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Keywords = intraday frequency

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23 pages, 2295 KiB  
Article
A Two-Stage Sustainable Optimal Scheduling Strategy for Multi-Contract Collaborative Distributed Resource Aggregators
by Lei Su, Wanli Feng, Cao Kan, Mingjiang Wei, Rui Su, Pan Yu and Ning Zhang
Sustainability 2025, 17(15), 6767; https://doi.org/10.3390/su17156767 - 25 Jul 2025
Viewed by 232
Abstract
To address the challenges posed by the instability of renewable energy output and load fluctuations on grid operations and to support the low-carbon sustainable development of the energy system, this paper integrates artificial intelligence technology to establish an economic stability dispatch framework for [...] Read more.
To address the challenges posed by the instability of renewable energy output and load fluctuations on grid operations and to support the low-carbon sustainable development of the energy system, this paper integrates artificial intelligence technology to establish an economic stability dispatch framework for distributed resource aggregators. A phased multi-contract collaborative scheduling model oriented toward sustainable development is proposed. Through intelligent algorithms, the model dynamically optimises decisions across the day-ahead and intraday phases: During the day-ahead scheduling phase, intelligent algorithms predict load demand and energy output, and combine with elastic performance-based response contracts to construct a user-side electricity consumption behaviour intelligent control model. Under the premise of ensuring user comfort, the model generates a 24 h scheduling plan with the objectives of minimising operational costs and efficiently integrating renewable energy. In the intraday scheduling phase, a rolling optimisation mechanism is used to activate energy storage capacity contracts and dynamic frequency stability contracts in real time based on day-ahead prediction deviations. This efficiently coordinates the intelligent frequency regulation strategies of energy storage devices and electric vehicle aggregators to quickly mitigate power fluctuations and achieve coordinated control of primary and secondary frequency regulation. Case study results indicate that the intelligent optimisation-driven multi-contract scheduling model significantly improves system operational efficiency and stability, reduces system operational costs by 30.49%, and decreases power purchase fluctuations by 12.41%, providing a feasible path for constructing a low-carbon, resilient grid under high renewable energy penetration. Full article
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18 pages, 773 KiB  
Article
Multi-Level Simulation Framework for Degradation-Aware Operation of a Large-Scale Battery Energy Storage Systems
by Leon Tadayon and Georg Frey
Energies 2025, 18(11), 2708; https://doi.org/10.3390/en18112708 - 23 May 2025
Viewed by 610
Abstract
The increasing integration of renewable energy sources necessitates efficient energy storage solutions, with large-scale battery energy storage systems (BESS) playing a key role in grid stabilization and time-shifting of energy. This study presents a multi-level simulation framework for optimizing BESS operation across multiple [...] Read more.
The increasing integration of renewable energy sources necessitates efficient energy storage solutions, with large-scale battery energy storage systems (BESS) playing a key role in grid stabilization and time-shifting of energy. This study presents a multi-level simulation framework for optimizing BESS operation across multiple markets while incorporating degradation-aware dispatch strategies. The framework integrates a day-ahead (DA) dispatch level, an intraday (ID) dispatch level, and a high-resolution simulation level to accurately model the impact of operational strategies on state of charge and battery degradation. A case study of BESS operation in the German electricity market is conducted, where frequency containment reserve provision is combined with DA and ID trading. The simulated revenue is validated by a battery revenue index. The study also compares full equivalent cycle (FEC)-based and state-of-health-based degradation models and discusses their application to cost estimation in dispatch optimization. The results emphasize the advantage of using FEC-based degradation costs for dispatch decision-making. Future research will include price forecasting and expanded market participation strategies to further improve and stabilize the profitability of BESS in multi-market environments. Full article
(This article belongs to the Special Issue Advances in Battery Energy Storage Systems)
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23 pages, 825 KiB  
Article
FinTech, Fractional Trading, and Order Book Dynamics: A Study of US Equities Markets
by Janhavi Shankar Tripathi and Erick W. Rengifo
FinTech 2025, 4(2), 16; https://doi.org/10.3390/fintech4020016 - 25 Apr 2025
Viewed by 1733
Abstract
This study investigates how the rise of commission-free FinTech platforms and the introduction of fractional trading (FT) have altered trading behavior and order book dynamics in the NASDAQ equity market. Leveraging high-frequency ITCH data from highly capitalized stocks—AAPL, AMZN, GOOG, and TSLA—we analyze [...] Read more.
This study investigates how the rise of commission-free FinTech platforms and the introduction of fractional trading (FT) have altered trading behavior and order book dynamics in the NASDAQ equity market. Leveraging high-frequency ITCH data from highly capitalized stocks—AAPL, AMZN, GOOG, and TSLA—we analyze market microstructure changes surrounding the implementation of FT. Our empirical findings show a statistically significant increase in price levels, average tick sizes, and price volatility in the post-FinTech-FT period, alongside elevated price impact factors (PIFs), indicating steeper and less liquid limit order books. These shifts reflect greater participation by non-professional investors with limited order placement precision, contributing to noisier price discovery and heightened intraday risk. The altered liquidity landscape and increased volatility raise important questions about the resilience and informational efficiency of modern equity markets under democratized access. Our findings contribute to the growing literature on retail trading and provide actionable insights for market regulators and exchanges evaluating the design and oversight of evolving trading mechanisms. Full article
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13 pages, 633 KiB  
Article
Sentiment Matters for Cryptocurrencies: Evidence from Tweets
by Radu Lupu and Paul Cristian Donoiu
Data 2025, 10(4), 50; https://doi.org/10.3390/data10040050 - 1 Apr 2025
Viewed by 5139
Abstract
This study provides empirical evidence that cryptocurrency market movements are influenced by sentiment extracted from social media. Using a high frequency dataset covering four major cryptocurrencies (Bitcoin, Ether, Litecoin, and Ripple) from October 2017 to September 2021, we apply state-of-the-art natural language processing [...] Read more.
This study provides empirical evidence that cryptocurrency market movements are influenced by sentiment extracted from social media. Using a high frequency dataset covering four major cryptocurrencies (Bitcoin, Ether, Litecoin, and Ripple) from October 2017 to September 2021, we apply state-of-the-art natural language processing techniques on tweets from influential Twitter accounts. We classify sentiment into positive, negative, and neutral categories and analyze its effects on log returns, liquidity, and price jumps by examining market reactions around tweet occurrences. Our findings show that tweets significantly impact trading volume and liquidity: neutral sentiment tweets enhance liquidity consistently, negative sentiments prompt immediate volatility spikes, and positive sentiments exert a delayed yet lasting influence on the market. This highlights the critical role of social media sentiment in influencing intraday market dynamics and extends the research on sentiment-driven market efficiency. Full article
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40 pages, 1215 KiB  
Article
Major Issues in High-Frequency Financial Data Analysis: A Survey of Solutions
by Lu Zhang and Lei Hua
Mathematics 2025, 13(3), 347; https://doi.org/10.3390/math13030347 - 22 Jan 2025
Cited by 5 | Viewed by 7252
Abstract
We review recent articles that focus on the main issues identified in high-frequency financial data analysis. The issues to be addressed include nonstationarity, low signal-to-noise ratios, asynchronous data, imbalanced data, and intraday seasonality. We focus on the research articles and survey papers published [...] Read more.
We review recent articles that focus on the main issues identified in high-frequency financial data analysis. The issues to be addressed include nonstationarity, low signal-to-noise ratios, asynchronous data, imbalanced data, and intraday seasonality. We focus on the research articles and survey papers published since 2020 on recent developments and new ideas that address the issues, while commonly used approaches in the literature are also reviewed. The methods for addressing the issues are mainly classified into two groups: data preprocessing methods and quantitative methods. The latter include various statistical, econometric, and machine learning methods. We also provide easy-to-read charts and tables to summarize all the surveyed methods and articles. Full article
(This article belongs to the Special Issue Recent Advances in Statistical Machine Learning)
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14 pages, 3157 KiB  
Article
An Advanced Time-Varying Capital Asset Pricing Model via Heterogeneous Autoregressive Framework: Evidence from the Chinese Stock Market
by Bohan Zhao, Hong Yin and Yonghong Long
Mathematics 2025, 13(1), 41; https://doi.org/10.3390/math13010041 - 26 Dec 2024
Viewed by 1128
Abstract
The capital asset pricing model (CAPM) is a foundational asset pricing model that is widely applied and holds particular significance in the globally influential Chinese stock market. This study focuses on the banking sector, enhancing the performance of the CAPM and further assessing [...] Read more.
The capital asset pricing model (CAPM) is a foundational asset pricing model that is widely applied and holds particular significance in the globally influential Chinese stock market. This study focuses on the banking sector, enhancing the performance of the CAPM and further assessing its applicability within the Chinese stock market context. This study incorporates a heterogeneous autoregressive (HAR) component into the CAPM framework, developing a CAPM-HAR model with time-varying beta coefficients. Empirical analysis based on high-frequency data demonstrates that the CAPM-HAR model not only enhances the capability of capturing market fluctuations but also significantly improves its applicability and predictive accuracy for stocks in the Chinese banking sector. Full article
(This article belongs to the Special Issue Mathematical Models and Applications in Finance)
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29 pages, 8143 KiB  
Article
Inner Multifractal Dynamics in the Jumps of Cryptocurrency and Forex Markets
by Haider Ali, Muhammad Aftab, Faheem Aslam and Paulo Ferreira
Fractal Fract. 2024, 8(10), 571; https://doi.org/10.3390/fractalfract8100571 - 29 Sep 2024
Cited by 5 | Viewed by 3110
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Complex Dynamics and Multifractal Analysis of Financial Markets)
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26 pages, 4975 KiB  
Article
The Intraday Dynamics Predictor: A TrioFlow Fusion of Convolutional Layers and Gated Recurrent Units for High-Frequency Price Movement Forecasting
by Ilia Zaznov, Julian Martin Kunkel, Atta Badii and Alfonso Dufour
Appl. Sci. 2024, 14(7), 2984; https://doi.org/10.3390/app14072984 - 2 Apr 2024
Viewed by 1938
Abstract
This paper introduces a novel deep learning approach for intraday stock price direction prediction, motivated by the need for more accurate models to enable profitable algorithmic trading. The key problems addressed are effectively modelling complex limit order book (LOB) and order flow (OF) [...] Read more.
This paper introduces a novel deep learning approach for intraday stock price direction prediction, motivated by the need for more accurate models to enable profitable algorithmic trading. The key problems addressed are effectively modelling complex limit order book (LOB) and order flow (OF) microstructure data and improving prediction accuracy over current state-of-the-art models. The proposed deep learning model, TrioFlow Fusion of Convolutional Layers and Gated Recurrent Units (TFF-CL-GRU), takes LOB and OF features as input and consists of convolutional layers splitting into three channels before rejoining into a Gated Recurrent Unit. Key innovations include a tailored input representation incorporating LOB and OF features across recent timestamps, a hierarchical feature-learning architecture leveraging convolutional and recurrent layers, and a model design specifically optimised for LOB and OF data. Experiments utilise a new dataset (MICEX LOB OF) with over 1.5 million LOB and OF records and the existing LOBSTER dataset. Comparative evaluation against the state-of-the-art models demonstrates significant performance improvements with the TFF-CL-GRU approach. Through simulated trading experiments, the model also demonstrates practical applicability, yielding positive returns when used for trade signals. This work contributes a new dataset, performance improvements for microstructure-based price prediction, and insights into effectively applying deep learning to financial time-series data. The results highlight the viability of data-driven deep learning techniques in algorithmic trading systems. Full article
(This article belongs to the Section Computing and Artificial Intelligence)
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21 pages, 739 KiB  
Article
A Noisy Fractional Brownian Motion Model for Multiscale Correlation Analysis of High-Frequency Prices
by Tim Leung and Theodore Zhao
Mathematics 2024, 12(6), 864; https://doi.org/10.3390/math12060864 - 15 Mar 2024
Viewed by 1717
Abstract
We analyze the multiscale behaviors of high-frequency intraday prices, with a focus on how asset prices are correlated over different timescales. The multiscale approach proposed in this paper is designed for the analysis of high-frequency intraday prices. It incorporates microstructure noise into the [...] Read more.
We analyze the multiscale behaviors of high-frequency intraday prices, with a focus on how asset prices are correlated over different timescales. The multiscale approach proposed in this paper is designed for the analysis of high-frequency intraday prices. It incorporates microstructure noise into the stochastic price process. We consider a noisy fractional Brownian motion model and illustrate its various statistical properties. This leads us to introduce new latent correlation and noise estimators. New numerical algorithms are developed for model estimation using empirical high-frequency data. For a collection of stocks and exchange-traded funds, examples are provided to illustrate the relationship between multiscale correlation and sampling frequency as well as the evolution of multiscale correlation over time. Full article
(This article belongs to the Special Issue Advanced Statistical Applications in Financial Econometrics)
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15 pages, 7574 KiB  
Article
100 Picosecond/Sub-10−17 Level GPS Differential Precise Time and Frequency Transfer
by Wei Song, Fu Zheng, Haoyuan Wang and Chuang Shi
Appl. Sci. 2023, 13(19), 10694; https://doi.org/10.3390/app131910694 - 26 Sep 2023
Cited by 1 | Viewed by 1350
Abstract
A Global Navigation Satellite System (GNSS) is a high-precision method for comparing clocks and time transfer. The GNSS carrier phase can provide more precise observable information than pseudorange. However, the carrier phase is ambiguous, and only pseudorange can provide the absolute time difference [...] Read more.
A Global Navigation Satellite System (GNSS) is a high-precision method for comparing clocks and time transfer. The GNSS carrier phase can provide more precise observable information than pseudorange. However, the carrier phase is ambiguous, and only pseudorange can provide the absolute time difference between two clocks. In our study, by taking full advantage of GNSS pseudorange and carrier-phase observables, a differential precise time transfer (DPT) method with a clustering constraint was employed to estimate the time difference between two clocks, aiming to achieve accurate and precise time and frequency transfer. Using this method, several time transfer results were analyzed for different baselines. For the common clock experiment, the time transfer results showed good consistency across different days, with an intra-day accuracy of within 20 ps. Furthermore, we evaluated the self-consistency of DPT using closure among three stations. DPT closure of the three stations had a peak-to-peak value of closure of about 25 ps. The closure did not change over time, indicating the self-consistency of the DPT processing in time transfer. Moreover, our results were compared to station clock solutions provided by the International GNSS Service (IGS), and the standard deviations (STDs) of the four baselines were all less than 100 ps within one month, confirming the time and frequency stability of the DPT method. In addition, we found that the time stability of DPT was less than 20 ps within one week. As for frequency stability, DPT achieved a 10−16 level of modified Allan deviation (MDEV) at an averaging time of about 1 day and a sub-10−17 level at an averaging time of one week. Full article
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36 pages, 5931 KiB  
Article
Tracking ‘Pure’ Systematic Risk with Realized Betas for Bitcoin and Ethereum
by Bilel Sanhaji and Julien Chevallier
Econometrics 2023, 11(3), 19; https://doi.org/10.3390/econometrics11030019 - 10 Aug 2023
Cited by 4 | Viewed by 6567
Abstract
Using the capital asset pricing model, this article critically assesses the relative importance of computing ‘realized’ betas from high-frequency returns for Bitcoin and Ethereum—the two major cryptocurrencies—against their classic counterparts using the 1-day and 5-day return-based betas. The sample includes intraday data from [...] Read more.
Using the capital asset pricing model, this article critically assesses the relative importance of computing ‘realized’ betas from high-frequency returns for Bitcoin and Ethereum—the two major cryptocurrencies—against their classic counterparts using the 1-day and 5-day return-based betas. The sample includes intraday data from 15 May 2018 until 17 January 2023. The microstructure noise is present until 4 min in the BTC and ETH high-frequency data. Therefore, we opt for a conservative choice with a 60 min sampling frequency. Considering 250 trading days as a rolling-window size, we obtain rolling betas < 1 for Bitcoin and Ethereum with respect to the CRIX market index, which could enhance portfolio diversification (at the expense of maximizing returns). We flag the minimal tracking errors at the hourly and daily frequencies. The dispersion of rolling betas is higher for the weekly frequency and is concentrated towards values of β > 0.8 for BTC (β > 0.65 for ETH). The weekly frequency is thus revealed as being less precise for capturing the ‘pure’ systematic risk for Bitcoin and Ethereum. For Ethereum in particular, the availability of high-frequency data tends to produce, on average, a more reliable inference. In the age of financial data feed immediacy, our results strongly suggest to pension fund managers, hedge fund traders, and investment bankers to include ‘realized’ versions of CAPM betas in their dashboard of indicators for portfolio risk estimation. Sensitivity analyses cover jump detection in BTC/ETH high-frequency data (up to 25%). We also include several jump-robust estimators of realized volatility, where realized quadpower volatility prevails. Full article
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14 pages, 839 KiB  
Article
Optimal Feature Analysis for Identification Based on Intracranial Brain Signals with Machine Learning Algorithms
by Ming Li, Yu Qi and Gang Pan
Bioengineering 2023, 10(7), 801; https://doi.org/10.3390/bioengineering10070801 - 4 Jul 2023
Cited by 2 | Viewed by 1557
Abstract
Biometrics, e.g., fingerprints, the iris, and the face, have been widely used to authenticate individuals. However, most biometrics are not cancellable, i.e., once these traditional biometrics are cloned or stolen, they cannot be replaced easily. Unlike traditional biometrics, brain biometrics are extremely difficult [...] Read more.
Biometrics, e.g., fingerprints, the iris, and the face, have been widely used to authenticate individuals. However, most biometrics are not cancellable, i.e., once these traditional biometrics are cloned or stolen, they cannot be replaced easily. Unlike traditional biometrics, brain biometrics are extremely difficult to clone or forge due to the natural randomness across different individuals, which makes them an ideal option for identity authentication. Most existing brain biometrics are based on an electroencephalogram (EEG), which typically demonstrates unstable performance due to the low signal-to-noise ratio (SNR). Thus, in this paper, we propose the use of intracortical brain signals, which have higher resolution and SNR, to realize the construction of a high-performance brain biometric. Significantly, this is the first study to investigate the features of intracortical brain signals for identification. Specifically, several features based on local field potential are computed for identification, and their performance is compared with different machine learning algorithms. The results show that frequency domain features and time-frequency domain features are excellent for intra-day and inter-day identification. Furthermore, the energy features perform best among all features with 98% intra-day and 93% inter-day identification accuracy, which demonstrates the great potential of intracraial brain signals to be biometrics. This paper may serve as a guidance for future intracranial brain researches and the development of more reliable and high-performance brain biometrics. Full article
(This article belongs to the Section Biosignal Processing)
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20 pages, 745 KiB  
Article
Multiscale Volatility Analysis for Noisy High-Frequency Prices
by Tim Leung and Theodore Zhao
Risks 2023, 11(7), 117; https://doi.org/10.3390/risks11070117 - 26 Jun 2023
Cited by 1 | Viewed by 2546
Abstract
We present a multiscale analysis of the volatility of intraday prices from high-frequency data. Our multiscale framework includes a fractional Brownian motion and microstructure noise as the building blocks. The proposed noisy fractional Brownian motion model is shown to possess a variety of [...] Read more.
We present a multiscale analysis of the volatility of intraday prices from high-frequency data. Our multiscale framework includes a fractional Brownian motion and microstructure noise as the building blocks. The proposed noisy fractional Brownian motion model is shown to possess a variety of volatility behaviors suitable for intraday price processes. Algorithms for numerical estimation from time series observations are then introduced, with a new Hurst exponent estimator proposed for the noisy fractional Brownian motion model. Using real-world high-frequency price data for a collection of U.S. stocks and ETFs, we estimate the parameters in the noisy fractional Brownian motion and illustrate how the volatility varies over different timescales. The Hurst exponent and noise level also exhibit an intraday pattern whereby the the noise ratio tends to be higher near market close. Full article
(This article belongs to the Special Issue Risks: Feature Papers 2023)
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16 pages, 6089 KiB  
Article
The Ocean Surface Current in the East China Sea Computed by the Geostationary Ocean Color Imager Satellite
by Youzhi Ma, Wenbin Yin, Zheng Guo and Jiliang Xuan
Remote Sens. 2023, 15(8), 2210; https://doi.org/10.3390/rs15082210 - 21 Apr 2023
Cited by 6 | Viewed by 4404
Abstract
High-frequency observations of surface current field data over large areas and long time series are imperative for comprehending sea-air interaction and ocean dynamics. Nonetheless, neither in situ observations nor polar-orbiting satellites can fulfill the requirements necessary for such observations. In recent years, geostationary [...] Read more.
High-frequency observations of surface current field data over large areas and long time series are imperative for comprehending sea-air interaction and ocean dynamics. Nonetheless, neither in situ observations nor polar-orbiting satellites can fulfill the requirements necessary for such observations. In recent years, geostationary satellite data with ultra-high temporal resolution have been increasingly utilized for the computation of surface flow fields. In this paper, the surface flow field in the East China Sea is estimated using maximum cross-correlation, which is the most widely used flow field computation algorithm, based on the total suspended solids (TSS) data acquired from the Geostationary Ocean Color Imager satellite. The inversion results were compared with the modeled tidal current data and the measured tidal elevation data for verification. The results of the verification demonstrated that the mean deviation of the long semiaxis of the tidal ellipse of the inverted M2 tide is 0.0335 m/s, the mean deviation of the short semiaxis is 0.0276 m/s, and the mean deviation of the tilt angle is 6.89°. Moreover, the spatially averaged flow velocity corresponds with the observed pattern of tidal elevation changes, thus showcasing the field’s significant reliability. Afterward, we calculated the sea surface current fields in the East China Sea for the years 2013 to 2019 and created distribution maps for both climatology and seasonality. The resulting current charts provide an intuitive display of the spatial structure and seasonal variations in the East China Sea circulation. Lastly, we performed a diagnostic analysis on the surface TSS variation mechanism in the frontal zone along the Zhejiang coast, utilizing inverted flow data collected on 3 August 2013, which had a high spatial coverage and complete time series. Our analysis revealed that the intraday variation in TSS in the local surface layer was primarily influenced by tide-induced vertical mixing. The research findings of this article not only provide valuable data support for the study of local ocean dynamics but also verify the reliability of short-period surface flow inversion of high-turbidity waters near the coast using geostationary satellites. Full article
(This article belongs to the Special Issue Recent Advancements in Remote Sensing for Ocean Current)
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15 pages, 1458 KiB  
Article
Daily Semiparametric GARCH Model Estimation Using Intraday High-Frequency Data
by Fangrou Chai, Yuan Li, Xingfa Zhang and Zhongxiu Chen
Symmetry 2023, 15(4), 908; https://doi.org/10.3390/sym15040908 - 13 Apr 2023
Cited by 5 | Viewed by 2458
Abstract
The GARCH model is one of the most influential models for characterizing and predicting fluctuations in economic and financial studies. However, most traditional GARCH models commonly use daily frequency data to predict the return, correlation, and risk indicator of financial assets, without taking [...] Read more.
The GARCH model is one of the most influential models for characterizing and predicting fluctuations in economic and financial studies. However, most traditional GARCH models commonly use daily frequency data to predict the return, correlation, and risk indicator of financial assets, without taking data with other frequencies into account. Hence, financial market information may not be sufficiently applied to the estimation of GARCH-type models. To partially solve this problem, this paper introduces intraday high-frequency data to improve estimation of the volatility function of a semiparametric GARCH model. To achieve this objective, a semiparametric volatility proxy model was proposed, which includes both symmetric and asymmetric cases. Under mild conditions, the asymptotic normality of estimators was established. Furthermore, we also discuss the impact of different volatility proxies on estimation precision. Both the simulation and empirical results showed that estimation of the volatility function could be improved by the introduction of high-frequency data. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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