# Proper test for time series

I have monthly precipitation for one site over a 35 year period. I would like to test if the total precipitation for a specific year is equal to the mean total precipitation for the rest of the period of record. In addition, I would also like to split the data into wet season [Nov-April] vs dry season [May-Oct].

I realize there are likely issues to take into account regarding auto-correlation, but I am yet analyze these types of data personally.

Would someone please be kind enough to point me in the right direction?

EDIT: IrishStat, The data is below, I use R, so below was created using dput(). I could figure this out, but it would take me a while and I would feel more confident if I were shown once.

PPT <- structure(list(YEAR = 1965:2010, JAN = c(3.15, 4.81, 4.94, 2.49, 4.26, 4.31, 3.96, 5.06, 3, 3.58, 6.27, 1.79, 6.14, 14.43, 5.04, 6.22, 0.94, 3.23, 6.6, 5.74, 5.08, 3.8, 6.35, 4.38, 0.87, 8.66, 15.48, 13.08, 7.52, 5.02, 5.45, 5.23, 5.88, 16.33, 5.18, 2.95, 3.33, 4.91, 2.49, 5.75, 2.08, 2.62, 4.38, 7.02, 2.16, 6.73), FEB = c(5.95, 9.37, 6.09, 2.84, 3.87, 4.27, 6.72, 3.78, 7.09, 6.56, 3.97, 2.39, 1.87, 4.48, 7.73, 1.46, 11.59, 9.58, 11.61, 5.95, 7.03, 4.66, 8.96, 6.4, 1.64, 10.3, 2.4, 8.06, 4.29, 1.88, 2.84, 6.36, 6.52, 6.19, 0.82, 0.87, 3.56, 2.65, 7.03, 8.12, 5.6, 4.22, 2.07, 4.88, 4.86, 5.43), MAR = c(4.89, 2.58, 0.69, 1.98, 8.94, 7.17, 4.9, 7.12, 8.23, 6.88, 6.83, 9.95, 4.73, 5.79, 9.29, 11.6, 2.59, 7.84, 9.38, 5.58, 5.06, 3.91, 5.43, 6.36, 4.32, 10.41, 4.89, 2.9, 7.31, 7.83, 10.03, 10.2, 3.75, 6.11, 5.79, 4.12, 8.73, 4.18, 5.2, 0.68, 4.28, 0.39, 0.54, 4.32, 14.43, 5.16), APR = c(0.61, 2.45, 1.68, 2.25, 4.55, 1.94, 0.48, 2.29, 9.22, 5.08, 6.84, 1.96, 1.83, 4.76, 5.77, 13.53, 1.24, 2.9, 12.97, 3.19, 1.01, 3.33, 0.63, 3.71, 2.88, 2.47, 9.1, 2.72, 2.55, 4.74, 6.71, 11.68, 6.32, 4.54, 0.11, 1.22, 0.3, 3.12, 3.06, 2.32, 20.5, 6.09, 3.43, 5.49, 2.07, 2.04), MAY = c(0.42, 5.66, 2.86, 2.7, 9.89, 2.86, 4.53, 7.8, 2.63, 3.4, 11.11, 10.7, 2.8, 10.71, 5.5, 9.61, 11.88, 1.64, 1.54, 2.49, 7.85, 5.35, 12.35, 0.31, 7, 4.94, 13.75, 2.4, 5.81, 2.89, 6.04, 0.48, 7.96, 0.81, 3.15, 0.65, 0.55, 2.34, 5.84, 2.03, 7.08, 3.44, 1.86, 9.26, 7.3, 7.03), JUN = c(15.43, 2.34, 4.33, 3.79, 4.27, 9.77, 2.82, 4.42, 6.91, 3.01, 3.52, 4.19, 0.71, 12.32, 3.37, 3.81, 2.51, 7.53, 6.6, 2.8, 8.08, 5.54, 15.58, 5.91, 18.52, 6.22, 5.85, 4.62, 3.38, 7.32, 4.53, 7.42, 5.88, 2.23, 8.32, 4.21, 13.77, 3.4, 9.54, 10.8, 10.44, 1.33, 6.35, 3.32, 3.69, 5.04), JUL = c(13.72, 7.6, 3.69, 6.95, 10.88, 1.93, 6.88, 6.56, 7.67, 5.29, 10.03, 8.03, 5.04, 23.67, 8.4, 4.06, 5.05, 11.94, 0.82, 3.9, 8, 4.29, 5.16, 4.8, 8.92, 5.84, 8.64, 5.03, 8.08, 10.94, 8.65, 6.55, 28.58, 6.17, 9.71, 3.21, 13.69, 10.76, 18.44, 4.73, 11.43, 5.34, 7.09, 5.4, 5.18, 2.18), AUG = c(10.1, 8.51, 7.88, 3.57, 12.91, 11.84, 6.4, 4.14, 4.3, 6.75, 8.69, 2.08, 9.33, 6.24, 5.41, 1.31, 6.28, 6.03, 6.08, 14.23, 5.78, 5.01, 12.48, 13.19, 2.17, 2, 6.74, 6.2, 7.57, 6.49, 9.61, 5.99, 1.25, 5.9, 5.99, 3.02, 11.02, 5.83, 5.2, 8.28, 11.4, 7.13, 5.96, 14.14, 6.24, 10.33), SEP = c(10.74, 3.43, 6.37, 4.19, 5.52, 5.53, 11.07, 1.01, 10.13, 7.6, 12.88, 6.48, 7.53, 3.97, 9.45, 5.28, 1.11, 2.31, 7.02, 0.57, 6.01, 3.07, 2.4, 16.12, 4.49, 1.64, 3.13, 1.44, 4.99, 1.46, 3.05, 7.72, 1.29, 24.11, 2.27, 9.53, 4.66, 14.77, 3.66, 12.63, 4.7, 5.25, 6.64, 7.67, 8.16, 6.33), OCT = c(2.57, 3.84, 7, 0.38, 1.75, 7.1, 0, 2.77, 1.28, 0.56, 5.52, 6.81, 4.13, 0, 0.69, 3.77, 1.9, 2.22, 5.53, 1.78, 13.08, 5.89, 0.23, 1.97, 2.28, 2.88, 2.28, 3.85, 5.3, 5.88, 12.13, 2.93, 4.88, 1.72, 4.42, 0.39, 3.43, 8.44, 2.68, 2.44, 0.25, 3.76, 8.78, 4.14, 7.67, 1.68), NOV = c(0.89, 4.88, 0.58, 3.83, 0.8, 2.69, 2.29, 4.64, 3.59, 3.05, 8.83, 5.57, 6.73, 3.76, 7.55, 5.43, 0.81, 4.78, 6.84, 2.26, 2.98, 6.06, 5.07, 4.44, 7.76, 1.84, 3.8, 11.7, 1.8, 3.71, 11.63, 2.13, 12.03, 6.27, 1.02, 7.34, 1.78, 5.69, 3.84, 11.64, 3.22, 3.7, 3.91, 3.47, 3.82, 4.73), DEC = c(3.87, 4.92, 5.77, 6.2, 8.27, 5.21, 5.93, 7.51, 7.07, 3.58, 3.63, 4.95, 4.76, 3.93, 1.96, 1.67, 5.88, 8.04, 5.5, 1.7, 3.29, 4.62, 5.3, 2.47, 5.21, 2.37, 2.87, 5.28, 4.73, 2.05, 4.84, 6.14, 2.52, 5.35, 5.4, 3.49, 2.74, 4.87, 4.13, 5.58, 3.9, 4.51, 5.77, 3.34, 13.52, 2.72)), .Names = c("YEAR", "JAN", "FEB", "MAR", "APR", "MAY", "JUN", "JUL", "AUG", "SEP", "OCT", "NOV", "DEC"), class = "data.frame", row.names = c(NA, -46L))

• OK, sorry about the data format and if I did something incorrect with regards to me editing the original post. I have emailed you a CSV file. – Patrick Aug 31 '11 at 20:40

Patrick, You could add a series 0,0,0,0,0,....1,1,1,1,1,1,1,1,1,1,1,1, ,0,0,0,0, where the 1's reflect the year you are trying to test. This variable would be the X variable in an ARMAX model and it's significance would test the hypothesis of that year being exceptional. Be careful to include both ARMA structure and any Level Shifts / Trends / Pulses and Seasonal PUlses that might also be necessary. If you don't have the software/smarts to be able to do this post the data to the web and I will look at it and report back and perhaps suggest some ways for you to do this yourself.

There is no reason to split the data as a forecast for May may depend on April. A comprehensive analysis will allow you to test myths.

• :chi It appears that you deleted my post as being inappropriate. Please let me know at dave@autobox.com why you did this so as I will not make the "mistake" again . Thanks – IrishStat Aug 31 '11 at 12:46
• :chi . Now I see ..It looks like a nearly identical answer ! Thanks for pulling it. – IrishStat Aug 31 '11 at 12:58
• Thanks for the offer IrishStat; I accept if you can find the time (I added the data above). I am really interested in whether the period of time from March 2009 to June 2010 is extreme (wetter or drier) relative to the rest of the period of record. I really appreciate this; I will use your suggestions as a starting point for my self-education on the topic. – Patrick Aug 31 '11 at 14:34

This is a second answer because there is not enough room in a "comment" . The data is monthly ( starting 1965/1) http://www.autobox.com/patrick/patrick.xls . After a brief review of the data it was apparent that there was no strong seasonal structure ! Oftentime there may be particular/individual months of the year that may have repetitive patterns thus we added 11 dummies to test the JAN-NOV effects. Additionally the OP had wanted to test the hypothesis that the period between 2009/3 and 2010/6 had "been statistically unusual". We show a test of that hypthesis ( t=-1.58 )at .suggesting non-significance. We found two months of the year to have statistically significant effects: (March (1.02) and October (-1.04)) . Additionally there were a number of months that had one-time effects (Pulses) and three Level Shifts in the series earmarking points in time that the mean of the series had shifted.. these three points in time were

   1975/12   -1.4
1985/4    +2.1
1999/7    -1.9


The final plot of residuals visually suggests randomness :

Finally the acf of the final model residuals supported the hypothesis if randomness . The very small acf values appear to be statistically significant but this is an artifact of the large sample size (n=552) this the standard error of the acf is approximately 1/sqrt(552) or about .04.

The bottom line , Patrick , is there is no assignable cause to the special period and that there are two unusual months of the year and there are essentially 4 regimes ( 3 level shifts ) besides some very unusual and unpredictable values/readings using the history of the series months. Precipitation might depend on something else besides it's history !.

I think that this data set should be scrutinized by readers of the list. Such scrutiny can only confirm/test the analysis conducted here. Different readers have access to different tools and sometimes these lead to different results/findings.