cfbd45bfc0
* Improve features * Add wasm32-wasi as a target * Update .github/workflows/ci.yml Co-authored-by: morenol <22335041+morenol@users.noreply.github.com>
542 lines
20 KiB
Rust
542 lines
20 KiB
Rust
//! # Ridge Regression
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//!
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//! [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\\)
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//! that assumes that there is approximately a linear relationship between \\(X\\) and \\(y\\).
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//! Ridge regression is an extension to linear regression that adds L2 regularization term to the loss function during training.
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//! This term encourages simpler models that have smaller coefficient values.
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//!
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//! In ridge regression coefficients \\(\beta_0, \beta_0, ... \beta_n\\) are are estimated by solving
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//!
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//! \\[\hat{\beta} = (X^TX + \alpha I)^{-1}X^Ty \\]
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//!
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//! 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.
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//! However, as \\(\alpha \rightarrow \infty\\), the impact of the shrinkage penalty grows, and the ridge regression coefficient estimates will approach zero.
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//!
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//! SmartCore uses [SVD](../../linalg/svd/index.html) and [Cholesky](../../linalg/cholesky/index.html) matrix decomposition to find estimates of \\(\hat{\beta}\\).
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//! The Cholesky decomposition is more computationally efficient and more numerically stable than calculating the normal equation directly,
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//! but does not work for all data matrices. Unlike the Cholesky decomposition, all matrices have an SVD decomposition.
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//!
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//! Example:
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//!
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//! ```
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//! use smartcore::linalg::basic::matrix::DenseMatrix;
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//! use smartcore::linear::ridge_regression::*;
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//!
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//! // Longley dataset (https://www.statsmodels.org/stable/datasets/generated/longley.html)
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//! let x = DenseMatrix::from_2d_array(&[
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//! &[234.289, 235.6, 159.0, 107.608, 1947., 60.323],
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//! &[259.426, 232.5, 145.6, 108.632, 1948., 61.122],
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//! &[258.054, 368.2, 161.6, 109.773, 1949., 60.171],
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//! &[284.599, 335.1, 165.0, 110.929, 1950., 61.187],
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//! &[328.975, 209.9, 309.9, 112.075, 1951., 63.221],
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//! &[346.999, 193.2, 359.4, 113.270, 1952., 63.639],
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//! &[365.385, 187.0, 354.7, 115.094, 1953., 64.989],
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//! &[363.112, 357.8, 335.0, 116.219, 1954., 63.761],
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//! &[397.469, 290.4, 304.8, 117.388, 1955., 66.019],
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//! &[419.180, 282.2, 285.7, 118.734, 1956., 67.857],
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//! &[442.769, 293.6, 279.8, 120.445, 1957., 68.169],
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//! &[444.546, 468.1, 263.7, 121.950, 1958., 66.513],
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//! &[482.704, 381.3, 255.2, 123.366, 1959., 68.655],
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//! &[502.601, 393.1, 251.4, 125.368, 1960., 69.564],
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//! &[518.173, 480.6, 257.2, 127.852, 1961., 69.331],
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//! &[554.894, 400.7, 282.7, 130.081, 1962., 70.551],
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//! ]);
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//!
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//! let y: Vec<f64> = vec![83.0, 88.5, 88.2, 89.5, 96.2, 98.1, 99.0,
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//! 100.0, 101.2, 104.6, 108.4, 110.8, 112.6, 114.2, 115.7, 116.9];
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//!
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//! let y_hat = RidgeRegression::fit(&x, &y, RidgeRegressionParameters::default().with_alpha(0.1)).
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//! and_then(|lr| lr.predict(&x)).unwrap();
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//! ```
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//!
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//! ## References:
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//!
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//! * ["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/)
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//! * ["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/)
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//!
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//! <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
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//! <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
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use std::fmt::Debug;
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use std::marker::PhantomData;
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#[cfg(feature = "serde")]
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use serde::{Deserialize, Serialize};
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use crate::api::{Predictor, SupervisedEstimator};
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use crate::error::Failed;
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use crate::linalg::basic::arrays::{Array1, Array2};
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use crate::linalg::traits::cholesky::CholeskyDecomposable;
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use crate::linalg::traits::svd::SVDDecomposable;
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use crate::numbers::basenum::Number;
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use crate::numbers::realnum::RealNumber;
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#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
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#[derive(Debug, Clone, Eq, PartialEq)]
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/// Approach to use for estimation of regression coefficients. Cholesky is more efficient but SVD is more stable.
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pub enum RidgeRegressionSolverName {
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/// Cholesky decomposition, see [Cholesky](../../linalg/cholesky/index.html)
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Cholesky,
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/// SVD decomposition, see [SVD](../../linalg/svd/index.html)
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SVD,
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}
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impl Default for RidgeRegressionSolverName {
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fn default() -> Self {
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RidgeRegressionSolverName::Cholesky
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}
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}
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/// Ridge Regression parameters
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#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
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#[derive(Debug, Clone)]
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pub struct RidgeRegressionParameters<T: Number + RealNumber> {
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/// Solver to use for estimation of regression coefficients.
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pub solver: RidgeRegressionSolverName,
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/// Controls the strength of the penalty to the loss function.
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pub alpha: T,
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/// If true the regressors X will be normalized before regression
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/// by subtracting the mean and dividing by the standard deviation.
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pub normalize: bool,
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}
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/// Ridge Regression grid search parameters
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#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
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#[derive(Debug, Clone)]
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pub struct RidgeRegressionSearchParameters<T: Number + RealNumber> {
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#[cfg_attr(feature = "serde", serde(default))]
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/// Solver to use for estimation of regression coefficients.
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pub solver: Vec<RidgeRegressionSolverName>,
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#[cfg_attr(feature = "serde", serde(default))]
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/// Regularization parameter.
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pub alpha: Vec<T>,
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#[cfg_attr(feature = "serde", serde(default))]
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/// If true the regressors X will be normalized before regression
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/// by subtracting the mean and dividing by the standard deviation.
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pub normalize: Vec<bool>,
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}
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/// Ridge Regression grid search iterator
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pub struct RidgeRegressionSearchParametersIterator<T: Number + RealNumber> {
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ridge_regression_search_parameters: RidgeRegressionSearchParameters<T>,
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current_solver: usize,
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current_alpha: usize,
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current_normalize: usize,
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}
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impl<T: Number + RealNumber> IntoIterator for RidgeRegressionSearchParameters<T> {
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type Item = RidgeRegressionParameters<T>;
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type IntoIter = RidgeRegressionSearchParametersIterator<T>;
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fn into_iter(self) -> Self::IntoIter {
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RidgeRegressionSearchParametersIterator {
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ridge_regression_search_parameters: self,
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current_solver: 0,
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current_alpha: 0,
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current_normalize: 0,
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}
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}
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}
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impl<T: Number + RealNumber> Iterator for RidgeRegressionSearchParametersIterator<T> {
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type Item = RidgeRegressionParameters<T>;
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fn next(&mut self) -> Option<Self::Item> {
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if self.current_alpha == self.ridge_regression_search_parameters.alpha.len()
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&& self.current_solver == self.ridge_regression_search_parameters.solver.len()
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{
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return None;
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}
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let next = RidgeRegressionParameters {
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solver: self.ridge_regression_search_parameters.solver[self.current_solver].clone(),
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alpha: self.ridge_regression_search_parameters.alpha[self.current_alpha],
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normalize: self.ridge_regression_search_parameters.normalize[self.current_normalize],
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};
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if self.current_alpha + 1 < self.ridge_regression_search_parameters.alpha.len() {
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self.current_alpha += 1;
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} else if self.current_solver + 1 < self.ridge_regression_search_parameters.solver.len() {
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self.current_alpha = 0;
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self.current_solver += 1;
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} else if self.current_normalize + 1
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< self.ridge_regression_search_parameters.normalize.len()
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{
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self.current_alpha = 0;
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self.current_solver = 0;
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self.current_normalize += 1;
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} else {
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self.current_alpha += 1;
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self.current_solver += 1;
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self.current_normalize += 1;
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}
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Some(next)
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}
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}
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impl<T: Number + RealNumber> Default for RidgeRegressionSearchParameters<T> {
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fn default() -> Self {
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let default_params = RidgeRegressionParameters::default();
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RidgeRegressionSearchParameters {
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solver: vec![default_params.solver],
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alpha: vec![default_params.alpha],
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normalize: vec![default_params.normalize],
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}
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}
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}
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/// Ridge regression
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#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
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#[derive(Debug)]
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pub struct RidgeRegression<
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TX: Number + RealNumber,
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TY: Number,
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X: Array2<TX> + CholeskyDecomposable<TX> + SVDDecomposable<TX>,
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Y: Array1<TY>,
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> {
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coefficients: Option<X>,
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intercept: Option<TX>,
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solver: Option<RidgeRegressionSolverName>,
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_phantom_ty: PhantomData<TY>,
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_phantom_y: PhantomData<Y>,
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}
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impl<T: Number + RealNumber> RidgeRegressionParameters<T> {
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/// Regularization parameter.
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pub fn with_alpha(mut self, alpha: T) -> Self {
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self.alpha = alpha;
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self
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}
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/// Solver to use for estimation of regression coefficients.
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pub fn with_solver(mut self, solver: RidgeRegressionSolverName) -> Self {
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self.solver = solver;
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self
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}
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/// If True, the regressors X will be normalized before regression by subtracting the mean and dividing by the standard deviation.
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pub fn with_normalize(mut self, normalize: bool) -> Self {
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self.normalize = normalize;
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self
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}
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}
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impl<T: Number + RealNumber> Default for RidgeRegressionParameters<T> {
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fn default() -> Self {
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RidgeRegressionParameters {
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solver: RidgeRegressionSolverName::default(),
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alpha: T::from_f64(1.0).unwrap(),
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normalize: true,
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}
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}
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}
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impl<
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TX: Number + RealNumber,
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TY: Number,
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X: Array2<TX> + CholeskyDecomposable<TX> + SVDDecomposable<TX>,
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Y: Array1<TY>,
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> PartialEq for RidgeRegression<TX, TY, X, Y>
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{
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fn eq(&self, other: &Self) -> bool {
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self.intercept() == other.intercept()
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&& self.coefficients().shape() == other.coefficients().shape()
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&& self
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.coefficients()
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.iterator(0)
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.zip(other.coefficients().iterator(0))
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.all(|(&a, &b)| (a - b).abs() <= TX::epsilon())
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}
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}
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impl<
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TX: Number + RealNumber,
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TY: Number,
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X: Array2<TX> + CholeskyDecomposable<TX> + SVDDecomposable<TX>,
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Y: Array1<TY>,
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> SupervisedEstimator<X, Y, RidgeRegressionParameters<TX>> for RidgeRegression<TX, TY, X, Y>
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{
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fn new() -> Self {
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Self {
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coefficients: Option::None,
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intercept: Option::None,
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solver: Option::None,
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_phantom_ty: PhantomData,
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_phantom_y: PhantomData,
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}
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}
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fn fit(x: &X, y: &Y, parameters: RidgeRegressionParameters<TX>) -> Result<Self, Failed> {
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RidgeRegression::fit(x, y, parameters)
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}
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}
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impl<
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TX: Number + RealNumber,
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TY: Number,
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X: Array2<TX> + CholeskyDecomposable<TX> + SVDDecomposable<TX>,
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Y: Array1<TY>,
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> Predictor<X, Y> for RidgeRegression<TX, TY, X, Y>
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{
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fn predict(&self, x: &X) -> Result<Y, Failed> {
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self.predict(x)
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}
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}
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impl<
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TX: Number + RealNumber,
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TY: Number,
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X: Array2<TX> + CholeskyDecomposable<TX> + SVDDecomposable<TX>,
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Y: Array1<TY>,
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> RidgeRegression<TX, TY, X, Y>
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{
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/// Fits ridge regression to your data.
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/// * `x` - _NxM_ matrix with _N_ observations and _M_ features in each observation.
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/// * `y` - target values
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/// * `parameters` - other parameters, use `Default::default()` to set parameters to default values.
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pub fn fit(
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x: &X,
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y: &Y,
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parameters: RidgeRegressionParameters<TX>,
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) -> Result<RidgeRegression<TX, TY, X, Y>, Failed> {
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//w = inv(X^t X + alpha*Id) * X.T y
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let (n, p) = x.shape();
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if n <= p {
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return Err(Failed::fit(
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"Number of rows in X should be >= number of columns in X",
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));
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}
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if y.shape() != n {
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return Err(Failed::fit("Number of rows in X should = len(y)"));
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}
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let y_column = X::from_iterator(
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y.iterator(0).map(|&v| TX::from(v).unwrap()),
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y.shape(),
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1,
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0,
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);
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let (w, b) = if parameters.normalize {
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let (scaled_x, col_mean, col_std) = Self::rescale_x(x)?;
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let x_t = scaled_x.transpose();
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let x_t_y = x_t.matmul(&y_column);
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let mut x_t_x = x_t.matmul(&scaled_x);
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for i in 0..p {
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x_t_x.add_element_mut((i, i), parameters.alpha);
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}
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let mut w = match parameters.solver {
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RidgeRegressionSolverName::Cholesky => x_t_x.cholesky_solve_mut(x_t_y)?,
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RidgeRegressionSolverName::SVD => x_t_x.svd_solve_mut(x_t_y)?,
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};
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for (i, col_std_i) in col_std.iter().enumerate().take(p) {
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w.set((i, 0), *w.get((i, 0)) / *col_std_i);
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}
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let mut b = TX::zero();
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for (i, col_mean_i) in col_mean.iter().enumerate().take(p) {
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b += *w.get((i, 0)) * *col_mean_i;
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}
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let b = TX::from_f64(y.mean_by()).unwrap() - b;
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(w, b)
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} else {
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let x_t = x.transpose();
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let x_t_y = x_t.matmul(&y_column);
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let mut x_t_x = x_t.matmul(x);
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for i in 0..p {
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x_t_x.add_element_mut((i, i), parameters.alpha);
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}
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let w = match parameters.solver {
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RidgeRegressionSolverName::Cholesky => x_t_x.cholesky_solve_mut(x_t_y)?,
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RidgeRegressionSolverName::SVD => x_t_x.svd_solve_mut(x_t_y)?,
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};
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(w, TX::zero())
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};
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Ok(RidgeRegression {
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intercept: Some(b),
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coefficients: Some(w),
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solver: Some(parameters.solver),
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_phantom_ty: PhantomData,
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_phantom_y: PhantomData,
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})
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}
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fn rescale_x(x: &X) -> Result<(X, Vec<TX>, Vec<TX>), Failed> {
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let col_mean: Vec<TX> = x
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.mean_by(0)
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.iter()
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.map(|&v| TX::from_f64(v).unwrap())
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.collect();
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let col_std: Vec<TX> = x
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.std_dev(0)
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.iter()
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.map(|&v| TX::from_f64(v).unwrap())
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.collect();
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for (i, col_std_i) in col_std.iter().enumerate() {
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if (*col_std_i - TX::zero()).abs() < TX::epsilon() {
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return Err(Failed::fit(&format!(
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"Cannot rescale constant column {}",
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i
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)));
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}
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}
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let mut scaled_x = x.clone();
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scaled_x.scale_mut(&col_mean, &col_std, 0);
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Ok((scaled_x, col_mean, col_std))
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}
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/// Predict target values from `x`
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/// * `x` - _KxM_ data where _K_ is number of observations and _M_ is number of features.
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pub fn predict(&self, x: &X) -> Result<Y, Failed> {
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let (nrows, _) = x.shape();
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let mut y_hat = x.matmul(self.coefficients());
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y_hat.add_mut(&X::fill(nrows, 1, self.intercept.unwrap()));
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Ok(Y::from_iterator(
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y_hat.iterator(0).map(|&v| TY::from(v).unwrap()),
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nrows,
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))
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}
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/// Get estimates regression coefficients
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pub fn coefficients(&self) -> &X {
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self.coefficients.as_ref().unwrap()
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}
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/// Get estimate of intercept
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pub fn intercept(&self) -> &TX {
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self.intercept.as_ref().unwrap()
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}
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}
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#[cfg(test)]
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mod tests {
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use super::*;
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use crate::linalg::basic::matrix::DenseMatrix;
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use crate::metrics::mean_absolute_error;
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#[test]
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fn search_parameters() {
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let parameters = RidgeRegressionSearchParameters {
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alpha: vec![0., 1.],
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..Default::default()
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};
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let mut iter = parameters.into_iter();
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assert_eq!(iter.next().unwrap().alpha, 0.);
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assert_eq!(
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iter.next().unwrap().solver,
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RidgeRegressionSolverName::Cholesky
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);
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assert!(iter.next().is_none());
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}
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#[cfg_attr(
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all(target_arch = "wasm32", not(target_os = "wasi")),
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wasm_bindgen_test::wasm_bindgen_test
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)]
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#[test]
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fn ridge_fit_predict() {
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let x = DenseMatrix::from_2d_array(&[
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&[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);
|
|
}
|
|
|
|
// TODO: implement serialization for new DenseMatrix
|
|
// #[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 = RidgeRegression::fit(&x, &y, Default::default()).unwrap();
|
|
|
|
// let deserialized_lr: RidgeRegression<f64, f64, DenseMatrix<f64>, Vec<f64>> =
|
|
// serde_json::from_str(&serde_json::to_string(&lr).unwrap()).unwrap();
|
|
|
|
// assert_eq!(lr, deserialized_lr);
|
|
// }
|
|
}
|