The information on physical quantities, such as wind pressures on buildings, sea wave heights and material microstructure features, consists of values recorded at finite sets of times and/or locations. For example, suppose that 𝑛 𝑠 independent samples of a real-valued stationary process 𝑋(𝑡) are recorded in a bounded time interval [0, 𝜏] at a time step 𝛥𝜏 = 𝜏∕𝑛, so that the data set {𝑥 𝑖 (𝑗 𝛥𝜏), 𝑗 = 0, 1, … , 𝑛}, 𝑖 = 1, … , 𝑛 𝑠 , provides 𝑛 𝑠 independent samples of the (𝑛 + 1)-dimensional random vector 𝑉 = ( 𝑋(𝑡 0 ), 𝑋(𝑡 1 ), … , 𝑋(𝑡 𝑛 ) )
. Generally, the data set is sufficient to estimate the mean vector, the correlation matrix and the marginal distribution of 𝑉 but is insufficient to characterize the extreme random variable max 0≤𝑗≤𝑛 |𝑉 𝑗 |. Hence, even if the random variables sup 𝑡∈[0,𝜏] |𝑋(𝑡)| and max 0≤𝑗≤𝑛 |𝑉 𝑗 | have similar distributions, the distribution of sup 𝑡∈[0,𝜏] |𝑋(𝑡)| cannot be approximated solely from data. The information provided by the data set has to be augmented to characterize extremes of 𝑋 (𝑡).
There are at least two approaches which can be employed to extend the available information on 𝑋(𝑡) and use it to characterize the extreme random variable sup 𝑡∈[0,𝜏] |𝑋(𝑡)|. A common step of these approaches is the estimation of the mean vector and correlation matrix of the random vector 𝑉 , which can be viewed as approximations of the mean and correlation functions of 𝑋(𝑡) provided that the time step 𝛥𝜏 is sufficiently small. The first approach constructs reduced order models {𝑉 𝑑 }, 𝑑 = 1, 2, …, of 𝑉 , see [1]. The models {𝑉 𝑑 } are truncated Karhunen-Loève (KL) representations of 𝑉 whose random entries are described by polynomial chaoses (PCs) and translation polynomial chaoses (PCTs) with coefficients which minimize objective functions quantifying the of sup 𝑡∈[0,𝜏] |𝑋 𝑑 (𝑡)| converges to that of sup 𝑡∈[0,𝜏] |𝑋(𝑡)| as 𝑑 → ∞ so that the distribution of the target quantity of interest sup 𝑡∈[0,𝜏] |𝑋 𝑑 (𝑡)| can be estimated from samples of 𝑋 𝑑 (𝑡) for a sufficiently large 𝑑. This property is essential in applications since samples of 𝑋 𝑑 (𝑡) can be generated by standard Monte Carlo algorithms while samples of 𝑋(𝑡) are not available since 𝑋(𝑡) has infinite stochastic dimension as an uncountable family of random variables indexed by 𝑡 ∈ [0, 𝜏].
Our objective is to construct simple probabilistic models for uni-and multi-variate wind pressure wind tunnel records which are capable to characterize accurately extremes of these records. It is assumed that the wind records are sufficiently long such that their mean, correlation and marginal distribution functions can be estimated accurately. Of the infinite family of processes which can be fitted to these estimates or only to some of them, we have selected one-and two-dimensional diffusion processes with linear drift coefficients and translations of Gaussian/non-Gaussian diffusion processes. The selected models fit exactly the marginal distribution of the target wind records. Their correlation function is selected to minimize an objective function which depends on the discrepancies between target and model correlations and extremes, so that it may or may not fit accurately the target correlation function. It is shown that even such simple models can provide accurate estimates of the distribution of extremes of wind pressure records.
The paper is organized as follows. We introduce the one-and twodimensional diffusion processes with linear drifts and construct their FD models in Section 2. Translation diffusion processes and their FD models are defined and examined in Section 3. The processes of the previous two sections are fitted to wind data in Section 4. The section also assess the performance of the resulting processes and of their FD models. Final comments are in Section 5.
We review briefly two of the diffusion processes constructed in [2,3] and give conditions under which the stochastic differential equations defining these processes admit unique strong solutions. These conditions illustrate some of the limitations of two-dimensional diffusion processes under consideration.
Let 𝑋(𝑡), 𝑡 ∈ [0, 𝜏], be a real-valued diffusion process defined by the stochastic differential equation
where 𝐵(𝑡) is the standard Brownian motion process and 𝜌 > 0. The stationary solution of this equation is a zero-mean non-Gaussian process with correlation function 𝑐(𝑠, 𝑡) = 𝐸[𝑋(𝑠)𝑋(𝑡)] = exp(-𝜌|𝑠 -𝑡|). Our objective is to estimate the distribution of the extreme sup 𝑡∈[0,𝜏] |𝑋(𝑡)| of the stationary process 𝑋(𝑡) during a time interval of length 𝜏 > 0.
The characterization of extremes of 𝑋(𝑡) requires detailed knowledge of the samples of this process, which is provided by the strong solution of (2.1), see [4] (Sect. 4.7). The following theorem gives conditions under which this solution exists and is unique.
The stochastic differential Eq. (2.1) admits a unique strong solution 𝑋(𝑡) with stationary marginal density function 𝑓 , if 𝑓 is continuously differentiable on R, lim 𝑥→∞ -2𝜌𝑥𝑓 (𝑥)∕𝑓 ′ (𝑥) < ∞ and
Proof. It has been shown in [2] that the stationary process 𝑋(𝑡) of (2.1) has a specified marginal density 𝑓 if its diffusion coefficient has the expression in (2.2).
The strong solutions of (2.1) exists and are unique if 𝐷(𝑥) is locally Lipschitz continuous and satisfies the growth condition [4,Chap. 4], i.e., if for each 𝜉 > 0, there exists 𝐶 𝜉 > 0 such that
and there exists 𝐾 > 0 such that
Since 𝑓 (𝑥) > 0 and ∫ 𝑥 -∞ 𝑢𝑓 (𝑢)𝑑𝑢 are continuous functions on R, then from (2.2), 𝐷(𝑥) 2 is also continuous on R. Therefore, for any |𝑥| ≤ 𝜉 with 0 < 𝜉 < ∞, 𝑓 (𝑥) > 0 so that 𝐷(𝑥) 2
by using l'Hopital rule. Since 𝐷(𝑥) 2 is bounded on R, then there exists a constant 𝐾 > 0 such that 𝐷(𝑥) 2 ≤ 𝐾(1 + 𝑥 2 ), which means that 𝐷(𝑥) satisfies condition (2.4). The differentiation of (2.2) gives
so that this function is continuous since 𝑓 (𝑥), 𝑓 ′ (𝑥) and 𝐷(𝑥) are continuous. Therefore, 𝐷(𝑥) is continuously differentiable so that it is locally Lipschitz continuous. This means that 𝐷(𝑥) satisfies condition (2.3). □
Let 𝑋(𝑡) = (𝑋 1 (𝑡), 𝑋 2 (𝑡)) 𝑇 , 𝑡 ∈ [0, 𝜏] be a diffusion process defined by the stochastic differential equations
where 𝐵 1 (𝑡) and 𝐵 2 (𝑡) are independent standard Brownian motion processes. The stationary solution 𝑋(𝑡) of ( 2 It was found in [3] that the stationary bivariate process 𝑋(𝑡) has the marginal density 𝑓 if the diffusion coefficients have the expressions
These results have been obtained by solving the time-invariant Fokker-Planck equation
) )
under the conditions
and
The condition of ( or 𝑎 12 = 𝑎 21 = 0. These assumptions impose rather severe restrictions on the form of the density 𝑓 and/or the correlation function.
As previously noted, our objective is to characterize extremes of the diffusion process 𝑋(𝑡) whose characterization requires detailed information on the sample properties of this process, which are provided by the strong solution of (2.5). The following theorem provides conditions under which (2.5) admits a unique strong solution.
where 𝑓 (𝑥 1 |𝑥 2 ) denotes the conditional density of 𝑋 1 (𝑡)|𝑋 2 (𝑡). For any fixed 𝑥 2 ∈ R,
and, since 1∕𝑓 (𝑥 1 |𝑥 2 ) increases much faster than 𝑥 2 1 as 𝑥 1 → ∞, the growth condition (2.11) is not satisfied. □
We conclude with the observation that the condition of (2.8) limits the use of 𝑋(𝑡) in (2.5) since the set of marginal distributions and correlation functions of 𝑋(𝑡) under this condition is small so that few data sets can be described by this process.
Let 𝑋(𝑡), 0 ≤ 𝑡 ≤ 𝜏, be a zero-mean, finite-variance R 𝑛 -valued stochastic process defined on a probability space ( 𝛺, , 𝑃 ) , which may or may not be stationary. The FD models {𝑋 𝑑 (𝑡)} of 𝑋(𝑡) are defined by
where 𝑤 denotes weak convergence,
This theorem is essential in applications since it shows that the distributions of the target extremes sup 𝑡∈[0,𝜏] |𝑋 𝑖 (𝑡)| and sup 𝑡∈[0,𝜏] ‖𝑋(𝑡)‖ can be estimated from samples of FD models of 𝑋(𝑡) provided that the stochastic dimensions {𝑑 𝑖 } are sufficiently large, which can be generated by standard Monte Carlo algorithms.
The following Corollary specializes the statement of the previous theorem to the case of real-valued stationary processes with exponential correlation function.
sup
Proof. The eigenfunctions of the exponential correlation function 𝑐(𝑠, 𝑡) = 𝑒 -𝜌|𝑠-𝑡| of 𝑋(𝑡) are given by (2.51) and (2.52) of [7]. They have the expressions
,
where 𝜉 𝑘 and 𝜂 𝑘 are solutions of
and
This corollary holds also for vector-valued processes provided that the components of these processes are exponentially correlated.
The stochastic process with exponential correlation function is not differentiable. As noted in [9,10], replacing the exponential correlation function by a smoother function can have certain advantages without necessarily compromising the modeling accuracy. The second derivative of these modified correlation functions exist and are continuous. The FD models constructed under these modified correlation functions converge to the target processes weakly under the sup-norm, see Theorem 3.2 in [6].
Let 𝑋(𝑡) = (𝑋 1 (𝑡), … , 𝑋 𝑛 (𝑡)) 𝑇 be an 𝑛-variate zero-mean stationary process with correlation function 𝑐(𝑢) = 𝐸[𝑋(𝑡)𝑋(𝑡 + 𝑢) 𝑇 ] and marginal distributions 𝐹 𝑖 , 𝑖 = 1, … , 𝑛 of the components of 𝑋(𝑡). We construct translation models for 𝑋(𝑡) by mapping stationary Gaussian and non-Gaussian processes into processes which match exactly the marginal distributions of the components of 𝑋(𝑡) and approximately the correlation functions of this process. We also examine the extremes of the resulting translation processes which are equivalent in the sense that they have the same marginal distributions and similar correlation functions.
The translation version 𝑋 𝑇 (𝑡) of 𝑋(𝑡) has the components The Gaussian process 𝐺(𝑡) can be defined as the stationary solution of the stochastic differential equation
where 𝛼 and 𝛽 are (𝑛, 𝑛) and (𝑛, 𝑚) real-valued matrices which need to be determined and 𝐵(𝑡) denotes an 𝑚-dimensional process whose components are independent standard Brownian motions. The above equation admits stationary solution if the real parts of the eigenvalues of 𝛼 are negative. Under this condition, the covariance matrix 𝛾 = 𝐸[𝐺(𝑡)𝐺(𝑡) 𝑇 ] and the correlation function of the stationary process 𝐺(𝑡) are such that
and 𝐸[𝐺(𝑡)𝐺(𝑡 + 𝑢) 𝑇 ] = exp(𝛼𝑢) 𝛾, where exp(⋅) denotes the matrix exponential function [4,Chap. 7].
The matrices 𝛼 and 𝛽 can be determined in two steps. First, the matrix 𝛼 is selected to minimize the objective function We conclude with the observation that the translation process
exists for any 𝛼 and 𝛽 under the stated conditions provided that the target distribution functions {𝐹 𝑖 } are strictly monotonically increasing and admit densities. The components of 𝑋 𝑇 (𝑡) also satisfy the stochastic differential equations, see [4,Chap. 4],
where ∇ℎ 𝑖 and ∇ 2 ℎ 𝑖 are the gradient and Hessian matrix of ℎ 𝑖 and Tr is the trace operator.
The construction of the previous subsection is extended to translation models which are obtained by memoryless transformations of non-Gaussian diffusion processes. The resulting translation models are equivalent in the sense that they have the same marginal distributions and similar correlation functions. Yet, they may have very different extremes.
Let 𝑋(𝑡) be the target process with zero-mean, correlation function 𝑐(𝑢) = 𝐸[𝑋(𝑡 + 𝑢) 𝑋(𝑡)] = 𝑒 -𝜌𝑢 , 𝜌 > 0, and marginal distribution 𝐹 . We construct translation models {𝑋 (𝑖) 𝑇 (𝑡)} of 𝑋(𝑡) from two (non-Gaussian) diffusion processes {𝑌 (𝑖) (𝑡)} defined by the stochastic differential equations
where 𝐷 1 (𝑥) = √ 2𝜌 and 𝐷 2 (𝑥) = (2𝜌(𝑥 + 2∕𝜃)∕𝜃) 1∕2 with 𝜌, 𝜃 > 0. The first two moments of these processes are 𝐸[𝑌 (𝑖) (𝑡)] = 0 and 𝐸[𝑌 (𝑖) (𝑡 + 𝑢) 𝑌 (𝑖) (𝑡)] = 𝑒 -𝜌𝑢 . Their marginal distributions are Gauss and Gamma with the densities 𝑓 1 (𝑥) = 𝜙(𝑥), 𝑥 ∈ R and 𝑓 2 (𝑥) = 𝜃(𝜃𝑥 + 2)𝑒 -𝜃𝑥+2 , 𝑥 > -2∕𝜃, where 𝜙 is the standard Gaussian density [12].
The stochastic differential Eq. (3.5) admits a unique strong solution since the diffusion coefficients 𝐷 𝑖 (𝑥), 𝑖 = 1, 2, are locally Lipschitz, a property which results directly from their expressions, and satisfy the growth condition in [4,Chap. 4] since there exist constants 𝐾 1 = 2𝜌 and 𝐾 2 = (2𝜌𝜃 + 4𝜌)∕𝜃 2 such that
The translation processes defined by the mapping 𝑋 (𝑖) 𝑇 (𝑡, 𝜔) = 𝐹 -1 •𝐹 𝑌 (𝑖) (𝑌 (𝑖) (𝑡, 𝜔)), 𝑖 = 1, 2, have the marginal distribution 𝐹 and their correlation functions are similar to the exponential correlation function of the processes 𝑌 (𝑖) (𝑡). Note also that the extremes of {𝑋 (𝑖) 𝑇 (𝑡)} occur at the same time as those of {𝑌 (𝑖) (𝑡)}.
Suppose that the target is a real-valued, zero-mean, stationary process 𝑋(𝑡) = (𝑋 1 (𝑡), 𝑋 2 (𝑡)) 𝑇 with correlation function 𝑐(𝑢) = 𝐸[𝑋(𝑡)𝑋(𝑡 + 𝑢) 𝑇 ] and component marginal distributions 𝐹 𝑖 , 𝑖 = 1, 2. We construct translation models of 𝑋(𝑡) from two diffusion processes 𝑌 (𝑖) (𝑡) = (𝑌 (𝑖) 1 (𝑡), 𝑌 (𝑖) 2 (𝑡)) 𝑇 𝑖 = 1, 2, which have the same correlation function as 𝑋(𝑡) by translating them such that their components match the marginal distributions {𝐹 𝑗 } of {𝑋 𝑗 (𝑡)}, 𝑗 = 1, 2.
The diffusion processes {𝑌 (𝑖) (𝑡)}, 𝑖 = 1, 2 are defined by the stochastic differential equations 𝑑𝑌 (𝑖) 1 (𝑡) = (𝑎 11 𝑌 (𝑖) 1 (𝑡) + 𝑎 12 𝑌 (𝑖) 2 (𝑡))𝑑𝑡 + 𝐷 (𝑖) 11 (𝑌 (𝑖) 1 (𝑡), 𝑌 (𝑖) 2 (𝑡))𝑑𝐵 1 (𝑡) +𝐷 (𝑖) 12 (𝑌 (𝑖) 1 (𝑡), 𝑌 (𝑖) 2 (𝑡))𝑑𝐵 2 (𝑡) 𝑑𝑌 (𝑖) 2 (𝑡) = (𝑎 21 𝑌 (𝑖) 1 (𝑡) + 𝑎 22 𝑌 (𝑖) 2 (𝑡))𝑑𝑡 + 𝐷 (𝑖) 21 (𝑌 (𝑖) 1 (𝑡), 𝑌 (𝑖) 2 (𝑡))𝑑𝐵 1 (𝑡) +𝐷 (𝑖) 22 (𝑌 (𝑖) 1 (𝑡), 𝑌 (𝑖) 2 (𝑡))𝑑𝐵 2 (𝑡),
where 𝐵 1 (𝑡) and 𝐵 2 (𝑡) are independent standard Brownian motion processes, 𝑎 11 < 0, 𝑎 22 < 0, 𝑎 12 𝑎 21 < 0 and the diffusion coefficients
with
The stationary solutions {𝑌 (𝑖) (𝑡)} of (3.7) are zero-mean processes with correlation function 𝑐(𝑠, 𝑡) = exp(𝐴|𝑡 -𝑠|)𝛾, where 𝐴 = [𝑎 11 𝑎 12 ; 𝑎 21 𝑎 22 ], exp(⋅) denotes the matrix exponential function [5, Sect. 1.5] and 𝛾 is the stationary covariance matrix of {𝑌 (𝑖) (𝑡)}, which exists if the real part of the eigenvalues of the matrix 𝐴 are negative. The process {𝑌 (1) (𝑡)} is Gaussian while {𝑌 (2) (𝑡)} is not. The joint densities of the two-dimensional vector ( 𝑌 (2) 1 (𝑡), 𝑌 (2) 2 (𝑡)
where 𝐶 1 is the normalizing constant.
For the diffusion coefficients under consideration, the stochastic differential Eq. (3.7) admits strong unique solution since the diffusion coefficients 𝐷 (𝑘) 𝑖𝑗 (𝑥 1 , 𝑥 2 ) are locally Lipschitz continuous and satisfy the growth condition (2.11). For example, the drift and diffusion coefficients of 𝑌 (2) (𝑡) are locally Lipschitz continuous since they are continuously differentiable by definition. The growth condition is satisfied since there exists 𝐾 2 = max{1, 𝜁}(𝜅 2 11 + 𝜅 2 12 + 𝜅 2 21 + 𝜅 2 22 )∕(4(𝜃 -1)) such that for any 𝑥 𝑖 ∈ R, 𝑖 = 1, 2, 𝐷 (2) 11 (𝑥 1 , 𝑥 2 ) 2 + 𝐷 (3) 12 (𝑥 1 , 𝑥 2 ) 2 + 𝐷 (3) 21 (𝑥 1 , 𝑥 2 ) 2 + 𝐷 (3) 22 (𝑥 As for the translation processes of the previous section, we note that the translation processes 𝑋 (𝑖) 𝑇 ,𝑗 (𝑡
have the marginal distributions {𝐹 𝑗 } and correlation functions similar to the exponential correlation function of the processes 𝑌 (𝑖) (𝑡), see [11,Sect. 3]. Note also that the extremes of {𝑋 (𝑖) 𝑇 ,𝑗 (𝑡)} and {𝑌 (𝑖) 𝑗 (𝑡)} occur at the same times.
The FD models {𝑋 𝑇 ,𝑑 (𝑡)} of the translation model 𝑋 𝑇 (𝑡) have the form
where 𝑌 𝑖,𝑑 𝑖 (𝑡) are finite dimensional models of the stationary stochastic process 𝑌 𝑖 (𝑡) of the type in (2.12) with marginal distribution 𝐹 𝑌 𝑖 , 𝑖 = 1, … , 𝑛. As previously, we are interested in extremes sup 𝑡∈[0,𝜏] |𝑋 𝑇 ,𝑖 (𝑡)| of 𝑋 𝑇 ,𝑖 (𝑡) in the stationary regime during a time interval of length 𝜏 > 0.
The following Corollary from [6] shows that the extremes of 𝑋 𝑇 ,𝑖,𝑑 𝑖 (𝑡) converges to those of 𝑋 𝑇 ,𝑖 (𝑡) in probability as 𝑑 𝑖 increases indefinitely if the input 𝑌 𝑖 is Gaussian. This means that the distribution of the target quantity of interest sup 𝑡∈[0,𝜏] |𝑋 𝑇 ,𝑖 (𝑡)| can be estimated from samples of 𝑋 𝑇 ,𝑖,𝑑 𝑖 (𝑡) for a sufficiently large 𝑑 𝑖 since convergence in probability implies convergence in distribution.
where 𝑝 denotes convergence in probability.
The random processes discussed in the previous sections are fitted to wind pressure time series recorded at the University of Florida boundary layer wind tunnel facility (UFBLWT). The data set is discussed extensively in [1]. It consists of wind pressure time series at 252 taps recorded under a steady fan speed of 1200 RPM.
The following subsection presents the data set and estimates the first two moments and the marginal distributions of the wind pressure time series. The subsequent sections fit the processes discussed in the first part of this study to these estimates, construct FD models for the resulting processes, referred to as target processes, and assess performance by comparing target and FD extremes.
The wind pressure data was recorded for 𝑇 = 180 seconds at a time step 𝛥𝑇 = 0.00164 second, so that it consists of 𝑛 = 𝑇 ∕𝛥𝑇 = 110, 000 readings. We consider records at two pressure taps and denote them by ( ỹ1 , ỹ2 , … , ỹ𝑛 )
, where ỹ𝑗 = ( ỹ1,𝑗 , ỹ2,𝑗 ) 𝑇 , 𝑗 = 1, … , 𝑛. The taps, labeled 120 and 126 in the experiment, have been selected since their skewness and kurtosis coefficients are 1.2138 and 5.8509 for tap 120 and 1.1926 and 5.9813 for tap 126. Segments of the record { ỹ𝑗 } of length 1000𝛥𝑇 are shown in the left and right panels of Fig. 1.
We assume that the wind records are samples of an ergodic process X(𝑡) = ( X1 (𝑡), X2 (𝑡)) 𝑇 and that these samples are sufficiently long to estimate the marginal distribution and the correlation function of this process. Let and
where 𝜙 denotes the density of the standard Gaussian variable and 𝜅 is a bandwidth parameter set equal to 𝜅 = (4∕3𝑛) 1∕5 [13, Chap.3]. These estimates are viewed as the actual correlation functions and marginal distributions of 𝑋(𝑡). The left and right panels of Fig. 2 show the histograms and the estimates f𝑖 (𝑥) of the normalized wind pressure records at taps 120 and 126. The plots show that the estimates of (4.2) are consistent with data. The estimates of the marginal distributions F𝑖 of 𝑋 𝑖 (𝑡) result from f𝑖 by integration, 𝑖 = 1, 2. The left and right panels of Fig. 3 show the marginal distributions obtained from data and the estimates F𝑖 (𝑥) of the normalized wind pressure records at taps 120 and 126 in logarithmic scale. We also estimate the distributions of the extremes sup
) ∕Std[ X𝑖 (𝑡)] of X(𝑡) by the following procedure. First, the wind record is partition in non-overlapping segments of duration 1000 𝛥𝑇 = 1.62 sec, so that there are 𝑞 = 1100 such segments. Second, the extremes of the individual segments are recorded and used to estimate the distribution of sup 𝑡∈[0,𝜏] |𝑋 𝑖 (𝑡)| at each pressure tap by using, e.g., the MATLAB ecdf function.
We fit the diffusion process 𝑋(𝑡) in (2.1) to the normalized wind pressure records at tap 120 by using the properties of this process summarized in Theorem 2.1. We construct FD models {𝑋 𝑑 (𝑡)} for fitted diffusion process 𝑋(𝑡) and assess their performance by comparing the distributions of sup 𝑡∈[0,𝜏] |𝑋(𝑡)| and sup 𝑡∈[0,𝜏] |𝑋 𝑑 (𝑡)| for 𝜏 = 0.162 sec., based on 5000 samples of 𝑋(𝑡) with the time step 𝛥𝜏 = 3.27 × 10 -5 sec. Note that this time step has no relation to that of the recorded wind pressure series. The diffusion process in (2.5) cannot be fitted to wind pressure records at two taps because of restrictions on its distributions, see Section 2.2.
The parameter 𝜌 of the exponential diffusion process in (2.1) is selected to minimize the objective function (3.3). Its diffusion coefficient 𝐷(𝑥) is calculated from (2.2) with 𝑓 set equal to the estimate f of the marginal distribution 𝑓 given by (4.2). We note that the defining equation of 𝑋(𝑡) admits a unique strong solutions by Theorem 2.1 since 𝑓 (𝑥) defined by (4.2) is continuously differentiable and lim 𝑥→∞ -2𝜌𝑥𝑓 (𝑥)∕𝑓 ′ (𝑥) = 2𝜌𝜅 2 < ∞.
The left panel of Fig. 4 shows the target correlation function ĉ(𝑘) for 𝑘 = 1, … , 100 estimated from the wind record at tap 120 (solid line) and the corresponding correlation functions of 𝑋(𝑡) obtained under the objective function (3.3) with (𝑤 1 = 1, 𝑤 2 = 0) (dotted line) and with (𝑤 1 = 1, 𝑤 2 = 10) (dashed line). We conclude with the observation that the diffusion process 𝑋(𝑡) in (2.1) can only fit arbitrary marginal distributions. Its extremes may or may not match target extremes depending on the objective function used to select the correlation parameter 𝜌 of 𝑋(𝑡). Correlation functions rather different from the target correlation function may yield accurate estimates of extremes. This is an example of a simple model which, although captures exactly a single feature of the target process 𝑋(𝑡), the marginal distribution of 𝑋(𝑡) in this example, characterizes the extremes of this process accurately. A possible explanation for this behavior is that (1) the fitted exponential correlation (objective function with (𝑤 1 = 1, 𝑤 2 = 0)) is too simplistic to capture the complex dependence of the wind time series and (2) the correlation
We develop FD models 𝑋 𝑑 (𝑡) for the process 𝑋(𝑡) constructed for the tap 120 under the objective function
We examine the potential of the FD models constructed in this study and in [6] for the special case in which data set consists of samples of a zero-mean stationary diffusion process 𝑌 (𝑡) with exponential correlation function 𝐸[𝑌 (𝑠)𝑌 (𝑡)] = 𝑒 -𝜅|𝑠-𝑡| and shifted Gamma marginal distribution with shape and scale parameters 2 and 𝜃. The process is defined by the stationary solution of the stochastic differential equation
with 𝜅 = 1 and 𝜃 = 1.
The following numerical results are based on 5000 independent samples of 𝑌 (𝑡) with 5000 time steps 𝛥𝜏 = 0.0002. The correlation ∑ 𝑑 𝑘=1 𝑍 PCT 𝑘 𝜑 𝑘 (𝑡) represents the random coefficients {𝑍 PCT 𝑘 } by PCT models, where 𝑚 denotes the degree of the underlying PC model, see [1] for details.
The left panel of Fig. 6 shows the correlation function of the target process 𝑌 (𝑡) (solid line) and the estimated correlation function of 𝑋(𝑡) (dotted line). The right panel of Fig. 6 shows with solid line in semi logarithmic scale an estimate of the target probability 𝑃 (sup 𝑡∈[0,𝜏] |𝑌 (𝑡)| > 𝑥). The dotted, thin dashed, heavy dashed and dotted-dashed lines are the corresponding probabilities of the diffusion processes 𝑋(𝑡), its FD models 𝑋 𝑑 (𝑡), the PCT models 𝑋 PCT,2
and 𝑌 (𝑡) is supported on [-2, ∞], the above two probabilities have different tails and 𝑃 (sup 𝑡∈[0,𝜏] 𝑌 (𝑡) < -𝑥) = 0 when 𝑥 ≥ 2.
We fit the diffusion process 𝑋 𝑇 (𝑡) of (3.4) to the wind pressure records at the taps 120 and 126, construct FD models for this process and compare extremes of 𝑋 𝑇 (𝑡) with those of their FD models 𝑋 𝑇 ,𝑑 (𝑡)
The Gaussian image of the translation processes of (3.4) for the wind records at taps 120 and 126 is constructed by using the objective function in (3. We conclude with the observation that the translation diffusion process 𝑋 𝑇 (𝑡) in (3.4) can fit exactly the marginal distribution of the target process 𝑋 𝑖 (𝑡) and approximately its correlation function. However, its extremes may or may not match target extremes depending on the objective function used to select the correlation parameter 𝛼 of 𝑋 𝑇 (𝑡). Correlation functions rather different from the target correlation may yield accurate estimates of extremes. This is another example of model which provides accurate estimates of a particular quantity of interest although it does not capture some target properties.
We develop FD models 𝑋 𝑇 ,𝑑 (𝑡) for the process 𝑋 𝑇 (𝑡) constructed for the taps 120 and 126 under the objective function (3.3)
Translation models in Section 4.3 can be extended to map non-Gaussian diffusion processes rather the Gaussian diffusion processes. The following two subsections show results for one-and two-dimensional translation non-Gaussian diffusion processes. We note that FD models for these processes can be constructed by the approach of Section 4.3 and are reported here.
Consider the one-dimensional translation diffusion processes
where F is the distribution of the estimated marginal density in (4.2) for the wind record at the tap 120, 𝑌 (𝑖) (𝑡) is the non-Gaussian diffusion 𝑇 (𝑡)}. We conclude with the observation that translation diffusion processes {𝑋 (𝑖) 𝑇 (𝑡)} based on the non-Gaussian, exponentially correlated diffusion processes of (3.5), which are equivalent in the sense that they have the marginal of the wind pressure time series recorded at a pressure tap and similar correlation functions, can have very different extremes. This is another illustration of the fact that extremes require information beyond the first two moments and marginal distributions.
Consider the two-dimensional translation diffusion processes 𝑗 (𝑡), 𝑖, 𝑗 = 1, 2. The process {𝑌 (1) 𝑗 (𝑡)} is Gaussian while {𝑌 (2) 𝑗 (𝑡)} is not. We fit the two dimensional translation diffusion processes {𝑋 (𝑖) 𝑇 ,𝑗 (𝑡)} in 𝑇 ,𝑗 (𝑡)}. We conclude with the observation that translation diffusion processes {𝑋 (𝑖) 𝑇 ,𝑗 (𝑡)} based on the non-Gaussian diffusion processes of (3.7), which are equivalent in the sense that they have the marginal of the wind pressure time series recorded at two pressure taps and similar correlation functions, can have very different extremes. This is another illustration of the fact that extremes require information beyond the first two moments and marginal distributions.
Two classes of random processes are constructed to describe wind pressures recorded at the University of Florida boundary layer wind tunnel facility (UFBLWT). The first consists of diffusion processes with linear drift. The functional form of the correlation functions of these processes is fixed, e.g., exponential correlation for one-dimensional processes. The second class consists of translations of Gaussian/non-Gaussian diffusion processes. These processes can characterize a broader set of wind data than those of the first class when dealing with multivariate time series.
It is shown that the processes of both classes can describe accurately the extremes of wind pressure records if their correlation functions are selected to minimize objective functions quantifying the discrepancy between correlations and extremes of these processes and of wind pressure records. It is also shown that the finite dimensional (FD) models developed for these processes can be used to estimate extreme wind pressures. The FD models are deterministic functions of time and finite sets of random variables.
Diffusion processes with linear drift and translations of Gaussian/non-Gaussian diffusion processes are fitted to wind pressure time series recorded at the University of Florida boundary layer wind tunnel facility (UFBLWT). The processes match exactly and approximately the marginal distributions and the correlation functions of these records. It is shown that (1) the extreme of these simple processes characterize accurately the extremes of the wind record if their correlation functions minimize objective functions which account for extremes and (2) the finite dimensional (FD) models of these processes can be used to estimate extremes of the wind records.