What are we trying to predict with ARIMA if we remove non-stationarity in data I am beginning to learn time series analysis and I read that for ARIMA models, one needs to have a stationary process...that would mean removing periodicities and trends in the data....but isn't that what we are trying to predict in the first place?
 That is, if there is a periodicity, i need to be able to use that fact in my prediction and my prediction should reflect that periodicity.  Similarly for any non stationary trend.
 If we remove the periodicities and trends by differencing or standardizing what are we predicting then in the analysis...what is the utility of such predictions where the underlying trends in the data are not utilized?
 What are the popular algorithms that can best handle non-stationary processes?(other than neural networks)
Thanks much
 A: In order to predict the observed series .......
The goal of ARIMA modeling is to separate the observed data to signal and noise 
.....this flowchart is useful to understand the why's and wherefores of ARIMA MODELLING.
https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf
The noise component is what has to be stationary i.e. invariant with constant expectation and variance. if not "THEN REMEDIAL ACTION IS REQUIRED TO MAKE IT SO OR APPROXIMATELY SO" by adding or subtracting structure/components.  
As the flow diagram indicates deterministic structure e.g. seasonal pulses , level/step shifts , time trends and depending upon the type of data daily effects, holiday effects et al need to be identified and used to adjust/condition the original data into a set of values that will be subsequently analyzed for auto-regressive structure. 
Quoting the reflection by AdamO  Interrupted Time Series Analysis - ARIMAX for High Frequency Biological Data? ..."The correlogram (ACF/PACF) should be calculated from residuals using a model that controls for intervention administration, otherwise the intervention effects are taken to be Gaussian noise, underestimating the actual autoregressive effect."
The aforementioned adjustments allow one to compute the acf and pacf ( in reality regression and partial regression coefficients) statistics in order to n impute the ARIMA model form via pattern recognition (decay,cutoff etc ) .
This is an iterative process much like peeling back an onion and correctly done can lead to a useful model. Now the forecast equation is used to project forward based not only on any needed deterministic structure BUT the stochastic ARIMA structure.
Hope this helps ....
