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Distribution of Residual in Autoregressive-Integrated Moving Average Time Series
Journal of the American Statistical Association
Abstract
Many statistical models, and in particular autoregressive—moving average time series models, can be regarded as means of transforming the data to white noise, that is, to an uncorrected sequence of errors. If the parameters are known exactly, this random sequence can be computed directly from the observations; when this calculation is made with estimates substituted for the true parameter values, the resulting sequence is referred to as the “residuals,” which can be regarded as estimates of the errors.If the appropriate model has been chosen, there will be zero autocorrelation in the errors. In checking adequacy of fit it is therefore logical to study the sample autocorrelation function of the residuals. For large samples the residuals from a correctly fitted model resemble very closely the true errors of the process; however, care is needed in interpreting the serial correlations of the residuals. It is shown here that the residual autocorrelations are to a close approximation representable as a singular linear transformation of the autocorrelations of the errors so that they possess a singular normal distribution. Failing to allow for this results in a tendency to overlook evidence of lack of fit. Tests of fit and diagnostic checks are devised which take these facts into account.
... Effectively analyzing such data requires an understanding of their intrinsic temporal relationships, which often pose significant challenges. The Autocorrelation Function (ACF) plays a pivotal role in this context [13][14][15][16][17][18][19][20][21][22][23][24], serving as a fundamental statistical tool to measure the degree of dependency between observations separated by different time lags. ...
... Actually, note that this result holds true for any time series with n ≥ 2, even for the nonstationary ones. Since a Gaussian variable N (µ, σ 2 ) can be equal to the constant value − 1 2 , only if µ = − 1 2 and in the degenerate case σ 2 = 0, Equations (12) and (13) imply that ...
... We recall that when constructing an ARMA(p, q) model, the Ljung-Box test must be applied to the residuals of the estimated model [13,14,56], and calculated mainly for h ranging from p + q + 1 to H = n/4 as suggested by [16]. Note that other upper bounds for h are suggested in other papers, such as H = min 20, n 4 [57], H = ln(n) [3], and more explicit bounds obtained form simulation procedures [58]. ...
The identification of the orders of time series models plays a crucial role in their accurate specification and forecasting. The Autocorrelation Function (ACF) is commonly used to identify the order q of Moving Average (MA(q)) models, as it theoretically vanishes for lags beyond q. This property is widely used in model selection, assuming the sample ACF follows an asymptotic normal distribution for robustness. However, our examination of the sum of the sample ACF reveals inconsistencies with these theoretical properties, highlighting a deviation from normality in the sample ACF for MA(q) processes. As a natural extension of the ACF, the Extended Autocorrelation Function (EACF) provides additional insights by facilitating the simultaneous identification of both autoregressive and moving average components. Using simulations, we evaluate the performance of q-order identification in MA(q) models, which is based on the properties of ACF. Similarly, for ARMA(p,q) models, we assess the (p,q)-order identification relying on EACF. Our findings indicate that both methods are effective for sufficiently long time series but may incorrectly favor an ARMA(p,q−1) model when the aq coefficient approaches zero. Additionally, if the cumulative sums of ACF (SACF) behave consistently and the Ljung–Box test validates the proposed model, it can serve as a strong candidate. The proposed models should then be compared based on their predictive performance. We illustrate our methodology with an application to wind speed data and sea surface temperature anomalies, providing practical insights into the relevance of our findings.
... This significant inverse problem has garnered Chaos ARTICLE pubs.aip.org/aip/cha the collective interest of specialists across multiple fields, including artificial intelligence, physics, and mathematics, [5][6][7][8][9] and has increasingly emerged as a focal topic of research. Numerous time-series models, including ARIMA, 10 RNN, 11 LSTM, 12 and SVR, 13 have been shown to be effective in representing and predicting the evolution of dynamical systems. These models have been extensively used in applied research within physics and engineering, [14][15][16] offering powerful tools for addressing intricate dynamical challenges. ...
... IV). DSTDEM was compared with six commonly used baseline methods, including traditional statistical regression methods, such as ARIMA 10 and VAR, 28 as well as machine learning approaches, such as SVR 41 and RBFN, 42 and deep learning frameworks, such as LSTM 43,44 and STICM. 22 The Root Mean Square Error (RMSE) and the Pearson Correlation Coefficient (PCC) were used to evaluate prediction performance. ...
Stochastic effects introduce significant uncertainty into dynamical systems, making the data-driven reconstruction and prediction of these systems highly complex. This study incorporates uncertainty learning into a deep learning model for time-series prediction, proposing a deep stochastic time-delay embedding model to improve prediction accuracy and robustness. First, this model constructs a deep probabilistic catcher to capture uncertainty information in the reconstructed mappings. These uncertainty representations are then integrated as meta-information into the reconstruction process of time-delay embedding, enabling it to fully capture system stochasticity and predict target variables over multiple time steps. Finally, the model is validated on both the Lorenz system and real-world datasets, demonstrating superior performance compared to existing methods, with robust results under noisy conditions.
... Above we have also introduced the discrete temporal index n ≤ N T and the time variable t n = n t. We first prescreen the data to filter out signals f k (n) that are not sufficiently different from statistical noise-an autocorrelation test, such as the Ljung-Box test [39,40], works very well in practice, requires linear computational time O(N o ) and can be completely parallelized requiring no communication between the classical processors, e.g., every CPU in a cluster receives a copy of the shadow data (a relatively small dataset) and processes it independently. We then define a data matrix D ∈ R N o ×N T that contains these signals [D] kn = f k (n) as row vectors and assume the column dimension is much larger than the row dimension N o N T due to our effective use of classical shadows as illustrated in Fig. 2. ...
... The observable expected values shown in Fig. 11 (left) were estimated the following way. First, an autocorrelation test (Ljung-Box test [39,40]) was performed on the timedependent Pauli expected values and the worst-scoring 100 observables were selected, i.e., the ones that are most well described by random fluctuations around the mean. This way we can select observables whose expected values are (nearly) the same at each time step. ...
We present shadow spectroscopy as a simulator-agnostic quantum algorithm for estimating energy gaps using very few circuit repetitions (shots) and no extra resources (ancilla qubits) beyond performing time evolution and measurements. The approach builds on the fundamental feature that every observable property of a quantum system must evolve according to the same harmonic components: we can reveal them by postprocessing classical shadows of time-evolved quantum states to extract a large number of time-periodic signals N o ∝ 10 8 , whose frequencies correspond to Hamiltonian energy differences with precision limited as ϵ ∝ 1 / T for simulation time T . We provide strong analytical guarantees that (a) quantum resources scale as O ( log N o ) , while the classical computational complexity is linear O ( N o ) , (b) the signal-to-noise ratio increases with the number of processed signals as ∝ N o , and (c) spectral peak positions are immune to reasonable levels of noise. We demonstrate our approach on model spin systems and the excited-state conical intersection of molecular CH 2 and verify that our method is indeed intuitively easy to use in practice, robust against gate noise, amiable to a new type of algorithmic-error mitigation technique, and uses relatively few shots given a reasonable initial state is supplied—we demonstrate that even 10 shots per time step can be sufficient. Finally, we measured a high-quality, experimental shadow spectrum of a spin chain on readily available IBM quantum computers, achieving the same precision as in noise-free simulations without using any advanced error mitigation, and verified scalability in tensor-network simulations of up to 100-qubit systems.
Published by the American Physical Society 2025
... This can be tested using the Ljung-Box statistic, of which the Null Hypothesis (H0) is "The data is uncorrelated". Therefore, it is desirable to have p-values above the chosen significance level since H0 should be accepted [29,30]. The Shapiro-Wilk test is used to evaluate whether a dataset is normally distributed (Null Hypothesis H0) or is not normally distributed (Alternative H1) and is calculated using the R stats package 4.2.1 [31]. ...
The digitalization of mechanical waste treatment can have a supporting effect in recovering valuable raw materials from mixed solid waste, achieving the EU’s recycling targets, and developing the waste industry into a circular economy. Therefore, influencing factors on machines and the entire plant must be known. Furthermore, optimization potentials can be developed with the help of digital approaches, such as dynamic control of machine and plant operation, to be able to control the heterogeneous material streams to enable optimized treatment and preserve valuable materials. To establish dynamic control in waste treatment machines, sensors are required to record the necessary variable parameters. In this work, the focus of those variables is placed on the volume flow leaving the shredding machine, where a volume flow sensor is used as a sensor system. To predict the volume flow of the shredder’s output to create a possible dynamic control, forecasting with ARIMA models is used. Initial results show that it is possible to predict a heterogeneous material stream with a selected model, currently over a time period of 10 s. However, the testing of additional prediction models still offers the opportunity of predicting the material stream over a longer period of time to enable dynamic control.
... 3 numaralı denklemdeki 2 test istatistiğinin değeri sayıca çok büyük olduğunda, testin temel hipotezi ret edildiği bildirilmiştir (Khuntia & Pattanayak, 2018). Box & Pierce (1970) tarafından ortaya koyulan portmanteau yöntemi, bir finansal zaman serisinin otokorelasyon katsayılarının anlamlı şekilde sıfırdan farklı olup olmadığını değerlendirmek için kullanılan önemli yöntemlerden biri olarak kabul edilmektedir. Bu yöntemin çalışma prensibi, 4 numaralı denklemde yer almaktadır: ...
Bu çalışma, BRICS-T ülkelerinin (Brezilya, Rusya, Hindistan, Çin, Güney Afrika ve Türkiye) finansal piyasalarını, Adaptif Piyasa Hipotezi (APH) çerçevesinde analiz ederek piyasa etkinliğinin zamanla değişen dinamiklerini ortaya koymayı amaçlamaktadır. Kasım 1997 - Kasım 2024 dönemine ait aylık borsa endeksi verileri kullanılarak APH, Genelleştirilmiş Spektral (GS) testi, Otomatik Portmanteau (AQ) testi ve Wild-Bootstrap Otomatik Varyans Oranı (WBAVR) testi ile sınanmıştır. Zamana bağlı adaptif davranışları incelemek için kayan pencereler tekniği uygulanmış ve APH’nin farklı dönemlerdeki adaptif piyasa özelliği sergileyip sergilemediği test edilmiştir. Ayrıca, jeopolitik risklerin piyasaların tahmin edilebilirliği üzerindeki etkileri doğrusal olmayan modelleme yapabilen ve makine öğrenimine dayalı bir yöntem olan KRLS (Kernel Regularized Least Squares) ile analiz edilmiştir. Elde edilen bulgulara göre, tüm dönemler baz alındığında hiçbir ülkenin adaptif piyasa özelliği göstermediği tespit edilmiştir. Ancak, kayan pancereler yöntemiyle ele alındığında brezilya ve Rusya 2008-2011 arasında güçlü, Hindistan 2006’da çok sınırlı, Çin 2014 ve 2015’te güçlü, Türkiye 2003,2007,2011’de çok sınırlı; 2023 ve 2024’te güçlü adaptif piyasa özelliği sergilemiştir. Güney Afrika’nın hiçbir dönemde adaptif piyasa özelliği sergilemediği görülmektedir. Ayrıca, KRLS dinamik olmayan sonuçları, Jeopolitik risklerin brezilya borsa tahmin edilebilirliğini artırdığını, diğer tüm ülkelerin borsa tahmin edilebilirliğini azalttığını ortaya çıkarmaktadır. Çalışmada, KRLS dinamik sonuçları ile jeopolitik risklerin marjinal etkilerinin ülke borsaları üzerindeki etkisi ortaya çıkarılmakta ve bu sonuçlara göre politika önerileri sunulmaktadır.
... Exponentially Weighted Moving Average (EWMA) is described within the works of (Hunter, 1986;Lucas & Saccucci, 1990;Cox, 1961). The Integrated Moving Average (ARIMA) method and its implications are outlined by (Box, & Pierce, 1970;Nelson, 1998). Semantic detection in the context of phishing attacks is assessed by the following studies: (Buchyk et al., 2024;Buchyk, Shutenko, & Toliupa, 2022;Toliupa et al., 2023). ...
B a c k g r o u n d . Nowadays, every critical sector of social institutions conducts its operations on top of distributed processing systems. Contemporary digital infrastructure heavily relies on user-provided data in its operation. As a result, distributed attacks based on botnets are in a continuous state of arms race with the protection methods that filtrate malicious data influx. A common method to do so often relies on heuristics and human-oriented verifications. As the new advancements in the field of artificial intelligence emerge, such attacks adopt new oblique paths towards achieving their goals. The successful execution of the said plan could lead to a gradual resource depletion on the target system. The purpose of this research is to address such threats with a combination of statistical and semantic approaches. M e t h o d s . The following research conducts a theoretical analysis and systematization of the distributed gradual degradation attack in distributed systems and its implication in the context of the evolving technologies of artificial intelligence. Mathematical modeling is leveraged to define the proposed model's properties and execution process. The proposed model heavily relies on statistical methods for analyzing time series data and its deviations, as well as classification neural networks for semantic detection of suspicious behavior. R e s u l t s . As a result of the following research, a new model is developed that leverages statistical and semantical verification for anomaly detection. The continuous monitoring and detection process is optimized towards highly loaded systems with a constant flurry of data streams. C o n c l u s i o n s . Since the distributed attacks could be potentially equipped with intelligent means to bypass existing security measures, the development of a protection model against potential resource leaks is gaining relevance. The recent success in the development of artificial generative intelligence leads to raising concerns about the safety and adequacy of the current security measures against automation-based distributed attack vectors. It is often a case that the protection models are inclined towards prevention of the attack rather than recovery. This approach, while targeting the source of risks, often leads to complacent design decisions without considering the potential outcomes of a successful breach. The proposed model provides a theoretical foundation for building systems that both react to the active execution of threats and perform recovery mechanisms, assuming that the attack may potentially bypass initial security measures.
... Several authors (Islam et al. 2019;Gowthaman et al. 2022;Al-Khateeb 2023;Wang et al. 2024;Zafra-Mejía et al. 2024) employed the ARIMA model for forecasting and predicting either water and sanitation status or water quality. The ARIMA model development process consists of four main steps: identification, estimation, diagnostic verification, and forecasting (Box & Pierce 1970). The ARIMA model is one of the most commonly used time series models, characterized by its parameters (p, d, q). ...
This study analyzes the trend and forecasts the future proportion of the people in Bangladesh using safely managed drinking water and sanitation services, aiming for progress toward SDGs. The annual datasets cover 2000–2022 and are sourced from the World Bank Databank, focusing on ‘Proportion of people using safely managed drinking water services’ and ‘Proportion of people using safely managed sanitation services.’ Initially, the Mann–Kendall test is applied to detect seasonal trends in the time series data. The ARIMA (0,1,0) model is identified as the optimal fit for forecasting drinking water services, while Holt's method is preferred for sanitation services. Results show an upward trend in both areas; however, the rates remain inadequate to meet SDG 6 targets. Projections indicate that by 2025, 60.6% of the population will have access to safely managed drinking water and 33.5% will have access to sanitation services, whereas the Bangladeshi government aims for 75 and 80%, respectively. Furthermore, by 2030, these proportions are expected to increase to 63.7% for drinking water and 37.2% for sanitation. This analysis suggests that, if current trends continue, the SDG targets 6.1 and 6.2 will not be achievable by 2030.
... Traditional statistical methods primarily rely on time-series analysis, with common models including Autoregressive Integrated Moving Average (ARIMA) and Multiple Linear Regression (MLR) [7]. The ARIMA model, as a classic time-series forecasting technique, effectively captures periodic fluctuations in air pollution data and is widely used for short-term predictions [8]. More sophisticated statistical approaches, such as the periodically correlated stationary processes model [9], offered enhanced capabilities in handling periodic patterns in time-series data. ...
This study presents an improved deep-learning model, termed Enhanced Time Series Mixer (E-TSMixer), for the prediction of particulate matter. By analyzing the temporal evolution of PM2.5 concentrations from multivariate monitoring data, the model demonstrates significant prediction capabilities while maintaining consistency with observed pollutant transport characteristics in the urban boundary layer. In E-TSMixer, a fully connected output layer is proposed to enhance the predictive capability for complex spatiotemporal dependencies. The relevant data on air quality and traffic flow are fused to achieve high-precision predictions of PM2.5 concentrations through a multivariate time-series forecasting model. An asymmetric penalty mechanism is added to dynamically optimize the loss function. Experimental results indicate that the proposed E-TSMixer model achieves higher accuracy for the prediction of PM2.5, which significantly outperforms the traditional models. Additionally, an intelligent dual regulation of fixed and dynamic threshold model is introduced and combined with E-TSMixer for the decision-making model of the real-time adjustments of the frequency, routes, and timing of water truck operation in practice.
... Detailed information about the dynamical system is usually unknown, making it difficult to obtain an accurate mathematical model describing the system's evolutionary pattern over time [16,17]. Therefore, various data-driven techniques have been designed to accomplish the prediction task [18][19][20][21][22]. Classical statistical regression methods are used to determine the consistency between output time series [23,24]; however, the correlation between time series is usually time-varying, leading to unstable parameters in the regression model. ...
Providing accurate predictions from known information to future states of nonlinear dynamical systems is challenging, and data-driven techniques have emerged as powerful tools for addressing this problem. In this work, we proposed a data-driven and model-free framework delayed coordinate mapping-reservoir computing (DCM-RC) for long-term accurate prediction of nonlinear dynamical systems through the synergy between DCM and a novel RC method. For dynamic systems, DCM-RC constructs delayed attractors conjugated to the original attractor topology by converting the spatial information of the dynamic system into the temporal dynamics of the target variables through the delayed embedding theorem. The basic architecture of multistep prediction is constructed by finding DCM, and RC with fused polynomial libraries is applied to cross-predict all states of the dynamical system, thus obtaining information about the future of the dynamical system. The effectiveness and accuracy of DCM-RC are demonstrated on the Lorenz-63 system and the Rőssler system.
... resulting in the residual series Z d , in which applies Cox-Stuart test for verifying trends, Fisher G test for verifying seasonality and Box e Pierce test (Box and Pierce, 1970) to verify whether residuals are independent and identically distributed, with zero mean and constant variance. ...
This study aimed to statistically evaluate the PM10 and TSP time series data in the RGV, between 2008 and 2017, verifying whether the series of each pollutant are generated by the same stochastic process. For that, the tests proposed by Coates and Diggle (1986), by Quenouille (1958) and the series difference procedure developed by Silva, Ferreira and Sáfadi (2000) were used. PM10 time series for Laranjeiras (E1), Carapina (E2), Jardim Camburi (E3), Enseada do Suá (E4), Vitória (E5), IBES (E6) and Cariacica (E8) stations were compared two by two, and for TPS time series of stations E3, E4, E5, E6 and E8 the same was done. Results indicate that, for a 5% significance level, stations E2, E3, E4, E5 and E6 for PM10 and, E3, E4, E5 and E6, for the TSP, present time series generated by the same stochastic process. Therefore, is considered that, the results obtained are indicative of the need to reformulation the initial RAMQAr project, which, if added to a pollutant dispersion study, can guarantee the network coverage area expansion, with emphasis in the existing stations re-spatialization, aiming to improve their data representativeness and installation of new stations in places still lacking monitoring.
... Various methods have been deployed to predict time series data. Statistical methods such as autoregressive integrated moving average (ARIMA) [9] and its variants as well as generalized autoregressive conditional heteroskedasticity (GARCH) [10] and its variants have widely been used in forecasting univariate time series data. While the ARIMA model focuses on modeling data that has a linear trend, this assumption, however, does not hold in real-world situations. ...
Accurate temperature forecasting is of paramount importance across various sectors, influencing decision-making processes and impacting numerous aspects of daily life. This study analyzes temperature time series data from the Nairobi County in Kenya, aiming to develop accurate hybrid time series forecasting models. Initial statistical tests revealed significant nonstationarity and nonlinearity in the data, prompting the adoption of specialized modeling techniques. Using variational mode decomposition (VMD), the raw time series was decomposed into interpretable components, enhancing feature representation and understanding of temperature dynamics. Hybrid forecasting models were then constructed by integrating VMD with both statistical (autoregressive integrated moving average [ARIMA]) and deep learning (gated recurrent unit [GRU], long short-term memory [LSTM], and Transformer) architectures. Evaluation metrics, including root mean squared error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and R-squared, highlighted the superiority of hybrid models over individual approaches, particularly those combining VMD with ARIMA, GRU, LSTM, and Transformer. The experimental results for temperature prediction show that the hybrid models combining VMD with statistical and deep learning networks achieved improved forecasting accuracy compared with baseline models. Specifically, the VMD–ARIMA–GRU model emerged as the top performer, demonstrating the lowest error metrics and highest explanatory power. With a low RMSE of 0.710090, MAE of 0.561726, and MAPE of 2.808193%, the model demonstrates remarkable accuracy in predicting temperature values. In addition, the high R-squared value of 0.779234 indicates that approximately 77.92% of the variance in the observed data is explained by the model, showcasing its robustness and effectiveness in capturing the underlying patterns in temperature time series data. Overall, this study underscores the importance of VMD in preprocessing data to enhance feature representation and forecasting accuracy. By combining statistical and deep learning methods, hybrid models incorporating VMD offer a comprehensive solution for accurate temperature prediction, with implications for climate modeling and environmental monitoring.
... where x i and y i are the latitude and longitude coordinates of the data, x and y are the arithmetic mean center, x t and y t are the mean center, and x t and y t are the difference between the two coordinates. The SARIMA model is a time series predicting method that extends the autoregressive integrated moving average (ARIMA) model [32,33] to account for seasonality. Implemented using the Box-Jenkins methodology for model identification, estimation, and prediction, the SARIMA model is particularly effective for real-time predictions and is known for its high accuracy. ...
As a comprehensive reflection of the thermal characteristics of the urban environment, the urban heat island (UHI) effect has triggered a series of ecological and environmental issues. Existing studies on the UHI effect in Shijiazhuang, the capital of Hebei Province, China, have primarily focused on spatial–temporal distribution characteristics and migration trends, with less focus on the influences of other contributing factors. This study focuses on Shijiazhuang city, using Landsat ETM+/OLI data from 2000 to 2020 to analyze the spatiotemporal traits of the UHI effect. The mono-window algorithm (MW) was used to retrieve land surface temperatures (LSTs), and the seasonal autoregressive integrated moving average (SARIMA) model was used to predict LST trends. Key factors such as the normalized difference vegetation index (NDVI), digital elevation model (DEM), population (POP), precipitation (PPT), impervious surface (IPS), potential evapotranspiration (PET), particulate matter 2.5 (PM2.5), and night light (NL) were analyzed using spatial autocorrelation to explore their dynamic relationship with the UHI. Specifically, a multi-scale analysis model was developed to search for the optimum urban spatial scale, enabling a comprehensive assessment of the spatiotemporal evolution and drivers of the UHI in Shijiazhuang. The UHI showed pronounced spatial clustering, expanding annually by 44.288 km², with a southeastward shift. Autumn exhibited the greatest reduction in UHI, while predictions suggested peak temperatures in summer 2027. According to the bivariate clustering analysis, the NDVI was the most influential factor in mitigating the UHI, while the IPS spatially showed the most significant enhancement in the UHI in the central urban areas. Other factors generally promoted the UHI after 2005. The multi-scale geographically weighted regression (MGWR) model was best fitted at a 3 km × 3 km scale. Considering the joint effects of multiple factors, the ranking of contributing factors to the model prediction is as follows: PET > DEM > NDVI > IPS > PPT > PM2.5 > NL > POP. The interactive effects, especially between the PET and DEM, reach a significant value of 0.72. These findings may address concerns regarding both future trends and mitigation indications for UHI variations in Shijiazhuang.
... In order to evaluate model validity, this study used three primary diagnostic assessments, which include the Portmanteau test for autocorrelations, serial correlation Lagrange multiplier (LM) testing, and a normality test. Firstly, the residual autocorrelations in time series models were assessed using the Portmanteau test (Box & Pierce, 1970). Secondly, the autocorrelation LM tests are intended to detect serial correlation in residuals (Breusch, 1978). ...
This study compared standard VAR, SVAR with short-run restrictions, and SVAR with long-run restrictions to investigate the effects of oil price shocks and the foreign exchange rate (ZAR/USD) on consumer prices in South Africa after the 2008 financial crisis. The standard VAR model revealed that consumer prices responded positively to oil price shocks in the short term, whereas the foreign exchange rate (ZAR/USD) revealed a fluctuating currency over time. That is, the South African rand (ZAR) initially appreciated against the US dollar (USD) in response to oil price shocks (periods 1:7), followed by a depreciation in periods 8:12. Imposing short-run restrictions on the SVAR model revealed that the foreign exchange rate (ZAR/USD) reacted to oil price shocks in a manner similar to the VAR model, with ZAR appreciating during the initial periods (1:7) and subsequently depreciating in the later periods (8:12). Consumer prices responded positively to oil price shocks, causing consumer prices to increase in the short run, which is consistent with the VAR findings. However, imposing long-run restrictions on our SVAR model yielded results that contrasted with those obtained under short-run restrictions and the standard VAR model. That is, oil price shocks had long-lasting effects on the foreign exchange rate, resulting in the depreciation of ZAR relative to USD over time. Additionally, oil price shocks reduced consumer prices, resulting in a deflationary effect in the long run. This study concluded that South Africa’s position as a net oil importer with a floating exchange rate renders the country vulnerable to short-term external shocks. Nonetheless, in the long term, the results indicated that the economy tends to adapt to oil price shocks over time.
... In theory, white noise is an idealized concept where each data point is assumed to be completely independent of the others, with no correlation between observations. It is typically modeled as a Gaussian process, implying a normal distribution with a constant mean and variance [16][17][18]. However, real-world noises are rarely entirely white because most real processes have some forms of correlation or dependency between different times [19][20][21]. ...
This paper contributes significantly to time series analysis by discussing the empirical properties of white noise and their implications for model selection. This paper illustrates the ways in which the standard assumptions about white noise typically fail in practice, with a special emphasis on striking differences in sample ACF and PACF. Such findings prove particularly important when assessing model adequacy and discerning between residuals of different models, especially ARMA processes. This study addresses issues involving testing procedures, for instance, the Ljung–Box test, to select the correct time series model determined in the review. With the improvement in understanding the features of white noise, this work enhances the accuracy of modeling diagnostics toward real forecasting practice, which gives it applied value in time series analysis and signal processing.
... The most commonly used tests for unit root testing are Augmented Dickey-Fuller (Said and Dickey, 1984), Phillips-Perron (Perron, 1988), kpps (Kwiatkowski et al., 1992) and Ljung-Box (Box and Pierce, 1970). In particular, the Ljung-Box test contrasts the null auto-correlation hypothesis of identically distributed Gaussian random variables, which is equivalent to test stationarity. ...
Accurate predictions of international trade flows form the basis for informed decision-making and strategic planning at both national and global levels. By offering reliable forecasts of market trends, these predictions allow policymakers, businesses, and economic institutions to anticipate shifts in supply and demand, adapt to evolving economic conditions, and mitigate potential risks. We applied a temporal fusion transformer-based (TFT) model with improved precision to predict international raw material trade flows. Our goal is to enhance prediction accuracy and robustness by leveraging the strengths of TFT in handling complex time series data, surpassing the performance of conventional machine learning techniques.Using an enriched dataset from the UN Comtrade database, the CEPII Gravity dataset, and the World Bank, our model achieves a 17% increase in compared to baseline models random forest and graph attention networks. Furthermore, the proposed model offers improved interpretability regarding feature importance, providing clearer insights into trade flow predictions.Our analysis demonstrates the TFT model’s ability to cope with economic disruptions such as COVID-19 and the Ukraine conflict, proving its reliability in volatile trade conditions. This work represents the first application of transformer-based methods to multi-horizon forecasting in raw material trade, offering novel insights into global economic trends.
The accurate prediction of sewage treatment indicators is crucial for optimizing management and supporting sustainable water use. This study proposes the KAN-LSTM model, a hybrid deep learning model combining Long short-term memory (LSTM) networks, Kolmogorov-Arnold Network (KAN) layers, and multi-head attention. The model effectively captures complex temporal dynamics and nonlinear relationships in sewage data, outperforming conventional methods. We applied correlation analysis with time-lag consideration to select key indicators. The KAN-LSTM model then processes them through LSTM layers for sequential dependencies, KAN layers for enhanced nonlinear modeling via learnable B-spline transformations, and multi-head attention for dynamic weighting of temporal features. This combination handles short-term patterns and long-range dependencies effectively. Experiments showed the model’s superior performance, achieving 95.13% R-squared score for FOss (final sedimentation basin outflow suspended solid, one indicator of our research predictions)and significantly improving prediction accuracy. These advancements in intelligent sewage treatment prediction modeling not only enhance water sustainability but also demonstrate the transformative potential of hybrid deep learning approaches. This methodology could be extended to optimize predictive tasks in sustainable aquaponic systems and other smart aquaculture applications.
Inflation is a critical global issue and also a significant challenge in Ethiopia. Despite its profound impact on the economy, research on inflation volatility in Ethiopia remains limited and insufficient. This paper aims to address these gaps by employing BEKK (Baba, Engle, Kraft, and Kroner) and DCC (Dynamic Conditional Correlation) - GARCH (Generalized Autoregressive Conditional Heteroscedasticity) models and analyze the characteristics of inflation trends, which supports informed economic decision making. We focus on four key inflation indicators: the Consumer Price Index (CPI), the Non-Food Price Index (NFPI), the Food Price Index (FPI), and the Exchange Rate (ER), which were compiled from the National Bank of Ethiopia (NBE) from January 2010 to December 2020. The study confirms inflation volatility, supported by the ARCH effect and Ljung-Box Q(m) statistics, along with conditional heteroscedasticity tests. This study demonstrates that, unlike previous approaches that neglected dynamic correlations in inflation volatility, the DCC-GARCH model decisively outperforms the BEKK-GARCH model in both parameter estimation and forecasting accuracy, as evidenced by significantly better Akaike Information Criterion (AIC), Schwarz Bayesian Information Criterion (SBIC), and Hannan-Quinn Information Criterion (HQIC) metrics. Our findings revealed that the DCC (1,1) model effectively captured volatility clustering without being persistent or explosive, as the sum of coefficients is less than 1, confirming mean reversion. In contrast to previous studies, our approach provided a more robust understanding of inflation dynamics, identifying CPI and FPI as the most volatile indicators. The study reveals significant correlations among inflation indicators-CPI, FPI, NFPI, and ER indicating a cohesive inflationary pattern. The coefficients show that past volatility and shocks persistently influence current volatility, underscoring their interdependence. The forecast from the best model reveals substantial instability is observed in CPI and FPI returns. It suggests a sharp increase in FPI and a rise in ER. The better method captured inflation volatility more effectively than other competent models. The DCC-GARCH model offered deeper insights into volatility dynamics, revealing the shortcomings of earlier time series models in addressing inflation volatility.
Time series anomaly detection is crucial in many fields due to the unique combinations and complex multi-scale time-varying features of time series data, which require accurate analysis. However, previous research has failed to adequately address these complexities, lacking effective decomposition and multi-scale modeling to comprehensively capture differences between normal and abnormal time points at various scales. To address this, our study aims to propose an innovative approach, the Multi-Scale Reconstruction Network (MSR-GAN). It features a Multi-Scale Decoupling Module (MTD) to separate input time series into different-scale components and models the reconstruction as parallel full-scale time series recovery. Furthermore, a Reconstructed Residual Collaborative Learning Module (RRCL) is constructed to perform inter-scale interactions by adaptively calculating importance scores for generator weight control. Extensive experiments demonstrate MSR-GAN’s state-of-the-art performance on multiple benchmark datasets for time series anomaly detection, thus providing a more effective solution, enhancing monitoring and handling of abnormal situations in related fields, and promoting the further development of time series analysis techniques.
Resampling approaches to approximate the sampling distribution of the statistic constructed from irregularly spaced data are still far from well-developed. We propose a novel approach, the Wiener-type integral approximation (WIA), as a complement to the existing approaches. It uses a Wiener-type integral (WI) to approximate the sampling distribution of a statistic. Meanwhile, the variance of the WI is consistent to that of the statistic. The construction of the WI involves integrating a localized version of the statistic with respect to standard Gaussian white noise, offering a general class of resampling estimators based on irregularly spaced spatial data. The proposed WIA imitates the second-order dependence of multiple statistics very well, enabling it to approximate the sampling distribution of multivariate statistics that can achieve the joint asymptotic normality. In this paper, we demonstrate the applicability of WIA to various important statistics, including the sample mean, sample variance, auto-covariance estimator, and discrete Fourier transform. Moreover, through simulation studies, the finite sample performance of the WIA is investigated, in comparison with some competitive approaches.
Optimizing inventory management in the pharmaceutical industry relies significantly on accurate sales forecasting for chain drugstores. Sales predictions for these stores, however, are affected by data quality and temporal characteristics, which limit the effectiveness of traditional statistical, machine learning, and ensemble learning methods. To address these challenges, this study introduces a sales forecasting model called TS-LGBM, which utilizes a sliding window approach to preserve the sequential integrity of sales data and integrates neural networks with the Light Gradient Boosting Machine (LightGBM). By incorporating a self-attention mechanism into LightGBM, the TS-LGBM model aims to enhance predictive accuracy. The model’s efficacy is validated using the Rossman dataset from Kaggle, followed by a case study with actual data from Z-chain retail drugstores. This study further refines data by factoring in temporal characteristics of various drugs, the density of nearby drugstores within a specified radius, and regional attributes associated with each drugstore. To evaluate performance, five models—TS-LGBM, TS-XGB, LightGBM, XGBoost and LSTM—are compared experimentally. Findings indicate that TS-LGBM achieves superior prediction accuracy compared to the other models. This study is intended for practical applications, as accurate sales forecasts for chain drugstores can enhance supply chain management efficiency and reduce operational costs.
Understanding the timely regulation of plant growth and phenology is crucial for assessing a terrestrial ecosystem’s productivity and carbon budget. The circadian clock, a system of genetic oscillators, acts as ‘Master of Ceremony’ during plant physiological processes. The mechanism is particularly elusive in trees despite its relevance. The primary and secondary tree growth, leaf senescence, bud set, and bud burst timing were investigated in 68 constructs transformed into Populus hybrids and compared with untransformed or transformed controls grown in natural or controlled conditions. The results were analyzed using generalized additive models with ordered-factor-smooth interaction smoothers. This meta-analysis shows that several genetic components are associated with the clock. Especially core clock-regulated genes affected tree growth and phenology in both controlled and field conditions. Our results highlight the importance of field trials and the potential of using the clock to generate trees with improved characteristics for sustainable silviculture (e.g., reprogrammed to new photoperiodic regimes and increased growth).
Cryptocurrencies have attracted significant attention from investors, regulators and the media since their emergence. In a world where digital advancements are increasingly included in everyday relations, studying the behaviour of cryptocurrencies and their impact on financial markets becomes a necessity. This paper introduces a comparative analysis towards a hybrid model combining classical and modern methods for predicting cryptocurrency prices. This study deals with everyday recordings of 10 cryptocurrencies that represent different technological innovations and use cases. Studying these cryptocurrencies can help understand volatility, volumes and price movements. We aim to develop a time series statistical model and to study the effectiveness of deep learning (DL) models, specifically long short-term memory (LSTM) model and the autoregressive integrated moving average (ARIMA) model, for predicting cryptocurrency prices accurately and forecasting stationary data. Combining ARIMA and LSTM, we managed to obtain a high value of R² for Binance Coin (BNB) cryptocurrency (0.936) with an average R² for all evaluated cryptocurrencies of 0.6555.
Multivariate time series (MTS) forecasting involves the use of multiple interrelated sequential data to predict future trends, necessitating the extraction of potential associative information from complex historical data. Currently, Transformers dominate the field of MTS prediction due to their core mechanism of self-attention, which effectively captures long-range dependencies. However, self-attention is inherently permutation-invariant, leading to the loss of sequential information. To address this issue, we propose the Dual-Branch Temporal Feature Fusion Network with Fast GRU (DBFF-GRU). In the feature fusion module, a dual-branch convolutional structure is employed to extract local and global features from the time series data separately, and a lightweight attention module is integrated into the global feature branch to capture dependencies among variables. Additionally, we introduce a fast iterative GRU structure to further capture long-term dependencies and enhance model efficiency. Extensive experiments on real-world data demonstrate the effectiveness of DBFF-GRU compared to state-of-the-art techniques.
Accurate time series forecasting is crucial in fields such as business, finance, and meteorology. To achieve more precise predictions and effectively capture the potential cycles and stochastic characteristics at different scales in time series, this paper optimizes the network structure of the Autoformer model. Based on multi-scale convolutional operations, a multi-scale feature fusion network is proposed, combined with date–time encoding to build the MD–Autoformer time series forecasting model, which enhances the model’s ability to capture information at different scales. In forecasting tasks across four fields—apparel sales, meteorology, finance, and disease—the proposed method achieved the lowest RMSE and MAE. Additionally, ablation experiments demonstrated the effectiveness and reliability of the proposed method. Combined with the TPE Bayesian optimization algorithm, the prediction error was further reduced, providing a reference for future research on time series forecasting methods.
We provide evidence that many narrative shocks used by prominent literature display some persistence. We show that the two leading methods to estimate impulse responses to an independently identified shock (local projections and distributed lag models) treat persistence differently, hence identifying different objects. We propose corrections to re‐establish the equivalence between local projections and distributed lag models, providing applied researchers with methods and guidance to estimate their desired object of interest. We apply these methods to well‐known empirical work and find that how persistence is treated has a sizable impact on the estimates of dynamic effects.
Large language models (LLMs) have been applied in many fields and have developed rapidly in recent years. As a classic machine learning task, time series forecasting has recently been boosted by LLMs. Recent works treat large language models as zero-shot time series reasoners without further fine-tuning, which achieves remarkable performance. However, some unexplored research problems exist when applying LLMs for time series forecasting under the zero-shot setting. For instance, the LLMs' preferences for the input time series are less understood. In this paper, by comparing LLMs with traditional time series forecasting models, we observe many interesting properties of LLMs in the context of time series forecasting. First, our study shows that LLMs perform well in predicting time series with clear patterns and trends but face challenges with datasets lacking periodicity. This observation can be explained by the ability of LLMs to recognize the underlying period within datasets, which is supported by our experiments. In addition, the input strategy is investigated, and it is found that incorporating external knowledge and adopting natural language paraphrases substantially improves the predictive performance of LLMs for time series. Our study contributes insight into LLMs' advantages and limitations in time series forecasting under different conditions.
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The primary aim of the paper is to place current methodological discussions in macroeconometric modeling contrasting the ‘theory first’ versus the ‘data first’ perspectives in the context of a broader methodological framework with a view to constructively appraise them. In particular, the paper focuses on Colander’s argument in his paper “Economists, Incentives, Judgement, and the European CVAR Approach to Macroeconometrics” contrasting two different perspectives in Europe and the US that are currently dominating empirical macroeconometric modeling and delves deeper into their methodological/philosophical underpinnings. It is argued that the key to establishing a constructive dialogue between them is provided by a better understanding of the role of data in modern statistical inference, and how that relates to the centuries old issue of the realisticness of economic theories.
Statistical Afodcls for Z'rcdiction and Control Models for Forecasting Seasonal and Nou-Seasonal Time Series Spectral Analysis of Time Serics
- G E Box
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