# PolynomialExtender

Polynomial feature extender.

Generate features consisting of all polynomial combinations of the features with degree less than or equal to the specified degree.

Be aware that the number of outputted features scales polynomially in the number of input features and exponentially in the degree. High degrees can cause overfitting.

## Parameters

- **degree**

  *Default* → `2`

  The maximum degree of the polynomial features.
- **interaction\_only**

  *Default* → `False`

  If `True` then only combinations that include an element at most once will be computed.
- **include\_bias**

  *Default* → `False`

  Whether or not to include a dummy feature which is always equal to 1.
- **bias\_name**

  *Default* → `bias`

  Name to give to the bias feature.

## Examples

```python
from river import feature_extraction as fx

X = [
    {'x': 0, 'y': 1},
    {'x': 2, 'y': 3},
    {'x': 4, 'y': 5}
]

poly = fx.PolynomialExtender(degree=2, include_bias=True)
for x in X:
    print(poly.transform_one(x))
```

```text
{'x': 0, 'y': 1, 'x*x': 0, 'x*y': 0, 'y*y': 1, 'bias': 1}
{'x': 2, 'y': 3, 'x*x': 4, 'x*y': 6, 'y*y': 9, 'bias': 1}
{'x': 4, 'y': 5, 'x*x': 16, 'x*y': 20, 'y*y': 25, 'bias': 1}
```

```python
X = [
    {'x': 0, 'y': 1, 'z': 2},
    {'x': 2, 'y': 3, 'z': 2},
    {'x': 4, 'y': 5, 'z': 2}
]

poly = fx.PolynomialExtender(degree=3, interaction_only=True)
for x in X:
    print(poly.transform_one(x))
```

```text
{'x': 0, 'y': 1, 'z': 2, 'x*y': 0, 'x*z': 0, 'y*z': 2, 'x*y*z': 0}
{'x': 2, 'y': 3, 'z': 2, 'x*y': 6, 'x*z': 4, 'y*z': 6, 'x*y*z': 12}
{'x': 4, 'y': 5, 'z': 2, 'x*y': 20, 'x*z': 8, 'y*z': 10, 'x*y*z': 40}
```

Polynomial features are typically used for a linear model to capture interactions between features. This may done by setting up a pipeline, as so:

```python
from river import datasets
from river import evaluate
from river import linear_model as lm
from river import metrics
from river import preprocessing as pp

dataset = datasets.Phishing()

model = (
    fx.PolynomialExtender() |
    pp.StandardScaler() |
    lm.LogisticRegression()
)

metric = metrics.Accuracy()

evaluate.progressive_val_score(dataset, model, metric)
```

```text
Accuracy: 88.88%
```

## Methods

> [!NOTE]
>
> **learn\_one**
>
> Update with a set of features `x`.
>
> A lot of transformers don't actually have to do anything during the `learn_one` step because they are stateless. For this reason the default behavior of this function is to do nothing. Transformers that however do something during the `learn_one` can override this method.
>
> **Parameters**
>
> - **x** — `dict[base.typing.FeatureName, Any]`

> [!NOTE]
>
> **transform\_one**
>
> Transform a set of features `x`.
>
> **Parameters**
>
> - **x** — `dict[base.typing.FeatureName, Any]`
>
> **Returns**
>
> `dict[base.typing.FeatureName, Any]`: The transformed values.