arimatime series
I am new to time series analysis, and I amm trying to understand the concepts behind this analysis.
But I am still confused about the methods used to do such analyss, and what is the difference between the decomposition of time series or using an ARIMA model.
Thank you in advance.
Matt, You are very right in the concerns that you have raised with respect to using unnecessary differencing structure . In order to identify an appropriate model 
for your data yielding significant structure while rendering a Gaussian Error process 
with an ACF of 
the Transfer Function Identification modelling process requires ( in this case ) suitable differencing to create surrogate series that are stationary and thus usable to IDENTIFY the relationshop. In this the differencing requirements for IDENTIFICATION were double differencing for the X and single differencing for the Y. Additionally an ARIMA filter for the doubly differenced X was found to be an AR(1). Applying this ARIMA filter ( for identification purposes only ! ) to both stationary series yielded the following cross-correlative structure .
suggesting a simple contemporaneous relationship.
. Note that while the original series exhibit non-stationarity this does not necessarily imply that differencing is needed in a causal model. The final model 
and final acf support this 
. In closing the final equation aside from the one empirically identified level shifts ( really intercept changes ) is
Y(t)=-4.78 + .192*X(t) - .177*X(t-1) which is NEARLY equal to
Y(t)=-4.78 + .192*[X(t)-X(t-1)] which means that changes in X effect the level of Y
Finally note the characteristics of the suggested model.
the Level Shift series (0,0,0,0,0,0,0,0,0,1,1,.........,1) suggests if left untreated the model residuals would exhibit a level shift at or around time period 10 THUS a test of the hypothesis of a common residual mean between the first 10 residuals and the last 42 would be significant at alpha=.0002 based upon a "t test of -4.10" . Note that the inclusion of a constant guarantees that the overall mean of the residuals does not differ significantly from zero BUT this is not necessarily for all subset time intervals. The following graph clearly shows this ( given that you were told to look ! ).The Actual/Fit/Forecast is quite illuminating 
. Statistics are like lampposts, some use them to lean on others use them for illumination.
Here's an open source ARIMA implementation. It's under GNU license. Not sure if they would mind you learning from it!
It's an ARIMA implementation in C++, windows executable and source available. I think going through it you should be able to learn ARIMA the quick way.
Best Answer
Time series decomposition generally involves partitioning a signal into seasonal, trend, residual and sometimes level, holiday etc. components, which assumes additive or multiplicative relationships.
seasonal_decomposemethod instatsmodel.tsalibrary is a simple application of this. A more advanced version, which applies Box-Cox transformation beforehand automatically and accepts multiple seasonality frequencies, ismstlin R. Facebook's Prophet also employs seasonal decomposition in itself. And, usually, a ARMA model on residuals followed by a decomposition step is a typical approach in analysis.In ARIMA, there isn't a decomposition of such type. It's a generalization of ARMA models, in which we first difference the series and fit an ARMA model. The differencing step is applied to make the signal more stationary, by eliminating trend and seasonality components. ARIMA models considering seasonality are called SARIMA, but it also doesn't involve decomposition.