# Logistic regression

A binomial model relates a linear predictor to a probability through the logistic function: 

$$

  \eta_i = \beta_0 + \widetilde x_i^\mathsf{T}\beta, \qquad p_i
         = \frac{1}{1 + e^{-\eta_i}}, \qquad \min_{\beta_0,\beta}\;\frac1n\sum_i[\log(1 + e^{\eta_i}) - y_i\eta_i]
           + R(\beta).

$$

Responses lie in $[0,1]$, with zeros and ones representing binary observations. The intercept is unpenalized. See [binomial datafits](<../datafits/binomial>) for fractional-response conventions. 

<a id="fit-ridge-logistic-regression"></a>

## Fit ridge logistic regression

```rust
use faer::Mat;
use shrinkage::{Binomial, Irls, Problem, Ridge, 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 = [0.0, 0.0, 0.0, 1.0, 1.0, 1.0];
    let fit = Problem::new(&x, Binomial::new(&y), Ridge::new(0.2)).fit_with(&Irls::new())?;
    assert_eq!(fit.termination(), Termination::Converged);
    let probabilities: Vec<_> = fit
        .predict(&x)?
        .iter()
        .map(|&eta| {
            if eta >= 0.0 {
                1.0 / (1.0 + (-eta).exp())
            } else {
                let e = eta.exp();
                e / (1.0 + e)
            }
        })
        .collect();
    println!("Probabilities: {probabilities:?}");
    Ok(())
}
```

`predict` returns logits. The two branches above convert them to probabilities without overflowing the exponential for extreme predictors. Shrinkage does not choose a classification threshold. 

[IRLS](<../reference/solvers/irls>) supports L1, ridge, and elastic net. [Proximal gradient](<../reference/solvers/proximal-gradient>) also supports the coupled penalties, while [hybrid](<../reference/solvers/hybrid>) uses weighted subproblems for SLOPE and the two fusion penalties. The [clustered lasso guide](<clustered-lasso#logistic-regression>) explains hybrid logistic budgets and diagnostics.