# Multivariate Gaussian

`MultivariateGaussian` fits $q$ real-valued responses with shared observations: 

$$

  F(Y,H) = \frac{\|Y - H\|_F^2}{2n}
         = \frac1{2n}\sum_{r=1}^q\sum_{i=1}^n(Y_{ir} - H_{ir})^2.

$$

The loss sums over responses and averages over observations. It never divides by $q$. Independent response fits therefore use the same penalty strengths as the corresponding single-response Gaussian problems. 

<a id="evaluate-the-loss"></a>

## Evaluate the loss

Store response columns contiguously. Entry $(i,r)$ has index `r * nobs + i`. The supplied dimensions must be positive and match the response length. 

```rust
use shrinkage::MultivariateGaussian;

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let y = [1.0, 3.0, 2.0, 4.0];
    let datafit = MultivariateGaussian::new(&y, 2, 2);
    let mut gradient = [0.0; 4];
    datafit.gradient(&[0.0; 4], &mut gradient)?;
    assert_eq!(gradient, [-0.5, -1.5, -1.0, -2.0]);
    Ok(())
}
```

Every response has a free, unpenalized intercept by default. `.fit_intercept(false)` disables all free intercepts. Predictors and predictions use the same response-major layout as the observations. 

Fitting uses [proximal gradient](<../reference/solvers/proximal-gradient>) and an [AxisPenalty](<../penalties/axis>) with an explicit feature-by-response `CoefficientLayout`. See [multivariate regression](<../models/multivariate>) for a complete example and feature selection across responses. 

API: [`shrinkage::MultivariateGaussian`](<https://shrinkage.cc/packages/rust/reference/7fdf5e23960fc73db0e87f24ddb2a99c31ff7cb273565e92aaae35a4fcc0bf0a>).