Increment-vector methodology: Transforming non-stationary series to stationary series

In time series analysis, it is well-known that the differencing operator del(d) may transform a non-stationary series, {Z(t)} say, to a stationary one, {W(t) = del(d)Z(t)}; and there are many procedures for analysing and modelling {Z(t)} which exploit this transformation. Rather differently, Matheron (1973) introduced a set of measures on R-n that transform an appropriate non-stationary spatial process to stationarity, and Cressie (1988) then suggested that specialized low-order analogues of these measures, called increment-vectors, be used in time series analysis. This paper develops a general theory of increment-vectors which provides a more powerful transformation tool than mere simple differencing. The methodology gives a handle on the second-moment structure and divergence behaviour of homogeneously non-stationary series which leads to many important applications such as determining the correct degree of differencing, forecasting and interpolation.