Merge pull request #23 from smartcorelib/ridge

Ridge regression

src/linalg/mod.rs

@@ -48,6 +48,7 @@ pub mod nalgebra_bindings;
 pub mod ndarray_bindings;
 /// QR factorization that factors a matrix into a product of an orthogonal matrix and an upper triangular matrix.
 pub mod qr;
+pub mod stats;
 /// Singular value decomposition.
 pub mod svd;
 
@@ -60,6 +61,7 @@ use cholesky::CholeskyDecomposableMatrix;
 use evd::EVDDecomposableMatrix;
 use lu::LUDecomposableMatrix;
 use qr::QRDecomposableMatrix;
+use stats::MatrixStats;
 use svd::SVDDecomposableMatrix;
 
 /// Column or row vector
@@ -168,6 +170,30 @@ pub trait BaseVector<T: RealNumber>: Clone + Debug {
     ///assert_eq!(a.unique(), vec![-7., -6., -2., 1., 2., 3., 4.]);
     /// ```
     fn unique(&self) -> Vec<T>;
+
+    /// Computes the arithmetic mean.
+    fn mean(&self) -> T {
+        self.sum() / T::from_usize(self.len()).unwrap()
+    }
+    /// Computes variance.
+    fn var(&self) -> T {
+        let n = self.len();
+
+        let mut mu = T::zero();
+        let mut sum = T::zero();
+        let div = T::from_usize(n).unwrap();
+        for i in 0..n {
+            let xi = self.get(i);
+            mu += xi;
+            sum += xi * xi;
+        }
+        mu /= div;
+        sum / div - mu * mu
+    }
+    /// Computes the standard deviation.
+    fn std(&self) -> T {
+        self.var().sqrt()
+    }
 }
 
 /// Generic matrix type.
@@ -515,6 +541,7 @@ pub trait Matrix<T: RealNumber>:
     + QRDecomposableMatrix<T>
     + LUDecomposableMatrix<T>
     + CholeskyDecomposableMatrix<T>
+    + MatrixStats<T>
     + PartialEq
     + Display
 {
@@ -550,3 +577,29 @@ impl<'a, T: RealNumber, M: BaseMatrix<T>> Iterator for RowIter<'a, T, M> {
         res
     }
 }
+
+#[cfg(test)]
+mod tests {
+    use crate::linalg::BaseVector;
+
+    #[test]
+    fn mean() {
+        let m = vec![1., 2., 3.];
+
+        assert_eq!(m.mean(), 2.0);
+    }
+
+    #[test]
+    fn std() {
+        let m = vec![1., 2., 3.];
+
+        assert!((m.std() - 0.81f64).abs() < 1e-2);
+    }
+
+    #[test]
+    fn var() {
+        let m = vec![1., 2., 3., 4.];
+
+        assert!((m.var() - 1.25f64).abs() < std::f64::EPSILON);
+    }
+}

src/linalg/naive/dense_matrix.rs

@@ -11,6 +11,7 @@ use crate::linalg::cholesky::CholeskyDecomposableMatrix;
 use crate::linalg::evd::EVDDecomposableMatrix;
 use crate::linalg::lu::LUDecomposableMatrix;
 use crate::linalg::qr::QRDecomposableMatrix;
+use crate::linalg::stats::MatrixStats;
 use crate::linalg::svd::SVDDecomposableMatrix;
 use crate::linalg::Matrix;
 pub use crate::linalg::{BaseMatrix, BaseVector};
@@ -443,6 +444,8 @@ impl<T: RealNumber> LUDecomposableMatrix<T> for DenseMatrix<T> {}
 
 impl<T: RealNumber> CholeskyDecomposableMatrix<T> for DenseMatrix<T> {}
 
+impl<T: RealNumber> MatrixStats<T> for DenseMatrix<T> {}
+
 impl<T: RealNumber> Matrix<T> for DenseMatrix<T> {}
 
 impl<T: RealNumber> PartialEq for DenseMatrix<T> {

src/linalg/nalgebra_bindings.rs

@@ -46,6 +46,7 @@ use crate::linalg::cholesky::CholeskyDecomposableMatrix;
 use crate::linalg::evd::EVDDecomposableMatrix;
 use crate::linalg::lu::LUDecomposableMatrix;
 use crate::linalg::qr::QRDecomposableMatrix;
+use crate::linalg::stats::MatrixStats;
 use crate::linalg::svd::SVDDecomposableMatrix;
 use crate::linalg::Matrix as SmartCoreMatrix;
 use crate::linalg::{BaseMatrix, BaseVector};
@@ -546,6 +547,11 @@ impl<T: RealNumber + Scalar + AddAssign + SubAssign + MulAssign + DivAssign + Su
 {
 }
 
+impl<T: RealNumber + Scalar + AddAssign + SubAssign + MulAssign + DivAssign + Sum + 'static>
+    MatrixStats<T> for Matrix<T, Dynamic, Dynamic, VecStorage<T, Dynamic, Dynamic>>
+{
+}
+
 impl<T: RealNumber + Scalar + AddAssign + SubAssign + MulAssign + DivAssign + Sum + 'static>
     SmartCoreMatrix<T> for Matrix<T, Dynamic, Dynamic, VecStorage<T, Dynamic, Dynamic>>
 {

src/linalg/ndarray_bindings.rs

@@ -53,6 +53,7 @@ use crate::linalg::cholesky::CholeskyDecomposableMatrix;
 use crate::linalg::evd::EVDDecomposableMatrix;
 use crate::linalg::lu::LUDecomposableMatrix;
 use crate::linalg::qr::QRDecomposableMatrix;
+use crate::linalg::stats::MatrixStats;
 use crate::linalg::svd::SVDDecomposableMatrix;
 use crate::linalg::Matrix;
 use crate::linalg::{BaseMatrix, BaseVector};
@@ -496,6 +497,11 @@ impl<T: RealNumber + ScalarOperand + AddAssign + SubAssign + MulAssign + DivAssi
 {
 }
 
+impl<T: RealNumber + ScalarOperand + AddAssign + SubAssign + MulAssign + DivAssign + Sum>
+    MatrixStats<T> for ArrayBase<OwnedRepr<T>, Ix2>
+{
+}
+
 impl<T: RealNumber + ScalarOperand + AddAssign + SubAssign + MulAssign + DivAssign + Sum> Matrix<T>
     for ArrayBase<OwnedRepr<T>, Ix2>
 {

src/linalg/stats.rs

@@ -0,0 +1,166 @@
+//! # Various Statistical Methods
+//!
+//! This module provides reference implementations for  various statistical functions.
+//! Concrete implementations of the `BaseMatrix` trait are free to override these methods for better performance.
+
+use crate::linalg::BaseMatrix;
+use crate::math::num::RealNumber;
+
+/// Defines baseline implementations for various statistical functions
+pub trait MatrixStats<T: RealNumber>: BaseMatrix<T> {
+    /// Computes the arithmetic mean along the specified axis.
+    fn mean(&self, axis: u8) -> Vec<T> {
+        let (n, m) = match axis {
+            0 => {
+                let (n, m) = self.shape();
+                (m, n)
+            }
+            _ => self.shape(),
+        };
+
+        let mut x: Vec<T> = vec![T::zero(); n];
+
+        let div = T::from_usize(m).unwrap();
+
+        for i in 0..n {
+            for j in 0..m {
+                x[i] += match axis {
+                    0 => self.get(j, i),
+                    _ => self.get(i, j),
+                };
+            }
+            x[i] /= div;
+        }
+
+        x
+    }
+
+    /// Computes variance along the specified axis.
+    fn var(&self, axis: u8) -> Vec<T> {
+        let (n, m) = match axis {
+            0 => {
+                let (n, m) = self.shape();
+                (m, n)
+            }
+            _ => self.shape(),
+        };
+
+        let mut x: Vec<T> = vec![T::zero(); n];
+
+        let div = T::from_usize(m).unwrap();
+
+        for i in 0..n {
+            let mut mu = T::zero();
+            let mut sum = T::zero();
+            for j in 0..m {
+                let a = match axis {
+                    0 => self.get(j, i),
+                    _ => self.get(i, j),
+                };
+                mu += a;
+                sum += a * a;
+            }
+            mu /= div;
+            x[i] = sum / div - mu * mu;
+        }
+
+        x
+    }
+
+    /// Computes the standard deviation along the specified axis.
+    fn std(&self, axis: u8) -> Vec<T> {
+        let mut x = self.var(axis);
+
+        let n = match axis {
+            0 => self.shape().1,
+            _ => self.shape().0,
+        };
+
+        for i in 0..n {
+            x[i] = x[i].sqrt();
+        }
+
+        x
+    }
+
+    /// standardize values by removing the mean and scaling to unit variance
+    fn scale_mut(&mut self, mean: &Vec<T>, std: &Vec<T>, axis: u8) {
+        let (n, m) = match axis {
+            0 => {
+                let (n, m) = self.shape();
+                (m, n)
+            }
+            _ => self.shape(),
+        };
+
+        for i in 0..n {
+            for j in 0..m {
+                match axis {
+                    0 => self.set(j, i, (self.get(j, i) - mean[i]) / std[i]),
+                    _ => self.set(i, j, (self.get(i, j) - mean[i]) / std[i]),
+                }
+            }
+        }
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::linalg::naive::dense_matrix::DenseMatrix;
+    use crate::linalg::BaseVector;
+
+    #[test]
+    fn mean() {
+        let m = DenseMatrix::from_2d_array(&[
+            &[1., 2., 3., 1., 2.],
+            &[4., 5., 6., 3., 4.],
+            &[7., 8., 9., 5., 6.],
+        ]);
+        let expected_0 = vec![4., 5., 6., 3., 4.];
+        let expected_1 = vec![1.8, 4.4, 7.];
+
+        assert_eq!(m.mean(0), expected_0);
+        assert_eq!(m.mean(1), expected_1);
+    }
+
+    #[test]
+    fn std() {
+        let m = DenseMatrix::from_2d_array(&[
+            &[1., 2., 3., 1., 2.],
+            &[4., 5., 6., 3., 4.],
+            &[7., 8., 9., 5., 6.],
+        ]);
+        let expected_0 = vec![2.44, 2.44, 2.44, 1.63, 1.63];
+        let expected_1 = vec![0.74, 1.01, 1.41];
+
+        assert!(m.std(0).approximate_eq(&expected_0, 1e-2));
+        assert!(m.std(1).approximate_eq(&expected_1, 1e-2));
+    }
+
+    #[test]
+    fn var() {
+        let m = DenseMatrix::from_2d_array(&[&[1., 2., 3., 4.], &[5., 6., 7., 8.]]);
+        let expected_0 = vec![4., 4., 4., 4.];
+        let expected_1 = vec![1.25, 1.25];
+
+        assert!(m.var(0).approximate_eq(&expected_0, std::f64::EPSILON));
+        assert!(m.var(1).approximate_eq(&expected_1, std::f64::EPSILON));
+    }
+
+    #[test]
+    fn scale() {
+        let mut m = DenseMatrix::from_2d_array(&[&[1., 2., 3.], &[4., 5., 6.]]);
+        let expected_0 = DenseMatrix::from_2d_array(&[&[-1., -1., -1.], &[1., 1., 1.]]);
+        let expected_1 = DenseMatrix::from_2d_array(&[&[-1.22, 0.0, 1.22], &[-1.22, 0.0, 1.22]]);
+
+        {
+            let mut m = m.clone();
+            m.scale_mut(&m.mean(0), &m.std(0), 0);
+            assert!(m.approximate_eq(&expected_0, std::f32::EPSILON));
+        }
+
+        m.scale_mut(&m.mean(1), &m.std(1), 1);
+        assert!(m.approximate_eq(&expected_1, 1e-2));
+    }
+}

src/linear/linear_regression.rs

@@ -154,8 +154,8 @@ impl<T: RealNumber, M: Matrix<T>> LinearRegression<T, M> {
     }
 
     /// Get estimates regression coefficients
-    pub fn coefficients(&self) -> M {
+    pub fn coefficients(&self) -> &M {
-        self.coefficients.clone()
+        &self.coefficients
     }
 
     /// Get estimate of intercept

src/linear/logistic_regression.rs

@@ -69,7 +69,8 @@ use crate::optimization::FunctionOrder;
 /// Logistic Regression
 #[derive(Serialize, Deserialize, Debug)]
 pub struct LogisticRegression<T: RealNumber, M: Matrix<T>> {
-    weights: M,
+    coefficients: M,
+    intercept: M,
     classes: Vec<T>,
     num_attributes: usize,
     num_classes: usize,
@@ -110,7 +111,7 @@ impl<T: RealNumber, M: Matrix<T>> PartialEq for LogisticRegression<T, M> {
                 }
             }
 
-            self.weights == other.weights
+            self.coefficients == other.coefficients && self.intercept == other.intercept
         }
     }
 }
@@ -237,27 +238,6 @@ impl<T: RealNumber, M: Matrix<T>> LogisticRegression<T, M> {
                 "incorrect number of classes: {}. Should be >= 2.",
                 k
             ))),
-            Ordering::Greater => {
-                let x0 = M::zeros(1, (num_attributes + 1) * k);
-
-                let objective = MultiClassObjectiveFunction {
-                    x,
-                    y: yi,
-                    k,
-                    phantom: PhantomData,
-                };
-
-                let result = LogisticRegression::minimize(x0, objective);
-
-                let weights = result.x.reshape(k, num_attributes + 1);
-
-                Ok(LogisticRegression {
-                    weights,
-                    classes,
-                    num_attributes,
-                    num_classes: k,
-                })
-            }
             Ordering::Equal => {
                 let x0 = M::zeros(1, num_attributes + 1);
 
@@ -269,8 +249,32 @@ impl<T: RealNumber, M: Matrix<T>> LogisticRegression<T, M> {
 
                 let result = LogisticRegression::minimize(x0, objective);
 
+                let weights = result.x;
+
                 Ok(LogisticRegression {
-                    weights: result.x,
+                    coefficients: weights.slice(0..1, 0..num_attributes),
+                    intercept: weights.slice(0..1, num_attributes..num_attributes + 1),
+                    classes,
+                    num_attributes,
+                    num_classes: k,
+                })
+            }
+            Ordering::Greater => {
+                let x0 = M::zeros(1, (num_attributes + 1) * k);
+
+                let objective = MultiClassObjectiveFunction {
+                    x,
+                    y: yi,
+                    k,
+                    phantom: PhantomData,
+                };
+
+                let result = LogisticRegression::minimize(x0, objective);
+                let weights = result.x.reshape(k, num_attributes + 1);
+
+                Ok(LogisticRegression {
+                    coefficients: weights.slice(0..k, 0..num_attributes),
+                    intercept: weights.slice(0..k, num_attributes..num_attributes + 1),
                     classes,
                     num_attributes,
                     num_classes: k,
@@ -285,22 +289,26 @@ impl<T: RealNumber, M: Matrix<T>> LogisticRegression<T, M> {
         let n = x.shape().0;
         let mut result = M::zeros(1, n);
         if self.num_classes == 2 {
-            let (nrows, _) = x.shape();
+            let y_hat: Vec<T> = x.matmul(&self.coefficients.transpose()).get_col_as_vec(0);
-            let x_and_bias = x.h_stack(&M::ones(nrows, 1));
+            let intercept = self.intercept.get(0, 0);
-            let y_hat: Vec<T> = x_and_bias
-                .matmul(&self.weights.transpose())
-                .get_col_as_vec(0);
             for i in 0..n {
                 result.set(
                     0,
                     i,
-                    self.classes[if y_hat[i].sigmoid() > T::half() { 1 } else { 0 }],
+                    self.classes[if (y_hat[i] + intercept).sigmoid() > T::half() {
+                        1
+                    } else {
+                        0
+                    }],
                 );
             }
         } else {
-            let (nrows, _) = x.shape();
+            let mut y_hat = x.matmul(&self.coefficients.transpose());
-            let x_and_bias = x.h_stack(&M::ones(nrows, 1));
+            for r in 0..n {
-            let y_hat = x_and_bias.matmul(&self.weights.transpose());
+                for c in 0..self.num_classes {
+                    y_hat.set(r, c, y_hat.get(r, c) + self.intercept.get(c, 0));
+                }
+            }
             let class_idxs = y_hat.argmax();
             for i in 0..n {
                 result.set(0, i, self.classes[class_idxs[i]]);
@@ -310,17 +318,13 @@ impl<T: RealNumber, M: Matrix<T>> LogisticRegression<T, M> {
     }
 
     /// Get estimates regression coefficients
-    pub fn coefficients(&self) -> M {
+    pub fn coefficients(&self) -> &M {
-        self.weights
+        &self.coefficients
-            .slice(0..self.num_classes, 0..self.num_attributes)
     }
 
     /// Get estimate of intercept
-    pub fn intercept(&self) -> M {
+    pub fn intercept(&self) -> &M {
-        self.weights.slice(
+        &self.intercept
-            0..self.num_classes,
-            self.num_attributes..self.num_attributes + 1,
-        )
     }
 
     fn minimize(x0: M, objective: impl ObjectiveFunction<T, M>) -> OptimizerResult<T, M> {
@@ -341,7 +345,9 @@ impl<T: RealNumber, M: Matrix<T>> LogisticRegression<T, M> {
 #[cfg(test)]
 mod tests {
     use super::*;
+    use crate::dataset::generator::make_blobs;
     use crate::linalg::naive::dense_matrix::*;
+    use crate::metrics::accuracy;
 
     #[test]
     fn multiclass_objective_f() {
@@ -471,6 +477,34 @@ mod tests {
         );
     }
 
+    #[test]
+    fn lr_fit_predict_multiclass() {
+        let blobs = make_blobs(15, 4, 3);
+
+        let x = DenseMatrix::from_vec(15, 4, &blobs.data);
+        let y = blobs.target;
+
+        let lr = LogisticRegression::fit(&x, &y).unwrap();
+
+        let y_hat = lr.predict(&x).unwrap();
+
+        assert!(accuracy(&y_hat, &y) > 0.9);
+    }
+
+    #[test]
+    fn lr_fit_predict_binary() {
+        let blobs = make_blobs(20, 4, 2);
+
+        let x = DenseMatrix::from_vec(20, 4, &blobs.data);
+        let y = blobs.target;
+
+        let lr = LogisticRegression::fit(&x, &y).unwrap();
+
+        let y_hat = lr.predict(&x).unwrap();
+
+        assert!(accuracy(&y_hat, &y) > 0.9);
+    }
+
     #[test]
     fn serde() {
         let x = DenseMatrix::from_2d_array(&[

src/linear/mod.rs

@@ -22,3 +22,4 @@
 
 pub mod linear_regression;
 pub mod logistic_regression;
+pub mod ridge_regression;

src/linear/ridge_regression.rs

@@ -0,0 +1,327 @@
+//! # Ridge Regression
+//!
+//! [Linear regression](../linear_regression/index.html) is the standard algorithm for predicting a quantitative response \\(y\\) on the basis of a linear combination of explanatory variables \\(X\\)
+//! that assumes that there is approximately a linear relationship between \\(X\\) and \\(y\\).
+//! Ridge regression is an extension to linear regression that adds L2 regularization term to the loss function during training.
+//! This term encourages simpler models that have smaller coefficient values.
+//!
+//! In ridge regression coefficients \\(\beta_0, \beta_0, ... \beta_n\\) are are estimated by solving
+//!
+//! \\[\hat{\beta} = (X^TX + \alpha I)^{-1}X^Ty \\]
+//!
+//! where \\(\alpha \geq 0\\) is a tuning parameter that controls strength of regularization. When \\(\alpha = 0\\) the penalty term has no effect, and ridge regression will produce the least squares estimates.
+//! However, as \\(\alpha \rightarrow \infty\\), the impact of the shrinkage penalty grows, and the ridge regression coefficient estimates will approach zero.
+//!
+//! SmartCore uses [SVD](../../linalg/svd/index.html) and [Cholesky](../../linalg/cholesky/index.html) matrix decomposition to find estimates of \\(\hat{\beta}\\).
+//! The Cholesky decomposition is more computationally efficient and more numerically stable than calculating the normal equation directly,
+//! but does not work for all data matrices. Unlike the Cholesky decomposition, all matrices have an SVD decomposition.
+//!
+//! Example:
+//!
+//! ```
+//! use smartcore::linalg::naive::dense_matrix::*;
+//! use smartcore::linear::ridge_regression::*;
+//!
+//! // 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 = RidgeRegression::fit(&x, &y, RidgeRegressionParameters {
+//!                        solver: RidgeRegressionSolverName::Cholesky,
+//!                        alpha: 0.1,
+//!                        normalize: true
+//! }).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/)
+//! * ["Numerical Recipes: The Art of Scientific Computing",  Press W.H., Teukolsky S.A., Vetterling W.T, Flannery B.P, 3rd ed., Section 15.4 General Linear Least Squares](http://numerical.recipes/)
+//!
+//! <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 serde::{Deserialize, Serialize};
+
+use crate::error::Failed;
+use crate::linalg::BaseVector;
+use crate::linalg::Matrix;
+use crate::math::num::RealNumber;
+
+#[derive(Serialize, Deserialize, Debug)]
+/// Approach to use for estimation of regression coefficients. Cholesky is more efficient but SVD is more stable.
+pub enum RidgeRegressionSolverName {
+    /// Cholesky decomposition, see [Cholesky](../../linalg/cholesky/index.html)
+    Cholesky,
+    /// SVD decomposition, see [SVD](../../linalg/svd/index.html)
+    SVD,
+}
+
+/// Ridge Regression parameters
+#[derive(Serialize, Deserialize, Debug)]
+pub struct RidgeRegressionParameters<T: RealNumber> {
+    /// Solver to use for estimation of regression coefficients.
+    pub solver: RidgeRegressionSolverName,
+    /// Controls the strength of the penalty to the loss function.
+    pub alpha: T,
+    /// If true the regressors X will be normalized before regression
+    /// by subtracting the mean and dividing by the standard deviation.
+    pub normalize: bool,
+}
+
+/// Ridge regression
+#[derive(Serialize, Deserialize, Debug)]
+pub struct RidgeRegression<T: RealNumber, M: Matrix<T>> {
+    coefficients: M,
+    intercept: T,
+    solver: RidgeRegressionSolverName,
+}
+
+impl<T: RealNumber> Default for RidgeRegressionParameters<T> {
+    fn default() -> Self {
+        RidgeRegressionParameters {
+            solver: RidgeRegressionSolverName::Cholesky,
+            alpha: T::one(),
+            normalize: true,
+        }
+    }
+}
+
+impl<T: RealNumber, M: Matrix<T>> PartialEq for RidgeRegression<T, M> {
+    fn eq(&self, other: &Self) -> bool {
+        self.coefficients == other.coefficients
+            && (self.intercept - other.intercept).abs() <= T::epsilon()
+    }
+}
+
+impl<T: RealNumber, M: Matrix<T>> RidgeRegression<T, M> {
+    /// Fits ridge 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: &M,
+        y: &M::RowVector,
+        parameters: RidgeRegressionParameters<T>,
+    ) -> Result<RidgeRegression<T, M>, Failed> {
+        //w = inv(X^t X + alpha*Id) * X.T y
+
+        let (n, p) = x.shape();
+
+        if n <= p {
+            return Err(Failed::fit(
+                "Number of rows in X should be >= number of columns in X",
+            ));
+        }
+
+        if y.len() != n {
+            return Err(Failed::fit("Number of rows in X should = len(y)"));
+        }
+
+        let y_column = M::from_row_vector(y.clone()).transpose();
+
+        let (w, b) = if parameters.normalize {
+            let (scaled_x, col_mean, col_std) = Self::rescale_x(x)?;
+            let x_t = scaled_x.transpose();
+            let x_t_y = x_t.matmul(&y_column);
+            let mut x_t_x = x_t.matmul(&scaled_x);
+
+            for i in 0..p {
+                x_t_x.add_element_mut(i, i, parameters.alpha);
+            }
+
+            let mut w = match parameters.solver {
+                RidgeRegressionSolverName::Cholesky => x_t_x.cholesky_solve_mut(x_t_y)?,
+                RidgeRegressionSolverName::SVD => x_t_x.svd_solve_mut(x_t_y)?,
+            };
+
+            for i in 0..p {
+                w.set(i, 0, w.get(i, 0) / col_std[i]);
+            }
+
+            let mut b = T::zero();
+
+            for i in 0..p {
+                b += w.get(i, 0) * col_mean[i];
+            }
+
+            let b = y.mean() - b;
+
+            (w, b)
+        } else {
+            let x_t = x.transpose();
+            let x_t_y = x_t.matmul(&y_column);
+            let mut x_t_x = x_t.matmul(x);
+
+            for i in 0..p {
+                x_t_x.add_element_mut(i, i, parameters.alpha);
+            }
+
+            let w = match parameters.solver {
+                RidgeRegressionSolverName::Cholesky => x_t_x.cholesky_solve_mut(x_t_y)?,
+                RidgeRegressionSolverName::SVD => x_t_x.svd_solve_mut(x_t_y)?,
+            };
+
+            (w, T::zero())
+        };
+
+        Ok(RidgeRegression {
+            intercept: b,
+            coefficients: w,
+            solver: parameters.solver,
+        })
+    }
+
+    fn rescale_x(x: &M) -> Result<(M, Vec<T>, Vec<T>), Failed> {
+        let col_mean = x.mean(0);
+        let col_std = x.std(0);
+
+        for i in 0..col_std.len() {
+            if (col_std[i] - T::zero()).abs() < T::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))
+    }
+
+    /// 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: &M) -> Result<M::RowVector, Failed> {
+        let (nrows, _) = x.shape();
+        let mut y_hat = x.matmul(&self.coefficients);
+        y_hat.add_mut(&M::fill(nrows, 1, self.intercept));
+        Ok(y_hat.transpose().to_row_vector())
+    }
+
+    /// Get estimates regression coefficients
+    pub fn coefficients(&self) -> &M {
+        &self.coefficients
+    }
+
+    /// Get estimate of intercept
+    pub fn intercept(&self) -> T {
+        self.intercept
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::linalg::naive::dense_matrix::*;
+    use crate::metrics::mean_absolute_error;
+
+    #[test]
+    fn ridge_fit_predict() {
+        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_cholesky = RidgeRegression::fit(
+            &x,
+            &y,
+            RidgeRegressionParameters {
+                solver: RidgeRegressionSolverName::Cholesky,
+                alpha: 0.1,
+                normalize: true,
+            },
+        )
+        .and_then(|lr| lr.predict(&x))
+        .unwrap();
+
+        assert!(mean_absolute_error(&y_hat_cholesky, &y) < 2.0);
+
+        let y_hat_svd = RidgeRegression::fit(
+            &x,
+            &y,
+            RidgeRegressionParameters {
+                solver: RidgeRegressionSolverName::SVD,
+                alpha: 0.1,
+                normalize: false,
+            },
+        )
+        .and_then(|lr| lr.predict(&x))
+        .unwrap();
+
+        assert!(mean_absolute_error(&y_hat_svd, &y) < 2.0);
+    }
+
+    #[test]
+    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 = RidgeRegression::fit(&x, &y, Default::default()).unwrap();
+
+        let deserialized_lr: RidgeRegression<f64, DenseMatrix<f64>> =
+            serde_json::from_str(&serde_json::to_string(&lr).unwrap()).unwrap();
+
+        assert_eq!(lr, deserialized_lr);
+    }
+}