EXPONENTIAL SMOOTHING AND NON‐NEGATIVE DATA

The most common forecasting methods in business are based on exponential smoothing, and the most common time series in business are inherently non‐negative. Therefore it is of interest to consider the properties of the potential stochastic models underlying exponential smoothing when applied to non‐negative data. We explore exponential smoothing state space models for non‐negative data under various assumptions about the innovations, or error, process. We first demonstrate that prediction distributions from some commonly used state space models may have an infinite variance beyond a certain forecasting horizon. For multiplicative error models that do not have this flaw, we show that sample paths will converge almost surely to zero even when the error distribution is non‐Gaussian. We propose a new model with similar properties to exponential smoothing, but which does not have these problems, and we develop some distributional properties for our new model. We then explore the implications of our results for inference, and compare the short‐term forecasting performance of the various models using data on the weekly sales of over 300 items of costume jewelry. The main findings of the research are that the Gaussian approximation is adequate for estimation and one‐step‐ahead forecasting. However, as the forecasting horizon increases, the approximate prediction intervals become increasingly problematic. When the model is to be used for simulation purposes, a suitably specified scheme must be employed.     

Evaluation of Recursive Detection Methods for Turning Points in Financial Time Series

Timely identification of turning points in economic time series is important for planning control actions and achieving profitability. This paper compares sequential methods for detecting peaks and troughs in stock values and deciding the time to trade. Three semi‐parametric methods are considered: double exponential smoothing, time‐varying parameters and prediction error statistics. These methods are widely used in monitoring, forecasting and control, and their common features are recursive computation and exponential weighting of observations. The novelty of this paper is the selection of smoothing and alarm coefficients for maximisation of the gain (the difference in level between subsequent peaks and troughs) of sample data. The methods are compared on applications to leading financial series and with simulation experiments.     

Adaptive exponentielle glättung erster ordnung

In the paper we introduce an adaptive procedure of the first order exponential smoothing. In this procedure we get a sequence of estimation converging in mean square to the unknown smoothing parameter and an asymptotically optimum prediction in the sense of the least square error. In case of breaks in the structure of time series we recommend a modification of the procedure.     

Incremental-decremental rolling window double exponential smoothing

Abstract

By modeling double exponential smoothing as a weighted directed acyclic graph, we design an implementation of rolling window double exponential smoothing which is incremental-decremental in the sense that points can be added to and removed from the window with overhead and computation independent of the window size. This has applications to real-time streaming analytics systems having certain universality and flexibility requirements.     

Modelling and Analysis of Data that Exhibit Temporal Decay

Two recent examples of data from statistical consulting work form the basis of this paper. Both data sets arose from situations where temporal decay was expected, and different models for this are discussed, in particular, variants on simple exponential decay: hyperexponential (or mixed exponential) decay and models based on differential equations. Nonparametric estimation, using local quadratic smoothing of the data, of components of these models is discussed and employed to help to check the appropriateness or otherwise of existing parametric analyses of the data. In one example, the suitability of the parametric analysis is confirmed; in the other, the parametric assumptions made at the time are shown to have some flaws, and an improved parametric analysis is provided.     

霍尔特指数平滑法参数的优选

霍尔特指数平滑方法有是一种高级的指数平滑方法,它有二个基本平滑公式和一个预测公式。霍尔特指数平滑方法进行预测时,最重要、而且最因难的工作是确定平滑参数的取值问题。目前,在有关参考文献中均未见到相关问题的论述。本文偿试使用运筹学的方法解决平滑参数的优选问题。为霍尔特指数平滑方法在实际中的应用提供一种有效的途径。     

Semiparametric Time-Varying Coefficients Regression Model for Longitudinal Data

Abstract.  In this paper, we consider a semiparametric time-varying coefficients regression model where the influences of some covariates vary non-parametrically with time while the effects of the remaining covariates follow certain parametric functions of time. The weighted least squares type estimators for the unknown parameters of the parametric coefficient functions as well as the estimators for the non-parametric coefficient functions are developed. We show that the kernel smoothing that avoids modelling of the sampling times is asymptotically more efficient than a single nearest neighbour smoothing that depends on the estimation of the sampling model. The asymptotic optimal bandwidth is also derived. A hypothesis testing procedure is proposed to test whether some covariate effects follow certain parametric forms. Simulation studies are conducted to compare the finite sample performances of the kernel neighbourhood smoothing and the single nearest neighbour smoothing and to check the empirical sizes and powers of the proposed testing procedures. An application to a data set from an AIDS clinical trial study is provided for illustration.     

An algorithm for generalized monotonic smoothing

In this paper, an algorithm for Generalized Monotonic Smoothing (GMS) is developed as an extension to exponential family models of the monotonic smoothing techniques proposed by Ramsay (1988, 1998a,b). A two-step algorithm is used to estimate the coefficients of bases and the linear term. We show that the algorithm can be embedded into the iterative re-weighted least square algorithm that is typically used to estimate the coefficients in Generalized Linear Models. Thus, the GMS estimator can be computed using existing routines in S-plus and other statistical software. We apply the GMS model to the Down's syndrome data set and compare the results with those from Generalized Additive Model estimation. The choice of smoothing parameter and testing of monotonicity are also discussed.     

Theory & Methods: Krige,Smooth, Both or Neither?

Both kriging and non-parametric regression smoothing can model a non-stationary regression function with spatially correlated errors. However comparisons have mainly been based on ordinary kriging and smoothing with uncorrelated errors. Ordinary kriging attributes smoothness of the response to spatial autocorrelation whereas non-parametric regression attributes trends to a smooth regression function. For spatial processes it is reasonable to suppose that the response is due to both trend and autocorrelation. This paper reviews methodology for non-parametric regression with autocorrelated errors which is a natural compromise between the two methods. Re-analysis of the one-dimensional stationary spatial data of Laslett (1994) and a clearly non-stationary time series demonstrates the rather surprising result that for these data, ordinary kriging outperforms more computationally intensive models including both universal kriging and correlated splines for spatial prediction. For estimating the regression function, non-parametric regression provides adaptive estimation, but the autocorrelation must be accounted for in selecting the smoothing parameter.     

A Note on Exponentially Smoothed Seasonal Differences

Filtering economic time series can be justified if variations within a certain frequency interval are relevant for the problem at hand. It is shown here that exponential smoothing of seasonal differences provides a simple means of damping short oscillations, including seasonal ones.     

Iterative weighted estimation based on variance modelling in linear regression models

ABSTRACT

The estimation of variance function plays an extremely important role in statistical inference of the regression models. In this paper we propose a variance modelling method for constructing the variance structure via combining the exponential polynomial modelling method and the kernel smoothing technique. A simple estimation method for the parameters in heteroscedastic linear regression models is developed when the covariance matrix is unknown diagonal and the variance function is a positive function of the mean. The consistency and asymptotic normality of the resulting estimators are established under some mild assumptions. In particular, a simple version of bootstrap test is adapted to test misspecification of the variance function. Some Monte Carlo simulation studies are carried out to examine the finite sample performance of the proposed methods. Finally, the methodologies are illustrated by the ozone concentration dataset.     

我国居民消费价格波动和预测：1997-2010

 CPI与人们的生活息息相关,同时也是经济分析和决策、价格监测和调控及国民经济核算的重要指标。本文以1997年1月至2009年12月我国衣食住行及城乡分类月度定基CPI数据为样本,在分析其波动特征及差异的基础上,通过指数平滑法的Holt-Winters模型将其分解为季节和趋势波动。结果表明,我国分类CPI各自具有明显的趋势和季节特征,并得出其波峰和波谷到达时间;模型对其有非常好的拟合效果,其MAPE依次为0.253%、0.816%、0.364%、0.359%、0.391%、0.338%。在此基础上对我国2010年各月的分类CPI进行了科学预测。     

Kernel smoothed prediction intervals for ARMA models

The procedures of estimating prediction intervals for ARMA processes can be divided into model based methods and empirical methods. Model based methods require knowledge of the model and the underlying innovation distribution. Empirical methods are based on sample forecast errors. In this paper we apply nonparametric quantile regression to empirical forecast errors using lead time as regressor. Using this method there is no need for a distributional assumption. But for the special data pattern in this application a double kernel method which allows smoothing in two directions is required. An estimation algorithm is presented and applied to some simulation examples.     

Robust non-parametric smoothing of non-stationary time series

Motivated by the need of extracting local trends and low frequency components in non-stationary time series, this paper discusses methods of robust non-parametric smoothing. Basic approach is the combination of the parametric M-estimation with kernel and local polynomial regression methods. The result is an iterative estimator that retains a linear structure, but has kernel weights also in the direction of the prediction errors. The design of smoothing coefficients is carried out with robust cross-validation criteria and rules of thumb. The method works well both to remove the influence of patches of outliers and to detect the local breaks and persistent structural change in time series.     

月度数据季节因素调整和预测

 内容提要：由于受气候条件、节假日、人们的风俗习惯、人口和国民经济增长等因素的影响,客运量呈现出周期性的增长趋势变化。为客运部门更好地安排客运计划,本文通过指数平滑法中的Holt-Winters模型将时间序列数据分解为季节波动和趋势波动。并对我国铁路、民航、水运和公路的2002-2009年的客运量数据进行拟合。结果表明,铁路和民航客运量数据具有明显的线性趋势和季节性特征,并进一步得出其波峰和波谷到达的时间;模型对铁路、民航、水运和公路客运量均有非常好的拟合效果,其平均绝对百分百误差（MAPE）依次为5.536%,7.49%、6.070%和3.633%。在此基础上对我国2010年各月份的客运量进行了科学预测。     

Kernel adjusted nonparametric regression

In this paper we propose and study a new kernel regression estimator in which the kernel is taken from a properly adapted location-scale family of the design distribution. We show that, while the original smoothing may be performed with sub-optimal bandwidths, adaptation of proper scale parameters yields overall optimal estimators. Unlike traditional smoothing methodology, our approach does not aim at estimating pivotal higher order derivatives.     

Thin plate regression splines

Summary. I discuss the production of low rank smoothers for d  ≥ 1 dimensional data, which can be fitted by regression or penalized regression methods. The smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to result in the minimum possible perturbation of the thin plate spline smoothing problem given the dimension of the basis used to construct the smoother. By making use of Lanczos iteration the basis change and truncation are computationally efficient. The smoothers allow the use of approximate thin plate spline models with large data sets, avoid the problems that are associated with 'knot placement' that usually complicate modelling with regression splines or penalized regression splines, provide a sensible way of modelling interaction terms in generalized additive models, provide low rank approximations to generalized smoothing spline models, appropriate for use with large data sets, provide a means for incorporating smooth functions of more than one variable into non-linear models and improve the computational efficiency of penalized likelihood models incorporating thin plate splines. Given that the approach produces spline-like models with a sparse basis, it also provides a natural way of incorporating unpenalized spline-like terms in linear and generalized linear models, and these can be treated just like any other model terms from the point of view of model selection, inference and diagnostics.     

ON SPLINE SMOOTHING WITH AUTOCORRELATED ERRORS

The generalised cross-validation criterion for choosing the degree of smoothing in spline regression is extended to accommodate an autocorrelated error sequence. It is demonstrated via simulation that the minimum generalised cross-validation smoothing spline is an inconsistent estimator in the presence of autocorrelated errors and that ignoring even moderate autocorrelation structure can seriously affect the performance of the cross-validated smoothing spline. The method of penalised maximum likelihood is used to develop an efficient algorithm for the case in which the autocorrelation decays exponentially. An application of the method to a published data-set is described. The method does not require the data to be equally spaced in time.     

A model-averaging approach for smoothing spline regression

ABSTRACT

This article considers nonparametric regression problems and develops a model-averaging procedure for smoothing spline regression problems. Unlike most smoothing parameter selection studies determining an optimum smoothing parameter, our focus here is on the prediction accuracy for the true conditional mean of Y given a predictor X. Our method consists of two steps. The first step is to construct a class of smoothing spline regression models based on nonparametric bootstrap samples, each with an appropriate smoothing parameter. The second step is to average bootstrap smoothing spline estimates of different smoothness to form a final improved estimate. To minimize the prediction error, we estimate the model weights using a delete-one-out cross-validation procedure. A simulation study has been performed by using a program written in R. The simulation study provides a comparison of the most well known cross-validation (CV), generalized cross-validation (GCV), and the proposed method. This new method is straightforward to implement, and gives reliable performances in simulations.     

Recursive and en-bloc approaches to signal extraction

In the literature on unobservable component models , three main statistical instruments have been used for signal extraction: fixed interval smoothing (FIS), which derives from Kalman's seminal work on optimal state-space filter theory in the time domain; Wiener-Kolmogorov-Whittle optimal signal extraction (OSE) theory, which is normally set in the frequency domain and dominates the field of classical statistics; and regularization , which was developed mainly by numerical analysts but is referred to as 'smoothing' in the statistical literature (such as smoothing splines, kernel smoothers and local regression). Although some minor recognition of the interrelationship between these methods can be discerned from the literature, no clear discussion of their equivalence has appeared. This paper exposes clearly the interrelationships between the three methods; highlights important properties of the smoothing filters used in signal extraction; and stresses the advantages of the FIS algorithms as a practical solution to signal extraction and smoothing problems. It also emphasizes the importance of the classical OSE theory as an analytical tool for obtaining a better understanding of the problem of signal extraction.