162bed2aa2
* test: run tests also in wasm targets * fix: install rand with wasm-bindgen por wasm targets * fix: use actual usize size to access buffer. * fix: do not run functions that create files in wasm. * test: do not run in wasm test that panics. Co-authored-by: Luis Moreno <morenol@users.noreply.github.com>
605 lines
21 KiB
Rust
605 lines
21 KiB
Rust
//! # PCA
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//!
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//! Principal components analysis (PCA) is a method that is used to select several linear combinations that capture most of the variation in your data.
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//! PCA is an unsupervised approach, since it involves only a set of features \\(X1, X2, . . . , Xn\\), and no associated response \\(Y\\).
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//! Apart from producing derived variables for use in supervised learning problems, PCA also serves as a tool for data visualization.
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//!
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//! PCA is scale sensitive. Before PCA is performed, the variables should be centered to have mean zero.
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//! Furthermore, the results obtained also depend on whether the variables have been individually scaled.
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//! Use `use_correlation_matrix` parameter to standardize your variables (to mean 0 and standard deviation 1).
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//!
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//! Example:
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//! ```
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//! use smartcore::linalg::naive::dense_matrix::*;
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//! use smartcore::decomposition::pca::*;
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//!
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//! // Iris data
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//! let iris = DenseMatrix::from_2d_array(&[
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//! &[5.1, 3.5, 1.4, 0.2],
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//! &[4.9, 3.0, 1.4, 0.2],
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//! &[4.7, 3.2, 1.3, 0.2],
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//! &[4.6, 3.1, 1.5, 0.2],
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//! &[5.0, 3.6, 1.4, 0.2],
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//! &[5.4, 3.9, 1.7, 0.4],
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//! &[4.6, 3.4, 1.4, 0.3],
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//! &[5.0, 3.4, 1.5, 0.2],
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//! &[4.4, 2.9, 1.4, 0.2],
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//! &[4.9, 3.1, 1.5, 0.1],
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//! &[7.0, 3.2, 4.7, 1.4],
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//! &[6.4, 3.2, 4.5, 1.5],
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//! &[6.9, 3.1, 4.9, 1.5],
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//! &[5.5, 2.3, 4.0, 1.3],
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//! &[6.5, 2.8, 4.6, 1.5],
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//! &[5.7, 2.8, 4.5, 1.3],
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//! &[6.3, 3.3, 4.7, 1.6],
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//! &[4.9, 2.4, 3.3, 1.0],
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//! &[6.6, 2.9, 4.6, 1.3],
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//! &[5.2, 2.7, 3.9, 1.4],
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//! ]);
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//!
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//! let pca = PCA::fit(&iris, PCAParameters::default().with_n_components(2)).unwrap(); // Reduce number of features to 2
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//!
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//! let iris_reduced = pca.transform(&iris).unwrap();
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//!
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//! ```
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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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#[cfg(feature = "serde")]
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use serde::{Deserialize, Serialize};
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use crate::api::{Transformer, UnsupervisedEstimator};
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use crate::error::Failed;
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use crate::linalg::Matrix;
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use crate::math::num::RealNumber;
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/// Principal components analysis algorithm
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#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
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#[derive(Debug)]
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pub struct PCA<T: RealNumber, M: Matrix<T>> {
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eigenvectors: M,
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eigenvalues: Vec<T>,
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projection: M,
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mu: Vec<T>,
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pmu: Vec<T>,
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}
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impl<T: RealNumber, M: Matrix<T>> PartialEq for PCA<T, M> {
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fn eq(&self, other: &Self) -> bool {
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if self.eigenvectors != other.eigenvectors
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|| self.eigenvalues.len() != other.eigenvalues.len()
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{
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false
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} else {
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for i in 0..self.eigenvalues.len() {
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if (self.eigenvalues[i] - other.eigenvalues[i]).abs() > T::epsilon() {
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return false;
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}
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}
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true
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}
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}
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}
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#[derive(Debug, Clone)]
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/// PCA parameters
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pub struct PCAParameters {
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/// Number of components to keep.
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pub n_components: usize,
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/// By default, covariance matrix is used to compute principal components.
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/// Enable this flag if you want to use correlation matrix instead.
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pub use_correlation_matrix: bool,
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}
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impl PCAParameters {
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/// Number of components to keep.
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pub fn with_n_components(mut self, n_components: usize) -> Self {
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self.n_components = n_components;
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self
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}
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/// By default, covariance matrix is used to compute principal components.
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/// Enable this flag if you want to use correlation matrix instead.
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pub fn with_use_correlation_matrix(mut self, use_correlation_matrix: bool) -> Self {
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self.use_correlation_matrix = use_correlation_matrix;
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self
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}
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}
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impl Default for PCAParameters {
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fn default() -> Self {
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PCAParameters {
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n_components: 2,
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use_correlation_matrix: false,
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}
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}
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}
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impl<T: RealNumber, M: Matrix<T>> UnsupervisedEstimator<M, PCAParameters> for PCA<T, M> {
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fn fit(x: &M, parameters: PCAParameters) -> Result<Self, Failed> {
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PCA::fit(x, parameters)
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}
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}
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impl<T: RealNumber, M: Matrix<T>> Transformer<M> for PCA<T, M> {
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fn transform(&self, x: &M) -> Result<M, Failed> {
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self.transform(x)
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}
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}
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impl<T: RealNumber, M: Matrix<T>> PCA<T, M> {
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/// Fits PCA to your data.
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/// * `data` - _NxM_ matrix with _N_ observations and _M_ features in each observation.
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/// * `n_components` - number of components to keep.
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/// * `parameters` - other parameters, use `Default::default()` to set parameters to default values.
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pub fn fit(data: &M, parameters: PCAParameters) -> Result<PCA<T, M>, Failed> {
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let (m, n) = data.shape();
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if parameters.n_components > n {
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return Err(Failed::fit(&format!(
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"Number of components, n_components should be <= number of attributes ({})",
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n
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)));
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}
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let mu = data.column_mean();
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let mut x = data.clone();
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for (c, mu_c) in mu.iter().enumerate().take(n) {
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for r in 0..m {
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x.sub_element_mut(r, c, *mu_c);
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}
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}
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let mut eigenvalues;
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let mut eigenvectors;
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if m > n && !parameters.use_correlation_matrix {
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let svd = x.svd()?;
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eigenvalues = svd.s;
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for eigenvalue in &mut eigenvalues {
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*eigenvalue = *eigenvalue * (*eigenvalue);
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}
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eigenvectors = svd.V;
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} else {
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let mut cov = M::zeros(n, n);
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for k in 0..m {
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for i in 0..n {
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for j in 0..=i {
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cov.add_element_mut(i, j, x.get(k, i) * x.get(k, j));
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}
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}
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}
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for i in 0..n {
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for j in 0..=i {
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cov.div_element_mut(i, j, T::from(m).unwrap());
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cov.set(j, i, cov.get(i, j));
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}
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}
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if parameters.use_correlation_matrix {
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let mut sd = vec![T::zero(); n];
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for (i, sd_i) in sd.iter_mut().enumerate().take(n) {
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*sd_i = cov.get(i, i).sqrt();
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}
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for i in 0..n {
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for j in 0..=i {
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cov.div_element_mut(i, j, sd[i] * sd[j]);
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cov.set(j, i, cov.get(i, j));
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}
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}
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let evd = cov.evd(true)?;
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eigenvalues = evd.d;
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eigenvectors = evd.V;
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for (i, sd_i) in sd.iter().enumerate().take(n) {
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for j in 0..n {
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eigenvectors.div_element_mut(i, j, *sd_i);
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}
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}
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} else {
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let evd = cov.evd(true)?;
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eigenvalues = evd.d;
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eigenvectors = evd.V;
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}
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}
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let mut projection = M::zeros(parameters.n_components, n);
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for i in 0..n {
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for j in 0..parameters.n_components {
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projection.set(j, i, eigenvectors.get(i, j));
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}
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}
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let mut pmu = vec![T::zero(); parameters.n_components];
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for (k, mu_k) in mu.iter().enumerate().take(n) {
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for (i, pmu_i) in pmu.iter_mut().enumerate().take(parameters.n_components) {
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*pmu_i += projection.get(i, k) * (*mu_k);
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}
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}
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Ok(PCA {
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eigenvectors,
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eigenvalues,
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projection: projection.transpose(),
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mu,
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pmu,
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})
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}
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/// Run dimensionality reduction for `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 transform(&self, x: &M) -> Result<M, Failed> {
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let (nrows, ncols) = x.shape();
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let (_, n_components) = self.projection.shape();
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if ncols != self.mu.len() {
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return Err(Failed::transform(&format!(
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"Invalid input vector size: {}, expected: {}",
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ncols,
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self.mu.len()
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)));
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}
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let mut x_transformed = x.matmul(&self.projection);
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for r in 0..nrows {
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for c in 0..n_components {
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x_transformed.sub_element_mut(r, c, self.pmu[c]);
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}
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}
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Ok(x_transformed)
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}
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/// Get a projection matrix
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pub fn components(&self) -> &M {
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&self.projection
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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::naive::dense_matrix::*;
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fn us_arrests_data() -> DenseMatrix<f64> {
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DenseMatrix::from_2d_array(&[
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&[13.2, 236.0, 58.0, 21.2],
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&[10.0, 263.0, 48.0, 44.5],
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&[8.1, 294.0, 80.0, 31.0],
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&[8.8, 190.0, 50.0, 19.5],
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&[9.0, 276.0, 91.0, 40.6],
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&[7.9, 204.0, 78.0, 38.7],
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&[3.3, 110.0, 77.0, 11.1],
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&[5.9, 238.0, 72.0, 15.8],
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&[15.4, 335.0, 80.0, 31.9],
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&[17.4, 211.0, 60.0, 25.8],
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&[5.3, 46.0, 83.0, 20.2],
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&[2.6, 120.0, 54.0, 14.2],
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&[10.4, 249.0, 83.0, 24.0],
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&[7.2, 113.0, 65.0, 21.0],
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&[2.2, 56.0, 57.0, 11.3],
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&[6.0, 115.0, 66.0, 18.0],
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&[9.7, 109.0, 52.0, 16.3],
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&[15.4, 249.0, 66.0, 22.2],
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&[2.1, 83.0, 51.0, 7.8],
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&[11.3, 300.0, 67.0, 27.8],
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&[4.4, 149.0, 85.0, 16.3],
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&[12.1, 255.0, 74.0, 35.1],
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&[2.7, 72.0, 66.0, 14.9],
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&[16.1, 259.0, 44.0, 17.1],
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&[9.0, 178.0, 70.0, 28.2],
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&[6.0, 109.0, 53.0, 16.4],
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&[4.3, 102.0, 62.0, 16.5],
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&[12.2, 252.0, 81.0, 46.0],
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&[2.1, 57.0, 56.0, 9.5],
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&[7.4, 159.0, 89.0, 18.8],
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&[11.4, 285.0, 70.0, 32.1],
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&[11.1, 254.0, 86.0, 26.1],
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&[13.0, 337.0, 45.0, 16.1],
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&[0.8, 45.0, 44.0, 7.3],
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&[7.3, 120.0, 75.0, 21.4],
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&[6.6, 151.0, 68.0, 20.0],
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&[4.9, 159.0, 67.0, 29.3],
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&[6.3, 106.0, 72.0, 14.9],
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&[3.4, 174.0, 87.0, 8.3],
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&[14.4, 279.0, 48.0, 22.5],
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&[3.8, 86.0, 45.0, 12.8],
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&[13.2, 188.0, 59.0, 26.9],
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&[12.7, 201.0, 80.0, 25.5],
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&[3.2, 120.0, 80.0, 22.9],
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&[2.2, 48.0, 32.0, 11.2],
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&[8.5, 156.0, 63.0, 20.7],
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&[4.0, 145.0, 73.0, 26.2],
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&[5.7, 81.0, 39.0, 9.3],
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&[2.6, 53.0, 66.0, 10.8],
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&[6.8, 161.0, 60.0, 15.6],
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])
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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 pca_components() {
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let us_arrests = us_arrests_data();
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let expected = DenseMatrix::from_2d_array(&[
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&[0.0417, 0.0448],
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&[0.9952, 0.0588],
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&[0.0463, 0.9769],
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&[0.0752, 0.2007],
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]);
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let pca = PCA::fit(&us_arrests, Default::default()).unwrap();
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assert!(expected.approximate_eq(&pca.components().abs(), 0.4));
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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 decompose_covariance() {
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let us_arrests = us_arrests_data();
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let expected_eigenvectors = DenseMatrix::from_2d_array(&[
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&[
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-0.0417043206282872,
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-0.0448216562696701,
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-0.0798906594208108,
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-0.994921731246978,
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],
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&[
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-0.995221281426497,
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-0.058760027857223,
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0.0675697350838043,
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0.0389382976351601,
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],
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&[
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-0.0463357461197108,
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0.97685747990989,
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0.200546287353866,
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-0.0581691430589319,
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],
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&[
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-0.075155500585547,
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0.200718066450337,
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-0.974080592182491,
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0.0723250196376097,
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],
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]);
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let expected_projection = DenseMatrix::from_2d_array(&[
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&[-64.8022, -11.448, 2.4949, -2.4079],
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&[-92.8275, -17.9829, -20.1266, 4.094],
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&[-124.0682, 8.8304, 1.6874, 4.3537],
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&[-18.34, -16.7039, -0.2102, 0.521],
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&[-107.423, 22.5201, -6.7459, 2.8118],
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&[-34.976, 13.7196, -12.2794, 1.7215],
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&[60.8873, 12.9325, 8.4207, 0.6999],
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&[-66.731, 1.3538, 11.281, 3.728],
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&[-165.2444, 6.2747, 2.9979, -1.2477],
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&[-40.5352, -7.2902, -3.6095, -7.3437],
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&[123.5361, 24.2912, -3.7244, -3.4728],
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&[51.797, -9.4692, 1.5201, 3.3478],
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&[-78.9921, 12.8971, 5.8833, -0.3676],
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&[57.551, 2.8463, -3.7382, -1.6494],
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&[115.5868, -3.3421, 0.654, 0.8695],
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&[55.7897, 3.1572, -0.3844, -0.6528],
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&[62.3832, -10.6733, -2.2371, -3.8762],
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&[-78.2776, -4.2949, 3.8279, -4.4836],
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&[89.261, -11.4878, 4.6924, 2.1162],
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&[-129.3301, -5.007, 2.3472, 1.9283],
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&[21.2663, 19.4502, 7.5071, 1.0348],
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&[-85.4515, 5.9046, -6.4643, -0.499],
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&[98.9548, 5.2096, -0.0066, 0.7319],
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&[-86.8564, -27.4284, 5.0034, -3.8798],
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&[-7.9863, 5.2756, -5.5006, -0.6794],
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&[62.4836, -9.5105, -1.8384, -0.2459],
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&[69.0965, -0.2112, -0.468, 0.6566],
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&[-83.6136, 15.1022, -15.8887, -0.3342],
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&[114.7774, -4.7346, 2.2824, 0.9359],
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&[10.8157, 23.1373, 6.3102, -1.6124],
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&[-114.8682, -0.3365, -2.2613, 1.3812],
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&[-84.2942, 15.924, 4.7213, -0.892],
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&[-164.3255, -31.0966, 11.6962, 2.1112],
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&[127.4956, -16.135, 1.3118, 2.301],
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&[50.0868, 12.2793, -1.6573, -2.0291],
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&[19.6937, 3.3701, 0.4531, 0.1803],
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&[11.1502, 3.8661, -8.13, 2.914],
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&[64.6891, 8.9115, 3.2065, -1.8749],
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&[-3.064, 18.374, 17.47, 2.3083],
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&[-107.2811, -23.5361, 2.0328, -1.2517],
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&[86.1067, -16.5979, -1.3144, 1.2523],
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&[-17.5063, -6.5066, -6.1001, -3.9229],
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&[-31.2911, 12.985, 0.3934, -4.242],
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&[49.9134, 17.6485, -1.7882, 1.8677],
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&[124.7145, -27.3136, -4.8028, 2.005],
|
|
&[14.8174, -1.7526, -1.0454, -1.1738],
|
|
&[25.0758, 9.968, -4.7811, 2.6911],
|
|
&[91.5446, -22.9529, 0.402, -0.7369],
|
|
&[118.1763, 5.5076, 2.7113, -0.205],
|
|
&[10.4345, -5.9245, 3.7944, 0.5179],
|
|
]);
|
|
|
|
let expected_eigenvalues: Vec<f64> = vec![
|
|
343544.6277001563,
|
|
9897.625949808047,
|
|
2063.519887011604,
|
|
302.04806302399646,
|
|
];
|
|
|
|
let pca = PCA::fit(&us_arrests, PCAParameters::default().with_n_components(4)).unwrap();
|
|
|
|
assert!(pca
|
|
.eigenvectors
|
|
.abs()
|
|
.approximate_eq(&expected_eigenvectors.abs(), 1e-4));
|
|
|
|
for i in 0..pca.eigenvalues.len() {
|
|
assert!((pca.eigenvalues[i].abs() - expected_eigenvalues[i].abs()).abs() < 1e-8);
|
|
}
|
|
|
|
let us_arrests_t = pca.transform(&us_arrests).unwrap();
|
|
|
|
assert!(us_arrests_t
|
|
.abs()
|
|
.approximate_eq(&expected_projection.abs(), 1e-4));
|
|
}
|
|
|
|
#[cfg_attr(target_arch = "wasm32", wasm_bindgen_test::wasm_bindgen_test)]
|
|
#[test]
|
|
fn decompose_correlation() {
|
|
let us_arrests = us_arrests_data();
|
|
|
|
let expected_eigenvectors = DenseMatrix::from_2d_array(&[
|
|
&[
|
|
0.124288601688222,
|
|
-0.0969866877028367,
|
|
0.0791404742697482,
|
|
-0.150572299008293,
|
|
],
|
|
&[
|
|
0.00706888610512014,
|
|
-0.00227861130898090,
|
|
0.00325028101296307,
|
|
0.00901099154845273,
|
|
],
|
|
&[
|
|
0.0194141494466002,
|
|
0.060910660326921,
|
|
0.0263806464184195,
|
|
-0.0093429458365566,
|
|
],
|
|
&[
|
|
0.0586084532558777,
|
|
0.0180450999787168,
|
|
-0.0881962972508558,
|
|
-0.0096011588898465,
|
|
],
|
|
]);
|
|
|
|
let expected_projection = DenseMatrix::from_2d_array(&[
|
|
&[0.9856, -1.1334, 0.4443, -0.1563],
|
|
&[1.9501, -1.0732, -2.04, 0.4386],
|
|
&[1.7632, 0.746, -0.0548, 0.8347],
|
|
&[-0.1414, -1.1198, -0.1146, 0.1828],
|
|
&[2.524, 1.5429, -0.5986, 0.342],
|
|
&[1.5146, 0.9876, -1.095, -0.0015],
|
|
&[-1.3586, 1.0889, 0.6433, 0.1185],
|
|
&[0.0477, 0.3254, 0.7186, 0.882],
|
|
&[3.013, -0.0392, 0.5768, 0.0963],
|
|
&[1.6393, -1.2789, 0.3425, -1.0768],
|
|
&[-0.9127, 1.5705, -0.0508, -0.9028],
|
|
&[-1.6398, -0.211, -0.2598, 0.4991],
|
|
&[1.3789, 0.6818, 0.6775, 0.122],
|
|
&[-0.5055, 0.1516, -0.2281, -0.4247],
|
|
&[-2.2536, 0.1041, -0.1646, -0.0176],
|
|
&[-0.7969, 0.2702, -0.0256, -0.2065],
|
|
&[-0.7509, -0.9584, 0.0284, -0.6706],
|
|
&[1.5648, -0.8711, 0.7835, -0.4547],
|
|
&[-2.3968, -0.3764, 0.0657, 0.3305],
|
|
&[1.7634, -0.4277, 0.1573, 0.5591],
|
|
&[-0.4862, 1.4745, 0.6095, 0.1796],
|
|
&[2.1084, 0.1554, -0.3849, -0.1024],
|
|
&[-1.6927, 0.6323, -0.1531, -0.0673],
|
|
&[0.9965, -2.3938, 0.7408, -0.2155],
|
|
&[0.6968, 0.2634, -0.3774, -0.2258],
|
|
&[-1.1855, -0.5369, -0.2469, -0.1237],
|
|
&[-1.2656, 0.194, -0.1756, -0.0159],
|
|
&[2.8744, 0.7756, -1.1634, -0.3145],
|
|
&[-2.3839, 0.0181, -0.0369, 0.0331],
|
|
&[0.1816, 1.4495, 0.7645, -0.2434],
|
|
&[1.98, -0.1428, -0.1837, 0.3395],
|
|
&[1.6826, 0.8232, 0.6431, 0.0135],
|
|
&[1.1234, -2.228, 0.8636, 0.9544],
|
|
&[-2.9922, -0.5991, -0.3013, 0.254],
|
|
&[-0.226, 0.7422, 0.0311, -0.4739],
|
|
&[-0.3118, 0.2879, 0.0153, -0.0103],
|
|
&[0.0591, 0.5414, -0.9398, 0.2378],
|
|
&[-0.8884, 0.5711, 0.4006, -0.3591],
|
|
&[-0.8638, 1.492, 1.3699, 0.6136],
|
|
&[1.3207, -1.9334, 0.3005, 0.1315],
|
|
&[-1.9878, -0.8233, -0.3893, 0.1096],
|
|
&[0.9997, -0.8603, -0.1881, -0.6529],
|
|
&[1.3551, 0.4125, 0.4921, -0.6432],
|
|
&[-0.5506, 1.4715, -0.2937, 0.0823],
|
|
&[-2.8014, -1.4023, -0.8413, 0.1449],
|
|
&[-0.0963, -0.1997, -0.0117, -0.2114],
|
|
&[-0.2169, 0.9701, -0.6249, 0.2208],
|
|
&[-2.1086, -1.4248, -0.1048, -0.1319],
|
|
&[-2.0797, 0.6113, 0.1389, -0.1841],
|
|
&[-0.6294, -0.321, 0.2407, 0.1667],
|
|
]);
|
|
|
|
let expected_eigenvalues: Vec<f64> = vec![
|
|
2.480241579149493,
|
|
0.9897651525398419,
|
|
0.35656318058083064,
|
|
0.1734300877298357,
|
|
];
|
|
|
|
let pca = PCA::fit(
|
|
&us_arrests,
|
|
PCAParameters::default()
|
|
.with_n_components(4)
|
|
.with_use_correlation_matrix(true),
|
|
)
|
|
.unwrap();
|
|
|
|
assert!(pca
|
|
.eigenvectors
|
|
.abs()
|
|
.approximate_eq(&expected_eigenvectors.abs(), 1e-4));
|
|
|
|
for i in 0..pca.eigenvalues.len() {
|
|
assert!((pca.eigenvalues[i].abs() - expected_eigenvalues[i].abs()).abs() < 1e-8);
|
|
}
|
|
|
|
let us_arrests_t = pca.transform(&us_arrests).unwrap();
|
|
|
|
assert!(us_arrests_t
|
|
.abs()
|
|
.approximate_eq(&expected_projection.abs(), 1e-4));
|
|
}
|
|
|
|
#[cfg_attr(target_arch = "wasm32", wasm_bindgen_test::wasm_bindgen_test)]
|
|
#[test]
|
|
#[cfg(feature = "serde")]
|
|
fn serde() {
|
|
let iris = DenseMatrix::from_2d_array(&[
|
|
&[5.1, 3.5, 1.4, 0.2],
|
|
&[4.9, 3.0, 1.4, 0.2],
|
|
&[4.7, 3.2, 1.3, 0.2],
|
|
&[4.6, 3.1, 1.5, 0.2],
|
|
&[5.0, 3.6, 1.4, 0.2],
|
|
&[5.4, 3.9, 1.7, 0.4],
|
|
&[4.6, 3.4, 1.4, 0.3],
|
|
&[5.0, 3.4, 1.5, 0.2],
|
|
&[4.4, 2.9, 1.4, 0.2],
|
|
&[4.9, 3.1, 1.5, 0.1],
|
|
&[7.0, 3.2, 4.7, 1.4],
|
|
&[6.4, 3.2, 4.5, 1.5],
|
|
&[6.9, 3.1, 4.9, 1.5],
|
|
&[5.5, 2.3, 4.0, 1.3],
|
|
&[6.5, 2.8, 4.6, 1.5],
|
|
&[5.7, 2.8, 4.5, 1.3],
|
|
&[6.3, 3.3, 4.7, 1.6],
|
|
&[4.9, 2.4, 3.3, 1.0],
|
|
&[6.6, 2.9, 4.6, 1.3],
|
|
&[5.2, 2.7, 3.9, 1.4],
|
|
]);
|
|
|
|
let pca = PCA::fit(&iris, Default::default()).unwrap();
|
|
|
|
let deserialized_pca: PCA<f64, DenseMatrix<f64>> =
|
|
serde_json::from_str(&serde_json::to_string(&pca).unwrap()).unwrap();
|
|
|
|
assert_eq!(pca, deserialized_pca);
|
|
}
|
|
}
|