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Simplify k-Means clustering algorithm.

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Oleksii Trekhleb 2020-12-19 21:13:11 +01:00
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2
README.md
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@ -147,7 +147,7 @@ a set of rules that precisely define a sequence of operations.
* **Machine Learning**
* `B` [NanoNeuron](https://github.com/trekhleb/nano-neuron) - 7 simple JS functions that illustrate how machines can actually learn (forward/backward propagation)
* `B` [k-NN](src/algorithms/ml/knn) - k-nearest neighbors classification algorithm
* `B` [k-Means](src/algorithms/ml/kmeans) - k-Means clustering algorithm
* `B` [k-Means](src/algorithms/ml/k-means) - k-Means clustering algorithm
* **Uncategorized**
* `B` [Tower of Hanoi](src/algorithms/uncategorized/hanoi-tower)
* `B` [Square Matrix Rotation](src/algorithms/uncategorized/square-matrix-rotation) - in-place algorithm

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src/algorithms/ml/kmeans/README.md → src/algorithms/ml/k-means/README.md
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@ -1,10 +1,10 @@
# k-Means Algorithm
The **k-Means algorithm** is an unsupervised Machine Learning algorithm. It's a clustering algorithm, which groups the sample data on the basis of similarity between dimentions of vectors.
The **k-Means algorithm** is an unsupervised Machine Learning algorithm. It's a clustering algorithm, which groups the sample data on the basis of similarity between dimensions of vectors.
In k-Means classification, the output is a set of classess asssigned to each vector. Each cluster location is continously optimized in order to get the accurate locations of each cluster such that they represent each group clearly.
In k-Means classification, the output is a set of classes assigned to each vector. Each cluster location is continuously optimized in order to get the accurate locations of each cluster such that they represent each group clearly.
The idea is to calculate the similarity between cluster location and data vectors, and reassign clusters based on it. [Euclidean distance](https://en.wikipedia.org/wiki/Euclidean_distance) is used mostly for this task.
The idea is to calculate the similarity between cluster location and data vectors, and reassign clusters based on it. [Euclidean distance](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/math/euclidean-distance) is used mostly for this task.
![Euclidean distance between two points](https://upload.wikimedia.org/wikipedia/commons/5/55/Euclidean_distance_2d.svg)
@ -13,9 +13,9 @@ _Image source: [Wikipedia](https://en.wikipedia.org/wiki/Euclidean_distance)_
The algorithm is as follows:
1. Check for errors like invalid/inconsistent data
2. Initialize the k cluster locations with initial/random k points
2. Initialize the `k` cluster locations with initial/random `k` points
3. Calculate the distance of each data point from each cluster
4. Assign the cluster label of each data point equal to that of the cluster at it's minimum distance
4. Assign the cluster label of each data point equal to that of the cluster at its minimum distance
5. Calculate the centroid of each cluster based on the data points it contains
6. Repeat each of the above steps until the centroid locations are varying
@ -23,9 +23,9 @@ Here is a visualization of k-Means clustering for better understanding:
![KNN Visualization 1](https://upload.wikimedia.org/wikipedia/commons/e/ea/K-means_convergence.gif)
_Image source: [Wikipedia](https://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm)_
_Image source: [Wikipedia](https://en.wikipedia.org/wiki/K-means_clustering)_
The centroids are moving continously in order to create better distinction between the different set of data points. As we can see, after a few iterations, the difference in centroids is quite low between iterations. For example between itrations `13` and `14` the difference is quite small because there the optimizer is tuning boundary cases.
The centroids are moving continuously in order to create better distinction between the different set of data points. As we can see, after a few iterations, the difference in centroids is quite low between iterations. For example between iterations `13` and `14` the difference is quite small because there the optimizer is tuning boundary cases.
## References

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src/algorithms/ml/kmeans/__test__/kmeans.test.js → src/algorithms/ml/k-means/__test__/kMeans.test.js
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import kMeans from '../kmeans';
import KMeans from '../kMeans';
describe('kMeans', () => {
it('should throw an error on invalid data', () => {
expect(() => {
kMeans();
}).toThrowError('Either dataSet or labels or toClassify were not set');
KMeans();
}).toThrowError('The data is empty');
});
it('should throw an error on inconsistent data', () => {
expect(() => {
kMeans([[1, 2], [1]], 2);
}).toThrowError('Inconsistent vector lengths');
KMeans([[1, 2], [1]], 2);
}).toThrowError('Matrices have different shapes');
});
it('should find the nearest neighbour', () => {
const dataSet = [[1, 1], [6, 2], [3, 3], [4, 5], [9, 2], [2, 4], [8, 7]];
const data = [[1, 1], [6, 2], [3, 3], [4, 5], [9, 2], [2, 4], [8, 7]];
const k = 2;
const expectedCluster = [0, 1, 0, 1, 1, 0, 1];
expect(kMeans(dataSet, k)).toEqual(expectedCluster);
const expectedClusters = [0, 1, 0, 1, 1, 0, 1];
expect(KMeans(data, k)).toEqual(expectedClusters);
expect(KMeans([[0, 0], [0, 1], [10, 10]], 2)).toEqual(
[0, 0, 1],
);
});
it('should find the clusters with equal distances', () => {
const dataSet = [[0, 0], [1, 1], [2, 2]];
const k = 3;
const expectedCluster = [0, 1, 2];
expect(kMeans(dataSet, k)).toEqual(expectedCluster);
expect(KMeans(dataSet, k)).toEqual(expectedCluster);
});
it('should find the nearest neighbour in 3D space', () => {
const dataSet = [[0, 0, 0], [0, 1, 0], [2, 0, 2]];
const k = 2;
const expectedCluster = [1, 1, 0];
expect(kMeans(dataSet, k)).toEqual(expectedCluster);
expect(KMeans(dataSet, k)).toEqual(expectedCluster);
});
});

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src/algorithms/ml/k-means/kMeans.js Normal file
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@ -0,0 +1,85 @@
import * as mtrx from '../../math/matrix/Matrix';
import euclideanDistance from '../../math/euclidean-distance/euclideanDistance';
/**
* Classifies the point in space based on k-Means algorithm.
*
* @param {number[][]} data - array of dataSet points, i.e. [[0, 1], [3, 4], [5, 7]]
* @param {number} k - number of clusters
* @return {number[]} - the class of the point
*/
export default function KMeans(
data,
k = 1,
) {
if (!data) {
throw new Error('The data is empty');
}
// Assign k clusters locations equal to the location of initial k points.
const dataDim = data[0].length;
const clusterCenters = data.slice(0, k);
// Continue optimization till convergence.
// Centroids should not be moving once optimized.
// Calculate distance of each candidate vector from each cluster center.
// Assign cluster number to each data vector according to minimum distance.
// Matrix of distance from each data point to each cluster centroid.
const distances = mtrx.zeros([data.length, k]);
// Vector data points' classes. The value of -1 means that no class has bee assigned yet.
const classes = Array(data.length).fill(-1);
let iterate = true;
while (iterate) {
iterate = false;
// Calculate and store the distance of each data point from each cluster.
for (let dataIndex = 0; dataIndex < data.length; dataIndex += 1) {
for (let clusterIndex = 0; clusterIndex < k; clusterIndex += 1) {
distances[dataIndex][clusterIndex] = euclideanDistance(
[clusterCenters[clusterIndex]],
[data[dataIndex]],
);
}
// Assign the closest cluster number to each dataSet point.
const closestClusterIdx = distances[dataIndex].indexOf(
Math.min(...distances[dataIndex]),
);
// Check if data point class has been changed and we still need to re-iterate.
if (classes[dataIndex] !== closestClusterIdx) {
iterate = true;
}
classes[dataIndex] = closestClusterIdx;
}
// Recalculate cluster centroid values via all dimensions of the points under it.
for (let clusterIndex = 0; clusterIndex < k; clusterIndex += 1) {
// Reset cluster center coordinates since we need to recalculate them.
clusterCenters[clusterIndex] = Array(dataDim).fill(0);
let clusterSize = 0;
for (let dataIndex = 0; dataIndex < data.length; dataIndex += 1) {
if (classes[dataIndex] === clusterIndex) {
// Register one more data point of current cluster.
clusterSize += 1;
for (let dimensionIndex = 0; dimensionIndex < dataDim; dimensionIndex += 1) {
// Add data point coordinates to the cluster center coordinates.
clusterCenters[clusterIndex][dimensionIndex] += data[dataIndex][dimensionIndex];
}
}
}
// Calculate the average for each cluster center coordinate.
for (let dimensionIndex = 0; dimensionIndex < dataDim; dimensionIndex += 1) {
clusterCenters[clusterIndex][dimensionIndex] = parseFloat(Number(
clusterCenters[clusterIndex][dimensionIndex] / clusterSize,
).toFixed(2));
}
}
}
// Return the clusters assigned.
return classes;
}

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src/algorithms/ml/kmeans/kmeans.js
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@ -1,98 +0,0 @@
/**
* Calculates calculate the euclidean distance between 2 vectors.
*
* @param {number[]} x1
* @param {number[]} x2
* @returns {number}
*/
function euclideanDistance(x1, x2) {
// Checking for errors.
if (x1.length !== x2.length) {
throw new Error('Inconsistent vector lengths');
}
// Calculate the euclidean distance between 2 vectors and return.
let squaresTotal = 0;
for (let i = 0; i < x1.length; i += 1) {
squaresTotal += (x1[i] - x2[i]) ** 2;
}
return Number(Math.sqrt(squaresTotal).toFixed(2));
}
/**
* Classifies the point in space based on k-nearest neighbors algorithm.
*
* @param {number[][]} dataSet - array of dataSet points, i.e. [[0, 1], [3, 4], [5, 7]]
* @param {number} k - number of nearest neighbors which will be taken into account (preferably odd)
* @return {number[]} - the class of the point
*/
export default function kMeans(
dataSetm,
k = 1,
) {
const dataSet = dataSetm;
if (!dataSet) {
throw new Error('Either dataSet or labels or toClassify were not set');
}
// starting algorithm
// assign k clusters locations equal to the location of initial k points
const clusterCenters = [];
const nDim = dataSet[0].length;
for (let i = 0; i < k; i += 1) {
clusterCenters[clusterCenters.length] = Array.from(dataSet[i]);
}
// continue optimization till convergence
// centroids should not be moving once optimized
// calculate distance of each candidate vector from each cluster center
// assign cluster number to each data vector according to minimum distance
let flag = true;
while (flag) {
flag = false;
// calculate and store distance of each dataSet point from each cluster
for (let i = 0; i < dataSet.length; i += 1) {
for (let n = 0; n < k; n += 1) {
dataSet[i][nDim + n] = euclideanDistance(clusterCenters[n], dataSet[i].slice(0, nDim));
}
// assign the cluster number to each dataSet point
const sliced = dataSet[i].slice(nDim, nDim + k);
let minmDistCluster = Math.min(...sliced);
for (let j = 0; j < sliced.length; j += 1) {
if (minmDistCluster === sliced[j]) {
minmDistCluster = j;
break;
}
}
if (dataSet[i].length !== nDim + k + 1) {
flag = true;
dataSet[i][nDim + k] = minmDistCluster;
} else if (dataSet[i][nDim + k] !== minmDistCluster) {
flag = true;
dataSet[i][nDim + k] = minmDistCluster;
}
}
// recalculate cluster centriod values via all dimensions of the points under it
for (let i = 0; i < k; i += 1) {
clusterCenters[i] = Array(nDim).fill(0);
let classCount = 0;
for (let j = 0; j < dataSet.length; j += 1) {
if (dataSet[j][dataSet[j].length - 1] === i) {
classCount += 1;
for (let n = 0; n < nDim; n += 1) {
clusterCenters[i][n] += dataSet[j][n];
}
}
}
for (let n = 0; n < nDim; n += 1) {
clusterCenters[i][n] = Number((clusterCenters[i][n] / classCount).toFixed(2));
}
}
}
// return the clusters assigned
const soln = [];
for (let i = 0; i < dataSet.length; i += 1) {
soln.push(dataSet[i][dataSet[i].length - 1]);
}
return soln;
}