A mixed portmanteau test for ARMA‐GARCH models by the quasi‐maximum exponential likelihood estimation approach

This paper investigates the joint limiting distribution of the residual autocorrelation functions and the absolute residual autocorrelation functions of ARMA‐GARCH models. This leads a mixed portmanteau test for diagnostic checking of the ARMA‐GARCH model fitted by using the quasi‐maximum exponential likelihood estimation approach in Zhu and Ling (2011) . Simulation studies are carried out to examine our asymptotic theory, and assess the performance of this mixed test and other two portmanteau tests in Li and Li (2008) . A real example is given.     

Evaluating Specification Tests for Markov‐Switching Time‐Series Models

Abstract. We evaluate the performance of several specification tests for Markov regime‐switching time‐series models. We consider the Lagrange multiplier (LM) and dynamic specification tests of Hamilton (1996) and Ljung–Box tests based on both the generalized residual and a standard‐normal residual constructed using the Rosenblatt transformation. The size and power of the tests are studied using Monte Carlo experiments. We find that the LM tests have the best size and power properties. The Ljung–Box tests exhibit slight size distortions, though tests based on the Rosenblatt transformation perform better than the generalized residual‐based tests. The tests exhibit impressive power to detect both autocorrelation and autoregressive conditional heteroscedasticity (ARCH). The tests are illustrated with a Markov‐switching generalized ARCH (GARCH) model fitted to the US dollar–British pound exchange rate, with the finding that both autocorrelation and GARCH effects are needed to adequately fit the data.     

On modelling and diagnostic checking of vector periodic autoregressive time series models

Abstract.  Vector periodic autoregressive time series models (PVAR) form an important class of time series for modelling data derived from climatology, hydrology, economics and electrical engineering, among others. In this article, we derive the asymptotic distributions of the least squares estimators of the model parameters in PVAR models, allowing the parameters in a given season to satisfy linear constraints. Residual autocorrelations from classical vector autoregressive and moving-average models have been found useful for checking the adequacy of a particular model. In view of this, we obtain the asymptotic distribution of the residual autocovariance matrices in the class of PVAR models, and the asymptotic distribution of the residual autocorrelation matrices is given as a corollary. Portmanteau test statistics designed for diagnosing the adequacy of PVAR models are introduced and we study their asymptotic distributions. The proposed test statistics are illustrated in a small simulation study, and an application with bivariate quarterly West German data is presented.     

Improved inference for first‐order autocorrelation using likelihood analysis

Abstract. Testing for first‐order autocorrelation in small samples using the standard asymptotic test can be seriously misleading. Recent methods in likelihood asymptotics are used to derive more accurate p‐value approximations for testing the autocorrelation parameter in a regression model. The methods are based on conditional evaluations and are thus specific to the particular data obtained. A numerical example and three simulations are provided to show that this new likelihood method provides higher order improvements and is superior in terms of central coverage even for autocorrelation parameter values close to unity.     

The Effect of the Estimation on Goodness‐of‐Fit Tests in Time Series Models

Abstract. We analyze, by simulation, the finite‐sample properties of goodness‐of‐fit tests based on residual autocorrelation coefficients (simple and partial) obtained using different estimators frequently used in the analysis of autoregressive moving‐average time‐series models. The estimators considered are unconditional least squares, maximum likelihood and conditional least squares. The results suggest that although the tests based on these estimators are asymptotically equivalent for particular models and parameter values, their sampling properties for samples of the size commonly found in economic applications can differ substantially, because of differences in both finite‐sample estimation efficiencies and residual regeneration methods.     

Local Gaussian Autocorrelation and Tests for Serial Independence

The traditional and most used measure for serial dependence in a time series is the autocorrelation function. This measure gives a complete characterization of dependence for a Gaussian time series, but it often fails for nonlinear time series models as, for instance, the generalized autoregressive conditional heteroskedasticity model (GARCH), where it is zero for all lags. The autocorrelation function is an example of a global measure of dependence. The purpose of this article is to apply to time series a well‐defined local measure of serial dependence called the local Gaussian autocorrelation. It generally works well also for nonlinear models, and it can distinguish between positive and negative dependence. We use this measure to construct a test of independence based on the bootstrap technique. This procedure requires the choice of a bandwidth parameter that is calculated using a cross validation algorithm. To ensure the validity of the test, asymptotic properties are derived for the test functional and for the bootstrap procedure, together with a study of its power for different models. We compare the proposed test with one based on the ordinary autocorrelation and with one based on the Brownian distance correlation. The new test performs well. Finally, there are also two empirical examples.     

A novel fault detection scheme for a nonlinear dynamic process based on generalized non-negative matrix projection-maximum mean discrepancy: Application on the DAMADICS benchmark process

In order to address the issue of minor fault detection in nonlinear dynamic processes, this paper proposes a fault detection method based on generalized non-negative matrix projection-maximum mean discrepancy (GNMP-MMD). Firstly, the GNMP is employed to acquire the residual scores of the samples. Subsequently, a sliding window approach is integrated with MMD for real-time monitoring of sample status within the residual subspace. In this study, GNMP is utilized to mitigate the impact of non-Gaussianity in data distribution, while MMD serves to alleviate autocorrelation among samples. A numerical case and experimental data collected from the DAMADICS process are utilized to simulate and validate the proposed method. Compared to traditional principal component analysis (PCA), dynamic principal component analysis (DPCA), dynamic kernel principal component analysis (DKPCA), non-negative matrix factorization (NMF), GNMP, and MMD, the experiment results clearly illustrate the feasibility of the proposed method.     

THE AUTOCORRELATION FUNCTION OF SEASONAL ARMA MODELS

Abstract. This note obtains the theoretical autocorrelation function of an ARMA model with multiplicative seasonality. It is shown that this function can be interpretated as the result of the interaction between the seasonal and regular autocorrelation patterns of the ARMA model. The use of this result makes easier the identification of the structure of the model, is helpful in choosing between a multiplicative or additive seasonal component and leads to a better understanding of the properties of the estimated autocorrelation function of scalar ARMA processes.     

REVERSED RESIDUALS IN AUTOREGRESSIVE TIME SERIES ANALYSIS

Abstract. Both linear and non-linear time series can have directional features which can be used to enhance the modelling and investigation of linear or non-linear autoregressive statistical models. For this purpose, reversed p th-order residuals are introduced. Cross-correlations of residuals and squared reversed residuals allow extensions of current model identification ideas. Quadratic types of partial autocorrelation functions are introduced to assess dependence associated with non-linear models which nevertheless have linear autoregressive correlation structures. The use of these residuals and their cross-correlation functions is exemplified empirically on some deseasonalized river flow data for which a first-order autoregressive model is a satisfactory second-order fit. Parallel theoretical computations are undertaken for the non-linear first-order random coefficient autoregressive model and comparisons are made. While the data are shown to be strongly non-linear, their correlational signatures are found to be convincingly different from those of a first-order autoregressive model with random coefficients.     

Residual stresses,shrinkage, and warpage of complex injection molded products: Numerical simulation and experimental validation

A numerical simulation model for predicting residual stresses and residual deformations which arise during the injection molding of thermoplastic polymers in the post-packing stage has been developed. A thermoviscoelastic model with volume relaxation is used for the calculation of residual stresses. The finite element method employed is based on the theory of shells as an assembly of flat elements. This theory is well suited for thin injection molded products of complex shape. The approach allows the prediction of residual deformations and residual stresses layer by layer like a truly three-dimensional calculation, while reducing the computational cost significantly. The hole drilling technique is used to measure the residual stresses across the thickness of the product. A three-dimensional laser digitizing system, an image processing technique and a dual displacement transducer system are used to measure the warpage. Experiments are carried out on polycarbonate and high density polyethylene parts. Numerical results are in qualitative agreement with experimental observations, i.e., the skin of the box is surrounded by a compressive region while the core region is in traction. The trend of both the experimental and the predicted residual stress profiles is close. Different examples are presented to illustrate the influence of the geometrical complexity of the shape on the final deformations and residual stresses. The influence of the mold temperature on residual stresses and warpage is also analyzed.     

ASYMPTOTIC RESULTS FOR PERIODIC AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. This paper is concerned with the derivation of asymptotic distributions for the sample autocovariance and sample autocorrelation functions of periodic autoregressive moving-average processes, which are useful in modelling periodically stationary time series. In an effort to obtain a parsimonious model representing a periodically stationary time series, the asymptotic properties of the discrete Fourier transform of the estimated periodic autocovariance and autocorrelation functions are presented. Application of the asymptotic results to some specific models indicates their usefulness for model identification analysis.     

A residual modified transformation formula from munsell to sRGB color system

A residual modified transformation formula from Munsell to sRGB color system is presented in this article. The development of the transformation formula is based on the 1625 Munsell color chips in the Munsell Renotation Data that could be displayed on the sRGB monitor. The developed transformation formula consists of two models, one is named as the corresponding matrix model and the other is the residual modified model. The corresponding matrix model was obtained using numerical analysis methods to map each chip color attribute values from Munsell to sRGB and then its corresponding matrix for each Hue was constructed. The residual modified model was obtained using the discrete cosine transform to construct a residual modified function, which was used to modify the transformation error of each chip after applying the corresponding matrix model. The transformed accuracy rate for the corresponding matrix model is 88.4% and for the residual modified model can be enhanced to 96.6% for all of the chips. The developed transformation formula can be applied to research in which Munsell colors are presented on the sRGB monitor. With the aid of these formulas, designers can show the advanced real‐time results on a sRGB monitor for the product's color planning based on Munsell color system. Therefore, this research has a great contribution on the practical application for color planning in product design. © 2014 Wiley Periodicals, Inc. Col Res Appl, 40, 243–255, 2015     

Temporal Aggregation of Stationary And Nonstationary Discrete‐Time Processes

Abstract. We study the autocorrelation structure and the spectral density function of aggregates from a discrete‐time process. The underlying discrete‐time process is assumed to be a stationary AutoRegressive Fractionally Integrated Moving‐Average (ARFIMA) process, after suitable number of differencing if necessary. We derive closed‐form expressions for the limiting autocorrelation function and the normalized spectral density of the aggregates, as the extent of aggregation increases to infinity. These results are then used to assess the loss of forecasting efficiency due to aggregation.     

Stochastic simulation of the influence of fibre path variability on the formation of residual stress and shape distortion

A stochastic cure simulation approach is developed and implemented to investigate the influence of fibre misalignment on cure. Image analysis is used to characterize fiber misalignment in a carbon non‐crimp fabric. It is found that variability in tow orientation is significant with a standard deviation of 1.2°. The autocorrelation structure is modeled using the Ornstein‐Uhlenbeck sheet and the stochastic problem is addressed by coupling a finite element model of cure with a Monte Carlo scheme. Simulation of the cure of an angle shaped carbon fiber‐epoxy component shows that fiber misalignment can cause considerable variability in the process outcome with a coefficient of variation in maximum residual stress up to approximately 2% (standard deviation of 1 MPa) and qualitative and quantitative variations in final distortion of the cured part with the standard deviation in twist and corner angle reaching values of 0.4° and 0.05° respectively. POLYM. COMPOS., 38:2642–2652, 2017. © 2015 The Authors Polymer Composites published by Wiley Periodicals, Inc. on behalf of Society of Plastics Engineers     

A COMPOSITE LINEAR MODEL GENERATING A STATIONARY STOCHASTIC PROCESS WITH GIVEN THIRD-ORDER AUTOCORRELATION FUNCTION

Abstract. A composite linear model is proposed which generates a non-Gaussian stationary stochastic process with a given third-order autocorrelation function and a white power spectrum. The design of the model is based on the fact that a type of finite-impulse-response linear system with a non-Gaussian white input series produces an output process whose third-order correlations exist only for special time lags. An arbitrary third-order autocorrelation function can be constructed by superposing output processes of this type. The model requires at most 2 L² + 4 L + 1 independent identically distributed (i.i.d.) input processes for the third-order autocorrelation function with the largest time lag L . Results of numerical experiments confirm the validity of the model.     

Stress models for electron beam-physical vapor deposition thermal barrier coatings using temperature-process-dependent model parameters

This article presents a new stress model for EB-PVD TBC to provide insight into TBC failure mechanisms. Cycling-induced and temperature-process dependent model parameters are incorporated into stress analysis of EB-PVD TBC and then used to simulate the variation of mechanical, thermal and inelastic behaviour from metallic bond coat to thermally grown oxide (TGO). This gives a smooth evolution of residual stresses and is more realistic than prior finite element (FE) work. Two types of interfacial roughness approximate profiles are presented and implemented in the FE model. Geometrical parameters are used to model the interface roughness, and residual stresses are then evaluated at specific positions within TBCs. This article's stress analysis establishes a link between stress distribution and the evolution of interfacial roughness during thermal cycles. Consequently, the results are expected to provide insight into the failure modes related to localized interfacial roughness evolution.     

A new viscoelastic phase-transformation model to simulate the injection-molding residual stress

The focus of this study is on thermally induced residual stress, which is the predominant cause for dimensional imperfections in unfilled injection-molded plastic products. A new viscoelastic phase-transformation model was proposed to simulate and predict the residual stress within injection-molded articles as induced during the cooling stage of the injection-molding cycle. The calculated results are in good agreement with the literature experimental data. Numerical simulation of a residual stress problem can be used to guide corrective measures if the problem arises and also to prevent a potential problem from occurring in the first place. © 1996 John Wiley & Sons, Inc.     

RECOGNIZING OVERDIFFERENCED TIME SERIES

Abstract. Differencing is often used to render a time series stationary. The decision of how much differencing to do is usually based on plots of data, the autocorrelation function or a statistical test. Hence, it may happen that an analyst mistakenly differences a stationary series. When that happens, the inverse autocorrelation function takes on a specific pattern. We characterize this pattern and discuss the behavior of sample estimates of the inverse autocorrelation function for such overdifferenced series.     

Process-induced residual stresses in compression molded UHMWPE

Non-isothermal cooling during processing causes the development of residual stresses, which are analyzed for compression molded UHMWPE, and affects the dimensional stability. The development of thermal residual stresses was predicted using an incremental stress analysis that included temperature-dependent material properties. Strain gauges were used to measure the residual stresses as layers were removed from a molded disk using a Process Simulated Laminate (PSL) approach. The PSL technique has not previously been applied to a compression molded neat polymer. For initial surface cooling rates of ～ 11°C/min, the model predicted a compressive stress at the bottom surface of 14 MPa and a tensile stress near the center of 2.5 MPa and matched the experimental distribution well. Because the compressive residual stress was 70% of the yield strength (～20 MPa), a lower cooling rate was also tested (2.6°C/min). The maximum tensile and compressive stresses for this cooling rate were, 0.91 MPa and 2.5 MPa, respectively. The model demonstrated its use for predicting thermal residual stresses in compression molded parts, instead of trial-and-error experimentation. UHMWPE is shown to develop residual stresses continually from ～ 120°C to 23°C.     

On Parameter Estimation for Exponential Dispersion Arma Models

Abstract. A class of autoregressive moving‐average (ARMA) models proposed by Jørgensen and Song [Journal of Applied Probability (1998), vol. 35, pp. 78–92] with exponential dispersion model margins are useful to deal with non‐normal stationary time series with high‐order autocorrelation. One property associated with the class of models is that the projection process takes the exact form of the classical Box and Jenkins ARMA representation, leading to considerable ease to establish theories. This paper focuses on the issue of parameter estimation for such models, which has not been thoroughly investigated in Jørgensen and Song's paper. The key of the proposed approach is to treat the residual process associated with the projection essentially as a measurement error, which enables us to formulate directly an ARMA representation for the observed time series. The parameter estimation therefore becomes straightforward using the existing methods for the Box and Jenkins ARMA models such as the quasi‐likelihood method. The approach is illustrated by simulation studies and by an analysis of myoclonic seizure counts.