# Feature crossing

> A synthetic feature formed by "crossing" categorical or bucketed features. Formally, a cross is a Cartesian product.[^1]

A [synthetic feature](https://wiki.g15e.com/pages/Synthetic%20feature.txt) formed by "crossing" [categorical](https://wiki.g15e.com/pages/Categorical%20data.txt) or [bucketed](https://wiki.g15e.com/pages/Bucketization.txt) features. Formally, a cross is a <Cartesian product>.[^1]

For example, consider a "mood forecasting" model that represents temperature in one of the following four buckets:

- `freezing`
- `chilly`
- `temperate`
- `warm`

And represents wind speed in one of the following three buckets:

- `still`
- `light`
- `windy`

Without feature crosses, the linear model trains independently on each of the preceding seven various buckets. So, the model trains on, for example, `freezing` independently of the training on, for example, `windy`. Alternatively, you could create a feature cross of temperature and wind speed. This synthetic feature would have the following 12 possible values:[^1]

- `freezing-still`
- `freezing-light`
- `freezing-windy`
- `chilly-still`
- `chilly-light`
- `chilly-windy`
- `temperate-still`
- `temperate-light`
- `temperate-windy`
- `warm-still`
- `warm-light`
- `warm-windy`

Thanks to feature crosses, the model can learn mood differences between a `freezing-windy` day and a `freezing-still` day.

Feature crosses are mostly used with linear models and are rarely used with [neural networks](https://wiki.g15e.com/pages/Artificial%20neural%20network.txt).

## Footnotes

[^1]: https://developers.google.com/machine-learning/glossary#feature_cross