feat: + ridge regression
This commit is contained in:
Volodymyr Orlov
@@ -22,3 +22,4 @@
|
||||
|
||||
pub mod linear_regression;
|
||||
pub mod logistic_regression;
|
||||
pub mod ridge_regression;
|
||||
|
||||
@@ -0,0 +1,323 @@
|
||||
//! # 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(&format!(
|
||||
"Number of rows in X should be >= number of columns in X"
|
||||
)));
|
||||
}
|
||||
|
||||
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.clone()
|
||||
}
|
||||
|
||||
/// 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);
|
||||
}
|
||||
}
|
||||
Reference in New Issue
Block a user