# Why and when stationarity is achieved by decomposition rather than differencing in ARIMA model

I would like to understand relationships between variables by which cross-correlation function, that means what is the extent one variable influence the other one.

ARMA model is used to fit two variables with time series structure. We know that the model requires the data to be stationary. Normally, we check the pattern by plotting the data. If a series is deemed nonstationary, we use differencing to make it stationary.

I also saw that there is a way to decompose the time series, and the residual time series from the seasonal decomposition is used as the starting point rather than the differenced series. See Worrall & Burt "Time series analysis of long-term river dissolved organic carbon records" (2004).

Any comments to help me understand the method will be appreciated.

Edited

I would like to spell the question with the example (Worrall.et.al 2004). Specifically, the step of the method and my understanding. If I were misunderstood, please feel free to correct me.

Step 1，decompose time series (TS) of color and TS of flow with function decompose() respectively, which to exact deflow$residuals, that the stationary of data is achieved. Step 2, derive ARMA model for deflow$residuals with function arma() , that is describe the behavior of the flow TS.

Step 3, the ARMA of color TS be used to filter the color TS, that is describe the behavior of the color TS. But I don't understand why do not directly derive ARMA model fitting decolor\$residuals? And which function for "filter"?

Step 4, calculating the residuals between of the floe and color with function cor(), which to represent how an unpredicted flow influences the color. Here confused, that the author expect influence on the color and why decompose color firstly?

Thanks for your patience upon my unmatured question.

• It is not obvious whether it is always possible to decompose a nonstationary time series into a finite number of stationary time series. For example, an integrated I(1) series cannot be additively decomposed into many I(0) series. It might be that after the original series is decomposed in its parts, at least one of the parts remains nonstationary. It would be easier to discuss your question given a particular example. However, you would need to spell it out (a title of a relevant article would generally not suffice). – Richard Hardy Apr 13 '15 at 18:16
• I have edited your post quite a bit, please check if I did not change the meaning. I hope it became easier to read. I still do not understand your first sentence. Also, if you mean seasonal decomposition in particular (rather than any decomposition), then normally the nonstationarity remains in the trend component, while the residual component is stationary. So the nonstationarity does not just disappear. Subject-matter consideration may lead to interest in the residual only; to that end the question is not purely statistical but depends on applications as well. – Richard Hardy Apr 13 '15 at 18:27
• Regarding seasonal decomposition itself, the idea is quite simple. The aim is to distinguish three unobservable components of the time series: a slowly moving trend component, a periodic seasonal component and a fast moving "noise", or residuals component. There are different methods to execute the decomposition, and they rely on different assumptions. Explaining these would take a long time. You could try looking in R documentation and the linked research papers, but that is time consuming. Perhaps someone else will suggest a shortcut. – Richard Hardy Apr 13 '15 at 18:32
• Thanks for your interpretation as well as let my expressions more understandable. I would like to spell the question with the example (Worrall.et.al 2004). Specifically, the step of the method and my understanding. If I were misunderstood, please feel free to correct me. – Rginner Apr 14 '15 at 8:30
• I doubt that many people would go and read the reference. That is very time consuming. Instead, you could present a "digested" version of it in your post. Then it is more likely that you will get help. Also, I encourage you to edit your first sentence; it is not clear what you mean there. – Richard Hardy Apr 14 '15 at 8:48