Volpi and coworkers discussed the idea of return period. Her statement is that independence is not a requirement for obtaining the classical equation of return period. This is inscribed in the old story of stochastic processes used as representation of phenomena that appear highly variable. The field needs some refreshment after the discovery of climate change. Volpi's et al. Has the merit to bring in some novelty. I was intrigued by the separation and relation between return period and waiting time.

1. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Independence is not a necessary condition
for the classical equation of return period
E. Volpi1, A. Fiori1, S. Grimaldi2, F. Lombardo1 and D. Koutsoyiannis3
1 Dipartimento di Ingegneria, Università Roma Tre (elena.volpi@uniroma3.it)
2 Dipartimento DIBAF, Università della Tuscia
3 Department of Water Resources and Environmental Engineering, National Technical
University of Athens, Greece
Criteri, metodi e modelli per l’analisi dei processi idrologici e la gestione delle acque
Metodi statistici per le applicazioni idrologiche

2. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Return period
• First introduced by Fuller (1914) – who pioneered statistical flood
frequency analysis in USA – to quantify hydrologic events rareness (e.g.
floods, draughts, etc.)
• Hypotheses commonly assumed in hydrology as necessary conditions
for conventional frequency analysis
1. Events arise from a stationary distribution
2. Events are independent of one another

3. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Return period
• First introduced by Fuller (1914) – who pioneered statistical flood
frequency analysis in USA – to quantify hydrologic events rareness (e.g.
floods, draughts, etc.)
• Hypotheses commonly assumed in hydrology as necessary conditions
for conventional frequency analysis
1. Events arise from a non-stationary distribution
2. Events are independent of one another
Salas and Obeysekera, Revisiting the
Concepts of Return Period and Risk for
Nonstationary Hydrologic Extreme Events
(JHE, 2014)
Read and Vogel, Reliability, return
periods, and risk under nonstationarity
(WRR, 2015)
Serinaldi and Kilsby, Stationary is undead:
Uncertainty dominates the distribution of
extremes (ADWR, 2015)
Du et al., Return period and risk analysis
of nonstationary low-flow series under
climate change (ADWR, 2015)
Read and Vogel, Hazard function analysis
for flood planning under nonstationarity
(WRR, 2016)

4. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Return period
• First introduced by Fuller (1914) – who pioneered statistical flood
frequency analysis in USA – to quantify hydrologic events rareness (e.g.
floods, draughts, etc.)
• Hypotheses commonly assumed in hydrology as necessary conditions
for conventional frequency analysis
1. Events arise from a stationary distribution
2. Events are not independent of one another
• Considerations
– Dependence has been recognized to be the rule rather than the exception
(e.g. Hurst, 1951; Mandelbrot, 1968)
– Non-stationarity may be confused with dependence in time (Montanari and
Koutsoyiannis, 2014)
– Stationarity should remain the default assumption (Gumbel, 1941; Serinaldi
and Kilsby, 2015)

5. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Definitions and properties
• Traditional methods define return period as the mean of
⇒ the mean of the waiting time to the next event
⇒ the mean of the interarrival time between successive events
−1 0 1 − 1
≤
= Pr =
>
= Pr = 1 −
−
: continuous and stationary
stochastic process in discrete time
present time

6. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Definitions and properties
• Traditional methods define return period as the mean of
⇒ the mean of the waiting time to the next event
⇒ the mean of the interarrival time between successive events
−1 0 1 − 1
≤
= Pr =
>
= Pr = 1 −
−
waiting time, =
present time

7. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Definitions and properties
• Traditional methods define return period as the mean of
⇒ the mean of the waiting time to the next event
⇒ the mean of the interarrival time between successive events
−1 0 1 − 1
≤
= Pr =
>
= Pr = 1 −
−
waiting time, =elapsing time,
interarrival time, = +

8. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Definitions and properties
• Traditional methods define return period as the mean of
⇒ the mean of the waiting time to the next event
⇒ the mean of the interarrival time between successive events
• Independent events: both definitions lead to the same formula
=
1
1 −
• Dependent events (Volpi et al., 2015)
1. Mean interarrival time: = whatever the time-dependence
structure of the process is
2. Mean waiting time: is affected by the autocorrelation structure of
the process
Volpi, E., A. Fiori, S. Grimaldi, F. Lombardo, and D. Koutsoyiannis (2015), One hundred years of return period:
Strengths and limitations, Water Resour. Res., 51, doi:10.1002/2015WR017820.

9. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
1 10 100 1000 104
1
10
100
1000
104
,
, two state Markov-dependent model, 2Mp
Pr( , ) = N , ;
, lag-1 correlation coefficient of the parent process
= =
independent case
r=0
r=0.25
r=0.5
r=0.75
r=0.95
r=0.99
Mean waiting time,

10. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Mean waiting time,
1 5 10 50 100 500 1000
0
1
2
3
4
10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
,
, fractionally integrated autoregressive process, FAR(1, )
Pr( , . . ) = N , ;
= 0.75, lag-1 correlation coefficient of the parent process
2MpAR(1), = 0.5
increasing
0.5 ≤ ≤ 0.9
autocorrelation function
parent process

11. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Mean waiting time,
1 10 100 1000 104
1
10
100
1000
104
, two state Markov-dependent model, 2Mp
Pr( , ) = G , ;
, lag-1 correlation coefficient of the parent process
ruled by the asymptotic
dependence of the joint
distribution
= N
= G
0
0.5
0.95
0.5
0.95
,
0 1 2 3 4
0
1
2
3
4
0 1 2 3 4
0
1
2
3
4
NG
joint pdfs

12. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Probability functions of and
= whatever the time-dependence structure of the process is
• Both the probability functions and are affected by the
autocorrelation structure of the process
1 10 100 1000
0.0
0.2
0.4
0.6
0.8
1.0
- interarrival time
= 0.9 ( = 10)
r=0
r=0.25
r=0.5
r=0.75
r=0.95
r=0.99
AR(1) process
1 5 10 50 100 500
0.0
0.2
0.4
0.6
0.8
1.0 - waiting time

13. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Probability of failure
= whatever the time-dependence structure of the process is
• Both the probability functions and are affected by the
autocorrelation structure of the process
• Probability of failure ( )
= Pr ≤ =
• , design life of the
structure/system
• Probability of failure in ,
~ 0.63
• for large T (independent case)
1 10 100 1000
0.0
0.2
0.4
0.6
0.8
1.0
- interarrival time
= 0.9 ( = 10)
~0.63 (independent case)
r=0
r=0.25
r=0.5
r=0.75
r=0.95
r=0.99
AR(1) process

14. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Equivalent return period (ERP)
1 10 100 1000
0.0
0.2
0.4
0.6
0.8
1.0
• : the period that would lead to the same probability of failure
pertaining to a given return period in the framework of classical
statistics (independent case)
= 0.9 ( = 10)
~0.63
r=0.75
independent case
persistent case
0
0.75
0.85
0.9
0.95
AR 1
2Mp

15. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Conclusions
• Return period properties are generally ruled by the joint probability
distribution in time and by the autocorrelation function of the parent
process
• The return period based on the concept of waiting time, effectively
accounts for the correlation structure of the hydrological process
• The return period (mean interarrival time) is not affected by the
time-dependence structure of the process
• The corresponding probability of failure, ( ), can be larger than
that pertaining to the independent case
• We propose the Equivalent Return Period ( ) for the time-
dependent context

16. XXXV CONVEGNO NAZIONALE DI IDRAULICA
E COSTRUZIONI IDRAULICHE
Bologna, 14-16 Settembre 2016
Main references
• Du, T., Xiong, L., Xu, C. Y., Gippel, C. J., Guo, S., and Liu, P. (2015). Return period and risk analysis of nonstationary low-flow
series under climate change. Journal of Hydrology, 527, 234-250.
• Fernández, B., and J. D. Salas (1999), Return period and risk of hydrologic events. I: mathematical formulation, Journal of
Hydrologic Engineering, 4(4), 308-316, doi:10.1061/(ASCE)1084-0699(1999)4:4(308).
• Fernández, B., and J. D. Salas (1999a), Return period and risk of hydrologic events. II: Applications, Journal of Hydrologic
Engineering, 4(4), 308-316, doi:10.1061/(ASCE)1084-0699(1999)4:4(308).
• Fuller, W. (1914), Flood flows, Transactions of the American Society of Civil Engineers, 77, 564-617.
• Gumbel, E. J. (1941), The return period of flood flows, Ann. Math. Stat., 12(2), 163–190.
• Gumbel, E. J. (1958), Statistics of Extremes, Columbia University Press, New York.
• Hurst, H. E. (1951), Long term storage capacities of reservoirs, Transactions of the American Society of Civil Engineers, 116(776-
808).
• Mandelbrot, B. B., and J. R. Wallis (1968), Noah, Joseph and operational hydrology, Water Resources Research, 4(5), 909-918.
• Montanari, A., and D. Koutsoyiannis (2014), Modeling and mitigating natural hazards: Stationarity is immortal!, Water Resources
Research, 50, 9748-9756, doi:10.1002/2014WR016092.
• Obeysekera, J., and Salas, J. D. (2016). Frequency of Recurrent Extremes under Nonstationarity. Journal of Hydrologic
Engineering, 21(5), 04016005.
• Read, L. K., and Vogel, R. M. (2016). Hazard function theory for nonstationary natural hazards. Natural Hazards and Earth
System Sciences, 16(4), 915.
• Salas, M., and J. Obeysekera (2014), Revisiting the Concepts of Return Period and Risk for Nonstationary Hydrologic Extreme
Events, Journal of Hydrologic Engineering, 19:554-568, doi:10.1061/(ASCE)HE.1943-5584.0000820.
• Serinaldi, F. (2014), Dismissing return periods!, Stochastic Environmental Research and Risk Assessment, pp. 1.11,
doi:10.1007/s00477-014-0916-1.
• Serinaldi, F., and C.G. Kilsby (2015), Stationary is undead: Uncertainty dominates the distribution of extremes, Advance in Water
Resources, 77, 17-36, doi:10.1016/j.advwatres.2014.12.013.
• Volpi, E., A. Fiori, S. Grimaldi, F. Lombardo, and D. Koutsoyiannis (2015), One hundred years of return period: Strengths and
limitations, Water Resour. Res., 51, doi:10.1002/2015WR017820.