 0, it is obvious

from (44) that +2 (0) = 0, which implies that M2 tends to a

higher-order function ofT. In fact, the limiting result from (40)

and (41) is

+2 () 4c0(c5)

3/25/2

3 2(c2)1/2(6c0c4 c22)

1/2. (45)

From (32) one then finds that M2 = O( T9/2), so that p2 and

the contribution of M2 to the variance of N are even smaller

than in the situation represented by (44). One can carry this idea

one step further by considering c5 also to be zero, as is required

for a process with a finite value of...X. This condition gives thestill smaller terms of

+2 () 21/2(5c2c6 2c

24)

3/24

3 2c22(6c4 c22)

1/2,

M2 21/2(5c2c6 2c

24)

3/2T6

45 2c22(6c4 c22)

1/2. (46)

Note that the limits in (46) depend only on even coefficients

in the expansion of (43). Thus, the limiting forms of +2 ()

and M2 in (46) will not be changed by setting more of

the odd coefficients in (43) to zero. In particular, a process

with an analytic autocovariance function will have only even


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Fig. 1. Covariance function for {X(t)}.

coefficients in the expansion of(43), but the limiting small-time

behavior of+2 () and M2 will be as given in (46).

6. Numerical examples based on dual occurrence rate

Let {X(t)} be mean-zero and have a spectral density

SX X() =1

2 (2 )1/2 (e(+0)

2

/(2

2

) + e(0)2

/(2

2

)) (47)

and autocovariance function

G() = e22/2 cos(0). (48)

Note that this has been normalized to give unit variance, so

the 0 correlation coefficient in (42) is identical to G().

Differentiation gives 1 and 2, and +1 and

+2 () are easily

evaluated. Since G() is analytic, only even coefficients appear

in the expansion of(43), and those needed for use in the small-

time expressions of(46) can be evaluated as

c2 = 2 + 2

02

, c4 =34 + 622

0+ 4

024

,

c6 = 156 + 45420 + 15

240 + 60

720.

(49)

The small-time dual crossing rate then has a limit from (46) of

+2 () 2(36 + 12420 + 9

240 + 2 60)

3/24

648 2(2 + 20)2(2 + 2 20)

1/2(50)

and M2 for small values ofT is similarly obtained.

Choosing 0 = 0 in (47) gives a simple bell-shaped

unimodal spectral density and 0 = 5 gives a moderately

narrowband process. For both of the processes, the values

of H20 and H21 from (33) are negative. The autocovariancefunctions for the two processes are shown in Fig. 1 and values of

+2 ()/(+1 )

2 are then shown in Fig. 2. Note that the normalized

form used for presenting +2 () is convenient in multiple ways.

Not only is it dimensionless, but also it can be viewed as a

normalized form of the conditional rate of up-crossings. In

particular, (18) gives +1 (t + |upcr. at t)/+1 =

+2 ()/(

+1 )

2.

Finally, the normalized form approaches unity for large values

of . Fig. 2 also confirms that +2 () agrees with the O(4)

behavior of(50) for small values of . The variance of N(T)

for the two sample situations is shown in Fig. 3. The plot also

gives the small-T Poisson approximation of+0 T and the large-

T approximation from (34). Numerical values for M2 are also

Fig. 2. Dual crossing rate.

Fig. 3. Variance ofN.

easily obtained, but are not shown here. For each of the spectral

densities, the M2 versus T curve tends smoothly from the

O(T6) limit of(46) for small T to the large-T parabola of(34).

The transitions of the numerical results in Figs. 2 and 3

between the small-time and the large-time limits are seento be smooth. It is noted, though, that the transitions are

quite different in form. In particular, the transitions for 0 =

0 are monotonic, while those for 0 = 5 approach thelarge-T asymptotes in an oscillatory manner, which loosely

correlates with the difference between the two autocovariance

functions. Note, in particular, that 0 = 5 sometimes gives

+2 () as significantly exceeding (

+1 )

2, most notably when approximates one period of the narrowband process. Viewing

this in terms of the conditional rate of occurrence of crossings

at time t + , this indicates that the presence of a peak at time

t significantly increases the likelihood of a crossing one periodlater. This is surely not unexpected for a narrowband process.

It may also be noted that the large-time rate of increase of

the variance of N(T) in Fig. 3 is significantly smaller than the

small-time value for both situations shown, but particularly for0 = 5. For 0 = 0, H20 in (34) is 71% of

+1 , which

results in the final rate of growth of the variance being 29% ofthe initial rate, while 0 = 5 gives H20 as 93% of

+1 so

that the rate of increase of the variance for large time is only

about 7% of the initial rate. These decreases in the variance of

the number of crossings, of course, mean that N(T, 0) has less

variability than the corresponding Poisson process, in the sense

that its coefficient of variation is less than (+0 T)1/2.

7. Summary

An investigation has been made of the number of times,

N, that a stochastic process has up-crossings of a fixed level


8/8


within a fixed time interval, T. Existing integral formulas

for the moments of N for a stationary Gaussian process

have been shown also to apply to processes that are neither

stationary nor Gaussian, and explicit formulas have been given

for approximating the probability distribution of N from the

moment formulas. The singularities of the multiple crossing

rates for equal time arguments have been investigated in somedetail for two to four crossings. Particular attention has been

given to the limiting situations of very small and very large

values of T, for which simplified results apply. For small T,

the number N approaches the well-known Poisson distribution,

but the results for large T are significantly different. For many

stationary processes it is shown that the variance ofN does tend

to grow linearly with T when T is very large, but the large-T

growth rate is sometimes much smaller than that of the small- T

Poisson process. The behavior of the skewness and kurtosis of

N has also been investigated, confirming the convergence of N

to a Gaussian distribution for large values of T.

In the special case of a stationary Gaussian process crossing

its own mean value, as in zero-crossings by a mean-zeroprocess, the general integral expressions become much simpler

and some of them can be evaluated in closed form. In particular,

the dual crossing rate can be obtained in closed form, and

this allows investigation of such quantities as the variance of

N and the probability of two crossings in a very small time

interval. Limited numerical results have been presented for this

situation to illustrate the general relationships developed in this

work.

References

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Selected papers on noise and stochastic processes. New York: Dover; 1954.

Reprinted from: Bell System Technical Journal 19441945: 2324.

[2] Cramer H, Leadbetter MR. The moments of the number of crossings of a

level by a stationary normal process. The Annals of Mathematical Statistics

1965;30(6):165663.

[3] Geman D. On the variance of the number of zeros of a stationary Gaussian

process. The Annals of Mathematical Statistics 1972;43(3):97782.

[4] Cuzick J. Conditions for finite moments of the number of zero crossings

for Gaussian processes. The Annals of Probability 1975;3(5):84958.

[5] Cuzick J. A central limit theorem for the number of zeros of a stationary

Gaussian process. The Annals of Probability 1976;4(4):54756.

[6] Cramer H, Leadbetter MR. Stationary and related stochastic processes:

Sample function properties and their applications. Mineola (NY): Dover;

2004. Reprinted from New York: Wiley; 1967.

[7] Leadbetter MR, Lindgren G, Rootzen H. Extremes and related properties

of random sequences and processes. New York: Springer-Verlag; 1983.

[8] Grigoriu M. Applied non-Gaussian processes. Englewood Cliffs (NJ):

Prentice Hall; 1995.

[9] Lin YK. Probabilistic theory of structural dynamics. Melbourne (FL):

Krieger Publishing; 1976. Reprinted from: New York: McGraw-Hill; 1967.

Counting level crossings by a stochastic process by Loren Lutes

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