Note

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# Fitting an Elastic Net with a precomputed Gram Matrix and Weighted Samples¶

The following example shows how to precompute the gram matrix while using weighted samples with an ElasticNet.

If weighted samples are used, the design matrix must be centered and then rescaled by the square root of the weight vector before the gram matrix is computed.

Note

`sample_weight`

vector is also rescaled to sum to`n_samples`

, see thedocumentation for the

`sample_weight`

parameter to`linear_model.ElasticNet.fit`

.

Let’s start by loading the dataset and creating some sample weights.

```
import numpy as np
from sklearn.datasets import make_regression
rng = np.random.RandomState(0)
n_samples = int(1e5)
X, y = make_regression(n_samples=n_samples, noise=0.5, random_state=rng)
sample_weight = rng.lognormal(size=n_samples)
# normalize the sample weights
normalized_weights = sample_weight * (n_samples / (sample_weight.sum()))
```

To fit the elastic net using the `precompute`

option together with the sample
weights, we must first center the design matrix, and rescale it by the
normalized weights prior to computing the gram matrix.

```
X_offset = np.average(X, axis=0, weights=normalized_weights)
X_centered = X - np.average(X, axis=0, weights=normalized_weights)
X_scaled = X_centered * np.sqrt(normalized_weights)[:, np.newaxis]
gram = np.dot(X_scaled.T, X_scaled)
```

We can now proceed with fitting. We must passed the centered design matrix to
`fit`

otherwise the elastic net estimator will detect that it is uncentered
and discard the gram matrix we passed. However, if we pass the scaled design
matrix, the preprocessing code will incorrectly rescale it a second time.

```
from sklearn.linear_model import ElasticNet
lm = ElasticNet(alpha=0.01, precompute=gram)
lm.fit(X_centered, y, sample_weight=normalized_weights)
```

**Total running time of the script:** ( 0 minutes 1.866 seconds)