I have a time series which has a very strong upward trend for the first half, then very strong downward for the second half and finishes pretty much back where it started. Should I split the data in two for analysis - or can I still account for a trend which is net neutral?
I think you can actually fit one model that can capture both upward and downward trends and there is no need to split your data (unnecessary) resulting in two models. In the following codes I have consiered an ARIMA model (red line) as well as plynomial regression of degree 2 (blue line). I fitted these model just as an example. Sometimes you actually need to split your data (See Nick's comment).
> x=0:40 > y=-(x-20)^2+rnorm(length(x),0,15) > plot(x,y) > library(forecast) > f=auto.arima(y) > summary(f) Series: y ARIMA(0,2,2) Coefficients: ma1 ma2 -1.4522 0.7282 s.e. 0.1240 0.1156 sigma^2 estimated as 451.2: log likelihood=-175.89 AIC=357.78 AICc=358.46 BIC=362.77 Training set error measures: ME RMSE MAE MPE MAPE MASE Training set -6.162264 20.71692 16.32124 30.21502 52.33051 0.1559912 > lines(fitted(f),col="red") > f2=lm(y~x+I(x^2)) > summary(f2) Call: lm(formula = y ~ x + I(x^2)) Residuals: Min 1Q Median 3Q Max -40.465 -8.068 2.406 9.482 23.517 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -401.88742 6.64150 -60.51 <2e-16 *** x 39.38738 0.76813 51.28 <2e-16 *** I(x^2) -0.97488 0.01857 -52.51 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 14.87 on 38 degrees of freedom Multiple R-squared: 0.9864, Adjusted R-squared: 0.9857 F-statistic: 1381 on 2 and 38 DF, p-value: < 2.2e-16 > lines(fitted(f2),col="blue") >
Such time series are usually hard to model using ARIMA. But I suggest you plot the ACf and PACF of the series and see. It may require differencing. You can also test to see if the variance is constant; if it is not you transform. Once the data is stationary you can make sense out of it.
As Nick has said splitting the series will generate two different processes and more information is needed.
The series might appear to be oscillating: if that's the case the best thing to do is to try spectral analysis.
I have had significant success in empirically identifying trend changes using AUTOBOX http://www.autobox.com/cms ,a commercially available piece of software that I have helped develop. You might want to look at stochastic vs deterministic trend/seasonality in time series forecasting for an interesting discussion.
A time trend model (deterministic in form) is as follows y(t)=a+bx1+cx2
etc where x1=1,2,3,4....t and x2=0,0,0,0,0,1,2,3,4 thus one trend applies to observations 1−t and a second trend applies to observations 6 to t.
AUTOBOX's automatic empirical procedures have been very successful BUT like all "new science" or "advanced innovative procedures" it needs to be constantly aggressively challenged. Test all things but hold fast to what you know to be true ! Please post your data or send it to me privately and I will use the data and report the results to the group. If you are skittish about releasing the data simply code it by adding/subtracting a constant. The procedures used to identify the number of trends and the length of each trend is based partially on the work of Tsay http://www.unc.edu/~jbhill/tsay.pdf. All of these advanced procedures are not currently available in the free software market.