Return period from a stochastic point of view

XXXV CONVEGNO NAZIONALE DI IDRAULICA E COSTRUZIONI IDRAULICHEBologna, 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 ([email protected])

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 acqueMetodi statistici per le applicazioni idrologiche


Return period• First introduced by Fuller (1914) – who pioneered statistical floodfrequency analysis in USA – to quantify hydrologic events rareness (e.g.floods, draughts, etc.)• Hypotheses commonly assumed in hydrology as necessary conditionsfor conventional frequency analysis1. Events arise from a stationary distribution2. Events are independent of one another


Return period• First introduced by Fuller (1914) – who pioneered statistical floodfrequency analysis in USA – to quantify hydrologic events rareness (e.g.floods, draughts, etc.)• Hypotheses commonly assumed in hydrology as necessary conditionsfor conventional frequency analysis1. Events arise from a non-stationary distribution2. Events are independent of one another Salas and Obeysekera, Revisiting the

Concepts of Return Period and Risk forNonstationary Hydrologic Extreme Events(JHE, 2014)

Read and Vogel, Reliability, returnperiods, and risk under nonstationarity(WRR, 2015)

Serinaldi and Kilsby, Stationary is undead:Uncertainty dominates the distribution ofextremes (ADWR, 2015)

Du et al., Return period and risk analysisof nonstationary low-flow series underclimate change (ADWR, 2015)

Read and Vogel, Hazard function analysisfor flood planning under nonstationarity(WRR, 2016)


Return period• First introduced by Fuller (1914) – who pioneered statistical floodfrequency analysis in USA – to quantify hydrologic events rareness (e.g.floods, draughts, etc.)• Hypotheses commonly assumed in hydrology as necessary conditionsfor conventional frequency analysis1. Events arise from a stationary distribution2. 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)


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



−1 0 1 − 1≤= Pr =>= Pr = 1 −

−waiting time, =present time



−1 0 1 − 1≤= Pr =>= Pr = 1 −

−waiting time, =elapsing time,

interarrival time, = +


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= 11 −• Dependent events (Volpi et al., 2015)1. Mean interarrival time: = whatever the time-dependence structure of the process is2. 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.


1 10 100 1000 1041

10

100

1000

104

,

, two state Markov-dependent model, 2MpPr( , ) = 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,


Mean waiting time,

1 5 10 50 100 500 10000

1

2

3

4

10 20 30 40 500.00.20.40.60.81.0,

, fractionally integrated autoregressive process, FAR(1, )Pr( , . . ) = N , ;= 0.75, lag-1 correlation coefficient of the parent process

2MpAR(1), = 0.5increasing 0.5 ≤ ≤ 0.9

autocorrelation functionparent process


Mean waiting time,

1 10 100 1000 1041

10

100

1000

104

, two state Markov-dependent model, 2MpPr( , ) = G , ; , lag-1 correlation coefficient of the parent processruled by the asymptoticdependence of the joint distribution

= N= G0

0.5

0.950.5

0.95

,

0 1 2 3 40

1

2

3

4

0 1 2 3 40

1

2

3

4 NG joint pdfs


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 10000.0

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 5000.0

0.2

0.4

0.6

0.8

1.0 - waiting time


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 10000.0

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


Equivalent return period (ERP)

1 10 100 10000.0

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)~0.63r=0.75

independent case

persistent case 0

0.75

0.85

0.9

0.95

AR 1

2Mp


Conclusions• Return period properties are generally ruled by the joint probabilitydistribution in time and by the autocorrelation function of the parentprocess• The return period based on the concept of waiting time, effectivelyaccounts for the correlation structure of the hydrological process• The return period (mean interarrival time) is not affected by thetime-dependence structure of the process• The corresponding probability of failure, ( ), can be larger thanthat pertaining to the independent case• We propose the Equivalent Return Period ( ) for the time-dependent context


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.

Return period from a stochastic point of view

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