Efficient Time-Domain Dimension Reduction Methods for Simulating Stationary Stochastic Processes

Abstract

The high-dimensional stochastic space caused by a large number of random variables remains a significant challenge hindering the practical application of stochastic process simulation in engineering. Although various dimension reduction techniques have been developed, their direct integration into time-domain simulation frameworks remains limited. To address this issue, this paper proposes two efficient time-domain dimension reduction methods for simulating stationary stochastic processes. The methods reduce the number of input random variables required for simulation to a single variable, while the randomness of the output stochastic process remains unchanged. The proposed methods are theoretically motivated by spectral decomposition of processes using two distinct strategies and explicitly incorporate the decay characteristics of the impulse response function associated with the stochastic process. Based on this, the random orthogonal functions can be naturally introduced to simulate the stationary stochastic process, which effectively resolves the high-dimensional random variables encountered in conventional time-domain simulations. Furthermore, the incorporation of a number-theoretic method enables uncertainty quantification of stochastic process samples. Numerical simulations demonstrate that the proposed methods reduce the random variable dimension from 2400 to 1 (99.95% reduction). Relative error of the simulated power spectral density remains below 2%, while computational time is reduced by approximately 4% compared with the conventional time-domain methods. These results demonstrate the effectiveness and practical applicability of the proposed approach in engineering stochastic process simulation.

1. Introduction

Strong winds, sea waves and other catastrophic effects, characterized by significant stochastic dynamic properties, can cause massive destruction to civil engineering structures within a short period of time, resulting in substantial casualties and economic losses [1,2,3,4]. Therefore, ensuring the serviceability and safety performance of engineering structures subjected to severe random dynamic excitations requires reliable modeling and simulation of the underlying stochastic processes. In practice, modeling and simulation of stochastic processes are commonly carried out using two main classes of approaches, namely frequency domain methods and time-domain methods [5]. Among these approaches, time-domain simulation methods are often considered attractive due to their direct generation of time histories and their convenient integration with time-domain solvers for stochastic dynamic response and reliability analysis [6,7,8]. They have therefore been widely adopted in the modeling and simulation of various stochastic hazard scenarios [9,10].

The time-domain approach based on digital filtering [11] can be traced back to Yule’s [12] seminal work on autoregressive (AR) modeling of sunspot records. Walker [13] later introduced the moving-average (MA) model to characterize large-scale atmospheric variability. Wold [14] subsequently synthesized the AR and MA formulations into the ARMA model for the representation of stationary stochastic processes, establishing a unified time-domain framework for stationary stochastic processes. These ARMA-type models established a conventional time-domain modeling framework for stochastic processes and have been widely used in practical simulations of random phenomena. In the field of civil engineering [15], this time-domain modeling framework has been extensively employed in the modeling and simulation of stochastic excitations associated with large-span spatial structures [16,17], offshore structures [18] and marine engineering [19], as well as earthquake engineering [20,21], covering various engineering scenarios. Here, the discussion is restricted to the time-domain approach based on digital filtering rather than the general history of stochastic simulation methods.

Despite the widespread use of the conventional time-domain modeling framework in stochastic engineering applications, certain inherent limitations remain [6,11]. First, the conventional time-domain modeling framework is typically based on empirical parametric modeling. Specifically, the determination of model parameters in conventional time-domain models often relies on heuristic criteria or statistical information criteria [11,22], such as AIC or BIC, which may not necessarily reflect the underlying physical characteristics of a specific stochastic process and may result in additional modeling uncertainty. Second, to simulate a stochastic process with a prescribed power spectral density (PSD) function, conventional time-domain methods, such as MA and ARMA models, typically require a large number of independent random variables as driving inputs [5,10]. This requirement inevitably leads to a high-dimensional representation, resulting in the well-known dimension dilemma [23,24]. As the stochastic dimension increases, the generation of sufficiently representative samples for uncertainty quantification becomes increasingly difficult and computationally expensive [25]. Consequently, conventional time-domain simulation methods face significant challenges when applied to modern uncertainty quantification frameworks, such as the widely accepted Probability Density Evolution Method (PDEM) [26], particularly for system-level reliability analysis of complex civil engineering structures aimed at ensuring structural safety under stochastic hazard loading.

To address the first issue, instead of relying on empirical parametric time-domain modeling, i.e., the conventional time-domain modeling framework, the present study follows a different theoretical route. Time-domain formulation is constructed based on the spectral representation of stationary stochastic processes, thereby providing a physically interpretable determination of model parameters through the impulse response function rather than empirical selection criteria. Based on this idea, two time-domain simulation strategies are developed. Specifically, the first strategy extends the previously developed non-stationary strategy [6] to stationary stochastic processes. The second strategy is newly proposed to improve computational efficiency.

To address high-dimensional challenges, i.e., the second issue associated with stochastic process simulation, various dimension reduction techniques have been developed in the literature. Chen et al. [27,28] propose stochastic harmonic function (SHF) methods based on spectral representation, in which a stochastic process is simulated using only a few randomly parameterized harmonic components, thereby significantly reducing the number of driving random variables required in the simulation. In a related study, Liu et al. [29,30] propose a random orthogonal function representation for stochastic processes, which further achieves dimension reduction within the spectral representation framework by enforcing orthogonality constraints among the random variables. These frequency domain-based approaches have been proven effective in alleviating the dimensionality issue inherent in spectral representation methods. However, existing dimension reduction research primarily focuses on frequency-domain simulation of stochastic processes and has reached a relatively mature stage [2], whereas comparable dimension reduction strategies for widely used time-domain simulation methods remain comparatively less explored. In particular, systematic resolution of the high-dimensional random variable issue in time-domain simulation methods has not been fully established.

Motivated by the current state of research, this study further extends Liu’s [2,29] orthogonal-function-based dimension reduction concept, originally developed for frequency domain stochastic process simulation, and adapts it to the proposed two time-domain simulation strategies. As a result, two time-domain dimension reduction methods are proposed, in which the stochastic process can be driven by only a single elementary random variable while preserving the prescribed PSD function. The first time-domain dimension reduction method was previously developed for the simulation of non-stationary seismic processes [6]. In the present study, the previously developed formulation of the first method is specialized to the stationary case by replacing the time-varying impulse response function with a time-invariant impulse response function, thereby enabling simulation of stationary stochastic processes. The second time-domain dimension reduction method is newly developed by integrating the proposed second strategy with the dimension reduction mechanism, resulting in a new time-domain dimension reduction simulation framework for stationary stochastic processes. In general, the two methods effectively mitigate the high-dimensional random-variable issue faced by conventional time-domain modeling frameworks.

The remainder of this paper is organized as follows. Section 2 formulates the two time-domain simulation methods. Section 3 presents the dimension reduction techniques based on random orthogonal functions. Section 4 provides numerical examples and comparative analyses to demonstrate the accuracy and efficiency of the proposed methods. Section 5 concludes the study.

2. Random Orthogonal Variable Representation of Stationary Processes in Time-Domain

2.1. Time-Domain Representation via Spectral Decomposition

According to the spectral decomposition theory for stochastic processes, a zero-mean real-valued stationary process $X\left(t\right)$ can be expressed in the following Fourier–Stieltjes integral form [31]:

$$X\left(t\right)={\displaystyle {\int }_{-\infty }^{\infty }{\mathrm{e}}^{\mathrm{i}\omega t}\mathrm{d}Z(\omega )}={\displaystyle {\int }_{-\infty }^{\infty }{\mathrm{e}}^{\mathrm{i}\omega t}H(\omega )\mathrm{d}U(\omega )}$$

(1)

where i denotes the imaginary unit, i.e., ${\mathrm{i}}^2=-1$. $\omega$ denotes the angular frequency. $Z(\omega )$ and $U(\omega )$ denote processes with orthogonal increments, whose increments $\mathrm{d}Z(\omega )$ and $\mathrm{d}U(\omega )$ satisfy the following conditions [2]:

$$E\left[\mathrm{d}Z(\omega )\right]=0, E\left[\mathrm{d}U(\omega )\right]=0$$

(2a)

$$E\left[\mathrm{d}Z(\omega )\right]=E\left[\mathrm{d}{Z}^{\ast }(-\omega )\right], E\left[\mathrm{d}U(\omega )\right]=E\left[\mathrm{d}{U}^{\ast }(-\omega )\right]$$

(2b)

$$E\left[\mathrm{d}Z(\omega )\mathrm{d}{Z}^{\ast }({\omega }^{\prime })\right]=S(\omega ){\mathsf{\delta }}_{\omega {\omega }^{\prime }}\mathrm{d}\omega$$

(2c)

$$E\left[\mathrm{d}U(\omega )\mathrm{d}{U}^{\ast }({\omega }^{\prime })\right]={\mathsf{\delta }}_{\omega {\omega }^{\prime }}\mathrm{d}\omega$$

(2d)

where $E\left[\cdot \right]$ denotes the mathematical expectation; superscript “$\ast$” denotes the complex conjugate. ${\mathsf{\delta }}_{\omega {\omega }^{\prime }}$ denotes the Kronecker delta, i.e., ${\mathsf{\delta }}_{\omega {\omega }^{\prime }}=1$ for $\omega ={\omega }^{\prime }$ and ${\mathsf{\delta }}_{\omega {\omega }^{\prime }}=0$ for $\omega \ne {\omega }^{\prime }$. $S(\omega )$ denotes the PSD function of the stationary stochastic process $X\left(t\right)$, satisfying $S(-\omega )=S(\omega )$.

Notably, $H(\omega )$ is a deterministic, complex-valued function satisfying the following condition [6,10]:

$$S(\omega )={\left|H(\omega )\right|}^2$$

(3)

Therefore, function $h(t)$ can be defined as the inverse Fourier transform of $H(\omega )$, as follows [5,10]:

$$h(t)={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_{-\infty }^{\infty }H(\omega )e^{i\omega t}}\mathrm{d}\omega$$

(4)

Then, by considering Equation (4), Equation (1) can be rewritten in time-domain form as follows [10]:

$$X\left(t\right)={\displaystyle {\int }_{-\infty }^{\infty }h\left(t-\tau \right)w(\tau )\mathrm{d}\tau }$$

(5)

where $w(\tau )$ denotes the white noise process with unit variance. It satisfies the following conditions [6]:

$$E\left[w(\tau )\right]=0, E\left[w(\tau )w({\tau }^{\prime })\right]=2\mathsf{\pi }\mathsf{\delta }(\tau -{\tau }^{\prime })$$

(6)

where $\mathsf{\delta }(\tau -{\tau }^{\prime })$ denotes the Dirac delta function.

Subsequently, under the assumptions that $X\left(t\right)=0$ for $t<0$, and $h\left(t-\tau \right)=0$ for $t<\tau$, Equation (6) can thus be rewritten as follows [10,32]:

$$X\left(t\right)={\displaystyle {\int }_0^{t}h(t-\tau )w(\tau )\mathrm{d}\tau }$$

(7)

Therefore, Equation (7) represents the time-domain formulation of the stationary process obtained via spectral decomposition given in Equation (1).

2.2. Discrete Time-Domain Representation for Strategy I

For particular numerical simulation, the continuous time-domain representation of stationary processes, i.e., Equation (7), must be discretized for simulation. In this regard, Equation (7) can be further rewritten as follows [5]:

$$X\left(t\right)={\displaystyle \sum _{i=1}^p{\displaystyle {\int }_{(i-1)\Delta t}^{i\Delta t}h(t-\tau )w(\tau )\mathrm{d}\tau }}, p=⌊t/\Delta t⌋$$

(8)

where $\Delta t$ denotes the simulation time interval; operator “$⌊\cdot ⌋$” denotes the floor-function operator; p denotes the simulation time step at fixed time t.

As a matter of fact, the typical simulation time interval $\Delta t$ used in engineering is small. Therefore, the impulse response function $h(t-\tau )$ for $\tau \in [(i-1)\Delta t, i\Delta t]$ within this small interval can be regarded as a constant, i.e., $h(t-\tau )=h(t-i\Delta t)$ for $\tau \in [(i-1)\Delta t, i\Delta t]$. Then, Equation (8) can be written as follows [6,10]:

$$X\left(t\right)={\displaystyle \sum _{i=1}^{p}h(t-i\Delta t){\displaystyle {\int }_{(i-1)\Delta t}^{i\Delta t}w(\tau )\mathrm{d}\tau }}, p=⌊t/\Delta t⌋$$

(9)

In the small interval, the integral over the interval can be regard as a variable, i.e., $V_i={\displaystyle {\int }_{(i-1)\Delta t}^{i\Delta t}w(\tau )\mathrm{d}\tau }$. It is easy to be proved [6]:

$$E\left[V_i\right]={\displaystyle {\int }_{(i-1)\Delta t}^{i\Delta t}E\left[w(\tau )\right]\mathrm{d}\tau }=0$$

(10a)

$$\begin{array}{l}E\left[V_i{V}_j\right]=E\left[{\displaystyle {\int }_{(i-1)\Delta t}^{i\Delta }w(\tau )\mathrm{d}\tau {\displaystyle {\int }_{(j-1)\Delta t}^{j\Delta }{w}^{\prime }(\tau )\mathrm{d}{\tau }^{\prime }}}\right]\\ ={\displaystyle {\int }_{(i-1)\Delta t}^{i\Delta }{\displaystyle {\int }_{(j-1)\Delta t}^{j\Delta }E\left[w(\tau )w^{\prime }(\tau )\right]\mathrm{d}\tau \mathrm{d}{\tau }^{\prime }}}\\ =2\mathsf{\pi }{\displaystyle {\int }_{(i-1)\Delta t}^{i\Delta }{\displaystyle {\int }_{(j-1)\Delta t}^{j\Delta }\mathsf{\delta }(\tau -{\tau }^{\prime })\mathrm{d}\tau \mathrm{d}{\tau }^{\prime }}}\\ =2\mathsf{\pi }\Delta t{\mathsf{\delta }}_{ij}\end{array}$$

(10b)

Then, the time-domain representation of the process, i.e., Equation (9), can be further expressed in a form with random orthogonal variables:

$$X\left(t\right)={\displaystyle \sum _{i=1}^{p}h(t-i\Delta t)V_i}, p=⌊t/\Delta t⌋$$

(11)

For simulating a process with the number of time steps N, Equation (11) can be particularly written in a matrix form:

$$X\left(t\right)=\left[\begin{array}{c}X\left(\Delta t\right)\\ X\left(2\Delta t\right)\\ ⋮\\ X\left(N\Delta t\right)\end{array}\right]=\left[\begin{array}{cccc}h(0)& & & \\ h(\Delta t)& h(0)& & \\ ⋮& ⋮& \ddots & \\ h\left((N-1)\Delta t\right)& h\left((N-2)\Delta t\right)& \cdots & h(0)\end{array}\right]\left[\begin{array}{c}{V}_1\\ V_2\\ ⋮\\ V_N\end{array}\right]$$

(12)

As shown in Equations (11) and (12), a process can be represented as a weighted sum of a series of random orthogonal variables. This is the time-domain simulation strategy based on random orthogonal variables.

2.3. Discrete Time-Domain Representation for Strategy II

In the above section, we developed a time-domain simulation strategy, which is explicitly derived from the discretized filtering white noise model. Therefore, the simulation strategy directly reflects the impulse response function $h(t-\tau )$ information in the time domain. However, for many physical processes, such as wind velocity processes, the function $h(t-\tau )$ rapidly decays as $\tau$ decreases at a fixed time t. Therefore, the $h(t-\tau )$ values at $\tau$ far from time t are sufficiently small to be neglected. As a result, to improve computational efficiency, an alternative time-domain simulation strategy can be proposed by preserving the relatively large values of $h(t-\tau )$. Specifically, Equation (7) can be split according to $\tau$, and the terms associated with small values of $\tau$ can be neglected as follows:

$$\begin{array}{l}X\left(t\right)={\displaystyle {\int }_0^{t-(q+1)\Delta t}h(t-\tau )w(\tau )\mathrm{d}\tau }+{\displaystyle {\int }_{t-(q+1)\Delta t}^{t}h(t-\tau )w(\tau )\mathrm{d}\tau }\\ \approx {\displaystyle {\int }_{t-(q+1)\Delta t}^{t}h(t-\tau )w(\tau )\mathrm{d}\tau }\end{array}$$

(13)

where q denotes the number of retained time steps after splitting.

Similar to the above-mentioned simulation strategy, Equation (13) is discretized, yielding:

$$X\left(t\right)={\displaystyle \sum _{j=0}^q{\displaystyle {\int }_{t-(j+1)\Delta t}^{t-j\Delta t}h(t-\tau )w(\tau )\mathrm{d}\tau }} ={\displaystyle \sum _{j=0}^{q}h(i\Delta t){\displaystyle {\int }_{t-(j+1)\Delta t}^{t-j\Delta t}w(\tau )\mathrm{d}\tau }}$$

(14)

In this case, the integral over $\tau \in \left[t-j\Delta t, t-(j+1)\Delta t\right]$ can also be regarded as a variable, i.e., $R_j={\displaystyle {\int }_{t-(j+1)\Delta t}^{t-j\Delta t}w(\tau )\mathrm{d}\tau }$. It readily satisfies the following conditions as well [29]:

$$E[R_j]=0, E\left[R_j{R}_i\right]={\mathsf{\delta }}_{ji}$$

(15)

Therefore, Equation (14) can be further rewritten as follows:

$$X\left(t\right)={\displaystyle \sum _{j=0}^{q}h(j\Delta t)R_j}$$

(16)

For simulating a process with the number of time steps N, Equation (16) can be particularly written in following matrix form:

$$X\left(t\right)=\left[\begin{array}{c}X\left(\Delta t\right)\\ X\left(2\Delta t\right)\\ ⋮\\ X\left(N\Delta t\right)\end{array}\right]=\left[\begin{array}{cccc}{R}_1& R_0& \cdots & R_{1-q}\\ R_2& R_1& \cdots & R_{2-q}\\ ⋮& ⋮& \ddots & ⋮\\ R_N& R_{N-1}& \cdots & R_{N-q}\end{array}\right]\left[\begin{array}{c}h(0)\\ h(\Delta t)\\ ⋮\\ h(q\Delta t)\end{array}\right]$$

(17)

Equations (16) and (17) are formulations of the alternative time-domain simulation strategy based on random orthogonal variables.

So far, the random orthogonal variable representation of stationary processes has been established, and two corresponding time-domain simulation strategies have been developed. The first strategy directly follows the discretized filtering white noise model, while the alternative strategy improves computational efficiency by exploiting the decay property of the impulse response function. These formulations provide a flexible and efficient foundation for numerical simulation in engineering. However, both simulation strategies involve a large number of random variables in the implementation of the stochastic simulation procedure. In conventional time-domain simulation methods, these numerous random variables are treated as mutually independent, and Monte Carlo techniques are therefore employed for sampling. In practice, however, pseudo-random number-based methods encounter significant difficulties when dealing with high-dimensional problems [33]. This limitation motivates the development of dimension reduction methods that can represent stochastic processes with a few random variables while preserving essential statistical characteristics [6,29]. These concepts will be introduced in detail in the next section.

3. Proposed Dimension Reduction Methods

3.1. Random Function Representation of Orthogonal Random Variables

The random orthogonal variable sets $\left\{V_i\right\}$ and $\left\{R_j\right\}$ involved in the above-mentioned two simulation strategies can be represented by random orthogonal function driven by a single elementary random variable. Such random function representation methods for dimension reduction have been successfully used in frequency domain simulations of stationary and non-stationary processes to address the high-dimensional random variables problem. As a result, the sets can be represented as follows [34]:

$$V_i=h_i(\mathsf{\Theta }), R_j=g_j(\mathsf{\Theta })$$

(18)

where $h_i(\cdot )$ and $g_i(\cdot )$ denote the random orthogonal functions. $\mathsf{\Theta }$ denotes the elementary random variable with probability density function (PDF) $p_{\mathsf{\Theta }}(\mathsf{\theta })$.

The random orthogonal functions in Equation (18) can take various forms. Therefore, the primary task is to identify appropriate random functions that satisfy the orthogonality conditions defined in Equations (10) and (15). This procedure can be carried out in two steps. The first step is to define sets of random functions as follows [29,35]:

Strategy I:

$${\overline{V}}_r=\sqrt{2}\cos \left(r\times \mathsf{\Theta }+\alpha \right), r=1,2,\cdots ,N$$

(19)

where $\mathsf{\Theta }$ is the elementary random variable uniformly distributed over the interval $\left[0, 2\mathsf{\pi }\right]$. $\alpha$ is a deterministic value in $[0, 2\mathsf{\pi }]$ and valued by $\mathsf{\pi }/4$.

Strategy II:

$${\overline{R}}_s=\sqrt{2}\cos \left(s\times \mathsf{\Theta }+\alpha \right), s=1,2,\cdots ,N+q$$

(20)

It is readily evident that the defined functions are mutually orthogonal. As an example, the proofs of Equation (19) are given as follows [36]:

$$E\left[{\overline{V}}_r\right]={\displaystyle {\int }_0^{2\mathsf{\pi }}\sqrt{2}\cos \left(r\times \theta +\mathsf{\pi }/4\right)\times \frac{1}{2\mathsf{\pi }}\mathrm{d}}\theta =0$$

(21a)

$$E\left[{\overline{V}}_r{\overline{V}}_s\right]={\displaystyle {\int }_0^{2\mathsf{\pi }}\sqrt{2}\cos \left(r\times \theta +\mathsf{\pi }/4\right)\times \sqrt{2}\cos \left(s\times \theta +\mathsf{\pi }/4\right)\times \frac{1}{2\mathsf{\pi }}\mathrm{d}}\theta ={\mathsf{\delta }}_{rs}$$

(21b)

The second step is to complete a determined mapping between defined random function sets and orthogonal random variables required in two simulation strategies, i.e., ${\overline{V}}_r\to V_i$ and ${\overline{R}}_s\to R_j$. This determined mapping relationship can be achieved by the MATLAB R2023b tool box functions rand(‘state’,0) and randperm$(\cdot )$. To be specific, the following MATLAB programming languages are used [35].

Strategy I:

$$\mathit{V}=\mathrm{zeros}(N,1); \mathrm{rand} (‘\mathrm{state}’,0); temp=\mathrm{randperm}(N);$$

$$\mathrm{for} l=1:N; \mathit{V}(l)=\overline{\mathit{V}}\left(temp(l)\right); \mathrm{end}$$

Strategy II:

$$\mathit{R}=\mathrm{zeros}(N+q,1); \mathrm{rand}(‘\mathrm{state}’,0); temp=\mathrm{randperm} (N+q);$$

$$\mathrm{for} l=1:N+q; \mathit{R}(l)=\overline{\mathit{R}}\left(temp(l)\right); \mathrm{end}$$

It should be noted that the determined mapping between the defined random function sets and the required orthogonal random variables is not unique. Many possible random seeds, i.e., different values of the ‘state’ parameter, can be selected in the pseudo-random number generator [5]; however, only one representative choice is illustrated in this article.

By the above procedure, Equations (11) and (16) can be driven by a single random variable, i.e., the dimension of the required random variables is reduced to one. On this basis, the well-defined assigned probability of the representative point can be obtained using various sampling techniques in one-dimensional space; such as number-theory methods [37].

3.2. Point-Selection Method

A representative sample function set generated by proposed dimension reduction methods can be interpreted from a probabilistic perspective. The procedure essentially divides the probability space of the single variable that drives the time-domain simulation formulation. Accordingly, the assigned probability of each sample function is the probability corresponding to the associated representative point. The uniformly distributed representative points can be obtained as follows [37]:

$${\theta }_k=(k-0.6)\times {\displaystyle \frac{2\mathsf{\pi }}{n_{\mathrm{sel}}}}$$

(22)

where ${\theta }_k$ denotes the k-th representative point. 0.6 denotes a fixed center shift to avoid boundary clustering. $n_{\mathrm{sel}}$ denotes the total number of selected representative points; further details can be found in Refs. [29,38].

The point-selection method, as shown in Equation (22), ensures that the representative points have well-defined assigned probabilities, i.e., $p_{{\theta }_k}=1/n_{\mathrm{sel}}$$(k=1,2,\cdots ,n_{\mathrm{sel}})$, and provides a uniform coverage of the whole one-dimensional probability space, i.e., ${\displaystyle {\sum }_{k=1}^{n_{\mathrm{sel}}}{p}_{{\theta }_k}=1}$. Therefore, the sample function corresponding to the representative points possesses the identical probability, i.e., $p_{X_k(t)}=1/n_{\mathrm{sel}}$, and the set constructed from the generated sample functions forms a complete probability set.

3.3. Implementation Procedure

The implementation procedure for generating the representative sample functions using the proposed time-domain dimension reduction methods is illustrated in Figure 1. The implementation procedure mainly involves five steps for both strategies I and II. The first step is identical for the two strategies, while differences arise from the second step onward. Specifically, the first step is to select an appropriate PSD function for the target stochastic process, from which the corresponding function $H(\omega )$ is obtained. Second, the two strategies differ due to the distinct discretization treatments of Equation (7), and therefore require different forms in the simulation. In particular, strategy I employs the impulse response function $h(t-i\Delta t)$, whereas strategy II uses the function $h(j\Delta t)$. Third, according to Equation (22), a set of representative points of the single elementary random variable is selected for the two proposed methods. Since both strategies involve only one elementary random variable, the same selection scheme is adopted for the representative points. Fourth, to complete the mapping from the single elementary random variable to random functions, the random orthogonal variables required in Equations (11) and (16) can be represented by random functions through Equation (18), i.e., ${\overline{V}}_r\to V_i$ and ${\overline{R}}_s\to R_j$. Finally, the random functions generated in the above step are substituted into simulation formulations, i.e., Equations (11) and (16), to generate the sample functions with well-defined assigned probabilities and to form a complete probabilistic set. Although the two strategies differ in formulation and implementation, their underlying principle is identical, i.e., the stochastic process is represented through a set of random functions with associated coefficients.

In this section, time-domain dimension reduction concepts are clearly defined by the above statements. It is worth noting that the proposed methods possess a few distinct features compared with conventional time-domain simulation methods and frequency domain dimension reduction methods. First, simulation strategies are derived from spectral decomposition formulation of the process. Therefore, the random variables required in the two strategies, i.e., in Equations (11) and (16), are mutually orthogonal, which is different from the widely accepted fact that random variables in conventional time-domain methods are mutually independent. In fact, the orthogonal constraint of variables in the proposed method is a sufficient condition for dimension reduction. Moreover, strategy II for the first time considers the physical characteristic of process, i.e., the rapidly decaying property of the impulse response function, which can greatly reduce the computational cost. In addition, the study is a successful expansion of dimension reduction concepts proposed by Liu et al. [2] from the frequency domain to the time domain. In practical engineering applications, the proposed methods make it possible to combine probabilistic methods, such as PDEM, to perform meticulous stochastic dynamic response analyses and reliability assessments of complex engineering structures.

4. Numerical Example

To verify the applicability of the proposed approaches to practical engineering, a numerical example based on turbulent wind velocity is considered. The wind velocity at point M, located 90 m above the mean sea level on an offshore wind turbine with a bucket foundation (tower height of 110 m), is selected for simulation. The location of point M is shown in Figure 2. The turbulent wind velocity is regarded as a stationary stochastic process in the present study.

The two-sided PSD function of the turbulent wind velocity is described by the kaimal spectrum, given as [39]:

$$S_{\mathrm{Kaimal}}(z,\omega )={\displaystyle \frac{50z{u}_{\ast }^2}{\mathsf{\pi }{v}_z{(1+\frac{50}{2\mathsf{\pi }{v}_z}z\omega )}^{5/3}}}$$

(23)

where $z$ denotes the height at the point M; $v_z$ denotes the mean wind velocity at the height of z; $u_{\ast }$ denotes the shear velocity of the flow, which is given as follows [40]:

$$u_{\ast }={\displaystyle \frac{\kappa v_z}{\ln (z/z_0)}}$$

(24)

where $\kappa =0.4$ denotes the von Kármán constant; $z_0$ denotes the surface roughness length.

The specific parameters used for the wind velocity simulations are listed in

Table 1. Simulation parameters for the two methods.

Parameters	Values
Duration time $T(s)$	600
Time interval $\Delta t(s)$	0.25
Number of time intervals $N$	2400
Number of the representative points $n_{\mathrm{sel}}$	233
Height of point M $z(\mathrm{m})$	90
Mean wind velocity $v_z(\mathrm{m}/\mathrm{s})$	42
Surface roughness length $z_0(\mathrm{m})$	0.0002
Order of the first method $q$	500

. Using these parameters, the stationary process is simulated through two dimension reduction methods (DRM) driven by a single variable and conventional methods (CM) driven by numerous random variables based on the Monte Carlo Simulation [41]. Figure 3 shows the sample functions generated by dimension reduction methods and conventional methods. From this figure, it is readily evident that these sample functions all exhibit stochastic and stationary characteristics, which indicates that the four methods, all derived from spectral decomposition, are capable of capturing the stochastic nature of turbulent wind velocity. This figure also demonstrates that the dimension reduction methods driven by a single random variable, consistent with the methods driven by numerous random variables, are able to effectively represent the stochastic characteristics of the random process.

Figure 4 shows a comparison of statistical characteristics of the mean and standard deviation between the simulated sample function sets and the target process. From the mean results, i.e., Figure 4a,b, the sample functions simulated by these methods exhibit pronounced zero-mean characteristics. Moreover, the DRMs show substantially smaller fluctuations compared to the CMs. For the standard results shown in Figure 4c,d, all methods exhibit good performance, although the DRMs do not perform as well as the mean case.

To further evaluate the performance of the proposed method, the PSD functions of the sample function sets can be calculated and compared. Figure 5 shows the comparison of the PSD function between the simulated samples and the target process. From this figure, several observations can be made. First, the four methods derived from the two strategies exhibit excellent agreement with the target PSD. Second, the two methods based on Strategy II show better performance than those based on Strategy I. In addition, it is worth noting that the DRMs driven by a single random variable exhibit excellent performance, which demonstrates the feasibility of the proposed dimension reduction approach.

In order to further quantify the simulation accuracy and compare the efficiency of the four methods,

Table 2. The relative errors and computational time for the two methods.

Simulation Methods		Mean Errors	Std. D Errors	PSD Errors	Computational Time (s)
Strategy I	CM-I	$5.95\times {10}^{-2}$	$4.33\times {10}^{-2}$	$1.97\times {10}^{-2}$	47.53
	DRM-I	$1.92\times {10}^{-5}$	$4.05\times {10}^{-2}$	$1.99\times {10}^{-2}$	45.73
Strategy II	CM-II	$5.87\times {10}^{-2}$	$3.41\times {10}^{-2}$	$1.97\times {10}^{-2}$	4.89
	DRM-II	$1.45\times {10}^{-5}$	$3.16\times {10}^{-2}$	$1.97\times {10}^{-2}$	4.51

summarizes the relative errors of the mean values, standard deviations, and PSDs, together with the corresponding computational times for all methods. As shown in Table 2, all statistical error measures of the four simulated methods are maintained at a low level, and the methods based on Strategy II consistently exhibit slightly smaller errors than those based on Strategy I, indicating that Strategy II achieves marginally better accuracy than Strategy I. In addition, with respect to computational efficiency, the methods under Strategy II (CM-II and DRM-II) require approximately 4.5 s, whereas those under Strategy I (CM-I and DRM-I) require approximately 47 s. This corresponds to nearly one order of magnitude improvement in computational efficiency, demonstrating that Strategy II is significantly more efficient than Strategy I. Furthermore, to evaluate the influence of incorporating dimension reduction techniques on each simulation strategy, the DRMs are compared with the corresponding CMs within both Strategy I and Strategy II. The computational time of the DRMs is approximately 4% lower than that of the corresponding CMs within the same strategy. The DRMs also exhibit slightly smaller errors in the mean and standard deviation. Importantly, the PSD accuracy remains below 2% when dimension reduction techniques are employed, indicating that the substantial reduction in the dimension of the input random variables does not compromise spectral accuracy. Overall, the results in Table 2 demonstrate that the proposed dimension reduction methods, particularly Strategy II, achieve a favorable balance between simulation accuracy and computational efficiency, making them well-suited for practical engineering applications.

Although the proposed methods demonstrate good accuracy and efficiency in the above examples, several disadvantages and limitations should be noted. First, the methods are developed for stationary stochastic processes with a prescribed PSD, therefore the performance may be affected when the PSD is not accurately known. Second, the simulation accuracy of the two methods is sensitive to the characteristics of the impulse response function. In particular, for Strategy II, the decay characteristic of the impulse response function further influences the determination of the required truncation order. Since different stochastic processes are characterized by different PSD functions, the associated impulse response functions exhibit distinct decay properties. Consequently, different stochastic processes may require different truncation orders in the simulation to achieve comparable accuracy. As a result, some prior understanding of the impulse response characteristics in practical engineering applications is necessary, which may increase the complexity of the simulation and limit its applicability. Finally, the proposed methods are applicable only to stationary stochastic processes. Therefore, the proposed methods are not suitable for strongly non-stationary processes with significant time–frequency variability. This limitation arises because the present formulations of the methods are constructed under the assumption of stationarity with a prescribed time-invariant PSD and impulse response function. For strongly non-stationary processes, the spectral characteristics evolve over time, and the corresponding impulse response function becomes time-dependent. In such cases, the time-invariant time-domain framework adopted in this study may no longer be applicable.

5. Conclusions

This study develops two time-domain methods which are theoretically motivated by spectral decomposition using two different strategies. Strategy I establishes the time-domain representation of stationary stochastic processes via the impulse response function. Strategy II further incorporates the decay characteristics of the impulse response function, resulting in improved computational efficiency relative to Strategy I. Dimension reduction techniques based on random orthogonal functions are incorporated into these time-domain simulation methods. Numerical analyses demonstrate the accuracy and computational efficiency of the proposed time-domain dimension reduction approaches. Some concluding remarks are drawn as follows:

• Time-domain simulation methods of stochastic processes are proposed based on spectral decomposition. The derivation clearly reveals that random variables involved in the time-domain formulations are non-independent, and the only constraint required is that they satisfy prescribed orthogonality conditions.
• The proposed time-domain dimension reduction methods reduce the stochastic dimension from 2400 to 1 (99.95% reduction), while the PSD error remains below 2%. Compared with conventional Monte Carlo-based time-domain methods, the computational time is reduced by approximately 4%. The methods effectively alleviate the high-dimensional random-variable problem in conventional time-domain methods, thereby improving the practicality of stochastic excitation modeling for dynamic response and reliability analysis in engineering applications.
• Compared with Strategy I, Strategy II more sufficiently accounts for the physical characteristics of the process, particularly the decay property of the impulse response function, which significantly improves computational efficiency. Numerical results indicate that both strategies maintain a similar PSD error below 2%, while the computational time of Strategy II is reduced by approximately one order of magnitude.
• The proposed framework integrating dimension reduction techniques into time-domain simulation methods generates sample functions with well-defined assigned probabilities, i.e., $p_{{\theta }_k}=1/n_{\mathrm{sel}}$$(k=1,2,\cdots ,n_{\mathrm{sel}})$. In addition, the resulting samples constitute a complete probabilistic information set of the stochastic process, i.e., ${\displaystyle {\sum }_{k=1}^{n_{\mathrm{sel}}}{p}_{{\theta }_k}=1}$. Therefore, these features enable accurate quantitative analysis of stochastic process-induced hazards.