Some results on the variogram in time series analysis

Abstract

Data used for monitoring and control of industrial processes are often best modeled as a time series. An important issue is to determine whether such time series are stationary. In this article we discuss the variogram-a graphical tool for assessing stationarity. We build on previous work and provide further details and more general results including analytical structures of variogram for various non-stationary processes, and illustrate with a number of examples of variograms using standard data sets from the literature and simulated data sets.

Asymptotic confidence intervals for variograms of stationary time series

“…However, it can be shown that not every non-stationary process possesses a variogram. For instance, see 7 for an example of ARIMA process for which variogram is not defined.…”

“…. Further, Box and Luceno 4 and Khachatryan and Bisgaard 7 provide analytic expressions for the standardized variogram for ARIMA processes. Specifically, Khachatryan and Bisgaard 7 show that the standardized variogram is always defined for autoregressive integrated moving average processes ARIMA(p, d, q) for any value of p and q and for d ≤ 1.…”

PM 10 Time Series Analysis Through Geostatistical Techniques

“…Besides the common second-order moments used to describe the random process {X t , t ∈ T}, such as the autocovariance function and the autocorrelation function, the variogram can also be considered and even preferred with respect to covariance function [10,19].…”

“…In this context, the use of geostatistical techniques could also be convenient, nevertheless these techniques are usually applied to analyze, through the variogram, spatial relationships among sample data measured at some locations in a domain and to predict the corresponding spatial phenomena [6,18,22,29]. In particular, the variogram could represent a complementary exploratory tool for assessing stationarity in time series [2,19] and it has the considerable advantage that it is defined in much wider circumstances than the autocovariance and the autocorrelation. Moreover, this analytical tool is appropriate to identify trends and periodicity exhibited by the data and to obtain kriging predictions of the variable under study, either for temporal intervals with missing values (interpolation mode) and in time points after the last available data (extrapolation mode).…”

“…Different studies have suggested the use of geostatistical methods in time domain [7,19]. In particular, De Iaco et al [12] illustrated the role of variogram in this context for different purposes.…”