Download Talk

Estimating exceedance probabilities for extreme ship
motions in irregular waves
Vladas Pipiras
University of North Carolina at Chapel Hill
David Taylor Model Basin - NSWCCD, July 1, 2014
Vladas Pipiras (UNC)
Exceedance probabilities
1 / 10
Basic problem
Suppose given a (time history) series of ship motion, say for 100 hours in
some condition, generated by an advanced hydrodynamics code or model
test. For example, the figure below depicts 5 minutes of the roll motion for
the ONR tumblehome top at the 45◦ heading, the speed of 6 knots, ...
generated by one of the simpler hydrodynamics codes.
Basic extrapolation problem: Estimate the probability of the roll angle
exceeding some large critical value, e.g. 60◦ (in either direction), and
provide a confidence interval (CI). Such exceedance would not be observed
in the given series, thus the term “extrapolation”.
Vladas Pipiras (UNC)
Exceedance probabilities
2 / 10
Goals, including validation
A proposed approach should be such that:
It is mathematically justified.
It passes a validation procedure.
Validation: A fast, simplified physics code is available to produce millions
of hours of ship motions that contain extreme events of interest. The
“true” exceedance probabilities can be deduced (estimated) directly from
the (time history) series.
A number of shorter records can be considered, not containing extreme
events of interest. The confidence intervals can be given in each record
according to the proposed method. The approach is then valid if the
number of these intervals containing the “true” value corresponds to the
confidence level.
E.g. with 95% confidence level and 100 records, about 95 of the CIs
should contain the “true” value.
Vladas Pipiras (UNC)
Exceedance probabilities
3 / 10
Proposed solution
Punchline: There is an approach which is mathematically justified and
passes validation.
The figure below depicts the performance of the confidence intervals
obtained through, what we call, lognormal method, for 100 records (of 100
hours each, under the earlier condition). The “true” probability is deduced
from the series of 115, 000 hours (containing 30 exceedances over 60◦ ).
Validation: The coverage frequency is around 95% as expected.
Vladas Pipiras (UNC)
Exceedance probabilities
4 / 10
How (and why) does this work?
Peaks-over-threshold (POT) approach: Let X denote the variable of
interest, e.g. the roll angle. Then, for the target x and a threshold u,
P(X > x) = P(X > u) · P(X > x|X > u).
The “non-rare” P(X > u) is estimated as the proportion of data above u.
Estimating the “rare” prob. is based on the fact: for large u and x > u,
P(X > x|X > u) ≈ 1 +
ξ(x−u)
σ
−1/ξ
= F u,ξ,σ (x),
where Fu,ξ,σ is the generalized Pareto distribution (GPD).
CIs: The CIs for P(X > x) are obtained by multiplying the respective
endpoints of the CIs for P(X > u) and P(X > x|X > u).
Vladas Pipiras (UNC)
Exceedance probabilities
5 / 10
Issues to consider
From previous slide:
P(X > x) = P(X > u)P(X > x|X > u) ≈ P(X > u) 1 +
ξ(x−u)
σ
−1/ξ
.
Most pressing issues:
What threshold u to choose? Or how to determine a range for GPD
fit? (Skipping)
What confidence intervals to use for exceedance probabilities in the
GPD framework?
How to deal with time dependence?
Other relevant issue:
Can the uncertainty (size) of confidence intervals be reduced?
Vladas Pipiras (UNC)
Exceedance probabilities
6 / 10
Time dependence and envelope peaks
The subsequent analysis is carried out on the envelope peaks only, the
so-called EPOT.
Vladas Pipiras (UNC)
Exceedance probabilities
7 / 10
GPD and confidence intervals
Generalized Pareto distribution (GPD): the complementary distribution
function has the form
 −1/ξ
ξ(x−u)


, u < x,
if ξ > 0,
1
+

σ

e−
F u,ξ,σ (x) =



 1+
x−u
σ
ξ(x−u)
σ
,
−1/ξ
u < x,
if ξ = 0,
, u < x < u + (− σξ ), if ξ < 0,
where ξ is the so-called shape parameter, σ is the scale parameter and u is
a threshold.
Estimating exceedance probability: with the ML estimators ξb and σ
b,
and for the target c units above threshold u,
b −1/ξb
ξc
b
pc = F u,ξ,b
.
b σ (u + c) = 1 +
σ
b
What confidence intervals should be used?
Vladas Pipiras (UNC)
Exceedance probabilities
8 / 10
GPD and confidence intervals
ξ0
−.1
.1
.3
true values
σ0
c
1
6.02
6.84
7.49
1
15.12
21.62
29.81
1
49.5
102.08
206.99
pc
10−4
10−5
10−6
10−4
10−5
10−6
10−4
10−5
10−6
norm
90.4
95.2
94
92.6
90.6
91.2
91.8
88.8
93.4
direct methods
logn
bound
boot
90.8
96.2
68.2
95.6
97.6
65.6
94.6
96
74.6
93.2
98
87
91.2
97.2
82.8
92.6
97.6
81.2
92.4
98
89
89.2
98.4
86.6
94.2
99
91.8
profl
76.2
78.8
80.8
98.4
97.6
97.8
97.2
97.6
98.6
quantile methods
logn
bound
profl
92
97
95
88.8
96.6
94.6
91.2
96.8
94.2
89.6
97
94.2
92.4
98.6
95.2
91.2
98.4
94.6
89.4
97.4
92.2
92.2
97.8
95
92.6
98.4
94.4
The sample size is n = 100.
And the winner is . . . the quantile-profile method!
Vladas Pipiras (UNC)
Exceedance probabilities
9 / 10
Uncertainty reduction
Several possibilities:
Fixing the upper bound for negative shape parameter. Seems to work
well for pitch motion.
Pooling data from different conditions by modeling, e.g. the scale
parameter as log σ = a + b × heading. Possible but not much of
uncertainty reduction. (Note that this is of independent interest, e.g.
in nonstationary modeling or interpolation.)
Simultaneous modeling of roll, pitch and/or other variables from the
same condition. Currently investigated. (This is also of independent
interest.)
Thank You!
Vladas Pipiras (UNC)
Exceedance probabilities
10 / 10