# Linear regression

A Gaussian linear model estimates a real-valued conditional mean. With a complete penalty $R$, its objective is 

$$

  \min_{\beta_0,\beta}\;\frac{\|y - \beta_0\mathbf{1} - \widetilde X\beta\|_2^2}{2n}
  + R(\beta).

$$

Choose [L1](<../penalties/l1>) for sparsity, [ridge](<../penalties/ridge>) for quadratic shrinkage, or another compatible penalty. The response must contain one finite real value per observation. 

<a id="fit-an-elastic-net"></a>

## Fit an elastic net

```rust
use faer::{Col, Mat};
use shrinkage::{ElasticNet, Gaussian, Problem, ProximalGradient, Termination};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let x = Mat::from_fn(6, 2, |i, j| if j == 0 { i as f64 } else { (i % 2) as f64 });
    let y = [1.0, 3.0, 4.0, 7.0, 8.0, 11.0];
    let fit = Problem::new(&x, Gaussian::new(&y), ElasticNet::new(0.1, 0.2))
        .fit_with(&ProximalGradient::<Col<f64>>::new())?;
    assert_eq!(fit.termination(), Termination::Converged);
    println!("Original-scale coefficients: {:?}", fit.coefficients());
    println!("Response means: {:?}", fit.predict(&x)?);
    Ok(())
}
```

The default fits an unpenalized intercept and standardizes features using training statistics. See [normalization](<../modeling#normalization-and-predictions>) to change these modeling choices. 

For Gaussian L1 problems, [coordinate descent](<../reference/solvers/coordinate-descent>) and the [Lasso builder](<lasso>) provide a specialized implementation. [IRLS](<../reference/solvers/irls>) also supports separable penalties; the [hybrid solver](<../reference/solvers/hybrid>) handles supported fusion and SLOPE models. The [compatibility table](<../reference/solvers/>) lists the available combinations.