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WATER RESOURCES RESEARCH, VOL. ???, XXXX, DOI:10.1002/,

100 years of Return Period: Strengths and

limitations

E. Volpi,1A. Fiori,1S. Grimaldi,23 F. Lombardo,1and D. Koutsoyiannis4

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Corresponding author: E. Volpi, Department of Engineering, University of Roma Tre, Via Vito

Volterra, 62, 00146 Roma, Italy (elena.volpi@uniroma3.it)

1Department of Engineering, University of

Roma Tre, Via V. Volterra, 62, 00146

Rome, Italy

2Department for Innovation in Biological,

Agro-food and Forest systems (DIBAF),

University of Tuscia, Via San Camillo De

Lellis snc, 01100 Viterbo, Italy

3Honors Center of Italian Universities

(H2CU), Sapienza University of Rome, Via

Eudossiana 18, 00184 Roma, Italy

4Department of Water Resources, Faculty

of Civil Engineering, National Technical

University of Athens, Heroon Polytechneiou

5, 15780 Zographou, Greece

D R A F T September 22, 2015, 2:23pm D R A F T

: 100 YEARS OF RETURN PERIOD X - 3

Abstract. 100 years from its original deﬁnition by Fuller [1914], the prob-1

abilistic concept of return period is widely used in hydrology as well as in2

other disciplines of geosciences to give an indication on critical event rareness.3

This concept gains its popularity, especially in engineering practice for de-4

sign and risk assessment, due to its ease of use and understanding; however,5

return period relies on some basic assumptions that should be satisﬁed for6

a correct application of this statistical tool. Indeed, conventional frequency7

analysis in hydrology is performed by assuming as necessary conditions that8

extreme events arise from a stationary distribution and are independent of9

one another. The main objective of this paper is to investigate the proper-10

ties of return period when the independence condition is omitted; hence, we11

explore how the diﬀerent deﬁnitions of return period available in literature12

aﬀect results of frequency analysis for processes correlated in time. We demon-13

strate that, for stationary processes, the independence condition is not nec-14

essary in order to apply the classical equation of return period (i.e. the in-15

verse of exceedance probability). On the other hand, we show that the time-16

correlation structure of hydrological processes modiﬁes the shape of the dis-17

tribution function of which the return period represents the ﬁrst moment.18

This implies that, in the context of time-dependent processes, the return pe-19

riod might not represent an exhaustive measure of the probability of failure,20

and that its blind application could lead to misleading results. To overcome21

this problem, we introduce the concept of Equivalent Return Period, which22

D R A F T September 22, 2015, 2:23pm D R A F T

X-4 : 100 YEARS OF RETURN PERIOD

controls the probability of failure still preserving the virtue of eﬀectively com-23

municating the event rareness.24

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: 100 YEARS OF RETURN PERIOD X - 5

1. Introduction

“The storm event had a return period of 30 years” or “this dam spillway was designed25

for a 1000-year return period discharge” are two classical statements that one could read26

or hear everyday. High-school students could read them in newspapers, housewives could27

hear them at the market or hydrologists could write them in a technical report. This simple28

example recalls that the return period is the most ubiquitous statistical concept adopted29

in hydrology but also in many other disciplines (seismology, oceanography, geology, etc...).30

It appears that the concept of return period was ﬁrst introduced by Fuller [1914] who31

pioneered statistical ﬂood frequency analysis in the USA. Return period ﬁnds wide pop-32

ularity mainly because it is a simple statistical tool taken from engineering practices33

[Gumbel, 1958]. For example, engineers who work on ﬂood control are interested in the34

expected time interval at which an event of given magnitude is exceeded for the ﬁrst time,35

which gives a deﬁnition of the return period. Another common deﬁnition is the average of36

the time intervals between two exceedances of a given threshold of river discharge. From37

a logical standpoint, the ﬁrst deﬁnition is as justiﬁable as the second one; they generally38

diﬀer, even though they become practically indistinguishable if consecutive events are39

independent in time. Both are used in hydrology [Fern´andez and Salas, 1999a, b] and, in40

this paper, we will show how they may aﬀect the frequency analysis applications under41

certain conditions.42

The return period is inversely related to the probability of exceedance of a speciﬁc43

value of the variable under consideration (e.g. river discharge). For example, the annual44

maximum ﬂood-ﬂow exceeded with a 1% probability in any year is called the 100-year45

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X-6 : 100 YEARS OF RETURN PERIOD

ﬂood. Therefore, a T-year return period does not mean that one and only one T-year46

event should occur every Tyears, but rather that the probability of the T-year ﬂood being47

exceeded is 1/T in every year [Stedinger et al., 1993].48

The traditional methods for determining the return period of extreme hydrologic events49

assume as key conditions that extreme events (i) arise from a stationary distribution, and50

(ii) are independent of one another. The hypotheses of stationarity and independence51

are commonly assumed as necessary conditions to proceed with conventional frequency52

analysis in hydrology [Chow et al., 1988]. Recently, the former assumption has been53

questioned by several researchers [e.g. Cooley, 2013; Salas and Obeysekera, 2014; Du54

et al., 2015; Read and Vogel, 2015]. However, we endorse herein the following important55

statement by Gumbel [1941] about the general validity of stationarity assumption. “In56

order to apply any theory we have to suppose that the data are homogeneous, i.e. that no57

systematical change of climate and no important change in the basin have occurred within58

the observation period and that no such changes will take place in the period for which59

extrapolations are made. It is only under these obvious conditions that forecasts can be60

made”. The reader is also referred to Koutsoyiannis and Montanari [2015] and Montanari61

and Koutsoyiannis [2014], where it can be noted that many have lately questioned the62

stationarity assumption, but careful investigation of claims made would reveal that they63

mostly arise from the confusion of dependence in time with nonstationarity.64

The purpose of this paper is to investigate the properties of return period when the65

independence condition is omitted. In hydrology, indeed, dependence has been recognized66

by many scientists to be the rule rather than the exception since a long time [e.g. Hurst,67

1951; Mandelbrot and Wallis, 1968]. The concept of dependence in extreme events relates68

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: 100 YEARS OF RETURN PERIOD X - 7

to the fact that the occurrence of a high or low value for the variable of interest (e.g.69

river discharge) has some inﬂuence on the value of any succeeding observation. Leadbetter70

[1983] found that the type of the limiting distribution for maxima is unaltered for weakly71

dependent occurrences of extreme events. We demonstrate that, under general depen-72

dence conditions, the classical relationship between the return period and the exceedance73

probability is again unaltered. On the other hand, we investigate the impact of the de-74

pendence structure on the shape of the distribution function of which the return period75

represents the ﬁrst moment.76

Based on the papers by Fern´andez and Salas [1999a], Sen [1999], and Douglas et al.77

[2002] we ﬁrst summarize in Section 2 the available deﬁnitions of return periods (aver-78

age occurrence interval - and - average recurrence interval) specifying the mass function79

equations and the related return period formulae. Moreover, in Section 2.2 and 2.3 the80

independent and time-dependent cases are analyzed in detail, while an Appendix provides81

the proof that the widely used return period equation (average recurrence interval) is not82

aﬀected by the dependence structure of the process of interest. However, in Section 2.383

it is pointed out that the time-dependence inﬂuences the shape of the interarrival time84

distribution function and the probability of failure.85

Two illustrative examples, i.e. using a two-state Markov process and an autoregressive86

process, are described in Section 3 and results are discussed in Section 4 in order to in-87

vestigate further the theoretical premises depicted in Sections 2.2 and 2.3. Besides, to88

overcome the diﬃculties that arise from the application of the return period concept in a89

time-dependent context, we propose in Section 4.1 the adoption of an Equivalent Return90

Period (ERP ), which resembles the classical deﬁnition of return period in the case of in-91

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X-8 : 100 YEARS OF RETURN PERIOD

dependence while it is able to control the probability of failure under the time-dependence92

condition. The ERP can be useful to avoid introducing the concept of probability of fail-93

ure in engineering practice. Indeed, the latter may not be as simple to understand as the94

return period, which is a well-established concept in applications, routinely employed by95

practitioners.96

Concluding remarks discuss the obtained results by stressing caution against using the97

concept of return period blindly given that multiple deﬁnitions exist. However, we conﬁrm98

the virtue of return period showing that the classical formulation is insensitive to the time-99

dependence condition.100

2. Return period and probability of failure

2.1. Mathematical framework

Let Z(τ) be a stochastic process that characterizes a natural process typically evolving101

in continuous time τ. As observations of Z(τ) are only made in discrete time, it is assumed102

here that the observations are made at constant time intervals ∆τ, and this interval is103

considered the unit of time. Hence, we consider the corresponding discrete-time process104

that is obtained by sampling Z(τ) at spacing ∆τ, i.e. Zj=Z(j∆τ) where j(= 1,2, ...)105

denotes discrete time. For convenience, herein we express discrete time as t=j−j0,106

where j0is the current time step; therefore the discrete-time process is indicated as Zt

107

and t= 0 denotes the present. We assume that Ztis a stationary process [Papoulis, 1991];108

thus, it is fully described up to the second order properties by its marginal probability109

function and its autocorrelation structure. Generally, in this paper we use upper case110

letters for random variables or events, and lower case letters for values, parameters, or111

constants.112

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We are interested in the occurrence of possible excursions of Ztabove/below a high/low113

level (threshold) z, which may determine the failure of a structure or system. In particular,114

we deﬁne a dangerous event as A={Z > z}, which is an extreme maximum; anyway,115

Acould be any type of extreme event, i.e. maximum or minimum. In the following we116

denote by pthe probability of the event B={Z≤z}, which is the complement of A; the117

probability of the event Ais given by 1 −p= Pr {Z > z}= Pr A.118

In hydrological applications, it is usually assumed that the event Awill occur on average119

once every return period T, where Tis a time interval and, for annual observations (i.e.,120

∆τ= 1 year), a number of years. In other words, the average time until the threshold z121

is exceeded equals Tyears [Stedinger et al., 1993], such as122

T

∆τ= E [X] =

∞

t=1

t fX(t) (1)

where Xis the number of discrete time steps to the occurrence of an event A,fX(t) =123

Pr {X=t}is its probability mass function (pmf) and E[.] denotes expectation. The124

deﬁnition of the return period leads to the formulation of the so-called probability of125

failure R(l) (also known in literature as ”risk”, even if it does not account for damages)126

which measures the probability that the event Aoccurs at least once over a speciﬁed127

period of time: the design life l(e.g. in years) of a system or structure, where l/∆τis a128

positive integer. Mathematically, we have129

R(l) = Pr{X≤l/∆τ}=

l/∆τ

t=1

fX(t) (2)

Thus, the probability of failure is nothing else than the distribution function FX(t) com-130

puted at t=l/∆τ.131

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As mentioned in the Introduction, two diﬀerent deﬁnitions of the return period are132

available in the hydrological literature [see, e.g., Fern´andez and Salas , 1999a and Douglas133

et al., 2002]. The return period Tmay be deﬁned as:134

(i) the mean time interval required to the ﬁrst occurrence of the event A,135

(ii) the mean time interval between any two successive occurrences of the event A.136

Deﬁnition (i) assumes that an event Aoccurred in the past (at t < 0); the discrete time137

elapsed since the last event Ato the current time step t= 0 is deﬁned as elapsing time and138

it is denoted here as te; the sketch in Figure 1 illustrates the variables used in the present139

analysis. In this work, we assume that time tecan be either deterministically known or140

unknown and investigate implications of both conditions on the analytical formulation of141

the return period. Under deﬁnition (i), the return period is based on the waiting time142

(W), i.e. the number of time steps between t= 0 and the next occurrence of A(see143

Figure 1). The sum of the waiting time and the elapsing time is denoted as interarrival144

time N=W+te.145

If we assume that teis unknown, the probability mass function of the waiting time146

is given by the joint probability of the sequence of events (B1, B2, ..Bt−1, At) (see, e.g.,147

Fern´andez and Salas , 1999a)148

fW(t) = Pr (B1, B2, ..Bt−1, At) (3)

where At(Bt) is the event A(B) occurred at time t. Instead, if teis determin-149

istically known, the pmf of the waiting time is given by the joint probability of150

the sequence of events (B1, B2, ..Bt−1, At) conditioned to the realization of the events151

(A−te, B−te+1, ..B−1, B0) occurred at t≤0, i.e.152

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: 100 YEARS OF RETURN PERIOD X - 11

fW|te(t) = Pr (B1, B2, ..Bt−1, At|A−te, B−te+1, ..B−1, B0) (4)

=Pr (A−te, B−te+1, ..B−1, B0, B1, B2, ..Bt−1, At)

Pr (A−te, B−te+1, ..B−1, B0)

Deﬁnition (ii) assumes that an event Ahas just occurred at t= 0. In such a case te= 0153

and the waiting time Wis identical to the interarrival time N. The pmf of the interarrival154

time fNis therefore a special case of equation (4), for te= 0, i.e.155

fN(t) = Pr (B1, B2, ..Bt−1, At|A0) (5)

=Pr (A0, B1, B2, ..Bt−1, At)

Pr A0

Note that Figure 1 depicts a more general case than the one represented by equation (5).156

In the Figure, we assume that two successive occurrences of the dangerous event Aare157

at times −teand t. Then, Nis the time elapsed between the two. As stated above, the158

speciﬁc case expressed by equation (5) can be obtained by setting te= 0. Moreover, we159

stress here that the relation N=W+tein the Figure holds only in the case the elapsing160

time teis known, i.e. when we account for the conditional waiting time W|te.161

It is interesting to note that the probability distributions of the unconditional (W, equa-162

tion (3)) and conditional (W|te, equation (4)) waiting time are interrelated. In Appendix163

A we derive some useful relations between the return periods TW,TW|teand TN.164

Substituting fW(equation (3)), fW|te(equation (4)) or fN(equation (5)) to fXin (1)165

and (2), we obtain the expressions of the return periods TW,TW|teand TNand of the166

corresponding probabilities of failure RW,RW|teand RN, respectively. In general, the167

probability mass functions given by equations (3) to (5) are expected to have diﬀerent168

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shapes, leading to diﬀerent values of the return period of the event A. In the following,169

we illustrate and discuss the diﬀerences among the above deﬁnitions when varying the170

correlation structure of the process Zt; speciﬁcally, we study ﬁrst the independent case,171

which is customary in hydrological applications, and then the more general case with some172

positive correlation in time (persistent case).173

2.2. Independent case

If Ztis a purely random process, then its random variables are mutually independent174

and their joint probability distribution equals the product of marginal ones. Therefore, we175

may write e.g. Pr (B0, B1, B2, ..Bt−1, At) = Pr B0Pr B1... Pr At. Substituting in equations176

(3), (4) and (5) the products of the marginal exceedance or non-exceedance probabilities177

and thanks to the stationarity assumption (that implies Pr At= 1 −pand Pr Bt=p178

for any t), we can derive the same geometric distribution in all cases. Therefore, fW=179

fW|te=fN=f, with180

f(t) = pt−1(1 −p) (6)

It follows from equation (1) that the return period T(T=TW=TW|te=TN) is given181

by182

T

∆τ=1

1−p(7)

while the variance of the pmf (6) is v=p/ (1 −p)2. From equation (6), it also follows183

that the probability of failure given by equation (2) becomes184

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: 100 YEARS OF RETURN PERIOD X - 13

R(l) = 1 −1−∆τ

Tl/∆τ

= 1 −pl/∆τ(8)

where again R=RW=RW|te=RN.185

Thus, for the independent case all the deﬁnitions of return period collapse to the same186

expression (7). This result, which is well known in the literature [e.g. Stedinger et al.,187

1993], builds on the fact that in the independent case the occurrence of an event at any188

time t≤0 does not inﬂuence what happens afterwards.189

2.3. Persistent case

Although independence of Ztis usually invoked for the derivation of equation (7) [e.g.190

Kottegoda and Rosso, 1997, p. 190], it is possible to show that the mean interarrival time191

TNis equal to (7) also in case of processes correlated in time; the general proof, which192

is given here for the ﬁrst time, is illustrated in detail in Appendix B. The same property193

was shown by Lloyd [1970] for the particular case of a Markov chain process. As shown194

in Appendix B, equation (7) for the mean interarrival time holds true, regardless of the195

type of the correlation structure of Zt.196

Even though the dependence structure of the process Ztdoes not aﬀect the expected197

value of N(i.e., TN), we show that this is not the case with its pmf fN(see equation198

(5)). Let us consider a process characterized by a positive correlation in time. If a199

dangerous event Aoccurs at t= 0, then the conditional probability of occurrence of200

another dangerous event at t= 1 will be greater than 1 −p(independent case); this201

yields that the probability mass function fN(t) will have a larger mass for t= 1 and a202

lower mass elsewhere with respect to the independent case (equation (6)). Hence, while203

the mean value remains the same, the variance of the interarrival time Nis larger than204

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X - 14 : 100 YEARS OF RETURN PERIOD

that of the independent case and it increases with the temporal correlation. This implies205

that the probability of failure RN(following equation (2)) is strongly aﬀected by the206

time-dependence structure of the process.207

Conversely, the return periods TWand TW|tedo account for the temporal correlation of208

Zt. Recalling that (1 −p) = 1/E [N] (see equations (7) and (1)), it can be shown that209

(see Appendix A, equation (A8))210

TW

TN

=1

2E [N2]

E [N]2+1

E [N](9)

Equation (9) shows that TWis greater than or equal to TN. It is easy to check that211

TW=TNfor independent processes, in line with the discussion reported under Section212

2.2. When the process is correlated in time, the term E [N2]/E [N]2is expected to increase213

with the autocorrelation of the process, thus resulting in the inequality TW> TN. Hence,214

the mean waiting time is generally larger than the mean interarrival time for temporally215

correlated processes.216

In the following Sections we will examine the pmfs of the waiting times Wand W|teand217

the interarrival time N, as well as their average values (TW,TW|teand TN), as functions218

of the temporal correlation of the process. To this end, we make use of two diﬀerent219

illustrative examples, the ﬁrst is based on a Markov chain, while the second uses an220

AR(1) model. For convenience - and without loss of generality - ∆τis set equal to one.221

3. Illustrative examples

3.1. Example 1: two state Markov-dependent process

We consider here a stochastic process Ztwhich is based on a Markov chain Yt. This222

process is considered here since it allows to easily derive the analytical expressions of223

D R A F T September 22, 2015, 2:23pm D R A F T

: 100 YEARS OF RETURN PERIOD X - 15

the probability mass functions of the waiting and interarrival times, as done in previous224

literature works by Lloyd [1970], Rosbjerg [1977] and Fern´andez and Salas [1999a]. The225

Markov chain Ythas two states, which here represent the events At={Zt> z}and226

Bt={Zt≤z}with probability 1 −pand p, respectively. For the Markov property, the227

probability of a state at a given time tdepends solely on the state at the previous time228

step t−1, e.g. Pr (Bt|Bt−1...B0) = Pr (Bt|Bt−1). Applying the chain rule to the Markov229

property (e.g. Papoulis, 1991, p. 636), it follows that the joint probability of a sequence230

of states, e.g. Pr (B1, B2, ..Bt) = Pr{Z1≤z, Z2≤z, ..., Zt≤z}, can be written as231

Pr (B1) Pr (B2|B1)... Pr (Bt|Bt−1) = Pr{Z1≤z}Pr{Z2≤z|Z1≤z}... Pr{Zt≤z|Zt−1≤232

z}.233

The process Ztdescribed above is indicated in the following as two state Markov-234

dependent process and denoted by 2Mp. For each value of p(i.e. of z)Ztis fully charac-235

terized by the marginal probabilities of the states Aand B(1 −pand p) and by the tran-236

sition probability matrix, M= [[Pr (At+1|At),Pr (At+1|Bt)],[Pr (Bt+1|At),Pr (Bt+1|Bt)]]237

where Pr (At+1|At) + Pr (Bt+1|At) = 1 and Pr (At+1 |Bt) + Pr (Bt+1|Bt) = 1. We denote238

by qthe joint probability of non-exceedance of the threshold value zfor two successive239

events, i.e. q= Pr (Bt+1 , Bt) for any t; it ensues that M= [[1 −(p−q)/(1 −p),1−240

q/p],[(p−q)/(1 −p), q/p]].241

The probability mass function of the unconditional waiting time fW(equation (3))242

becomes243

fW(t) = 1−p(t= 1)

pq

pt−21−q

p(t≥2) (10)

with mean given by244

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X - 16 : 100 YEARS OF RETURN PERIOD

TW= 1 + p2

(p−q)(11)

and variance var [W] = p2(p−p2+q)/(p−q)2. After substituting equation (10) in (2),245

the probability of failure in a period of length lis given by246

RW(l) = 1 −pq

pl−1

(12)

while the pmf of the conditional waiting time fW|te(equation (4)) for te>0 reduces to247

fW|te(t) = q

pt−11−q

p(13)

with mean248

TW|te=p

(p−q)(14)

and variance var [W|te] = pq/(p−q)2. The probability of failure based on the conditional249

waiting time is given by250

RW|te(l) = 1 −q

pl−1

(15)

Equation (14) shows how for the 2Mp model the mean waiting time distribution is not251

aﬀected by the value of te. This builds upon the fact that the conditional non-exceedance252

probability at tdepends only on that at t−1, due to the property of the Markov chain.253

Finally, the pmf of the interarrival time N(equation (5)) assumes the following expres-254

sion255

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: 100 YEARS OF RETURN PERIOD X - 17

fN(t) = 1−(p−q)/(1 −p) (t= 1)

(p−q)

(1−p)q

pt−21−q

p(t≥2) (16)

while its mean is given by equation (7) with ∆τ= 1 (following the general proof given in256

Appendix B), and the variance is equal to var [N] = p(p−2p2+q)/[(p−1)2(p−q)]. The257

probability of failure in a period of length lis given by258

RN(l) = 1 −p−q

1−pq

pl−1

(17)

The joint probability qmay assume values in the range [max(2p−1,0) , p]: the lower259

and upper bounds correspond to perfect negative and positive correlations in time, respec-260

tively; in the independent case, q=p2. We consider here only processes positively corre-261

lated (i.e. persistent), as it is commonly the case in hydrology (e.g. rainfall and discharge);262

thus, q∈[p2, p]. Furthermore, we assume that Ztis a standard Gaussian process and that263

the joint probability qis ruled by a bivariate Gaussian distribution; under the latter264

assumption, qcan be described in terms of the lag-1 autocorrelation coeﬃcient ρ. Speciﬁ-265

cally, qis computed as q= Pr {Zt+1 ≤z, Zt≤z}=z

−∞ z

−∞ fZ(ztzt+1;0,Σ2)dzt+1dzt

266

where fZis the probability density function of the bivariate Gaussian distribution267

N2(Z;0,Σ2) with zero mean and Σ2={{1, ρ},{ρ, 1}}, with ρ∈[0,1]. Note that ρ268

denotes the correlation in the parent process Ztand not that between the events exceed-269

ing the threshold, i.e. A={Z > z}. The correlation between the extremes is ruled by270

the shape of the parent bivariate distribution, which is assumed here to be Gaussian; the271

latter assumption implies that the correlation between the events Ais negligible to null272

for high threshold values, since the Gaussian process is asymptotic independent.273

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3.2. Example 2: AR(1) process

We now assume that Ztfollows an AR(1) process (ﬁrst-order autoregressive process), i.e.274

Zt=ρZt−1+αtwhere ρis the lag-1 correlation coeﬃcient and αt∼ N 0,1−ρ2, such275

that the process is characterized by a multivariate Gaussian distributions Nt(Z;0,Σt)276

with Z={Z1, Z2..Zt}and Σt=ρ|i−k|,i, k = 1..t. We assume again ρ∈[0,1].277

Even if conceptually simple and similar to the 2Mp (see e.g. Saldarriaga and Yevjevich,278

1970), AR(1) is rather diﬀerent in terms of the pmfs fW,fW|teand fN. Both the processes279

are based on the Markov property; however, in AR(1) the Markov property applies to the280

continuos random variable Zand not to the state Y={Z≶z}. It means that in AR(1)281

the joint probability fZ(z1, z2..zt) can be expressed as fZ(z1)fZ(z2|z1)..fZ(zt|zt−1), while282

the same simpliﬁcation cannot apply to the joint probability of a sequence of states, e.g.283

Pr (B1, B2, ..Bt) = Pr{Z1≤z, Z2≤z, ..., Zt≤z}, as for 2Mp. The joint probability of284

any sequence can be estimated by proper integration of the joint pdf of the multivariate285

Gaussian distribution Nt. This entails that the pmfs fW,fW|teand fN, given by equations286

(3), (4) and (5) respectively, cannot be simpliﬁed as in the case of 2Mp, but they can be287

written as288

fW(t) = z

−∞ z

−∞

.. +∞

z

fZ(z1, z2..zt;0,Σt)dz1dz2..dzt(18)

fW|te(t) = +∞

zz

−∞ .. +∞

zfZ(z−tez−te+1..zt;0,Σt+te)dz−tedz−te+1..dzt

+∞

zz

−∞ .. z

−∞ fZ(z−tez−te+1..z0;0,Σte)dz−te..dz0

(19)

while fNcan be derived from the latter under the assumption te= 0. Finally, substituting289

the previous expressions in (1) and (2) we get the corresponding return periods and290

probabilities of failure.291

D R A F T September 22, 2015, 2:23pm D R A F T

: 100 YEARS OF RETURN PERIOD X - 19

Interestingly enough, unlike the 2Mp, fW|te(19) depends on te, i.e. the elapsing292

time. This relies on the fact that the conditional non-exceedance probability at t, i.e.293

Pr (Bt|Bt−1...B0), generally depends on the whole sequence of previous events for AR(1),294

while it only depends on that at t−1 for the 2Mp. In such a sense, AR(1) is more295

correlated than 2Mp.296

4. Results and discussion

We start this Section by discussing the eﬀects of temporal correlation on the probability297

mass functions fW(equation (10)) and fN(equation (16)), and the related return periods298

TW,TN(equation (11) and (7) with ∆τ= 1, respectively) for the two state Markov-299

dependent process (2Mp).300

Figure 2 illustrates TWand TNas functions of the independent return period T(i.e. of301

the non-exceedance marginal probability p) for several values of the correlation coeﬃcient302

ρ. It is seen that TNequals T, being independent of ρas demonstrated in Appendix B;303

for ρ= 0 (black line) it is always TN=TW=T= (1 −p)−1. Conversely, the mean304

waiting time TWincreases with ρ(equation (9)); TWis always greater than the mean305

interarrival time TN, which thus represents a lower bound for the return period (Figure306

2a). Speciﬁcally, for values of Taround 5, TWis roughly eight times larger than TNfor307

ρ= 0.99 and about twice for ρ= 0.75; for small and very large values of T(i.e. for308

small and high values of the threshold z, respectively) TWtends to the independent limit309

T= (1 −p)−1(Figure 2b).310

As discussed in Section 2.3, although TN=Tfor any ρ, the pmf fN(as well as fW) may311

be signiﬁcantly inﬂuenced by the correlation structure of the Ztprocess. The distribution312

functions of Wand Nare illustrated in Figure 3, for various values of ρand p= 0.9.313

D R A F T September 22, 2015, 2:23pm D R A F T

X - 20 : 100 YEARS OF RETURN PERIOD

The mean values for each distribution (i.e. the return periods normalized with respect to314

∆τ= 1) are denoted by the vertical dashed lines. The broadness of both distributions315

increases with ρ, as also indicated by the increase of their variance and skewness (not316

shown).317

Figure 3a shows that the distribution function computed at TW, which corresponds to318

the probability of failure in the period TW(see equation (2)), is independent of ρtaking319

approximately the value 0.63 for high values of p[Stedinger et al., 1993].320

On the other hand, FNchanges dramatically when increasing temporal correlation ρ.321

This may result in very high values of the probability of failure for the same TN, even322

for small time intervals t(Figure 3b). Thus, although the return period TNremains the323

same for correlated and independent processes (all the vertical dashed lines corresponding324

to the diﬀerent values of ρcollapse into a unique line, depicted in black), the probability325

that the threshold zis exceeded in the period TNcan be much larger for the former than326

for the latter (up to about 0.9 for the limit case ρ= 0.99).327

We now illustrate and discuss the probability functions for W,W|teand Nfor the AR(1)328

process, as well as the corresponding mean values, as functions of the lag-1 autocorrelation329

coeﬃcient ρ. Results are compared to those obtained for the previously analyzed 2Mp330

case.331

The probability mass functions fW(equations (18)) and fN(equation (19) for te= 0)332

for AR(1) are similar to those for 2Mp, even if they are characterized by a much larger333

dispersion, and thus they are not shown here. Their averages TWand TNare depicted334

in Figure 4, as function of the independent return period T, for ρ= 0.75 and ρ= 0.99.335

TWand TNfor AR(1) (continuos lines) are also compared to those pertaining to the 2Mp336

D R A F T September 22, 2015, 2:23pm D R A F T

: 100 YEARS OF RETURN PERIOD X - 21

(dashed lines). The mean waiting times TWfor the two models are similar, although TW

337

is generally larger for AR(1); since the two processes have the same ρ, this result is a338

direct consequence of the stronger correlation of AR(1) with respect to 2Mp, as explained339

in previous Section. Larger diﬀerences are expected for even more persistent processes,340

i.e. processes characterized by a longer range persistence with respect to the AR(1).341

As mentioned in the previous Section, the stronger correlation of AR(1) also inﬂuences342

the mean conditional waiting time TW|te, which depends on the elapsing time tein contrast343

to that of 2Mp. TW|te(te) is illustrated in Figure 5 for p= 0.9 and for a few values of the344

correlation coeﬃcient ρ. For each value of ρ,TW|teis by deﬁnition equal to the mean inter-345

arrival time TNfor te= 0 (see equation (4)); TW|teincreases with tetending to an asymp-346

totic value that is greater than TW(dashed lines). This behaviour arises from the fact that347

the conditional non-exceedance probability (B1, B2, ..Bt−1, At|A−te, B−te+1, ..B−1, B0) (eq.348

4) depends on the whole sequence of previous events. However, as tebecomes very high349

the previous dangerous event A−tehas occurred too distant in time to signiﬁcantly aﬀect350

the realization of the next event at time t; the latter is mainly controlled by a sequence351

of antecedent events whose length strictly depends on the shape of the autocorrelation352

function of the underling process Zt. Due to the exponential shape of the AR(1) auto-353

correlation function, i.e. ρt(t) = ρt,TW|teis expected to approach the asymptotic value354

when tebecomes larger than the integral scale of the process, λ(ρ) = 1/(1 −ρ).355

Conversely, TW|tefor 2Mp maintains a constant value for any te>0 since the con-356

ditional joint probability in equation (4) Pr(B1, B2, ..Bt−1, At|A−te, B−te+1, ..B−1, B0) de-357

pends only on the state at t= 0, due to the Markov property of the Ytchain (as already358

D R A F T September 22, 2015, 2:23pm D R A F T

X - 22 : 100 YEARS OF RETURN PERIOD

discussed in Section 3.1); moreover, being inﬂuenced by a longer sequence of safe events359

(B), both TW|teand TWof AR(1) are larger than those of 2Mp (results not shown).360

We ﬁnally explore how the probabilities of failure RW(TW), RW|teTW|teand RN(TN)361

behave as functions of the correlation coeﬃcient ρ; results are summarized in Figure 6 for362

the processes 2Mp and AR(1) and compared to the independent case. For both processes,363

the probability of failure based on the interarrival time (N) may assume values much364

larger than the independent case; RN(TN) signiﬁcantly increases with the autocorrelation365

of the process ρ, (compare e.g. 2Mp for ρ= 0.75 and ρ= 0.99) and, more generally, with366

the correlation structure of the process (compare AR(1) and 2Mp for the same value of ρ).367

On the contrary, when we consider the waiting time W(conditional and unconditional),368

the probability of failure is less than the independent case. This reduction is signiﬁcant369

when we account for the elapsing time te, thus when we add information about the last370

dangerous event occurred in the past. Note that Figure 6 speciﬁcally refers to the cases371

te= 10 for AR(1) while it is representative of any te>0 for 2Mp. As for AR(1),372

RW|teTW|tereduces with respect to the independent case when teis much larger than373

the integral scale of the process, i.e. te> λ when ρ= 0.75 (λ= 4) (Fig. 6a); conversely,374

when the event Ahas happened in the recent past (when ρ= 0.99, we have te< λ with375

λ= 100), the conditional waiting time for high phas a behaviour which approaches that376

of the interarrival time (i.e. with higher probability of failure than the independent case,377

as in Figure 6b).378

4.1. Equivalent Return Period (ERP)

The return period is a means of expressing the exceedance probability. Despite being379

a standard term in engineering applications (in engineering hydrology in particular), the380

D R A F T September 22, 2015, 2:23pm D R A F T

: 100 YEARS OF RETURN PERIOD X - 23

concept of return period is not always an adequate measure of the probability of failure381

and has been sometimes incorrectly understood and misused [Serinaldi, 2014]. The results382

discussed in previous Section strengthen the above message, extending it to correlated Zt

383

ﬁelds (with Markovian dependence); for the cases examined here, the statistics of the384

waiting or interarrival time show negligible diﬀerences with respect to the independent385

case for small values of ρ, while they are strongly aﬀected by the autocorrelation when386

ρ&0.5 (see Figures 2 and 5). Consequently, using directly the probability of failure387

in engineering practice could be a better choice under the latter condition. However,388

although more eﬀective and appropriate, the probability of failure may not be as simple389

to understand as the return period, which is already an established concept in applications390

and routinely employed by practitioners.391

To overcome this problem, we introduce the concept of ”equivalent” return period392

(ERP ). Its aim is to retain the relative simplicity of the return period concept and393

extend it to temporally correlated hydrological variables; for correlated processes, ERP394

is deﬁned to be the period that would lead to the same probability of failure pertaining to395

a given return period Tin the framework of classical statistics (independent case). Hence,396

ERP resembles the classical deﬁnition of return period in the case of independence, thus397

preserving its simplicity and strength in indicating the event rareness; in addition it is398

able to control the probability of failure under the time-dependence condition.399

ERP can be deﬁned starting from the concept of interarrival time (N) or waiting400

time (W). Practitioners should adopt the most appropriate deﬁnition according to the401

circumstances, the task and the data available. If the time teelapsed since the last402

dangerous event is known, it could be adopted the deﬁnition based on the conditional403

D R A F T September 22, 2015, 2:23pm D R A F T

X - 24 : 100 YEARS OF RETURN PERIOD

waiting time, or that based on the interarrival time in the case te= 0; the latter could404

be the case where an existing structure failed because of an event Aand the immediate405

construction of another structure is needed (as discussed by Fern´andez and Salas [1999a]).406

In the case we are accounting for the interarrival time (N), ERP can be calculated407

assuming RN(ERP ) = R(T) where RNis the probability of failure based on the inter-408

arrival time (equation (2) for fX=fN), while R(T) is given by equation (8) for l=T.409

For the 2Mp RNis given by equation (17) (where ∆τ= 1) when l=ERP ; thus, the410

analytical formulation of ERP can be easily derived as411

ERP = 1 + ln 1−p

p−q+1

1−pln p

ln q

p

(20)

For the AR(1), RNcan be numerically computed by substituting equation (19) in (2).412

In the case of more complex models for the simulation of hydrological quantities, ERP413

could be computed directly by numerical Monte Carlo simulations.414

Figure 7 depicts the behaviour of ERP as function of T, for both the AR(1) (continuous415

lines) and 2Mp processes (dashed lines; equation (20) with p= 1 −1/T ). The ﬁgure416

shows that the values of ERP and Ttend to coincide asymptotically; this is especially417

so for small correlation coeﬃcients. For a given T, the value of ERP is always smaller418

(sometimes much smaller) than T; diﬀerences increase with the correlation coeﬃcient ρ419

and with the correlation structure of the process (compare AR(1) to 2Mp). Recalling that420

T= 1/(1 −p), Figure 7 can be used either to determine ERP when the p-th quantile zis

421

known (i.e., for a given event A={Z > z}that will be exceeded with probability 1 −p)422

in risk assessment problems, or to determine the design variable (i.e. the threshold z) in423

terms of ponce the ERP is ﬁxed in design problems; in the latter case we choose ERP424

D R A F T September 22, 2015, 2:23pm D R A F T

: 100 YEARS OF RETURN PERIOD X - 25

and then calculate the design variable z, such that the probability of failure is equal to425

that we should have in the independent case.426

We emphasize that results shown here are obtained under several assumptions, such as427

the type of temporal correlation, bivariate Gaussian distribution, etc.; this implies that, for428

example, a diﬀerent distribution may result in larger diﬀerences between the independent429

and time-correlated conditions (due, e.g., to asymptotic dependence). Hence, further work430

is needed to generalize the above results.431

5. Conclusions

The return period is a critical parameter largely adopted in hydrology for risk assess-432

ment and design. It is deﬁned as the mean value of the waiting time to the next dangerous433

event (TW) or the interarrival time between successive dangerous events (TN). As shown434

in previous literature, both deﬁnitions lead to the same result in the case of time inde-435

pendence of the underlying process. However, in cases of time-persistent processes the436

two deﬁnitions lead to diﬀerent expressions. Hence, we reexamine herein the above deﬁ-437

nitions in the context of temporally correlated processes; furthermore, by making use of438

two illustrative examples we discuss the eﬀects of the temporal correlation ρof the parent439

process on the return period and the probability of failure. The examples proposed here440

are based on a two state Markov-dependent process (2Mp), and an AR(1) process; even441

if the two processes share the Markov property, they are characterized by rather diﬀerent442

time distributions.443

The main conclusions drawn in this paper are listed below.444

•We provide a unitary framework for the estimation of the return periods TW,TN

445

and the related probabilities of failure RW,RNin the context of persistent processes:446

D R A F T September 22, 2015, 2:23pm D R A F T

X - 26 : 100 YEARS OF RETURN PERIOD

we provide general relationships for the probability functions of the waiting time W(un-447

conditional and conditional on the time teelapsed since the last dangerous event) and448

the interarrival time N. The choice between Wand Nin applications depends on the449

available information on past events and the type of structure.450

•We demonstrate that the mean interarrival time TNis not aﬀected by the time-451

dependence structure of the process, e.g. the correlation coeﬃcient ρ. Thus, the well452

known formula for independent processes is valid for any process, temporally correlated453

or not.454

•Although TNis not aﬀected by ρ, for persistent processes the corresponding proba-455

bility of failure can be much larger than that pertaining to the independent case, which is456

itself not negligible. Hence, the mean interarrival time TNcan easily provide a biased and457

wrong perception of the risk of failure, especially in the presence of temporally correlated458

hydrological variables.459

•On the other hand, the mean waiting times eﬀectively account for the correlation460

structure of the hydrological process. TWis always larger than the mean interarrival461

time TN, which acts as a lower bound. If the time tefrom the last dangerous event is462

deterministically known, we can use that information to condition the waiting time Wto463

the next occurrence.464

•The return periods TWand TW|tetypically increase with the correlation ρ. Specif-465

ically, they depend on the overall correlation structure of the process, as highlighted by466

comparing results for 2Mp and AR(1); in the case of processes characterized by a longer467

range persistence with respect to the AR(1), we may expect even stronger diﬀerences.468

D R A F T September 22, 2015, 2:23pm D R A F T

: 100 YEARS OF RETURN PERIOD X - 27

•The analyses carried out here provide some further insight into the overall meaning469

and signiﬁcance of the return period, especially in view of hydrological applications, but470

also in other geophysical ﬁelds. Despite being a simple and easy to implement metric, the471

return period should be used with caution in the presence of time-correlated processes.472

Indeed, the probability of failure depends on the whole shape of the probability function,473

which in turn may strongly depend on ρ, and the return period is just the ﬁrst order474

moment; the latter may not be relevant when in presence of asymmetric and skewed475

distributions, like e.g. some of those displayed in Figure 3.476

•To partially overcome the above limitations, we propose to adopt in the time-477

dependent context the Equivalent Return Period (ERP ), which preserves the virtue of the478

classical return period of eﬀectively communicating the event rareness. ERP resembles479

the classical deﬁnition of return period in the case of independence, while it is able to480

control the probability of failure under the time-dependence condition.481

We conclude with a note on the practical implications of the present analysis. Results482

shown here highlight that the independence condition is not necessary for the application483

of the classical return period equation; notwithstanding this, practitioners should take484

care of the time-persistence structure of the process when estimating risk from data, to485

correctly evaluate the probability of failure (e.g. through E RP ). However, it is interesting486

to stress that the diﬀerences between the correlated and uncorrelated case are small to487

negligible when ρ.0.5. Thus, the temporal correlation of the process may be safely488

disregarded in such cases, as far as the return period is concerned.489

Acknowledgments. We thank the Editor and the three anonymous Reviewers for490

their thoughtful comments. The research has been partially funded by the Italian Min-491

D R A F T September 22, 2015, 2:23pm D R A F T

X - 28 : 100 YEARS OF RETURN PERIOD

istry of University and Research through the projects PRIN 2010JHF437 and PRIN492

20102AXKAJ. No data was used in producing this manuscript.493

D R A F T September 22, 2015, 2:23pm D R A F T

: 100 YEARS OF RETURN PERIOD X - 29

Appendix A: General relationships between fW,fW|teand fN

Since we can write that Pr (A−te, B−te+1 , ..B−1, B0) = Pr (B0, B1, ..Bte−1, Ate) =494

fW(te+ 1), the probability mass function of the conditional waiting time, fW|te(t), can495

be expressed as function of fWand fNas in the following496

fW|te(t) = Pr (A−te, B−te+1, ..B−1, B0, B1, B2, ..Bt−1, At)

Pr (A−te, B−te+1, ..B−1, B0)(A1)

=Pr (B−te+1, ..B−1, B0, B1, B2, ..Bt−1, At|A−te) Pr A−te

Pr (A−te, B−te+1, ..B−1, B0)

=(1 −p)

fW(te+ 1)fN(t+te)

By making use of the simple identity Pr(C) = Pr(AC) + Pr(BC), which is valid for any497

events Aand C(with Balways denoting the complement of A), fWcan be expressed as498

function of fN

499

fW(t) = Pr (B1, ..Bt−1, At) (A2)

= Pr (A0, B1, ..Bt−1, At) + Pr (B0, B1, ..Bt−1, At)

= Pr (B1, ..Bt−1, At|A0) Pr A0+ Pr (B0, B1, ..Bt−1, At)

=fN(t) (1 −p) + fW(t+ 1)

by solving equation (A2) for fNand substituting the resulting expression in (A1) we500

obtain501

fW|te(t) = 1

fW(te+ 1) [fW(t+te)−fW(t+te+ 1)] (A3)

Since fNis a special case of fW|te, when te= 0 equation (A3)502

D R A F T September 22, 2015, 2:23pm D R A F T

X - 30 : 100 YEARS OF RETURN PERIOD

fN(t) = 1

1−p[fW(t)−fW(t+ 1)] (A4)

Moreover, if we exploit the recursive property of equation (A2), we can write503

fW(2) = fW(1) −(1 −p)fN(1) (A5)

fW(3) = fW(2) −(1 −p)fN(2)

=fW(1) −(1 −p)fN(1) −(1 −p)fN(2)

fW(4) = fW(3) −(1 −p)fN(3)

=fW(1) −(1 −p)fN(1) −(1 −p)fN(2) −(1 −p)fN(3)

...

thus obtaining504

fW(t+ 1) = fW(1) −(1 −p)

t

k=1

fN(k) (A6)

= (1 −p)1−

t

k=1

fN(k)

= (1 −p) [1 −FN(t)]

= (1 −p)FN(t)

where we used fW(1) = Pr A1= 1 −pand the survival function of N, i.e. FN(t) =505

1−FN(t) = 1 −t

k=1fN(k) = ∞

k=t+1fN(k). The relationship between fW|teand fNis506

obtained by substituting equations (A4) and (A6) into (A3)507

fW|te(t) = fN(t+te)

FN(te)(A7)

D R A F T September 22, 2015, 2:23pm D R A F T

: 100 YEARS OF RETURN PERIOD X - 31

We adopt equation (A6) to derive the analytical expression of the return period TWas508

function of fN

509

TW

∆τ=∞

t=1t(1 −p)FN(t−1) = (1 −p)∞

t=1tF N(t−1) (A8)

= (1 −p)∞

t=1t∞

k=tfN(k) = (1 −p)∞

k=1fN(k)k

t=1t

= (1 −p)∞

k=1

k(k+ 1)

2fN(k)

= (1 −p)∞

k=1

k2

2fN(k) + ∞

k=1

k

2fN(k)

=(1 −p)

2EN2+ E [N]

Finally, substituting equation (A7) into (1) we obtain TW|teas function of fN

510

TW|te

∆τ=∞

t=1tfN(t+te)

FN(te)(A9)

=1

FN(te)∞

t=1 [(t+te)fN(t+te)−tefN(t+te)]

=1

FN(te)∞

k=te+1kfN(k)−te∞

k=te+1fN(k)

=1

FN(te)∞

k=te+1kfN(k)−teFN(te)

=∞

k=te+1

kfN(k)

FN(te)−te

Appendix B: Mean interarrival time, TN

Substituting equation (5), which is of general validity, in (1) we have511

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X - 32 : 100 YEARS OF RETURN PERIOD

TN

∆τ=

∞

t=1

t fN(t) = 1 Pr{N= 1}+ 2 Pr{N= 2}+... (B1)

= Pr (A1|A0) + 2 Pr (B1, A2|A0) + 3 Pr (B1, B2, A3|A0) + ...

=1

Pr A0

[Pr (A0, A1) + 2 Pr (A0, B1, A2) + 3 Pr (A0, B1, B2, A3) + ...]

=1

1−p[Pr (A0, A1) + 2 Pr (A0, B1, A2) + 3 Pr (A0, B1, B2, A3) + ...]

By making use again of the identity Pr(CA) = Pr(C)−Pr(CB), where Balways denotes512

the opposite event of A, we obtain513

TN

∆τ=1

1−p[(Pr A0−Pr (A0, B1)) + 2 (Pr(A0, B1)−Pr (A0, B1, B2)) (B2)

+3 (Pr (A0, B1, B2)−Pr (A0, B1, B2, B3)) + ..]

=1

1−p[Pr A0+ Pr (A0, B1) + Pr (A0, B1, B2) + Pr (A0, B1, B2, B3) + ..]

Using once more the same identity, we ﬁnd514

TN

∆τ=1

1−p[(1 −Pr B0) + (Pr B1−Pr (B0, B1)) (B3)

+ (Pr (B1, B2)−Pr (B0, B1, B2)) + ..]

=1

1−p

which proves to be valid because of stationarity, i.e. Pr B0= Pr B1, Pr (B0, B1) =515

Pr (B1, B2), etc.516

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: 100 YEARS OF RETURN PERIOD X - 33

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D R A F T September 22, 2015, 2:23pm D R A F T

X - 36 : 100 YEARS OF RETURN PERIOD

List of Figures572

1. Illustrative sketch of the quantities involved in the deﬁnitions of the return period:573

excursions of the Ztprocess above/below a threshold level zdeﬁning the dangerous (At)574

and safe (Bt) events.575

2. Two state Markov-dependent process (2Mp): return periods TWand TNas function576

of Tfor several values of the correlation coeﬃcient ρin absolute value (a) and normalized577

with respect to the independent value T(b). Note that TN=Tfor every value of ρ, while578

TW=TN=Tfor ρ= 0 (black line).579

3. Two state Markov-dependent process (2Mp): distribution functions of the waiting580

time, FW(a) and of the interarrival time, FN(b) for p= 0.9 and for several values of the581

correlation coeﬃcient ρ; the averages of the distributions (return periods) are indicated by582

the vertical dashed lines. For the sake of clarity, the distribution functions of the discrete583

random variables Wand Nare represented as continuous functions.584

4. Return periods TWand TNas function of Tand for two values of the correlation585

coeﬃcient ρfor the AR(1) process (continuous lines) compared to the two state Markov-586

dependent process (2Mp, dashed lines). Note that TN=Tfor every value of ρ, while587

TW=TN=Tfor ρ= 0 (black line).588

5. AR(1) process: mean conditional waiting time TW|te(continuous lines) as function589

of the elapsing time tefor p= 0.9 and for several values of the correlation coeﬃcient ρ;590

the corresponding mean unconditional waiting times TW(dashed lines) are depicted as591

reference.592

6. Probabilities of failure RW(TW) (continuous lines), RW|teTW|te(dot-dashed lines)593

or RN(TN) (dashed lines) as functions of pfor both AR(1) (a, b) and 2Mp (c, d); graphs594

D R A F T September 22, 2015, 2:23pm D R A F T

: 100 YEARS OF RETURN PERIOD X - 37

refer to the cases ρ= 0.75 (a, c) and ρ= 0.99 (b, d). Note that RW|teTW|teof 2Mp is595

valid for any te>0. Results are compared to the independent case (black line).596

7. Equivalent Return Period (ERP ), based on the interarrival time N, as function of597

the independent return period Tfor several values of the lag-1 correlation coeﬃcient ρ;598

curves for ρ < 0.75 are not shown because the diﬀerences between ERP and Tare small599

to negligible.600

D R A F T September 22, 2015, 2:23pm D R A F T

−

−1 01−1

waiting time,

elapsing time,

interarrival time, =+

={>}, dangerous event

={≤}, safe event

110 100 1000 10

4

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=AR(1)

=2Mp

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Probability of failure

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cd

ab

||

||

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