A computer program for time series forecasting using single and double exponential smoothing techniques

This paper presents a microcomputer program for time series forecasting. The program has been developed in GW-BASIC for Zenith 150 microcomputers which are IBM PC compatible. It utilizes Single exponential smoothing, Adaptive-response-rate single exponential smoothing, and Brown's double exponential smoothing methods to forecast the future values of a given time series. The program produces plots of the original time series and forecasted series as well as forecasting errors. It computes 90% and 95% confidence intervals for forecasted values and calculates the following statistics: Mean squared error, Mean absolute percentage error, Mean absolute error, Durbin-Watson statistic, and Theil's U statistic.     

Different Approaches to Forecast Interval Time Series: A Comparison in Finance

An interval time series (ITS) is a time series where each period is described by an interval. In finance, ITS can describe the temporal evolution of the high and low prices of an asset throughout time. These price intervals are related to the concept of volatility and are worth considering in order to place buy or sell orders. This article reviews two approaches to forecast ITS. On the one hand, the first approach consists of using univariate or multivariate forecasting methods. The possible cointegrating relation between the high and low values is analyzed for multivariate models and the equivalence of the VAR models is shown for the minimum and the maximum time series, as well as for the center and radius time series. On the other hand, the second approach adapts classic forecasting methods to deal with ITS using interval arithmetic. These methods include exponential smoothing, the k-NN algorithm and the multilayer perceptron. The performance of these approaches is studied in two financial ITS. As a result, evidences of the predictability of the ITS are found, especially in the interval range. This fact opens a new path in volatility forecasting.     

Forecasting irregularly spaced data: An extension of double exponential smoothing

Automated forecasts are often required, in practice, using data series from which certain points are missing and from data occurring at completely irregular time intervals. For instance, in computerised inventory control, fast methods of dealing with such data are required. There is an almost complete absence in the literature of computationally efficient methods for such a situation. This paper gives an extension of single and double exponential smoothing adapted to data occurring at irregular time intervals. These extensions are shown to have modest computational requirements and little sensitivity to initial conditions. Results of tests on sample data series are given showing only a minor decrease in accuracy with missing data, and indicating the appropriate method of choosing the smoothing parameter. Application of this method to published government time series is illustrated by two examples, firstly, to river water quality data originating from samples taken at irregular time intervals and, secondly, to divorce rate statistics from which certain points are missing due to summarizing the data. Successive summarizing of these series is found to have a negligible effect on forecast accuracy implying attractive cost saving possibilities in data collection and publication.     

针对PM2.5单时间序列数据的动态调整预测模型

针对细颗粒物PM2.5的浓度预测,本文提出了基于单时间序列数据的动态调整模型.在动态指数平滑算法中,指数平滑次数与参数基于样本数据并借助二分查找进行调整.在动态马尔科夫模型中,马尔科夫链的残差状态数、隐马尔科夫模型的隐状态数、连续样本数和阈值参数都通过训练数据加以调整.动态调整模型将指数平滑法和马尔科夫模型有效结合起来,指数平滑法得到的预测值由马尔科夫模型进行校正,从而提高预测准确度.基于大量实际PM2.5数据进行测试,验证了算法的有效性.并与其他现有的灰色模型、人工神经网络、自回归滑动平均模型、支持向量机等方法进行了对比,表明所提模型能够得到精度更高的预测结果.本文模型不局限于PM2.5数据,还可应用于其他类型的数据预测.     

Assessing the performance of smoothing functions to estimate land surface phenology on temperate grassland

NDVI (Normalized Difference Vegetation Index) time-series have been used for permitting a land surface phenology retrieval but these time series are affected by clouds and aerosols, which add noise to the signal sensor. In this sense, several smoothing functions are used to remove noise introduced by undetected clouds and poor atmospheric conditions, but a comparison between methods is still necessary due to disagreements about its performance in the literature. The application of a smoothing function is a necessarily previous step to describe land surface phenology in different ecosystems. The aims of this research were to evaluate the consistency of different smoothing functions from TIMESAT software and their impacts on phenological attributes of temperate grassland – a complex mosaic of land uses with natural vegetated and agricultural regions using NDVI-MODIS time series. An adaptive Savitzky–Golay (SG) filter, Asymmetric Gaussian (AG) and Double Logistic (DL) functions to fitting NDVI data were used and their performances were assessed using the measures root mean square error (RMSE), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and bias. Besides, differences on the estimation of the start of the growing season (SOS) and the length of the growing season (LOS) were obtained. High and low RMSE over croplands and grassland were observed for the three smoothing functions; in the rest of the region, the SG filter showed more reliable results. Patterns of difference on the estimation of SOS and LOS between SG filter and the other two models were randomly distributed, where differences of 20–50 days were found. This study demonstrated that methods from TIMESAT software are robust and spatially consistent but must be carefully used.     

High granular and short term time series forecasting of $$\hbox {PM}_{2.5}$$ PM 2.5 air pollutant - a comparative review

Forecasting time series has acquired immense research importance and has vast applications in the area of air pollution monitoring. This work attempts to investigate the abilities of various existing techniques when applied for short term, high granular time series forecasting of PM_2.5. More specifically, a comparative study has been provided, taking into account both popularly used models and lesser-used models in this area. The study has been carried out considering ten well defined models that are ARIMA (auto-regressive integrated moving average), SARIMA (seasonal ARIMA), SES (single exponential smoothing), DES (double exponential smoothing), TES (triple exponential smoothing), ANN (artificial neural network), DT (decision tree), kNN (k-nearest neighbor), LSTM (long short-term memory) and MCFO (markov chain first order). A framework has been built that categories the models, implements them under identical execution environment and forecasts succeeding values. Implementation has been carried out over five data sets of real-world air pollution time series, that are collected from five differently located government setup monitoring stations over a period of 1 year (July 2018-June 2019). Rigorous statistical analysis has been performed that yields an insight to the nature and variability of these time series data. Forecasting has been carried out on short term basis, focusing on high granularity whereas, three different lengths of forecast horizon (1 day, 1 week, and 1 month) have been tested. Eventually, the models have been compared in terms of their associated performance measuring units namely, RMSE (root mean of squared error), MAE (mean absolute error) and MAPE (mean absolute percentage error). The comparative results verified with multiple datasets show that all the models posses less error for a shorter forecast horizon, where LSTM providing the best performance. Superiority of machine learning and deep learning models are found in case of longer length of forecast horizon with kNN achieving best accuracy whereas, significant performance degradation of ARIMA is found for longer forecast horizon. Moreover, TES, DT, kNN, LSTM, MCFO are found to be well adopted in relation with shape and variability of the data. Note that the performance on various length of high granular forecast horizon have been studied over multiple datasets that give an added value to this work.

     

A decision support system methodology for forecasting of time series based on soft computing

Exponential procedures are widely used as forecasting techniques for inventory control and business planning. A number of modifications to the generalized exponential smoothing (Holt-Winters) approach to forecasting univariate time series is presented, which have been adapted into a tool for decision support systems. This methodology unifies the phases of estimation and model selection into just one optimization framework which permits the identification of robust solutions. This procedure may provide forecasts from different versions of exponential smoothing by fitting the updated formulas of Holt-Winters and selects the best method using a fuzzy multicriteria approach. The elements of the set of local minima of the non-linear programming problems allow us to build the membership functions of the conflicting objectives. It is compared to other forecasting methods on the 111 series from the M-competition.     

基于长短期神经网络的遥测参数趋势预测

时间序列分析的主要目的是根据已有的历史数据对未来进行预测。传统的时间序列预测主要依靠基于模型的方法，比如季节性差分整合移动平均自回归模型（SARIMA）和指数平滑法（EXP）等。此类方法的参数选择严重依赖于专家经验，适用性并不广泛。针对周期性遥测参数，采用长短期记忆网络（LSTM），学习长时序依赖关系并给出多步预测值。试验通过将预测问题转化为监督学习问题建立半实时仿真环境，并重点研究了观测窗口、预测窗口、网络结构等对性能指标的影响。对比LSTM、SARIMA、EXP，结果表明LSTM具备优异的线性拟合能力和良好的非线性关系映射能力。LSTM预测方法摆脱了传统方法受制于专家经验和模型精度低等问题，为开展实时遥测参数预测奠定了基础。     

ESPSA: A prediction-based algorithm for streaming time series segmentation

Streaming time series segmentation is one of the major problems in streaming time series mining, which can create the high-level representation of streaming time series, and thus can provide important supports for many time series mining tasks, such as indexing, clustering, classification, and discord discovery. However, the data elements in streaming time series, which usually arrive online, are fast-changing and unbounded in size, consequently, leading to a higher requirement for the computing efficiency of time series segmentation. Thus, it is a challenging task how to segment streaming time series accurately under the constraint of computing efficiency. In this paper, we propose exponential smoothing prediction-based segmentation algorithm (ESPSA). The proposed algorithm is developed based on a sliding window model, and uses the typical exponential smoothing method to calculate the smoothing value of arrived data element of streaming time series as the prediction value of the future data. Besides, to determine whether a data element is a segmenting key point, we study the statistical characteristics of the prediction error and then deduce the relationship between the prediction error and the compression rate. The extensive experiments on both synthetic and real datasets demonstrate that the proposed algorithm can segment streaming time series effectively and efficiently. More importantly, compared with candidate algorithms, the proposed algorithm can reduce the computing time by orders of magnitude.     

基于Holt-Winters的时间序列预测在大数据监控系统中的分析与应用

针对大数据监控系统对时间序列预测准确性和实时性的需求,以及大数据监控系统中时间序列呈现趋势性和季节性变化的特点,选择Holt-Winters算法建立时间序列预测模型。首先介绍时间序列的概念和特点,然后分析Holt-Winters算法的原理以及预测条件。选取合适的平滑系数是影响Holt-Winters算法预测准确性的关键,结合L-BFGS算法在不同时间区间求最优解,实现动态平滑系数的选取。最后以用户2天的页面访问量作为实验数据,通过相对误差指标的比较分析,验证该算法能满足大数据监控系统对时间序列预测的需求,具有较好的实际应用效果。     

基于指数平滑的云服务器请求量集成预测模型

针对现有弹性云服务器(elastic cloud server,ECS)未来请求量预测模型准确度低、稳定性差等问题,提出一种基于指数平滑的Stacking集成预测模型。以多个二次指数平滑模型作为基础模型,将线性回归模型作为集成模型对多组指数平滑预测值进行最终拟合;预测过程中使用多组二次指数平滑模型对ECS的历史请求时序进行构造集成模型训练数据集并加入平滑系数的动态优化。与传统单一模型的对比实验结果表明,该模型在实际云服务器请求量预测过程中具有更好的准确性和稳定性。     

A model updating strategy for predicting time series with seasonal patterns

Traditional methodologies for time series prediction take the series to be predicted and split it into training, validation, and test sets. The first one serves to construct forecasting models, the second set for model selection, and the third one is used to evaluate the final model. Different time series approaches such as ARIMA and exponential smoothing, as well as regression techniques such as neural networks and support vector regression, have been successfully used to develop forecasting models. A problem that has not yet received proper attention, however, is how to update such forecasting models when new data arrives, i.e. when a new event of the considered time series occurs.This paper presents a strategy to update support vector regression based forecasting models for time series with seasonal patterns. The basic idea of this updating strategy is to add the most recent data to the training set every time a predefined number of observations takes place. This way, information in new data is taken into account in model construction. The proposed strategy outperforms the respective static version in almost all time series studied in this work, considering three different error measures.     

基于投影寻踪的时间序列多属性决策研究

针对属性权重完全未知且属性值为多维时间序列的评价决策问题,提出一种基于加速遗传算法-投影寻踪和多属性决策的混杂评价决策模型方法.该方法将首先利用投影寻殊方法对多维时间序列数据按照属性进行降维处理,以解决数据处理过程中"维数灾难"带来的影响,并使用加速遗传算法确定最佳投影方向作为属性权重;对于得到的具有时间序列特性的决策...     

Time Series Data Cleaning under Multi-Speed Constraints

As the basis of data management and analysis, data quality issues have increasingly become a research hotspot in related fields, which contributes to optimization of big data and artificial intelligence technology. Generally, physical failures or technical defects in data collectors and recorders cause anomalies in collected data. These anomalies will strongly impact on subsequent data analysis and artificial intelligence processes; thus, data should be processed and cleaned accordingly before application. Existing repairing methods based on smoothing will cause a large number of originally correct data points being over-repaired into wrong values. The constraint-based methods such as sequential dependency and SCREEN cannot accurately repair data under complex conditions since the constraints are relatively simple. A time series data repairing method under multi-speed constraints is further proposed based on the principle of minimum repairing. Then, dynamic programming is used to solve the problem of data anomalies with optimal repairing. Specifically, multiple speed intervals are set to constrain time series data, and a series of candidate repairing points are formed for each data point according to the speed constraints. Next, the optimal repair solution is selected from these candidates based on the dynamic programming method. With regard to the feasibility study of this method, an artificial dataset, two real datasets, and another real dataset with real anomalies are employed for experiments in case of different rates of anomalies and data sizes. Experimental results demonstrate that, compared with the existing methods based on smoothing or constraints, the proposed method has better performance in terms of RMS errors and time cost. In addition, the investigation of clustering and classification accuracy with several datasets reveals the impact of data quality on subsequent data analysis and artificial intelligence. The proposed method can improve the quality of data analysis and artificial intelligence results.     

一种指数平滑预测的参数优化方法及实现

时间序列预测法在各种基于时态数据库的计算中有着广泛的应用前景。文中介绍了时间序列预测法中的单指数平滑、双指数平滑和三指数平滑三种指数平滑预测方法，不同的预测方法适合于对不同时间特性的数据、平稳性数据、趋势性数据或季节波动性数据进行预测，使用相应的预测方法均达到很好的平滑效果。同时还介绍了如何应用IGS算法对指数平滑的参数进行优化，从而得到更好的平滑效果和预测结果，使之在社会实际当中发挥更好的作用。     

Damp trend Grey Model forecasting method for airline industry

This paper presents a modification of the Grey Model (GM) to forecast routes passenger demand growth in the air transportation industry. Forecast methods like Holt-Winters, autoregressive models, exponential smoothing, neural network, fuzzy logic, GM model calculate very high airlines routes pax growth. For this reason, a modification has been done to the GM model to damp trend calculations as time grows. The simulation results show that the modified GM model reduces the model exponential estimations grow. It allows the GM model to forecast reasonable routes passenger demand for long lead-times forecasts. It makes this model an option to calculate airlines routes pax flow when few data points are available.The United States domestic air transport market data are used to compare the performance of the GM model with the proposed model.     

On allocation of resources during testing phase of a modular software

To provide reliable software, it is tested over a wide range of testing environment. In the process, testing resources such as time, testing personnel etc. are used. These resources are not infinitely large and therefore need to be used judiciously. In this article, we discuss the testing resource allocation problem among modules to maximize the total fault removal from software consisting of several independent components (modules). For the resulting optimization problem, we define marginal testing effort function (MTEF), where the testing resource consumption is represented in terms of fault removal. The three MTEFs proposed in this article account for both exponential and S-shaped growth curves, which are commonly used in software reliability analysis. Results are illustrated numerically using different data sets.     

Robust exponential smoothing of multivariate time series

Multivariate time series may contain outliers of different types. In the presence of such outliers, applying standard multivariate time series techniques becomes unreliable. A robust version of multivariate exponential smoothing is proposed. The method is affine equivariant, and involves the selection of a smoothing parameter matrix by minimizing a robust loss function. It is shown that the robust method results in much better forecasts than the classic approach in the presence of outliers, and performs similarly when the data contain no outliers. Moreover, the robust procedure yields an estimator of the smoothing parameter less subject to downward bias. As a byproduct, a cleaned version of the time series is obtained, as is illustrated by means of a real data example.     

指数平滑与自回归融合预测模型及实证

针对银行业务中的隔夜头寸预测问题,融合指数平滑与自回归预测的思想方法,提出了一个时间序列一步预测的微分动力学方程,证明了方程的离散化结构同无隐层BP神经网络的等价性。讨论了模型预测有效性问题,并进行了实证分析。对12个样本预报偏差的拟合优度检验表明,绝对预报偏差近似服从指数分布;实证分析了环比波动特征同模型预报误差以及预报同态度的关系,表明在一定条件下模型的预报是有效的,同NLP框架下的LSTM和GRU进行了对比实验表明该模型有更好的表现;定义了稳态指数和转折指数来描述时间序列的环比波动特征,分析表明稳态指数和转折指数可以预估模型预报的误差水平和同态度水平;研究了模型前端降噪对预报结果的影响,结果表明前端降噪可以抑制模型预报的“过敏”行为,有利于对时间序列变化趋势的动态跟踪。     

Visualization and inference based on wavelet coefficients, SiZer and SiNos

SiZer (SIgnificant ZERo crossing of the derivatives) and SiNos (SIgnificant NOn-Stationarities) are scale-space based visualization tools for statistical inference. They are used to discover meaningful structure in data through exploratory analysis involving statistical smoothing techniques. Wavelet methods have been successfully used to analyze various types of time series. In this paper, we propose a new time series analysis approach, which combines the wavelet analysis with the visualization tools SiZer and SiNos. We use certain functions of wavelet coefficients at different scales as inputs, and then apply SiZer or SiNos to highlight potential non-stationarities. We show that this new methodology can reveal hidden local non-stationary behavior of time series, that are otherwise difficult to detect.