In line with critical duration concept, the recommended approach suggests taking an envelope of model results to determine the rainfall duration that results in the maximum value--rather than treating the duration as a
stochastic variable. More guidance can be found in Chapter 4 of ARR (Nathan and Ling 2016; Nathan and Weinmann 2016), resulting from an earlier discussion paper by Nathan and Weinmann (2013); however, other reviews of previous applications and theory are also available.
Notations [R.sup.n]: n-dimensional Euclidean space Superscript T: Transpose E (x): Mathematical expectation of random variable x tr([omicron]): Trace of matrix [omicron] [[delta].sub.tj]: Kronecker delta function [I.sub.n]: n by n identity matrix [perpendicular to]: Orthogonality Prob(*): Probability of the occurrence of the event * [??]([omicron]|*): Estimate of the
stochastic variable x([omicron]) based on measurements before time *, i.e., the projection of x([omicron]) on the linear space generated by the measurements before time * [mathematical: Estimation error.
The
stochastic variable [[??].sub.ip] for each input can be expressed as [[??].sub.ip] = [[bar.x].sub.ip] + [a.sub.ip][xi], in which p has variation [1,b] and i has variation [1,n].
In the GBM, variable P follows a lognormal distribution, since the percentage rate of variation of the
stochastic variable (dP/P) follows a normal distribution with the mean and variance shown below:
Most of the existing works usually consider the trust values at the current time slot, and model trust as the
stochastic variable. However, in fact, trust evolves over time, and trust is a stochastic process.
It is proved that the state x(k) is uncorrelated with the
stochastic variable sequence [{w(k)}.sub.k [greater than or equal to] 0].
Step 2.The second step is to calculate the random component of each
stochastic variable. The random component is simply the residual from the predicted or non-random component of the variable.
This paper will focus mainly on the analytical evaluation of the average number of individuals <x(t)>, exploiting the properties of the correlation function of the
stochastic variable [xi](t) without using the probability density p(x,t).
E(x) denotes the expectation of
stochastic variable. [I.sub.N]/[I.sub.m] = {m + 1, ..., N}.
The following theorem characterizes the distribution of the
stochastic variable X([theta]).