Editing Linear trend estimation (section)

==Advanced models==
Thus far, the data have been assumed to consist of the trend plus noise, with the noise at each data point being [[independent and identically distributed random variables]] with a normal distribution. Real data (for example, climate data) may not fulfill these criteria. This is important, as it makes an enormous difference to the ease with which the statistics can be analyzed so as to extract maximum information from the data series. If there are other non-linear effects that have a [[correlation]] to the independent variable (such as cyclic influences), the use of least-squares estimation of the trend is not valid. Also, where the variations are significantly larger than the resulting straight line trend, the choice of start and end points can significantly change the result. That is, the model is mathematically [[Statistical model specification|misspecified]]. Statistical inferences (tests for the presence of a trend, confidence intervals for the trend, etc.) are invalid unless departures from the standard assumptions are properly accounted for, for example, as follows:

*Dependence: autocorrelated time series might be modelled using [[autoregressive moving average model]]s.
*Non-constant variance: in the simplest cases, [[weighted least squares]] might be used.
*Non-normal distribution for errors: in the simplest cases, a [[generalised linear model|generalized linear model]] might be applicable.
*[[Unit root]]: taking first (or occasionally second) differences of the data, with the level of differencing being identified through various unit root tests.<ref name=":2">{{cite book|url=https://www.otexts.org/fpp/8/1|title=Forecasting: principles and practice|date=20 September 2014|access-date=May 17, 2015}}</ref>
In [[R (programming language)|R]], the linear trend in data can be estimated by using the 'tslm' function of the 'forecast' package.