Time Series Models for Distributional Data in Bayes Space
To solve the problem of non-closed linear operations in distributional data,this paper proposes time series models for distributional data in Bayes space.Under the framework of symbolic data analysis,distributional data are known as numerical modal data whose realizations can be histograms,empirical distributions or empirical estimates of parametric distributions.Since the elements of the data table are probability density functions,cumulative distribution functions or quantile functions which carry information with constraints,standard methods are not appro-priate for their statistical processing.In this paper,the specific features of density functions are accounted for in Bayes space whose linear operations are closed,and the space of density functions form a complete inner product space with good alge-braic properties.To build up a concise methodology for distributional time series,numeric characteristics of distributional time series,the difference operator and the lag operator are first defined by linear operations and inner products of probability density functions in Bayes space.Furthermore,the methods for model specification and parameter estimation of the distributional AR model,MA model,ARMA model and the distributional ARIMA model are deduced with a complete modelling scheme.Finally,two series of simulation experiments and a real data analysis demonstrate the usefulness and effectiveness of the proposed methods for distributional time series.
Symbolic datadistributional dataBayes spacetime series
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