An Examination of the Use of Adaptive Filtering in Forecasting

Adaptive filtering is a technique for preparing short- to medium-term forecasts based on the weighting of historical observations, in a similar way to moving average and exponential smoothing. However, adaptive filtering, as it has been developed in electrical engineering, attempts to distinguish a signal pattern from random noise, rather than simply smoothing the noise of past data. This paper reviews the technique of adaptive filtering and investigates its applications and limitations for the forecasting practitioner. This is done by looking at the performance of adaptive filtering in forecasting a number of time series and by comparing it with other forecasting techniques.     

Adaptive Exponential Smoothing Revisited

Adaptive exponential smoothing models are designed to improve performance by letting the smoothing parameter vary according to the most recent forecasting accuracy. This paper argues that the constant exponential smoothing results used in two comparative studies are inadequate as benchmarks. A reexamination does not indicate that adaptive exponential smoothing methods provide superior forecasts compared to those obtainable from constant exponential smoothing with a considerate choice of the smoothing constant. No support was found for the alleged advantages of the Dennis run based adaptive procedure.     

Multi-item sales forecasting with total and split exponential smoothing

Efficient supply chain management relies on accurate demand forecasting. Typically, forecasts are required at frequent intervals for many items. Forecasting methods suitable for this application are those that can be relied upon to produce robust and accurate predictions when implemented within an automated procedure. Exponential smoothing methods are a common choice. In this empirical case study paper, we evaluate a recently proposed seasonal exponential smoothing method that has previously been considered only for forecasting daily supermarket sales. We term this method ‘total and split’ exponential smoothing, and apply it to monthly sales data from a publishing company. The resulting forecasts are compared against a variety of methods, including several available in the software currently used by the company. Our results show total and split exponential smoothing outperforming the other methods considered. The results were also impressive for a method that trims outliers and then applies simple exponential smoothing.     

A Note on Forecasting with Adaptive Filtering

Adaptive filtering has been suggested as a short-term-forecasting technique that is superior to other time series analysis methods. This note discusses several aspects of adaptive filtering and indicates its equivalence to an autoregressive process. The performance of adaptive filtering and two other forecasting techniques are compared for a particular time series.     

The Application of Kernel Smoothing to Time Series Data

There are already a lot of models to fit a set of stationary time series, such as AR, MA, and ARMA models. For the non-stationary data, an ARIMA or seasonal ARIMA models can be used to fit the given data. Moreover, there are also many statistical softwares that can be used to build a stationary or non-stationary time series model for a given set of time series data, such as SAS, SPLUS, etc. However, some statistical softwares wouldn＇t work well for small samples with or without missing data, especially for small time series data with seasonal trend. A nonparametric smoothing technique to build a forecasting model for a given small seasonal time series data is carried out in this paper. And then, both the method provided in this paper and that in SAS package are applied to the modeling of international airline passengers data respectively, the comparisons between the two methods are done afterwards. The results of the comparison show us the method provided in this paper has superiority over SAS＇s method.     

Dynamic Detection of Change Points in Long Time Series

We consider the problem of detecting change points (structural changes) in long sequences of data, whether in a sequential fashion or not, and without assuming prior knowledge of the number of these change points. We reformulate this problem as the Bayesian filtering and smoothing of a non standard state space model. Towards this goal, we build a hybrid algorithm that relies on particle filtering and Markov chain Monte Carlo ideas. The approach is illustrated by a GARCH change point model.     

CUSUM vs Smoothed-Error Forecast Monitoring Schemes: Some Simulation Results

This paper compares the performance of CUSUM and smoothed-error tracking signals for monitoring the adequacy of exponential smoothing forecasts. Previous research has favoured the CUSUM. However, there is some evidence that the performance of the smoothed-error signal can be improved by a simple modification in its application: the use of different smoothing parameters in the tracking signal and the forecasting model. The effects of this modification are tested using simulated time series. We conclude that the CUSUM is robust to the choice of forecasting parameter, while the smoothed-error signal is not. The CUSUM is also more responsive to small changes in the time series, regardless of the parameters used.     

Adaptive Filtering Revisited

This paper shows that the adaptive filtering and forecasting techniques proposed by Makridakis and Wheelwright can be viewed as approximations to a more precise filtering method in which the Kalman filter is applied to a dynamic autoregressive model which is a special case of the models of Harrison and Stevens. The correct "learning" or "training factors" are shown to be data-dependent matrices rather than scalar constants.     

Forecasting daily supermarket sales using exponentially weighted quantile regression

Inventory control systems typically require the frequent updating of forecasts for many different products. In addition to point predictions, interval forecasts are needed to set appropriate levels of safety stock. The series considered in this paper are characterised by high volatility and skewness, which are both time-varying. These features motivate the consideration of forecasting methods that are robust with regard to distributional assumptions. The widespread use of exponential smoothing for point forecasting in inventory control motivates the development of the approach for interval forecasting. In this paper, we construct interval forecasts from quantile predictions generated using exponentially weighted quantile regression. The approach amounts to exponential smoothing of the cumulative distribution function, and can be viewed as an extension of generalised exponential smoothing to quantile forecasting. Empirical results are encouraging, with improvements over traditional methods being particularly apparent when the approach is used as the basis for robust point forecasting.     

Improving demand forecasting accuracy using nonlinear programming software

We address the problem of forecasting real time series with a proportion of zero values and a great variability among the nonzero values. In order to calculate forecasts for a time series, the model coefficients must be estimated. The appropriate choice of values for the smoothing parameters in exponential smoothing methods relies on the minimization of the fitting errors of historical data. We adapt the generalized Holt–Winters formulation so that it can consider the starting values of the local components of level, trend and seasonality as decision variables of the nonlinear programming problem associated with this forecasting procedure. A spreadsheet model is used to solve the problems of optimization efficiently. We show that our approach produces accurate forecasts with little data per product.     

Progressive Tuning of Simple Exponential Smoothing Forecasts

The paper outlines a finite sample version of exponential smoothing, and proposes a formula for estimating the smoothing parameter. The resulting method, which can be implemented on a recursive basis over time, is compared with alternative approaches, such as progressive numerical optimization of the smoothing parameter and adaptive forecasting on both synthetic and real data.     

拟马尔可夫过程的磨光优化算法及应用

通过对磨光法及马尔可夫过程的研究,马氏过程作为区间预测的一种方法,在很大程度上约束了它预测的科学性,另外,磨光法本身也是一种迭代的方法,对于拟合的精度还是难于控制,通过拟马尔可夫矩阵与磨光法相结合及优化工具,得到拟马尔可夫过程的磨光优化算法,实例表明:拟马尔可夫过程的磨光优化算法使修正磨光后的值逼近原数据值的程度较其它算法更好,而且,拟马尔可夫矩阵反应了从一种状态到另一种状态的转移程度,并且这种算法具有更好的推广和应用。     

Potential inventory cost reductions using advanced time series forecasting techniques

This paper compares demand forecasts computed using the time series forecasting techniques of vector autoregression (VAR) and Bayesian VAR (BVAR) with forecasts computed using exponential smoothing and seasonal decomposition. These forecasts for three demand data series were used to determine three inventory management policies for each time series. The inventory costs associated with each of these policies were used as a further basis for comparison of the forecasting techniques. The results show that the BVAR technique, which uses mixed estimation, is particularly useful in reducing inventory costs in cases where the limited historical data offer little useful information for forecasting. The BVAR technique was effective in improving forecast accuracy and reducing inventory costs in two of the three cases tested. In the third case, unrestricted VAR and exponential smoothing produced the lowest experimental forecast errors and computed inventory costs. Furthermore, this research illustrates that improvements in demand forecasting can provide better cost reductions than relying on stochastic inventory models to provide cost reductions.     

Monitoring an Exponential Smoothing Forecasting System

The first part of this paper is concerned with the variance of the smoothed error when the forecasting system being used is exponential smoothing. The expression derived for this variance involves the variance of the noise, the smoothing constant and a sum of squared binomial coefficients. It is also shown that the variance of the sum of errors equals the variance of the smoothed error for one less degree of freedom divided by the square of the smoothing constant.The second part of the paper considers the practical application of the above result and also gives values for the tracking signal limits, obtained by simulation, which could be used in the automatic monitoring of a forecasting system.     

Integrating noisy data

Suppose the parametric form of a curve is not known, but only a set of observations. Quadrature formulae can be used to integrate a function only known from a set of data points. However, the results will be unreliable if the data contains measurement errors (noise). The method presented here fits an even degree piecewise polynomial to the data where all the data points are being used as knot points and the smoothing parameter is optimal for the indefinite integral of the curve which happens to be a smoothing spline. After the smoothing parameter has been chosen, this approach is less computationally expensive than fitting a smoothing spline and integrating.     

组合预测模型在能源消费预测中的应用

能源的需求预测是一个复杂的非线形系统,其发展变化具有增长性和波动性,组合预测对于信息不完备的复杂经济系统具有一定的实用性.本文利用我国能源消费的历史数据,采用灰色预测的GM(1,1)模型、BP神经网络模型和三次指数平滑模型进行优化组合,建立了能源消费组合预测模型,实证分析结果表明预测值和实际结果有很好的一致性,可以作为能源消费预测的有效工具.     

A new method to mitigate data fluctuations for time series prediction

Although the classic exponential-smoothing models and grey prediction models have been widely used in time series forecasting, this paper shows that they are susceptible to fluctuations in samples. A new fractional bidirectional weakening buffer operator for time series prediction is proposed in this paper. This new operator can effectively reduce the negative impact of unavoidable sample fluctuations. It overcomes limitations of existing weakening buffer operators, and permits better control of fluctuations from the entire sample period. Due to its good performance in improving stability of the series smoothness, the new operator can better capture the real developing trend in raw data and improve forecast accuracy. The paper then proposes a novel methodology that combines the new bidirectional weakening buffer operator and the classic grey prediction model. Through a number of case studies, this method is compared with several classic models, such as the exponential smoothing model and the autoregressive integrated moving average model, etc. Values of three error measures show that the new method outperforms other methods, especially when there are data fluctuations near the forecasting horizon. The relative advantages of the new method on small sample predictions are further investigated. Results demonstrate that model based on the proposed fractional bidirectional weakening buffer operator has higher forecasting accuracy.     

The Monitoring of Exponentially Weighted Forecasts

By considering equivalences between various forecasting systems, the exact stochastic process followed by the one-step-ahead errors may be found. This process, the error process, is important for any monitoring scheme, and is a function of the forecasting system and the underlying data process. The error process is obtained for the most general form of exponential smoothing systems used in optimal conditions. The statistical properties are derived. In particular, the approximate variance of Trigg's smoothed error tracking signal is obtained explicitly for several exponential smoothing systems, and a procedure is given for obtaining it numerically for any such system. The use of different smoothing constants in the forecasting system and the tracking signal is discussed and it is found that suitable choice of the tracking signal constant greatly improves the performance of the signal, making it more comparable with CUSUM schemes.     

Some results on Tchebycheffian spline functions

This report derives explicit solutions to problems involving Tchebycheffian spline functions. We use a reproducing kernel Hilbert space which depends on the smoothness criterion, but not on the form of the data, to solve explicitly Hermite-Birkhoff interpolation and smoothing problems. Sard's best approximation to linear functionals and smoothing with respect to linear inequality constraints are also discussed. Some of the results are used to show that spline interpolation and smoothing is equivalent to prediction and filtering on realizations of certain stochastic processes.     

Forecasting sales of slow and fast moving inventories

Traditional computerised inventory control systems usually rely on exponential smoothing to forecast the demand for fast moving inventories. Practices in relation to slow moving inventories are more varied, but the Croston method is often used. It is an adaptation of exponential smoothing that (1) incorporates a Bernoulli process to capture the sporadic nature of demand and (2) allows the average variability to change over time. The Croston approach is critically appraised in this paper. Corrections are made to underlying theory and modifications are proposed to overcome certain implementation difficulties. A parametric bootstrap approach is outlined that integrates demand forecasting with inventory control. The approach is illustrated on real demand data for car parts.