# When to use Fourier ARIMA models

Are you wondering whether you should use a Fourier ARIMA model for your next time series project? Or maybe you want to learn more about the advantages and disadvantages of Fourier ARIMA models? Well then you are in the right place! In this article, we tell you everything you need to know to understand when to use Fourier ARIMA models.

We will start out by discussing what types of datasets Fourier ARIMA models are used for. After that, we will discuss the main differences between Fourier ARIMA models and standard ARIMA models. Next, we will discuss some of the main advantages and disadvantages of Fourier ARIMA models. Finally, we will provide examples of situations where you should and should not use Fourier ARIMA models.

Note. In this article, we are specifically discussing the practice of introducing Fourier terms into an ARIMA model as explanatory variables to model seasonal effects. We want to clarify this as there are other types of models that combine elements of Fourier analysis with ARIMA models.

## What kind of data should you use Fourier ARIMA models for?

What kind of datasets are Fourier ARIMA models used for? Fourier ARIMA models are used when you have time series data. Time series data is data where you have repeated measurements that are taken on the same quantity at regular intervals. For example, if you measured the temperature at the top of the Empire State Building every morning at 9am, this would be an example of time series data. You would have data on the same quantity (temperature) that is measured at regular intervals (every 24 hours) over a prolonged period of time.

One important thing to note about ARIMA models is that they can also handle covariate data. That means that this is a good family of models to look into if you have a dataset with both repeated measurements on both your outcome variable and repeated measurements on covariates that are related to your outcome variable. This is not necessarily true of all models that are used on time series data. There are many time series models that are not capable of handling covariate data.

## Differences between ARIMA models and Fourier ARIMA models

What are the main differences between standard ARIMA models and Fourier ARIMA models? The main differences between standard ARIMA models and Fourier ARIMA models is that while standard ARIMA models can only incorporate information on a single seasonal pattern, Fourier ARIMA models can incorporate information on multiple different seasonal patterns. For example, if you had a dataset with daily measurements that had both weekly seasonality and yearly seasonality, then you might opt for a Fourier ARIMA model over a standard ARIMA model.

What are the main advantages and disadvantages of Fourier ARIMA models? In this section, we will discuss some of the main advantages and disadvantages of Fourier ARIMA models. This will provide useful context that will help you understand when Fourier ARIMA models should and should not be used.

### Advantages of Fourier ARIMA models

What are some of the main advantages of Fourier ARIMA models? Here are the main advantages of Fourier ARIMA models compared to other time series models.

• Can handle multiple seasonality natively. One of the main advantages of Fourier ARIMA models is that they can handle multiple seasonality. This is a great option to reach for if you have a dataset that you suspect has multiple different seasonal patterns in it. Additional seasonal patterns in Fourier ARIMA models are introduced as covariates that are presented to the model, so there is no limit to the number of seasonal patterns you can capture.
• Can handle covariates. Another advantage of Fourier ARIMA models is that like traditional ARIMA models, they are able to handle covariate information. This is a very important quality if you have a dataset with multiple covariates that need to be incorporated into the model.
• Flexible model specification. Another advantage of Fourier ARIMA models is that like traditional ARIMA models, they allow for flexible model specification. That means that the models can be adapted to be used on many different types of time series datasets.
• Reliable performance. ARIMA family models generally have more reliable and consistent performance than some other time series models. That means that even if ARIMA models do not produce the absolute best performance possible, you can generally rely on them to do reasonably well in situations where all model assumptions are met. The same is true of Fourier ARIMA models.
• Can handle missing data. ARIMA family models are also generally able to accommodate missing data.

### Disadvantages of Fourier ARIMA models

What are some of the main disadvantages of Fourier ARIMA models? Here are some of the main disadvantages of Fourier ARIMA models compared to other time series models.

• Not as broadly known as standard ARIMA models. One of the main disadvantages of Fourier ARIMA models is that they are not as common or well understood as standard ARIMA models. That means that it may be more difficult to find a collaborator who can give you meaningful feedback on your model. It also means that stakeholders who are skeptical of methods they are not familiar with may provide more pushback.
• Require larger datasets than standard ARIMA models. ARIMA family models in general can be trained on smaller datasets than models that make use of complex neural network architectures. That being said, Fourier ARIMA models generally require larger sample sizes than standard ARIMA models. This is especially true if you are including seasonal patterns that span over a long period of time.
• Slightly less interpretable than standard ARIMA models. While ARIMA models are generally fairly interpretable, the Fourier terms that are used in Fourier ARIMA models can be a little more difficult to interpret. This means that Fourier ARIMA models may be a little less interpretable than standard ARIMA models.
• Can struggle with mean shifts. Like standard ARIMA models, Fourier ARIMA models can struggle in situations where there are large mean shifts in your data. If your data has large mean shifts, you may be better off looking into other options.
• Only intended for univariate time series. Like standard ARIMA models, Fourier ARIMA models are only intended to be used in situations where you have a single time series you want to model. If you have multiple time series that you want to model simultaneously, you may be better off looking into univariate time series models.
• Can not model nonlinear dependencies over time. Like standard ARIMA models, Fourier ARIMA models struggle to model nonlinear dependencies over time. For example, if the relationship between the current value of your outcome and the future value of your outcome is quadratic, ARIMA models may struggle to capture this relationship.
• Sensitive to outliers. Like standard ARIMA models, Fourier ARIMA models can be sensitive to outliers.
• Specifying parameters is more of an art than a science. As with standard ARIMA models, specifying the parameters of a Fourier ARIMA model is more of an art than a science. This is especially true when you consider the additional complexity introduced by the Fourier terms. This means that you will generally need to spend some time tuning your model and ensuring that it has an appropriate fit.
• Can have high time complexity. As with standard ARIMA models, Fourier ARIMA models can have high time complexity depending on the model specification. The amount of time and computational resources that are required to train these types of models can vary a lot based on the parameters that are selected.
• Requires that data can be made stationary by differencing. Like other ARIMA family models, Fourier ARIMA models rely on the assumption that your data is stationary or can be made stationary by differencing. If you have data that is not stationary and cannot be made stationary by differencing, you are better off looking for another model.

## When to use Fourier ARIMA models

When does it make sense to use Fourier ARIMA models to model your dataset? In this section, we will describe situations where it makes sense to use Fourier ARIMA models.

• When there are multiple seasonal patterns in your dataset. Fourier ARIMA models are a great option to turn to when there are multiple seasonal patterns in your dataset and you want to be able to include all of these seasonal patterns in your model. This is especially true if you have additional covariates that you want to include in your model. Some of the other common statistical models that can handle multiple seasonality cannot handle covariate information.

## When not to use Fourier ARIMA models

When should you avoid using Fourier ARIMA models for your time series data? Here are some examples of situations where you should avoid using Fourier ARIMA models.

• When there is a large mean shift or disruption in your data. Fourier ARIMA models and standard ARIMA models can both struggle in situations where there is a large mean shift or disruption in your data. In these cases, you may be better off using a model that was specifically designed to accommodate these situations such as Facebook Prophet.
• When you need to forecast multiple time series simultaneously. Fourier ARIMA models are univariate time series models that can only handle one time series at a time. If you are in a situation where you need to jointly forecast multiple time series models and it would not make sense to forecast each time series individually without information from the other time series, then you should look for a method that was specifically designed for multivariate time series analysis.
• When you have a very small dataset. If you have a very small dataset that does not have many data points, then you may be better off using an ARIMA model. Even if you believe that your data has multiple seasonal patterns in it, it does not make sense to try to capture all of those different patterns if you do not have enough data to do so. In these situations, you are often best off choosing the most pronounced seasonal pattern and building a standard ARIMA model that only considers that component.

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