Merge potential next release v0.4 (#187) Breaking Changes
* First draft of the new n-dimensional arrays + NB use case * Improves default implementation of multiple Array methods * Refactors tree methods * Adds matrix decomposition routines * Adds matrix decomposition methods to ndarray and nalgebra bindings * Refactoring + linear regression now uses array2 * Ridge & Linear regression * LBFGS optimizer & logistic regression * LBFGS optimizer & logistic regression * Changes linear methods, metrics and model selection methods to new n-dimensional arrays * Switches KNN and clustering algorithms to new n-d array layer * Refactors distance metrics * Optimizes knn and clustering methods * Refactors metrics module * Switches decomposition methods to n-dimensional arrays * Linalg refactoring - cleanup rng merge (#172) * Remove legacy DenseMatrix and BaseMatrix implementation. Port the new Number, FloatNumber and Array implementation into module structure. * Exclude AUC metrics. Needs reimplementation * Improve developers walkthrough New traits system in place at `src/numbers` and `src/linalg` Co-authored-by: Lorenzo <tunedconsulting@gmail.com> * Provide SupervisedEstimator with a constructor to avoid explicit dynamical box allocation in 'cross_validate' and 'cross_validate_predict' as required by the use of 'dyn' as per Rust 2021 * Implement getters to use as_ref() in src/neighbors * Implement getters to use as_ref() in src/naive_bayes * Implement getters to use as_ref() in src/linear * Add Clone to src/naive_bayes * Change signature for cross_validate and other model_selection functions to abide to use of dyn in Rust 2021 * Implement ndarray-bindings. Remove FloatNumber from implementations * Drop nalgebra-bindings support (as decided in conf-call to go for ndarray) * Remove benches. Benches will have their own repo at smartcore-benches * Implement SVC * Implement SVC serialization. Move search parameters in dedicated module * Implement SVR. Definitely too slow * Fix compilation issues for wasm (#202) Co-authored-by: Luis Moreno <morenol@users.noreply.github.com> * Fix tests (#203) * Port linalg/traits/stats.rs * Improve methods naming * Improve Display for DenseMatrix Co-authored-by: Montana Low <montanalow@users.noreply.github.com> Co-authored-by: VolodymyrOrlov <volodymyr.orlov@gmail.com>
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Lorenzo
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//! # QR Decomposition
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//!
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//! Any real square matrix \\(A \in R^{n \times n}\\) can be decomposed as a product of an orthogonal matrix \\(Q\\) and an upper triangular matrix \\(R\\):
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//!
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//! \\[A = QR\\]
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//!
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//! Example:
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//! ```
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//! use smartcore::linalg::basic::matrix::DenseMatrix;
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//! use smartcore::linalg::traits::qr::*;
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//!
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//! let A = DenseMatrix::from_2d_array(&[
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//! &[0.9, 0.4, 0.7],
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//! &[0.4, 0.5, 0.3],
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//! &[0.7, 0.3, 0.8]
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//! ]);
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//!
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//! let qr = A.qr().unwrap();
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//! let orthogonal: DenseMatrix<f64> = qr.Q();
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//! let triangular: DenseMatrix<f64> = qr.R();
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//! ```
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//!
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//! ## References:
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//! * ["No bullshit guide to linear algebra", Ivan Savov, 2016, 7.6 Matrix decompositions](https://minireference.com/)
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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., 2.10 QR Decomposition](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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#![allow(non_snake_case)]
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use std::fmt::Debug;
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use crate::error::Failed;
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use crate::linalg::basic::arrays::Array2;
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use crate::numbers::basenum::Number;
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use crate::numbers::realnum::RealNumber;
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#[derive(Debug, Clone)]
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/// Results of QR decomposition.
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pub struct QR<T: Number + RealNumber, M: Array2<T>> {
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QR: M,
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tau: Vec<T>,
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singular: bool,
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}
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impl<T: Number + RealNumber, M: Array2<T>> QR<T, M> {
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pub(crate) fn new(QR: M, tau: Vec<T>) -> QR<T, M> {
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let mut singular = false;
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for tau_elem in tau.iter() {
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if *tau_elem == T::zero() {
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singular = true;
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break;
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}
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}
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QR { QR, tau, singular }
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}
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/// Get upper triangular matrix.
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pub fn R(&self) -> M {
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let (_, n) = self.QR.shape();
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let mut R = M::zeros(n, n);
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for i in 0..n {
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R.set((i, i), self.tau[i]);
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for j in i + 1..n {
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R.set((i, j), *self.QR.get((i, j)));
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}
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}
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R
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}
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/// Get an orthogonal matrix.
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pub fn Q(&self) -> M {
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let (m, n) = self.QR.shape();
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let mut Q = M::zeros(m, n);
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let mut k = n - 1;
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loop {
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Q.set((k, k), T::one());
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for j in k..n {
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if self.QR.get((k, k)) != &T::zero() {
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let mut s = T::zero();
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for i in k..m {
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s += *self.QR.get((i, k)) * *Q.get((i, j));
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}
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s = -s / *self.QR.get((k, k));
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for i in k..m {
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Q.add_element_mut((i, j), s * *self.QR.get((i, k)));
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}
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}
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}
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if k == 0 {
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break;
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} else {
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k -= 1;
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}
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}
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Q
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}
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fn solve(&self, mut b: M) -> Result<M, Failed> {
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let (m, n) = self.QR.shape();
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let (b_nrows, b_ncols) = b.shape();
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if b_nrows != m {
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panic!(
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"Row dimensions do not agree: A is {} x {}, but B is {} x {}",
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m, n, b_nrows, b_ncols
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);
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}
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if self.singular {
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panic!("Matrix is rank deficient.");
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}
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for k in 0..n {
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for j in 0..b_ncols {
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let mut s = T::zero();
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for i in k..m {
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s += *self.QR.get((i, k)) * *b.get((i, j));
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}
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s = -s / *self.QR.get((k, k));
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for i in k..m {
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b.add_element_mut((i, j), s * *self.QR.get((i, k)));
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}
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}
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}
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for k in (0..n).rev() {
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for j in 0..b_ncols {
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b.set((k, j), *b.get((k, j)) / self.tau[k]);
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}
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for i in 0..k {
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for j in 0..b_ncols {
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b.sub_element_mut((i, j), *b.get((k, j)) * *self.QR.get((i, k)));
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}
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}
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}
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Ok(b)
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}
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}
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/// Trait that implements QR decomposition routine for any matrix.
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pub trait QRDecomposable<T: Number + RealNumber>: Array2<T> {
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/// Compute the QR decomposition of a matrix.
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fn qr(&self) -> Result<QR<T, Self>, Failed> {
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self.clone().qr_mut()
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}
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/// Compute the QR decomposition of a matrix. The input matrix
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/// will be used for factorization.
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fn qr_mut(mut self) -> Result<QR<T, Self>, Failed> {
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let (m, n) = self.shape();
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let mut r_diagonal: Vec<T> = vec![T::zero(); n];
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for (k, r_diagonal_k) in r_diagonal.iter_mut().enumerate().take(n) {
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let mut nrm = T::zero();
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for i in k..m {
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nrm = nrm.hypot(*self.get((i, k)));
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}
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if nrm.abs() > T::epsilon() {
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if self.get((k, k)) < &T::zero() {
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nrm = -nrm;
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}
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for i in k..m {
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self.div_element_mut((i, k), nrm);
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}
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self.add_element_mut((k, k), T::one());
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for j in k + 1..n {
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let mut s = T::zero();
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for i in k..m {
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s += *self.get((i, k)) * *self.get((i, j));
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}
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s = -s / *self.get((k, k));
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for i in k..m {
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self.add_element_mut((i, j), s * *self.get((i, k)));
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}
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}
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}
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*r_diagonal_k = -nrm;
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}
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Ok(QR::new(self, r_diagonal))
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}
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/// Solves Ax = b
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fn qr_solve_mut(self, b: Self) -> Result<Self, Failed> {
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self.qr_mut().and_then(|qr| qr.solve(b))
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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 approx::relative_eq;
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#[cfg_attr(target_arch = "wasm32", wasm_bindgen_test::wasm_bindgen_test)]
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#[test]
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fn decompose() {
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let a = DenseMatrix::from_2d_array(&[&[0.9, 0.4, 0.7], &[0.4, 0.5, 0.3], &[0.7, 0.3, 0.8]]);
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let q = DenseMatrix::from_2d_array(&[
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&[-0.7448, 0.2436, 0.6212],
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&[-0.331, -0.9432, -0.027],
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&[-0.5793, 0.2257, -0.7832],
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]);
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let r = DenseMatrix::from_2d_array(&[
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&[-1.2083, -0.6373, -1.0842],
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&[0.0, -0.3064, 0.0682],
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&[0.0, 0.0, -0.1999],
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]);
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let qr = a.qr().unwrap();
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assert!(relative_eq!(qr.Q().abs(), q.abs(), epsilon = 1e-4));
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assert!(relative_eq!(qr.R().abs(), r.abs(), epsilon = 1e-4));
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}
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#[cfg_attr(target_arch = "wasm32", wasm_bindgen_test::wasm_bindgen_test)]
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#[test]
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fn qr_solve_mut() {
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let a = DenseMatrix::from_2d_array(&[&[0.9, 0.4, 0.7], &[0.4, 0.5, 0.3], &[0.7, 0.3, 0.8]]);
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let b = DenseMatrix::from_2d_array(&[&[0.5, 0.2], &[0.5, 0.8], &[0.5, 0.3]]);
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let expected_w = DenseMatrix::from_2d_array(&[
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&[-0.2027027, -1.2837838],
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&[0.8783784, 2.2297297],
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&[0.4729730, 0.6621622],
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]);
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let w = a.qr_solve_mut(b).unwrap();
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assert!(relative_eq!(w, expected_w, epsilon = 1e-2));
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}
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}
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