Hurst Exponent Research Articles

System and transcript dynamics of cells infected with severe acute respiratory syndrome virus 2 (SARS-CoV-2)

Statistical laws arise in many complex systems and can be explored to gain insights into their structure and behavior. Here, we investigate the dynamics of cells infected with severe acute respiratory syndrome virus 2 (SARS-CoV-2) at the system and individual gene levels; and demonstrate that the statistical frameworks used here are robust in spite of the technical noise associated with single-cell RNA sequencing (scRNA-seq) data. A biphasic fit to Taylor’s power law was observed, and it is likely associated with the larger sampling noise inherent to the measure of less expressed genes. The type of the distribution of the system, as assessed by Taylor’s parameters, varies along the course of infection in a cell type-dependent manner, but also sampling noise had a significant influence on Taylor’s parameters. At the individual gene level, we found that genes that displayed signals of punctual rank stability and/or long-range dependence behavior, as measured by Hurst exponents, were associated with translation, cellular respiration, apoptosis, protein-folding, virus processes, and immune response. Those genes were analyzed in the context of a protein-protein interaction network to find possible therapeutic targets.

Fractal Analysis of Mining Wastewater Time Series Parameters: Balkhash Urban Region and Sayak Ore District

The population life and health quality are significantly reduced due to water resources pollution caused by heavy metals, especially in urban agglomerations located close to metal ore mining and processing facilities. The greatest environmental pollution occurs during the extraction of Cu, Zn, and Pb. In this study, a fractal R/S analysis of wastewater discharge indicators time series from a metal ore mining facility located in the Sayak ore district in the Republic of Kazakhstan (turbidity, electrical conductivity, flow magnitude, and pH level) was carried out. A sharp increase in the flow rate was recorded from 10 to 15 July 2024 and an increase in the electrical conductivity from 4 to 26 July 2024. However, the latest type of indicator assessment does not exceed the critical level for life. The presence of electrical conductivity indicators time series long-term memory and persistence was also recorded (the Hurst exponent for the electrical conductivity time series is fixed in the 0.56 to 0.59 range and does not go below the threshold value for randomness according to the Anis-Lloyd formula). Thus, the value-changing process is controlled and stable, and minor changes in turbidity indicate that these releases do not significantly harm the environment. Despite this, the results obtained do not allow for a comprehensive analysis of the state of releases as the data from all deposits is not available. Therefore, due to the time constraints of the data provided for analysis, it is difficult to fully assess the impact of specific metal ore mining facilities on the environmental safety of the Balkhash urban region. In addition, many studies indicate very high risks of chronic diseases for the population living in this region. The findings of this study enable us to conclude that the application of fractal analysis and the calculation of fractal characteristics for time series of emissions can serve as an indicator of the environmental status within the given area. This information can be used by environmental services to build reliable environmental pollution monitoring systems.

Dynamic Black–Litterman Portfolios Incorporating Asymmetric Fractal Uncertainty

This study investigates the profitability of portfolios that integrate asymmetric fractality within the Black–Litterman (BL) framework. It predicts 10-day-ahead exchange-traded fund (ETF) prices using recurrent neural networks (RNNs) based on historical price information and technical indicators; these predictions are utilized as BL views. While constructing the BL portfolio, the Hurst exponent obtained from the asymmetric multifractal detrended fluctuation analysis is employed to determine the uncertainty associated with the views. The Hurst exponent describes the long-range persistence in time-series data, which can also be interpreted as the uncertainty in time-series predictions. Additionally, uncertainty is measured using asymmetric fractality to account for the financial time series’ asymmetric characteristics. Then, backtesting is conducted on portfolios comprising 10 countries’ ETFs, rebalanced on a 10-day basis. While benchmarking to a Markowitz portfolio and the MSCI world index, profitability is assessed using the Sharpe ratio, maximum drawdown, and sub-period analysis. The results reveal that the proposed model enhances the overall portfolio return and demonstrates particularly strong performance during negative trends. Moreover, it identifies ongoing investment opportunities, even in recent periods. These findings underscore the potential of fractality in adjusting uncertainty for diverse portfolio optimization applications.

Solar Wind Turbulence and Complexity Probed with Rank-Ordered Multifractal Analysis (ROMA)

The Rank-Ordered Multifractal Analysis (ROMA) is a tool designed to characterize scale (in)variance and multifractality based on rank ordering the fluctuations in “groups” characterized by the same mono-fractal behavior (Hurst exponent). A range-limited structure-function analysis provides the mono-fractal index for each rank-ordered range of fluctuations. We discuss here two examples of multi-scale solar wind turbulence and complexity where ROMA is applied on the following: (a) data collected by Ulysses spacecraft in the fast solar wind, outside the ecliptic, between 25 and 31 January 2007, at roughly 2.5 Astronomical Units (AU) from the Sun, in the Southern heliosphere, at latitudes between −76.5 and −77.3 degrees, and (b) slow solar wind data collected in the ecliptic plane by Venus Express spacecraft, at 0.72 AU, on 28 January 2007. The ROMA spectrum of fast solar wind derived from ULYSSES data shows a scale-dependent structure of fluctuations: (1) at the smallest/kinetic range of scales (800 to 3200 km), persistent fluctuations are dominant, and (2) at the inertial range of scales (104 to 2 × 105 km), anti-persistent fluctuations are dominant, but less clearly developed and possibly indicative for the development of instabilities with cross-over behavior. The ROMA spectrum of the slow solar wind derived from Venus Express data, suggests a different structure of turbulence: (1) fully developed multifractal turbulence across scales between 5 × 104 and 4 × 105 km, with the Hurst index changing from anti-persistent to persistent values for the larger amplitude magnetic fluctuations; (2) at the smallest scales (400 to 6400 km), fluctuations are mainly anti-persistent, and the ROMA spectrum indicates a tendency towards mono-fractal behavior.

Identification of ocular artifact in EEG signals using VMD and Hurst exponent.

Electroencephalographic (EEG) readings are usually infected with unavoidable artifacts, especially physiological ones. One such physiological artifact is the ocular artifacts (OAs) that are generally related to eyes and are characterized by high magnitude and a specific spike pattern in the prefrontal region of the brain. During the long-duration EEG acquisition, the retrieval of important information becomes quite complicated in prefrontal regions as ocular artifacts dominate the EEG recorded, making it difficult to discern underlying brain activity. With the progress and development in signal processing techniques, artifact handling has become a progressive field of investigation. This paper presents a framework for the detection and correction of ocular artifacts. This study emphasizes improving the quality and reducing the time complexity by using higher-order statistics (HOS) for artifact identification and variational mode decomposition (VMD) for OA correction. An overall SNR of 14 dB, MAE of 0.09, and PSNR of 33.59 dB has been attained by the proposed framework. It was observed that the proposed HOS-VMD surpassed the state-of-the-art mode decomposition techniques.

Chaos in Bitcoin Cryptocurrency Metrics: Analysis and Forecasts

AbstractCryptocurrencies, particularly Bitcoin have attracted a lot of attention in the last decades of humanity. Analyzing cryptocurrencies algorithmic differences, chaotic behavior and self-similarity in cryptocurrency metrics might give significant insights for identifying risks and opportunities. Determining the degree of chaos in crypto metrics is critical for understanding complexity, improving prediction capabilities, and supporting decision-making. This study focuses on the analysis of chaos and self-similarity in Bitcoin dynamics for predictability perspective. Return, rate of return and volume quantities in different scales are analyzed with using rescaled range method to reveal the degree of self-similarity. Hurst parameter extracts a comprehensive summary providing information on how current values depend on previous ones to reveal any persistence in Bitcoin metrics. Daily rate of return and return give Hurst degree around 0.64 while they are in between 0.52–0.55 for minutely and hourly based prices. However, an increasing persistence is observed with the increasing time window. Although the largest Lyapunov exponents stay in the positive region for prices and returns of Bitcoin, they are approximately zero for inspected statistics. Periodic characteristics of Bitcoin are also investigated to reveal any dependencies on halving mechanism of Bitcoin. Detailed self-similarity analysis on specific periods shows that bull and bear market seasons don’t make any significant effect on the degree of Hurst parameter. Due to nonlinear and unpredictable characteristics of Bitcoin metrics, distribution fittings are applied to characterize BTC return and rate of return. While Wakeby distribution gives best fitting for daily return, Cauchy distribution gives best for hourly returns.

Fractional Brownian motion in confining potentials: non-equilibrium distribution tails and optimal fluctuations

Abstract At long times, a fractional Brownian particle in a confining external potential reaches a non-equilibrium (non-Boltzmann)
steady state. Here we consider scale-invariant power-law potentials $V(x)\sim |x|^m$, where $m>0$, and employ the optimal fluctuation method (OFM) to determine the large-$|x|$ tails of the steady-state probability distribution $\mathcal{P}(x)$ of the particle position. The calculations involve finding the optimal (that is, the most likely) path of the particle, which determines these tails, via a minimization of the exact action functional for this system, which has recently become available. Exploiting dynamical scale invariance of the model in conjunction with the OFM ansatz, we establish the large-$|x|$ tails of $\ln \mathcal{P}(x)$ up to a dimensionless factor $\alpha(H,m)$, where $0<H<1$ is the Hurst exponent. We determine $\alpha(H,m)$ analytically (i) in the limits of $H\to 0$ and $H\to 1$, and (ii) for $m=2$ and arbitrary $H$, corresponding to the fractional Ornstein-Uhlenbeck (fOU) process. Our results for the fOU process are in agreement with the previously known exact $\mathcal{P}(x)$ and autocovariance. The form of the tails of $\mathcal{P}(x)$ yields exact conditions, in terms of $H$ and $m$, for the particle confinement in the potential.
For $H\neq 1/2$, the tails encode the non-equilibrium character of the steady state distribution, and we observe violation of time reversibility of the system except for $m=2$. To compute the optimal paths and the factor $\alpha(H,m)$ for arbitrary permissible $H$ and $m$, one needs to solve an (in general nonlinear) integro-differential equation. To this end we develop a specialized numerical iteration algorithm which
accounts analytically for an intrinsic cusp singularity of the optimal paths for $H<1/2$.


Hurst Exponent and Event-by-Event Fluctuations in Relativistic Nucleus–Nucleus Collisions

A joint study of multi-particle pseudo-rapidity correlations and event-by-event fluctuations in the distributions of secondary particles and fragments of the target nucleus and the projectile nucleus was carried out in order to search for correlated clusters of secondary particles. An analysis of the collisions of the sulfur nucleus with photoemulsion nuclei at an energy of 200 A·GeV is presented based on experimental data obtained at the SPS at CERN. The analysis of multi-particle correlations was performed using the Hurst method. A detailed analysis of each individual event showed that in events of complete destruction of a projectile nucleus with a high multiplicity of secondary particles, long-distance multi-particle pseudo-rapidity correlations are observed. The distribution of average pseudo-rapidity in such events differs significantly from others, as it is much narrower, and its average value is noticeably shifted towards lower values <η>.

Texture-Image-Oriented Coverless Data Hiding Based on Two-Dimensional Fractional Brownian Motion

In an AI-immersing age, scholars look for new possibilities of employing AI technology to their fields, and how to strengthen security and protect privacy is no exception. In a coverless data hiding domain, the embedding capacity of an image generally depends on the size of a chosen database. Therefore, choosing a suitable database is a critical issue in coverless data hiding. A novel coverless data hiding approach is proposed by applying deep learning models to generate texture-like cover images or code images. These code images are then used to construct steganographic images to transmit covert messages. Effective mapping tables between code images in the database and hash sequences are established during the process. The cover images generated by a two-dimensional fractional Brownian motion (2D FBM) are simply called fractional Brownian images (FBIs). The only parameter, the Hurst exponent, of the 2D FBM determines the patterns of these cover images, and the seeds of a random number generator determine the various appearances of a pattern. Through the 2D FBM, we can easily generate as many FBIs of multifarious sizes, patterns, and appearances as possible whenever and wherever. In the paper, a deep learning model is treated as a secret key selecting qualified FBIs as code images to encode corresponding hash sequences. Both different seeds and different deep learning models can pick out diverse qualified FBIs. The proposed coverless data hiding scheme is effective when the amount of secret data is limited. The experimental results show that our proposed approach is more reliable, efficient, and of higher embedding capacity, compared to other coverless data hiding methods.

Ethereum futures and the efficiency of cryptocurrency spot markets

This paper examines the impact of the introduction of Ethereum futures contracts on the market efficiency of major cryptocurrency (Bitcoin, Ethereum, Ripple, Litecoin, and Dogecoin) spot prices. Using a multifractality-based approach and daily data from September 4, 2017 to February 16, 2024, the main results show a slight improvement in market efficiency. Specifically, the degree of multifractality persistence decreases, implying reduced market inefficiencies in major cryptocurrencies. The temporal linear correlation effect and thick tail effect are less pronounced post-launch. The asymmetry of the generalized Hurst exponent increases after the launch of Ethereum futures, with a higher persistence under the downward trend of cryptocurrencies noted. This downward trend emerges after the launch of Ethereum futures, coinciding with the final stage of the COVID-19 pandemic. Additional analysis shows that fat tails and temporal linear correlations are the main sources of multifractality. The results highlight the influence of introducing financial derivatives into the relatively new and volatile cryptocurrency area, which should concern traders, hedgers, investors, and regulators.

Using Short Time Series of Monofractal Synthetic Fluctuations to Estimate the Foreign Exchange Rate: The Case of the US Dollar and the Chilean Peso (USD–CLP)

Short time series are fundamental in the foreign exchange market due to their ability to provide real-time information, allowing traders to react quickly to market movements, thus optimizing profits and mitigating risks. Economic transactions show a strong connection to foreign currencies, making exchange rate prediction challenging. In this study, the exchange rate estimation between the US dollar (USD) and the Chilean peso (CLP) for a short period, from 2 August 2021 to 31 August 2022, is modeled using the nonlinear Schrödinger equation (NLSE) and calculated with the fourth-order Runge–Kutta method, respectively. Additionally, the daily fluctuations of the current exchange rate are characterized using the Hurst exponent, H, and later used to generate short synthetic fluctuations to predict the USD–CLP exchange rate. The results show that the USD–CLP exchange rate can be estimated with an error of less than 5%, while when using short synthetic fluctuations, the exchange rate shows an error of less than 10%.

Anomaly Detection in Fractal Time Series with LSTM Autoencoders

This study explores the application of neural networks for anomaly detection in time series data exhibiting fractal properties, with a particular focus on changes in the Hurst exponent. The objective is to investigate whether changes in fractal properties can be identified by transitioning from the analysis of the original time series to the analysis of the sequence of Hurst exponent estimates. To this end, we employ an LSTM autoencoder neural network, demonstrating its effectiveness in detecting anomalies within synthetic fractal time series and real EEG signals by identifying deviations in the sequence of estimates. Whittle’s method was utilized for the precise estimation of the Hurst exponent, thereby enhancing the model’s ability to differentiate between normal and anomalous data. The findings underscore the potential of machine learning techniques for robust anomaly detection in complex datasets.

Impact of surface-roughness and fractality on electrical conductivity of SnS thin films

Mono- and multi-fractal geometry have been used to explore the surface characteristics of scanning electron microscopy (SEM) micrographs of the SnS films with thicknesses of 100nm (SnS1) to 600nm (SnS4), respectively. For this investigation, the SnS thin films have been grown on fluorine-doped tin oxide (FTO)-coated glass substrate through the thermal evaporation route, and surface morphologies are captured by SEM. Two-dimensional multi-fractal detrended fluctuation analysis (MFDFA) based on the partition function is used to examine whether the surfaces have a multi-fractal nature or not. The partition function is applied to extract the generalized Hurst exponent from the segment size. It has been found that surfaces with higher surface roughness induce substantial nonlinearity and a wider width of the multi-fractal spectrum. The multi-fractal spectrum acquired from the analysis of the geometry and shape of the singularity spectrum is used to quantify the irregularity and complexity of surfaces. Minkowski functionals (MFs) parameters such as volume, boundary, and connectivity were measured for each thin film. Moreover, we tried to correlate the electrical conductivity with the mono- and multi-fractal parameters such as fractal dimension (Df), singularity strength function (Δα), singularity spectrum Δf(α), and it is observed that the conductivity of a thin film decreases with decreasing fractal dimension. The minimum (maximum) resistivity (conductivity) was observed for the surface having a larger fractal dimension. The present investigation suggests that such SnS surfaces, having minimal resistivity and maximum conductivity on the roughest surface, indicate enhanced light trapping capacity and can be utilized as active layers for advanced optoelectronics devices.

INTEGRATED MODEL FOR FORECASTING TIME SERIES OF ENVIRONMENTAL POLLUTION PARAMETERS

The quality of life in large urban areas is considerably diminished by air pollution, with major contributors being motor vehicles, industrial activities, and fossil fuel combustion. A major contributor to air pollution is coal-fired and thermal power plants, which are commonly found in emerging markets. In Astana, Kazakhstan, a rapidly expanding city's significant reliance on coal for heating and considerable building exacerbate air pollution. This research is essential for improving urban development practices that support sustainable growth in rapidly expanding cities. Using time series data from four monitoring stations in Astana using fractal R/S analysis, the study looks at long-term patterns in air pollutant levels, especially PM10 and PM2.5. The stations' Hurst exponents were determined to be 0.723, 0.548, 0.442, and 0.462. Additionally, the flow window method was used to study the Hurst exponent's dynamic behavior. The findings showed that one station's pollution levels had long-term memory, which suggests that the time series is persistent. While anti-persistence was noted in the third and fourth sites, data from the second station indicated nearly random behavior. The Hurst exponent values explain the October 2021 spike in pollution levels, which is probably caused by thermal power plants close to the city. The fractal analysis of time series could serve as an indicator of environmental conditions in a given region, with persistent pollution trends potentially aiding in predicting critical pollution events. Anti-persistence or temporary pollution spikes may be influenced by the observation station's proximity to pollution sources. Overall, the findings suggest that fractal time series analysis can act as a valuable tool for monitoring environmental health in urban areas.

Automated recognition of mental cognitive workload through nonlinear EEG analysis

Nowadays, with the remarkable advancements in detection instruments and artificial intelligence, there has been extensive utilization of human mental state monitoring in various domains. Few studies have explored how nonlinear analysis methods can detect cognitive workload despite the complex nature of EEG signals and advancements in signal processing techniques. In addition, the fuzziness of human mental conditions makes the need to use fuzzy engineering tools tangible in this field. Therefore, this investigation aimed to develop a decision support algorithm to improve previous efforts for the classification of task EEG and resting through machine learning algorithms. Various nonlinear features were calculated from all 19 EEG channels: Hurst exponent, Lempel–Ziv complexity, detrended fluctuation analysis, Higuchi fractal dimension, Katz fractal dimension, permutation entropy, singular value decomposition entropy, Petrosian fractal dimension, sample entropy, and Lyapunov exponent. During the classification step, a newly developed EPC-FC (Expert per Class Fuzzy Classifier) is introduced, utilizing an ensemble framework with specialized sub-classifiers for identifying a particular condition. By training sub-classifiers with the negative correlation learning (NCL) approach, the EPC-FC is designed to be exceptionally adaptable. Additionally, the separation of sub-classifiers within each class provides versatility and clarity to the system’s design. The proposed approach based on fuzzy systems and nonlinear analyses was applied to EEG data for mental workload recognition, which provides an excellent accuracy of 98.50% and an F1-score of 98.56% which is much higher than previous findings in this field. Also, the obtained results indicate that utilizing the proposed EPC-FC classifier maintains a consistently high accuracy exceeding 90% across various levels of SNRs. The obtained results proved the high potential of nonlinear analysis to detect cognitive states of the brain, which is consistent with the nonlinear and fuzzy nature of EEG data. Other nonlinear approaches should be considered for future studies to improve the current results.

Multifractal Analysis of the Impact of Fuel Cell Introduction in the Korean Electricity Market

This study employs multifractal detrended fluctuation analysis to investigate the impact of fuel cell introduction in the Korean electricity market via the lens of multifractal scaling behavior. Using multifractal analysis, the research delineates discrepancies between peak and off-peak hours, accounting for the daily cyclicity of the electricity market, and proposes a crossover point detection method based on the Chow test. Furthermore, the impacts of fuel cell introduction are evidenced through various methods that encompass multifractal spectra and market efficiency. The findings initially indicate a higher degree of multifractality during off-peak hours relative to peak hours. In particular, the crossover points emerged solely during off-peak hours, unveiling short- and long-term dynamics predicated on a near-annual cycle. Additionally, the average Hurst exponent for the short-term was 0.542, while the average for the long-term was 0.098, representing a notable discrepancy. The introduction of fuel cells attenuated the heterogeneity in the scaling behavior, which is potentially attributable to the decreased volatility in both the supply and demand spectra. Remarkably, after the introduction of fuel cells, there was a discernible decrease in the influence of long-range correlation within multifractality, and the market exhibited an increased propensity toward random-walk behavior. This phenomenon was also detected in the market deficiency measure, from an average of 0.536, prior to the introduction, to an average of 0.267, following the introduction, signifying an improvement in market efficiency. This implies that the introduction of fuel cells into the market engendered increased supply stability and a consistent increase in demand, mitigating volatility on both the supply and demand sides, thus increasing market efficiency.

Inner Multifractal Dynamics in the Jumps of Cryptocurrency and Forex Markets

Jump dynamics in financial markets exhibit significant complexity, often resulting in increased probabilities of subsequent jumps, akin to earthquake aftershocks. This study aims to understand these complexities within a multifractal framework. To do this, we employed the high-frequency intraday data from six major cryptocurrencies (Bitcoin, Ethereum, Litecoin, Dashcoin, EOS, and Ripple) and six major forex markets (Euro, British pound, Canadian dollar, Australian dollar, Swiss franc, and Japanese yen) between 4 August 2019 and 4 October 2023, at 5 min intervals. We began by extracting daily jumps from realized volatility using a MinRV-based approach and then applying Multifractal Detrended Fluctuation Analysis (MFDFA) to those jumps to explore their multifractal characteristics. The results of the MFDFA—especially the fluctuation function, the varying Hurst exponent, and the Renyi exponent—confirm that all of these jump series exhibit significant multifractal properties. However, the range of the Hurst exponent values indicates that Dashcoin has the highest and Litecoin has the lowest multifractal strength. Moreover, all of the jump series show significant persistent behavior and a positive autocorrelation, indicating a higher probability of a positive/negative jump being followed by another positive/negative jump. Additionally, the findings of rolling-window MFDFA with a window length of 250 days reveal persistent behavior most of the time. These findings are useful for market participants, investors, and policymakers in developing portfolio diversification strategies and making important investment decisions, and they could enhance market efficiency and stability.

Application of Chelyshkov polynomials in solving stochastic model with fractional Brownian motion

Abstract This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.

Anti-Persistent Values of the Hurst Exponent Anticipate Mean Reversion in Pairs Trading: The Cryptocurrencies Market as a Case Study

Pairs trading is a short-term speculation trading strategy based on matching a long position with a short position in two assets in the hope that their prices will return to their historical equilibrium. In this paper, we focus on identifying opportunities where mean reversion will happen quickly, as the commission costs associated with keeping the positions open for an extended period of time can eliminate excess returns. To this end, we propose the use of the local Hurst exponent as a signal to open trades in the cryptocurrencies market. We conduct a natural experiment to show that the spread of pairs with anti-persistent values of Hurst revert to their mean significantly faster. Next, we verify that this effect is universal across pairs with different levels of co-movement. Finally, we back-test several pairs trading strategies that include H<0.5 as an indicator and check that all of them result in profits. Hence, we conclude that the Hurst exponent represents a meaningful indicator to detect pairs trading opportunities in the cryptocurrencies market.

Local powers of least‐squares‐based test for panel fractional Ornstein–Uhlenbeck process

In recent years, significant advancements have been made in the field of identifying financial asset price bubbles, particularly through the development of time‐series unit‐root tests featuring fractionally integrated errors and panel unit‐root tests. This study introduces an innovative approach for assessing the sign of the persistence parameter () within a panel fractional Ornstein‐Uhlenbeck process, based on the least squares estimator of . This method incorporates three distinct test statistics based on the Hurst parameter (), which can take values in the range of , be equal to , or fall within the interval of . The null hypothesis corresponds to . Based on a panel of continuous records of observations, the null asymptotic distributions are obtained when the time span () is fixed and the number of cross sections () goes to infinity. The power function of the tests is obtained under the local alternative where is close to zero in the order of . This alternative covers the departure from the unit root hypothesis from the explosive side, enabling the calculation of lower power in bubble tests. The hypothesis testing problem and the local power function are also considered when a panel of discrete‐sampled observations is available under a sequential limit, that is, the sampling interval shrinks to zero followed by the goes to infinity.