4.2.4 Lead time contribution in ePAD
4.2.4.2 Value of information model
For many EEW systems based on a regional network, a warning is continually updated at some fixed time interval, ∆t. In this case, the decision problem need no longer be a single decision in time because one may choose not to initiate any action at time t, then another decision may be made at t+ ∆t when the updated EEW information may have less uncertainty. A more general version of this problem can arise in optimal policy searching for control or robotics problems, or optimal stopping time for option pricing. A standard methodology is to develop a dynamic model for the evolution of the state of interest and pick a policy that optimizes a predetermined cost function. However, in the case of EEW, because of the extremely short lead time, there will be
no turning back once an action is initiated. Hence, except a0 (no action initiated), all actions will lead to a state termination. This simplifies the problem to identify the extra benefit (or cost) ofa0
due to a possibly better (or worse) decision made att+ ∆tbased on all future EEW information, rather than a full optimal policy searching algorithm. In other words, the same decision criterion as shown in Equation 4.7 can be used, but E[DF|D(t), a0], which is originally equal to 0 because not taking any action induces zero benefit and cost, has an added value due to the potential change of EEW information in the future:
E[DF|D(t), a0] =V oI (4.13) where, motivated by the theory of the value of information (Howard, 1966), I model the added value V oI as the expected performance of the decision criterion over the domain of p(IM, Tlead|D(t+ n∆t)) for alln >0 minus the original expected performance ofa0 (which equals zero).
Applying the theory of value of information (Section 2.6) to the case of multiple EEW warning, the gain function G(π) is related to DF(IM, Tlead, a), where the decision policy π is simply the chosen decision criteria. The information I is related to the future updated EEW information D(t+n∆t) for integern >0. Technically,I ={D(t+n∆t)|∀n >0}:
E[G(πI)|I] = max
a∈Ωa
{E[DF|a, D(t), D(t+ ∆t), ...]} (4.14) p(I) =p(D(t+ ∆t), D(t+ 2∆t), ...|D(t)) (4.15) However, modeling of p(I) in this case refers to modeling the probability of all possible fu- ture seismic data based on the current seismic data, which is extremely difficult. An alternative is to choose a model for estimating future EEW information. For example, if one models the EEW informationp(IM, Tlead|D(t)) as a product of two independent log-normal PDFs with means µE(t), µT(t), and standard deviations σE(t), σT(t), an intuitive choice of modeling future EEW information can be made by fixing the mean values but reducing the standard deviation values,
that is for some real valueν and for alln >0:
µE(t+n∆t) =µE(t), µT(t+n∆t) =µT(t) σE(t+n∆t) = σE(t)
νn , σT(t+n∆t) = σT(t) νn
In general, for any model chosen for estimating future EEW information with a vector of parameters ν:
E[G(πI)] = Z
E[DF|ν]p(ν|D(t)) dν (4.16) Many models can be used to estimatep(I), but most of them will lead to a complexE[DF|ν] that becomes impossible to evaluate. In this study, I propose a simple model based on the assumption of perfect information.
The most ideal assumption one can make on future EEW information is to assume that in the next EEW update, D(t+ ∆t) will provide enough information to perfectly determine IM (or M andR depending on the EEW system). Although this is an impractical assumption that can never be achieved, it provides important insights into the influence of uncertain lead time on decision making for EEW applications. Insights and possible improvement of this model for practical use are discussed in the example in Section 4.4. The perfect information assumption implies that I ={IM ,ˆ Tˆlead}, where ˆIM and ˆTlead represent the trueIM and Tlead values, respectively, and:
p(I) =p( ˆIM ,Tˆlead) =p(IM, Tlead|D(t)) (4.17) which is the current EEW information because it reflects the current belief on the true IM and Tlead values. Meanwhile:
E[G(πI)|I] = max
a∈ΩaE[DF|IM, Tlead−∆t, a] (4.18) Note that ∆tis subtracted fromTleadbecauseE[G(πI)|I] refers to the expected value under decision made at t+ ∆t, where the current estimate of Tlead will be reduced by ∆t.
As E[G(π)] = E[DF|D(t), a0] = 0 because it represents the expected value of not taking any action at timetwhen no future information is available, the added value due to the uncertain lead
time equals V oI:
V oI =E[G(πI)]−0
= Z Z ∞
∆t
i∈[0,n]max E[DF|IM, Tlead−∆t, ai]p(IM, Tlead|D(t)) dTleaddIM
(4.19)
where fori∈[1, n]:
E[DF|IM, Tlead−∆t, ai] =DF(IM, Tlead−∆t, ai) (4.20) because expected value of DF is simply DF when IM and Tlead are given.
For the case i= 0:
E[DF|IM, Tlead−∆t, a0] =DF(IM, A0) = 0 (4.21) Note that whenTleadis between 0 and ∆t, no new decision can be made, meaning thatE[G(πI)|I] = 0. Therefore the integral in Equation 4.19 excludesTlead values between 0 and ∆t.
In summary, this model can be simply represented by a single DF:
DF(IM, Tlead−∆t, a0)
=
maxi∈[1,n]{0, E[DF|IM, Tlead−∆t, ai]}, Tlead >∆t
0, Tlead ≤∆t
(4.22)
In practice, large earthquakes usually induce losses that are substantially larger than the cost of a mitigation action. Therefore, a decision maker is likely to prefer avoiding a missed alarm to a false alarm. DF without a lead time model, which is the basic cost-benefit analysis case, will usually give a substantially larger risk of a false alarm than a missed alarm. In this sense, the value of information model aims at reducing the false alarm rate while maintaining a low missed alarm rate. However, the perfect information model may suppress the false alarm too much so that it leads to an unacceptable increase in the missed alarm rate. To compensate the ”overconfidence”
from assuming perfect information at the next EEW update, ∆t could be artificially increased to reduce the influence of theV oI model (as shown in Section 4.4.2.2). This is equivalent to assuming perfect information at a larger ∆tvalue, but the decision will be re-done before the assumed perfect
information is obtained. Hence, the decision is made based on nearly perfect information instead of perfect information, which indirectly compensates for the overconfidence while maintaining a relatively low computational effort for practical purposes. Other practical models for estimating future EEW information to calculate Equation 4.16 are left for future research.