Embedding a Gaussian discrete‐time autoregressive moving average process in a Gaussian continuous‐time autoregressive moving average process

We review some applications of continuous-time ARMA processes in the modelling of time series observed at discrete times t 1,…, t N . The problem of finding a continuous-time ARMA process in which a given discrete-time ARMA process can be “embedded” is discussed and some of the properties of a family of continuous-time analogues of Tong’s SETARMA models are considered. An approach to inference for such models is described and illustrated with reference to a variety of data sets.KeywordsAutocorrelation FunctionStochastic Differential EquationTime Series AnalysisARMA ModelTransition DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Continuous‐time autoregressive moving average (CARMA) processes have recently been used widely in the modelling of non‐uniformly spaced data and as a tool for dealing with high‐frequency data of the form ,n = 0, 1, 2,…, where Δ is small and positive. Such data occur in many fields of application, particularly in finance and in the study of turbulence. This article is concerned with the characteristics of the process , when Δ is small and the underlying continuous‐time process is a specified CARMA process.

USING CONTINUOUS-TIME MODELING FOR ERRORS-IN-VARIABLES IDENTIFICATION

Considering absolute log returns as a proxy for stochastic volatility, the influence of explanatory variables on absolute log returns of ultra high frequency data is analysed. The irregular time structure and time dependency of the data is captured by utilizing a continuous time ARMA(p,q) process. In particular, we propose a mixed effect model class for the absolute log returns. Explanatory variable information is used to model the fixed effects, whereas the error is decomposed in a non-negative Lévy driven continuous time ARMA(p,q) process and a market microstructure noise component. The parameters are estimated in a state space approach. In a small simulation study the performance of the estimators is investigated. We apply our model to IBM trade data and quantify the influence of bid-ask spread and duration on a daily basis. To verify the correlation in irregularly spaced data we use the variogram, known from spatial statistics. Copyright © 2006 John Wiley & Sons, Ltd.

On non-negative modeling with CARMA processes

On continuous-time threshold ARMA processes

A general approach for modeling the volatility process in continuous-time is based on the convolution of a kernel with a non-decreasing Levy process, which is non-negative if the kernel is non-negative. Within the framework of Continuous-time Auto-Regressive Moving-Average (CARMA) processes, we derive a necessary condition for the kernel to be non-negative, and propose a numerical method for checking the non-negativity of a kernel function. These results can be lifted to solving a similar problem with another approach to modeling volatility via the COntinuous-time Generalized Auto-Regressive Conditional Heteroscedastic (COGARCH) processes.

The ability to reason over partially observable networks of interacting states is a fundamental competency in probabilistic robotics. While the well-known factor graph and Gaussian process models provide flexible and computationally efficient solutions for this inference problem in the special cases in which all of the hidden states are either finite-dimensional parameters or real-valued functions, respectively, in many cases we are interested in reasoning about heterogeneous networks whose hidden states are comprised of both finite-dimensional parameters and functions. To that end, in this paper we propose a novel probabilistic generative model that incorporates both factor graphs and Gaussian processes to model these heterogeneous systems. Our model improves upon prior approaches to inference within these networks by removing the assumption of any specific set of conditional independences amongst the modeled states, thereby significantly expanding the class of systems that can be represented. Furthermore, we show that inference within this model can always be performed by means of a two-stage procedure involving inference within a factor graph followed by inference over a Gaussian process; by exploiting fast inference methods for the individual factor graph and Gaussian process models to solve each of these subproblems in succession, we thus obtain a general framework for computationally efficient inference over heterogeneous finite-/infinite-dimensional systems.

This article presents a comprehensive study of sales predictions using time series analysis, focusing on a case study of Walmart sales data. The aim of this study is to evaluate the effectiveness of various time series forecasting techniques in predicting weekly sales data for Walmart stores. Leveraging a dataset from Kaggle comprising weekly sales data from various Walmart stores around the United States, this study explores the effectiveness of time series analysis in forecasting future sales trends. Various time series analysis techniques, including Auto Regressive Integrated Moving Average (ARIMA), Seasonal Auto Regressive Integrated Moving Average (SARIMA), Prophet, Exponential Smoothing, and Gaussian Processes, are applied to model and forecast Walmart sales data. By comparing the performance of these models, the study seeks to identify the most accurate and reliable methods for forecasting retail sales, thereby providing valuable insights for improving sales predictions in the retail sector. The study includes an extensive exploratory data analysis (EDA) phase to preprocess the data, detect outliers, and visualize sales trends over time. Additionally, the article discusses the partitioning of data into training and testing sets for model evaluation. Performance metrics such as Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) are utilized to compare the accuracy of different time series models. The results indicate that Gaussian Processes outperform other models in terms of accuracy, with an RMSE of 34,116.09 and an MAE of 25,495.72, significantly lower than the other models evaluated. For comparison, ARIMA and SARIMA models both yielded an RMSE of 555,502.2 and an MAE of 462,767.3, while the Prophet model showed an RMSE of 567,509.2 and an MAE of 474,990.8. Exponential Smoothing also performed well with an RMSE of 555,081.7 and an MAE of 464,110.5. These findings suggest the potential of Gaussian Processes for accurate sales forecasting. However, the study also highlights the strengths and weaknesses of each forecasting methodology, emphasizing the need for further research to refine existing techniques and explore novel modeling approaches. Overall, this study contributes to the understanding of time series analysis in retail sales forecasting and provides insights for improving future forecasting endeavors.

We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of nonlinear Fourier analysis (NLFA) methods (Osborne 2010) to analyze the data set: The approach spectrally decomposes a nonlinear wave field into sine waves, Stokes waves, and phase-locked Stokes waves otherwise known as breather trains. Breathers are nonlinear beats, or packets which “breathe” up and down smoothly over cycle times of minutes to hours. The maximum amplitudes of the packets during the cycle have a largest central wave whose properties are often associated with the study of “rogue waves.” The mathematical physics of the nonlinear Schrodinger (NLS) equation is assumed and the methods of algebraic geometry are applied to give the nonlinear spectral representation. The distinguishing characteristic of the NLFA method is its ability to spectrally decompose a time series into its nonlinear coherent structures (Stokes waves and breathers) rather than just sine waves. This is done by the implementation of multidimensional, quasi-periodic Fourier series, rather than ordinary Fourier series. To determine preliminary estimates of nonlinearity, we use the significant wave height Hs, the peak period Tp, and the length of the time series T. The time series analyzed here have 8192 points and T =1677.72 s = 27.96 min. Near the peak of the storm, we find Hs ≈ 0.55 m, Tp ≈ 2.4 s so that for the wave steepness of a near Gaussian process, \({S} = \left (\pi ^{5/2}/g\right )H_{s}/{T}_{p}^{2}\), we find S ≈ 0.17, quite high for ocean waves. Likewise, we estimate the Benjamin-Feir (BF) parameter for a near Gaussian process, \({I_{BF}} = \left (\pi ^{5/2}/g \right ) H_{s} T/{T}_{p}^{3}\), and we find IBF ≈ 119. Since the BF parameter describes the nonlinear behavior of the modulational instability, leading to the formation of breather packets in a measured wave train, we find the IBF for these storm waves to be a surprisingly high number. This is because IBF, as derived here, roughly estimates the number of breather trains in a near Gaussian time series. The BF parameter suggests that there are roughly 119 breather trains in a time series of length 28 min near the peak of the storm, meaning that we would have average breather packets of about 14 s each with about 5-6 waves in each packet. Can these surprising results, estimated from simple parameters, be true from the point of view of the complex nonlinear wave dynamics of the BF instability and the NLS equation? We analyze the data set with the NLFA to verify, from a nonlinear spectral point of view, the presence of large numbers of breather trains and we determine many of their properties, including the rise time for the breathers to grow to their maximum amplitudes from a quiescent initial state. Energetically, about 95% of the NLFA components are found to consist of breather trains; the remaining small amplitude components are sine and Stokes waves. The presence of a large number of densely packed breather trains suggests an interpretation of the data in terms of breather turbulence, highly nonlinear integrable turbulence theoretically predicted for the NLS equation, providing an interesting paradigm for the nonlinear wave motion, in contrast to the random phase Gaussian approximation often considered in the analysis of data.

Reviewed by: Performing Korea by Patrice Pavis J. Hwang Yuh PERFORMING KOREA. By Patrice Pavis. London and New York: Palgrave, 2017. 276 pp. Hardcover, $109.00. Patrice Pavis' current book, entitled Performing Korea, is a record of contemporary Korean performances, examining what the author "perceived" as Korean performance from the viewpoint of a foreign spectator. This experience resulted from "seeing Korean dance and theatre almost every day" (p. 249) during the author's stay in South Korea between 2011 and 2012. This book's greatest asset is how it provides insight into the contemporary landscape of Korean theatre. In this book, Pavis explores and analyzes "everyday practices drawn from his own experience of wandering" Korea (p. xi), of which his viewpoint is also partly inspired by Roland Barthes' Empire of Signs. With a diversity of artistic practices from Korean performances and dances to K-pop and Pansori adaptations the author shows and (re)arranges contemporary theatre practices by writing about emerging and "unknown" directors' works into the very contemporary context of Korean theatre. This approach thus offers a new understanding of Korean theatre, which can be replicated outside of Korea in the global context of contemporary theatre. Performing Korea has five parts consisting of 16 chapters. In part, titled "Introduction to the Examples and to the Method," the author looks at "theatre and theatre research in Korea and elsewhere" (chapter 1), "Globalization in a few Korean performances" (chapter 2), and "Mise en Scène made in Korea" (chapter 3). By delineating the feature of Korean practitioners that "they place themselves, without regrets or hang-ups, in a multipolar, globalized world" (p. 11), Pavis states that [End Page 506] "[m]y examples will not be Korean traditional forms, but practices involving Euro-American texts, performances, and forms, performed by Korean artists and best explained from their point of view, with their understanding of interculturalism, of Koreanization" (p. 30). Korean stage productions are understood "in no way standardized, which should prompt the critic to analyze with more care what kind of Koreanization is taking place in each specific case" (p. 53). The author defines the opaque and complex term Koreanization as one which "means adapting the play to the special needs of a new audience" (pp. 63–64). From this point of view, Pavis argues, Korean theatre practitioners' "styles and methods of showing Koreanness do not differ radically from the Koreanization of a foreign play" (p. 65) as a product of "an idealized and stereotypical view of the past . . . and . . . a self- and state-induced way of staging accessible and exportable texts and performances" (p. 65). Part 2, titled "On a Few Theatre Productions," describes contemporary practices on Korean stage, ranging from well-known theatre directors' works to emerging directors' such as Lee Young-Seok's Coming Up Air and Kim Minseung's I, Na Hyeseok, the Undesirable. This second part of the book also includes a comparison of French and Korean playwrights' use of dramaturgical structure and theme. What makes part 2 interesting is that the author analyzes, in a very meticulous way, which performances embody "telling style and technique," as a new attempt to aestheticize contemporary Korean theatre. In this regard, this book challenges a dominant narrative of contemporary Korean theatre that focuses more on famous directors' productions in terms of popularity in favor of highlighting some emergent artists. In my view, part 3, "On a Few Theatre-Dance and Dance Performances" and part 4 "On a Few Cultural Performances" are the most interesting sections of the book. These two parts contain work on K-pop and installation art, thereby broadening this book's exploration of Korean culture and theatre. By bringing up margins of contemporary Korean theatre narrative into one of symptomatic phenomena, this book articulates how the diversity of Korean practices can be understood without "the risk of ethnocentrism and cultural prejudice or appropriation" (p. 31). This is best highlighted by the fact that the author "had ample access to the empirical and naïve dimension of everyday existence" of Korean people (p. 258). In part 5, titled "Epilogue," the author writes, "[m]y only ambition in Korea was to attend a few shows, to do micro-analyses...

We study an infinite system of independent symmetric random walks on a hierarchical group, in particular, the <em>c</em>-random walks. Such walks are used, e.g., in mathematical physics and population biology. The number variance problem consists in investigating if the variance of the number of “particles” $N_n(L)$ lying in the ball of radius $L$<em></em> at a given step $n$<em></em> remains bounded, or even better, converges to a finite limit, as $L\\to\\infty$. We give a necessary and sufficient condition and discuss its relationship to transience/recurrence property of the walk. Next we consider normalized fluctuations of $N_n(L)$ around the mean as $n\\to\\infty$ and $L$<em></em> is increased in an appropriate way. We prove convergence of finite dimensional distributions to a Gaussian process whose properties are discussed. As the $c$<em></em>-random walks mimic symmetric stable processes on $\\mathbb{R}$, we compare our results with those obtained by Hambly and Jones (2007, 2009), who studied the number variance problem for an infinite system of independent symmetric stable processes on $\\mathbb{R}$. Since the hierarchical group is an ultrametric space, corresponding results for symmetric stable processes and hierarchical random walks may be analogous or quite different, as has been observed in other contexts. An example of a difference in the present context is that for the stable processes a fluctuation limit process is a Gaussian process which is not Markovian and has long range dependent stationary increments, but the counterpart for hierarchical random walks is Markovian, and in a special case it has independent increments.

This paper addresses a single-machine scheduling problem with nonresumable jobs and flexible maintenance activities (SJM) and its online version SJMO. It is showed that the competitive ratio of the classical list scheduling (LS) algorithm is 2 for SJMO, and it is the best possible polynomial time approximation algorithm for SJMO from both the competitive ratio point of view unless P = NP and the computational time complexity point of view. Based on the relationship between SJM and SJMO, it is showed that the LS algorithm is the best possible polynomial time approximation algorithm for SJM from both the worst-case ratio point of view unless P = NP and the computational time complexity point of view, which is a significant improvement of some of the existing results on SJM and its special case in the literature. Note that the worst-case ratio of the classical longest processing time first (LPT) algorithm is also 2 for SJM. In order to show which algorithm is better for SJM on average, numerical experiments are conducted. Computational results show that the LPT algorithm performs better than the LS algorithm in all combinations from an average-case error point of view.

We propose a parsimonious stochastic model for characterising the distributional and temporal properties of rainfall. The model is based on an integrated Ornstein–Uhlenbeck process driven by the Hougaard Lévy process. We derive properties of this process and propose an extended model which generalises the Ornstein–Uhlenbeck process to the class of continuous-time ARMA processes. The model is illustrated by fitting it to empirical rainfall data on both daily and hourly time scales. It is shown that the model is sufficiently flexible to capture important features of the rainfall process across locations and time scales. Finally, we study an application to the pricing of rainfall derivatives which introduces the market price of risk via the Esscher transform. We first give a result specifying the risk-neutral expectation of a general moving average process. Then we illustrate the pricing method by calculating futures prices based on empirical daily rainfall data, where the rainfall process is specified by our model.

Subluminal Lorentz transformations in six-dimensional space-time have been derived in different special cases by using the relative matrix velocities between the frames and it has been shown that the Thomas precession is shown only in the row and column generalizations of relative velocity and that these cases are more akin to the spirit of relativity than others. Mean square speeds of spatial origins of frames have been calculated in all cases by an averaging process and it has been shown that no scheme of six-dimensional space-time gives the correct length contraction and time dilation suspecting the six-dimensional space-time formulation for subluminal world in spite of the appeal of its symmetry. Transformations of frequencies in all the cases are shown to lead to a correct longitudinal Doppler effect only but not the transverse one, contrary to the well-established experimental results, and it has been concluded that there cannot be a viable scheme of subluminal Lorentz transformations in six-dimensional space-time.