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use crate::k_means::errors::{KMeansError, Result}; use crate::k_means::helpers::IncrementalMean; use crate::k_means::hyperparameters::{KMeansHyperParams, KMeansHyperParamsBuilder}; use linfa::{ dataset::{DatasetBase, Targets}, traits::*, Float, }; use ndarray::{s, Array1, Array2, ArrayBase, Axis, Data, DataMut, Ix1, Ix2, Zip}; use ndarray_rand::rand; use ndarray_rand::rand::Rng; use ndarray_stats::DeviationExt; use rand_isaac::Isaac64Rng; use std::collections::HashMap; #[cfg(feature = "serde")] use serde_crate::{Deserialize, Serialize}; #[cfg_attr( feature = "serde", derive(Serialize, Deserialize), serde(crate = "serde_crate") )] #[derive(Clone, Debug, PartialEq)] /// K-means clustering aims to partition a set of unlabeled observations into clusters, /// where each observation belongs to the cluster with the nearest mean. /// /// The mean of the points within a cluster is called *centroid*. /// /// Given the set of centroids, you can assign an observation to a cluster /// choosing the nearest centroid. /// /// We provide an implementation of the _standard algorithm_, also known as /// Lloyd's algorithm or naive K-means. /// /// More details on the algorithm can be found in the next section or /// [here](https://en.wikipedia.org/wiki/K-means_clustering). /// /// ## The algorithm /// /// K-means is an iterative algorithm: it progressively refines the choice of centroids. /// /// It's guaranteed to converge, even though it might not find the optimal set of centroids /// (unfortunately it can get stuck in a local minimum, finding the optimal minimum if NP-hard!). /// /// There are three steps in the standard algorithm: /// - initialisation step: how do we choose our initial set of centroids? /// - assignment step: assign each observation to the nearest cluster /// (minimum distance between the observation and the cluster's centroid); /// - update step: recompute the centroid of each cluster. /// /// The initialisation step is a one-off, done at the very beginning. /// Assignment and update are repeated in a loop until convergence is reached (either the /// euclidean distance between the old and the new clusters is below `tolerance` or /// we exceed the `max_n_iterations`). /// /// ## Parallelisation /// /// The work performed by the assignment step does not require any coordination: /// the closest centroid for each point can be computed independently from the /// closest centroid for any of the remaining points. /// /// This makes it a good candidate for parallel execution: `KMeans::fit` parallelises the /// assignment step thanks to the `rayon` feature in `ndarray`. /// /// The update step requires a bit more coordination (computing a rolling mean in /// parallel) but it is still parallelisable. /// Nonetheless, our first attempts have not improved performance /// (most likely due to our strategy used to split work between threads), hence /// the update step is currently executed on a single thread. /// /// ## Tutorial /// /// Let's do a walkthrough of a training-predict-save example. /// /// ``` /// use linfa::DatasetBase; /// use linfa::traits::{Fit, Predict}; /// use linfa_clustering::{KMeansHyperParams, KMeans, generate_blobs}; /// use ndarray::{Axis, array, s}; /// use ndarray_rand::rand::SeedableRng; /// use rand_isaac::Isaac64Rng; /// use approx::assert_abs_diff_eq; /// /// // Our random number generator, seeded for reproducibility /// let seed = 42; /// let mut rng = Isaac64Rng::seed_from_u64(seed); /// /// // `expected_centroids` has shape `(n_centroids, n_features)` /// // i.e. three points in the 2-dimensional plane /// let expected_centroids = array![[0., 1.], [-10., 20.], [-1., 10.]]; /// // Let's generate a synthetic dataset: three blobs of observations /// // (100 points each) centered around our `expected_centroids` /// let observations = DatasetBase::from(generate_blobs(100, &expected_centroids, &mut rng)); /// /// // Let's configure and run our K-means algorithm /// // We use the builder pattern to specify the hyperparameters /// // `n_clusters` is the only mandatory parameter. /// // If you don't specify the others (e.g. `n_runs`, `tolerance`, `max_n_iterations`) /// // default values will be used. /// let n_clusters = expected_centroids.len_of(Axis(0)); /// let model = KMeans::params(n_clusters) /// .tolerance(1e-2) /// .fit(&observations) /// .expect("KMeans fitted"); /// /// // Once we found our set of centroids, we can also assign new points to the nearest cluster /// let new_observation = DatasetBase::from(array![[-9., 20.5]]); /// // Predict returns the **index** of the nearest cluster /// let dataset = model.predict(new_observation); /// // We can retrieve the actual centroid of the closest cluster using `.centroids()` /// let closest_centroid = &model.centroids().index_axis(Axis(0), dataset.targets()[0]); /// ``` /// /*/// /// // The model can be serialised (and deserialised) to disk using serde /// // We'll use the JSON format here for simplicity /// let filename = "k_means_model.json"; /// let writer = std::fs::File::create(filename).expect("Failed to open file."); /// serde_json::to_writer(writer, &model).expect("Failed to serialise model."); /// /// let reader = std::fs::File::open(filename).expect("Failed to open file."); /// let loaded_model: KMeans<f64> = serde_json::from_reader(reader).expect("Failed to deserialise model"); /// /// assert_abs_diff_eq!(model.centroids(), loaded_model.centroids(), epsilon = 1e-10); /// assert_eq!(model.hyperparameters(), loaded_model.hyperparameters()); /// ``` */ pub struct KMeans<F: Float> { centroids: Array2<F>, } impl<F: Float> KMeans<F> { pub fn params(nclusters: usize) -> KMeansHyperParamsBuilder<F, Isaac64Rng> { KMeansHyperParams::new(nclusters) } pub fn params_with_rng<R: Rng + Clone>( nclusters: usize, rng: R, ) -> KMeansHyperParamsBuilder<F, R> { KMeansHyperParams::new_with_rng(nclusters, rng) } /// Return the set of centroids as a 2-dimensional matrix with shape /// `(n_centroids, n_features)`. pub fn centroids(&self) -> &Array2<F> { &self.centroids } } impl<'a, F: Float, R: Rng + Clone, D: Data<Elem = F>, T: Targets> Fit<'a, ArrayBase<D, Ix2>, T> for KMeansHyperParams<F, R> { type Object = Result<KMeans<F>>; /// Given an input matrix `observations`, with shape `(n_observations, n_features)`, /// `fit` identifies `n_clusters` centroids based on the training data distribution. /// /// An instance of `KMeans` is returned. /// fn fit(&self, dataset: &DatasetBase<ArrayBase<D, Ix2>, T>) -> Self::Object { let mut rng = self.rng(); let observations = dataset.records().view(); let mut min_inertia = F::infinity(); let mut best_centroids = None; let mut best_iter = None; let mut memberships = Array1::zeros(observations.dim().0); let n_runs = self.n_runs(); for _ in 0..n_runs { let mut inertia = min_inertia; let mut centroids = get_random_centroids(self.n_clusters(), &observations, &mut rng); let mut converged_iter: Option<u64> = None; for n_iter in 0..self.max_n_iterations() { update_cluster_memberships(¢roids, &observations, &mut memberships); let new_centroids = compute_centroids(self.n_clusters(), &observations, &memberships); inertia = compute_inertia(&new_centroids, &observations, &memberships); let distance = centroids .sq_l2_dist(&new_centroids) .expect("Failed to compute distance"); centroids = new_centroids; if distance < self.tolerance() { converged_iter = Some(n_iter); break; } } // We keep the centroids which minimize the inertia (defined as the sum of // the squared distances of the closest centroid for all observations) // over the n runs of the KMeans algorithm. if inertia < min_inertia { min_inertia = inertia; best_centroids = Some(centroids.clone()); best_iter = converged_iter; } } match best_iter { Some(_n_iter) => match best_centroids { Some(centroids) => Ok(KMeans { centroids }), _ => Err(KMeansError::InertiaError( "No inertia improvement (-inf)".to_string(), )), }, None => Err(KMeansError::NotConverged(format!( "KMeans fitting algorithm {} did not converge. Try different init parameters, \ or increase max_n_iterations, tolerance or check for degenerate data.", (n_runs + 1) ))), } } } impl<'a, F: Float, R: Rng + Clone, D: Data<Elem = F>, T: Targets> Fit<'a, ArrayBase<D, Ix2>, T> for KMeansHyperParamsBuilder<F, R> { type Object = Result<KMeans<F>>; fn fit(&self, dataset: &DatasetBase<ArrayBase<D, Ix2>, T>) -> Self::Object { self.build().fit(dataset) } } impl<F: Float, D: Data<Elem = F>> Predict<&ArrayBase<D, Ix2>, Array1<usize>> for KMeans<F> { /// Given an input matrix `observations`, with shape `(n_observations, n_features)`, /// `predict` returns, for each observation, the index of the closest cluster/centroid. /// /// You can retrieve the centroid associated to an index using the /// [`centroids` method](#method.centroids). fn predict(&self, observations: &ArrayBase<D, Ix2>) -> Array1<usize> { compute_cluster_memberships(&self.centroids, observations) } } impl<F: Float, D: Data<Elem = F>, T: Targets> Predict<DatasetBase<ArrayBase<D, Ix2>, T>, DatasetBase<ArrayBase<D, Ix2>, Array1<usize>>> for KMeans<F> { fn predict( &self, dataset: DatasetBase<ArrayBase<D, Ix2>, T>, ) -> DatasetBase<ArrayBase<D, Ix2>, Array1<usize>> { let predicted = self.predict(dataset.records()); dataset.with_targets(predicted) } } /// We compute inertia defined as the sum of the squared distances /// of the closest centroid for all observations. fn compute_inertia<F: Float>( centroids: &ArrayBase<impl Data<Elem = F> + Sync, Ix2>, observations: &ArrayBase<impl Data<Elem = F>, Ix2>, cluster_memberships: &ArrayBase<impl Data<Elem = usize>, Ix1>, ) -> F { let mut dists = Array1::<F>::zeros(observations.nrows()); Zip::from(observations.genrows()) .and(cluster_memberships) .and(&mut dists) .par_apply(|observation, &cluster_membership, d| { *d = centroids .row(cluster_membership) .sq_l2_dist(&observation) .expect("Failed to compute distance"); }); dists.sum() } /// K-means is an iterative algorithm. /// We will perform the assignment and update steps until we are satisfied /// (according to our convergence criteria). /// /// If you check the `compute_cluster_memberships` function, /// you can see that it expects to receive centroids as a 2-dimensional array. /// /// `compute_centroids` wraps our `compute_centroids_hashmap` to return a 2-dimensional array, /// where the i-th row corresponds to the i-th cluster. fn compute_centroids<F: Float>( // The number of clusters could be inferred from `centroids_hashmap`, // but it is indeed possible for a cluster to become empty during the // multiple rounds of assignment-update optimisations // This would lead to an underestimate of the number of clusters // and several errors down the line due to shape mismatches n_clusters: usize, // (n_observations, n_features) observations: &ArrayBase<impl Data<Elem = F>, Ix2>, // (n_observations,) cluster_memberships: &ArrayBase<impl Data<Elem = usize>, Ix1>, ) -> Array2<F> { let centroids_hashmap = compute_centroids_hashmap(&observations, &cluster_memberships); let (_, n_features) = observations.dim(); let mut centroids: Array2<F> = Array2::zeros((n_clusters, n_features)); for (centroid_index, centroid) in centroids_hashmap.into_iter() { centroids .slice_mut(s![centroid_index, ..]) .assign(¢roid.current_mean); } centroids } /// Iterate over our observations and capture in a HashMap the new centroids. /// The HashMap is a (cluster_index => new centroid) mapping. fn compute_centroids_hashmap<F: Float>( // (n_observations, n_features) observations: &ArrayBase<impl Data<Elem = F>, Ix2>, // (n_observations,) cluster_memberships: &ArrayBase<impl Data<Elem = usize>, Ix1>, ) -> HashMap<usize, IncrementalMean<F>> { let mut new_centroids: HashMap<usize, IncrementalMean<F>> = HashMap::new(); Zip::from(observations.genrows()) .and(cluster_memberships) .apply(|observation, cluster_membership| { if let Some(incremental_mean) = new_centroids.get_mut(cluster_membership) { incremental_mean.update(&observation); } else { new_centroids.insert( *cluster_membership, IncrementalMean::new(observation.to_owned()), ); } }); new_centroids } /// Given a matrix of centroids with shape (n_centroids, n_features) /// and a matrix of observations with shape (n_observations, n_features), /// update the 1-dimensional `cluster_memberships` array such that: /// /// membership[i] == closest_centroid(¢roids, &observations.slice(s![i, ..]) /// fn update_cluster_memberships<F: Float>( centroids: &ArrayBase<impl Data<Elem = F> + Sync, Ix2>, observations: &ArrayBase<impl Data<Elem = F> + Sync, Ix2>, cluster_memberships: &mut ArrayBase<impl DataMut<Elem = usize>, Ix1>, ) { Zip::from(observations.axis_iter(Axis(0))) .and(cluster_memberships) .par_apply(|observation, cluster_membership| { *cluster_membership = closest_centroid(¢roids, &observation) }); } /// Given a matrix of centroids with shape (n_centroids, n_features) /// and a matrix of observations with shape (n_observations, n_features), /// return a 1-dimensional `membership` array such that: /// /// membership[i] == closest_centroid(¢roids, &observations.slice(s![i, ..]) /// fn compute_cluster_memberships<F: Float>( // (n_centroids, n_features) centroids: &ArrayBase<impl Data<Elem = F>, Ix2>, // (n_observations, n_features) observations: &ArrayBase<impl Data<Elem = F>, Ix2>, ) -> Array1<usize> { observations.map_axis(Axis(1), |observation| { closest_centroid(¢roids, &observation) }) } /// Given a matrix of centroids with shape (n_centroids, n_features) and an observation, /// return the index of the closest centroid (the index of the corresponding row in `centroids`). fn closest_centroid<F: Float>( // (n_centroids, n_features) centroids: &ArrayBase<impl Data<Elem = F>, Ix2>, // (n_features) observation: &ArrayBase<impl Data<Elem = F>, Ix1>, ) -> usize { let mut iterator = centroids.genrows().into_iter().peekable(); let first_centroid = iterator .peek() .expect("There has to be at least one centroid"); let (mut closest_index, mut minimum_distance) = ( 0, first_centroid .sq_l2_dist(&observation) .expect("Failed to compute distance"), ); for (centroid_index, centroid) in iterator.enumerate() { let distance = centroid .sq_l2_dist(&observation) .expect("Failed to compute distance"); if distance < minimum_distance { closest_index = centroid_index; minimum_distance = distance; } } closest_index } fn get_random_centroids<F: Float, D: Data<Elem = F>>( n_clusters: usize, observations: &ArrayBase<D, Ix2>, rng: &mut impl Rng, ) -> Array2<F> { let (n_samples, _) = observations.dim(); let indices = rand::seq::index::sample(rng, n_samples, n_clusters).into_vec(); observations.select(Axis(0), &indices) } #[cfg(test)] mod tests { use super::*; use approx::assert_abs_diff_eq; use ndarray::{array, stack, Array, Array1, Array2, Axis}; use ndarray_rand::rand::SeedableRng; use ndarray_rand::rand_distr::Uniform; use ndarray_rand::RandomExt; fn function_test_1d(x: &Array2<f64>) -> Array2<f64> { let mut y = Array2::zeros(x.dim()); Zip::from(&mut y).and(x).apply(|yi, &xi| { if xi < 0.4 { *yi = xi * xi; } else if xi >= 0.4 && xi < 0.8 { *yi = 3. * xi + 1.; } else { *yi = f64::sin(10. * xi); } }); y } #[test] fn test_n_runs() { let mut rng = Isaac64Rng::seed_from_u64(42); let xt = Array::random_using(50, Uniform::new(0., 1.), &mut rng).insert_axis(Axis(1)); let yt = function_test_1d(&xt); let data = stack(Axis(1), &[xt.view(), yt.view()]).unwrap(); // First clustering with one iteration let dataset = DatasetBase::from(data); let model = KMeans::params_with_rng(3, rng.clone()) .n_runs(1) .fit(&dataset) .expect("KMeans fitted"); let clusters = model.predict(dataset); let inertia = compute_inertia(model.centroids(), &clusters.records, &clusters.targets); // Second clustering with 10 iterations (default) let dataset2 = DatasetBase::from(clusters.records().clone()); let model2 = KMeans::params_with_rng(3, rng) .fit(&dataset2) .expect("KMeans fitted"); let clusters2 = model2.predict(dataset2); let inertia2 = compute_inertia(model2.centroids(), &clusters2.records, &clusters2.targets); // Check we improve inertia assert!(inertia2 < inertia); } #[test] fn compute_centroids_works() { let cluster_size = 100; let n_features = 4; // Let's setup a synthetic set of observations, composed of two clusters with known means let cluster_1: Array2<f64> = Array::random((cluster_size, n_features), Uniform::new(-100., 100.)); let memberships_1 = Array1::zeros(cluster_size); let expected_centroid_1 = cluster_1.mean_axis(Axis(0)).unwrap(); let cluster_2: Array2<f64> = Array::random((cluster_size, n_features), Uniform::new(-100., 100.)); let memberships_2 = Array1::ones(cluster_size); let expected_centroid_2 = cluster_2.mean_axis(Axis(0)).unwrap(); // `stack` combines arrays along a given axis: https://docs.rs/ndarray/0.13.0/ndarray/fn.stack.html let observations = stack(Axis(0), &[cluster_1.view(), cluster_2.view()]).unwrap(); let memberships = stack(Axis(0), &[memberships_1.view(), memberships_2.view()]).unwrap(); // Does it work? let centroids = compute_centroids(2, &observations, &memberships); assert_abs_diff_eq!( centroids.index_axis(Axis(0), 0), expected_centroid_1, epsilon = 1e-5 ); assert_abs_diff_eq!( centroids.index_axis(Axis(0), 1), expected_centroid_2, epsilon = 1e-5 ); assert_eq!(centroids.len_of(Axis(0)), 2); } #[test] // An observation is closest to itself. fn nothing_is_closer_than_self() { let n_centroids = 20; let n_features = 5; let mut rng = Isaac64Rng::seed_from_u64(42); let centroids: Array2<f64> = Array::random_using( (n_centroids, n_features), Uniform::new(-100., 100.), &mut rng, ); let expected_memberships: Vec<usize> = (0..n_centroids).into_iter().collect(); assert_eq!( compute_cluster_memberships(¢roids, ¢roids), Array1::from(expected_memberships) ); } #[test] fn oracle_test_for_closest_centroid() { let centroids = array![[0., 0.], [1., 2.], [20., 0.], [0., 20.],]; let observations = array![[1., 0.5], [20., 2.], [20., 0.], [7., 20.],]; let memberships = array![0, 2, 2, 3]; assert_eq!( compute_cluster_memberships(¢roids, &observations), memberships ); } }