## 2 Basic Principles and Concepts in Time Series Analysis
### ARIMA Time Series Modeling  
In 1970, [PERSON] and [PERSON] [7] popularized the _autoregressive integrated moving average_ (ARIMA) model, which is based on the general _autoregressive integrated moving average_ (ARMA) model described by [PERSON] [24].  
\[y_{t}=\sum_{i=1}^{p}\phi_{i}y_{t-i}+\sum_{j=1}^{q}\theta_{j}w_{t-j}+w_{t} \tag{2}\]  
In Equation (2), the first term represents the autoregressive (AR) part, and the second term represents the moving average (MA) part. The \(w_{k}\) terms represent white noise. Typically, \(w_{k}\) are assumed to be random variables that come from a normal distribution \(N(0,\sigma_{w}^{2})\). The parameters \(\phi_{i}\) and \(\theta_{j}\) are estimated from the historical values of the time series. We can estimate \(w_{t}\) at time \(t_{k}\) as \(\hat{w}_{t_{k}}=y_{t_{k}}-\hat{y}_{t_{k}}\).  
The ARIMA model is an improvement on the ARMA model for dealing with non-stationary time series with trends. The SARIMA model is an extension of the ARIMA model for dealing with data with seasonal patterns; for more information, please consult [PERSON] and [PERSON] [21].  
Models from the ARIMA family suppose that the time series is generated from linear, time-invariant processes; this assumption is not valid for many situations. Nevertheless, up to now, the ARIMA model and its variants have continued to be very popular; for instance, they are used by most official statistical agencies around the world as an essential part of their modeling strategy when working with macroeconomic or ecological/environmental temporal data. The popularity of ARIMA modeling stems from the following facts:  
1. Linear modeling is always at the forefront of the literature;
2. Linear models are easy to learn and implement;
3. Interpretation of results coming from linear models relies on well-defined, well-developed, standardized, mechanized procedures with solid theoretical foundations (e.g., there are established procedures that help us build confidence intervals associated with point forecasts, founded on statistical and probabilistic theory centered around the normal distribution).  
On the other hand, when phenomena require the investigation of alternative nonlinear time series models, it is useful to consider the following comment by [PERSON]:  
_Since all models are wrong, the scientist cannot obtain a \"correct\" one by excessive elaboration; on the contrary, following [PERSON], he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist, so over-elaboration and over-parametrization is often the mark of mediocrity._  
([PERSON] [25], 1976)  
Occam's razor (a principle also known as parsimony) is often used to avoid the danger of overfitting the training data, that is, to choose a model that perfectly fits the time series data but is very complex and hence often does not generalize well. Time series analysis sometimes follows this principle in that many ARIMA or SARIMA models perform increasingly worse as the number of historical values used to forecast \(y_{t+1}\) increases [26].