Some Nonlinear Exponential Smoothing Models are Unstable

This paper discusses the instability of eleven nonlinear state space models that underly exponential smoothing. Hyndman et al. (2002) proposed a framework of 24 state space models for exponential smoothing, including the well-known simple exponential smoothing, Holt's linear and Holt-Winters' additive and multiplicative methods. This was extended to 30 models with Taylor's (2003) damped multiplicative methods. We show that eleven of these 30 models are unstable, having infinite forecast variances. The eleven models are those with additive errors and either multiplicative trend or multiplicative seasonality, as well as the models with multiplicative errors, multiplicative trend and additive seasonality. The multiplicative Holt-Winters' model with additive errors is among the eleven unstable models. We conclude that: (1) a model with a multiplicative trend or a multiplicative seasonal component should also have a multiplicative error; and (2) a multiplicative trend should not be mixed with additive seasonality.Exponential smoothing, forecast variance, nonlinear models, prediction intervals, stability, state space models.

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 which 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 three hundred 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.forecasting; time series; exponential smoothing; positive-valued processes; seasonality; state space models.

Photon phonon entanglement in coupled optomechanical arrays

We consider an array of three optomechanical cavities coupled either reversibly or irreversibly to each other and calculate the amount of entanglement between the different optical and mechanical modes. We show the composite system exhibits intercavity photon-phonon entanglement.Comment: Restructured paper after referee comments, Published versio

Invertibility Conditions for Exponential Smoothing Models

In this article we discuss invertibility conditions for some state space models, including the models that underly simple exponential smoothing, Holt's linear method, Holt-Winters' additive method and damped trend versions of Holt's and Holt-Winters' methods. The parameter space for which the model is invertible is compared to the usual parameter regions. We find that the usual parameter restrictions (requiring all smoothing parameters to lie between 0 and 1) do not always lead to invertible models. Conversely, some invertible models have parameters which lie outside the usual region. We also find that all seasonal exponential smoothing methods are non-invertible when the usual equations are used. However, this does not affect the forecast mean. Alternative models are presented which solve the problem while retaining the basic exponential smoothing ideas.exponential smoothing, invertibility, state space models.