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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::naive::dense_matrix::*; //! 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 serde::{Deserialize, Serialize}; use crate::api::{Predictor, SupervisedEstimator}; use crate::error::Failed; use crate::linalg::Matrix; use crate::math::num::RealNumber; #[derive(Serialize, Deserialize, Debug, Clone)] /// 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, /// SVD decomposition, see [SVD](../../linalg/svd/index.html) SVD, } /// Linear Regression parameters #[derive(Serialize, Deserialize, Debug, Clone)] pub struct LinearRegressionParameters { /// Solver to use for estimation of regression coefficients. pub solver: LinearRegressionSolverName, } /// Linear Regression #[derive(Serialize, Deserialize, Debug)] pub struct LinearRegression<T: RealNumber, M: Matrix<T>> { coefficients: M, intercept: T, solver: LinearRegressionSolverName, } impl LinearRegressionParameters { /// Solver to use for estimation of regression coefficients. pub fn with_solver(mut self, solver: LinearRegressionSolverName) -> Self { self.solver = solver; self } } impl Default for LinearRegressionParameters { fn default() -> Self { LinearRegressionParameters { solver: LinearRegressionSolverName::SVD, } } } impl<T: RealNumber, M: Matrix<T>> PartialEq for LinearRegression<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>> SupervisedEstimator<M, M::RowVector, LinearRegressionParameters> for LinearRegression<T, M> { fn fit( x: &M, y: &M::RowVector, parameters: LinearRegressionParameters, ) -> Result<Self, Failed> { LinearRegression::fit(x, y, parameters) } } impl<T: RealNumber, M: Matrix<T>> Predictor<M, M::RowVector> for LinearRegression<T, M> { fn predict(&self, x: &M) -> Result<M::RowVector, Failed> { self.predict(x) } } impl<T: RealNumber, M: Matrix<T>> LinearRegression<T, M> { /// 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: &M, y: &M::RowVector, parameters: LinearRegressionParameters, ) -> Result<LinearRegression<T, M>, Failed> { let y_m = M::from_row_vector(y.clone()); let b = y_m.transpose(); 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".to_string(), )); } let a = x.h_stack(&M::ones(x_nrows, 1)); let w = match parameters.solver { LinearRegressionSolverName::QR => a.qr_solve_mut(b)?, LinearRegressionSolverName::SVD => a.svd_solve_mut(b)?, }; let wights = w.slice(0..num_attributes, 0..1); Ok(LinearRegression { intercept: w.get(num_attributes, 0), coefficients: wights, solver: parameters.solver, }) } /// 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::*; #[test] fn ols_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_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)); } #[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 = LinearRegression::fit(&x, &y, Default::default()).unwrap(); let deserialized_lr: LinearRegression<f64, DenseMatrix<f64>> = serde_json::from_str(&serde_json::to_string(&lr).unwrap()).unwrap(); assert_eq!(lr, deserialized_lr); } }