3 Trend Estimation
3.1 Time series replicates
Consider the trend estimation problem when the observations are serially correlated, i.e., when one has time series data. Such data are very common in many areas of applications, and examples include climate, geophysics, ecology, engineering, medicine, economics, and so on. A time series is generated when a process is monitored over time and observations are recorded. The most important feature of such data is that the observations are serially correlated. In some cases, the observed series may be treated as a time series, as, for instance, in the case of observations from deep ice cores or from a horizontal track in an ecological study.
More generally, several series may be available, as, for instance, in Figure 3.2a–c, and the problem may be to obtain an estimate of the common trend. Thus suppose that k series are available, the length of each series being n. The problem is estimation of the common mean. When one has exactly one series, then k = 1 (see, for instance, Figure 3.1). To formulate the problem, one writes down a nonparametric regression model, where the regression function is the trend function of interest. Statistical properties of the error term are decisive of the properties of the estimated trend.
Figure 3.1 Yearly means of daily precipitation totals (05:40 am–05:40 am following day in mm) in Arosa (1840 m asl; ...