In the previous lesson, you learned a bit about time series forecasting and loaded a dataset showing the fluctuations of electrical load over a time period.
In this lesson, you will discover a specific way to build models with [ARIMA: *A*uto*R*egressive *I*ntegrated *M*oving *A*verage](https://wikipedia.org/wiki/Autoregressive_integrated_moving_average). ARIMA models are particularly suited to fit data that shows [non-stationarity](https://wikipedia.org/wiki/Stationary_process).
- 🎓 **Stationarity**. From a statistical context, stationarity refers to data whose distribution does not change when shifted in time. Non-stationary data, then, shows fluctuations due to trends that must be transformed to be analyzed. Seasonality, for example, can introduce fluctuations in data and can be eliminated by a process of 'seasonal-differencing'.
- 🎓 **[Differencing](https://wikipedia.org/wiki/Autoregressive_integrated_moving_average#Differencing)**. Differencing data, again from a statistical context, refers to the process of transforming non-stationary data to make it stationary by removing its non-constant trend. "Differencing removes the changes in the level of a time series, eliminating trend and seasonality and consequently stabilizing the mean of the time series." [Paper by Shixiong et al](https://arxiv.org/abs/1904.07632)
- **AR - for AutoRegressive**. Autoregressive models, as the name implies, look 'back' in time to analyze previous values in your data and make assumptions about them. These previous values are called 'lags'. An example would be data that shows monthly sales of pencils. Each month's sales total would be considered an 'evolving variable' in the dataset. This model is built as the "evolving variable of interest is regressed on its own lagged (i.e., prior) values." [wikipedia](https://wikipedia.org/wiki/Autoregressive_integrated_moving_average)
- **I - for Integrated**. As opposed to the similar 'ARMA' models, the 'I' in ARIMA refers to its *[integrated](https://wikipedia.org/wiki/Order_of_integration)* aspect. The data is 'integrated' when differencing steps are applied so as to eliminate non-stationarity.
- **MA - for Moving Average**. The [moving-average](https://wikipedia.org/wiki/Moving-average_model) aspect of this model refers to the output variable that is determined by observing the current and past values of lags.
Open the [_/working_](https://github.com/microsoft/ML-For-Beginners/tree/main/7-TimeSeries/2-ARIMA/working) folder in this lesson and find the [_notebook.ipynb_](https://github.com/microsoft/ML-For-Beginners/blob/main/7-TimeSeries/2-ARIMA/working/notebook.ipynb) file.
Now your data is loaded, so you can separate it into train and test sets. You'll train your model on the train set. As usual, after the model has finished training, you'll evaluate its accuracy using the test set. You need to ensure that the test set covers a later period in time from the training set to ensure that the model does not gain information from future time periods.
1. Allocate a two-month period from September 1 to October 31, 2014 to the training set. The test set will include the two-month period of November 1 to December 31, 2014:
Since this data reflects the daily consumption of energy, there is a strong seasonal pattern, but the consumption is most similar to the consumption in more recent days.
Now, you need to prepare the data for training by performing filtering and scaling of your data. Filter your dataset to only include the time periods and columns you need, and scaling to ensure the data is projected in the interval 0,1.
> 🎓 What are all these parameters for? In an ARIMA model there are 3 parameters that are used to help model the major aspects of a time series: seasonality, trend, and noise. These parameters are:
`p`: the parameter associated with the auto-regressive aspect of the model, which incorporates *past* values.
`d`: the parameter associated with the integrated part of the model, which affects the amount of *differencing* (🎓 remember differencing 👆?) to apply to a time series.
`q`: the parameter associated with the moving-average part of the model.
> Note: If your data has a seasonal aspect - which this one does - , we use a seasonal ARIMA model (SARIMA). In that case you need to use another set of parameters: `P`, `D`, and `Q` which describe the same associations as `p`, `d`, and `q`, but correspond to the seasonal components of the model.
Selecting the best values for an ARIMA model's parameters can be challenging as it's somewhat subjective and time intensive. You might consider using an `auto_arima()` function from the [`pyramid` library](https://alkaline-ml.com/pmdarima/0.9.0/modules/generated/pyramid.arima.auto_arima.html),
To evaluate your model, you can perform the so-called `walk forward` validation. In practice, time series models are re-trained each time a new data becomes available. This allows the model to make the best forecast at each time step.
Starting at the beginning of the time series using this technique, train the model on the train data set. Then make a prediction on the next time step. The prediction is evaluated against the known value. The training set is then expanded to include the known value and the process is repeated.
> Note: You should keep the training set window fixed for more efficient training so that every time you add a new observation to the training set, you remove the observation from the beginning of the set.
This process provides a more robust estimation of how the model will perform in practice. However, it comes at the computation cost of creating so many models. This is acceptable if the data is small or if the model is simple, but could be an issue at scale.
Walk-forward validation is the gold standard of time series model evaluation and is recommended for your own projects.
> [MAPE](https://www.linkedin.com/pulse/what-mape-mad-msd-time-series-allameh-statistics/) is used to show prediction accuracy as a ratio defined by the above formula. The difference between actual<sub>t</sub> and predicted<sub>t</sub> is divided by the actual<sub>t</sub>. "The absolute value in this calculation is summed for every forecasted point in time and divided by the number of fitted points n." [wikipedia](https://wikipedia.org/wiki/Mean_absolute_percentage_error)
Dig into the ways to test the accuracy of a Time Series Model. We touch on MAPE in this lesson, but are there other methods you could use? Research them and annotate them. A helpful document can be found [here](https://otexts.com/fpp2/accuracy.html)
This lesson touches on only the basics of Time Series Forecasting with ARIMA. Take some time to deepen your knowledge by digging into [this repository](https://microsoft.github.io/forecasting/) and its various model types to learn other ways to build Time Series models.
