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I am working on measuring variability of geotechnical data. I see in the literature it is mentioned that, non-stationary data should be first converted into stationary data (for example by trend removal), so that further statistical analysis can be done on the detrended stationary data. Also, it is mentioned that most of the statistical tools assume that the data is stationary, that is why it is important to make the non-stationary data stationary. My question is exactly where is this stationarity assumption is required, i.e., if my data is non-stationary and I still use statistical analysis which result is going to be in error? For example I'm finding the correlation coefficient of the data. If I go on using the non-stationary data will I get wrong value? Why?

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    $\begingroup$ I don't have time for a full response right now, but the basic issue is akin to an omitted variables problem. $\endgroup$ – gregmacfarlane Apr 8 '13 at 14:21
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    $\begingroup$ So that you can be assured that the parameters you are trying to estimate are not changing over time, for then, what would parameter estimates mean? $\endgroup$ – tchakravarty Apr 8 '13 at 18:49
  • $\begingroup$ That depends on your model. Sometimes non-stationarity is built in the model (see ARIMA vs ARMA). Here are also good answers: stats.stackexchange.com/questions/19715/… $\endgroup$ – Marius Hofert Jan 14 at 13:45
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First let's decide what form of stationarity you are asking about. There are two types:

(1) strict stationarity: All aspects of a time series behavior are not dependent on time. i.e. for every m & n the distribtions of $\newcommand{\Cov}{\operatorname{Cov}}Z_t, Z_{t1}, \dots, Z_{t+m+n}$ are the same.

(2) weak stationarity (sometimes called covariance stationary): if $\mu, \sigma^2,\gamma$ are unchanged by shifts in time.

-strict stationarity and weak stationarity are equivalent for Gaussian processes, since a normal distribution is uniquely characterized by its first two moments.

Let's assume that we're talking about weak stationarity here because you seem to be asking about correlation.

Formulas for covariance:

$\gamma = \Cov(Z_t,Z_{t+k})$ = $E[Z_t -\mu]E[Z_{t+k}-\mu]$

$\rho = \frac{\Cov(Z_t,Z_{t+k})}{\sigma_{Z(t)}\sigma_{Z(t+k)}}$

So covariance and correlation are both functions of the mean and variance.

So for example, if the series is consistently increasing over time, the sample mean and variance will grow with the size of the sample, and they will always underestimate the mean and variance in future periods. And if the mean and variance of a series are not well-defined, then neither are its correlations with other variables. For this reason you should also be cautious about trying to extrapolate regression models fitted to nonstationary data.

Most statistical forecasting methods are based on the assumption that the time series can be rendered approximately stationary (i.e., "stationarized") through the use of mathematical transformations. A stationarized series is relatively easy to predict: you simply predict that its statistical properties will be the same in the future as they have been in the past!

I stole the last two paragraphs from this article from Duke University's website. They described it better than I could have.

http://people.duke.edu/~rnau/411diff.htm

There are multiple methods for making a time series stationary, which include detrending and first differencing. You can find many descriptions of both on websites. The Duke U link above explains first differencing. Forecasts and means/variances can be made with the detrended model, and then the model can be converted back to the original.

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  • $\begingroup$ Dear Clark, Thanks so much for taking the time to answer my query. $\endgroup$ – Naveed Apr 8 '13 at 20:54
  • $\begingroup$ I believe I understand when there is an increasing or decreasing trend the mean is changing, and to deal with forecasting related works we don't want that. However, for my work we are not dealing with forecasting at all. We have thousands of data on a very small depth interval and we want to find the correlation length of the data. We don't worry about how the data would forecast. We have a fixed amount of data. So, now even if the data is non-stationary, the mean is NOT meaningless. However, still in literature people are detrending to make data stationary. I wonder Why? $\endgroup$ – Naveed Apr 8 '13 at 21:05
  • $\begingroup$ To get to the correlation length, I first need to calculate correlation coefficient. $\endgroup$ – Naveed Apr 8 '13 at 21:08
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I am new in Time series analysis (TSA), but my basic understanding is that the main aim of TSA is forecasting. A time series is made up of several components: Trend, Seasonal, cyclical, and irregular. By transforming or 'stationarizing' the series, its statistical properties (mean, variance) are easily forecasted as they remain fixed. The trend and seasonal components remain unchanged. The analysis is then mainly concentrated on forecasting the irregular component. Feel free to correct me if I am wrong.

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