Simulating nonstationary non-Gaussian vector process based on continuous wavelet transform

doi:10.1016/j.ymssp.2021.108340

Mechanical Systems and Signal Processing

Volume 165, 15 February 2022, 108340

https://doi.org/10.1016/j.ymssp.2021.108340 Get rights and content

Highlights

•
Proposed a new algorithm to simulate nonstationary non-Gaussian vector process.
•
Used continuous wavelet transform with Littlewood-Paley & generalized Morse wavelets.
•
Emphasized the impact of non-redundant aspect of continuous wavelet transform..
•
Validated algorithm using numerical examples for ground motions and wind speeds.

Abstract

In the present study, we propose a new iterative algorithm based on discretized continuous wavelet transform (CWT) to simulate nonstationary non-Gaussian vector process for prescribed marginal probability distribution functions and the time-scale power spectral density function matrix. The algorithm is designed for a CWT pair within the wavelet analysis framework. It can be used with analytical wavelets satisfying the admissibility condition and can cope with time-dependent and time-independent coherence. The proposed algorithm is applied to generate nonstationary downburst winds and seismic ground motions at multiple sites within the wavelet analysis framework. The numerical validation analysis confirms that the proposed algorithm can lead to the simulated records matching the prescribed marginal probability distribution, marginal time-scale power spectral density function and coherence function.

Introduction

A nonstationary vector process, such as the seismic ground motions and downburst winds can be modelled and simulated using several approaches. The most popular approach is based on the assumption that the vector process can be modelled based on the evolutionary process that is characterized by the evolutionary power spectral density (PSD) function [[1], [2]]. It was suggested [1] that the evolutionary PSD function for given signals could be evaluated by using the windowed Fourier transform (i.e., short-time Fourier transform (STFT)). However, the application of the STFT could provide a good time localized high-frequency resolution or good low-frequency resolution but not both [[3], [4], [5], [6]]. Moreover, the use of the evolutionary vector process cannot deal with the possible time-dependent coherence between the stochastic processes [2]. This could be overcome by considering that each nonstationary process is represented as a sum of multiple evolutionary processes [[7], [8], [9], [10], [11]].

The simulation of the evolutionary vector process can be carried out using the spectral representation method (SRM) [[12], [13]]. The samples generated by directly applying SRM are Gaussian. However, the time history of the winds, wind pressures acted on the structure, and seismic ground motions may exhibit non-Gaussian behaviour. An approach to simulate a non-Gaussian process is to apply the translation process [[14], [15]] and SRM. This application requires the use of an iterative procedure to find the underlying Gaussian evolutionary PSD function for the non-Gaussian evolutionary process that is to be simulated [[16], [17], [18], [19]].

To have a better time–frequency or the time-scale representation of a given signal, instead of STFT, the wavelet transforms [[20], [21]] could be applied. The use of the wavelet transforms to model the seismic ground motions was considered by several researchers, including Basu and Gupta [22], Iyama and Kuwamura [23], Gurley and Kareem [24], Spanos and Failla [6], Spanos et al. [25], and Sarkar et al. [26]. The application of the continuous wavelet transform (CWT) to characterize and approximately evaluate the evolutionary PSD function was described in [6]. This allows the nonstationary process to still be modelled as an evolutionary process and simulated by using SRM. The approach of using the results from CWT to define the evolutionary PSD function was considered by Huang and Chen [9] in dealing with the vector process for downburst winds; it was also considered in [10], where each nonstationary process is represented as the sum of multiple evolutionary processes (i.e., sigma oscillatory process). One of the disadvantages of using CWT is associated with its redundant representation. For example, given a signal that is digitized at N discrete points in time, the signal is represented by the wavelet coefficients at k number of scales and N discrete points in time. The signal can be reconstructed using k × N wavelet coefficients. However, one could construct a signal by using the inverse CWT (ICWT) for arbitrarily assigned k × N wavelet coefficients, but the application of CWT to the constructed signal does not necessarily lead to the originally assigned k × N wavelet coefficients. This non-preserving property of ICWT implies that the signals for a nonstationary vector process can be constructed by using the arbitrarily assigned wavelet coefficients based on the target wavelet spectra. However, the application of CWT to the constructed signals may lead to the wavelet coefficients differ from the assigned values. This results in the wavelet spectra of the constructed signals deviate from the target wavelet spectra. It is noted that an algorithm based on CWT with the modified Littlewood-Paley wavelets was proposed in [26] to simulate the nonstationary vector process. This algorithm requires the use of a seed signal. The algorithm assigns wavelet coefficients based on the prescribed condition in the wavelet domain and applies the inverse CWT to the assigned wavelet coefficients with additional scale-independent random phase angles to sample the vector process. The algorithm with the modified Littlewood-Paley wavelets is adequate, at least for a single process, because of its orthogonality property in frequency for properly selected parameters. However, in general, it may not be applied with other analytical wavelets satisfying the admissibility condition because of the non-preserving property of ICWT. This will be elaborated on in the following sections.

An iterative power and amplitude correction (IPAC) algorithm was proposed in Hong et al. [27] to simulate a nonstationary non-Gaussian process based on the time–frequency or time-scale transform (see also [28]). The algorithm combines and extends the concept that a sample of the stochastic process can be defined in the wavelet domain [29] and the iterative amplitude adjusted Fourier transform (IAAFT) algorithm for generating surrogates [30]. They showed that the IPAC algorithm can be used efficiently and effectively to simulate the nonstationary Gaussian or non-Gaussian process with the prescribed marginal probability distribution function and the time-scale (dependent) PSD function. The iteration ensures that the problem associated with the non-preserving property of ICWT is coped with.

In the present study, we propose a new algorithm to simulate nonstationary non-Gaussian vector process by applying (discretized) CWT pairs. The new algorithm can be viewed as an extension to the IPAC algorithm, but take into account the time-dependent or time-independent coherence for pairs of nonstationary processes. For the simulation, we consider that the time-scale PSD function and marginal probability distribution function of each process, as well as the coherence function, are prescribed. Unlike the case for a single process, the phase modification during the iteration must be synchronized to maintain coherence while satisfying the prescribed marginal probability distribution function and the PSD function for each process. Although only the generalized Morse wavelets [31] and the modified Littlewood-Paley wavelets [26] are considered for the numerical analysis shown in the present study, the algorithm is equally applicable by considering other analytical wavelets satisfying the admissibility condition. Numerical examples are used to validate and illustrate the proposed algorithm by simulating seismic ground motions and downburst wind velocity.

Section snippets

Continuous wavelet transform and its inverse – a brief description

The basic elements of CWT that are to be used in the present study are summarized in this section. The term CWT is also used for its discretized version, as well, which should be distinguished from the discrete wavelet transform. CWT provides the time-scale decomposition of a signal and is defined as [[20], [21]], $x_{W} (s, τ) = W T (x (t)) = \frac{1}{\sqrt{|s|}} \int_{- \infty}^{\infty} x (t) ψ * (\frac{t - τ}{s}) d t,$ where $x_{W} (s, τ)$ is the wavelet coefficient; WT() denotes the CWT; ψ(⋅) is the mother wavelet; s and τ are the scale and translation parameters; and *

Implication of non-preserving property of the inverse wavelet transform for arbitrarily assigned wavelet coefficients

To see the implication of the non-preserving property of ICWT, consider a seismic ground motion record illustrated in Fig. 2a, which is a horizontal record component from the Northridge earthquake with ID 1007 and the event ID 127 in http://peer.berkeley.edu/nga/search.html. By applying the WT with MLPW, the obtained $|x_{W} (s, τ)|$ is shown in Fig. 2b by using ς = 2^1/4 and s₀ = 2^1/4 as suggested in Sarkar et al. [26] and $c_{0} = Δ_{t}$ . These constants are used throughout the present study for MLPW. Note that

Proposed algorithm for simulating vector process based on CWT

In this section, we propose to extend the IPAC algorithm [27] to simulate a vector nonstationary non-Gaussian process based on CWT with a coherence function that depends on time or scale or both. The IPAC algorithm was given to simulate a single nonstationary non-Gaussian process for the prescribed marginal probability distribution (PDF) $F_{X, t} (x (t))$ and one-sided TSPSD function $S_{W, X X} (s, τ)$ , where the time-dependent standard deviation for the marginal distribution, σ(t), is given by, $σ^{2} (τ) = \int_{0}^{\infty} S_{W, X X} ($

Example application of the proposed algorithm

Numerical applications and examples illustrating the proposed algorithm to simulate the vector nonstationary and non-Gaussian process are presented in this section for the simulation of the wind velocity time history and seismic ground motions. We note that, for ground motions, there are many available proposed parametric time–frequency dependent PSD functions (for review see [43]), time-dependent coherence models (see [[44], [45], [46], [47], [48]]), and time-dependent parametric coherence

Conclusions

A new iterative algorithm to simulate the nonstationary vector process is proposed within the wavelet analysis framework. The algorithm is based on continuous wavelet transform and can be used to simulate the nonstationary Gaussian or non-Gaussian vector process with equal ease. The use of the continuous wavelet transform leads to a very refined time-scale representation.

The proposed algorithm is an extension of the iterative power and amplitude correction (IPAC) algorithm for a single

CRediT authorship contribution statement

H.P. Hong: Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Resources, Supervision, Writing – original draft, Writing - review & editing. X.Z. Cui: Conceptualization, Investigation, Methodology, Formal analysis, Writing – original draft, Writing - review & editing. D. Qiao: Conceptualization, Investigation, Formal analysis, Writing – original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

We gratefully acknowledge the financial support received from the Natural Sciences and Engineering Research Council of Canada, Canada, (RGPIN-2016-04814, for HPH).

References (58)

G. Huang et al.
Wavelets-based estimation of multivariate evolutionary spectra and its application to nonstationary downburst winds
Eng. Struct.
(2009)
G. Huang
An efficient simulation approach for multivariate nonstationary process: hybrid of wavelet and spectral representation method
Probab. Eng. Mech.
(2014)
M. Shinozuka et al.
Digital simulation of random processes and its applications
J. Sound Vib.
(1972)
G. Deodatis
Non-stationary stochastic vector processes: seismic ground motion applications
Probab. Eng. Mech.
(1996)
F.J. Ferrante et al.
Stochastic simulation of non-Gaussian/non-stationary properties in a functionally graded plate
Comput. Methods Appl. Mech. Eng.
(2005)
M.D. Shields et al.
A simple and efficient methodology to approximate a general non-Gaussian stationary stochastic vector process by a translation process with applications in wind velocity simulation
Probab. Eng. Mech.
(2013)
P.D. Spanos et al.
Stochastic processes evolutionary spectrum estimation via harmonic wavelets
Comput. Methods Appl. Mech. Eng.
(2005)
K. Sarkar et al.
Wavelet-based generation of spatially correlated accelerograms
Soil Dyn. Earthquake Eng.
(2016)
P.C. Liu
Wavelet spectrum analysis and ocean wind waves
T. Schreiber et al.
Surrogate time series
Phys. D
(2000)

K.D. Orwig et al.

Near-surface wind characteristics of extreme thunderstorm outflows

J. Wind Eng. Ind. Aerodyn.

(2007)

Y. Su et al.

Derivation of time-varying mean for non-stationary downburst winds

J. Wind Eng. Ind. Aerodyn.

(2015)

L. Wang et al.

A data-driven approach for simulation of full-scale downburst wind speeds

J. Wind Eng. Ind. Aerodyn.

(2013)

M.B. Priestley

Evolutionary spectra and non-stationary processes

J. R. Stat Soc B

(1965)

M.B. Priestley et al.

On the analysis of bivariate non-stationary processes

J R Stat Soc: Ser B (Methodological)

(1973)

L. Cohen

Time-frequency Analysis

(1995)

H.P. Hong et al.

Time–frequency spectral representation models to simulate nonstationary processes and their use to generate ground motions

J. Eng. Mech.

(2020)

A. Kareem et al.

Numerical simulation of wind effects

P.D. Spanos et al.

Evolutionary spectra estimation using wavelets

J. Eng. Mech.

(2004)

F. Battaglia

Some extensions in the evolutionary spectral analysis of a stochastic process

Bolletino della Unione Matematica Italiana

(1979)

J.P. Conte et al.

Fully nonstationary analytical earthquake ground-motion model

J. Eng. Mech.

(1997)

L. Peng et al.

Evolutionary spectra-based time-varying coherence function and application in structural response analysis to downburst winds

J. Struct. Eng.

(2018)

M. Grigoriu

Crossings of non-Gaussian translation processes

J. Eng. Mech.

(1984)

M. Grigoriu

Simulation of stationary non-Gaussian translation processes

J. Eng. Mech.

(1998)

G. Deodatis et al.

Simulation of highly skewed non-Gaussian stochastic processes

J. Eng. Mech.

(2001)

Y. Wu et al.

Simulation of spatially varying non-Gaussian and nonstationary seismic ground motions by the spectral representation method

J. Eng. Mech.

(2018)

I. Daubechies

D.B. Percival et al.

Wavelet Methods for Time Series Analysis

(2006)

B. Basu et al.

Seismic response of SDOF systems by wavelet modeling of nonstationary processes

J. Eng. Mech.

(1998)

Cited by (26)

Efficacy assessment for multi-vehicle formations based on data augmentation considering reliability
2024, Advanced Engineering Informatics
Nowadays, aerial vehicle swarm (AVS) formations have been widely applied to military actions. Meanwhile, assessing their efficacy has also received increasing attention due to the significance for offline tactic planning and online formation switching, which places extra emphasis on the combination of empirical knowledge and experimental data, the transparency of assessment process and the explainability of assessment results. In practical engineering, the efficacy data collected from AVS formations are often incomplete due to limited experiments, and unreliable due to multi-source uncertainties, which poses a great challenge to the development of efficacy assessment system that balances reliability, explainability and accuracy. Oriented to the efficacy assessment of AVS formations under incomplete and unreliable efficacy data, the solution based on belief rule base (BRB) is developed. The reliable heterogeneous data augmentation (RHDA) method is first proposed to improve the availability of efficacy data collected from AVS formation with destination configuration by those collected from AVS formation with source configuration. Then, the feature reliability calculation method based on configuration similarity is designed to avoid the interference of subjective intention. Besides, the configuration importance measurement method based on first-order traceability analysis is put forward to provide decision-makers with references for selecting the most important source configuration when multiple ones exist. The feasibility and effectiveness of developed solution are verified by appropriate simulation experiments.
Neural network-aided simulation of non-Gaussian stochastic processes
2024, Reliability Engineering and System Safety
Simulation of non-Gaussian stochastic processes is of paramount importance to dynamic reliability assessment of structures driven by non-Gaussian loads. In this paper, a novel translation model based on neural network is proposed to convert the target non-Gaussian power spectrum to the underlying Gaussian power spectrum for simulating non-Gaussian stochastic processes. First, a valid neural network model, whose structure is similar to the transformation from a non-Gaussian power spectrum to its underlying Gaussian power spectrum, is established. Then, starting with an initial underlying Gaussian power spectrum, the non-Gaussian power spectrum corresponding to the target non-Gaussian distribution is obtained through the Wiener-Khintchine transform and translation process. After that, the discrete data of non-Gaussian power spectrum is employed as the input layer in the neural network above, while the discrete data of Gaussian power spectrum serves as the output layer. With the trained neural network, the required underlying Gaussian power spectrum can be directly obtained by inputting the target non-Gaussian power spectrum. Finally, Gaussian samples can be generated by using the spectral representation method, which are then converted into non-Gaussian samples via the translation process. The effectiveness, accuracy, and applicability of the proposed method are demonstrated through two numerical examples.
An enhanced cyclostationary method and its application on the incipient fault diagnosis of induction motors
2023, Measurement: Journal of the International Measurement Confederation
The cyclostationary analysis techniques have been extensively explored for the purpose of fault detection in rotating machinery. However, there are still huge challenges because of both limited detection frequency range and low fault identification accuracy. This paper proposes an improved cyclostationary method to enhance incipient fault features. Firstly, the continuous wavelet transform is used to accurately locate important frequency bands, and the fault modulation mechanism or fast kurtogram can be adopted to design the optimal wavelet transform scale factor. Secondly, the Teager-Kaiser energy operator is improved to be used in the frequency domain for the weak fault feature enhancement. Finally, fault features are presented in the cyclic frequency domain through spectral coherence and enhanced envelope spectrum. The proposed method is verified through both numerical simulation and experiments, including incipient half-broken rotor bar, and rolling bearing outer race faults in induction motors.
An iterative multi-fidelity scheme for simulating multi-dimensional non-Gaussian random fields
2023, Mechanical Systems and Signal Processing
This paper presents an efficient multi-fidelity scheme to simulate multi-dimensional non-Gaussian random fields that are specified by covariance functions and marginal probability distribution functions. To develop the multi-fidelity scheme, two numerical algorithms are proposed in turn. The first algorithm is used to simulate random samples of the random field. In this algorithm, initial random samples are first generated to meet the marginal distribution and an iterative procedure is adopted to change the ranking of the random samples to match the target covariance function. By using the above algorithm, random samples satisfying the target covariance function and the target marginal distribution are obtained, but it is computationally intensive and is not suitable to simulate large-scale random fields. By taking advantage of Karhunen–Loève expansion, a multi-fidelity algorithm is then proposed to reduce the computational effort. The random variables in Karhunen–Loève expansion are calculated via performing the first algorithm on the low-fidelity model and the deterministic functions in Karhunen–Loève expansion are solved on the high-fidelity model. In this way, the proposed method has low computational effort and a high fidelity simultaneously. Three numerical examples, including two- and three-dimensional non-Gaussian random fields, are used to verify the good performance of the proposed algorithms.
Analyzing, modelling, and simulating nonstationary thunderstorm winds in two horizontal orthogonal directions at a point in space
2023, Journal of Wind Engineering and Industrial Aerodynamics
Thunderstorm winds in the horizontal plane are time-varying, resulting in the time-varying alongwind direction and mean wind speed in the alongwind direction. The fluctuating winds in the alongwind and crosswind directions are time-varying as well. The characterization of the mean and fluctuating winds for their simulation in two horizontal orthogonal directions are incomplete or unavailable in the literature. In the present study, we analyze and model the thunderstorm winds with respect to their time- and directional-variation of the alongwind speed and direction, as well as the fluctuating wind components in the alongwind and crosswind directions. For the analysis, we use the continuous harmonic wavelet. We show that the power spectral density (PSD) functions of the fluctuating winds depend on the time-varying mean wind speed. Based on the time-frequency spectral analysis, we propose the PSD functions and lagged coherence functions for the fluctuating thunderstorm winds in the alongwind and crosswind directions. We also present new empirical models for the time-varying alongwind direction and mean wind speed in the alongwind direction. Finally, we suggest the steps to simulate the wind speed in the alongwind and crosswind directions with time-varying directions and provide numerical examples to illustrate the effectiveness of our proposed models.
Regional differences in China's electric vehicle sales forecasting: Under supply-demand policy scenarios
2023, Energy Policy
Policy incentives are the key driving force for the electric vehicle (EV) market cultivation. During the EV market cultivation in different regions, the supply-side and demand-side policies have different effects. Accurately forecasting the stage characteristics and response sensitivity of EV sales under supply-demand side policy scenarios, which is crucial to the EV promotion policies design. This study selects the EV sales in 31 provinces with data available, as the basis for decision-making, and proposes a multi-factor prediction model integrating grey relation analysis (GRA), discrete wavelet transform (DWT), and bidirectional long short-term memory. Combined with the development difference of 31 provinces, the penetration of China's EV market under the benchmark, supply, demand, and ideal scenarios are verified. The experimental results show that the average Mean Absolute Percentage Error (MAPE) of the GRA-DWT-BiLSTM model is 9.884, and the 31 samples show good applicability for EV sales forecasting. In 2027, the growth rate of China's EV sales in the demand-side scenario will exceed the supply-side scenario. Under the ideal scenario, China's EV penetration rate will reach 27.31%, 42.40%, and 52.97% in 2025, 2030, and 2035 respectively. The forecast results provide a decision-making basis for China's EV market sequential supply-demand side policies.

View all citing articles on Scopus

View full text

Simulating nonstationary non-Gaussian vector process based on continuous wavelet transform

Highlights

Abstract

Introduction

Section snippets

Continuous wavelet transform and its inverse – a brief description

Implication of non-preserving property of the inverse wavelet transform for arbitrarily assigned wavelet coefficients

Proposed algorithm for simulating vector process based on CWT

Example application of the proposed algorithm

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgment

Eng. Struct.

Probab. Eng. Mech.

J. Sound Vib.

Probab. Eng. Mech.

Comput. Methods Appl. Mech. Eng.

Probab. Eng. Mech.

Comput. Methods Appl. Mech. Eng.

Soil Dyn. Earthquake Eng.

Phys. D

J. Wind Eng. Ind. Aerodyn.

J. Wind Eng. Ind. Aerodyn.

J. Wind Eng. Ind. Aerodyn.

Evolutionary spectra and non-stationary processes

J. R. Stat Soc B

On the analysis of bivariate non-stationary processes

J R Stat Soc: Ser B (Methodological)

Time-frequency Analysis

Time–frequency spectral representation models to simulate nonstationary processes and their use to generate ground motions

J. Eng. Mech.

Numerical simulation of wind effects

Evolutionary spectra estimation using wavelets

J. Eng. Mech.

Some extensions in the evolutionary spectral analysis of a stochastic process

Bolletino della Unione Matematica Italiana

Fully nonstationary analytical earthquake ground-motion model

J. Eng. Mech.

Evolutionary spectra-based time-varying coherence function and application in structural response analysis to downburst winds

J. Struct. Eng.

Crossings of non-Gaussian translation processes

J. Eng. Mech.

Simulation of stationary non-Gaussian translation processes

J. Eng. Mech.

Simulation of highly skewed non-Gaussian stochastic processes

J. Eng. Mech.

Simulation of spatially varying non-Gaussian and nonstationary seismic ground motions by the spectral representation method

J. Eng. Mech.

Wavelet Methods for Time Series Analysis

Seismic response of SDOF systems by wavelet modeling of nonstationary processes

J. Eng. Mech.