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//! # Linear Regression
//!
//! Linear regression is a very straightforward approach for predicting a quantitative response \\(y\\) on the basis of a linear combination of explanatory variables \\(X\\).
//! Linear regression assumes that there is approximately a linear relationship between \\(X\\) and \\(y\\). Formally, we can write this linear relationship as
//!
//! \\[y \approx \beta_0 + \sum_{i=1}^n \beta_iX_i + \epsilon\\]
//!
//! where \\(\epsilon\\) is a mean-zero random error term and the regression coefficients \\(\beta_0, \beta_0, ... \beta_n\\) are unknown, and must be estimated.
//!
//! While regression coefficients can be estimated directly by solving
//!
//! \\[\hat{\beta} = (X^TX)^{-1}X^Ty \\]
//!
//! the \\((X^TX)^{-1}\\) term is both computationally expensive and numerically unstable. An alternative approach is to use a matrix decomposition to avoid this operation.
//! `smartcore` uses [SVD](../../linalg/svd/index.html) and [QR](../../linalg/qr/index.html) matrix decomposition to find estimates of \\(\hat{\beta}\\).
//! The QR 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 QR decomposition, all matrices have an SVD decomposition.
//!
//! Example:
//!
//! ```
//! use smartcore::linalg::basic::matrix::DenseMatrix;
//! use smartcore::linear::linear_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 lr = LinearRegression::fit(&x, &y,
//! LinearRegressionParameters::default().
//! with_solver(LinearRegressionSolverName::QR)).unwrap();
//!
//! let y_hat = lr.predict(&x).unwrap();
//! ```
//!
//! ## References:
//!
//! * ["Pattern Recognition and Machine Learning", C.M. Bishop, Linear Models for Regression](https://www.microsoft.com/en-us/research/uploads/prod/2006/01/Bishop-Pattern-Recognition-and-Machine-Learning-2006.pdf)
//! * ["An Introduction to Statistical Learning", James G., Witten D., Hastie T., Tibshirani R., 3. Linear Regression](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 std::marker::PhantomData;
#[cfg(feature = "serde")]
use serde::{Deserialize, Serialize};
use crate::api::{Predictor, SupervisedEstimator};
use crate::error::Failed;
use crate::linalg::basic::arrays::{Array1, Array2};
use crate::linalg::traits::qr::QRDecomposable;
use crate::linalg::traits::svd::SVDDecomposable;
use crate::numbers::basenum::Number;
use crate::numbers::realnum::RealNumber;
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
#[derive(Debug, Default, Clone, Eq, PartialEq)]
/// Approach to use for estimation of regression coefficients. QR is more efficient but SVD is more stable.
pub enum LinearRegressionSolverName {
/// QR decomposition, see [QR](../../linalg/qr/index.html)
QR,
#[default]
/// SVD decomposition, see [SVD](../../linalg/svd/index.html)
SVD,
}
/// Linear Regression parameters
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
#[derive(Debug, Clone)]
pub struct LinearRegressionParameters {
#[cfg_attr(feature = "serde", serde(default))]
/// Solver to use for estimation of regression coefficients.
pub solver: LinearRegressionSolverName,
}
impl Default for LinearRegressionParameters {
fn default() -> Self {
LinearRegressionParameters {
solver: LinearRegressionSolverName::SVD,
}
}
}
/// Linear Regression
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
#[derive(Debug)]
pub struct LinearRegression<
TX: Number + RealNumber,
TY: Number,
X: Array2<TX> + QRDecomposable<TX> + SVDDecomposable<TX>,
Y: Array1<TY>,
> {
coefficients: Option<X>,
intercept: Option<TX>,
_phantom_ty: PhantomData<TY>,
_phantom_y: PhantomData<Y>,
}
impl LinearRegressionParameters {
/// Solver to use for estimation of regression coefficients.
pub fn with_solver(mut self, solver: LinearRegressionSolverName) -> Self {
self.solver = solver;
self
}
}
/// Linear Regression grid search parameters
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
#[derive(Debug, Clone)]
pub struct LinearRegressionSearchParameters {
#[cfg_attr(feature = "serde", serde(default))]
/// Solver to use for estimation of regression coefficients.
pub solver: Vec<LinearRegressionSolverName>,
}
/// Linear Regression grid search iterator
pub struct LinearRegressionSearchParametersIterator {
linear_regression_search_parameters: LinearRegressionSearchParameters,
current_solver: usize,
}
impl IntoIterator for LinearRegressionSearchParameters {
type Item = LinearRegressionParameters;
type IntoIter = LinearRegressionSearchParametersIterator;
fn into_iter(self) -> Self::IntoIter {
LinearRegressionSearchParametersIterator {
linear_regression_search_parameters: self,
current_solver: 0,
}
}
}
impl Iterator for LinearRegressionSearchParametersIterator {
type Item = LinearRegressionParameters;
fn next(&mut self) -> Option<Self::Item> {
if self.current_solver == self.linear_regression_search_parameters.solver.len() {
return None;
}
let next = LinearRegressionParameters {
solver: self.linear_regression_search_parameters.solver[self.current_solver].clone(),
};
self.current_solver += 1;
Some(next)
}
}
impl Default for LinearRegressionSearchParameters {
fn default() -> Self {
let default_params = LinearRegressionParameters::default();
LinearRegressionSearchParameters {
solver: vec![default_params.solver],
}
}
}
impl<
TX: Number + RealNumber,
TY: Number,
X: Array2<TX> + QRDecomposable<TX> + SVDDecomposable<TX>,
Y: Array1<TY>,
> PartialEq for LinearRegression<TX, TY, X, Y>
{
fn eq(&self, other: &Self) -> bool {
self.intercept == other.intercept
&& self.coefficients().shape() == other.coefficients().shape()
&& self
.coefficients()
.iterator(0)
.zip(other.coefficients().iterator(0))
.all(|(&a, &b)| (a - b).abs() <= TX::epsilon())
}
}
impl<
TX: Number + RealNumber,
TY: Number,
X: Array2<TX> + QRDecomposable<TX> + SVDDecomposable<TX>,
Y: Array1<TY>,
> SupervisedEstimator<X, Y, LinearRegressionParameters> for LinearRegression<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: LinearRegressionParameters) -> Result<Self, Failed> {
LinearRegression::fit(x, y, parameters)
}
}
impl<
TX: Number + RealNumber,
TY: Number,
X: Array2<TX> + QRDecomposable<TX> + SVDDecomposable<TX>,
Y: Array1<TY>,
> Predictor<X, Y> for LinearRegression<TX, TY, X, Y>
{
fn predict(&self, x: &X) -> Result<Y, Failed> {
self.predict(x)
}
}
impl<
TX: Number + RealNumber,
TY: Number,
X: Array2<TX> + QRDecomposable<TX> + SVDDecomposable<TX>,
Y: Array1<TY>,
> LinearRegression<TX, TY, X, Y>
{
/// Fits Linear 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: LinearRegressionParameters,
) -> Result<LinearRegression<TX, TY, X, Y>, Failed> {
let b = X::from_iterator(
y.iterator(0).map(|&v| TX::from(v).unwrap()),
y.shape(),
1,
0,
);
let (x_nrows, num_attributes) = x.shape();
let (y_nrows, _) = b.shape();
if x_nrows != y_nrows {
return Err(Failed::fit(
"Number of rows of X doesn\'t match number of rows of Y",
));
}
let a = x.h_stack(&X::ones(x_nrows, 1));
let w = match parameters.solver {
LinearRegressionSolverName::QR => a.qr_solve_mut(b)?,
LinearRegressionSolverName::SVD => a.svd_solve_mut(b)?,
};
let weights = X::from_slice(w.slice(0..num_attributes, 0..1).as_ref());
Ok(LinearRegression {
intercept: Some(*w.get((num_attributes, 0))),
coefficients: Some(weights),
_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 bias = X::fill(nrows, 1, *self.intercept());
let mut y_hat = x.matmul(self.coefficients());
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()
}
}
#[cfg(test)]
mod tests {
use super::*;
use crate::linalg::basic::matrix::DenseMatrix;
#[test]
fn search_parameters() {
let parameters = LinearRegressionSearchParameters {
solver: vec![
LinearRegressionSolverName::QR,
LinearRegressionSolverName::SVD,
],
};
let mut iter = parameters.into_iter();
assert_eq!(iter.next().unwrap().solver, LinearRegressionSolverName::QR);
assert_eq!(iter.next().unwrap().solver, LinearRegressionSolverName::SVD);
assert!(iter.next().is_none());
}
#[cfg_attr(
all(target_arch = "wasm32", not(target_os = "wasi")),
wasm_bindgen_test::wasm_bindgen_test
)]
#[test]
fn ols_fit_predict() {
let x = DenseMatrix::from_2d_array(&[
&[234.289, 235.6, 159.0, 107.608, 1947., 60.323],
&[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],
&[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,
];
let y_hat_qr = LinearRegression::fit(
&x,
&y,
LinearRegressionParameters {
solver: LinearRegressionSolverName::QR,
},
)
.and_then(|lr| lr.predict(&x))
.unwrap();
let y_hat_svd = LinearRegression::fit(&x, &y, Default::default())
.and_then(|lr| lr.predict(&x))
.unwrap();
assert!(y
.iter()
.zip(y_hat_qr.iter())
.all(|(&a, &b)| (a - b).abs() <= 5.0));
assert!(y
.iter()
.zip(y_hat_svd.iter())
.all(|(&a, &b)| (a - b).abs() <= 5.0));
}
// 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 = LinearRegression::fit(&x, &y, Default::default()).unwrap();
// let deserialized_lr: LinearRegression<f64, f64, DenseMatrix<f64>, Vec<f64>> =
// serde_json::from_str(&serde_json::to_string(&lr).unwrap()).unwrap();
// assert_eq!(lr, deserialized_lr);
// let default = LinearRegressionParameters::default();
// let parameters: LinearRegressionParameters = serde_json::from_str("{}").unwrap();
// assert_eq!(parameters.solver, default.solver);
// }
}