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#![allow(clippy::needless_range_loop)]
//! # Elastic Net
//!
//! Elastic net is an extension of [linear regression](../linear_regression/index.html) that adds regularization penalties to the loss function during training.
//! Just like in ordinary linear regression you assume a linear relationship between input variables and the target variable.
//! Unlike linear regression elastic net adds regularization penalties to the loss function during training.
//! In particular, the elastic net coefficient estimates \\(\beta\\) are the values that minimize
//!
//! \\[L(\alpha, \beta) = \vert \boldsymbol{y} - \boldsymbol{X}\beta\vert^2 + \lambda_1 \vert \beta \vert^2 + \lambda_2 \vert \beta \vert_1\\]
//!
//! where \\(\lambda_1 = \\alpha l_{1r}\\), \\(\lambda_2 = \\alpha (1 - l_{1r})\\) and \\(l_{1r}\\) is the l1 ratio, elastic net mixing parameter.
//!
//! In essense, elastic net combines both the [L1](../lasso/index.html) and [L2](../ridge_regression/index.html) penalties during training,
//! which can result in better performance than a model with either one or the other penalty on some problems.
//! The elastic net is particularly useful when the number of predictors (p) is much bigger than the number of observations (n).
//!
//! Example:
//!
//! ```
//! use smartcore::linalg::basic::matrix::DenseMatrix;
//! use smartcore::linear::elastic_net::*;
//!
//! // Longley dataset (https://www.statsmodels.org/stable/datasets/generated/longley.html)
//! let x = DenseMatrix::from_2d_array(&[
//! &[234.289, 235.6, 159.0, 107.608, 1947., 60.323],
//! &[259.426, 232.5, 145.6, 108.632, 1948., 61.122],
//! &[258.054, 368.2, 161.6, 109.773, 1949., 60.171],
//! &[284.599, 335.1, 165.0, 110.929, 1950., 61.187],
//! &[328.975, 209.9, 309.9, 112.075, 1951., 63.221],
//! &[346.999, 193.2, 359.4, 113.270, 1952., 63.639],
//! &[365.385, 187.0, 354.7, 115.094, 1953., 64.989],
//! &[363.112, 357.8, 335.0, 116.219, 1954., 63.761],
//! &[397.469, 290.4, 304.8, 117.388, 1955., 66.019],
//! &[419.180, 282.2, 285.7, 118.734, 1956., 67.857],
//! &[442.769, 293.6, 279.8, 120.445, 1957., 68.169],
//! &[444.546, 468.1, 263.7, 121.950, 1958., 66.513],
//! &[482.704, 381.3, 255.2, 123.366, 1959., 68.655],
//! &[502.601, 393.1, 251.4, 125.368, 1960., 69.564],
//! &[518.173, 480.6, 257.2, 127.852, 1961., 69.331],
//! &[554.894, 400.7, 282.7, 130.081, 1962., 70.551],
//! ]);
//!
//! let y: Vec<f64> = vec![83.0, 88.5, 88.2, 89.5, 96.2, 98.1, 99.0,
//! 100.0, 101.2, 104.6, 108.4, 110.8, 112.6, 114.2, 115.7, 116.9];
//!
//! let y_hat = ElasticNet::fit(&x, &y, Default::default()).
//! and_then(|lr| lr.predict(&x)).unwrap();
//! ```
//!
//! ## References:
//!
//! * ["An Introduction to Statistical Learning", James G., Witten D., Hastie T., Tibshirani R., 6.2. Shrinkage Methods](http://faculty.marshall.usc.edu/gareth-james/ISL/)
//! * ["Regularization and variable selection via the elastic net", Hui Zou and Trevor Hastie](https://web.stanford.edu/~hastie/Papers/B67.2%20(2005)%20301-320%20Zou%20&%20Hastie.pdf)
//!
//! <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
//! <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
use std::fmt::Debug;
use std::marker::PhantomData;
#[cfg(feature = "serde")]
use serde::{Deserialize, Serialize};
use crate::api::{Predictor, SupervisedEstimator};
use crate::error::Failed;
use crate::linalg::basic::arrays::{Array, Array1, Array2, MutArray};
use crate::numbers::basenum::Number;
use crate::numbers::floatnum::FloatNumber;
use crate::numbers::realnum::RealNumber;
use crate::linear::lasso_optimizer::InteriorPointOptimizer;
/// Elastic net parameters
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
#[derive(Debug, Clone)]
pub struct ElasticNetParameters {
#[cfg_attr(feature = "serde", serde(default))]
/// Regularization parameter.
pub alpha: f64,
#[cfg_attr(feature = "serde", serde(default))]
/// The elastic net mixing parameter, with 0 <= l1_ratio <= 1.
/// For l1_ratio = 0 the penalty is an L2 penalty.
/// For l1_ratio = 1 it is an L1 penalty. For 0 < l1_ratio < 1, the penalty is a combination of L1 and L2.
pub l1_ratio: f64,
#[cfg_attr(feature = "serde", serde(default))]
/// If True, the regressors X will be normalized before regression by subtracting the mean and dividing by the standard deviation.
pub normalize: bool,
#[cfg_attr(feature = "serde", serde(default))]
/// The tolerance for the optimization
pub tol: f64,
#[cfg_attr(feature = "serde", serde(default))]
/// The maximum number of iterations
pub max_iter: usize,
}
/// Elastic net
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
#[derive(Debug)]
pub struct ElasticNet<TX: FloatNumber + RealNumber, TY: Number, X: Array2<TX>, Y: Array1<TY>> {
coefficients: Option<X>,
intercept: Option<TX>,
_phantom_ty: PhantomData<TY>,
_phantom_y: PhantomData<Y>,
}
impl ElasticNetParameters {
/// Regularization parameter.
pub fn with_alpha(mut self, alpha: f64) -> Self {
self.alpha = alpha;
self
}
/// The elastic net mixing parameter, with 0 <= l1_ratio <= 1.
/// For l1_ratio = 0 the penalty is an L2 penalty.
/// For l1_ratio = 1 it is an L1 penalty. For 0 < l1_ratio < 1, the penalty is a combination of L1 and L2.
pub fn with_l1_ratio(mut self, l1_ratio: f64) -> Self {
self.l1_ratio = l1_ratio;
self
}
/// If True, the regressors X will be normalized before regression by subtracting the mean and dividing by the standard deviation.
pub fn with_normalize(mut self, normalize: bool) -> Self {
self.normalize = normalize;
self
}
/// The tolerance for the optimization
pub fn with_tol(mut self, tol: f64) -> Self {
self.tol = tol;
self
}
/// The maximum number of iterations
pub fn with_max_iter(mut self, max_iter: usize) -> Self {
self.max_iter = max_iter;
self
}
}
impl Default for ElasticNetParameters {
fn default() -> Self {
ElasticNetParameters {
alpha: 1.0,
l1_ratio: 0.5,
normalize: true,
tol: 1e-4,
max_iter: 1000,
}
}
}
/// ElasticNet grid search parameters
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
#[derive(Debug, Clone)]
pub struct ElasticNetSearchParameters {
#[cfg_attr(feature = "serde", serde(default))]
/// Regularization parameter.
pub alpha: Vec<f64>,
#[cfg_attr(feature = "serde", serde(default))]
/// The elastic net mixing parameter, with 0 <= l1_ratio <= 1.
/// For l1_ratio = 0 the penalty is an L2 penalty.
/// For l1_ratio = 1 it is an L1 penalty. For 0 < l1_ratio < 1, the penalty is a combination of L1 and L2.
pub l1_ratio: Vec<f64>,
#[cfg_attr(feature = "serde", serde(default))]
/// If True, the regressors X will be normalized before regression by subtracting the mean and dividing by the standard deviation.
pub normalize: Vec<bool>,
#[cfg_attr(feature = "serde", serde(default))]
/// The tolerance for the optimization
pub tol: Vec<f64>,
#[cfg_attr(feature = "serde", serde(default))]
/// The maximum number of iterations
pub max_iter: Vec<usize>,
}
/// ElasticNet grid search iterator
pub struct ElasticNetSearchParametersIterator {
lasso_regression_search_parameters: ElasticNetSearchParameters,
current_alpha: usize,
current_l1_ratio: usize,
current_normalize: usize,
current_tol: usize,
current_max_iter: usize,
}
impl IntoIterator for ElasticNetSearchParameters {
type Item = ElasticNetParameters;
type IntoIter = ElasticNetSearchParametersIterator;
fn into_iter(self) -> Self::IntoIter {
ElasticNetSearchParametersIterator {
lasso_regression_search_parameters: self,
current_alpha: 0,
current_l1_ratio: 0,
current_normalize: 0,
current_tol: 0,
current_max_iter: 0,
}
}
}
impl Iterator for ElasticNetSearchParametersIterator {
type Item = ElasticNetParameters;
fn next(&mut self) -> Option<Self::Item> {
if self.current_alpha == self.lasso_regression_search_parameters.alpha.len()
&& self.current_l1_ratio == self.lasso_regression_search_parameters.l1_ratio.len()
&& self.current_normalize == self.lasso_regression_search_parameters.normalize.len()
&& self.current_tol == self.lasso_regression_search_parameters.tol.len()
&& self.current_max_iter == self.lasso_regression_search_parameters.max_iter.len()
{
return None;
}
let next = ElasticNetParameters {
alpha: self.lasso_regression_search_parameters.alpha[self.current_alpha],
l1_ratio: self.lasso_regression_search_parameters.alpha[self.current_l1_ratio],
normalize: self.lasso_regression_search_parameters.normalize[self.current_normalize],
tol: self.lasso_regression_search_parameters.tol[self.current_tol],
max_iter: self.lasso_regression_search_parameters.max_iter[self.current_max_iter],
};
if self.current_alpha + 1 < self.lasso_regression_search_parameters.alpha.len() {
self.current_alpha += 1;
} else if self.current_l1_ratio + 1 < self.lasso_regression_search_parameters.l1_ratio.len()
{
self.current_alpha = 0;
self.current_l1_ratio += 1;
} else if self.current_normalize + 1
< self.lasso_regression_search_parameters.normalize.len()
{
self.current_alpha = 0;
self.current_l1_ratio = 0;
self.current_normalize += 1;
} else if self.current_tol + 1 < self.lasso_regression_search_parameters.tol.len() {
self.current_alpha = 0;
self.current_l1_ratio = 0;
self.current_normalize = 0;
self.current_tol += 1;
} else if self.current_max_iter + 1 < self.lasso_regression_search_parameters.max_iter.len()
{
self.current_alpha = 0;
self.current_l1_ratio = 0;
self.current_normalize = 0;
self.current_tol = 0;
self.current_max_iter += 1;
} else {
self.current_alpha += 1;
self.current_l1_ratio += 1;
self.current_normalize += 1;
self.current_tol += 1;
self.current_max_iter += 1;
}
Some(next)
}
}
impl Default for ElasticNetSearchParameters {
fn default() -> Self {
let default_params = ElasticNetParameters::default();
ElasticNetSearchParameters {
alpha: vec![default_params.alpha],
l1_ratio: vec![default_params.l1_ratio],
normalize: vec![default_params.normalize],
tol: vec![default_params.tol],
max_iter: vec![default_params.max_iter],
}
}
}
impl<TX: FloatNumber + RealNumber, TY: Number, X: Array2<TX>, Y: Array1<TY>> PartialEq
for ElasticNet<TX, TY, X, Y>
{
fn eq(&self, other: &Self) -> bool {
if self.intercept() != other.intercept() {
return false;
}
if self.coefficients().shape() != other.coefficients().shape() {
return false;
}
self.coefficients()
.iterator(0)
.zip(other.coefficients().iterator(0))
.all(|(&a, &b)| (a - b).abs() <= TX::epsilon())
}
}
impl<TX: FloatNumber + RealNumber, TY: Number, X: Array2<TX>, Y: Array1<TY>>
SupervisedEstimator<X, Y, ElasticNetParameters> for ElasticNet<TX, TY, X, Y>
{
fn new() -> Self {
Self {
coefficients: Option::None,
intercept: Option::None,
_phantom_ty: PhantomData,
_phantom_y: PhantomData,
}
}
fn fit(x: &X, y: &Y, parameters: ElasticNetParameters) -> Result<Self, Failed> {
ElasticNet::fit(x, y, parameters)
}
}
impl<TX: FloatNumber + RealNumber, TY: Number, X: Array2<TX>, Y: Array1<TY>> Predictor<X, Y>
for ElasticNet<TX, TY, X, Y>
{
fn predict(&self, x: &X) -> Result<Y, Failed> {
self.predict(x)
}
}
impl<TX: FloatNumber + RealNumber, TY: Number, X: Array2<TX>, Y: Array1<TY>>
ElasticNet<TX, TY, X, Y>
{
/// Fits elastic net regression to your data.
/// * `x` - _NxM_ matrix with _N_ observations and _M_ features in each observation.
/// * `y` - target values
/// * `parameters` - other parameters, use `Default::default()` to set parameters to default values.
pub fn fit(
x: &X,
y: &Y,
parameters: ElasticNetParameters,
) -> Result<ElasticNet<TX, TY, X, Y>, Failed> {
let (n, p) = x.shape();
if y.shape() != n {
return Err(Failed::fit("Number of rows in X should = len(y)"));
}
let n_float = n as f64;
let l1_reg = TX::from_f64(parameters.alpha * parameters.l1_ratio * n_float).unwrap();
let l2_reg =
TX::from_f64(parameters.alpha * (1.0 - parameters.l1_ratio) * n_float).unwrap();
let y_mean = TX::from_f64(y.mean_by()).unwrap();
let (w, b) = if parameters.normalize {
let (scaled_x, col_mean, col_std) = Self::rescale_x(x)?;
let (x, y, gamma) = Self::augment_x_and_y(&scaled_x, y, l2_reg);
let mut optimizer = InteriorPointOptimizer::new(&x, p);
let mut w = optimizer.optimize(
&x,
&y,
l1_reg * gamma,
parameters.max_iter,
TX::from_f64(parameters.tol).unwrap(),
)?;
for i in 0..p {
w.set(i, gamma * *w.get(i) / col_std[i]);
}
let mut b = TX::zero();
for i in 0..p {
b += *w.get(i) * col_mean[i];
}
b = y_mean - b;
(X::from_column(&w), b)
} else {
let (x, y, gamma) = Self::augment_x_and_y(x, y, l2_reg);
let mut optimizer = InteriorPointOptimizer::new(&x, p);
let mut w = optimizer.optimize(
&x,
&y,
l1_reg * gamma,
parameters.max_iter,
TX::from_f64(parameters.tol).unwrap(),
)?;
for i in 0..p {
w.set(i, gamma * *w.get(i));
}
(X::from_column(&w), y_mean)
};
Ok(ElasticNet {
intercept: Some(b),
coefficients: Some(w),
_phantom_ty: PhantomData,
_phantom_y: PhantomData,
})
}
/// Predict target values from `x`
/// * `x` - _KxM_ data where _K_ is number of observations and _M_ is number of features.
pub fn predict(&self, x: &X) -> Result<Y, Failed> {
let (nrows, _) = x.shape();
let mut y_hat = x.matmul(self.coefficients.as_ref().unwrap());
let bias = X::fill(nrows, 1, self.intercept.unwrap());
y_hat.add_mut(&bias);
Ok(Y::from_iterator(
y_hat.iterator(0).map(|&v| TY::from(v).unwrap()),
nrows,
))
}
/// Get estimates regression coefficients
pub fn coefficients(&self) -> &X {
self.coefficients.as_ref().unwrap()
}
/// Get estimate of intercept
pub fn intercept(&self) -> &TX {
self.intercept.as_ref().unwrap()
}
fn rescale_x(x: &X) -> Result<(X, Vec<TX>, Vec<TX>), Failed> {
let col_mean: Vec<TX> = x
.mean_by(0)
.iter()
.map(|&v| TX::from_f64(v).unwrap())
.collect();
let col_std: Vec<TX> = x
.std_dev(0)
.iter()
.map(|&v| TX::from_f64(v).unwrap())
.collect();
for (i, col_std_i) in col_std.iter().enumerate() {
if (*col_std_i - TX::zero()).abs() < TX::epsilon() {
return Err(Failed::fit(&format!("Cannot rescale constant column {i}")));
}
}
let mut scaled_x = x.clone();
scaled_x.scale_mut(&col_mean, &col_std, 0);
Ok((scaled_x, col_mean, col_std))
}
fn augment_x_and_y(x: &X, y: &Y, l2_reg: TX) -> (X, Vec<TX>, TX) {
let (n, p) = x.shape();
let gamma = TX::one() / (TX::one() + l2_reg).sqrt();
let padding = gamma * l2_reg.sqrt();
let mut y2 = Vec::<TX>::zeros(n + p);
for i in 0..y.shape() {
y2.set(i, TX::from(*y.get(i)).unwrap());
}
let mut x2 = X::zeros(n + p, p);
for j in 0..p {
for i in 0..n {
x2.set((i, j), gamma * *x.get((i, j)));
}
x2.set((j + n, j), padding);
}
(x2, y2, gamma)
}
}
#[cfg(test)]
mod tests {
use super::*;
use crate::linalg::basic::matrix::DenseMatrix;
use crate::metrics::mean_absolute_error;
#[test]
fn search_parameters() {
let parameters = ElasticNetSearchParameters {
alpha: vec![0., 1.],
max_iter: vec![10, 100],
..Default::default()
};
let mut iter = parameters.into_iter();
let next = iter.next().unwrap();
assert_eq!(next.alpha, 0.);
assert_eq!(next.max_iter, 10);
let next = iter.next().unwrap();
assert_eq!(next.alpha, 1.);
assert_eq!(next.max_iter, 10);
let next = iter.next().unwrap();
assert_eq!(next.alpha, 0.);
assert_eq!(next.max_iter, 100);
let next = iter.next().unwrap();
assert_eq!(next.alpha, 1.);
assert_eq!(next.max_iter, 100);
assert!(iter.next().is_none());
}
#[cfg_attr(
all(target_arch = "wasm32", not(target_os = "wasi")),
wasm_bindgen_test::wasm_bindgen_test
)]
#[test]
fn elasticnet_longley() {
let x = DenseMatrix::from_2d_array(&[
&[234.289, 235.6, 159.0, 107.608, 1947., 60.323],
&[259.426, 232.5, 145.6, 108.632, 1948., 61.122],
&[258.054, 368.2, 161.6, 109.773, 1949., 60.171],
&[284.599, 335.1, 165.0, 110.929, 1950., 61.187],
&[328.975, 209.9, 309.9, 112.075, 1951., 63.221],
&[346.999, 193.2, 359.4, 113.270, 1952., 63.639],
&[365.385, 187.0, 354.7, 115.094, 1953., 64.989],
&[363.112, 357.8, 335.0, 116.219, 1954., 63.761],
&[397.469, 290.4, 304.8, 117.388, 1955., 66.019],
&[419.180, 282.2, 285.7, 118.734, 1956., 67.857],
&[442.769, 293.6, 279.8, 120.445, 1957., 68.169],
&[444.546, 468.1, 263.7, 121.950, 1958., 66.513],
&[482.704, 381.3, 255.2, 123.366, 1959., 68.655],
&[502.601, 393.1, 251.4, 125.368, 1960., 69.564],
&[518.173, 480.6, 257.2, 127.852, 1961., 69.331],
&[554.894, 400.7, 282.7, 130.081, 1962., 70.551],
]);
let y: Vec<f64> = vec![
83.0, 88.5, 88.2, 89.5, 96.2, 98.1, 99.0, 100.0, 101.2, 104.6, 108.4, 110.8, 112.6,
114.2, 115.7, 116.9,
];
let y_hat = ElasticNet::fit(
&x,
&y,
ElasticNetParameters {
alpha: 1.0,
l1_ratio: 0.5,
normalize: false,
tol: 1e-4,
max_iter: 1000,
},
)
.and_then(|lr| lr.predict(&x))
.unwrap();
assert!(mean_absolute_error(&y_hat, &y) < 30.0);
}
#[cfg_attr(
all(target_arch = "wasm32", not(target_os = "wasi")),
wasm_bindgen_test::wasm_bindgen_test
)]
#[test]
fn elasticnet_fit_predict1() {
let x = DenseMatrix::from_2d_array(&[
&[0.0, 1931.0, 1.2232755825400514],
&[1.0, 1933.0, 1.1379726120972395],
&[2.0, 1920.0, 1.4366265120543429],
&[3.0, 1918.0, 1.206005737827858],
&[4.0, 1934.0, 1.436613542400669],
&[5.0, 1918.0, 1.1594588621640636],
&[6.0, 1933.0, 1.19809994745985],
&[7.0, 1918.0, 1.3396363871645678],
&[8.0, 1931.0, 1.2535342096493207],
&[9.0, 1933.0, 1.3101281563456293],
&[10.0, 1922.0, 1.3585833349920762],
&[11.0, 1930.0, 1.4830786699709897],
&[12.0, 1916.0, 1.4919891143094546],
&[13.0, 1915.0, 1.259655137451551],
&[14.0, 1932.0, 1.3979191428724789],
&[15.0, 1917.0, 1.3686634746782371],
&[16.0, 1932.0, 1.381658454569724],
&[17.0, 1918.0, 1.4054969025700674],
&[18.0, 1929.0, 1.3271699396384906],
&[19.0, 1915.0, 1.1373332337674806],
]);
let y: Vec<f64> = vec![
1.48, 2.72, 4.52, 5.72, 5.25, 4.07, 3.75, 4.75, 6.77, 4.72, 6.78, 6.79, 8.3, 7.42,
10.2, 7.92, 7.62, 8.06, 9.06, 9.29,
];
let l1_model = ElasticNet::fit(
&x,
&y,
ElasticNetParameters {
alpha: 1.0,
l1_ratio: 1.0,
normalize: true,
tol: 1e-4,
max_iter: 1000,
},
)
.unwrap();
let l2_model = ElasticNet::fit(
&x,
&y,
ElasticNetParameters {
alpha: 1.0,
l1_ratio: 0.0,
normalize: true,
tol: 1e-4,
max_iter: 1000,
},
)
.unwrap();
let mae_l1 = mean_absolute_error(&l1_model.predict(&x).unwrap(), &y);
let mae_l2 = mean_absolute_error(&l2_model.predict(&x).unwrap(), &y);
assert!(mae_l1 < 2.0);
assert!(mae_l2 < 2.0);
assert!(l1_model.coefficients().get((0, 0)) > l1_model.coefficients().get((1, 0)));
assert!(l1_model.coefficients().get((0, 0)) > l1_model.coefficients().get((2, 0)));
}
// TODO: serialization for the new DenseMatrix needs to be implemented
// #[cfg_attr(all(target_arch = "wasm32", not(target_os = "wasi")), wasm_bindgen_test::wasm_bindgen_test)]
// #[test]
// #[cfg(feature = "serde")]
// fn serde() {
// let x = DenseMatrix::from_2d_array(&[
// &[234.289, 235.6, 159.0, 107.608, 1947., 60.323],
// &[259.426, 232.5, 145.6, 108.632, 1948., 61.122],
// &[258.054, 368.2, 161.6, 109.773, 1949., 60.171],
// &[284.599, 335.1, 165.0, 110.929, 1950., 61.187],
// &[328.975, 209.9, 309.9, 112.075, 1951., 63.221],
// &[346.999, 193.2, 359.4, 113.270, 1952., 63.639],
// &[365.385, 187.0, 354.7, 115.094, 1953., 64.989],
// &[363.112, 357.8, 335.0, 116.219, 1954., 63.761],
// &[397.469, 290.4, 304.8, 117.388, 1955., 66.019],
// &[419.180, 282.2, 285.7, 118.734, 1956., 67.857],
// &[442.769, 293.6, 279.8, 120.445, 1957., 68.169],
// &[444.546, 468.1, 263.7, 121.950, 1958., 66.513],
// &[482.704, 381.3, 255.2, 123.366, 1959., 68.655],
// &[502.601, 393.1, 251.4, 125.368, 1960., 69.564],
// &[518.173, 480.6, 257.2, 127.852, 1961., 69.331],
// &[554.894, 400.7, 282.7, 130.081, 1962., 70.551],
// ]);
// let y = vec![
// 83.0, 88.5, 88.2, 89.5, 96.2, 98.1, 99.0, 100.0, 101.2, 104.6, 108.4, 110.8, 112.6,
// 114.2, 115.7, 116.9,
// ];
// let lr = ElasticNet::fit(&x, &y, Default::default()).unwrap();
// let deserialized_lr: ElasticNet<f64, f64, DenseMatrix<f64>, Vec<f64>> =
// serde_json::from_str(&serde_json::to_string(&lr).unwrap()).unwrap();
// assert_eq!(lr, deserialized_lr);
// }
}