MODELING LONG-MEMORY PROCESSES FOR OPTIMAL LONG-RANGE PREDICTION

Abstract. We look at the implications of modeling observations from a fractionally differenced noise process using an approximating AR ( p ) model. The approximation is used because of computational difficulties in the estimation of the differencing parameter of the fractional noise model. Because the fractional noise process is long-range dependent, we assess the applicability of the approximating autoregressive (AR) model based on its long-range forecasting accuracy compared with that of the fractional noise model. We derive the asymptotic k -step-ahead prediction error for a fractional noise process modeled as an AR( p ) process and compare it with the k -step-ahead prediction error obtained when the model for the observed series is correctly specified as a fractional noise process and the fractional differencing parameter d is either known or estimated. We also assess the validity of the asymptotic results for a finite sample size via simulation. We see that AR models can be useful for long-range forecasting of long-memory data, provided that consideration is given to the forecast horizon when choosing an approximating model.     

THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS

Abstract. The definitions of fractional Gaussian noise and integrated (or fractionally differenced) series are generalized, and it is shown that the two concepts are equivalent. A new estimator of the long memory parameter in these models is proposed, based on the simple linear regression of the log periodogram on a deterministic regressor. The estimator is the ordinary least squares estimator of the slope parameter in this regression, formed using only the lowest frequency ordinates of the log periodogram. Its asymptotic distribution is derived, from which it is evident that the conventional interpretation of these least squares statistics is justified in large samples. Using synthetic data the asymptotic theory proves to be reliable in samples of 50 observations or more. For three postwar monthly economic time series, the estimated integrated series model provides more reliable out-of-sample forecasts than do more conventional procedures.     

基于LSTM-RNN模型的铁水硅含量预测

针对高炉炼铁是一个动态过程,具有大延迟,工况复杂的特性。采用LSTM-RNN模型进行硅含量预测,充分发挥了其处理时间序列时挖掘前后关联信息的优势。首先根据时间序列趋势及相关系数选择自变量,并采用复杂工况的实际生产数据进行验证。然后用程序自动求解最优参数进行硅含量预测。最后将LSTM-RNN模型与PLS模型及RNN模型的结果进行对比,验证该方法的优势。研究发现LSTM-RNN模型预测误差稳定,预测精度较高,比传统的统计学及神经网络方法取得了更好的预测精度。     

Prediction Interval for Autoregressive Time Series via Oracally Efficient Estimation of Multi‐Step‐Ahead Innovation Distribution Function

A kernel distribution estimator (KDE) is proposed for multi‐step‐ahead prediction error distribution of autoregressive time series, based on prediction residuals. Under general assumptions, the KDE is proved to be oracally efficient as the infeasible KDE and the empirical cumulative distribution function (cdf) based on unobserved prediction errors. Quantile estimator is obtained from the oracally efficient KDE, and prediction interval for multi‐step‐ahead future observation is constructed using the estimated quantiles and shown to achieve asymptotically the nominal confidence levels. Simulation examples corroborate the asymptotic theory.     

An Efficient Taper for Potentially Overdifferenced Long-memory Time Series

We propose a new complex-valued taper and derive the properties of a tapered Gaussian semiparametric estimator of the long-memory parameter d ε (−0.5, 1.5). The estimator and its accompanying theory can be applied to generalized unit root testing. In the proposed method, the data are differenced once before the taper is applied. This guarantees that the tapered estimator is invariant with respect to deterministic linear trends in the original series. Any detrimental leakage effects due to the potential noninvertibility of the differenced series are strongly mitigated by the taper. The proposed estimator is shown to be more efficient than existing invariant tapered estimators. Invariance to k th order polynomial trends can be attained by differencing the data k times and then applying a stronger taper, which is given by the k th power of the proposed taper. We show that this new family of tapers enjoys strong efficiency gains over comparable existing tapers. Analysis of both simulated and actual data highlights potential advantages of the tapered estimator of d compared with the nontapered estimator.     

Recursive adjustment,unit root tests and structural breaks

In this paper, we introduce unit root tests for time series with a potential structural break computed from test regressions in which the deterministic components have been recursively adjusted. We present finite sample critical values as well as Monte Carlo results on the size and power performance of the new procedures, and compare these with other available tests in the literature, such as OLS and quasi‐differenced based tests (see, for instance, Perron, (1997) Perron and Rodriguez, (2003) and Carrion‐i‐Silvestre et al. (2009) ). The small sample behaviour of the tests is evaluated in a known and an unknown break date context allowing for negligible and non‐negligible initial conditions. In the unknown break date case, two break date estimation procedures are considered, one based on the minimum unit root t‐statistic and the other based on the minimum sum of squared residuals obtained from a regression on a set of deterministic variables. The size and power performance of the recursive adjustment based procedure in the unknown break date case is encouraging. A further result of this paper relates to the aditional finite sample evidence on the performance of quasi‐differenced unit root tests, complementing the results in Perron and Rodriguez (2003) .     

colour and constitution of some N-phenylpyrrolidinylazo dyes: application of the PPP molecular orbital method

Electronic absorption spectral data for a series of monoazo dyes derived from N-phenylpyrrolidine have been calculated by the PPP molecular orbital method. In order to optimise the agreement between experimental and calculated results, certain parameters were reassessed by trial and error calculations. New parameters for some sulphur-containing heterocyclic systems permit the satisfactory prediction of λ_max values for the derived dyes. The relative intensities of the various heterocyclic azo dyes are also reasonably well accounted for by the PPP method.     

Influence of Missing Values on the Prediction of a Stationary Time Series

Abstract. The influence of missing observations on the linear prediction of a stationary time series is investigated. Simple bounds for the prediction error variance and asymptotic behaviours for short and long‐memory processes respectively are presented.     

Gaussian Semi-parametric Estimation of Fractional Cointegration

Abstract. We analyse consistent estimation of the memory parameters of a nonstationary fractionally cointegrated vector time series. Assuming that the cointegrating relationship has substantially less memory than the observed series, we show that a multi-variate Gaussian semi-parametric estimate, based on initial consistent estimates and possibly tapered observations, is asymptotically normal. The estimates of the memory parameters can rely either on original (for stationary errors) or on differenced residuals (for nonstationary errors) assuming only a convergence rate for a preliminary slope estimate. If this rate is fast enough, semi-parametric memory estimates are not affected by the use of residuals and retain the same asymptotic distribution as if the true cointegrating relationship were known. Only local conditions on the spectral densities around zero frequency for linear processes are assumed. We concentrate on a bivariate system but discuss multi-variate generalizations and show the performance of the estimates with simulated and real data.     

ESTIMATION OF COEFFICIENTS OF TIME SERIES REGRESSION WITH A NONSTATIONARY ERROR PROCESS

Abstract. We treat a problem of estimating unknown coefficients of a time series regression when the variance of the error changes with time, i.e. when a process which the error term obeys is nonstationary. First, we show the weak consistency of the ordinary least squares estimator for the coefficients of a polynomial regression under some assumptions on the covariance structure of the error process. Next, we propose a nonparametric method for estimating the variance of the error process and a weighted least squares estimator of the regression coefficients, which is constructed by using the estimator of the variance. We investigate statistical properties of our proposed estimator in the following way. We consider the prediction of a future value of a linear trend by using our proposed estimator and evaluate its prediction error. By simulation studies, we compare the prediction error of the predictor constructed by using our proposed estimator with the prediction errors obtained for other estimators including the ordinary least squares estimator when the variance of the error process increases with time and the sample sizes are small. As a result, our proposed estimator seems to be reasonable.     

KERNEL REGRESSION SMOOTHING OF TIME SERIES

Abstract. A class of non-parametric regression smoothers for times series is defined by the kernel method. The kernel approach allows flexible modelling of a time series without reference to a specific parametric class. The technique is applicable to detection of non-linear dependences in time series and to prediction in smooth regression models with serially correlated observations.
In practice these estimators are to be tuned by a smoothing parameter. A data-driven selector for this smoothing parameter is presented that asymptotically minimizes a squared error measure. We prove asymptotic optimality of this selector. We illustrate the technique with a simulated example and by constructing a smooth prediction curve for the variation of gold prices. In both cases the non-parametric method proves to be useful in uncovering non-linear structure.     

Prediction model of slurry pH based on mechanism and error compensation for mineral flotation process

A suitable pH value of the slurry is a key to efficient mineral flotation. Considering the control delay problem of pH value caused by offline pH measurement, an integrated prediction model for pH value in bauxite froth flotation is proposed, which considers the effect of ore compositions on pH value. Firstly, a regression model is obtained for alkali (Na₂CO₃) consumed by the reaction between ore and alkali. According to the first-order hydrolysis of the remaining alkali, a mechanism-based prediction model is presented for the pH value. Then, considering the complexity of the flotation mechanism, an error prediction model which uses time series of the error of the mechanism model as inputs is presented based on autoregressive moving average (ARMA) method to compensate the mechanism model. Finally, expert rules are established to correct the error compensation direction, which could reflect the dynamic changes during the process accurately and effectively. Simulation results using industrial data show that the presented model meets the needs of the industrial process, which laid the foundation for predictive control of pH regulator.     

An Improvement of Akaike's FPE Criterion to Reduce its Variability

The selection of the number of past observations to be included in a linear predictor (the order of the predictor) should be done with minimum variability, since it is not taken into account in the inference stage. For finite time series, there is a trade-off between variability and optimality (in the sense of mean squared prediction error). The widely used Akaike criteria, FPE and AIC, lead to highly variable estimated orders, whereas consistent criteria are downward biased when the optimal order increases with the sample size. In this paper we propose the use of a sequence of tests to analyse the order selected with FPE. The result is a new identification criterion. In a simulation study, this criterion is shown to reach a compromise with respect to the above-mentioned trade-off. Some asymptotic properties are also derived.     

基于混合评价指标的自组织模糊神经网络设计研究

针对在无增长和修剪阈值时模糊神经网络结构难以自适应问题,提出一种基于混合评价指标(hybrid evaluation index, HEI)的结构设计方法。首先,通过模糊C均值聚类算法(fuzzy C-means clustering, FCM)确定初始规则层神经元数目及其中心与宽度。其次,基于戴维森堡丁指数(Davies bouldin index, DBI)和邓恩指数(Dunn index, DI)提出一种新的相关性评价指标(relevance evaluation index, REI)来计算规则层各神经元输出之间的相关性,同时根据训练过程中网络输出均方根误差(root mean square error, RMSE)的变化情况来确定网络的学习能力,然后基于REI和RMSE提出了HEI。通过HEI来调整模糊神经网络的拓扑结构,有效解决了在无增长和修剪阈值时网络结构难以动态自调整的问题且避免了网络结构冗余。最后,通过对Mackey-Glass时间序列预测、非线性系统辨识和大气中PM_2.5浓度预测,证明了该结构设计方法的可行性和有效性。     

A GENERALIZED VARIANCE RATIO TEST OF ARIMA (p, 1, q) MODEL SPECIFICATION

Abstract. The variance ratio test is often used as a check of the hypothesis that a time series is generated by a random walk. A natural extension of the test is developed to cover the case where the assumed model is ARIMA(p, 1, q), with unknown parameters. Small sample properties of the generalized test are investigated, and the test is applied to a frequently analysed data set on US quarterly real gross national product. In effect, we are testing for low frequency misspecification in assumed autoregressive moving-average (ARMA) models for a differenced series.     

一种基于时序误差补偿的动态软测量建模方法

针对目前静态软测量建模方法无法反映工业过程动态信息,造成预测模型精度低、鲁棒性差等问题,提出了一种基于最小二乘支持向量机(LS-SVM)和自回归-滑动平均模型(ARMA)的软测量建模方法。首先,建立了基于LS-SVM的软测量模型,利用ARMA模型对预测误差的动态估计,通过增加动态校正环节,实现了对静态模型的动态校正以改善系统动态响应特性。最后将上述方法用于乙烯精馏过程中乙烷浓度的软测量建模,仿真结果表明:与单一使用LSSVM模型相比,该方法具有跟踪性能好、泛化能力强等优点,是一种有效的软测量建模方法。     

Improved prediction limits for a general class of Gaussian models

In this article we consider the problem of prediction for a general class of Gaussian models, which includes, among others, autoregressive moving average time‐series models, linear Gaussian state space models and Gaussian Markov random fields. Using an idea presented in Sjöstedt‐De Luna and Young (2003) , in the context of spatial statistics, we discuss a method for obtaining prediction limits for a future random variable of interest, taking into account the uncertainty introduced by estimating the unknown parameters. The proposed prediction limits can be viewed as a modification of the estimative prediction limit, with unconditional, and eventually conditional, coverage error of smaller asymptotic order. The modifying term has a quite simple form and it involves the bias and the mean square error of the plug‐in estimators for the conditional expectation and the conditional variance of the future observation. Applications of the results to Gaussian time‐series models are presented.     

COMPARISON OF TIME AND CROSS-SECTIONAL AGGREGATION UNDER A TIME SERIES RANDOM COMPONENT MODEL

Abstract. Small-area estimation under a stationary time series random component model is considered. Cross-sectional aggregation and varying degrees of time aggregation are treated as competing prediction methods. An estimated mean-squared prediction error criterion is used to compare these methods. Some exact and asymptotic properties of this criterion are developed, a consistent estimator of the associated asymptotic variance is presented and simultaneous approximate confidence intervals for the mean-squared prediction errors are discussed. Time aggregation of a single series is considered as a special case. In addition, an extension to the assessment of mean-squared prediction errors of synthetic small-area predictors is outlined.     

SEARCHING FOR ADDITIVE OUTLIERS IN NONSTATIONARY TIME SERIES*

Abstract. Recently, Vogelsang (1999) proposed a method to detect outliers which explicitly imposes the null hypothesis of a unit root. It works in an iterative fashion to select multiple outlier in a given series. We show, via simulations, that, under the null hypothesis of no outliers, it has the right size in finite samples to detect a single outlier but, when applied in an iterative fashion to select multiple outliers, it exhibits severe size distortions towards finding an excessive number of outliers. We show that his iterative method is incorrect and derive the appropriate limiting distribution of the test at each step of the search. Whether corrected or not, we also show that the outliers need to be very large for the method to have any decent power. We propose an alternative method based on first‐differenced data that has considerably more power. We also show that our method to identify outliers leads to unit root tests with more accurate finite sample size and robustness to departures from a unit root. The issues are illustrated using two US/Finland real‐exchange rate series.