Random Level-Shift Time Series Models,ARIMA Approximations,and Level-Shift Detection

The main purpose of this article is to assess the performance of autoregressive integrated moving average (ARIMA) models when occasional level shifts occur in the time series under study. A random level-shift time series model that allows the level of the process to change occasionally is introduced. Between two consecutive changes, the process behaves like the usual autoregressive moving average (ARMA) process. In practice, a series generated from a random level-shift ARMA (RLARMA) model may be misspecified as an ARIMA process. The efficiency of this ARIMA approximation with respect to estimation of current level and forecasting is investigated. The results of examining a special case of an RLARMA model indicate that the ARIMA approximations are inadequate for estimating the current level, but they are robust for forecasting future observations except when there is a very low frequency of level shifts or when the series are highly negatively correlated. A level-shift detection procedure is presented to handle the low-frequency level-shift phenomena, and its usefulness in building models for forecasting is demonstrated.     

A Benchmarking Approach to Forecast Combination

This article is concerned with the development of a statistical model-based approach to optimally combine forecasts derived from an extrapolative model, such as an autoregressive integrated moving average (ARIMA) time series model, with forecasts of a particular characteristic of the same series obtained from independent sources. The methods derived combine the strengths of all forecasting approaches considered in the combination scheme. The implications of the general theory are investigated in the context of some commonly encountered seasonal ARIMA models. An empirical example to illustrate the method is included.     

The Seasonal Adjustment Procedures for the Consumer Price Indexes: Some Empirical Results

Beginning with January of 1987, the consumer price indexes (CPI's) have been seasonally adjusted by the X-11 autoregressive integrated moving average (ARIMA) procedure. This modification of the X-11 procedure was introduced following an empirical investigation into three aspects of seasonal adjustment methodology as applied to several CPI series—the choice of ARIMA models to fit and forecast those series, the improvements made by the ARIMA modification in terms of revision and smoothness of the seasonally adjusted series, and the effect on the quality of seasonal adjustment and the identifiability of seasonality due to the ARIMA modification. This article reports the results of that investigation, in addition, a brief account is given of the U.S. Bureau of Labor Statistics procedures relating to the projected seasonal factors, seasonally adjusted aggregate series, and the revisions of the seasonally adjusted series.     

Effect of systematic sampling on arima models

Given a general homogeneous non-stationary autoregressive integrated moving average process ARIMA(p,d,q), the corresponding model for the subseries obtained by a systematic sampling is derived. The article then shows that the sampled subseries approaches approximately to an integrated moving average process IMA(d,l), l≤(d-l), regardless of the autoregressive and moving average structures in the original series. In particular, the sampled subseries from an ARIMA (p,l,q) process approaches approximately to a simple random walk model.     

A recursive approach for estimating missing observations in an univariate time series

A linear recursive technique that does not use the Kalman filter approach is proposed to estimate missing observations in an univariate time series. It is assumed that the series follows an invertible ARIMA model. The procedure is based on the restricted forecasting approach, and the recursive linear estimators are optimal in terms of minimum mean-square error.     

ARIMA modelling of birth data

In this article the fitting of ARIMA models to time series relating to the births at Edendale Hospital in Natal, South Africa, over a 16-year period is discussed. The model (011)X(011)12 provides andexcellent fit to the monthly totals of mothers delivered but serious discrepancies between estimates of the moving average parameters obtained by the method of unconditional least squares using various statistical computer packages were observed. These can be ascribed to the fact that certain of the packages use the method of backcasting to calculate the unconditional sum-of-squares function in the estimation procedure and this approach breaks down for values of the moving average parameters close to the invertibility boundary. A simulation study to compare the different methods of estimation for the model (011) X (011)12 and to assess the seriousness of discrepancies in the ULS estimates is described. ARIMA models which provided the best fit to the series of monthly totals of caesarean sections performed, breechs births and instrumental deliveries are non-seasonal and reflect a dependence of present on past observations. In contrast the series involving stillbirths and neonatal deaths are white noise indicating that perinatal deaths are random events.     

Comparison of data analysis procedures for real-time nanoparticle sampling data using classical regression and ARIMA models

Real-time monitoring is necessary for nanoparticle exposure assessment to characterize the exposure profile, but the data produced are autocorrelated. This study was conducted to compare three statistical methods used to analyze data, which constitute autocorrelated time series, and to investigate the effect of averaging time on the reduction of the autocorrelation using field data. First-order autoregressive (AR(1)) and autoregressive-integrated moving average (ARIMA) models are alternative methods that remove autocorrelation. The classical regression method was compared with AR(1) and ARIMA. Three data sets were used. Scanning mobility particle sizer data were used. We compared the results of regression, AR(1), and ARIMA with averaging times of 1, 5, and 10?min. AR(1) and ARIMA models had similar capacities to adjust autocorrelation of real-time data. Because of the non-stationary of real-time monitoring data, the ARIMA was more appropriate. When using the AR(1), transformation into stationary data was necessary. There was no difference with a longer averaging time. This study suggests that the ARIMA model could be used to process real-time monitoring data especially for non-stationary data, and averaging time setting is flexible depending on the data interval required to capture the effects of processes for occupational and environmental nano measurements.     

Statistical analysis of error for fourth-order ordinary differential equation solvers

We develop an autoregressive integrated moving average (ARIMA) model to study the statistical behavior of the numerical error generated from three fourth-order ordinary differential equation solvers: Milne's method, Adams–Bashforth method and a new method that randomly switches between the Milne and Adams–Bashforth methods. With the actual error data based on three differential equations, we desire to identify an ARIMA model for each data series. Results show that some of the data series can be described by ARIMA models but others cannot. Based on the mathematical form of the numerical error, other statistical models should be investigated in the future. Finally, we assess the multivariate normality of the sample mean error generated by the switching method.     

Predictive Accuracy Gain From Disaggregate Sampling in ARIMA Models

We compare the forecast accuracy of autoregressive integrated moving average (ARIMA) models based on data observed with high and low frequency, respectively. We discuss how, for instance, a quarterly model can be used to predict one quarter ahead even if only annual data are available, and we compare the variance of the prediction error in this case with the variance if quarterly observations were indeed available. Results on the expected information gain are presented for a number of ARIMA models including models that describe the seasonally adjusted gross national product (GNP) series in the Netherlands. Disaggregation from annual to quarterly GNP data has reduced the variance of short-run forecast errors considerably, but further disaggregation from quarterly to monthly data is found to hardly improve the accuracy of monthly forecasts.     

Forecasting Study of Shanghai’s and Shenzhen’s Stock Markets Using a Hybrid Forecast Method

In this article, a novel hybrid method to forecast stock price is proposed. This hybrid method is based on wavelet transform, wavelet denoising, linear models (autoregressive integrated moving average (ARIMA) model and exponential smoothing (ES) model), and nonlinear models (BP Neural Network and RBF Neural Network). The wavelet transform provides a set of better-behaved constitutive series than stock series for prediction. Wavelet denoising is used to eliminate some slight random fluctuations of stock series. ARIMA model and ES model are used to forecast the linear component of denoised stock series, and then BP Neural Network and RBF Neural Network are developed as tools for nonlinear pattern recognition to correct the estimation error of the prediction of linear models. The proposed method is examined in the stock market of Shanghai and Shenzhen and the results are compared with some of the most recent stock price forecast methods. The results show that the proposed hybrid method can provide a considerable improvement for the forecasting accuracy. Meanwhile, this proposed method can also be applied to analysis and forecast reliability of products or systems and improve the accuracy of reliability engineering.     

A Survey of Recent Developments in Seasonal Adjustment

In recent years there have been notable advances in the methodology for analyzing seasonal time series. This paper summarizes some recent research on seasonal adjustment problems and procedures. Included are signal-extraction methods based on autoregressive integrated moving average (ARIMA) models, improvements in X–11, revisions in preliminary seasonal factors, regression and other model-based methods, robust methods, seasonal model identification, aggregation, interrelating seasonally adjusted series, and causal approaches to seasonal adjustment.     

Fractionally integrated ARMA for crude palm oil prices prediction: case of potentially overdifference

Dealing with stationarity remains an unsolved problem. Some of the time series data, especially crude palm oil (CPO) prices persist towards nonstationarity in the long-run data. This dilemma forces the researchers to conduct first-order difference. The basic idea is that to obtain the stationary data that is considered as a good strategy to overcome the nonstationary counterparts. An opportune remark as it is, this proxy may lead to overdifference. The CPO prices trend elements have not been attenuated but nearly annihilated. Therefore, this paper presents the usefulness of autoregressive fractionally integrated moving average (ARFIMA) model as the solution towards the nonstationary persistency of CPO prices in the long-run data. In this study, we employed daily historical Free-on-Board CPO prices in Malaysia. A comparison was made between the ARFIMA over the existing autoregressive-integrated moving average (ARIMA) model. Here, we employed three statistical evaluation criteria in order to measure the performance of the applied models. The general conclusion that can be derived from this paper is that the usefulness of the ARFIMA model outperformed the existing ARIMA model.     

An optimal k of kth MA-ARIMA models under AR(p) models

In this article, we discuss finding the optimal k of (i) kth simple moving average, (ii) kth weighted moving average, and (iii) kth exponential weighted moving average based on simulated autoregressive AR(p) model. We run a simulation using the three above examining method under specific conditions. The main finding is that the optimal k = 4 and then k = 3. Especially, the fourth WMA ARIMA model, fourth EWMA ARIMA model, and third EWMA ARIMA model are the best forecasting models among others, respectively. For all the six real data reveal the similar results of simulation study.     

A prediction approach for precise marketing based on ARIMA-ARCH Model: A case of China Mobile

The autoregressive integrated moving average (ARIMA) model presents improved performance in forecasting short-term trends because it considers the dependence of time series and the interference of stochastic volatility. Thus, in this study, we establish ARIMA(0, 2, 1) based on the historical data of large-scale online marketing promotions to realize precise marketing of China Mobile's Ling Xi Voice app in the communication market. We eliminate the auto-regression effect of residual series by establishing the ARIMA model combined with the autoregressive conditional heteroskedasticity (ARCH) model denoted as ARIMA(0, 2, 1) ? ARCH(1), the ARIMA model combined with the generalized ARCH (GARCH) model denoted as ARIMA(0, 2, 1) ? GARCH(1, 1), and the ARIMA model combined with the threshold GARCH model denoted as ARIMA(0, 2, 1) ? TGARCH(2, 1). The performance of the aforementioned models is then compared for validation. Considering the characteristics of the communication markets and the attractive statistical properties of ARIMA, we apply ARIMA(0, 2, 1) to forecast the cumulative number of Ling Xi Voice app users for precise marketing that offers reliable agreement for China Mobile to further advertise and study the market demand. Our analysis contributes toward the development of the current knowledge on forecasting the number of app users in the communication market and provides a new idea to increase the market share for communication operators.     

Outlier Detection in Time Series Models Using Local Influence Method

We propose a new procedure for detecting a patch of outliers or influential observations for autoregressive integrated moving average (ARIMA) model using local influence analysis. It is shown that the dependency aspects of time series data gives rise to masking or smearing effects when the local influence analysis is performed using current perturbation schemes. We suggest a new perturbation scheme to take into account the dependent structure of time series data, and employ the stepwise local influence method to give a diagnostic procedure. We show that the new perturbation scheme can avoid the smearing effects, and the stepwise technique of local influence can successfully deal with masking effects. Various simulation studies are performed to show the efficiency of proposed methodology and a real example is used for illustrations.     

A Note on Overdifferencing and the Equivalence of Seasonal Time Series Models With Monthly Means and Models With (0, 1, 1)12 Seasonal Parts When ⊖ = 1

Two general models for monthly seasonal time series are considered, one in which seasonality is modeled with monthly means and another in which seasonality is modeled with a (0, 1, 1)₁₂ ARIMA structure. The models are shown to be equivalent if the seasonal moving average parameter (?) is 1 and if the same assumptions about the 12 initial observations are made for both models. The role of the assumptions about the initial observations is analyzed, and it is argued that for practical purposes the two models can be regarded as equivalent when ? = 1. It is observed that the result extends easily to more general models involving overdifferencing.     

Monitoring the parameter changes in general ARIMA time series models

We propose methods for monitoring the residuals of a fitted ARIMA or an autoregressive fractionally integrated moving average (ARFIMA) model in order to detect changes of the parameters in that model. We extend the procedures of Box & Ramirez (1992) and Ramirez (1992) and allow the differencing parameter, d to be fractional or integer. Test statistics are approximated by Wiener processes. We carry out simulations and also apply our method to several real time series. The results show that our method is effective for monitoring all parameters in ARFIMA models.     

Short-term forecasting with a computationally efficient nonparametric transfer function model

In this paper a semi-parametric approach is developed to model non-linear relationships in time series data using polynomial splines. Polynomial splines require very little assumption about the functional form of the underlying relationship, so they are very flexible and can be used to model highly non-linear relationships. Polynomial splines are also computationally very efficient. The serial correlation in the data is accounted for by modelling the noise as an autoregressive integrated moving average (ARIMA) process, by doing so, the efficiency in nonparametric estimation is improved and correct inferences can be obtained. The explicit structure of the ARIMA model allows the correlation information to be used to improve forecasting performance. An algorithm is developed to automatically select and estimate the polynomial spline model and the ARIMA model through backfitting. This method is applied on a real-life data set to forecast hourly electricity usage. The non-linear effect of temperature on hourly electricity usage is allowed to be different at different hours of the day and days of the week. The forecasting performance of the developed method is evaluated in post-sample forecasting and compared with several well-accepted models. The results show the performance of the proposed model is comparable with a long short-term memory deep learning model.     

About estimation of ARIMA process with strong mixing MA part

A central limit theorem is provided for the least squares estimates of the autoregressive parameters in an ARIMA process with strong mixing moving average part.     

A recursive approach to parameter estimation in regression and time series models

In this paper we discuss the recursive (or on line) estimation in (i) regression and (ii) autoregressive integrated moving average (ARIMA) time series models. The adopted approach uses Kalman filtering techniques to calculate estimates recursively. This approach is used for the estimation of constant as well as time varying parameters. In the first section of the paper we consider the linear regression model. We discuss recursive estimation both for constant and time varying parameters. For constant parameters, Kalman filtering specializes to recursive least squares. In general, we allow the parameters to vary according to an autoregressive integrated moving average process and update the parameter estimates recursively. Since the stochastic model for the parameter changes will "be rarely known, simplifying assumptions have to be made. In particular we assume a random walk model for the time varying parameters and show how to determine whether the parameters are changing over time. This is illustrated with an example.