Abstract: India is witnessing tremendous growth in dairy industry. The milk production has increased from 20 million tonnes in 1961 to 132 million tonnes in 2012-13. India has been retaining its number one position in milk production for many years. Dairy Industry in India is growing at the rate of 10% per annum. Considering this, it is essential to know the future production to improve and sustain the growth and development of sector. The objective of the study is to find out most suitable forecasting method for milk production for sustainable future production and policy implications. The data used in study is secondary data, collected from FAOSTAT (1961 to 2012) and NDDB (1991 to 2012). Stationarity of data was checked with Autocorrelation Function (ACF) and Partial autocorrelation function (PACF), after confirming the stationarity, Autoregressive Integrated Moving Average (ARIMA) and Vector Autoregression (VAR) models were used. Akaike Information Criteria (AIC), Schwartz Bayesian Criteria (SBC), Mean Absolute Percentage Error (MAPE), R square and RMSE were used to test reliability of model. The results indicate that ARIMA (1, 1, 1) is more suitable method with the use of SPSS software package for forecasting of milk. Milk production is expected to be 160 million tonnes by 2017.

Abstract: Forecasting the number of epidemic disease is very important for CDC (center for disease control and prevention). To improve the forecast accuracy, an ARIMA (autoregressive integrated moving average) based model is proposed in this paper. First, autocorrelation (AC) and partial autocorrelation (PAC) analysis are introduced to establish a stationary time series, where the autocorrelation order, moving average order and difference order are estimated. Secondly, least square s method (LS) is employed to estimate the parameters of the prediction model. Finally, the real data between Jan. and Aug. 2014 coming from a CDC are fed into the proposed model and the forecast accuracy obtained is 92.1%, which significantly outperforms the simple moving average method currently used in the CDC.

Abstract: Modeling nonstationary processes is of paramount importance to many scientific disciplines including environmental science, ecology, and finance, among others. Consequently, flexible methodology that provides accurate estimation across a wide range of processes is a subject of ongoing interest. We propose a novel approach to model-based time–frequency estimation using time-varying autoregressive models. In this context, we take a fully Bayesian approach and allow both the autoregressive coefficients and innovation variance to vary over time. Importantly, our estimation method uses the lattice filter and is cast within the partial autocorrelation domain. The marginal posterior distributions are of standard form and, as a convenient by-product of our estimation method, our approach avoids undesirable matrix inversions. As such, estimation is extremely computationally efficient and stable. To illustrate the effectiveness of our approach, we conduct a comprehensive simulation study that compares our method with other competing methods and find that, in most cases, our approach performs superior in terms of average squared error between the estimated and true time-varying spectral density. Lastly, we demonstrate our methodology through three modeling applications; namely, insect communication signals, environmental data (wind components), and macroeconomic data (US gross domestic product (GDP) and consumption).

Abstract: The paper introduces the generalised partial autocorrelation (GPAC) coefficients of a stationary stochastic process. The latter are related to the generalised autocovariances, the inverse Fourier transform coefficients of a power transformation of the spectral density function. By interpreting the generalised partial autocorrelations as the partial autocorrelation coefficients of an auxiliary process, we derive their properties and relate them to essential features of the original process. Based on a parameterisation suggested by [1] and on Whittle likelihood, we develop an estimation strategy for the GPAC coefficients. We further prove that the GPAC coefficients can be used to estimate the mutual information between the past and the future of a time series.

Abstract: A central issue in population ecology is to determine the structure of negative feedback-density depend process which regulates population dynamics and seasonal fluctuations In this work the incidence of population density dependences and seasonality was examined in fruit orchards of three closely related pest species (Adoxophyes orana, Anarsia lineatella and (Grapholita) Grapholitha molesta) Analysis included 13 moth population time series during 2003–2011 Additionally, considering that time lags and seasonality are fundamental characteristics of ecological organisation and pest management, the work aimed to introduce a step wise algorithm to detect significant population feedbacks, moth seasonality and population synchronisation of nearby locations In the proposed procedure, each population-time series was first analysed on the basis of autocorrelation and partial autocorrelation Moreover, assuming that each of the ecological variable, observed at successive time points, consist of a stochastic process, autoregressive moving average ARMA(p,q) models and seasonal autoregressive moving average models SARMA(p,q)x(P,Q) S were fitted on data The Akaike information criteria was further used by the stepwise algorithm for parameter optimization and model improvement Model construction is accompanied by a presentation of the fitting results and a discussion of the heuristic benchmarks used to assess the forecasting performance of the models Life cycles of populations belonging to same species appeared to synchronise by terms of their autocorrelation functions Delayed density dependence and order was in most cases of lag:1 and 2, while lag >3 was not found more frequently as expected by chance In A orana and A lineatella moth species lag = 1 delayed density dependence was significantly more frequent and in particular in nearby locations However, the structure of the fitted models varied with respect to species and observation region In some cases, seasonal models were considered to be more accurate in simulating moth population dynamics Finally, to provide means in forecasting moth emergence and abundance, utile in pest management, the models were trained using 2003–2009 data sets and their forecasting performance were validated for each case using data sets of 2010–2011 In most cases, the constructed stochastic linear autoregressive models simulated the population outbreaks very well Describing and forecasting stochastic population fluctuations is a basic tenet of theoretical and applied ecology, while detecting the relative roles of exogenous and endogenous mechanisms can partly describe the phenomenological behavior of pest population time series data and improve pest management

Abstract: This work proposes a method to find the set of the most influential lags and the rule structure of a Takagi---Sugeno---Kang (TSK) fuzzy model for time series applications. The proposed method resembles the techniques that prioritize lags, evaluating the proximity of nearby samples in the input space using the closeness of the corresponding target values. Clusters of samples are generated, and the consistency of the mapping between the predicted variable and the set of candidate past values is evaluated. A TSK model is established, and possible redundancies in the rule base are avoided. The proposed method is evaluated using simulated and real data. Several simulation experiments were conducted for five synthetic nonlinear autoregressive processes, two nonlinear vector autoregressive processes and eight benchmark time series. The results show a competitive performance in the mean square error and a promising ability to find a proper set of lags for a given autoregressive process.

Abstract: This study specifically fitted an appropriate ARIMAX model for the Nigerian non-oil export using exchange rate (in dollars) as the exogenous variable by adopting the Box-Jenkins iterative three-stage modelling approach – identification, estimation and diagnostic checking. The time plot of the two series at level showed that the mean and variance are not constant but variant with time. The Augumented Dickey-Fuller (ADF) tests confirmed both series are non stationarity hence the two series were differenced once. Results of the unit roots tests in the first difference rejected the null hypotheses at 5% level of significance of non stationarity after first difference. The Autocorrelation function (ACF) and Partial autocorrelation function (PACF) combined patterns suggested AR(2) and MA(6) respectively. By comparing their various Akaike information criteria (AIC), the parsimonious model was estimated as ARIMAX (2,1,5). The goodness of fit test confirmed the adequacy of the estimated model. All the roots of the estimated ARIMAX process lie inside the unit circle. The plots of the residuals are mostly uncorrelated, the actual and the fitted agreed very closely. In addition, Q-statistics and LM test both indicate serial non-correlation. Result from the estimated model implies that exchange rate has no impact on Nigerian non-oil as exchange rate failed to be significant. Keywords : Non-oil export, ARIMAX model, exogenous variable.

Abstract: A new version of the partial autocorrelation plot and a new family of subset autoregressive models are introduced. A comprehensive approach to model identification, estimation and diagnostic checking is developed for these models. These models are better suited to efficient model building of high-order autoregressions with long time series. Several illustrative examples are given.

Abstract: In this paper, the Box-Jenkins modelling procedure is used to determine an ARIMA model and go further to forecasting. The mobile cellular subscription data for the study were taken from the administrative data submitted to the Zambia Information and Communications Technology Authority (ZICTA) as quarterly returns by all three mobile network operators Airtel Zambia, MTN Zambia and Zamtel. The time series of annual figures for mobile cellular subscription for all mobile network operators is from 2000 to 2014 and has a total of 15 observations. Results show that the ARIMA (1, 2, 1) is an adequate model which best fits the mobile cellular subscription time series and is therefore suitable for forecasting subscription. The model predicts a gradual rise in mobile cellular subscription in the next 5 years, culminating to about 9.0% cumulative increase in 2019.

Abstract: Understanding the regulation of natural populations has been a long-standing problem in ecology. Here we analyze the population dynamics of 17 species of saproxylic beetles in Shizuoka Prefecture, Japan collected over 11–12 years using autoregressive integrated moving average (ARIMA) models. We first examined the dynamics for indications of the order of the ARIMA models and evaluated the time series to determine that it was not simply a random, white noise sequence. All species dynamics were not mere random noise, and ARIMA models up to lag 3 were considered. The best model was selected from the possible ones using several criteria: model convergence, weak residual autocorrelation, the small sample AIC must be among the smallest that were not significantly different, and the lag indicated by the cutoff values in the detrended partial autocorrelation function. We found significant and nearly significant direct density-dependence for 14 of the 17 species, varying from −0.709 and stronger. The characteristic return rates were strong and only one species had a weak return rate (>0.9), implying that these species were strongly regulated by density-dependent factors. We found that populations with higher order ARIMA models (lag 2 and 3) had weaker return rates than populations with ARIMA models with only one lag, suggesting that species with more complex dynamics were more weakly regulated. These results contrast with previous suggestions that 20+ years are needed to detect density dependence from population time series and that most populations are weakly regulated.

Abstract: The paper describes an empirical study of modeling and forecasting time series data of groundnut production in India. Yearly groundnut production data for the period of 1950–1951 to 2013–2014 of India were analyzed by time-series methods. Autocorrelation and partial autocorrelation functions were calculated for the data. The Box Jenkins ARIMA methodology has been used for forecasting. The diagnostic checking has shown that ARIMA (0, 1, 1) is appropriate. The forecasts from 2015–2016 to 2024–2025 are calculated based on the selected model. The forecasting power of autoregressive integrated moving average model was used to forecast groundnut production for ten leading years. These forecasts would be helpful for the policy makers to foresee ahead of time the future requirements of groundnut seed, import and/or export and adopt appropriate measures in this regard.

Abstract: The SARIMA time series model is fitted to the monthly average maximum and minimum temperature data sets collected at Giridih, India for the years 1990-2011. From the time-series plots, we observe that the patterns of both the series are quite different; maximum temperature series contain sharp peaks in almost all the years while it is not true for the minimum temperature series and hence both the series are modeled separately (also for the sake of simplicity). SARIMA models are selected based on observing autocorrelation function (ACF) and partial autocorrelation function (PACF) of the monthly temperature series. The model parameters are obtained by using maximum likelihood method with the help of three tests [i.e., standard error, ACF and PACF of residuals and Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC) and corrected Akaike Information Criteria (AICc)]. Adequacy of the selected models is determined using diagnostic checking with the standardized residuals, ACF of residuals, normal Q-Q plot of the standardized residuals and p-values of the Ljung-Box statistic. The models ARIMA (1; 0; 2) × (0; 1; 1)12 and ARIMA (0; 1; 1) × (1; 1; 1)12 are finally selected for forecasting of monthly average maximum and minimum temperature values respectively for the eastern plateau region of India.

Abstract: Regression model assumes that the error terms are non-correlated. This is not always true with time series data as observations at one point in time often tend to be correlated with nearby observations. Durbin Watson test was carried out to determine whether or not the error terms are autocorrelated. Where autocorrelation existed, various models for estimating time series with autocorrelation were employed to model the series. The models used are Autoregressive model (AR), Moving Average model (MA), The Autoregressive Moving Average model (ARMA) and Integrated Moving Average model (ARIMA). These models were used on the exchange rate of Naira per Dollar at various market segments and then compared using the standard error and the ratio of coefficient to standard error criteria to obtain the model that best fit. Results however suggest that mixed models at smaller lags should often be adopted in modeling financial time series with evidence of autocorrelated error terms.

Abstract: This work modelled road traffic accidents that exhibit seasonal behaviour and non-statioarity in variance. The modelling procedure was demonstrated using monthly road traffic accident data (1999-2014) obtained from federal road safety commission in Port Harcourt, Nigeria. Seasonality and non-stationarity in variance were detected from the series raw plot, autocorrelation and partial autocorrelation functions. To obtain stability in variance and level of the series, some transformation techniques were applied. Seasonal component was incorporated into the Box and Jenkins (1970) ARIMA model to cater for the periodic nature of the series. The resulting estimated seasonal-ARIMA model was subjected to different diagnostic checks, and was found to be adequate. The proposed model was then used to generate forecasts of road traffic accidents for the next thirty months. Keywords: Stationarity, Seasonality, Transformation, Autocorrelation and Partial autocorrelation.

Abstract: In the previous chapters we have seen in which way the mean μ, and, more importantly, the autocovariance function, γ(h), h = 0, ±1, ±2, …, of a stationary stochastic process {X t } characterize its dynamic properties, at least if we restrict ourself to the first two moments. In particular, we have investigated how the autocovariance function is related to the coefficients of the corresponding ARMA process. Thus the estimation of the ACF is not only interesting for its own sake, but also for the specification and identification of appropriate ARMA models. It is therefore of outmost importance to have reliable (consistent) estimators for these entities. Moreover, we want to test specific features for a given time series. This means that we have to develop corresponding testing theory. As the small sample distributions are hard to get, we rely for this purpose on asymptotic theory.

Abstract: The paper deals with a forecast of developments in alcohol consumption based on current alcohol consumption per capita (expressed in litres of pure alcohol), and time series extrapolations. Alcohol consumption is to be considered from the vantage point of knowing the specifics of the product and the consequences of its excessive consumption. The predictive methodology makes use of the Box-Jenkins method; the ARIMA model, taking into account the autocorrelation and partial autocorrelation process, which is a prerequisite for the successful identification of a time series model; model parameter estimation; appropriate transformations of time series; determining the order of differentiation and subsequent verification of the model. The chosen methodology for future trends in alcohol consumptions is a prerequisite for the proposed optional measures to control alcohol consumption in the Czech Republic. Due to the long term nature of the process to draw up and implement alcohol consumption regulation measures, the forecast covers the forthcoming 10 years.

Abstract: Following recent research work on the autocorrelation of autoregressive Hilbertian discrete time processes, we likewise give an estimator for the autocorrelation of periodically correlated autoregressive Hilbertian processes and then prove the strong consistency of the estimator.

Abstract: Forecasting exchange rate requires a model that can capture the non-stationary and non-linearity of the exchange rate data. In this paper, empirical mode decomposition (EMD) is combines with least squares support vector machine (LSSVM) model in order to forecast daily USD/TWD exchange rate. EMD is used to decompose exchange rate data behaviors which are non-linear and nonstationary. LSSVM has been successfully used in non-linear regression estimation problems and pattern recognition. However, its input number selection is not based on any theories or techniques. In this proposed model, the exchange rate is decompose first by using EMD into several simple intrinsic mode oscillations called intrinsic mode function (IMF) and a residual. Permutation distribution clustering (PDC) is used to cluster the IMF and the residual into few groups according to their similarities in order to improve the LSSVM input. After that, LSSVM is used to forecast each of the groups and all the forecasted value is sum up in order to obtain the final exchange rate forecasting value where the best number of input for the LSSVM is determine by using partial autocorrelation function (PACF). The result shows that the modified EMD-LSSVM (MEMD-LSSVM) outperforms single LSSVM and hybrid model of EMD-LSSVM.

Abstract: Divergently different time series are considered in this article. The monthly passengers traffic at Cross lines limited, Calabar from 1990 to 2015 and monthly incidence of tuberculosis diseases at University of Calabar Teaching Hospital based from 1990-2015. The research adopted the statistical models based on time series analysis by Box and Jenkins methodology via the autocorrelation and the partial autocorrelation functions which showed that the two series are not stationary. Logarithm transformation was used to stabilize the variances of the two series and the residual autocorrelation and the partial autocorrelation functions is made stationary. Both regular and seasonal differencing was applied to the two-transformed set of data to obtain stationary series. The study employed ARIMA model on the classes of the two series, and the parameters of the identified model were estimated by the use of SPSS. The two models so chosen were ARMA (2,1,0) x (1,1,1)12 for passengers’ traffic and ARMA (1,0,1) x (1,1,2)12 for tuberculosis cases and forecasts was done for 12 months for the two series. The adequacy of the model was achieved and model fit for passengers traffic yields R-square, RMSE and MAPE of 0.876, 9.137, and 27.479 respectively and for tuberculosis cases yields R-square, RMSE and MAPE of 0.614, 6.785 and 26.522 respectively, recommendation and conclusion was made for the area of study.

Abstract: In this research, a vector bilinear autoregressive time series model was proposed and used to model three revenue series ( X 1 , X 2 , X 3 ) . The “orders” of the three series were identified on the basis of the distribution of autocorrelation and partial autocorrelation functions and were used to construct the vector bilinear models. The estimates obtained from the bilinear fits were compared graphically with those obtained from fitting linear (autoregressive) models. Residual variance and Box-Ljung Q statistic comparisons were also made. The result showed that vector bilinear autoregressive (BIVAR) models provide better estimates than the long embraced linear models. Keywords: Linear time series, Autoregressive process, Autocorrelation function, Partial autocorrelation function, Vector time series and bilinear vector process

Abstract: Soft computing models like Artificial Neural Network (ANN) have been widely used to model complex hydrological processes, such as rainfall-runoff and have been reported to be one of the promising tools in hydrology. In this paper, the influences of back propagation algorithm and their efficiencies which affect the input dimensions on rainfall runoff model have been demonstrated. The capability of the Artificial Neural Network with different input dimensions have been attempted and demonstrated with a case study on Sarada River Basin. The developed ANN models were able to map relationship between input and output data sets used. The developed model on rainfall and runoff pattern have been calibrated and validated. The significant input variables for training of ANN models were selected based on statistical parameters viz. cross-correlation, autocorrelation, and partial autocorrelation function. Various combinations were attempted and six combinations were selected based on the statistics of these functions. It was found those models considering rainfall lag rainfall and lag discharge as inputs were performing better than those considering rainfall alone. It was found that the neural network model developed is performing well. It can be inferred from the developed model, Neural Network model was able to predict runoff from rain fall data fairly well for a small semi-arid catchment area considered in the present study.

Abstract: Figures presenting autocorrelation (ACF) and partial autocorrelation (PACF) analyses of differenced ring width (RWd) chronologies.

Abstract: Additional file 6: Annex 1. Right side: Autocorrelation (ACF) and partial autocorrelation (PACF) functions of the residuals from ARIMA model (1, 0, 1) Ă (1, 0, 1)12 on log-transformed, differenced data. Left side: ACF and PACF of the residuals from ARIMA model (4, 0, 1) Ă (1, 0, 1)12 on log-transformed, differenced data.

Abstract: Accurate wind speed forecasting is a fundamental element of wind power prediction. Thus, a new hybrid wind speed forecasting model, using variational mode decomposition (VMD), the partial autocorrelation function (PACF), and weighted regularized extreme learning machine (WRELM), is proposed to improve the accuracy of wind speed forecasting. First, the historic wind speed time series is decomposed into several intrinsic mode functions (IMFs). Second, the partial correlation of each IMF sequence is analyzed using PACF to select the optimal subfeature set for particular predictors of each IMF. Then, the predictors of each IMF are constructed in order to enhance its strength using WRELM. Finally, wind speed is obtained by adding up all the predictors. The experiment, using real wind speed data, verified the effectiveness and advancement of the new approach.

Abstract: The central message of this article is that for robust and efficient damage detection the damage sensitive features should be selected optimally in a systematic way such that only these features that contribute the most to damage detectability be retained. Furthermore, suitable transformations of the original features may also enhance damage detectability. We explore these principles using data from a wind turbine blade. Several damage extent scenarios are introduced non-destructively. Partial autocorrelation coefficients are proposed as vibration-based damage sensitive features. Scores calculated with principal component analysis of partial autocorrelation coefficients are the transformed damage sensitive features. Statistical distances between the damage sensitive feature subsets estimated from the healthy and a reference damage state are calculated with respect to a statistical threshold as a measure of optimality. The fast forward method and a genetic algorithm are used to optimize the detectability o...