The Variance of Lead-Time Demand

Many short-term forecasting systems are based on exponentially weighted moving averages. It is usual to forecast the cumulative demand over a lead time or production horizon, and to describe this forecast in terms of its mean and variance. When the forecast horizon is fixed, the variance is often taken as the product of the number of periods and the variance per period. This is a serious error and typically underestimates the variance by a factor of about two. This paper details the need for a proper awareness of the correction factors.     

A robust forecasting system,based on the combination of two simple moving averages

For series with negligible growth and seasonality, simple moving averages are frequently used to estimate the current level of a process, and the resultant value projected as a forecast for future observations. This paper shows that a linear combination of two simple moving averages (SMA) can provide an improved estimate of the underlying level of the process. The proposition is demonstrated by simulation, and good combinations are listed. The theory underlying the improvement is developed. The general rules are then illustrated through an application in an inventory situation.     

Optimality and robustness of combinations of moving averages

A combination of moving averages has been shown previously to be more accurate than simple moving averages, under certain conditions, and to be more robust to non-optimal parameter specification. However, the use of the method depends on specification of three parameters: length of greater moving average, length of shorter moving average, and the weighting given to the former. In this paper, expressions are derived for the optimal values of the three parameters, under the conditions of a steady state model. These expressions reduce a three-parameter search to a single-parameter search. An expression is given for the variance of the sampling error of the optimal combination of moving averages and this is shown to be marginally greater than that for exponentially weighted moving averages (EWMA). Similar expressions for optimal parameters and the resultant variance are derived for equally weighted combinations. The sampling variance of the mean of such combinations is shown to be almost identical to the optimal general combination, thus simplifying the use of combinations further. It is demonstrated that equal weight combinations are more robust than EWMA to noise to signal ratios lower than expected, but less robust to noise to signal ratios higher than expected.     

Adaptive Filtering: Once More with Gusto

Adaptive filtering, when used as a forecasting method, proposes to be able to distinguish a "signal pattern" of a time series instead of just smoothing out the random noise introduced by the data. Adaptive filtering is claimed by its creators to "...always do as well if not better than either moving averages, exponential smoothing,...". In order to see whether this claim could be substantiated, the author has taken the approach of a casual user of forecasting methods and has sought to determine whether adaptive filtering is useful, or not, as a forecasting method. The method was used to compute forecasts for ten sets of data on monthly insurance payments in a Finnish insurance company, and the experience gained from this work is compared with criticisms of the method expressed by a number of writers. It is shown that the method performs quite well for practical purposes, despite the fact that it has some major theoretical shortcomings.     

Optimal continued fractions and the moving average ergodic theorem

We use the moving average ergodic theorem of A. Bellow, R. Jones and J. Rosenblatt to derive various results in metric number theory primarily concerning moving averages of various sequences attached to the optimal continued fraction expansion of a real number.     

Semiparametric Bootstrap Approach to Hypothesis Tests and Confidence Intervals for the Hurst Coefficient

A major application of rescaled adjusted range analysis (R–S analysis) is to the study of price fluctuations in financial markets. There, the value of the Hurst constant, H, in a time series may be interpreted as an indicator of the irregularity of the price of a commodity, currency or similar quantity. Interval estimation and hypothesis testing for H are central to comparative quantitative analysis. In this paper we propose a new bootstrap, or Monte Carlo, approach to such problems. Traditional bootstrap methods in this context are based on fitting a process chosen from a wide but relatively conventional range of discrete time series models, including autoregressions, moving averages, autoregressive moving averages and many more. By way of contrast we suggest simulation using a single type of continuous-time process, with its fractal dimension. We provide theoretical justification for this method, and explore its numerical properties and statistical performance by application to real data on commodity prices and exchange rates. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stable mixed moving averages

Summary The class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable mixed moving averages. Their distribution determines a certain combination of the filter function and the mixing measure, leading to a generalization of a theorem of Kanter (1973) for usual moving averages. Stable mixed moving averages contain sums of independent stable moving averages, are ergodic and are not harmonizable. Also a class of stable mixed moving averages is constructed with the reflection positivity property.Research supported by AFSOR Contract 91-0030Research also supported by ARO DAAL-91-G-0176Research also supported by AFOSR 90-0168Research also supported by ONR N00014-91-J-0277     

Some properties of a simple moving average when applied to forecasting a time series

Simple (equally weighted) moving averages are frequently used to estimate the current level of a time series, with this value being projected as a forecast for future observations. A key measure of the effectiveness of the method is the sampling error of the estimator, which this paper defines in terms of characteristics of the data. This enables the optimal length of the average for any steady state model to be established and the lead time forecast error derived. A comparison of the performance of a simple moving average (SMA) with an exponentially weighted moving average (EWMA) is made. It is shown that, for a steady state model, the variance of the forecast error is typically less than 3% higher than the appropriate EWMA. This relatively small difference may explain the inconclusive results from the empirical studies about the relative predictive performance of the two methods.     

Exponential moving average versus moving?exponential?average

In this note we discuss the mathematical tools to define trend indicators which are used to describe market trends. We explain the relation between averages and moving averages on the one hand and the so called exponential moving average (EMA) on the other hand. We present a lot of examples and give the definition of the most frequently used trend indicator, the MACD.     

Exponential moving average versus moving exponential average

In this note we discuss the mathematical tools to define trend indicators which are used to describe market trends. We explain the relation between averages and moving averages on the one hand and the so called exponential moving average (EMA) on the other hand. We present a lot of examples and give the definition of the most frequently used trend indicator, the MACD.     

移动平均线方法的最佳步长组合的确定

移动平均是股票技术分析中最常用的方法之一，但现在没有多少文章来讨论窨应用步长为几的移动平均为好，即赚钱最多或赚钱的机会最高，该文提出三种确定最佳步长的方法以供参考。     

A numerical method to compute solidification and melting processes

This paper deals with thermodynamically consistent numerical predictions of solidification and melting processes of pure materials using moving grids. Till date, enthalpy-porosity-based formulations of numerical codes have been generally the popular choice, although because of an artificial numerical smearing of the interface, it is virtually impossible to reproduce a sharp melting/solidification front that is supposed to exist for phase changes of pure substances. Numerical techniques based on moving grid methods have been relatively less used as they rely on complex and time-consuming adaptive grid generations. Using the moving grid approach, the authors present a method to solve solidification and melting problems. A simple linear interpolation is used to slide grid nodes along the interface to handle the otherwise obtained grid skewness near the interface. The numerical approach employed is validated with standard test cases available in the literature. The capability of capturing very complex flow field structures and the superiority of the present approach over enthalpy-porosity-based formulations is discussed. The authors also demonstrate the ability of the set-up computer code to solve complex thermofluid processes such as occur during crystal growth in Czochralski reactors.     

Limit theorems for additive statistics based on moving average samples

We study statistics based on samples of moving averages generated by stationary sequence of random variables. The central limit theorem (CLT) is proved for sequences of observations defined by an analytic function of moving averages under consideration. For U- and V -statistics with canonical (degenerate) kernels, the limit distributions are studied.     

生产安全事故宏观发展态势的计量经济学研究

为了研究工矿商贸就业人员10万人生产安全事故死亡率时间序列变化特征,基于我国行业生产安全事故死亡人数及第二、三产业就业人员数量等2方面年度统计数据,通过研究事故死亡率时间序列的自回归移动平均过程,论文构建了事故死亡率时间序列的分阶段自回归移动平均模型.研究表明:工矿商贸行业10万人事故死亡率变化趋势具有明显的分阶段波动特征,事故死亡率序列均为趋势平稳过程;序列当期观测值与滞后1期观测值具有显著的自相关性;各阶段事故率自回归移动平均模型结构不尽相同;特征描述模型为正确把握我国安全生产状况及趋势提供理论依据.     

The impact of special days in call arrivals forecasting: A neural network approach to modelling special days

A key challenge for call centres remains the forecasting of high frequency call arrivals collected in hourly or shorter time buckets. In addition to the complex intraday, intraweek and intrayear seasonal cycles, call arrival data typically contain a large number of anomalous days, driven by the occurrence of holidays, special events, promotional activities and system failures. This study evaluates the use of a variety of univariate time series forecasting methods for forecasting intraday call arrivals in the presence of such outliers. Apart from established, statistical methods, we consider artificial neural networks (ANNs). Based on the modelling flexibility of the latter, we introduce and evaluate different methods to encode the outlying periods. Using intraday arrival series from a call centre operated by one of Europe’s leading entertainment companies, we provide new insights on the impact of outliers on the performance of established forecasting methods. Results show that ANNs forecast call centre data accurately, and are capable of modelling complex outliers using relatively simple outlier modelling approaches. We argue that the relative complexity of ANNs over standard statistical models is offset by the simplicity of coding multiple and unknown effects during outlying periods.     

How does innovation''''s tail risk determine marginal tail risk of a stationary financial time series?

We discuss the relationship between the marginal tail risk probability and theinnovation's tail risk probability for some stationary financial time series models. We firstgive the main results on the tail behavior of a class of infinite weighted sums of randomvariables with heavy-tailed probabilities. And then, the main results are applied to threeimportant types of time series models; infinite order moving averages, the simple bilineartime series and the solutions of stochastic difference equations. The explicit formulasare given to describe how the marginal tail probabilities come from the innovation's tailprobabilities for these time series. Our results can be applied to the tail estimation of timeseries and are useful for risk analysis in finance.     

Asymptotic distributions of nonparametric regression estimators for longitudinal or functional data

The estimation of a regression function by kernel method for longitudinal or functional data is considered. In the context of longitudinal data analysis, a random function typically represents a subject that is often observed at a small number of time points, while in the studies of functional data the random realization is usually measured on a dense grid. However, essentially the same methods can be applied to both sampling plans, as well as in a number of settings lying between them. In this paper general results are derived for the asymptotic distributions of real-valued functions with arguments which are functionals formed by weighted averages of longitudinal or functional data. Asymptotic distributions for the estimators of the mean and covariance functions obtained from noisy observations with the presence of within-subject correlation are studied. These asymptotic normality results are comparable to those standard rates obtained from independent data, which is illustrated in a simulation study. Besides, this paper discusses the conditions associated with sampling plans, which are required for the validity of local properties of kernel-based estimators for longitudinal or functional data.     

Small and Large Scale Behavior of the Poissonized Telecom Process

The stable Telecom process has infinite variance and appears as a limit of renormalized renewal reward processes. We study its Poissonized version where the infinite variance stable measure is replaced by a Poisson point measure. We show that this Poissonized version converges to the stable Telecom process at small scales and to the Gaussian fractional Brownian motion at large scales. This process is therefore locally as well as asymptotically self-similar. The value of the self-similarity parameter at large scales, namely the self-similarity parameter of the limit fractional Brownian motion, depends on the form the Poissonized Telecom process. The Poissonized Telecom process is a Poissonized mixed moving average. We investigate more general Poissonized mixed moving averages as well.     

Berry-Esséen Bounds for Long Memory Moving Averages via Stein's Method and Malliavin Calculus

Using the Stein method on Wiener chaos introduced in Nourdin and Peccati [10], we prove Berry-Esséen bounds for long memory moving averages.     

Discovering metamodels’ quality-of-fit for simulation via graphical techniques

Metamodels are used in many disciplines to replace simulation models of complex multivariate systems. To discover metamodels ‘quality-of-fit’ for simulation, simple information returned by average-based statistics, such as root-mean-square error RMSE, are often used. The sample of points used in determining these averages is restricted in size, especially for simulation models of complex multivariate systems. Obviously, decisions made based on average values can be misleading when the sample size is not adequate, and contributions made by each individual data point in such samples need to be examined. This paper presents methods that can be used to discover metamodels quality-of-fit graphically by means of two-dimensional plots. Three plot types are presented; these are the so-called circle plots, marksman plots, and ordinal plots. Such plots can be used to facilitate visual inspection of the effect on metamodel accuracy of each individual point in the data sample used for metamodel validation. The proposed methods can be used to complement quantitative validation statistics; in particular, for situations where there is not enough validation data or the validation data is too expensive to generate.