If I were you, I would question why we haven’t discussed AR or MA in isolation before turning to S/ARIMA. This is due to the fact that ARIMA is merely a combination of them, and you can quickly convert an ARIMA to an AR by offsetting the parameter for the MA and vice versa. Before we get started, let’s take a brief look back. We have seen how the ARIMA model operates and how to manually determine the proper method for determining the model parameters through trial and error. I have two questions for you: is there a better way to go about doing this? Secondly, we have seen models like AR, MA, and their combination ARIMA, how do you choose the best model?
Simply put, an MA model uses past forecast errors (residuals) to predict future values, whereas an AR model uses past values of the time series to do so. It assumes that future values are a linear combination of past values, with coefficients representing the weights of each past value. It makes the assumption that the future values are the linear sum of the forecast errors from the past.
Choosing Between AR and MA Models:
Understanding the type of data is necessary in order to select between AR and MA models. An AR model could be appropriate if the data shows distinct trends, but MA models are better at capturing transient fluctuations. Model order selection entails temporal dependency analysis using statistical tools such as the Partial AutoCorrelation Function (PACF) and AutoCorrelation Function (ACF). Exploring both AR and MA models and contrasting their performance using information criteria (AIC, BIC) and diagnostic tests may be part of the iterative process.
A thorough investigation of the features of a given dataset, such as temporal dependencies, trends, and fluctuations, is essential for making the best decision. Furthermore, taking into account ARIMA models—which integrate both AR and MA components—offers flexibility for a variety of time series datasets.
In order to produce the most accurate and pertinent model, the selection process ultimately entails a nuanced understanding of the complexities of the data and an iterative refinement approach.
Let’s get back to our data from “Analyze Boston”. To discern the optimal AutoRegressive (AR) and Moving Average (MA) model orders, Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots are employed.
Since the first notable spike in our plots occurs at 1, we tried to model the AR component with an order of 1. And guess what? After a brief period of optimism, the model stalled at zero. The MA model, which had the same order and a flat line at 0, produced results that were comparable. For this reason, in order to identify the optimal pairing of AR and MA orders, a comprehensive search for parameters is necessary.
As demonstrated in my earlier post, these plots can provide us with important insights and aid in the development of a passable model. However, we can never be too certain of anything, and we don’t always have the time to work through the labor-intensive process of experimenting with the parameters. Why not just have our code handle it.
To maximise the model fit, the grid search methodically assessed different orders under the guidance of the Akaike Information Criterion (AIC). As a result of this rigorous analysis, the most accurate AR and MA models were found, and they were all precisely calibrated and predicted. I could thus conclude that (0,0,7) and (8,0,0) were the optimal orders for MA and AR respectively.
