On the conditional distribution of velocity for a stochastic process

Journal

of Sound

ON

and Vibration (1985) 98(2), 309-31 I

THE

CONDITIONAL DISTRIBUTION OF VELOCITY STOCHASTIC PROCESS

FOR

A

1. INTRODUCTION

This note concerns the distribution of velocity X(t) at some specified level of a stochastic process X(t). The distribution is found by two methods: firstly the ensemble distribution is found from the usual definition of a conditional probability density [l], and secondly the temporal distribution is found from a consideration of the crossings of a specified level. In general, these two distributions are different and the reason for this is discussed together with possible applications. 2.

ENSEMBLE

DISTRIBUTION

Let the joint probability density function for the two-dimensional random process (X(f) X(t)) at some particular time t = t, be written as pxi(xl, i,, t,), and the density function for the random process X(t) at this time be written as px(x,, t,). In many applications these two density functions are abbreviated to p,&x, i) and px(x), but in the present case it is important that the notation used be as precise as possible. The conditional velocity distribution at time t = t, can be written as [l] PX,x(%, t,lx,, 4) =Px&,

f,, 4)lPx(%

fl).

(1)

Since the above distribution refers to the time t = t,, it gives the conditional distribution of X(t) across the ensemble. For the case of a stationary Gaussian process, equation ( 1) gives ~x~~(%, 4/x,, 4) = (1/427r ok) exp c-f (Wo*)2] (2) where a2 is the r.m.s. velocity. Equation (2) states that the velocity has a Gaussian distribution which is independent of the value of X(f). 3.

TEMPORAL

DISTRIBUTION

Given a time history of the random process X(t) it is possible to define the quantity N(x,, t,) such that the average number of times the process crosses the level x, with positive velocity in a time T is 1: N(x,, t,) dt,. It can be shown that [l] N(% 4) =

I0

CCI ipxk(x,,

-Is,t,) dzi.

(3)

It is also possible to define the quantity N(x,, ii, ti) such that the average number of times the process crosses the level x, with velocity greater than 1i in a time T is jt N(x,, ii, t,) dt,. A slight modification of the analysis leading to equation (3) gives

J

N(x,, I,, 4) = ,D k&x(x,,

% t,) di.

(4)

N(x,, &, t,) can be interpreted as the mean rate with which X(t) crosses the level x, with velocity greater than i,. Given that X(t) crosses the level x1 at time t = t,, the 309 0022-46OX/SS/O20309+03

%03.00/O

@ 1985 Academic

Press Inc. (London)

Limited

310

LETTERS

TO THE

EDITOR

probability that the velocity will be less than I, can be written as PX,,(f,,

r,lx,, r,) = 1 - N(x,, I,, t,)lN(x,,

-co, t,).

(5)

Differentiating equation (5) with respect to i, gives the density function of velocity at a crossing of the level X(t) = x, as pX,JJf,,

r,lx,, t,) = I~llPXAX,, f,, 4VNx,,

--co, tl).

(6)

Since the above distribution has been derived by considering the crossings of the level X(t) = x, in a particular time history of X(r), it represents a temporal distribution. When X(t) is a stationary Gaussian process, equation (6) yields pXlx(%, Ux,, ri) = ((%1/2a)

exp I- t (%/oZ),

which states that the velocity has a Rayleigh distribution value of X(t). 4.

which is independent

(7) of the

DISCUSSION

Clearly the density functions given by equations (1) and (6) are different, and thus the conditional density function for velocity is not ergodic. Perhaps the best explanation of this_is afforded by firstly considering a situation in which the conditional density function is ergodic. The random process X(t) has the parameter time, which is in reality continuous, and therefore forms an uncountably infinite indexing set. However, if one considers the case where the process X(t) is sampled at evenly spaced discrete times, then each sample time can be related to an integer by a numbering scheme such as t,,, t,, rZ, t,, . . . , etc. The number of sample times is then countably infinite since this is true of the integers, and time then forms a countably infinite indexing set. The concept of a countably infinite number of sample points in the temporal sense can be related to a countably infinite number of sample functions in the ensemble, and thus for a stationary process the joint density function pxx(x,, iI, t,) can be applied to the temporal sample points if the dependence on t is dropped. The temporal conditional density function for velocity will then be given by equation (1). The question then arises as to why equation (6) is not obtained for the temporal conditional density function in this case. The reason is that between any two sampling points, say t, and t2, there is an uncountably infinite number of times which are not included in the sample, and thus it can readily be seen that not all times at which a crossing of the level xi occurs are included in the discrete sample points. However, in the derivation of equation (6) all such times are considered, and thus a different conditional density function is obtained. It may at first be thought that the true temporal distribution for continuous time can be extrapolated from the above by allowing the interval between succesive sampling times (At say) to tend to zero. This is not the case however since for any non-zero At, no matter how small, there still exists an uncountably infinite number of times between successive sampling points, and a sample which contains all crossings of a certain level can never be obtained. For this reason it is not surprising that equation (6) gives a different result from that obtained for the temporal distribution with discrete times, and thus to that obtained across the ensemble, equation ( 1) . An interesting point about equations (2) and (7) is that they yield results which are independent of the value of X(t), whereas intuitively it might be expected that velocities would decrease with increasing X(t) as in the case of a sine curve. This can be explained by considering crossing of two levels x2 and x, with x2 > x1 > 0. A trajectory of X(t) starting at X(r) = 0 with positive velocity can reach a peak below xi, between xl and x2

LETI-ERS TO

THE

311

EDITOR

or above x2. Those trajectories which peak above x2 can be expected to cross the level x1 with a greater velocity than the level x2, as in the case of a sine curve. Considering these trajectories alone would lead to the conclusion that velocity decreases as the level increases. Those trajectories which peak between x, and x2 will not affect the velocity statistics at the level x2 but they will modify the results at x1. In fact, since these trajectories have lower peaks they will in general have lower velocities, and their inclusion will lower the expected velocity at the level x,. In this way it is conceivable that the distribution of velocity at x, and x2 will be the same. Equation (6) gives the distribution of the velocity with which the stochastic process X(t) crosses the level x,. If X(t) represents the motion of some object and x, is some physical barrier, then this equation can be used to predict the velocity of impact, which is of interest in assessing impact damage. One example of this is where X(t) represents the surface elevation of the sea and x, represents the location of a bracing member of an offshore oil platform. Offshore Structures Group, CranJield Institute of Technology, Cranjeld MK43 OAL, England (Received 3 1 October

R. S.

LANGLEY

1984) REFERENCE

1. Y. K. LIN

Company.

1967 Probabilistic Theory of Structural Dynamics. New York:

McGraw-Hill

Book