Nonlinear Autoregressive Model Research Articles

A hybrid network of NARX and DS-attention applied for the state estimation of lithium-ion batteries

Monitoring the state of the battery, including the state of charge (SOC) and state of health (SOH), is crucial for ensuring the safety and reliability of electrical equipment. The paper presents a novel hybrid network that combines nonlinear autoregressive model with exogenous inputs (NARX) and DS-attention. The proposed DS-attention method establishes a robust mapping relationship between inputs and outputs, it is a specialized method of the recurrent neural network that enhances the estimation performance by incorporating division function and self-adaptive function into the attention mechanism. The division function is designed to efficiently differentiate between exogenous inputs and state outputs, thereby minimizing the potential for cross-interference between them, and the self-adaptive function optimizes the query features within the attention mechanism. The results demonstrate that the hybrid method of NARX and DS-attention achieves higher prediction accuracy for SOH and SOC estimation of lithium-ion batteries. Specifically, compared with the best-performing similar methods, the proposed hybrid method enhanced the prediction accuracy by 22.9%, 60.7%, and 51.2% across three different SOH datasets, respectively. In terms of long sequence SOC forecasting, the improvements in prediction accuracy under two different working conditions are 82.5% and 60.5%, respectively. The proposed algorithm optimizes the attention mechanism based on the characteristics of the NARX, resulting in higher estimation accuracy for predicting the state of lithium batteries.

Data-driven modeling of the aerodynamic deformation and drag for a freely moving drop in the sub-critical Weber number regime

Accurate prediction of the dynamics and deformation of freely moving drops is crucial for numerous droplet applications. When the Weber number is finite but below a critical value, the drop deviates from its spherical shape and deforms as it is accelerated by the gas stream. Since aerodynamic drag on the drop depends on its shape oscillation, accurately modeling the drop shape evolution is essential for predicting the drop’s velocity and position. In this study, 2D axisymmetric interface-resolved simulations were performed to provide a comprehensive dataset for developing a data-driven model. Parametric simulations were conducted by systematically varying the drop diameter and free-stream velocity, achieving wide ranges of Weber and Reynolds numbers. The instantaneous drop shapes obtained in simulations are characterized by spherical harmonics. Temporal data of the drag and modal coefficients are collected from the simulation data to train a Nonlinear Auto-Regressive models with eXogenous inputs (NARX) neural network model. The overall model consists of two multi-layer perceptron networks, which predict the modal coefficients and the drop drag, respectively. The drop shape can be reconstructed with the predicted modal coefficients. The model predictions are validated against the simulation data in the testing set, showing excellent agreement for the evolutions of both the drop shape and drag.

Asymmetric impact of climatic parameters on hemorrhagic fever with renal syndrome in Shandong using a nonlinear autoregressive distributed lag model

Hemorrhagic fever with renal syndrome (HFRS) poses a major threat in Shandong. This study aimed to investigate the long- and short-term asymmetric effects of meteorological factors on HFRS and establish an early forecasting system using autoregressive distributed lag (ARDL) and nonlinear ARDL (NARDL) models. Between 2004 and 2019, HFRS exhibited a declining trend (average annual percentage change = − 9.568%, 95% CI − 16.165 to − 2.451%) with a bimodal seasonality. A long-term asymmetric influence of aggregate precipitation (AP) (Wald long-run asymmetry [WLR] = − 2.697, P = 0.008) and aggregate sunshine hours (ASH) (WLR = 2.561, P = 0.011) on HFRS was observed. Additionally, a short-term asymmetric impact of AP (Wald short-run symmetry [WSR] = − 2.419, P = 0.017), ASH (WSR = 2.075, P = 0.04), mean wind velocity (MWV) (WSR = − 4.594, P < 0.001), and mean relative humidity (MRH) (WSR = − 2.515, P = 0.013) on HFRS was identified. Also, HFRS demonstrated notable variations in response to positive and negative changes in ∆MRH(−), ∆AP(+), ∆MWV(+), and ∆ASH(−) at 0–2 month delays over the short term. In terms of forecasting, the NARDL model demonstrated lower error rates compared to ARDL. Meteorological parameters have substantial long- and short-term asymmetric and/or symmetric impacts on HFRS. Merging NARDL model with meteorological factors can enhance early warning systems and support proactive measures to mitigate the disease's impact.

An analysis of the geopolitical and economics influence on tourist arrivals in Russia using a nonlinear autoregressive distributed lag model

PurposeThe COVID-19 pandemic and the onset of the conflict in Ukraine led to a sustained downturn in tourist arrivals (TA) in Russia. This paper aims to explore the influence of geopolitical risk (GPR) and other indices on TA over 1995–2023.Design/methodology/approachWe employ a nonlinear autoregressive distributed lag (NARDL) model to analyze the effects, capturing both the positive and negative shocks of these variables on TA.FindingsOur research demonstrates that the NARDL model is more effective in elucidating the complex dynamics between macroeconomic factors and TA. Both an increase and a decrease in GPR lead to an increase in TA. A 1% negative shock in GPR leads to an increase in TA by 1.68%, whereas a 1% positive shock in GPR also leads to an increase in TA by 0.5%. In other words, despite the increase in GPR, the number of tourists coming to Russia increases by 0.5% for every 1% increase in that risk. Several explanations could account for this phenomenon: (1) risk-tolerant tourists: some tourists might be less sensitive to GPR or they might find the associated risks acceptable; (2) economic incentives: increased risk might lead to a depreciation in the local currency and lower costs, making travel to Russia more affordable for international tourists; (3) niche tourism: some tourists might be attracted to destinations experiencing turmoil, either for the thrill or to gain firsthand experience of the situation; (4) lagged effects: there might be a time lag between the increase in risk and the actual impact on tourist behavior, meaning the effects might be observed differently over a longer period.Originality/valueOur study, employing the NARDL model and utilizing a dataset spanning from 1995 to 2023, investigates the impact of GPR, gross domestic product (GDP), real effective exchange rate (REER) and economic policy uncertainty (EPU) on TA in Russia. This research is unique because the dataset was compiled by the authors. The results show a complex relationship between GPR and TA, indicating that factors influencing TA can be multifaceted and not always intuitive.

Asymmetric effects of oil demand and oil supply shocks on New Zealand's trade

We analyze how different types of oil price shocks affect trade and whether these effects are asymmetric. Our focus is on New Zealand, which has historically been a net oil importer but recently transitioned to an import-only model to meet its needs for finished petroleum products. The analysis proceeds in two stages. First, we identify oil supply shocks, aggregate demand shocks, and oil-specific demand shocks using a recursive structural vector autoregression model. Then, we study whether the effects of these shocks on New Zealand's exports to and imports from its key trading partners—Australia, China, and the United States—are asymmetric using nonlinear autoregressive distributed lag models. The results show differences in the effects of different types of oil price shocks on exports and imports in the short and long run; however, there is limited evidence of asymmetries in these effects, especially in the case of exports. Exports to Australia and China, the most critical markets for New Zealand's exporters, are largely insensitive to oil price shocks.

A wind speed forecasting model using nonlinear auto-regressive model optimized by the hybrid chaos-cloud salp swarm algorithm

Wind energy forecasting is significantly affected by the strong volatility, intermittency and variability of the wind speed sequence itself. Therefore, to improve forecast accuracy and stability, based on nonlinear auto-regressive model with exogenous inputs (NARX) and the hybrid chaos-cloud salp swarm algorithm (CC-SSA), a short-term wind speed prediction method is proposed. Firstly, to reduce the complexity of the original wind speed data and generate subcomponents with different patterns and low complexity, a mixed modal decomposition method is carried out by combining variational modal decomposition (VMD) based on Pearson correlation coefficient and generalized S-transform (GST) based on adaptive sample entropy. Therefore, the complementary advantages of different mode decompositions are obtained by combining the two different subcomponents into a mixed component. Secondly, by using cloud model and chaotic map, the improved CC-SSA algorithm is proposed to improve the convergence performance of salp swarm algorithm (SSA). Finally, by using CC-SSA algorithm to optimize the weights of NARX, the CC-SSA-NARX forecasting model is established to predict wind speed. The experimental results show that for the 1-Step, 2-Step, and 3-Step datasets of the actual wind speed in the studied region, the MAE, MAPE, RMSE, and R2 of the CC-SSA-NARX model are 0.23, 6 %, 0.27, and 0.97 respectively. The proposed model present the highest prediction index among the other 7 comparative models, showing high accuracy and generalization ability in short-term wind speed prediction, it can provide a certain reference for enhancing the stability of wind power generation and promoting the sustainable development of the wind power industry to some extent.

Using nonlinear auto-regressive with exogenous input neural network (NNARX) in blood glucose prediction

BackgroundPredicting of future blood glucose (BG) concentration is important for diabetes control. Many automatic BG monitoring or controlling systems use BG predictors. The accuracy of the prediction for long prediction time is a major factor affecting the performance of the control system. The predicted BG can be used for glycemia management in the form of early hypoglycemic/hyperglycemic alarms or adjusting insulin injections. Recent developments in continuous glucose monitoring (CGM) devices open new opportunities for glycemia management of diabetic patients. Many of those systems need prediction for long prediction horizons to avoid going through hypo or hyperglycemia.MethodsIn this article a nonlinear autoregressive exogenous input neural network (NNARX) is proposed to predict the glucose concentration for longer prediction horizons (PHs) than that was obtained previously with an established recurrent neural network (RNN). The proposed NNARX is a modified version from our previously published RNN with different initialization and building technique but has the same architecture. The modification is based on starting with building nonlinear autoregressive exogenous input model using MATLAB and train it, then close the loop to get NNARX network.ResultsThe results of using the proposed NNARX indicate that the proposed NNARX is better in prediction and stability than unmodified RNN as PH becomes higher than 45 minutes.ConclusionsModification in RNN building extends the ability of the prediction till 100 minutes. It performs statistically significant improvements in the FIT and RMSE values for 100 minutes prediction. It also decreases root mean squared error (RMSE) for both 45 and 60 minutes of prediction.

Unlocking the dynamic linkages between sustainable equity investment and economic policy uncertainty: An empirical analysis for G‐20 countries

AbstractThe study examines the dynamic relationships between sustainable equity investment and economic policy uncertainty of G‐20 countries using monthly environmental, social, and governance (ESG) equity and economic policy uncertainty (EPU) indices. Using the autoregressive distributed lag model and nonlinear autoregressive distributed lag model models, the study finds a negative long‐run relationship between sustainable equity investments and economic policy uncertainties in Australia, Canada, the USA, Brazil, Mexico, Germany, Italy, and Japan. Investors in these G20 countries may perceive that companies with higher ESG performance are more likely to face regulatory scrutiny, legal action, or reputational damage if they are associated with high levels of economic policy uncertainties. As a result, ESG market indices may underperform when EPU is high and vice versa. It supports prospect theory and suggests that individuals are more sensitive to potential losses than gains. On the contrary, the relationship is positive in the case of the USA, Brazil, China, and to some extent India. This might be because firms with high ESG performance could manage risks better and seize opportunities associated with EPU, which helps ESG market indices to outperform when EPU is high. It is supported by the legitimacy theory that says to maintain the legitimacy and credibility of the company, the investment must be invested in ESG initiatives, which can lead to improved long‐term financial performance and market value.

Modeling the influence of mineral rents and low‐carbon energy on environmental quality: New insights from a sustainability perspective

AbstractIn recent decades, the detrimental impacts of climate change have been increasingly felt worldwide. This is due to the extreme consumption of natural resources to fuel economic activity. Confronting a widening ecological deficit, humanity must urgently accelerate its journey toward sustainable development goals (SDGs). This demands active efforts to curb environmental pollution and protect natural resources, safeguarding the planet for future generations. Since Brazil has high biocapacity and ecological areas, the impact of energy resources on ecological sustainability policies cannot be excluded. Moreover, Brazil holds a vital position in the world for global foreign direct investment flows. In this regard, this paper considers the asymmetric impact of mineral rents and foreign direct investments when investigating Brazil's environmental sustainability. In addition, it evaluates elements like economic growth and low‐carbon energy consumption from 1970 to 2021. To this end, the research applies the Fourier nonlinear autoregressive distributed lag model and draws three significant conclusions. First, resource extraction undermines environmental sustainability. Second, the inflow of foreign direct investment reduces the load capacity factor, implying that the pollution haven hypothesis holds for Brazil. Third, low‐carbon energy consumption contributes to environmental sustainability. Our findings highlight the critical role of regulating mineral resource rents in achieving SDGs in Brazil. Moreover, increasing investment in clean energy sources and transitioning toward low‐carbon energy can promote sustainable development in Brazil. The Brazilian government should abandon mineral extraction, focus on low‐carbon energy consumption, and prevent companies from destroying nature by introducing strict environmental regulations for foreign investment flows.

Does State Dependence Matter in Relation to Oil Price Shocks on Global Economic Conditions?

Abstract A common thread in the literature shows that an oil price shock can have a major impact on global economic conditions. We examine the global dimensions of changes to the global oil price and world economic uncertainty using three model types: ordinary least square (OLS); general additive model (GAM); and non-linear vector autoregression (VAR) model with local projections (LP). Our study highlights a positive and statistically significant effect of oil prices on economic uncertainty during non-expansionary periods, yet the impact is negative on economic uncertainty during periods of economic growth. Using a VAR-LP we analyze the global dimensions of a world oil price shock on global economic conditions and investigate whether there is consistency in how an oil price shock influences economic growth, consumer prices and economic uncertainty based on the state of economic conditions. The empirical evidence shows that during an expansionary (a non-expansionary) period, the impact of an oil price shock lowers (elevates) economic uncertainty. The empirical evidence from the three model types taken together indicate a presence of state dependence on the influence of an oil price shock.

Bootstrap prediction inference of nonlinear autoregressive models

The nonlinear autoregressive (NLAR) model plays an important role in modeling and predicting time series. One‐step ahead prediction is straightforward using the NLAR model, but the multi‐step ahead prediction is cumbersome. For instance, iterating the one‐step ahead predictor is a convenient strategy for linear autoregressive (LAR) models, but it is suboptimal under NLAR. In this article, we first propose a simulation and/or bootstrap algorithm to construct optimal point predictors under an or loss criterion. In addition, we construct bootstrap prediction intervals in the multi‐step ahead prediction problem; in particular, we develop an asymptotically valid quantile prediction interval as well as a pertinent prediction interval for future values. To correct the undercoverage of prediction intervals with finite samples, we further employ predictive – as opposed to fitted – residuals in the bootstrap process. Simulation and empirical studies are also given to substantiate the finite sample performance of our methods.

Natural resource development, economic policy uncertainty and urban residents’ consumption in China: a nonlinear ARDL and time-varying parameter vector autoregressive with stochastic volatility approach

AbstractIncreased risks of economic policy uncertainty and overexploitation of natural resources exist in China. At the same time, the growth rate of urban residents’ consumption has generally declined. The paper analyses the role of economic policy uncertainty (EPU) and natural resource exploitation on the urban residents’ consumption in China. Based on the data from the first quarter of 2002 to the third quarter of 2021, the paper uses a nonlinear autoregressive distributed lag model to verify the asymmetric effects. Then the paper constructs a time-varying parameter vector autoregressive model with stochastic volatility term to analyze the nonlinear responses. Impulse response analysis was used to further explain the relationship between the three. The negative impact of rising EPU on urban residents’ consumption is larger than its reduction. Negative shocks to natural resource development increase the urban residents’ consumption. Positive shocks reduce the urban residents’ consumption. There is a time-varying non-linear effect of EPU and natural resource development on urban residents’ consumption in China. The negative impact of EPU on urban consumption has been further exacerbated by major crises such as the financial crisis, COVID-19 and the post-crisis period. The negative impact of natural resource development diminished after the government introduced industrial upgrading policies and environmental regulations. This study provides constructive suggestions for the optimization of economic policies and the improvement of urban consumption. This study also enriches consumer theory and provides new evidence for the resource curse hypothesis.

Importance of natural energy, and blue resources to address environmental challenges in view of the Saudi Vision 2030

AbstractThe purpose of this study is to scrutinize the impact of natural resources, energy, and blue economic indicators on the ecological footprint of Saudi Arabia over the period from 1993 to 2022 by employing a nonlinear autoregressive distributed lag model. We have simulated the missing data using the Markov Chain Monte Carlo algorithm. The study encompasses the effects of post‐Vision 2030 and post‐COVID‐19 policies and their assistance in minimizing the environmental footprint. The observed results determine that non‐renewable energy consumption increases the ecological footprint, whereas natural resources and biocapacity drop the ecological footprints in the case of pre‐ and post‐Vision 2030. For variable fisheries production, the post‐ and pre‐Vision 2030 results demonstrate a boosted ecological footprint in Saudi Arabia, with the highest coefficient among all results. This research offers valuable insights into how Saudi Arabia's energy consumption and natural resource management impact its ecological footprint, highlighting the effectiveness of Vision 2030 and post‐COVID policies in promoting environmental sustainability. The study's findings provide crucial guidance for policy making to reduce environmental impact while considering the role of fisheries and biocapacity in ecological balance. Based on empirical findings, this study commends some policy understandings that assist in being effectively implemented towards a sustainable environment.

Relationship between Energy and Economic Growth: Evidence from a Panel Nonlinear ARDL Model

In this article, we study the heterogeneous relationship among energy consumption and economic growth by incorporation gross fixed capital formation, foreign direct investment, international trade and finance development as control variables in a selected sample of 11 countries of the Middle East and North Africa over the period 1980-2020 using a panel nonlinear autoregressive distributed lag model. Empirical results show the presence of an asymmetric long run relationship between energy consumption and economic growth in the group of countries. The Pooled Mean Group estimates indicate that positive and negative changes in energy consumption have positive and significant effects on output growth in the long term, whereas both Mean Group and Difference Fixed Effect estimates report that only negative changes in energy consumption have positive and significant effect on long-term economic growth in the selected sample of countries. In addition, the short run individual effects of energy use on economic growth are positive and significant for only three countries (Jordan, Saudi Arabia, and Tunisia). These results are very important for the design of energy policies and sustainable economic growth. Energy-saving policies are suitable for long-run economic growth for the entire sample, while they are only suitable for eight countries out of eleven in the short-run.

Asymmetric Effect of Oil Price on Economic Activity: Evidence from Lebanon Using NARDL Model

The aim of this paper is to investigate the impact of oil prices on the real economic activity of Lebanon, represented by the Gross Domestic Product (GDP) while controlling for inflation. For this purpose, this study employs a non-linear auto-regressive distributed lag model (NARDL) that enables to take into consideration the asymmetrical nature of the relationship between these two variables. The empirical findings confirm the asymmetric impact of changes in oil prices on economic activity. Notably, reductions in oil prices positively influence GDP in the long term and exert a more pronounced fluctuating effect compared to increases in oil prices, which are statistically insignificant. In other words, the country’s GDP, serving as a measure of the total value of finished goods and services, does not respond significantly to oil price rises but is influenced by oil price reductions. This empirical analysis focuses particularly on Lebanon, and time series data spanning from 1988 to 2021 are used to explore this relationship. This study guides Lebanon's policymakers, revealing the impact of declining oil prices on economic activity. It highlights growth opportunities amidst challenges like economic crises and corruption. Advocating for prudent policies and strategies, it offers a pathway for sustainable development.

Economic growth, structural changes and employment linkages in India

PurposeIndia started economic reforms at a rapid pace to catch up the world economy by following the services-led-growth model during the post-liberalisation period. Over the years, the growing unemployment rate posits a re-look into the dynamics of growth model for wider work force participation. In this backdrop, the paper aims to examine the dynamics of structural changes in employment pattern in view of economic growth led by services-led growth model in India.Design/methodology/approachThe study employs a non-linear autoregressive model (NARDL) to examine the effect of the growth rates in three broad economic sectors namely agriculture and allied, services and industry on work force participation representing the employment opportunities in India.FindingsThe results highlight that the rapid expansion of the service sector has not occurred with enough employment opportunities by the same rate. By contrast, the growth in the industrial sector significantly creates employment opportunities in the short and long run. These results support the industry led growth model over the services for sustainable and inclusive economic growth in the country.Research limitations/implicationsThe study relies on combined labour force participation rates rather than gender-specific rates. Further, the regulatory, working conditions and economic incentives may affect the gender-specific engagement of the labour force in three broad sectors.Practical implicationsThe results offer important insight into changing patterns in employment with policy lessons. A wider workforce force participation calls for expansion of manufacturing activities through pro-industry programmes.Originality/valueThe study makes pioneer efforts to examine the dynamics of labour force participation with respect to the growth of three broad economic sectors of the Indian economy. The results provide new insights with policy implications for the changing employment pattern and policy response.

Platinum and palladium price forecasting through neural networks

To many commodity market participants, forecasts of price series represent a critical task. In this work, nonlinear autoregressive neural network models’ potential is explored for forecasting daily prices series of platinum and palladium over about a fifty-year period. For this purpose, one hundred and twenty model settings are examined, including different training algorithms, numbers of hidden neurons and delays, and ratios used to segment the data. With the analysis, two models leading to stable and accurate forecast results are constructed for the prices of the two commodities. In particular, the models’ performance in terms of the relative root mean square error is 1.86% and 3.61% for platinum and palladium, respectively, for the overall sample. Results in this work could help technical forecasts and policy analysis. The forecast framework might be extended to other different commodities.

Prediction of SO2 emission concentration in industrial flue gas based on deep learning: The ammonia desulfurization system of the Yunnan aluminum carbon plant as the research object

The desulfurization efficiency of flue gas desulfurization (FGD) systems is affected by many operation parameters, and predicting SO2 emission concentration through mechanism models requires a significant number of resources, limiting the process optimization of the overall desulfurization efficiency. Taking the desulfurization system of Yunnan aluminum carbon plant as the research object, based on the inlet flue gas, process control parameters, and pollutants of the discharged flue gas, we constructed the NARX (nonlinear auto-regressive model with external inputs) neural network and TCN (temporal convolutional network) models. Predicted the concentration of SO2 in the discharged flue gas in the next moment from the above three types of historical data, optimized the industrial control parameters, and reduced the concentration of the discharged SO2. The results show that both TCN and NARX models have strong nonlinear dynamic descriptive ability, and both are capable of accurately predicting the SO2 concentration in the emitted flue gas. The TCN neural network model can control the mean absolute percentage error and root mean square error of the dataset within 1.46% and 1.86, respectively. The response relationship established by the model has an important role in guiding the regulation of SO2 emission and process operation parameters, such as reducing the flue gas pressure of the absorber tower inlet or increasing the pressure of the concentration section of the absorber tower, which can effectively improve the desulfurization rate.

Predicting the Torque Demand of a Battery Electric Vehicle for Real-World Driving Maneuvers Using the NARX Technique

An identification technique is proposed to create a relation between the accelerator pedal position and the corresponding driving moment. This step is beneficial to replace the complex physical model of the vehicle control unit, especially when the sufficient information needed to model certain functionalities of the vehicle control unit are unavailable. We utilized the nonlinear autoregressive exogenous model to regenerate the electric motor torque demand, given the accelerator pedal position, the motor’s angular speed, and the vehicle’s speed. This model proved to be extremely efficient in representing this highly complex relationship. The data employed for the identification process were chosen from an actual three-dimensional route with sudden changes of a dynamic nature in the driving mode, different speed limits, and elevations, as an attempt to thoroughly cover the driving moment scope based on the alternation of the given inputs. Analyzing the selected route data points showed the widespread coverage of the motor’s operational scope compared to a standard driving cycle. The training outcome revealed that linear modeling is inadequate for identifying the targeted system, and has a substantial estimation error. Adding the nonlinearity feature to the model led to an exceptionally high accuracy for the estimation and validation datasets. The main finding of this work is that the combined model from the nonlinear autoregressive exogenous and the sigmoid network enables the accurate modeling of highly nonlinear dynamic systems. Accordingly, the maximum absolute estimation error for the motor’s moment was less than 10 Nm during the real-world driving maneuver. The highest errors are found around the maximum motor’s moment. Finally, the model is validated with measurements from an actual field test maneuver. The identified model predicted the driving moment with a correlation of 0.994.

Defect detection and localisation using guided wave images from array data processed by nonlinear autoregressive exogenous model and Gamma statistical operator

Guided wave structural health monitoring (GWSHM) systems, using the delay-and-sum imaging algorithm, are an efficient solution to detect and localise defects in industrial structures. However, the image artifacts caused by either imperfect detection or sensor lay-out limitations make it difficult to identify and locate defects accurately. In order to enhance the performance of defect detection and localisation in GWSHM systems, this paper proposes a three-step procedure for post-processing guided wave signals prior to image formation. In the first step, the signals are processed using the nonlinear autoregressive exogenous model to suppress noise from benign features. The second step calculates the probability of defect presence based on the rescaled Gamma cumulative distribution function. This probabilistic threshold is then determined from the quantile mapping. Finally, a guide wave image is formed using the delay-and-sum imaging algorithm. The experimental validation was performed to inspect a 6 mm-diameter through-thickness circular hole on an aluminium plate and the defects were further scaled as simulated datasets to test its detectability under various amplitudes. In the second procedure step, the detection and localisation performance of the proposed procedure was compared with that of using the signal difference coefficient and the Rayleigh maximum likelihood estimator. It is shown that the proposed procedure can enhance the contrast between damaged and undamaged regions, providing more reliable and accurate guided wave images.