پیش بینی سری زمانی ترکیبی: یک روش فضای حالت  Forecasting compositional time series: A state space approach

1. Introduction The need for forecasts of proportions arises in a wide variety of areas, such as business, economics, demography, and political science. Specific examples include the market shares of competing products, the proportions of jobs in different sectors of the economy, and the age composition of a population. In some cases, only measurements of the proportions are available; in others, such as mar-ket share data, both total sales and proportions are available, although the analysis typically focuses on the latter. Statistical methods for the analysis of data on proportions are known as compositional time series methods, and one essential component is some form of transformation to ensure that the specified random variables are non-negative and defined on a simplex so that they sum to one. The monograph by Aitchison (1986) is a key reference for compositional data analysis. His analysis shows that we can draw on a wide variety of established and well-understood statistical methods by using the log-ratio transformation to map the proportions onto the real line. This transformation has become relatively standard over the years for both cross-sectional and time series analysis (Aitchison & Egozcue, 2005; Brundson & Smith, 1998; Quintana & West, 1988), and is adopted in our paper. Akey advantage of the transformation is that it enables us to develop the analysis on the whole real line, allowing parameter estimation to be based on least squares (minimum generalized variance) arguments. Of course, the construction of prediction intervals requires explicit distributional assumptions, with the multivariate normal distribution being the most common choice. Once the log-ratio transformation has been applied, other forms of analysis become feasible, such as functional data analysis (cf. Hyndman & Booth, 2008), which enables the consideration of more general time-dependent mean and variance structures. This paper does not consider this possibility or the other approaches in the literature, based on the Dirichlet distribution (Grunwald, Raftery, & Guttorp, 1993) or the hyper-spherical transformation (Mills, 2010). The novel feature of this paper is its coupling of the logratio transformation with linear innovations state space models and the associated technique of vector exponential smoothing (De Silva, Hyndman, & Snyder, 2009, 2010; Hyndman, Koehler, Ord, & Snyder, 2008). The rationale for this shift in emphasis lies in the nature of market share data, where new brands may be launched or old brands fade away. State space models can be structured easily to allow for the intrinsically non-stationary nature of a start-up, so that varying numbers of series (‘‘births’’ and ‘‘deaths’’) may be considered. In contrast, vector ARIMA models (Barceló-Vidal, Aguilar, & Martín-Fernández, 2011; Brundson & Smith, 1998) assume that series are stationary after suitable differencing, and can notionally extend to the ‘‘infinite past’’. These authors do not consider births or deaths.