Time-series decomposition in Mathematica

Based on @b.gatessucks answer and on @RahulNarain comment tip, I created this functions for the multiplicative decompose case. I changed @b.gatessucks method for seasonality to keep it closer from R method, and used TemporalData to easily handle time interval.

decompose[data_,startDate_]:=Module[{dateRange,plot,plotOptions,observedData,observedPlot,trendData,trendPlot,dataDetrended,seasonalData,seasonalPlot,randomData,randomPlot},

    dateRange={{startDate,1},Automatic,"Month"};

    (*Setting Plot Options*)
    plotOptions={AspectRatio -> 0.2,ImageSize-> 600,ImagePadding->{{30,10},{10,5}},Joined->True};

    (*Observed data*)
    observedData=TemporalData[data,dateRange]["Path"];
    observedPlot=DateListPlot[observedData,Sequence@plotOptions];

    (*Extracting trend component*)
    trendData=Interpolation[MovingAverage[observedData,12],InterpolationOrder->1];
    trendPlot=DateListPlot[{#,trendData[#]}&/@observedData[[All,1]],Sequence@plotOptions]//Quiet;
    dataDetrended=N@{#[[1]],#[[2]]/trendData[#[[1]]]}&/@observedData//Quiet;

    (*Extracting seasonal component*)
    seasonalData=Mean/@Flatten[Partition[N@dataDetrended[[All,2]],12,12,1,{}],{2}];
    seasonalData=TemporalData[PadRight[seasonalData,Length[data],seasonalData],dateRange]["Path"];
    seasonalPlot=DateListPlot[seasonalData,Sequence@plotOptions];

    (*Extracting random component*)
    randomData=TemporalData[dataDetrended[[All,2]]/seasonalData[[All,2]],dateRange]["Path"];
    randomPlot=DateListPlot[randomData,Sequence@plotOptions];


    (*Plotting data*)
    plot=Labeled[
            Grid[Transpose[{Rotate[Style[#,15,Bold],90\[Degree]]&/@{"observed","trend","seasonal","random"}
                          ,{observedPlot,trendPlot,seasonalPlot,randomPlot}}
                          ]
                ]
            ,Style["Decomposition of multiplicative time series",Bold,17]
            ,Top
            ]

]

Using the functions like this:

rawData = Import["http://www.massey.ac.nz/~pscowper/ts/cbe.dat"][[2 ;;, 3]];
decompose[rawData, 1958]

We get:

Almost exactly as in R!

I say "almost" because R don't use interpolation, so the MovingAverage lost 12 point in R that we don't lose in this function due to interpolation method. I prefer to keep the ticks in each plot, I find it's better to read. It's a question of personal options.

While you come back with a version 9 solution here is an old school approach :

The first entry is labels so I removed it :

rawData = Import["http://www.massey.ac.nz/~pscowper/ts/cbe.dat"][[2 ;;]];

Added the dates to the imported data : they are monthly dates starting {1958, 1, 1} :

data = Transpose[{NestList[DatePlus[#, {1, "Month"}] &, {1958, 1, 1}, Length[rawData] - 1], 
                  rawData[[All, 3]]}];

plotData = DateListPlot[data, Joined -> True, PlotLabel -> "Data"];

The trend is a 12 - point moving average, found by trial and error and common sense :

mA = 12;
trend = Interpolation[Transpose[{AbsoluteTime[#] & /@ data[[1 ;; -mA, 1]], 
                      MovingAverage[data[[All, 2]], mA]}], InterpolationOrder -> 0];

plotTrend =  DateListPlot[{#, trend[AbsoluteTime[#]]} & /@ data[[1 ;; -mA, 1]], PlotLabel -> "Trend"];

Since we're doing a multiplicative decomposition we divide the starting data by its trend :

dataDetrended = {#[[1]], #[[2]]/trend[AbsoluteTime[#[[1]]]]} & /@ data[[1 ;; -mA]];

Seasonality is just a sine with a 1-year period :

seasonality = NonlinearModelFit[{AbsoluteTime[#[[1]]], #[[2]]} & /@ dataDetrended, 
                 a + b Sin[2 \[Pi] t/(365.25 86400 ) + f], {a, b, f}, t];

plotSeasonality = DateListPlot[{#, seasonality[AbsoluteTime[#]]} & /@ data[[1 ;; -mA, 1]], Joined -> True, PlotLabel -> "Seasonality"];

The random part is what is left after diving the de-trended data by the seasonal factor.

random = {#[[1]], #[[2]]/seasonality[AbsoluteTime[#[[1]]]]} & /@ dataDetrended[[1 ;; -mA]];

plotRandom = DateListPlot[random, Joined -> True, PlotLabel -> "Random"];

Final result :

GraphicsColumn[{plotData, plotTrend, plotSeasonality, plotRandom}, ImageSize -> 350]

In my experience economists tend not to write time-series decomposition algorithms of the their own but rather use well established ones.

Among them two most well known are ARIMA-X13 and TRAMO-SEATS. Both are implemented by the US Census Bureau and executables are available here.

I've tried and implemented a simple package that calls the CB's files in the background and returns seasonally adjusted series. It very much follows the strategy that I learned by reading this answer on exporting multipage pdfs in mma on SE.

To use ARIMA-X13 in mma you should:

1. Download Economica package here and put it into your $UserBaseDirectory.
2. Download X13AS.EXE file from the CB's site and put it into "C:\Windows\System32"

Then you might extract the trend component of your series:

unemp = Rest@
 QuandlData[
  "ILOSTAT/UNE_TUNE_NB_SEX_T_AGE_AGGREGATE_TOTAL_Q_RUS", {{2001, 1, 
 1}, {2014, 1, 1}}]

unempSA = SeasonalAdjustmentX13[Sort@unemp]

DateListPlot[{unemp, unempSA}]

ARIMA-X13 allows also to extract cyclical component and noise, however I've yet to add them to that package.