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Technical note: On the long-term statistics of extremes
Technical Note: On the long-term statistics of extremes ARVID NAESS
Norwegian Hydrodynamic Laboratories, Ship and Ocean Laboratory, P.O. Box 4118- Valentinlyst, N-7001 Trondheim, Norway In a previous note 1 in this Journal the author advocated the use of an upcrossing analysis technique in order to investigate the extreme values of a stationary stochastic process. This method is preferable simply because it requires the least amount of knowledge necessary to carry out an extreme value analysis, and also from a methodological point of view is it preferable as a general technique (when correlation effects are neglected). The basic assumption underlying this approach is that the upcrossings of high levels can be considered statistically independent (i.e. neglect of correlation effects), which is equivalent to the assumption of Poisson-distributed upcrossings. Let the (real) stationary stochastic process under study be denoted by X(t), where t denotes time. X(t) is assumed to have sufficiently regular sample functions to justify the subsequent analysis. For a discussion of suitable regularity conditions, reference is made to the appropriate literature? '3 This point is treated lightly here, because in the practical cases of interest to this study, such regularity conditions are almost invariably satisfied. The following equation may then be written down: ~ Prob {X(t)~<~; O~
(1)
where v~ denotes the mean frequency of ~-upcrossings by
X(t). It is given by 4' s
~=
7
~1xYxx(~, e) d~
(2)
o
where fx;?(x, de) denotes the joint probability density function of X = X(t) and J" = X(t)= (d/dt)X(t). By the assumption of stationarity, v~ is independent of t. Defining X(T)=max(X(t); O<~t
(3)
where FX(7)(~) denotes the probability distribution function of the stochastic variable X(T), which is the extreme value of X(t) during the interval (0, T). From equation (3) may be derived the extreme value statistics required, under the assumption that X(t) can be considered to coincide with a stationary process throughout the interval (0, T). This situation is frequently violated in practice. For instance, when X(t) represents the response of some structure subjected to the ocean environment, and the length of the time interval (0, T) is days, weeks, years or perhaps even the entire expected lifetime of the structure, then X(t) can certainly not be considered to coincide with a stationary process over (0, T). In the case of the ocean environment, the intervals of 'stationarity' are of the Acccpted Junc 1984. Discussion closes December 1984. 0141-1187/84/040227-02 $2.00 © 1984 CML Publications
order of hours. Hence, in the long-term cases mentioned above, equation (3) can no longer be applied to investigate the extreme values over periods of time of such length. However, equation (3) is easily generalised to the nonstationary case by retaining the assumption of independent upcrossings of high levels. The corresponding expression in this case is4 T
where the mean frequency of ~-upcrossings, as determined by equation (2), is no longer independent of t. It will be shown subsequently that by exploiting equation (4), i.e. by using the upcrossing analysis technique, a very neat and compact expression is obtained for the long-term probability distribution function of the extreme value. This is in contrast to the more or less complex expressions that appear from time to time in the periodicals, expressions that are generally based on peak value analyses, which will not in any case give better extreme value estimates than the upcrossing analysis technique. ~ In order to apply equation (4) in the study of longterm statistics of extremes, we shall have to incorporate information about the variability in time of the statistical properties of X(t). This is done by assuming that the properties of X(t) locally are uniquely specified by environmental parameters, like, for example, significant wave height H s and mean zero-upcrossing period T~, which are then considered to be stochastic variables6-9 or rather, stochastic processes. This approach is based on the assumption that X(t) can be considered locally stationary; i.e. to coincide with a stationary process over time intervals of restricted length. This is equivalent to assuming that the environmental parameters in question can be considered constant over these same time intervals. In the sequel W(t) denotes the stochastic vector process of the environmental parameters, e.g. W(t) = (Hs(t), Tz(t)). To be able to incorporate the statistical variability of W(t) into equation (4), it is necessary to impose a particular condition on the process W(t). This concerns the possibility of deriving estimates of statistical parameters from time averages, i.e. ergodicity (in some specific sense). Although not generally stated explicitly, the assumption of an ergodic property on W(t) is, in fact, basic to the concept of long-term statistics. Regarding the problem of estimating the probability distribution functions of the stochastic process W(t), all we have available i~ part of a realization of W(t). To be able to use this limited information to estimate the probabilistic structure of the process W(t), requires the assumption that W(t) is ergodic (in distribution).
Applied OceanResearch, 1984, Vol. 6, No. 4
227
Technical note The following development will serve to illustrate how the assumption of ergodicity is exploited. A stochastic process Z(t) is introduced by defining
where v~ =
-+ f v~(w)fw(w ) dw
Z(t) = 1, W(t) < w = 0, if not
w
(5)
where w is arbitrary, but fixed. The short-hand notation W(t) < w means componentwise inequalities. From equation (6) follows immediately that E [Z(t)] = Fw(w)
(6)
where F w denotes the probability distribution function of W(t). We now form the time average T
Zr=TfZ(t)
dt
(7)
Fw(w)
(8)
0
Clearly,
E [ZT]
=
E [Z(t)]
=
The assumption of ergodicity ensures the validity of the following equalities lim Z T = E [ZT] = Fw(W ) w.p. 1 (9) T~ '~ where w.p. 1 denotes with probability one. 2"10 A consequence of this is that for almost every realization w(t) o f W(t), the length of time, dTw, that w(t) spends within the interval (w, w + dw) throughout (0, T) is given by dTw = Tfw(w) dw,
T ~ oo
(15)
In order to calculate statistical moments like E[,~(T)] and Var D((T)], a method developed by the author u may be used. It requires that at least an asymptotic expression is available for P~(~ -> oo), and that this asymptotic function satisfies certain conditions, which it seems to do in most practical cases. It should perhaps be stated explicitly that equation (14) requires T to be large compared to the scales of variation of W(t). Since W(t). is known by observation to contain seasonal variation, i.e. to contain a scale of variation of the order of months, T in equation (14) should therefore be correspondingly large. This comment immediately points to another problem connected to the modelling of W(t) as an ergodic process. It is hard to imagine that the seasonal variation of the sea state as it is observed, i.e. mild summer breeze versus strong winter gale, is just accidental, as it should be if our model assumptions had been correct. However, the question of whether equation (14) is acceptable or not, does not entirely depend on whether we accept or reject the model assumptions, but rather on whether we can agree on an averaging function to use in equation (14). Then, an equation like (14) may be used also for, say, a particular month in the winter, provided an appropriate averaging function (corresponding to a conditional probability density function) can be given
(10)
where fw denotes the joint probability density function of W(t). fw is independent of t since W(t) was assumed stationary. From equation (10) it then follows that for almost every realization w(t) o f W(t)
ACKNOWLEDGEMENTS
This work has been financially supported by the Royal Norwegian Council for Scientific and Industrial Research (NTNF) and the Norwegian State Oil Company (Statoil).
T
0
REFERENCES
w
where v~(w(t)) denotes the ~-upcrossing frequency of the stochastic process X(t) given that W ( t ) = w(t) (t fixed). Equation (11) may also be derived directly by assuming W(t) to be strictly ergodic. 2' ~o Then (provided both sides of" the equation are well-defined and finite) T
lira T . - . + oo
;-f
v~(W(t))dt=E[v~(W(O))] w.p. 1. (12)
0
Since /" ElY,(W(0))] = J v~(w)fw(w) dw
(13)
w
it follows that equation (11)holds for almost every realization w(t). Assuming now that equation (11) holds for the particular realization at hand, it is seen from equations (4) and (11) that
F£(r)(~)=exp {-- f v~(w) Tfw(w) dw} w
= exp {-- p~ T},
228
T ~ oo
Applied Ocean Research, 1984, Vol. 6, No. 4
(14)
l
Naess, A. On a rational approach to extreme value analysis,
Applied Ocean Research 1984, 6 (3) 2 Cram6r, H. and Leadbetter, M. R. Stationary and Related Stochastic Processes, John Wiley and Sons, Inc. New York, 1967 3 Adler, R. J. The Geometry of Random Fields, John Wiley and Sons, Inc. New York, 1981 4 Lm, Y. K. Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, 1967 5 Rice, S. O. Mathematical analysis of random noise. In Selected Papers on Noise and Stochastic Processes, Wax, N. ed., Dover Publications, New York, 1954 6 Nordenstr~m, N. Methods for predicting long term distributions of wave loads and probability of failure for ships. Appendix I Long term distributions of wave height and period, 1969, DnV Report No. 69-21-$. Det norske Veritas, Oslo 7 St Denis, M. On statistical techniques for predicting the extreme dimension of ocean waves and of amplitudes of ~hip responses. Proc. First STAR Symposium, The Society of Naval Architects and Marine Engineers, New York, 1975 8 Ochi, M. K. Wave statistics for the design of ships and ocean structures, SNAME Transactions 1978, 86, 47 9 Battjes, J. A. Long-term wave height distributions at seven stations around the British Isles. Deutsche Hydrographische Zeitschrfft 1972, 25 (4) 10 Dooh, J. L. Stochastic Processes, John Wiley and Sons, New York, 1953 11 Naess, A. The joint crossing frequency of stochastic processes and its application to wave theory. To appear in Applied Ocean
Research.
Norwegian Hydrodynamic Laboratories, Ship and Ocean Laboratory, P.O. Box 4118- Valentinlyst, N-7001 Trondheim, Norway In a previous note 1 in this Journal the author advocated the use of an upcrossing analysis technique in order to investigate the extreme values of a stationary stochastic process. This method is preferable simply because it requires the least amount of knowledge necessary to carry out an extreme value analysis, and also from a methodological point of view is it preferable as a general technique (when correlation effects are neglected). The basic assumption underlying this approach is that the upcrossings of high levels can be considered statistically independent (i.e. neglect of correlation effects), which is equivalent to the assumption of Poisson-distributed upcrossings. Let the (real) stationary stochastic process under study be denoted by X(t), where t denotes time. X(t) is assumed to have sufficiently regular sample functions to justify the subsequent analysis. For a discussion of suitable regularity conditions, reference is made to the appropriate literature? '3 This point is treated lightly here, because in the practical cases of interest to this study, such regularity conditions are almost invariably satisfied. The following equation may then be written down: ~ Prob {X(t)~<~; O~
(1)
where v~ denotes the mean frequency of ~-upcrossings by
X(t). It is given by 4' s
~=
7
~1xYxx(~, e) d~
(2)
o
where fx;?(x, de) denotes the joint probability density function of X = X(t) and J" = X(t)= (d/dt)X(t). By the assumption of stationarity, v~ is independent of t. Defining X(T)=max(X(t); O<~t
(3)
where FX(7)(~) denotes the probability distribution function of the stochastic variable X(T), which is the extreme value of X(t) during the interval (0, T). From equation (3) may be derived the extreme value statistics required, under the assumption that X(t) can be considered to coincide with a stationary process throughout the interval (0, T). This situation is frequently violated in practice. For instance, when X(t) represents the response of some structure subjected to the ocean environment, and the length of the time interval (0, T) is days, weeks, years or perhaps even the entire expected lifetime of the structure, then X(t) can certainly not be considered to coincide with a stationary process over (0, T). In the case of the ocean environment, the intervals of 'stationarity' are of the Acccpted Junc 1984. Discussion closes December 1984. 0141-1187/84/040227-02 $2.00 © 1984 CML Publications
order of hours. Hence, in the long-term cases mentioned above, equation (3) can no longer be applied to investigate the extreme values over periods of time of such length. However, equation (3) is easily generalised to the nonstationary case by retaining the assumption of independent upcrossings of high levels. The corresponding expression in this case is4 T
where the mean frequency of ~-upcrossings, as determined by equation (2), is no longer independent of t. It will be shown subsequently that by exploiting equation (4), i.e. by using the upcrossing analysis technique, a very neat and compact expression is obtained for the long-term probability distribution function of the extreme value. This is in contrast to the more or less complex expressions that appear from time to time in the periodicals, expressions that are generally based on peak value analyses, which will not in any case give better extreme value estimates than the upcrossing analysis technique. ~ In order to apply equation (4) in the study of longterm statistics of extremes, we shall have to incorporate information about the variability in time of the statistical properties of X(t). This is done by assuming that the properties of X(t) locally are uniquely specified by environmental parameters, like, for example, significant wave height H s and mean zero-upcrossing period T~, which are then considered to be stochastic variables6-9 or rather, stochastic processes. This approach is based on the assumption that X(t) can be considered locally stationary; i.e. to coincide with a stationary process over time intervals of restricted length. This is equivalent to assuming that the environmental parameters in question can be considered constant over these same time intervals. In the sequel W(t) denotes the stochastic vector process of the environmental parameters, e.g. W(t) = (Hs(t), Tz(t)). To be able to incorporate the statistical variability of W(t) into equation (4), it is necessary to impose a particular condition on the process W(t). This concerns the possibility of deriving estimates of statistical parameters from time averages, i.e. ergodicity (in some specific sense). Although not generally stated explicitly, the assumption of an ergodic property on W(t) is, in fact, basic to the concept of long-term statistics. Regarding the problem of estimating the probability distribution functions of the stochastic process W(t), all we have available i~ part of a realization of W(t). To be able to use this limited information to estimate the probabilistic structure of the process W(t), requires the assumption that W(t) is ergodic (in distribution).
Applied OceanResearch, 1984, Vol. 6, No. 4
227
Technical note The following development will serve to illustrate how the assumption of ergodicity is exploited. A stochastic process Z(t) is introduced by defining
where v~ =
-+ f v~(w)fw(w ) dw
Z(t) = 1, W(t) < w = 0, if not
w
(5)
where w is arbitrary, but fixed. The short-hand notation W(t) < w means componentwise inequalities. From equation (6) follows immediately that E [Z(t)] = Fw(w)
(6)
where F w denotes the probability distribution function of W(t). We now form the time average T
Zr=TfZ(t)
dt
(7)
Fw(w)
(8)
0
Clearly,
E [ZT]
=
E [Z(t)]
=
The assumption of ergodicity ensures the validity of the following equalities lim Z T = E [ZT] = Fw(W ) w.p. 1 (9) T~ '~ where w.p. 1 denotes with probability one. 2"10 A consequence of this is that for almost every realization w(t) o f W(t), the length of time, dTw, that w(t) spends within the interval (w, w + dw) throughout (0, T) is given by dTw = Tfw(w) dw,
T ~ oo
(15)
In order to calculate statistical moments like E[,~(T)] and Var D((T)], a method developed by the author u may be used. It requires that at least an asymptotic expression is available for P~(~ -> oo), and that this asymptotic function satisfies certain conditions, which it seems to do in most practical cases. It should perhaps be stated explicitly that equation (14) requires T to be large compared to the scales of variation of W(t). Since W(t). is known by observation to contain seasonal variation, i.e. to contain a scale of variation of the order of months, T in equation (14) should therefore be correspondingly large. This comment immediately points to another problem connected to the modelling of W(t) as an ergodic process. It is hard to imagine that the seasonal variation of the sea state as it is observed, i.e. mild summer breeze versus strong winter gale, is just accidental, as it should be if our model assumptions had been correct. However, the question of whether equation (14) is acceptable or not, does not entirely depend on whether we accept or reject the model assumptions, but rather on whether we can agree on an averaging function to use in equation (14). Then, an equation like (14) may be used also for, say, a particular month in the winter, provided an appropriate averaging function (corresponding to a conditional probability density function) can be given
(10)
where fw denotes the joint probability density function of W(t). fw is independent of t since W(t) was assumed stationary. From equation (10) it then follows that for almost every realization w(t) o f W(t)
ACKNOWLEDGEMENTS
This work has been financially supported by the Royal Norwegian Council for Scientific and Industrial Research (NTNF) and the Norwegian State Oil Company (Statoil).
T
0
REFERENCES
w
where v~(w(t)) denotes the ~-upcrossing frequency of the stochastic process X(t) given that W ( t ) = w(t) (t fixed). Equation (11) may also be derived directly by assuming W(t) to be strictly ergodic. 2' ~o Then (provided both sides of" the equation are well-defined and finite) T
lira T . - . + oo
;-f
v~(W(t))dt=E[v~(W(O))] w.p. 1. (12)
0
Since /" ElY,(W(0))] = J v~(w)fw(w) dw
(13)
w
it follows that equation (11)holds for almost every realization w(t). Assuming now that equation (11) holds for the particular realization at hand, it is seen from equations (4) and (11) that
F£(r)(~)=exp {-- f v~(w) Tfw(w) dw} w
= exp {-- p~ T},
228
T ~ oo
Applied Ocean Research, 1984, Vol. 6, No. 4
(14)
l
Naess, A. On a rational approach to extreme value analysis,
Applied Ocean Research 1984, 6 (3) 2 Cram6r, H. and Leadbetter, M. R. Stationary and Related Stochastic Processes, John Wiley and Sons, Inc. New York, 1967 3 Adler, R. J. The Geometry of Random Fields, John Wiley and Sons, Inc. New York, 1981 4 Lm, Y. K. Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, 1967 5 Rice, S. O. Mathematical analysis of random noise. In Selected Papers on Noise and Stochastic Processes, Wax, N. ed., Dover Publications, New York, 1954 6 Nordenstr~m, N. Methods for predicting long term distributions of wave loads and probability of failure for ships. Appendix I Long term distributions of wave height and period, 1969, DnV Report No. 69-21-$. Det norske Veritas, Oslo 7 St Denis, M. On statistical techniques for predicting the extreme dimension of ocean waves and of amplitudes of ~hip responses. Proc. First STAR Symposium, The Society of Naval Architects and Marine Engineers, New York, 1975 8 Ochi, M. K. Wave statistics for the design of ships and ocean structures, SNAME Transactions 1978, 86, 47 9 Battjes, J. A. Long-term wave height distributions at seven stations around the British Isles. Deutsche Hydrographische Zeitschrfft 1972, 25 (4) 10 Dooh, J. L. Stochastic Processes, John Wiley and Sons, New York, 1953 11 Naess, A. The joint crossing frequency of stochastic processes and its application to wave theory. To appear in Applied Ocean
Research.