K-means++ clustering: Difference between revisions
(→{{header|JavaScript}}: live demo link broken) |
|||
| Line 1,850: | Line 1,850: | ||
=={{header|JavaScript}}== |
=={{header|JavaScript}}== |
||
{{improve|JavaScript|live demo link broken/404}} |
|||
'''Solution''' |
'''Solution''' |
||
Revision as of 03:32, 22 January 2022
You are encouraged to solve this task according to the task description, using any language you may know.
K-means++ clustering a classification of data, so that points assigned to the same cluster are similar (in some sense). It is identical to the K-means algorithm, except for the selection of initial conditions.
The task is to implement the K-means++ algorithm. Produce a function which takes two arguments: the number of clusters K, and the dataset to classify. K is a positive integer and the dataset is a list of points in the Cartesian plane. The output is a list of clusters (related sets of points, according to the algorithm).
For extra credit (in order):
- Provide a function to exercise your code, which generates a list of random points.
- Provide a visualization of your results, including centroids (see example image).
- Generalize the function to polar coordinates (in radians).
- Generalize the function to points in an arbitrary N space (i.e. instead of x,y pairs, points are an N-tuples representing coordinates in ℝN).
If this is different or more difficult than the [naive] solution for ℝ2, discuss what had to change to support N dimensions.
Extra credit is only awarded if the examples given demonstrate the feature in question. To earn credit for 1. and 2., visualize 6 clusters of 30,000 points in ℝ2. It is not necessary to provide visualization for spaces higher than ℝ2 but the examples should demonstrate features 3. and 4. if the solution has them.
C
Output is in EPS. 100,000 point EPS file can take quite a while to display.
To extend the code to handle dimensions higher than 2, make POINT have more coordinates, change the dist2 distance function, and change the finding of centroids in the lloyd K-Means function. Multidimensional scaling will be needed to visualize the output.
This code uses the function kppAllinger to find the initial centroids. It is faster than the original function kpp, especially with large data sets. The function kppAllinger uses an array to keep track of the shortest distance from each point to the previously selected centroids. It also uses a bisection search to select the points. It doesn't use the function nearestDistance. The original functions kpp and nearestDistance are included here for comparison.
<lang c># define NUMBER_OF_POINTS 100000
- define NUMBER_OF_CLUSTERS 11
- define MAXIMUM_ITERATIONS 100
- define RADIUS 10.0
- include <stdio.h>
- include <stdlib.h>
- include <math.h>
typedef struct {
double x;
double y;
int group;
} POINT;
/*-------------------------------------------------------
gen_xy
This function allocates a block of memory for data points, gives the data points random values and returns a pointer to them. The data points fall within a circle of the radius passed to the function. This does not create a uniform 2-dimensional distribution.
*/
POINT * gen_xy(int num_pts, double radius) { int i; double ang, r; POINT * pts;
pts = (POINT*) malloc(sizeof(POINT) * num_pts);
for ( i = 0; i < num_pts; i++ ) { ang = 2.0 * M_PI * rand() / (RAND_MAX - 1.); r = radius * rand() / (RAND_MAX - 1.); pts[i].x = r * cos(ang); pts[i].y = r * sin(ang); } return pts; }
/*------------------------------------------------------- dist2
This function returns the squared euclidean distance between two data points.
*/
double dist2(POINT * a, POINT * b) { double x = a->x - b->x; double y = a->y - b->y; return x*x + y*y; }
/*------------------------------------------------------ nearest
This function returns the index of the cluster centroid nearest to the data point passed to this function.
*/
int nearest(POINT * pt, POINT * cent, int n_cluster) { int i, clusterIndex; double d, min_d;
min_d = HUGE_VAL; clusterIndex = pt->group; for (i = 0; i < n_cluster; i++) { d = dist2(¢[i], pt); if ( d < min_d ) { min_d = d; clusterIndex = i; } } return clusterIndex; }
/*------------------------------------------------------ nearestDistance
This function returns the distance of the cluster centroid nearest to the data point passed to this function.
*/
double nearestDistance(POINT * pt, POINT * cent, int n_cluster) { int i; double d, min_d;
min_d = HUGE_VAL; for (i = 0; i < n_cluster; i++) { d = dist2(¢[i], pt); if ( d < min_d ) { min_d = d; } }
return min_d; }
/*----------------------------------------------------------------------
bisectionSearch This function makes a bisectional search of an array of values that are ordered in increasing order, and returns the index of the first element greater than the search value passed as a parameter. This code is adapted from code by Andy Allinger given to the public domain.
Input:
x A pointer to an array of values in increasing order to be searched. n The number of elements in the input array x. v The search value.
Output:
Returns the index of the first element greater than the search value, v.
*/
int bisectionSearch(double *x, int n, double v) { int il, ir, i;
if (n < 1) {
return 0;
}
/* If v is less than x(0) or greater than x(n-1) */
if (v < x[0]) {
return 0;
}
else if (v > x[n-1]) {
return n - 1;
}
/*bisection search */ il = 0; ir = n - 1;
i = (il + ir) / 2; while ( i != il ) { if (x[i] <= v) { il = i; } else { ir = i; } i = (il + ir) / 2; }
if (x[i] <= v) i = ir; return i; } /* end of bisectionSearch */
/*------------------------------------------------------- kppAllinger
This function uses the K-Means++ method to select the cluster centroids.
This code is adapted from code by Andy Allinger given to the public domain.
Input: pts A pointer to an array of data points. num_pts The number of points in the pts array. centroids A pointer to an array to receive the centroids. num_clusters The number of clusters to be found.
Output: centroids A pointer to the array of centroids found.
*/
void kppAllinger(POINT * pts, int num_pts, POINT * centroids, int num_clusters) { int j; int selectedIndex; int cluster; double sum; double d; double random; double * cumulativeDistances; double * shortestDistance;
cumulativeDistances = (double*) malloc(sizeof(double) * num_pts);
shortestDistance = (double*) malloc(sizeof(double) * num_pts);
/* Pick the first cluster centroids at random. */
selectedIndex = rand() % num_pts;
centroids[0] = pts[ selectedIndex ];
for (j = 0; j < num_pts; ++j) shortestDistance[j] = HUGE_VAL;
/* Select the centroids for the remaining clusters. */ for (cluster = 1; cluster < num_clusters; cluster++) {
/* For each point find its closest distance to any of the previous cluster centers */ for ( j = 0; j < num_pts; j++ ) { d = dist2(&pts[j], ¢roids[cluster-1] );
if (d < shortestDistance[j]) shortestDistance[j] = d; }
/* Create an array of the cumulative distances. */ sum = 0.0; for (j = 0; j < num_pts; j++) { sum += shortestDistance[j]; cumulativeDistances[j] = sum; }
/* Select a point at random. Those with greater distances have a greater probability of being selected. */ random = (float) rand() / (float) RAND_MAX * sum; selectedIndex = bisectionSearch(cumulativeDistances, num_pts, random);
/* assign the selected point as the center */ centroids[cluster] = pts[selectedIndex]; }
/* Assign each point the index of it's nearest cluster centroid. */ for (j = 0; j < num_pts; j++) pts[j].group = nearest(&pts[j], centroids, num_clusters);
free(shortestDistance); free(cumulativeDistances);
return; } /* end, kppAllinger */
/*------------------------------------------------------- kpp
This function uses the K-Means++ method to select the cluster centroids.
*/
void kpp(POINT * pts, int num_pts, POINT * centroids, int num_clusters) { int j; int cluster; double sum; double * distances;
distances = (double*) malloc(sizeof(double) * num_pts);
/* Pick the first cluster centroids at random. */ centroids[0] = pts[ rand() % num_pts ];
/* Select the centroids for the remaining clusters. */
for (cluster = 1; cluster < num_clusters; cluster++) {
/* For each data point find the nearest centroid, save its distance in the distance array, then add it to the sum of total distance. */ sum = 0.0; for ( j = 0; j < num_pts; j++ ) { distances[j] = nearestDistance(&pts[j], centroids, cluster); sum += distances[j]; }
/* Find a random distance within the span of the total distance. */ sum = sum * rand() / (RAND_MAX - 1);
/* Assign the centroids. the point with the largest distance will have a greater probability of being selected. */ for (j = 0; j < num_pts; j++ ) { sum -= distances[j]; if ( sum <= 0) { centroids[cluster] = pts[j]; break; } } }
/* Assign each observation the index of it's nearest cluster centroid. */ for (j = 0; j < num_pts; j++) pts[j].group = nearest(&pts[j], centroids, num_clusters);
free(distances);
return; } /* end, kpp */
/*-------------------------------------------------------
lloyd
This function clusters the data using Lloyd's K-Means algorithm after selecting the intial centroids using the K-Means++ method. It returns a pointer to the memory it allocates containing the array of cluster centroids.
*/
POINT * lloyd(POINT * pts, int num_pts, int num_clusters, int maxTimes) { int i, clusterIndex; int changes; int acceptable = num_pts / 1000; /* The maximum point changes acceptable. */
if (num_clusters == 1 || num_pts <= 0 || num_clusters > num_pts )
return 0;
POINT * centroids = (POINT *)malloc(sizeof(POINT) * num_clusters);
if ( maxTimes < 1 ) maxTimes = 1;
/* Assign initial clustering randomly using the Random Partition method for (i = 0; i < num_pts; i++ ) { pts[i].group = i % num_clusters; }
- /
/* or use the k-Means++ method */
/* Original version kpp(pts, num_pts, centroids, num_clusters);
- /
/* Faster Allinger version */ kppAllinger(pts, num_pts, centroids, num_clusters);
do { /* Calculate the centroid of each cluster. ----------------------------------------*/
/* Initialize the x, y and cluster totals. */ for ( i = 0; i < num_clusters; i++ ) { centroids[i].group = 0; /* used to count the cluster members. */ centroids[i].x = 0; /* used for x value totals. */ centroids[i].y = 0; /* used for y value totals. */ }
/* Add each observation's x and y to its cluster total. */ for (i = 0; i < num_pts; i++) { clusterIndex = pts[i].group; centroids[clusterIndex].group++; centroids[clusterIndex].x += pts[i].x; centroids[clusterIndex].y += pts[i].y; }
/* Divide each cluster's x and y totals by its number of data points. */ for ( i = 0; i < num_clusters; i++ ) { centroids[i].x /= centroids[i].group; centroids[i].y /= centroids[i].group; }
/* Find each data point's nearest centroid */ changes = 0; for ( i = 0; i < num_pts; i++ ) { clusterIndex = nearest(&pts[i], centroids, num_clusters); if (clusterIndex != pts[i].group) { pts[i].group = clusterIndex; changes++; } }
maxTimes--; } while ((changes > acceptable) && (maxTimes > 0));
/* Set each centroid's group index */ for ( i = 0; i < num_clusters; i++ ) centroids[i].group = i;
return centroids; } /* end, lloyd */
/*------------------------------------------------------- print_eps
this function prints the results.
*/
void print_eps(POINT * pts, int num_pts, POINT * centroids, int num_clusters) {
- define W 400
- define H 400
int i, j; double min_x, max_x, min_y, max_y, scale, cx, cy; double *colors = (double *) malloc(sizeof(double) * num_clusters * 3);
for (i = 0; i < num_clusters; i++) { colors[3*i + 0] = (3 * (i + 1) % 11)/11.; colors[3*i + 1] = (7 * i % 11)/11.; colors[3*i + 2] = (9 * i % 11)/11.; }
max_x = max_y = - HUGE_VAL; min_x = min_y = HUGE_VAL; for (j = 0; j < num_pts; j++) { if (max_x < pts[j].x) max_x = pts[j].x; if (min_x > pts[j].x) min_x = pts[j].x; if (max_y < pts[j].y) max_y = pts[j].y; if (min_y > pts[j].y) min_y = pts[j].y; }
scale = W / (max_x - min_x); if (scale > H / (max_y - min_y)) scale = H / (max_y - min_y); cx = (max_x + min_x) / 2; cy = (max_y + min_y) / 2;
printf("%%!PS-Adobe-3.0\n%%%%BoundingBox: -5 -5 %d %d\n", W + 10, H + 10); printf( "/l {rlineto} def /m {rmoveto} def\n" "/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def\n" "/s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath " " gsave 1 setgray fill grestore gsave 3 setlinewidth" " 1 setgray stroke grestore 0 setgray stroke }def\n" );
for (i = 0; i < num_clusters; i++) {
printf("%g %g %g setrgbcolor\n",
colors[3*i], colors[3*i + 1], colors[3*i + 2]);
for (j = 0; j < num_pts; j++) { if (pts[j].group != i) continue; printf("%.3f %.3f c\n", (pts[j].x - cx) * scale + W / 2, (pts[j].y - cy) * scale + H / 2); } printf("\n0 setgray %g %g s\n", (centroids[i].x - cx) * scale + W / 2, (centroids[i].y - cy) * scale + H / 2); } printf("\n%%%%EOF");
free(colors);
return; } /* end print_eps */
/*------------------------------------------------------- main
*/
int main() { int num_pts = NUMBER_OF_POINTS; int num_clusters = NUMBER_OF_CLUSTERS; int maxTimes = MAXIMUM_ITERATIONS; double radius = RADIUS; POINT * pts; POINT * centroids;
/* Generate the observations */ pts = gen_xy(num_pts, radius);
/* Cluster using the Lloyd algorithm and K-Means++ initial centroids. */ centroids = lloyd(pts, num_pts, num_clusters, maxTimes);
/* Print the results */ print_eps(pts, num_pts, centroids, num_clusters);
free(pts); free(centroids);
return 0; }</lang>
D
<lang d>import std.stdio, std.math, std.random, std.typecons, std.algorithm;
// On Windows this uses the printf from the Microsoft C runtime, // that doesn't handle real type and some of the C99 format // specifiers, but it's faster for bulk printing. extern(C) nothrow int printf(const char*, ...);
struct Point {
immutable double x, y; // Or float. size_t cluster;
}
Point[] generatePoints(in size_t nPoints,
in double radius,
ref Xorshift rnd)
in {
assert(nPoints > 0); assert(radius > 0);
} out(result) {
assert(result.length == nPoints);
foreach (const ref p; result) {
assert(p.cluster == 0);
assert(!p.x.isNaN && !p.y.isNaN);
}
} body {
Point[] points; points.reserve(nPoints);
// This is not a uniform 2D distribution.
foreach (immutable i; 0 .. nPoints) {
immutable r = uniform(0.0, radius, rnd);
immutable ang = uniform(0.0, 2 * PI, rnd);
points ~= Point(r * ang.cos, r * ang.sin); // Sincos?
}
return points;
}
struct ClusterCenter {
double x, y;
void opAssign(in ref Point p) pure nothrow @nogc {
this.x = p.x;
this.y = p.y;
}
}
const(ClusterCenter)[] lloyd(Point[] points,
in size_t nclusters,
ref Xorshift rnd)
in {
assert(points.length >= nclusters);
assert(nclusters > 0);
foreach (const ref p; points)
assert(!p.x.isNaN && !p.y.isNaN);
} out(result) {
assert(result.length == nclusters);
foreach (const ref cc; result)
assert(!cc.x.isNaN && !cc.y.isNaN);
} body {
/// Distance and index of the closest cluster center.
static Tuple!(size_t, double)
nearestClusterCenter(in ref Point point,
in ClusterCenter[] centers) pure nothrow @nogc
in {
assert(centers.length > 0);
} out(result) {
assert(result[0] < centers.length);
immutable ClusterCenter c = centers[result[0]];
immutable d = (c.x - point.x) ^^ 2 + (c.y - point.y) ^^ 2;
assert(feqrel(result[1], d) > 45); // Arbitrary.
} body {
static double sqrDistance2D(in ref ClusterCenter a,
in ref Point b) pure nothrow @nogc{
return (a.x - b.x) ^^ 2 + (a.y - b.y) ^^ 2;
}
size_t minIndex = point.cluster;
double minDist = double.max;
foreach (immutable i, const ref cc; centers) {
immutable d = sqrDistance2D(cc, point);
if (minDist > d) {
minDist = d;
minIndex = i;
}
}
return tuple(minIndex, minDist); }
static void kMeansPP(Point[] points,
ClusterCenter[] centers,
ref Xorshift rnd)
in {
assert(points.length >= centers.length);
assert(centers.length > 0);
} body {
centers[0] = points[uniform(0, $, rnd)];
auto d = new double[points.length];
foreach (immutable i; 1 .. centers.length) {
double sum = 0;
foreach (immutable j, const ref p; points) {
d[j] = nearestClusterCenter(p, centers[0 .. i])[1];
sum += d[j];
}
sum = uniform(0.0, sum, rnd);
foreach (immutable j, immutable dj; d) {
sum -= dj;
if (sum > 0)
continue;
centers[i] = points[j];
break;
}
}
foreach (ref p; points)
// Implicit cast of Hconst!ClusterCenter
// to ClusterCenter[].
p.cluster = nearestClusterCenter(p, centers)[0];
}
auto centers = new ClusterCenter[nclusters]; kMeansPP(points, centers, rnd); auto clusterSizes = new size_t[centers.length];
size_t changed;
do {
// Find clusters centroids.
centers[] = ClusterCenter(0, 0);
clusterSizes[] = 0;
foreach (immutable i, const ref p; points)
with (centers[p.cluster]) {
clusterSizes[p.cluster]++;
x += p.x;
y += p.y;
}
foreach (immutable i, ref cc; centers) {
cc.x /= clusterSizes[i];
cc.y /= clusterSizes[i];
}
// Find closest centroid of each point.
changed = 0;
foreach (ref p; points) {
immutable minI = nearestClusterCenter(p, centers)[0];
if (minI != p.cluster) {
changed++;
p.cluster = minI;
}
}
// Stop when 99.9% of points are good.
} while (changed > (points.length >> 10));
return centers;
}
void printEps(in Point[] points, in ClusterCenter[] centers,
in size_t W = 400, in size_t H = 400) nothrow
in {
assert(points.length >= centers.length);
assert(centers.length > 0);
assert(W > 0 && H > 0);
foreach (const ref p; points)
assert(!p.x.isNaN && !p.y.isNaN);
foreach (const ref cc; centers)
assert(!cc.x.isNaN && !cc.y.isNaN);
} body {
auto findBoundingBox() nothrow @nogc {
double min_x, max_x, min_y, max_y;
max_x = max_y = -double.max;
min_x = min_y = double.max;
foreach (const ref p; points) {
if (max_x < p.x) max_x = p.x;
if (min_x > p.x) min_x = p.x;
if (max_y < p.y) max_y = p.y;
if (min_y > p.y) min_y = p.y;
}
assert(max_x > min_x && max_y > min_y);
return tuple(min(W / (max_x - min_x), H / (max_y - min_y)),
(max_x + min_x) / 2, (max_y + min_y) / 2);
}
//immutable (scale, cx, cy) = findBoundingBox();
immutable sc_cx_cy = findBoundingBox();
immutable double scale = sc_cx_cy[0];
immutable double cx = sc_cx_cy[1];
immutable double cy = sc_cx_cy[2];
static immutable struct Color { immutable double r, g, b; }
immutable size_t k = centers.length;
Color[] colors;
colors.reserve(centers.length);
foreach (immutable i; 0 .. centers.length)
colors ~= Color((3 * (i + 1) % k) / double(k),
(7 * i % k) / double(k),
(9 * i % k) / double(k));
printf("%%!PS-Adobe-3.0\n%%%%BoundingBox: -5 -5 %d %d\n",
W + 10, H + 10);
printf("/l {rlineto} def /m {rmoveto} def\n" ~
"/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def\n" ~
"/s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath " ~
" gsave 1 setgray fill grestore gsave 3 setlinewidth" ~
" 1 setgray stroke grestore 0 setgray stroke }def\n");
foreach (immutable i, const ref cc; centers) {
printf("%g %g %g setrgbcolor\n", colors[i].tupleof);
foreach (const ref p; points) {
if (p.cluster != i)
continue;
printf("%.3f %.3f c\n",
(p.x - cx) * scale + W / 2,
(p.y - cy) * scale + H / 2);
}
printf("\n0 setgray %g %g s\n",
(cc.x - cx) * scale + W / 2,
(cc.y - cy) * scale + H / 2);
}
"\n%%%%EOF".printf;
}
void main() {
enum size_t nPoints = 100_000; enum size_t nClusters = 11; // k. auto rnd = 1.Xorshift; // For speed and repeatability.
auto points = generatePoints(nPoints, 10, rnd); const clusterCenters = lloyd(points, nClusters, rnd); printEps(points, clusterCenters);
}</lang> Compiled with ldc2 it's about as fast as the C entry.
Euler Math Toolbox
<lang Euler Math Toolbox> >type kmeanscluster
function kmeanscluster (x: numerical, k: index)
n=rows(x); m=cols(x);
i=floor((0:k)/k*(n-1))+1;
means=zeros(k,m);
loop 1 to k;
means[#]=sum(x[i[#]:(i[#+1]-1)]')'/(i[#+1]-i[#]);
end;
j=1:n;
loop 1 to n;
d=sum((x[#]-means)^2);
j[#]=extrema(d')[2];
end;
repeat
loop 1 to k;
i=nonzeros(j==#);
if cols(i)==0 then means[#]=1;
else means[#]=(sum(x[i]')/cols(i))';
endif;
end;
jold=j;
loop 1 to n;
d=sum((x[#]-means)^2);
j[#]=extrema(d')[2];
end;
if all(jold==j) then break; endif;
end
return j
endfunction
</lang>
Let us apply to random data.
<lang Euler Math Toolbox> >load clustering.e
Functions for clustering data.
>np=5; m=3*normal(np,2); % Spread n points randomly around these points. >n=5000; x=m[intrandom(1,n,np)]+normal(n,2); % The function kmeanscluster contains the algorithm. It returns the % indices of the clusters the points contain to. >j=kmeanscluster(x,np); % We plot each point with a color representing its cluster. >P=x'; ... > plot2d(P[1],P[2],r=totalmax(abs(m))+2,color=10+j,points=1,style="."); ... > loop 1 to k; plot2d(m[#,1],m[#,2],points=1,style="o#",add=1); end; ... > insimg; </lang>
Fortran
<lang Fortran>
- KMPP - K-Means++ - Traditional data clustering with a special initialization
- Public Domain - This program may be used by any person for any purpose.
- Origin:
- Hugo Steinhaus, 1956
- Refer to:
- "kmeans++: the advantages of careful seeding"
- David Arthur and Sergei Vassilvitskii
- Proceedings of the eighteenth annual ACM-SIAM symposium
- on Discrete algorithms, 2007
- ____Variable_______I/O_______Description___________________Type_______
- X(P,N) In Data points Real
- P In Dimension of the data Integer
- N In Number of points Integer
- K In # clusters Integer
- C(P,K) Out Center points of clusters Real
- Z(N) Out What cluster a point is in Integer
- WORK(N) Neither Real
- IFAULT Out Error code Integer
SUBROUTINE KMPP (X, P, N, K, C, Z, WORK, IFAULT)
IMPLICIT NONE
INTEGER P, N, K, Z, IFAULT
REAL X, C, WORK
DIMENSION X(P,N), C(P,K), Z(N), WORK(N)
- constants
INTEGER ITER ! maximum iterations
REAL BIG ! arbitrary large number
PARAMETER (ITER = 1000,
$ BIG = 1E33)
- local variables
INTEGER
$ H, ! count iterations
$ I, ! count points
$ I1, ! point marked as initial center
$ J, ! count dimensions
$ L, ! count clusters
$ L0, ! present cluster ID
$ L1 ! new cluster ID
REAL
$ BEST, ! shortest distance to a center
$ D2, ! squared distance
$ TOT, ! a total
$ W ! temp scalar
LOGICAL CHANGE ! whether any points have been reassigned
- Begin.
IFAULT = 0
IF (K < 1 .OR. K > N) THEN ! K out of bounds
IFAULT = 3
RETURN
END IF
DO I = 1, N ! clear Z
Z(I) = 0
END DO
- initial centers
DO I = 1, N
WORK(I) = BIG
END DO
CALL RANDOM_NUMBER (W)
I1 = MIN(INT(W * FLOAT(N)) + 1, N) ! choose first center at random
DO J = 1, P
C(J,1) = X(J,I1)
END DO
DO L = 2, K ! initialize other centers
TOT = 0.
DO I = 1, N ! measure from each point
BEST = WORK(I)
D2 = 0. ! to prior center
DO J = 1, P
D2 = D2 + (X(J,I) - C(J,L-1)) **2 ! Squared Euclidean distance
IF (D2 .GE. BEST) GO TO 10 ! needless to add to D2
END DO ! next J
IF (D2 < BEST) BEST = D2 ! shortest squared distance
WORK(I) = BEST
10 TOT = TOT + BEST ! cumulative squared distance
END DO ! next data point
- Choose center with probability proportional to its squared distance
- from existing centers.
CALL RANDOM_NUMBER (W)
W = W * TOT ! uniform at random over cumulative distance
TOT = 0.
DO I = 1, N
I1 = I
TOT = TOT + WORK(I)
IF (TOT > W) GO TO 20
END DO ! next I
20 CONTINUE
DO J = 1, P ! assign center
C(J,L) = X(J,I1)
END DO
END DO ! next center to initialize
- main loop
DO H = 1, ITER
CHANGE = .FALSE.
- find nearest center for each point
DO I = 1, N
L0 = Z(I)
L1 = 0
BEST = BIG
DO L = 1, K
D2 = 0.
DO J = 1, P
D2 = D2 + (X(J,I) - C(J,L)) **2
IF (D2 .GE. BEST) GO TO 30
END DO
30 CONTINUE
IF (D2 < BEST) THEN ! new nearest center
BEST = D2
L1 = L
END IF
END DO ! next L
IF (L0 .NE. L1) THEN
Z(I) = L1 ! reassign point
CHANGE = .TRUE.
END IF
END DO ! next I
IF (.NOT. CHANGE) RETURN ! success
- find cluster centers
DO L = 1, K ! zero population
WORK(L) = 0.
END DO
DO L = 1, K ! zero centers
DO J = 1, P
C(J,L) = 0.
END DO
END DO
DO I = 1, N
L = Z(I)
WORK(L) = WORK(L) + 1. ! count
DO J = 1, P
C(J,L) = C(J,L) + X(J,I) ! add
END DO
END DO
DO L = 1, K
IF (WORK(L) < 0.5) THEN ! empty cluster check
IFAULT = 1 ! fatal error
RETURN
END IF
W = 1. / WORK(L)
DO J = 1, P
C(J,L) = C(J,L) * W ! multiplication is faster than division
END DO
END DO
END DO ! next H
IFAULT = 2 ! too many iterations
RETURN
END ! of KMPP
- test program (extra credit #1)
PROGRAM TPEC1
IMPLICIT NONE
INTEGER N, P, K
REAL TWOPI
PARAMETER (N = 30 000,
$ P = 2,
$ K = 6,
$ TWOPI = 6.2831853)
INTEGER I, L, Z(N), IFAULT
REAL X(P,N), C(P,K), R, THETA, W, WORK(N)
- Begin
CALL RANDOM_SEED()
DO I = 1, N ! random points over unit circle
CALL RANDOM_NUMBER (W)
R = SQRT(W) ! radius
CALL RANDOM_NUMBER (W)
THETA = W * TWOPI ! angle
X(1,I) = R * COS(THETA) ! Cartesian coordinates
X(2,I) = R * SIN(THETA)
END DO
- Call subroutine
CALL KMPP (X, P, N, K, C, Z, WORK, IFAULT)
PRINT *, 'kmpp returns with error code ', IFAULT
- Print lists of points in each cluster
DO L = 1, K
PRINT *, 'Cluster ', L, ' contains points: '
10 FORMAT (I6, $)
20 FORMAT ()
DO I = 1, N
IF (Z(I) .EQ. L) PRINT 10, I
END DO
PRINT 20
END DO
- Write CSV file with Y-coordinates in different columns by cluster
OPEN (UNIT=1, FILE='tpec1.csv', STATUS='NEW', IOSTAT=IFAULT)
IF (IFAULT .NE. 0) PRINT *, 'tpec1: trouble opening file'
30 FORMAT (F8.4, $)
40 FORMAT (',', $)
50 FORMAT (F8.4)
DO I = 1, N
WRITE (UNIT=1, FMT=30, IOSTAT=IFAULT) X(1,I)
IF (IFAULT .NE. 0) PRINT *, 'tpec1: trouble writing X-coord'
DO L = 1, Z(I) ! one comma per cluster ID
WRITE (UNIT=1, FMT=40, IOSTAT=IFAULT)
IF (IFAULT .NE. 0) PRINT *, 'tpec1: trouble writing comma'
END DO
WRITE (UNIT=1, FMT=50, IOSTAT=IFAULT) X(2,I)
IF (IFAULT .NE. 0) PRINT *, 'tpec1: trouble writing Y-coord'
END DO
- Write the centroids in the far column
DO L = 1, K
WRITE (UNIT=1, FMT=30, IOSTAT=IFAULT) C(1,L)
IF (IFAULT .NE. 0) PRINT *, 'tpec1: trouble writing X-coord'
DO I = 1, K+1
WRITE (UNIT=1, FMT=40, IOSTAT=IFAULT)
IF (IFAULT .NE. 0) PRINT *, 'tpec1: trouble writing comma'
END DO
WRITE (UNIT=1, FMT=50, IOSTAT=IFAULT) C(2,L)
IF (IFAULT .NE. 0) PRINT *, 'tpec1: trouble writing Y-coord'
END DO
CLOSE (UNIT=1)
END ! of test program
</lang>
Uniform random points over the unit circle (compare to the solution in C)
(External image)
The points with clusters marked by color: (External image)
Go
<lang go>package main
import (
"fmt" "image" "image/color" "image/draw" "image/png" "math" "math/rand" "os" "time"
)
type r2 struct {
x, y float64
}
type r2c struct {
r2 c int // cluster number
}
// kmpp implements K-means++, satisfying the basic task requirement func kmpp(k int, data []r2c) {
kMeans(data, kmppSeeds(k, data))
}
// kmppSeeds is the ++ part. // It generates the initial means for the k-means algorithm. func kmppSeeds(k int, data []r2c) []r2 {
s := make([]r2, k)
s[0] = data[rand.Intn(len(data))].r2
d2 := make([]float64, len(data))
for i := 1; i < k; i++ {
var sum float64
for j, p := range data {
_, dMin := nearest(p, s[:i])
d2[j] = dMin * dMin
sum += d2[j]
}
target := rand.Float64() * sum
j := 0
for sum = d2[0]; sum < target; sum += d2[j] {
j++
}
s[i] = data[j].r2
}
return s
}
// nearest finds the nearest mean to a given point. // return values are the index of the nearest mean, and the distance from // the point to the mean. func nearest(p r2c, mean []r2) (int, float64) {
iMin := 0
dMin := math.Hypot(p.x-mean[0].x, p.y-mean[0].y)
for i := 1; i < len(mean); i++ {
d := math.Hypot(p.x-mean[i].x, p.y-mean[i].y)
if d < dMin {
dMin = d
iMin = i
}
}
return iMin, dMin
}
// kMeans algorithm. Lloyd's func kMeans(data []r2c, mean []r2) {
// initial assignment
for i, p := range data {
cMin, _ := nearest(p, mean)
data[i].c = cMin
}
mLen := make([]int, len(mean))
for {
// update means
for i := range mean {
mean[i] = r2{}
mLen[i] = 0
}
for _, p := range data {
mean[p.c].x += p.x
mean[p.c].y += p.y
mLen[p.c]++
}
for i := range mean {
inv := 1 / float64(mLen[i])
mean[i].x *= inv
mean[i].y *= inv
}
// make new assignments, count changes
var changes int
for i, p := range data {
if cMin, _ := nearest(p, mean); cMin != p.c {
changes++
data[i].c = cMin
}
}
if changes == 0 {
return
}
}
}
// parameters for extra credit exercises type ecParam struct {
k int nPoints int xBox, yBox int stdv int
}
// extra credit 1 and 2: func main() {
ec := &ecParam{6, 30000, 300, 200, 30}
origin, data := genECData(ec)
vis(ec, data, "origin")
fmt.Println("Data set origins:")
fmt.Println(" x y")
for _, o := range origin {
fmt.Printf("%5.1f %5.1f\n", o.x, o.y)
}
kmpp(ec.k, data)
fmt.Println(
"\nCluster centroids, mean distance from centroid, number of points:")
fmt.Println(" x y distance points")
cent := make([]r2, ec.k)
cLen := make([]int, ec.k)
inv := make([]float64, ec.k)
for _, p := range data {
cent[p.c].x += p.x
cent[p.c].y += p.y
cLen[p.c]++
}
for i, iLen := range cLen {
inv[i] = 1 / float64(iLen)
cent[i].x *= inv[i]
cent[i].y *= inv[i]
}
dist := make([]float64, ec.k)
for _, p := range data {
dist[p.c] += math.Hypot(p.x-cent[p.c].x, p.y-cent[p.c].y)
}
for i, iLen := range cLen {
fmt.Printf("%5.1f %5.1f %8.1f %6d\n",
cent[i].x, cent[i].y, dist[i]*inv[i], iLen)
}
vis(ec, data, "clusters")
}
// genECData generates random data for extra credit tasks. // k origin points are randomly selected in a bounding box. // nPoints/k coordinates are then generated for each origin point. // The x and y coordinates of the data are normally distributed // with standard deviation stdv. Thus data coordinates are not // constrained to the origin box; they can range to +/- max float64. func genECData(ec *ecParam) (orig []r2, data []r2c) {
rand.Seed(time.Now().UnixNano())
orig = make([]r2, ec.k)
data = make([]r2c, ec.nPoints)
for i, n := 0, 0; i < ec.k; i++ {
x := rand.Float64() * float64(ec.xBox)
y := rand.Float64() * float64(ec.yBox)
orig[i] = r2{x, y}
for j := ec.nPoints / ec.k; j > 0; j-- {
data[n].x = rand.NormFloat64()*float64(ec.stdv) + x
data[n].y = rand.NormFloat64()*float64(ec.stdv) + y
data[n].c = i
n++
}
}
return
}
// vis writes a .png for extra credit 2. func vis(ec *ecParam, data []r2c, fn string) {
colors := make([]color.NRGBA, ec.k)
for i := range colors {
i3 := i * 3
third := i3 / ec.k
frac := uint8((i3 % ec.k) * 255 / ec.k)
switch third {
case 0:
colors[i] = color.NRGBA{frac, 255 - frac, 0, 255}
case 1:
colors[i] = color.NRGBA{0, frac, 255 - frac, 255}
case 2:
colors[i] = color.NRGBA{255 - frac, 0, frac, 255}
}
}
bounds := image.Rect(-ec.stdv, -ec.stdv, ec.xBox+ec.stdv, ec.yBox+ec.stdv)
im := image.NewNRGBA(bounds)
draw.Draw(im, bounds, image.NewUniform(color.White), image.ZP, draw.Src)
fMinX := float64(bounds.Min.X)
fMaxX := float64(bounds.Max.X)
fMinY := float64(bounds.Min.Y)
fMaxY := float64(bounds.Max.Y)
for _, p := range data {
imx := math.Floor(p.x)
imy := math.Floor(float64(ec.yBox) - p.y)
if imx >= fMinX && imx < fMaxX && imy >= fMinY && imy < fMaxY {
im.SetNRGBA(int(imx), int(imy), colors[p.c])
}
}
f, err := os.Create(fn + ".png")
if err != nil {
fmt.Println(err)
return
}
err = png.Encode(f, im)
if err != nil {
fmt.Println(err)
}
err = f.Close()
if err != nil {
fmt.Println(err)
}
}</lang> Text output:
Data set origins:
x y
256.8 188.6
91.7 51.2
201.8 100.2
161.6 102.8
78.9 152.9
97.8 17.4
Cluster centroids, mean distance from centroid, number of points:
x y distance points
152.4 102.1 30.9 5654
104.8 8.7 31.4 4947
211.3 99.4 32.0 4961
78.3 57.7 29.4 4817
257.7 191.4 36.5 4915
76.9 156.5 35.0 4706
Visualization. Original clusters on left, discovered clusters on right.
Haskell
Solution Uses Map for clusterization and MonadRandom library for random sampling. Vectors are represented as lists, so the solution could be extended to any space dimension.
<lang haskell>{-# LANGUAGE Strict,FlexibleInstances #-} module KMeans where
import Control.Applicative import Control.Monad.Random import Data.List (minimumBy, genericLength, transpose) import Data.Ord (comparing) import qualified Data.Map.Strict as M
type Vec = [Float]
type Cluster = [Vec]
kMeansIteration :: [Vec] -> [Vec] -> [Cluster] kMeansIteration pts = clusterize . fixPoint iteration
where
iteration = map centroid . clusterize
clusterize centroids = M.elems $ foldr add m0 pts
where add x = M.insertWith (++) (centroids `nearestTo` x) [x]
m0 = M.unions $ map (`M.singleton` []) centroids
nearestTo :: [Vec] -> Vec -> Vec nearestTo pts x = minimumBy (comparing (distance x)) pts
distance :: Vec -> Vec -> Float distance a b = sum $ map (^2) $ zipWith (-) a b
centroid :: [Vec] -> Vec centroid = map mean . transpose
where mean pts = sum pts / genericLength pts
fixPoint :: Eq a => (a -> a) -> a -> a fixPoint f x = if x == fx then x else fixPoint f fx where fx = f x
-- initial sampling
kMeans :: MonadRandom m => Int -> [Vec] -> m [Cluster] kMeans n pts = kMeansIteration pts <$> take n <$> randomElements pts
kMeansPP :: MonadRandom m => Int -> [Vec] -> m [Cluster] kMeansPP n pts = kMeansIteration pts <$> centroids
where centroids = iterate (>>= nextCentroid) x0 !! (n-1)
x0 = take 1 <$> randomElements pts
nextCentroid cs = (: cs) <$> fromList (map (weight cs) pts)
weight cs x = (x, toRational $ distance x (cs `nearestTo` x))
randomElements :: MonadRandom m => [a] -> m [a] randomElements pts = map (pts !!) <$> getRandomRs (0, length pts)
-- sample cluster generation
instance (RandomGen g, Monoid m) => Monoid (Rand g m) where
mempty = pure mempty mappend = liftA2 mappend
mkCluster n s m = take n . transpose <$> mapM randomsAround m
where randomsAround x0 = map (\x -> x0+s*atanh x) <$> getRandomRs (-1,1)</lang>
Examples
<lang haskell>module Main where
import Graphics.EasyPlot import Data.Monoid
import KMeans
test = do datum <- mkCluster 1000 0.5 [0,0,1]
<> mkCluster 2000 0.5 [2,3,1]
<> mkCluster 3000 0.5 [2,-3,0]
cls <- kMeansPP 3 datum
mapM_ (\x -> print (centroid x, length x)) cls
main = do datum <- sequence [ mkCluster 30100 0.3 [0,0]
, mkCluster 30200 0.4 [2,3]
, mkCluster 30300 0.5 [2,-3]
, mkCluster 30400 0.6 [6,0]
, mkCluster 30500 0.7 [-3,-3]
, mkCluster 30600 0.8 [-5,5] ]
cls <- kMeansPP 6 (mconcat datum)
plot (PNG "plot1.png") $ map listPlot cls
where
listPlot = Data2D [Title "",Style Dots] [] . map (\(x:y:_) -> (x,y))
</lang> Result: all centroids and clusters are found.
λ> test ([3.161875e-3,-3.096125e-3,0.99095285],1002) ([2.004138,2.9655986,1.0139971],1999) ([2.006579,-2.9902787],2999)
Huginn
<lang huginn>#! /bin/sh exec huginn -E "${0}" "${@}"
- ! huginn
import Algorithms as algo; import Mathematics as math; import OperatingSystem as os;
class Color { r = 0.; g = 0.; b = 0.; } class Point { x = 0.; y = 0.; group = -1; }
k_means_initial_centroids( points_, clusterCount_ ) { centroids = []; discreteRng = math.Randomizer( math.Randomizer.DISTRIBUTION.DISCRETE, 0, size( points_ ) - 1 ); uniformRng = math.Randomizer( math.Randomizer.DISTRIBUTION.UNIFORM, 0.0, 1.0 ); centroids.push( copy( points_[discreteRng.next()] ) ); for ( i : algo.range( clusterCount_ - 1 ) ) { distances = []; sum = 0.0; for ( p : points_ ) { shortestDist = math.INFINITY; for ( c : centroids ) { dx = c.x - p.x; dy = c.y - p.y; d = dx * dx + dy * dy; if ( d < shortestDist ) { shortestDist = d; } } distances.push( ( shortestDist, p ) ); sum += shortestDist; } sum *= uniformRng.next(); for ( d : distances ) { sum -= d[0]; if ( sum <= 0.0 ) { centroids.push( copy( d[1] ) ); break; } } } for ( i, c : algo.enumerate( centroids ) ) { c.group = i; } return ( centroids ); }
k_means( points_, clusterCount_, maxError_ = 0.001, maxIter_ = 100 ) { centroids = k_means_initial_centroids( points_, clusterCount_ ); pointCount = real( size( points_ ) ); for ( iter : algo.range( maxIter_ ) ) { updated = 0.0; for ( p : points_ ) { shortestDist = math.INFINITY; g = 0; for ( c : centroids ) { dx = c.x - p.x; dy = c.y - p.y; dist = dx * dx + dy * dy; if ( dist < shortestDist ) { shortestDist = dist; g = c.group; } } if ( p.group != g ) { p.group = g; updated += 1.0; } } for ( c : centroids ) { n = 0; c.x = 0.; c.y = 0.; for ( p : points_ ) { if ( p.group == c.group ) { c.x += p.x; c.y += p.y; n += 1; } } if ( n > 0 ) { c.x /= real( n ); c.y /= real( n ); } } err = updated / pointCount; os.stderr().write_line( "err = {}\n".format( err ) ); if ( err < maxError_ ) { os.stderr().write_line( "done in {} iterations\n".format( iter ) ); break; } } return ( centroids ); }
gen_points( numPoints_ ) { phiGen = math.Randomizer( math.Randomizer.DISTRIBUTION.UNIFORM, 0., 2. * math.pi( real ) ); rGen = math.Randomizer( math.Randomizer.DISTRIBUTION.TRIANGLE, 0., 1., 1. ); points = []; for ( i : algo.range( numPoints_ ) ) { phi = phiGen.next(); r = rGen.next(); points.push( Point( r * math.cosinus( phi ), r * math.sinus( phi ) ) ); } return ( points ); }
import ProgramOptions as po;
main( argv_ ) { poh = po.Handler( "k-means++", "k-means++ clustering algorithm demo" ); poh.add_option( name: "numPoints,N", requirement: po.VALUE_REQUIREMENT.REQUIRED, help: "number of points", conversion: integer, valueName: "num", defaultValue: 30000 ); poh.add_option( name: "numClusters,C", requirement: po.VALUE_REQUIREMENT.REQUIRED, help: "number of custers", conversion: integer, valueName: "num", defaultValue: 7 ); poh.add_option( name: "maxIterations,I", requirement: po.VALUE_REQUIREMENT.REQUIRED, help: "maximum number of iterations for the algorithm to run", conversion: integer, valueName: "num", defaultValue: 100 ); poh.add_option( name: "maxInvalidRatio,R", requirement: po.VALUE_REQUIREMENT.REQUIRED, help: "maximum ratio of points that are still assigned to invalid centroids", conversion: real, valueName: "num", defaultValue: 0.001 ); poh.add_option( name: "help,H", requirement: po.VALUE_REQUIREMENT.NONE, help: "show help information and stop" ); poh.add_option( name: "verbose,v", requirement: po.VALUE_REQUIREMENT.NONE, help: "show more info about program execution" ); parsed = poh.command_line( argv_ ); if ( parsed == none ) { return ( 1 ); } if ( parsed.options["help"] ) { print( poh.help_string() + "\n" ); return ( 0 ); } if ( parsed.options["verbose"] ) { os.stderr().write_line( string( parsed ) + "\n" ); } points = gen_points( parsed.options["numPoints"] ); print_eps( points, k_means( points, parsed.options["numClusters"], parsed.options["maxInvalidRatio"], parsed.options["maxIterations"] ) ); }
print_eps( points, cluster_centers, W = 400, H = 400 ) { colors = []; for ( i : algo.range( size( cluster_centers ) ) ) { ii = real( i ); colors.push( Color( ( 3. * ( ii + 1. ) % 11. ) / 11.0, ( 7. * ii % 11. ) / 11.0, ( 9. * ii % 11. ) / 11.0 ) ); } max_x = max_y = - math.INFINITY; min_x = min_y = math.INFINITY; for ( p : points ) { if ( max_x < p.x ) { max_x = p.x; } if ( min_x > p.x ) { min_x = p.x; } if ( max_y < p.y ) { max_y = p.y; } if ( min_y > p.y ) { min_y = p.y; } } scale = math.min( real( W ) / ( max_x - min_x ), real( H ) / ( max_y - min_y ) ); cx = ( max_x + min_x ) / 2.; cy = ( max_y + min_y ) / 2.; print( "%!PS-Adobe-3.0\n%%BoundingBox: -5 -5 {} {}\n".format( W + 10, H + 10 ) ); print( "/l {rlineto} def /m {rmoveto} def\n" "/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def\n" "/s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath " " gsave 1 setgray fill grestore gsave 3 setlinewidth" " 1 setgray stroke grestore 0 setgray stroke }def\n" ); for ( i, cc : algo.enumerate( cluster_centers ) ) { print( "{} {} {} setrgbcolor\n".format( colors[i].r, colors[i].g, colors[i].b ) ); for ( p : points ) { if ( p.group != i ) { continue; } print( "{:.3f} {:.3f} c\n".format( ( p.x - cx ) * scale + real( W ) / 2., ( p.y - cy ) * scale + real( H ) / 2. ) ); } print("\n0 setgray {} {} s\n".format( ( cc.x - cx ) * scale + real( W ) / 2., ( cc.y - cy ) * scale + real( H ) / 2. ) ); } print( "\n%%%%EOF\n" ); }</lang>
J
Solution:<lang j> NB. Selection of initial centroids, per K-means++
initialCentroids =: (] , randomCentroid)^:(<:@:]`(,:@:seedCentroid@:[))~
seedCentroid =: {~ ?@#
randomCentroid =: [ {~ [: wghtProb [: <./ distance/~
distance =: +/&.:*:@:-"1 NB. Extra credit #3 (N-dimensional is the same as 2-dimensional in J)
wghtProb =: 1&$: : ((%{:)@:(+/\)@:] I. [ ?@$ 0:)"0 1 NB. Due to Roger Hui http://j.mp/lj5Pnt
NB. Having selected the initial centroids, the standard K-means algo follows
centroids =: ([ mean/.~ closestCentroid)^:(]`_:`initialCentroids)
closestCentroid =: [: (i.<./)"1 distance/
mean =: +/ % #</lang>
Extra credit:<lang j> randMatrix =: ?@$&0 NB. Extra credit #1
packPoints =: <"1@:|: NB. Extra credit #2: Visualization code due to Max Harms http://j.mp/l8L45V plotClusters =: dyad define NB. as is the example image in this task
require 'plot' pd 'reset;aspect 1;type dot;pensize 2' pd@:packPoints&> y pd 'type marker;markersize 1.5;color 0 0 0' pd@:packPoints x pd 'markersize 0.8;color 255 255 0' pd@:packPoints x pd 'show' )
NB. Extra credit #4: Polar coordinates are not available in this version NB. but wouldn't be hard to provide with &.cartToPole .</lang>
Example:<lang j> plotRandomClusters =: 3&$: : (dyad define) dataset =. randMatrix 2 {. y,2
centers =. x centroids dataset clusters =. centers (closestCentroid~ </. ]) dataset centers plotClusters clusters )
plotRandomClusters 300 NB. 300 points, 3 clusters 6 plotRandomClusters 30000 NB. 3e5 points, 6 clusters 10 plotRandomClusters 17000 5 NB. 17e3 points, 10 clusters, 5 dimensions</lang>
Java
<lang java> import java.util.Random;
public class KMeansWithKpp{ // Variables Needed public Point[] points; public Point[] centroids; Random rand; public int n; public int k;
// hide default constructor private KMeansWithKpp(){ }
KMeansWithKpp(Point[] p, int clusters){ points = p; n = p.length; k = Math.max(1, clusters); centroids = new Point[k]; rand = new Random(); }
private static double distance(Point a, Point b){
return (a.x - b.x) * (a.x - b.x) + (a.y - b.y) * (a.y - b.y);
}
private static int nearest(Point pt, Point[] others, int len){ double minD = Double.MAX_VALUE; int index = pt.group; len = Math.min(others.length, len); double dist; for (int i = 0; i < len; i++) { if (minD > (dist = distance(pt, others[i]))) { minD = dist; index = i; } } return index; }
private static double nearestDistance(Point pt, Point[] others, int len){ double minD = Double.MAX_VALUE; len = Math.min(others.length, len); double dist; for (int i = 0; i < len; i++) { if (minD > (dist = distance(pt, others[i]))) { minD = dist; } } return minD; }
private void kpp(){ centroids[0] = points[rand.nextInt(n)]; double[] dist = new double[n]; double sum = 0; for (int i = 1; i < k; i++) { for (int j = 0; j < n; j++) { dist[j] = nearestDistance(points[j], centroids, i); sum += dist[j]; } sum = (sum * rand.nextInt(Integer.MAX_VALUE)) / Integer.MAX_VALUE; for (int j = 0; j < n; j++) { if ((sum -= dist[j]) > 0) continue; centroids[i].x = points[j].x; centroids[i].y = points[j].y; } } for (int i = 0; i < n; i++) { points[i].group = nearest(points[i], centroids, k); } }
public void kMeans(int maxTimes){ if (k == 1 || n <= 0) { return; } if(k >= n){ for(int i =0; i < n; i++){ points[i].group = i; } return; } maxTimes = Math.max(1, maxTimes); int changed; int bestPercent = n/1000; int minIndex; kpp(); do { for (Point c : centroids) { c.x = 0.0; c.y = 0.0; c.group = 0; } for (Point pt : points) { if(pt.group < 0 || pt.group > centroids.length){ pt.group = rand.nextInt(centroids.length); } centroids[pt.group].x += pt.x; centroids[pt.group].y = pt.y; centroids[pt.group].group++; } for (Point c : centroids) { c.x /= c.group; c.y /= c.group; } changed = 0; for (Point pt : points) { minIndex = nearest(pt, centroids, k); if (k != pt.group) { changed++; pt.group = minIndex; } } maxTimes--; } while (changed > bestPercent && maxTimes > 0); } }
// A class for point(x,y) in plane
class Point{ public double x; public double y; public int group;
Point(){ x = y = 0.0; group = 0; }
/* Generates a random points on 2D Plane within given X-axis and Y-axis */ public Point[] getRandomPlaneData(double minX, double maxX, double minY, double maxY, int size){ if (size <= 0) return null; double xdiff, ydiff; xdiff = maxX - minX; ydiff = maxY - minY; if (minX > maxX) { xdiff = minX - maxX; minX = maxX; } if (maxY < minY) { ydiff = minY - maxY; minY = maxY; } Point[] data = new Point[size]; Random rand = new Random(); for (int i = 0; i < size; i++) { data[i].x = minX + (xdiff * rand.nextInt(Integer.MAX_VALUE)) / Integer.MAX_VALUE; data[i].y = minY + (ydiff * rand.nextInt(Integer.MAX_VALUE)) / Integer.MAX_VALUE; } return data; }
/* Generate Random Polar Coordinates within given radius */ public Point[] getRandomPolarData(double radius, int size){ if (size <= 0) { return null; } Point[] data = new Point[size]; double radi, arg; Random rand = new Random(); for (int i = 0; i < size; i++) { radi = (radius * rand.nextInt(Integer.MAX_VALUE)) / Integer.MAX_VALUE; arg = (2 * Math.PI * rand.nextInt(Integer.MAX_VALUE)) / Integer.MAX_VALUE; data[i].x = radi * Math.cos(arg); data[i].y = radi * Math.sin(arg); } return data; }
} </lang>
JavaScript
Solution
Live Demo (Extra Credit #2) KMeans++ in JavaScript
<lang javascript> /**
* kmeans module * * cluster(model, k, converged = assignmentsConverged) * distance(p, q), * distanceSquared(p, q), * centroidsConverged(delta) * assignmentsConverged(model, newModel) * assignmentsToClusters(model) */
define(function () {
"use strict";
/**
* @public
* Calculate the squared distance between two vectors.
*
* @param [number] p vector with same dimension as q
* @param [number] q vector with same dimension as p
* @return {number} the distance between p and q squared
*/
function distanceSquared(p, q) {
const d = p.length; // dimension of vectors
if(d !== q.length) throw Error("p and q vectors must be the same length")
let sum = 0;
for(let i = 0; i < d; i += 1) {
sum += (p[i] - q[i])**2
}
return sum;
}
/**
* @public
* Calculate the distance between two vectors of the same dimension.
*
* @param [number] p vector of same dimension as q
* @param [number] q vector of same dimension as p
* @return the distance between vectors p and q
*/
function distance(p, q) {
return Math.sqrt(distanceSquared(p, q));
}
/**
* @private
* find the closest centroid for the given observation and return it's index.
*
* @param number centroids - array of k vectors, each vector with same dimension as observations.
* these are the center of the k clusters
* @param number observation - vector with same dimension as centroids.
* this is the observation to be clustered.
* @return {number} the index of the closest centroid in centroids
*/
function findClosestCentroid(centroids, observation) {
const k = centroids.length; // number of clusters/centroids
let centroid = 0;
let minDistance = distance(centroids[0], observation);
for(let i = 1; i < k; i += 1) {
const dist = distance(centroids[i], observation);
if(dist < minDistance) {
centroid = i;
minDistance = dist;
}
}
return centroid;
}
/**
* @private
* Calculate the centroid for the given observations.
* This takes the average of all observations (at each dimension).
* This average vector is the centroid for those observations.
*
* @param number observations - array of observations (each observatino is a vectors)
* @return [number] centroid for given observations (vector of same dimension as observations)
*/
function calculateCentroid(observations) {
const n = observations.length; // number of observations
const d = observations[0].length; // dimension of vectors
// create zero vector of same dimension as observation
let centroid = [];
for(let i = 0; i < d; i += 1) {
centroid.push(0.0);
}
//
// sum all observations at each dimension
//
for(let i = 0; i < n; i += 1) {
//
// add the observation to the sum vector, element by element
// to prepare to calculate the average at each dimension.
//
for(let j = 0; j < d; j += 1) {
centroid[j] += observations[i][j];
}
}
//
// divide each dimension by the number of observations
// to create the average vector.
//
for(let j = 0; j < d; j += 1) {
centroid[j] /= n;
}
return centroid; }
/**
* @private
* calculate the cluster assignments for the observations, given the centroids.
*
* @param number centroids - list of vectors with same dimension as observations
* @param number observations - list of vectors with same dimension as centroids
* @return [number] list of indices into centroids; one per observation.
*/
function assignClusters(centroids, observations) {
const n = observations.length; // number of observations
const assignments = [];
for(let i = 0; i < n; i += 1) {
assignments.push(findClosestCentroid(centroids, observations[i]));
}
return assignments; // centroid index for each observation }
/**
* @private
* calculate one step of the k-means algorithm;
* - assign each observation to the nearest centroid to create clusters
* - calculate a new centroid for each cluster given the observations in the cluster.
*
* @param number centroids - list of vectors with same dimension as observations
* @param number observations - list of vectors with same dimension as centroids
* @return a new model with observations, centroids and assignments
*/
function kmeansStep(centroids, observations) {
const k = centroids.length; // number of clusters/centroids
// assign each observation to the nearest centroid to create clusters
const assignments = assignClusters(centroids, observations); // array of cluster indices that correspond observations
// calculate a new centroid for each cluster given the observations in the cluster
const newCentroids = [];
for(let i = 0; i < k; i += 1) {
// get the observations for this cluster/centroid
const clusteredObservations = observations.filter((v, j) => assignments[j] === i);
// calculate a new centroid for the observations
newCentroids.push(calculateCentroid(clusteredObservations));
}
return {'observations': observations, 'centroids': newCentroids, 'assignments': assignments }
}
/**
* @public
* Run k-means on the given model until each centroid converges to with the given delta
* The initial model is NOT modified by the algorithm, rather a new model is returned.
*
* @param {*} model - object with
* observations: array, length n, of data points; each datapoint is
* itself an array of numbers (a vector).
* The length each datapoint (d) vector should be the same.
* centroids: array of data points.
* The length of the centroids array indicates the number of
* of desired clusters (k).
* each datapoint is array (vector) of numbers
* with same dimension as the datapoints in observations.
* assignments: array of integers, one per observation,
* with values 0..centroids.length - 1
* @param number delta - the maximum difference between each centroid in consecutive runs for convergence
* @return {*} - result with
* model: model, as described above, with updated centroids and assignments,
* iterations: number of iterations,
* durationMs: elapsed time in milliseconds
*/
function kmeans(model, maximumIterations = 200, converged = assignmentsConverged) {
const start = new Date();
// calculate new centroids and cluster assignments
let newModel = kmeansStep(model.centroids, model.observations);
// continue until centroids do not change (within given delta)
let i = 0;
while((i < maximumIterations) && !converged(model, newModel)) {
model = newModel; // new model is our model now
// console.log(model);
// calculate new centroids and cluster assignments
newModel = kmeansStep(model.centroids, model.observations);
i += 1;
}
// console.log(newModel);
const finish = new Date();
return {'model': newModel, 'iterations': i, 'durationMs': (finish.getTime() - start.getTime())};
}
/**
* @public
* Return a function that determines convergence based on the centroids.
* If two consecutive sets of centroids remain within a given delta,
* then the algorithm is converged.
*
* @param number delta, the maximum difference between each centroid in consecutive runs for convergence
* @return function to use as the converged function in kmeans call.
*/
function centroidsConverged(delta) {
/**
* determine if two consecutive set of centroids are converged given a maximum delta.
*
* @param number centroids - list of vectors with same dimension as observations
* @param number newCentroids - list of vectors with same dimension as observations
* @param number delta - the maximum difference between each centroid in consecutive runs for convergence
*/
return function(model, newModel) {
const centroids = model.centroids;
const newCentroids = newModel.centroids;
const k = centroids.length; // number of clusters/centroids
for(let i = 0; i < k; i += 1) {
if(distance(centroids[i], newCentroids[i]) > delta) {
return false;
}
}
return true;
}
}
/**
* @public
* determine if two consecutive set of clusters are converged;
* the clusters are converged if the cluster assignments are the same.
*
* @param {*} model - object with observations, centroids, assignments
* @param {*} newModel - object with observations, centroids, assignments
* @param number delta - the maximum difference between each centroid in consecutive runs for convergence
*/
function assignmentsConverged(model, newModel) {
function arraysEqual(a, b) {
if (a === b) return true;
if (a === undefined || b === undefined) return false;
if (a === null || b === null) return false;
if (a.length !== b.length) return false;
// If you don't care about the order of the elements inside
// the array, you should sort both arrays here.
for (var i = 0; i < a.length; ++i) {
if (a[i] !== b[i]) return false;
}
return true;
}
return arraysEqual(model.assignments, newModel.assignments);
}
/**
* Use the model assignments to create
* array of observation indices for each centroid
*
* @param {object} model with observations, centroids and assignments
* @reutrn number array of observation indices for each cluster
*/
function assignmentsToClusters(model) {
//
// put offset of each data points into clusters using the assignments
//
const n = model.observations.length;
const k = model.centroids.length;
const assignments = model.assignments;
const clusters = [];
for(let i = 0; i < k; i += 1) {
clusters.push([])
}
for(let i = 0; i < n; i += 1) {
clusters[assignments[i]].push(i);
}
return clusters; }
//
// return public methods
//
return {
'cluster': kmeans,
'distance': distance,
'distanceSquared': distanceSquared,
'centroidsConverged': centroidsConverged,
'assignmentsConverged': assignmentsConverged,
"assignmentsToClusters": assignmentsToClusters
};
});
/**
* kmeans++ initialization module */
define(function (require) {
"use strict";
const kmeans = require("./kmeans");
/**
* @public
* create an initial model given the data and the number of clusters.
*
* This uses the kmeans++ algorithm:
* 1. Choose one center uniformly at random from among the data points.
* 2. For each data point x, compute D(x), the distance between x and
* the nearest center that has already been chosen.
* 3. Choose one new data point at random as a new center,
* using a weighted probability distribution where a point x is chosen with probability proportional to D(x)^2.
* 4. Repeat Steps 2 and 3 until k centers have been chosen.
* 5. Now that the initial centers have been chosen, proceed using
* standard k-means clustering.
*
* @param {[float]} observations the data as an array of number
* @param {integer} k the number of clusters
*/
return function(observations, k) {
/**
* given a set of n weights,
* choose a value in the range 0..n-1
* at random using weights as a distribution.
*
* @param {*} weights
*/
function weightedRandomIndex(weights, normalizationWeight) {
const n = weights.length;
if(typeof normalizationWeight !== 'number') {
normalizationWeight = 0.0;
for(let i = 0; i < n; i += 1) {
normalizationWeight += weights[i];
}
}
const r = Math.random(); // uniformly random number 0..1 (a probability)
let index = 0;
let cumulativeWeight = 0.0;
for(let i = 0; i < n; i += 1) {
//
// use the uniform probability to search
// within the normalized weighting (we divide by totalWeight to normalize).
// once we hit the probability, we have found our index.
//
cumulativeWeight += weights[i] / normalizationWeight;
if(cumulativeWeight > r) {
return i;
}
}
throw Error("algorithmic failure choosing weighted random index");
}
const n = observations.length;
const distanceToCloseCentroid = []; // distance D(x) to closest centroid for each observation
const centroids = []; // indices of observations that are chosen as centroids
//
// keep list of all observations' indices so
// we can remove centroids as they are created
// so they can't be chosen twice
//
const index = [];
for(let i = 0; i < n; i += 1) {
index[i] = i;
}
//
// 1. Choose one center uniformly at random from among the data points.
//
let centroidIndex = Math.floor(Math.random() * n);
centroids.push(centroidIndex);
for(let c = 1; c < k; c += 1) {
index.slice(centroids[c - 1], 1); // remove previous centroid from further consideration
distanceToCloseCentroid[centroids[c - 1]] = 0; // this effectively removes it from the probability distribution
//
// 2. For each data point x, compute D(x), the distance between x and
// the nearest center that has already been chosen.
//
// NOTE: we used the distance squared (L2 norm)
//
let totalWeight = 0.0;
for(let i = 0; i < index.length; i += 1) {
//
// if this is the first time through, the distance is undefined, so just set it.
// Otherwise, choose the minimum of the prior closest and this new centroid
//
const distanceToCentroid = kmeans.distanceSquared(observations[index[i]], observations[centroids[c - 1]]);
distanceToCloseCentroid[index[i]] =
(typeof distanceToCloseCentroid[index[i]] === 'number')
? Math.min(distanceToCloseCentroid[index[i]], distanceToCentroid)
: distanceToCentroid;
totalWeight += distanceToCloseCentroid[index[i]];
}
//
// 3. Choose one new data point at random as a new center,
// using a weighted probability distribution where a point x is chosen with probability proportional to D(x)^2.
//
centroidIndex = index[weightedRandomIndex(distanceToCloseCentroid, totalWeight)];
centroids.push(centroidIndex);
// 4. Repeat Steps 2 and 3 until k centers have been chosen.
}
//
// 5. Now that the initial centers have been chosen, proceed using
// standard k-means clustering. Return the model so that
// kmeans can continue.
//
return {
'observations': observations,
'centroids': centroids.map(x => observations[x]), // map centroid index to centroid value
'assignments': observations.map((x, i) => i % centroids.length) // distribute among centroids
}
}
});
/**
* Extra Credit #1 * module for creating random models for kmeans clustering */
define(function (require) {
"use strict";
const kmeans = require("./kmeans");
/**
* @return a random, normally distributed number
*/
function randomNormal() {
// n = 6 gives a good enough approximation
return ((Math.random() + Math.random() + Math.random() + Math.random() + Math.random() + Math.random()) - 3) / 3;
}
/**
* Generate a uniform random unit vector
*
* @param {Integer} d dimension of data
* @return n random datapoints of dimension d with length == 1
*/
function randomUnitVector(d) {
const range = max - min;
let magnitude = 0.0;
const observation = [];
// uniform random for each dimension
for(let j = 0; j < d; j += 1) {
const x = Math.random();
observation[j] = x;
magnitude = x * x;
}
// normalize
const magnitude = Math.sqrt(magnitude);
for(let j = 0; j < d; j += 1) {
observation[j] /= magnitude;
}
return observation; }
/**
* Generate a uniform random unit vectors for clustering
*
* @param {Integer} n number of data points
* @param {Integer} d dimension of data
* @return n random datapoints of dimension d with length == 1
*/
function randomUnitVectors(n, d) {
// create n random observations, each of dimension d
const observations = [];
for(let i = 0; i < n; i += 1) {
// create random observation of dimension d
const observation = randomUnitVector(d);
observations.push(observation);
}
return observations; }
/**
* Generate a spherical random vector
*
* @param {Integer} n number of data points
* @param {Integer} d dimension of data
* @param {Number} r radium from center for data point
* @return n random datapoints of dimension d
*/
function randomSphericalVector(d, r) {
const observation = [];
let magnitude = 0.0;
for(let j = 0; j < d; j += 1)
{
const x = randomNormal();
observation[j] = x;
magnitude += x * x;
}
// normalize
magnitude = Math.sqrt(magnitude);
for(let j = 0; j < d; j += 1) {
observation[j] = observation[j] * r / magnitude;
}
return observation; }
/**
* Generate a spherical random vectors
*
* @param {Integer} n number of data points
* @param {Integer} d dimension of data
* @param {Number} max radius from center for data points
* @return n random datapoints of dimension d
*/
function randomSphericalVectors(n, d, r) {
// create n random observations, each of dimension d
const observations = [];
for(let i = 0; i < n; i += 1) {
// create random observation of dimension d with random radius
const observation = randomSphericalVector(d, Math.random() * r);
observations.push(observation);
}
return observations; }
/**
* Generate a uniform random model for clustering
*
* @param {Integer} n number of data points
* @param {Integer} d dimension of data
* @param {Number} radius of sphere
* @return n random datapoints of dimension d
*/
function randomVectors(n, d, min, max) {
const range = max - min;
// create n random observations, each of dimension d
const observations = [];
for(let i = 0; i < n; i += 1) {
// create random observation of dimension d
const observation = randomVector(d, min, max);
observations.push(observation);
}
return observations; }
/**
* Generate a uniform random model for clustering
*
* @param {Integer} d dimension of data
* @param {Number} radius of sphere
* @return n random datapoints of dimension d
*/
function randomVector(d, min, max) {
// create random observation of dimension d
const range = max - min;
const observation = [];
for(let j = 0; j < d; j += 1) {
observation.push(min + Math.random() * range);
}
return observation; }
return {
'randomVector': randomVector,
'randomUnitVector': randomUnitVector,
'randomSphericalVector': randomSphericalVector,
'randomVectors': randomVectors,
'randomUnitVectors': randomUnitVectors,
'randomSphericalVectors': randomSphericalVectors
}
});
/**
* Extra Credit #4 * Application to cluster random data using kmeans++ * * cluster(k, n, d) - cluster n data points of dimension d into k clusters * plot(canvas, result) - plot the results of cluster() to the given html5 canvas using clusterjs */
define(function (require) {
"use strict";
const kmeans = require("./kmeans/kmeans");
const kmeanspp = require("./kmeans/kmeanspp");
const randomCentroidInitializer = require("./kmeans/randomCentroidInitializer");
const kmeansRandomModel = require("./kmeans/kmeansRandomModel");
/**
* @public
* Load iris dataset and run kmeans on it given the number of clusters
*
* @param {integer} k number of clusters to create
*/
function cluster(k, n, d) {
//
// map iris data rows from dictionary to vector (array), leaving out the label
//
const observations = kmeansRandomModel.randomSphericalVectors(n, d, 10.0);
//
// create the intial model and run it
//
// const initialModel = randomCentroidInitializer(observations, k);
const initialModel = kmeanspp(observations, k);
//
// cluster into given number of clusters
//
const results = kmeans.cluster(initialModel);
//
// do this for the convenience of the plotting functions
//
results.clusters = kmeans.assignmentsToClusters(results.model);
return results; }
const clusterColor = ['red', 'green', 'blue', 'yellow', 'purple', 'cyan', 'magenta', 'pink', 'brown', 'black']; let chart = undefined;
/**
* plot the clustred iris data model.
*
* @param {object} results of cluster(), with model, clusters and clusterCompositions
* @param {boolean} showClusterColor true to show learned cluster points
* @param {boolean} showSpeciesColor true to show known dataset labelled points
*/
function plot(canvas, results) {
//
// map iris data rows from dictionary to vector (array), leaving out the label
//
const model = results.model;
const observations = model.observations;
const assignments = model.assignments;
const centroids = model.centroids;
const d = observations[0].length;
const n = observations.length;
const k = centroids.length;
//
// put offset of each data points into clusters using the assignments
//
const clusters = results.clusters;
//
// plot the clusters
//
const chartData = {
// for the purposes of plotting in 2 dimensions, we will use
// x = dimension 0 and y = dimension 1
datasets: clusters.map(function(c, i) {
return {
label: "cluster" + i,
data: c.map(d => ({'x': observations[d][0], 'y': observations[d][1]})),
backgroundColor: clusterColor[i % clusterColor.length],
pointBackgroundColor: clusterColor[i % clusterColor.length],
pointBorderColor: clusterColor[i % clusterColor.length]
};
})
};
const chartOptions = {
responsive: true,
maintainAspectRatio: false,
title: {
display: true,
text: 'Random spherical data set (d=$d, n=$n) clustered using K-Means (k=$k)'
.replace("$d", d)
.replace('$n', n)
.replace('$k', k)
},
legend: {
position: 'bottom',
display: true
},
scales: {
xAxes: [{
type: 'linear',
position: 'bottom',
scaleLabel: {
labelString: 'x axis',
display: false,
}
}],
yAxes: [{
type: 'linear',
position: 'left',
scaleLabel: {
labelString: 'y axis',
display: false
}
}]
}
};
//
// we need to destroy the previous chart so it's interactivity
// does not continue to run
//
if(undefined !== chart) {
chart.destroy()
}
chart = new Chart(canvas, {
type: 'scatter',
data: chartData,
options: chartOptions,
});
}
return {'cluster': cluster, 'plot': plot};
});
</lang>
jq
Adapted from #Wren
The jq program shown here generates PostScript.
Since jq does not currently include a PRNG, we will use /dev/urandom as a source of entropy as per the following bash script, which invoces jq to generate 30,000 points in accordance with the task description. The result of one run of this script is shown at https://imgur.com/gallery/F0q1yO6 <lang sh>#!/bin/bash export LC_ALL=C
- Generate $1 pseudo-random integers of width $2.
- The integers may have leading 0s
function prng {
cat /dev/urandom | tr -cd '0-9' | fold -w "$2" | head -n "$1"
}
PTS=30000 prng $((4 * PTS)) 3 | jq -nr --argjson PTS $PTS -f k-means++-clustering.jq </lang>
k-means++-clustering.jq <lang jq>
- A Point is represented by a JSON array: [x, y, group]
def hugeVal: infinite;
def RAND_MAX : 1000;
def PTS: $PTS; def K : 6; def W : 400; def H : 400;
def rand: input | tonumber; def randf(m): m * rand / (RAND_MAX - 1);
def pi: 1 | atan * 4;
- Pseudo-randomly generate `count` points in a circle with the given radius
def genXY(count; radius):
# note: this is not a uniform 2-d distribution
pi as $pi
| reduce range(0; count) as $i ([];
.[$i] = [0, 0, 0]
| randf(2 * $pi) as $ang
| randf(radius) as $r
| .[$i][0] = $r * ($ang|cos)
| .[$i][1] = $r * ($ang|sin) ) ;
def dist2($a; $b):
($a[0] - $b[0]) as $x | ($a[1] - $b[1]) as $y | $x * $x + $y * $y;
- output: {minD, minI}
def nearest($pt; $cent; $nCluster):
{ minD : hugeVal,
minI : $pt[2] }
| reduce range(0; $nCluster) as $i (.;
dist2($cent[$i]; $pt) as $d
| if .minD > $d
then .minD = $d
| .minI = $i
else .
end ) ;
- input: {pts, cent}
- output: ditto
def kpp(len):
(.cent|length) as $nCent
| .cent[0] = .pts[rand % len]
| . + { d: []}
| reduce range(1; $nCent) as $nCluster (.;
.sum = 0
| reduce range(0; len) as $j (.;
.d[$j] = nearest(.pts[$j]; .cent; $nCluster).minD
| .sum += .d[$j] )
| .sum = randf(.sum)
| label $out
| .emit = null
# Be sure to emit something:
| foreach (range(0; len), null) as $j (.;
if $j == null then .emit = .
else .sum -= .d[$j]
| if .sum > 0
then .
else .cent[$nCluster] = .pts[$j]
| .emit = .
end
end;
select(.emit) | (.emit, break $out)
) )
| reduce range(0; len) as $j (.;
.pts[$j][2] = nearest(.pts[$j]; .cent; $nCent).minI ) ;
- Input: an array of Point (.pts)
- Output: {pts, cent}
def lloyd(len; nCluster):
{pts: .,
cent: [range(0; nCluster) | [0,0,0]]}
| kpp(len)
# stop when > 99.9% of points are good | until( .changed > (len / 1E4);
# group elements of centroids are used as counters
reduce range(0; nCluster) as $i (.; .cent[$i] = [0,0,0])
| reduce range(0; len) as $j (.;
.pts[$j] as $p
| .cent[$p[2]] |= [ .[0]+$p[0], .[1]+$p[1], .[2] + 1] )
| reduce range(0; nCluster) as $i (.;
(.cent[$i][2] | if . == 0 then 0.0001 else . end) as $divisor
| .cent[$i] |= [ (.[0]/$divisor), (.[1]/$divisor), .[2] ] )
| .changed = 0
# find closest centroid of each point
| reduce range(0; len) as $j (.;
.pts[$j] as $p
| nearest($p; .cent; nCluster).minI as $minI
| if $minI != $p[2]
then .changed += 1
| .pts[$j][2] = $minI
else .
end) )
| .cent |= reduce range(0; nCluster) as $i (.; .[$i][2] = $i ) ;
def printEps($pts; len; cent; nCluster):
def f1($x;$y;$z): "\($x) \($y) \($z) setrgbcolor"; def f2($x;$y) : "\($x) \($y) c"; def f3($x;$y) : "\n0 setgray \($x) \($y) s";
def colors:
reduce range(0; nCluster) as $i ([];
.[3 * $i + 0] = (3 * ($i + 1) % 11) / 11
| .[3 * $i + 1] = (7 * $i % 11) / 11
| .[3 * $i + 2] = (9 * $i % 11) / 11 );
{colors: colors}
| .minX = hugeVal
| .minY = hugeVal
| .maxX = -hugeVal
| .maxY = -hugeVal
| reduce range(0; len) as $j (.;
$pts[$j] as $p
| if .maxX < $p[0] then .maxX = $p[0] else . end
| if .minX > $p[0] then .minX = $p[0] else . end
| if .maxY < $p[1] then .maxY = $p[1] else . end
| if .minY > $p[1] then .minY = $p[1] else . end )
| ([((W / (.maxX - .minX))), ((H / (.maxY - .minY)))] | min) as $scale
| ( (.maxX + .minX) / 2) as $cx
| ( (.maxY + .minY) / 2) as $cy
| "%!PS-Adobe-3.0\n%%BoundingBox: -5 -5 \(W + 10) \(H + 10)",
"/l {rlineto} def /m {rmoveto} def",
"/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def",
"/s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath ",
" gsave 1 setgray fill grestore gsave 3 setlinewidth",
" 1 setgray stroke grestore 0 setgray stroke }def",
(range(0; nCluster) as $i
| cent[$i] as $c
| f1(.colors[3 * $i]; .colors[3 * $i + 1]; .colors[3 * $i + 2]),
( range(0; len) as $j
| $pts[$j] as $p
| if $p[2] == $i
then f2( ($p[0] - $cx) * $scale + W / 2; ($p[1] - $cy) * $scale + H / 2)
else empty
end ),
f3( ($c[0] - $cx) * $scale + W / 2; ($c[1] - $cy) * $scale + H / 2)),
"\n%%EOF"
- The required clustering function with two arguments.
- It returns {pts, cent}
- where .pts and .cent are arrays of Points,
- with the cluster id as the third item.
def cluster($nCluster; $points):
$points | lloyd( length; $nCluster);
- The task:
genXY(PTS; 10) | lloyd(PTS; K) as { pts: $pts, cent: $cent } | printEps($pts; PTS; $cent; K)</lang>
- Output:
See https://imgur.com/gallery/F0q1yO6
Julia
<lang julia># run via Julia REPL using Clustering, Plots, DataFrames, RDatasets
const iris = dataset("datasets", "iris") plt1 = plot(title= "Species Classification", xlabel = "Sepal Width", ylabel = "Sepal length") plt2 = plot(title= "Kmeans++ Classification", xlabel = "Sepal Width", ylabel = "Sepal length")
for (i, sp) in enumerate(unique(iris[!, :Species]))
idx = iris[!, :Species] .== sp spwidth, splength = iris[idx, :SepalWidth], iris[idx, :SepalLength] scatter!(plt1, spwidth, splength, color = [:red, :green, :blue][i], legend = false)
end
features = collect(Matrix(iris[!, 1:4])')
- K Means ++
result = kmeans(features, 3, init = KmppAlg()) # set to 3 clusters with kmeans++
for center in unique(result.assignments)
idx = result.assignments .== center spwidth, splength = iris[idx, :SepalWidth], iris[idx, :SepalLength] scatter!(plt2, spwidth, splength, color = [:green, :red, :blue][center], legend = false)
end
plot(plt1, plt2) </lang>
Kotlin
The terminal output should, of course, be redirected to an .eps file so that it can be viewed with (for instance) Ghostscript.
As in the case of the C example, the data is partitioned into 11 clusters though, unlike C (which doesn't use srand), the output will be different each time the program is run. <lang scala>// version 1.2.21
import java.util.Random import kotlin.math.*
data class Point(var x: Double, var y: Double, var group: Int)
typealias LPoint = List<Point> typealias MLPoint = MutableList<Point>
val origin get() = Point(0.0, 0.0, 0) val r = Random() val hugeVal = Double.POSITIVE_INFINITY
const val RAND_MAX = Int.MAX_VALUE const val PTS = 100_000 const val K = 11 const val W = 400 const val H = 400
fun rand() = r.nextInt(RAND_MAX)
fun randf(m: Double) = m * rand() / (RAND_MAX - 1)
fun genXY(count: Int, radius: Double): LPoint {
val pts = List(count) { origin }
/* note: this is not a uniform 2-d distribution */
for (i in 0 until count) {
val ang = randf(2.0 * PI)
val r = randf(radius)
pts[i].x = r * cos(ang)
pts[i].y = r * sin(ang)
}
return pts
}
fun dist2(a: Point, b: Point): Double {
val x = a.x - b.x val y = a.y - b.y return x * x + y * y
}
fun nearest(pt: Point, cent: LPoint, nCluster: Int): Pair<Int, Double> {
var minD = hugeVal
var minI = pt.group
for (i in 0 until nCluster) {
val d = dist2(cent[i], pt)
if (minD > d) {
minD = d
minI = i
}
}
return minI to minD
}
fun kpp(pts: LPoint, len: Int, cent: MLPoint) {
val nCent = cent.size
val d = DoubleArray(len)
cent[0] = pts[rand() % len].copy()
for (nCluster in 1 until nCent) {
var sum = 0.0
for (j in 0 until len) {
d[j] = nearest(pts[j], cent, nCluster).second
sum += d[j]
}
sum = randf(sum)
for (j in 0 until len) {
sum -= d[j]
if (sum > 0.0) continue
cent[nCluster] = pts[j].copy()
break
}
}
for (j in 0 until len) pts[j].group = nearest(pts[j], cent, nCent).first
}
fun lloyd(pts: LPoint, len: Int, nCluster: Int): LPoint {
val cent = MutableList(nCluster) { origin }
kpp(pts, len, cent)
do {
/* group element for centroids are used as counters */
for (i in 0 until nCluster) {
with (cent[i]) { x = 0.0; y = 0.0; group = 0 }
}
for (j in 0 until len) {
val p = pts[j]
val c = cent[p.group]
with (c) { group++; x += p.x; y += p.y }
}
for (i in 0 until nCluster) {
val c = cent[i]
c.x /= c.group
c.y /= c.group
}
var changed = 0
/* find closest centroid of each point */
for (j in 0 until len) {
val p = pts[j]
val minI = nearest(p, cent, nCluster).first
if (minI != p.group) {
changed++
p.group = minI
}
}
}
while (changed > (len shr 10)) /* stop when 99.9% of points are good */
for (i in 0 until nCluster) cent[i].group = i return cent
}
fun printEps(pts: LPoint, len: Int, cent: LPoint, nCluster: Int) {
val colors = DoubleArray(nCluster * 3)
for (i in 0 until nCluster) {
colors[3 * i + 0] = (3 * (i + 1) % 11) / 11.0
colors[3 * i + 1] = (7 * i % 11) / 11.0
colors[3 * i + 2] = (9 * i % 11) / 11.0
}
var minX = hugeVal
var minY = hugeVal
var maxX = -hugeVal
var maxY = -hugeVal
for (j in 0 until len) {
val p = pts[j]
if (maxX < p.x) maxX = p.x
if (minX > p.x) minX = p.x
if (maxY < p.y) maxY = p.y
if (minY > p.y) minY = p.y
}
val scale = minOf(W / (maxX - minX), H / (maxY - minY))
val cx = (maxX + minX) / 2.0
val cy = (maxY + minY) / 2.0
print("%%!PS-Adobe-3.0\n%%%%BoundingBox: -5 -5 %${W + 10} ${H + 10}\n")
print("/l {rlineto} def /m {rmoveto} def\n")
print("/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def\n")
print("/s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath ")
print(" gsave 1 setgray fill grestore gsave 3 setlinewidth")
print(" 1 setgray stroke grestore 0 setgray stroke }def\n")
val f1 = "%g %g %g setrgbcolor"
val f2 = "%.3f %.3f c"
val f3 = "\n0 setgray %g %g s"
for (i in 0 until nCluster) {
val c = cent[i]
println(f1.format(colors[3 * i], colors[3 * i + 1], colors[3 * i + 2]))
for (j in 0 until len) {
val p = pts[j]
if (p.group != i) continue
println(f2.format((p.x - cx) * scale + W / 2, (p.y - cy) * scale + H / 2))
}
println(f3.format((c.x - cx) * scale + W / 2, (c.y - cy) * scale + H / 2))
}
print("\n%%%%EOF")
}
fun main(args: Array<String>) {
val v = genXY(PTS, 10.0) val c = lloyd(v, PTS, K) printEps(v, PTS, c, K)
}</lang>
Lua
<lang lua>
local function load_data(npoints, radius)
-- Generate random data points
--
local data = {}
for i = 1,npoints do
local ang = math.random() * (2.0 * math.pi)
local rad = math.random() * radius
data[i] = {x = math.cos(ang) * rad, y = math.sin(ang) * rad}
end
return data
end
local function print_eps(data, nclusters, centers, cluster)
local WIDTH = 400 local HEIGHT = 400
-- Print an EPS file with clustered points
--
local colors = {}
for k = 1,nclusters do
colors[3*k + 0] = (3 * k % 11) / 11.0
colors[3*k + 1] = (7 * k % 11) / 11.0
colors[3*k + 2] = (9 * k % 11) / 11.0
end
local max_x, max_y, min_x, min_y = -math.maxinteger, -math.maxinteger, math.maxinteger, math.maxinteger for i = 1,#data do if max_x < data[i].x then max_x = data[i].x end if min_x > data[i].x then min_x = data[i].x end if max_y < data[i].y then max_y = data[i].y end if min_y > data[i].y then min_y = data[i].y end end
local scale = WIDTH / (max_x - min_x) if scale > HEIGHT / (max_y - min_y) then scale = HEIGHT / (max_y - min_y) end
local cx = (max_x + min_x) / 2.0 local cy = (max_y + min_y) / 2.0
print(string.format("%%!PS-Adobe-3.0\n%%%%BoundingBox: -5 -5 %d %d",
WIDTH + 10, HEIGHT + 10))
print(string.format("/l {rlineto} def /m {rmoveto} def\n/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def\n/s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath gsave 1 setgray fill grestore gsave 3 setlinewidth 1 setgray stroke grestore 0 setgray stroke }def"
))
-- print(string.format("%g %g %g setrgbcolor\n", 1, 2, 3))
for k = 1,nclusters do
print(string.format("%g %g %g setrgbcolor",
colors[3*k], colors[3*k + 1], colors[3*k + 2]))
for i = 1,#data do
if cluster[i] == k then
print(string.format("%.3f %.3f c",
(data[i].x - cx) * scale + WIDTH / 2.0,
(data[i].y - cy) * scale + HEIGHT / 2.0))
end
end
print(string.format("0 setgray %g %g s",
(centers[k].x - cx) * scale + WIDTH / 2.0,
(centers[k].y - cy) * scale + HEIGHT / 2.0))
end
print(string.format("\n%%%%EOF"))
end
local function kmeans(data, nclusters, init)
-- K-means Clustering -- assert(nclusters > 0) assert(#data > nclusters) assert(init == "kmeans++" or init == "random")
local diss = function(p, q)
-- Computes the dissimilarity between points 'p' and 'q'
--
return math.pow(p.x - q.x, 2) + math.pow(p.y - q.y, 2)
end
-- Initialization
--
local centers = {} -- clusters centroids
if init == "kmeans++" then
local K = 1
-- take one center c1, chosen uniformly at random from 'data'
local i = math.random(1, #data)
centers[K] = {x = data[i].x, y = data[i].y}
local D = {}
-- repeat until we have taken 'nclusters' centers
while K < nclusters do
-- take a new center ck, choosing a point 'i' of 'data' with probability
-- D(i)^2 / sum_{i=1}^n D(i)^2
local sum_D = 0.0
for i = 1,#data do
local min_d = D[i]
local d = diss(data[i], centers[K])
if min_d == nil or d < min_d then
min_d = d
end
D[i] = min_d
sum_D = sum_D + min_d
end
sum_D = math.random() * sum_D
for i = 1,#data do
sum_D = sum_D - D[i]
if sum_D <= 0 then
K = K + 1
centers[K] = {x = data[i].x, y = data[i].y}
break
end
end
end
elseif init == "random" then
for k = 1,nclusters do
local i = math.random(1, #data)
centers[k] = {x = data[i].x, y = data[i].y}
end
end
-- Lloyd K-means Clustering
--
local cluster = {} -- k-partition
for i = 1,#data do cluster[i] = 0 end
local J = function()
-- Computes the loss value
--
local loss = 0.0
for i = 1,#data do
loss = loss + diss(data[i], centers[cluster[i]])
end
return loss
end
local updated = false
repeat
-- update k-partition
--
local card = {}
for k = 1,nclusters do
card[k] = 0.0
end
updated = false
for i = 1,#data do
local min_d, min_k = nil, nil
for k = 1,nclusters do
local d = diss(data[i], centers[k])
if min_d == nil or d < min_d then
min_d, min_k = d, k
end
end
if min_k ~= cluster[i] then updated = true end
cluster[i] = min_k
card[min_k] = card[min_k] + 1.0
end
-- print("update k-partition: ", J())
-- update centers
--
for k = 1,nclusters do
centers[k].x = 0.0
centers[k].y = 0.0
end
for i = 1,#data do
local k = cluster[i]
centers[k].x = centers[k].x + (data[i].x / card[k])
centers[k].y = centers[k].y + (data[i].y / card[k])
end
-- print(" update centers: ", J())
until updated == false
return centers, cluster, J()
end
------------------------------------------------------------------------------ ---- MAIN --------------------------------------------------------------------
local N_POINTS = 100000 -- number of points local N_CLUSTERS = 11 -- number of clusters
local data = load_data(N_POINTS, N_CLUSTERS) centers, cluster, loss = kmeans(data, N_CLUSTERS, "kmeans++") -- print("Loss: ", loss) -- for k = 1,N_CLUSTERS do -- print("center.x: ", centers[k].x, " center.y: ", centers[k].y) -- end print_eps(data, N_CLUSTERS, centers, cluster) </lang>
Mathematica/Wolfram Language
Solution - Initial kmeans code comes from http://mathematica.stackexchange.com/questions/7441/k-means-clustering, now extended to kmeans++ by introducing the function initM.
Was not able to upload pictures of the result...:
<lang>initM[list_List, k_Integer, distFunc_Symbol] :=
Module[{m = {RandomChoice[list]}, n, d},
While[Length[m] < k,
n = RandomChoice@Nearest[m, #] & /@ list;
d = Apply[distFunc, Transpose[{n, list}], {1}];
m = Append[m, RandomChoice[d -> list]]
];
m
];
kmeanspp[list_, k_,
opts : OptionsPattern[{DistanceFunction ->
SquaredEuclideanDistance, "RandomSeed" -> {}}]] :=
BlockRandom[SeedRandom[OptionValue["RandomSeed"]];
Module[{m = initM[list, k, OptionValue[DistanceFunction]], update,
partition, clusters}, update[] := m = Mean /@ clusters;
partition[_] := (clusters =
GatherBy[list,
RandomChoice@
Nearest[m, #, (# -> OptionValue[#] &@DistanceFunction)] &];
update[]);
FixedPoint[partition, list];
{clusters, m}
]
];</lang>
Extra credit:
1. no changes required for N dimensions, it juts works.
2. random data can be generated with
dim = 3;
points = 3000;
l = RandomReal[1, {points, dim}];
or
l = Select[ RandomReal[{-1, 1}, {points,2}],
EuclideanDistance[#, {0, 0}] <= 1 &];
or
x1 = RandomVariate[MultinormalDistribution[{0, 0}, {{1, 0}, {0, 20}}],
points];
x2 =
RandomVariate[MultinormalDistribution[{10, 0}, {{1, 0}, {0, 20}}],
points];
l = Join[x1, x2];
3. data can be visualized with 2D:
dim = 2;
points = 30000;
l = RandomReal[1, {points, dim}];
k = 6
r1 = kmeanspp[l, k];
p1 = ListPlot[r1[[1]]];
p2 = ListPlot[r1[[2]],PlotMarkers -> {"#"}];
Show[{p1, p2}]
3D:
dim = 3;
points = 3000;
l = RandomReal[1, {points, dim}];
k = 6
r1 = kmeanspp[l, k];
p1 = ListPointPlot3D[r1[[1]]];
p2 = ListPointPlot3D[r1[[2]]];
Show[{p1, p2}]
Another version
KMeans[k_, data_] :=
Module[{Renew, Label, Iteration},
clusters = RandomSample[data, k];
Label[clusters_] :=
Flatten[Table[
Ordering[
Table[EuclideanDistance[datai, clustersj], {j,
Length[clusters]}], 1], {i, Length[data]}]];
Renew[labels_] :=
Module[{position},
position = PositionIndex[labels];
Return[Table[Mean[data[[positioni]]], {i, Length[position]}]]];
Iteration[labels_, clusters_] :=
Module[{newlabels, newclusters},
newclusters = Renew[labels];
newlabels = Label[newclusters];
If[newlabels == labels, labels,
Iteration[newlabels, newclusters]]];
Return[Iteration[clusters, Label[clusters]]]]
Nim
<lang nim>#
- compile:
- nim c -d:release kmeans.nim
- and pipe the resultant EPS output to a file, e.g.
- kmeans > results.eps
import random, math, strutils
const
FloatMax = 1.0e100 nPoints = 100_000 nClusters = 11
type
Point = object x, y: float group: int Points = seq[Point] ClusterDist = tuple[indx: int, dist: float] ColorRGB = tuple[r, g, b: float]
proc generatePoints(nPoints: int, radius: float): Points =
result.setLen(nPoints)
for i in 0..<nPoints:
let
r = rand(1.0) * radius
ang = rand(1.0) * 2 * PI
result[i] = Point(x: r * cos(ang),
y: r * sin(ang),
group: 0)
proc nearestClusterCenter(point: Point, cluster_centers: Points): ClusterDist =
# Distance and index of the closest cluster center proc sqrDistance2D(a, b: Point): float = result = (a.x - b.x) ^ 2 + (a.y - b.y) ^ 2
result = (indx: point.group, dist: FLOAT_MAX)
for i, cc in pairs(cluster_centers):
let d = sqrDistance2D(cc, point)
if result.dist > d:
result.dist = d
result.indx = i
proc kpp(points: var Points, clusterCenters: var Points) =
let choice = points[rand(points.high)] clusterCenters[0] = choice
var d: seq[float] sum = 0.0
d.setLen(points.len)
for i in 1..clusterCenters.high:
sum = 0.0
for j, p in pairs(points):
d[j] = nearestClusterCenter(p, cluster_centers[0..i])[1]
sum += d[j]
sum *= rand(1.0)
for j, di in pairs(d):
sum -= di
if sum > 0.0:
continue
clusterCenters[i] = points[j]
break
for _, p in mpairs(points): p.group = nearestClusterCenter(p, clusterCenters)[0]
proc lloyd(points: var Points, nclusters: int): Points =
#result is the cluster_centers let lenpts10 = points.len shr 10 var changed = 0 minI = 0 result.setLen(nclusters)
# call k++ init kpp(points, result)
while true:
# group element for centroids are used as counters
for _, cc in mpairs(result):
cc.x = 0.0
cc.y = 0.0
cc.group = 0
for p in points:
let i = p.group
result[i].group += 1
result[i].x += p.x
result[i].y += p.y
for _, cc in mpairs(result):
cc.x /= cc.group.float
cc.y /= cc.group.float
# find closest centroid of each PointPtr changed = 0
for _, p in mpairs(points):
minI = nearest_cluster_center(p, result)[0]
if minI != p.group:
changed += 1
p.group = minI
# stop when 99.9% of points are good
if changed <= lenpts10:
break
for i, cc in mpairs(result):
cc.group = i
proc printEps(points: Points, cluster_centers: Points, W: int = 400, H: int = 400) =
var colors: seq[ColorRGB]
colors.setLen(clusterCenters.len)
#assert((3.0 * 5.0) mod 11.0 == 4.0)
#assert(3.0 * 5.0 mod 11.0 == 4.0)
#assert((3.0 * 5.0 mod 11.0) / 2.0 == 2.0)
#assert(3.0 * 5.0 mod 11.0 / 2.0 == 2.0)
for i in 0..<clusterCenters.len:
let
f1 = i.float
f2 = (i + 1).float
colors[i] = (r: (3.0 * f2) mod 11.0 / 11.0,
g: (7.0 * f1) mod 11.0 / 11.0,
b: (9.0 * f1) mod 11.0 / 11.0 )
var max_x = -FLOAT_MAX max_y = -FLOAT_MAX min_x = FLOAT_MAX min_y = FLOAT_MAX
for p in points: if max_x < p.x: max_x = p.x if min_x > p.x: min_x = p.x if max_y < p.y: max_y = p.y if min_y > p.y: min_y = p.y
let
scale = min(W.float / (max_x - min_x),
H.float / (max_y - min_y))
cx = (max_x + min_x) / 2.0
cy = (max_y + min_y) / 2.0
echo "%!PS-Adobe-3.0\n%%BoundingBox: -5 -5 $1 $2" % [$(W + 10), $(H + 10)]
echo """/l {rlineto} def /m {rmoveto} def
/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def /s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath
gsave 1 setgray fill grestore gsave 3 setlinewidth 1 setgray stroke grestore 0 setgray stroke }def"""
for i, cc in pairs(clusterCenters):
echo "$1 $2 $3 setrgbcolor" %
[formatFloat(colors[i].r, ffDecimal, 6),
formatFloat(colors[i].g, ffDecimal, 6),
formatFloat(colors[i].b, ffDecimal, 6)]
for p in points:
if p.group != i:
continue
echo "$1 $2 c" % [formatFloat( ((p.x - cx) * scale + W / 2), ffDecimal, 3),
formatFloat( ((p.y - cy) * scale + H / 2), ffDecimal, 3)]
echo "\n0 setgray $1 $2 s" % [formatFloat( ((cc.x - cx) * scale + W / 2), ffDecimal, 3),
formatFloat( ((cc.y - cy) * scale + H / 2), ffDecimal, 3)]
echo "\nshowpage\n%%EOF"
proc main() =
randomize()
var points = generatePoints(nPoints, 10.0) let clusterCentrs = lloyd(points, nClusters) printEps(points, clusterCentrs)
main()</lang>
Phix
I nicked the initial dataset creation from Go, as an alternative <lang Phix>-- demo\rosetta\K_means_clustering.exw -- Press F5 to restart include pGUI.e
Ihandle dlg, canvas, timer cdCanvas cddbuffer, cdcanvas
constant TITLE = "K-means++ clustering"
constant useGoInitialData = false -- (not very well centered)
constant N = 30000, -- number of points
K = 16 -- number of clusters
sequence {Px, Py, Pc} @= repeat(0,N), -- coordinates of points and their cluster
{Cx, Cy} @= repeat(0,K) -- coordinates of centroid of cluster
constant colours = {CD_RED, CD_DARK_RED, CD_BLUE, CD_DARK_BLUE, CD_CYAN, CD_DARK_CYAN,
CD_GREEN, CD_DARK_GREEN, CD_MAGENTA, CD_DARK_MAGENTA, CD_YELLOW,
CD_DARK_YELLOW, CD_DARK_ORANGE, CD_INDIGO, CD_PURPLE, CD_DARK_GREY}
if length(colours)<K then ?9/0 end if
function Centroid() -- Find new centroids of points grouped with current centroids bool change = false
for c=1 to K do -- for each centroid...
integer x=0, y=0, count:= 0; -- find new centroid
for i=1 to N do -- for all points
if Pc[i] = c then -- grouped with current centroid...
x += Px[i]
y += Py[i]
count += 1
end if
end for
if count!=0 then
x = floor(x/count)
y = floor(y/count)
if Cx[c]!=x
or Cy[c]!=y then
Cx[c] = x
Cy[c] = y
change:= true
end if
end if
end for
return change
end function
function sq(atom x) return x*x end function
procedure Voronoi() -- Group points with their nearest centroid
integer d2, -- distance squared,
min_d2 -- minimum distance squared
for i=1 to N do -- for each point...
min_d2 := #3FFFFFFF -- find closest centroid
for c=1 to K do
d2 := sq(Px[i]-Cx[c]) + sq(Py[i]-Cy[c])
if d2<min_d2 then
min_d2 := d2
Pc[i] := c -- update closest centroid
end if
end for
end for
end procedure
function rand_xy() -- Return random X,Y biased for polar coordinates
atom d := rand(240)-1, -- distance: 0..239
a := rnd()*2*PI -- angle: 0..2pi
integer x:= floor(d*cos(a))+320, -- rectangular coords centered on screen
y:= floor(d*sin(a))+240 -- (that is, assuming 640x480)
return {x,y}
end function
--This little bit is copied from/based on Go: constant k = K,
nPoints = N,
xBox = 300,
yBox = 200,
stdv = 30
function genECData()
sequence orig = repeat({0,0}, k),
data = repeat({0,0,0}, nPoints)
integer n = 0, nk = k
for i=1 to k do
integer x := rand(xBox)+320,
y := rand(yBox)+240
orig[i] = {x, y}
for j=1 to floor((nPoints-n)/nk) do
n += 1
atom d := rand(stdv)-1, -- distance: 0..239
a := rnd()*2*PI -- angle: 0..2pi
integer nx:= floor(d*cos(a))+x, -- rectangular coords centered on screen
ny:= floor(d*sin(a))+y -- (that is, assuming 640x480)
data[n] = {nx,ny,i}
end for
nk -= 1
end for
if n!=nPoints then ?9/0 end if
return {orig, data}
end function --</Go ends>
integer iteration = 0
function redraw_cb(Ihandle /*ih*/, integer /*posx*/, integer /*posy*/)
integer {w, h} = IupGetIntInt(canvas, "DRAWSIZE")
cdCanvasActivate(cddbuffer)
if iteration=0 then
if useGoInitialData then
sequence {origins,data} = genECData()
{Px, Py, Pc} = columnize(data)
{Cx, Cy} = columnize(origins)
else
for i=1 to N do {Px[i],Py[i]} = rand_xy() end for -- random set of points
for i=1 to K do {Cx[i],Cy[i]} = rand_xy() end for -- random set of cluster centroids
end if
end if
sequence {r,g,b} @ = repeat(0,w*h)
Voronoi()
bool change := Centroid()
for i=1 to N do
integer idx = Px[i]+(Py[i]-1)*w
{r[idx],g[idx],b[idx]} = cdDecodeColor(colours[Pc[i]])
end for
for i=1 to K do
integer idx = Cx[i]+(Cy[i]-1)*w
{r[idx],g[idx],b[idx]} = cdDecodeColor(CD_WHITE)
end for
cdCanvasPutImageRectRGB(cddbuffer, w, h, {r,g,b})
cdCanvasFlush(cddbuffer)
if change then
iteration += 1
IupSetStrAttribute(dlg, "TITLE", "%s (iteration %d)",{TITLE,iteration})
else
IupSetInt(timer,"RUN",0) -- (stop timer)
IupSetStrAttribute(dlg, "TITLE", TITLE)
end if
return IUP_DEFAULT
end function
function timer_cb(Ihandle /*ih*/)
IupUpdate(canvas) return IUP_IGNORE
end function
function map_cb(Ihandle ih)
cdcanvas = cdCreateCanvas(CD_IUP, ih) cddbuffer = cdCreateCanvas(CD_DBUFFER, cdcanvas) return IUP_DEFAULT
end function
function esc_close(Ihandle /*ih*/, atom c)
if c=K_ESC then return IUP_CLOSE end if
if c=K_F5 then
iteration = 0
IupSetInt(timer,"RUN",1) -- (restart timer)
end if
return IUP_CONTINUE
end function
procedure main()
IupOpen()
canvas = IupCanvas(NULL)
IupSetAttribute(canvas, "RASTERSIZE", "640x480")
IupSetCallback(canvas, "MAP_CB", Icallback("map_cb"))
IupSetCallback(canvas, "ACTION", Icallback("redraw_cb"))
timer = IupTimer(Icallback("timer_cb"), 100)
dlg = IupDialog(canvas,"DIALOGFRAME=YES")
IupSetAttribute(dlg, "TITLE", TITLE)
IupSetCallback(dlg, "K_ANY", Icallback("esc_close"))
IupShow(dlg) IupSetAttribute(canvas, "RASTERSIZE", NULL) IupMainLoop() IupClose()
end procedure
main()</lang> Probably the hardest part of handling more than 2 dimensions would be deleteing all the GUI code, or modifying it to produce an n-dimensional representation. Obviously you would need Pz and Cz, or replace them with n-tuples, and to replace rand_xy().
Python
<lang python>from math import pi, sin, cos from collections import namedtuple from random import random, choice from copy import copy
try:
import psyco psyco.full()
except ImportError:
pass
FLOAT_MAX = 1e100
class Point:
__slots__ = ["x", "y", "group"]
def __init__(self, x=0.0, y=0.0, group=0):
self.x, self.y, self.group = x, y, group
def generate_points(npoints, radius):
points = [Point() for _ in xrange(npoints)]
# note: this is not a uniform 2-d distribution
for p in points:
r = random() * radius
ang = random() * 2 * pi
p.x = r * cos(ang)
p.y = r * sin(ang)
return points
def nearest_cluster_center(point, cluster_centers):
"""Distance and index of the closest cluster center"""
def sqr_distance_2D(a, b):
return (a.x - b.x) ** 2 + (a.y - b.y) ** 2
min_index = point.group min_dist = FLOAT_MAX
for i, cc in enumerate(cluster_centers):
d = sqr_distance_2D(cc, point)
if min_dist > d:
min_dist = d
min_index = i
return (min_index, min_dist)
def kpp(points, cluster_centers):
cluster_centers[0] = copy(choice(points)) d = [0.0 for _ in xrange(len(points))]
for i in xrange(1, len(cluster_centers)):
sum = 0
for j, p in enumerate(points):
d[j] = nearest_cluster_center(p, cluster_centers[:i])[1]
sum += d[j]
sum *= random()
for j, di in enumerate(d):
sum -= di
if sum > 0:
continue
cluster_centers[i] = copy(points[j])
break
for p in points:
p.group = nearest_cluster_center(p, cluster_centers)[0]
def lloyd(points, nclusters):
cluster_centers = [Point() for _ in xrange(nclusters)]
# call k++ init kpp(points, cluster_centers)
lenpts10 = len(points) >> 10
changed = 0
while True:
# group element for centroids are used as counters
for cc in cluster_centers:
cc.x = 0
cc.y = 0
cc.group = 0
for p in points:
cluster_centers[p.group].group += 1
cluster_centers[p.group].x += p.x
cluster_centers[p.group].y += p.y
for cc in cluster_centers:
cc.x /= cc.group
cc.y /= cc.group
# find closest centroid of each PointPtr
changed = 0
for p in points:
min_i = nearest_cluster_center(p, cluster_centers)[0]
if min_i != p.group:
changed += 1
p.group = min_i
# stop when 99.9% of points are good
if changed <= lenpts10:
break
for i, cc in enumerate(cluster_centers):
cc.group = i
return cluster_centers
def print_eps(points, cluster_centers, W=400, H=400):
Color = namedtuple("Color", "r g b");
colors = []
for i in xrange(len(cluster_centers)):
colors.append(Color((3 * (i + 1) % 11) / 11.0,
(7 * i % 11) / 11.0,
(9 * i % 11) / 11.0))
max_x = max_y = -FLOAT_MAX min_x = min_y = FLOAT_MAX
for p in points:
if max_x < p.x: max_x = p.x
if min_x > p.x: min_x = p.x
if max_y < p.y: max_y = p.y
if min_y > p.y: min_y = p.y
scale = min(W / (max_x - min_x),
H / (max_y - min_y))
cx = (max_x + min_x) / 2
cy = (max_y + min_y) / 2
print "%%!PS-Adobe-3.0\n%%%%BoundingBox: -5 -5 %d %d" % (W + 10, H + 10)
print ("/l {rlineto} def /m {rmoveto} def\n" +
"/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def\n" +
"/s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath " +
" gsave 1 setgray fill grestore gsave 3 setlinewidth" +
" 1 setgray stroke grestore 0 setgray stroke }def")
for i, cc in enumerate(cluster_centers):
print ("%g %g %g setrgbcolor" %
(colors[i].r, colors[i].g, colors[i].b))
for p in points:
if p.group != i:
continue
print ("%.3f %.3f c" % ((p.x - cx) * scale + W / 2,
(p.y - cy) * scale + H / 2))
print ("\n0 setgray %g %g s" % ((cc.x - cx) * scale + W / 2,
(cc.y - cy) * scale + H / 2))
print "\n%%%%EOF"
def main():
npoints = 30000 k = 7 # # clusters
points = generate_points(npoints, 10) cluster_centers = lloyd(points, k) print_eps(points, cluster_centers)
main()</lang>
Racket
The k-means clustering: <lang racket>
- lang racket
(require racket/dict
math/distributions)
- Divides the set of points into k clusters
- using the standard k-means clustering algorithm
(define (k-means data k #:initialization (init k-means++))
(define (iteration centroids) (map centroid (clusterize data centroids))) (fixed-point iteration (init data k) #:same-test small-shift?))
- Finds the centroid for a set of points
(define (centroid pts)
(vector-map (curryr / (length pts))
(for/fold ([sum (car pts)]) ([x (in-list (cdr pts))])
(vector-map + x sum))))
- Divides the set of points into clusters
- using given centroids
(define (clusterize data centroids)
(for*/fold ([res (map list centroids)]) ([x (in-list data)]) (define c (argmin (distanse-to x) centroids)) (dict-set res c (cons x (dict-ref res c)))))
- Stop criterion
- all centroids change their positions
- by less then 0.1% of the minimal distance between centroids.
(define (small-shift? c1 c2)
(define min-distance
(apply min
(for*/list ([x (in-list c2)]
[y (in-list c2)] #:unless (equal? x y))
((metric) x y))))
(for/and ([a (in-list c1)] [b (in-list c2)])
(< ((metric) a b) (* 0.001 min-distance))))
</lang>
Initialization methods
<lang racket>
- picks k points from a dataset randomly
(define (random-choice data k)
(for/list ([i (in-range k)]) (list-ref data (random (length data)))))
- uses k-means++ algorithm
(define (k-means++ data k)
(for/fold ([centroids (random-choice data 1)]) ([i (in-range (- k 1))])
(define weights
(for/list ([x (in-list data)])
(apply min (map (distanse-to x) centroids))))
(define new-centroid
(sample (discrete-dist data weights)))
(cons new-centroid centroids)))
</lang>
Different metrics
<lang racket> (define (euclidean-distance a b)
(for/sum ([x (in-vector a)] [y (in-vector b)]) (sqr (- x y))))
(define (manhattan-distance a b)
(for/sum ([x (in-vector a)] [y (in-vector b)]) (abs (- x y))))
(define metric (make-parameter euclidean-distance)) (define (distanse-to x) (curry (metric) x)) </lang>
The fixed point operator
<lang racket> (define (fixed-point f x0 #:same-test [same? equal?])
(let loop ([x x0] [fx (f x0)]) (if (same? x fx) fx (loop fx (f fx)))))
</lang>
Creating sample clusters
<lang racket>
(define (gaussian-cluster N
#:stdev (σ 1)
#:center (r0 #(0 0))
#:dim (d 2))
(for/list ([i (in-range N)])
(define r (for/vector ([j (in-range d)]) (sample (normal-dist 0 σ))))
(vector-map + r r0)))
(define (uniform-cluster N
#:radius (R 1)
#:center (r0 #(0 0)))
(for/list ([i (in-range N)])
(define r (* R (sqrt (sample (uniform-dist)))))
(define φ (* 2 pi (sample (uniform-dist))))
(vector-map + r0 (vector (* r (cos φ)) (* r (sin φ))))))
</lang>
Visualization
<lang racket> (require plot)
(define (show-clustering data k #:method (method k-means++))
(define c (k-means data k #:initialization method))
(display
(plot
(append
(for/list ([d (clusterize data c)]
[i (in-naturals)])
(points d #:color i #:sym 'fullcircle1))
(list (points c
#:sym 'fullcircle7
#:fill-color 'yellow
#:line-width 3)))
#:title (format "Initializing by ~a" (object-name method)))))
</lang>
Testing
<lang racket> (module+ test
(define circle (uniform-cluster 30000)) ; using k-means++ method (show-clustering circle 6) ; using standard k-means method (show-clustering circle 6 #:method random-choice) ; using manhattan distance (parameterize ([metric manhattan-distance]) (show-clustering circle 6)))
</lang>
The difficult case.
<lang racket> (module+ test
(define clouds
(append
(gaussian-cluster 1000 #:stdev 0.5 #:center #(0 0))
(gaussian-cluster 1000 #:stdev 0.5 #:center #(2 3))
(gaussian-cluster 1000 #:stdev 0.5 #:center #(2.5 -1))
(gaussian-cluster 1000 #:stdev 0.5 #:center #(6 0))))
; using k-means++ method
(show-clustering clouds 4)
; using standard k-means method
(show-clustering clouds 4 #:method random-choice))
</lang>
Multi-dimensional case.
<lang racket> (module+ test
(define 5d-data
(append
(gaussian-cluster 1000 #:dim 5 #:center #(2 0 0 0 0))
(gaussian-cluster 1000 #:dim 5 #:center #(0 2 0 0 0))
(gaussian-cluster 1000 #:dim 5 #:center #(0 0 2 0 0))
(gaussian-cluster 1000 #:dim 5 #:center #(0 0 0 2 0))
(gaussian-cluster 1000 #:dim 5 #:center #(0 0 0 0 2))))
(define centroids (k-means 5d-data 5))
(map (curry vector-map round) centroids))
</lang> Output shows that centroids were found correctly.
(#(-0.0 2.0 -0.0 0.0 0.0) #(0.0 0.0 -0.0 2.0 -0.0) #(2.0 -0.0 -0.0 -0.0 -0.0) #(-0.0 -0.0 2.0 0.0 0.0) #(-0.0 -0.0 0.0 0.0 2.0))
Raku
(formerly Perl 6)
We use Complex numbers to represent points in the plane. We feed the algorithm with three artificially made clouds of points so we can easily see if the output makes sense. <lang perl6>sub postfix:«-means++»(Int $K) {
return sub (@data) {
my @means = @data.pick;
until @means == $K {
my @cumulD2 = [\+] @data.map: -> $x {
min @means.map: { abs($x - $_)**2 }
}
my $rand = rand * @cumulD2[*-1];
@means.push: @data[
(^@data).first: { @cumulD2[$_] > $rand }
];
}
sub cluster { @data.classify: -> $x { @means.min: { abs($_ - $x) } } }
loop (
my %cluster;
$*TOLERANCE < [+] (@means Z- keys (%cluster = cluster))».abs X** 2;
@means = %cluster.values.map( { .elems R/ [+] @$_ } )
) { ; }
return @means;
}
} my @centers = 0, 5, 3 + 2i; my @data = flat @centers.map: { ($_ + .5 - rand + (.5 - rand) * i) xx 100 } @data.=pick(*); .say for 3-means++(@data);</lang>
- Output:
5.04622376429502+0.0145269848483031i 0.0185674577571743+0.0298199687431731i 2.954898072093+2.14922298688815i
Rust
(the initial point selection part)
<lang rust>extern crate csv; extern crate getopts; extern crate gnuplot; extern crate nalgebra; extern crate num; extern crate rand; extern crate rustc_serialize; extern crate test;
use getopts::Options; use gnuplot::{Axes2D, AxesCommon, Color, Figure, Fix, PointSize, PointSymbol}; use nalgebra::{DVector, Iterable}; use rand::{Rng, SeedableRng, StdRng}; use rand::distributions::{IndependentSample, Range}; use std::f64::consts::PI; use std::env;
type Point = DVector<f64>;
struct Cluster<'a> {
members: Vec<&'a Point>, center: Point,
}
struct Stats {
centroids: Vec<Point>, mean_d_from_centroid: DVector<f64>,
}
/// DVector doesn't implement BaseFloat, so a custom distance function is required. fn sqdist(p1: &Point, p2: &Point) -> f64 {
(p1.clone() - p2.clone()).iter().map(|x| x * x).fold(0f64, |a, b| a + b)
}
/// Returns (distance^2, index) tuple of winning point. fn nearest(p: &Point, candidates: &Vec<Point>) -> (f64, usize) {
let (dsquared, the_index) = candidates.iter()
.enumerate()
.fold((sqdist(p, &candidates[0]), 0),
|(d, index), next| {
let dprime = sqdist(p, &candidates[next.0]);
if dprime < d {
(dprime, next.0)
} else {
(d, index)
}
});
(dsquared, the_index)
}
/// Computes starting centroids and makes initial assignments. fn kpp(points: &Vec<Point>, k: usize, rng: &mut StdRng) -> Stats {
let mut centroids: Vec<Point> = Vec::new(); // Random point for first centroid guess: centroids.push(points[rng.gen::<usize>() % points.len()].clone()); let mut dists: Vec<f64> = vec![0f64; points.len()];
for _ in 1..k {
let mut sum = 0f64;
for (j, p) in points.iter().enumerate() {
let (dsquared, _) = nearest(&p, ¢roids);
dists[j] = dsquared;
sum += dsquared;
}
// This part chooses the next cluster center with a probability proportional to d^2
sum *= rng.next_f64();
for (j, d) in dists.iter().enumerate() {
sum -= *d;
if sum <= 0f64 {
centroids.push(points[j].clone());
break;
}
}
}
let clusters = assign_clusters(points, ¢roids); compute_stats(&clusters)
}
fn assign_clusters<'a>(points: &'a Vec<Point>, centroids: &Vec<Point>) -> Vec<Cluster<'a>> {
let mut clusters: Vec<Cluster> = Vec::new();
for _ in 0..centroids.len() {
clusters.push(Cluster {
members: Vec::new(),
center: DVector::new_zeros(points[0].len()),
});
}
for p in points.iter() {
let (_, nearest_index) = nearest(p, centroids);
clusters[nearest_index].center = clusters[nearest_index].center.clone() + p.clone();
clusters[nearest_index].members.push(p);
}
for i in 0..clusters.len() {
clusters[i].center = clusters[i].center.clone() / clusters[i].members.len() as f64;
}
clusters
}
/// Computes centroids and mean-distance-from-centroid for each cluster. fn compute_stats(clusters: &Vec<Cluster>) -> Stats {
let mut centroids = Vec::new(); let mut means_vec = Vec::new();
for c in clusters.iter() {
let pts = &c.members;
let seed: DVector<f64> = DVector::new_zeros(pts[0].len());
let centroid = pts.iter().fold(seed, |a, &b| a + b.clone()) / pts.len() as f64;
means_vec.push(pts.iter().fold(0f64, |acc, pt| acc + sqdist(pt, ¢roid).sqrt()) /
pts.len() as f64);
centroids.push(centroid);
}
Stats {
centroids: centroids,
mean_d_from_centroid: DVector::from_slice(means_vec.len(), means_vec.as_slice()),
}
}
fn lloyd<'a>(points: &'a Vec<Point>,
k: usize,
stoppage_delta: f64,
max_iter: u32,
rng: &mut StdRng)
-> (Vec<Cluster<'a>>, Stats) {
let mut clusters = Vec::new(); // Choose starting centroids and make initial assignments let mut stats = kpp(points, k, rng);
for i in 1..max_iter {
let last_means: DVector<f64> = stats.mean_d_from_centroid.clone();
clusters = assign_clusters(points, &stats.centroids);
stats = compute_stats(&clusters);
let err = sqdist(&stats.mean_d_from_centroid, &last_means).sqrt();
if err < stoppage_delta {
println!("Stoppage condition reached on iteration {}", i);
return (clusters, stats);
}
// Console output
print!("Iter {}: ", i);
for (cen, mu) in stats.centroids.iter().zip(stats.mean_d_from_centroid.iter()) {
print_dvec(cen);
print!(" {:1.2} | ", mu);
}
print!("{:1.5}\n", err);
}
println!("Stoppage condition not reached by iteration {}", max_iter);
(clusters, stats)
}
/// Uniform sampling on the unit disk. fn generate_points(n: u32, rng: &mut StdRng) -> Vec<Point> {
let r_range = Range::new(0f64, 1f64); let theta_range = Range::new(0f64, 2f64 * PI); let mut points: Vec<Point> = Vec::new();
for _ in 0..n {
let root_r = r_range.ind_sample(rng).sqrt();
let theta = theta_range.ind_sample(rng);
points.push(DVector::<f64>::from_slice(2, &[root_r * theta.cos(), root_r * theta.sin()]));
}
points
}
// Plot clusters (2d only). Closure idiom allows us to borrow and mutate the Axes2D. fn viz(clusters: Vec<Cluster>, stats: Stats, k: usize, n: u32, e: f64) {
let mut fg = Figure::new();
{
let prep = |fg: &mut Figure| {
let axes: &mut Axes2D = fg.axes2d();
let title: String = format!("k = {}, n = {}, e = {:4}", k, n, e);
let centroids_x = stats.centroids.iter().map(|c| c[0]);
let centroids_y = stats.centroids.iter().map(|c| c[1]);
for cluster in clusters.iter() {
axes.points(cluster.members.iter().map(|p| p[0]),
cluster.members
.iter()
.map(|p| p[1]),
&[PointSymbol('O'), PointSize(0.25)]);
}
axes.set_aspect_ratio(Fix(1.0))
.points(centroids_x,
centroids_y,
&[PointSymbol('o'), PointSize(1.5), Color("black")])
.set_title(&title[..], &[]);
};
prep(&mut fg);
}
fg.show();
}
fn print_dvec(v: &DVector<f64>) {
print!("(");
for elem in v.at.iter().take(v.len() - 1) {
print!("{:+1.2}, ", elem)
}
print!("{:+1.2})", v.at.iter().last().unwrap());
}
fn print_usage(program: &str, opts: Options) {
let brief = format!("Usage: {} [options]", program);
print!("{}", opts.usage(&brief));
}
fn main() {
let args: Vec<String> = env::args().collect(); let mut k: usize = 7; let mut n: u32 = 30000; let mut e: f64 = 1e-3; let max_iterations = 100u32;
let mut opts = Options::new();
opts.optflag("?", "help", "Print this help menu");
opts.optopt("k",
"",
"Number of clusters to assign (default: 7)",
"<clusters>");
opts.optopt("n",
"",
"Operate on this many points on the unit disk (default: 30000)",
"<pts>");
opts.optopt("e",
"",
"Min delta in norm of successive cluster centroids to continue (default: 1e-3)",
"<eps>");
opts.optopt("f", "", "Read points from file (overrides -n)", "<csv>");
let program = args[0].clone();
let matches = match opts.parse(&args[1..]) {
Ok(m) => m,
Err(f) => panic!(f.to_string()),
};
if matches.opt_present("?") {
print_usage(&program, opts);
return;
}
match matches.opt_str("k") {
None => {}
Some(x) => k = x.parse::<usize>().unwrap(),
};
match matches.opt_str("n") {
None => {}
Some(x) => n = x.parse::<u32>().unwrap(),
};
match matches.opt_str("e") {
None => {}
Some(x) => e = x.parse::<f64>().unwrap(),
};
let seed: &[_] = &[1, 2, 3, 4]; let mut rng: StdRng = SeedableRng::from_seed(seed);
let mut points: Vec<Point>;
match matches.opt_str("f") {
None => {
// Proceed with random 2d data
points = generate_points(n, &mut rng)
}
Some(file) => {
points = Vec::new();
let mut rdr = csv::Reader::from_file(file.clone()).unwrap();
for row in rdr.records().map(|r| r.unwrap()) {
// row is Vec<String>
let floats: Vec<f64> = row.iter().map(|s| s.parse::<f64>().unwrap()).collect();
points.push(DVector::<f64>::from_slice(floats.len(), floats.as_slice()));
}
assert!(points.iter().all(|v| v.len() == points[0].len()));
n = points.len() as u32;
println!("Read {} points from {}", points.len(), file.clone());
}
};
assert!(points.len() >= k); let (clusters, stats) = lloyd(&points, k, e, max_iterations, &mut rng);
println!(" k centroid{}mean dist pop",
std::iter::repeat(" ").take((points[0].len() - 2) * 7 + 7).collect::<String>());
println!("=== {} =========== =====",
std::iter::repeat("=").take(points[0].len() * 7 + 2).collect::<String>());
for i in 0..clusters.len() {
print!(" {:>1} ", i);
print_dvec(&stats.centroids[i]);
print!(" {:1.2} {:>4}\n",
stats.mean_d_from_centroid[i],
clusters[i].members.len());
}
if points[0].len() == 2 {
viz(clusters, stats, k, n, e)
}
} </lang> [Plots exist but file upload is broken at the moment.]
Output of run on 30k points on the unit disk:
Stoppage condition reached on iteration 10 k centroid mean dist pop === ================ =========== ===== 0 (+0.34, -0.61) 0.27 4425 1 (+0.70, -0.01) 0.26 4293 2 (-0.37, -0.59) 0.27 4319 3 (+0.35, +0.61) 0.26 4368 4 (-0.00, +0.01) 0.25 4095 5 (-0.34, +0.62) 0.26 4190 6 (-0.71, +0.04) 0.26 4310
Extra credit 4: Use of the DVector type in the nalgebra crate gives some arithmetic vector operations for free, and generalizes to n dimensions with no work. Here is the output of running this program on the 4-D Fisher Iris data (I don't think this data clusters well):
k centroid mean dist pop === ============================== =========== ===== 0 (+5.00, +3.43, +1.46, +0.25) 0.49 49 1 (+5.88, +2.74, +4.39, +1.43) 0.73 61 2 (+6.85, +3.08, +5.72, +2.05) 0.73 39
Scheme
The eps output is translated from the C version. The 'tester' functions demonstrate the unit square and the unit circle, with eps graphical output, and a 5D unit square, with text-only output. Nothing special is needed to handle multiple dimensions: all points are represented as lists, which the euclidean distance function works through in a loop.
<lang scheme> (import (scheme base) ; headers for R7RS Scheme
(scheme file)
(scheme inexact)
(scheme write)
(srfi 1 lists)
(srfi 27 random-bits))
- calculate euclidean distance between points, any dimension
(define (euclidean-distance pt1 pt2)
(sqrt (apply + (map (lambda (x y) (square (- x y))) pt1 pt2))))
- input
- - K
- the target number of clusters K
- - data
- a list of points in the Cartesian plane
- output
- - a list of K centres
(define (kmeans++ K data)
(define (select-uniformly data)
(let loop ((index (random-integer (length data))) ; uniform selection of index
(rem data)
(front '()))
(if (zero? index)
(values (car rem) (append (reverse front) (cdr rem)))
(loop (- index 1) (cdr rem) (cons (car rem) front)))))
;
(define (select-weighted centres data)
(define (distance-to-nearest datum)
(apply min (map (lambda (c) (euclidean-distance c datum)) centres)))
;
(let* ((weights (map (lambda (d) (square (distance-to-nearest d))) data))
(target-weight (* (apply + weights) (random-real))))
(let loop ((rem data)
(front '())
(weight-sum 0.0)
(wgts weights))
(if (or (>= weight-sum target-weight) (null? (cdr rem)))
(values (car rem) (append (reverse front) (cdr rem)))
(loop (cdr rem)
(cons (car rem) front)
(+ weight-sum (car wgts))
(cdr weights))))))
;
(let-values (((pt rem) (select-uniformly data)))
(let loop ((centres (list pt))
(items rem))
(if (= (length centres) K)
centres
(let-values (((pt rem) (select-weighted centres items)))
(loop (cons pt centres)
rem))))))
- assign a point into a cluster
- input
- a point and a list of cluster centres
- output
- index of cluster centre
(define (assign-cluster pt centres)
(let* ((distances (map (lambda (centre) (euclidean-distance centre pt)) centres))
(smallest (apply min distances)))
(list-index (lambda (d) (= d smallest)) distances)))
- input
- - num
- the number of clusters K
- - data
- a list of points in the Cartesian plane
- output
- - list of K centres
(define (cluster K data)
(define (centroid-for-cluster i assignments)
(let* ((cluster (map cadr (filter (lambda (a-d) (= (car a-d) i)) (zip assignments data))))
(length-cluster (length cluster)))
; compute centroid for cluster
(map (lambda (vals) (/ (apply + vals) length-cluster)) (apply zip cluster))))
;
(define (update-centres assignments)
(map (lambda (i) (centroid-for-cluster i assignments)) (iota K)))
;
(let ((initial-centres (kmeans++ K data)))
(let loop ((centres initial-centres)
(assignments (map (lambda (datum) (assign-cluster datum initial-centres)) data)))
(let* ((new-centres (update-centres assignments))
(new-assignments (map (lambda (datum) (assign-cluster datum new-centres)) data)))
(if (equal? assignments new-assignments)
new-centres
(loop new-centres new-assignments))))))
- using eps output, based on that in C - only works for 2D points
(define (save-as-eps filename data clusters K)
(when (file-exists? filename) (delete-file filename))
(with-output-to-file
filename
(lambda ()
(let* ((W 400)
(H 400)
(colours (make-vector (* 3 K) 0.0))
(max-x (apply max (map car data)))
(min-x (apply min (map car data)))
(max-y (apply max (map cadr data)))
(min-y (apply min (map cadr data)))
(scale (min (/ W (- max-x min-x))
(/ H (- max-y min-y))))
(cx (/ (+ max-x min-x) 2))
(cy (/ (+ max-y min-y) 2)))
;; set up colours
(for-each
(lambda (i)
(vector-set! colours (+ (* i 3) 0) (inexact (/ (modulo (* 3 (+ i 1)) 11) 11)))
(vector-set! colours (+ (* i 3) 1) (inexact (/ (modulo (* 7 i) 11) 11)))
(vector-set! colours (+ (* i 3) 2) (inexact (/ (modulo (* 9 i) 11) 11))))
(iota K))
(display ;; display header
(string-append
"%!PS-Adobe-3.0\n%%BoundingBox: -5 -5 "
(number->string (+ 10 W)) " " (number->string (+ 10 H)) "\n"
"/l {rlineto} def /m {rmoveto} def\n"
"/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def\n"
"/s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath "
" gsave 1 setgray fill grestore gsave 3 setlinewidth"
" 1 setgray stroke grestore 0 setgray stroke }def\n"))
;; display points
(for-each ; top loop runs over the clusters
(lambda (i)
(display
(string-append (number->string (vector-ref colours (* i 3)))
" "
(number->string (vector-ref colours (+ (* i 3) 1)))
" "
(number->string (vector-ref colours (+ (* i 3) 2)))
" setrgbcolor\n"))
(for-each ;loop over points in cluster
(lambda (pt)
(when (= i (assign-cluster pt clusters))
(display
(string-append (number->string (+ (* (- (car pt) cx) scale) (/ W 2)))
" "
(number->string (+ (* (- (cadr pt) cy) scale) (/ H 2)))
" c\n"))))
data)
(let ((center (list-ref clusters i))) ; display cluster centre
(display
(string-append "\n0 setgray "
(number->string (+ (* (- (car center) cx) scale) (/ W 2)))
" "
(number->string (+ (* (- (cadr center) cy) scale) (/ H 2)))
" s\n"))))
(iota K))
(display "\n%%EOF")))))
- extra credit 1
- creates a list of n random points in n-D unit square
(define (make-data num-points num-dimensions)
(random-source-randomize! default-random-source) (map (lambda (i) (list-tabulate num-dimensions (lambda (i) (random-real)))) (iota num-points)))
- extra credit 2, uses eps visualisation to display result
(define (tester-1 num-points K)
(let ((data (make-data num-points 2))) (save-as-eps "clusters-1.eps" data (cluster K data) K)))
- extra credit 3
- uses radians instead to make data
(define (tester-2 num-points K radius)
(random-source-randomize! default-random-source)
(let ((data (map (lambda (i)
(let ((ang (* (random-real) 2 (* 4 (atan 1))))
(rad (* radius (random-real))))
(list (* rad (cos ang)) (* rad (sin ang)))))
(iota num-points))))
;; extra credit 2, uses eps visualisation to display result
(save-as-eps "clusters-2.eps" data (cluster K data) K)))
- extra credit 4
- arbitrary dimensions - already handled, as all points are lists
(define (tester-3 num-points K num-dimensions)
(display "Results:\n") (display (cluster K (make-data num-points num-dimensions))) (newline))
(tester-1 30000 6) (tester-2 30000 6 10) (tester-3 30000 6 5) </lang>
Images in eps files are output for the 2D unit square and unit circle.
Text output for the 5D centres:
Results: ((0.2616723761604841 0.6134082964889989 0.29284958577190745 0.5883330600440337 0.2701242883590077) (0.4495151954110258 0.7213650269267102 0.4785552477630192 0.2520793123281655 0.73785249828929) (0.6873676767669482 0.3228592693134481 0.4713526933057497 0.23850999205524145 0.3104607677290796) (0.6341937732424933 0.36435831485631176 0.2760548254423423 0.7120766805103155 0.7028127288541974) (0.2718747392615238 0.2743005712228975 0.7515030778279079 0.5424997615106112 0.5849261595501698) (0.6882031980026069 0.7048387370769692 0.7373477088448752 0.6859917992267395 0.4027193966445248))
SequenceL
<lang sequencel> import <Utilities/Sequence.sl>; import <Utilities/Random.sl>; import <Utilities/Math.sl>; import <Utilities/Conversion.sl>;
Point ::= (x : float, y : float); Pair<T1, T2> ::= (first : T1, second : T2);
W := 400; H := 400;
// ------------ Utilities -------------- distance(a, b) := (a.x-b.x)^2 + (a.y-b.y)^2;
nearestDistance(point, centers(1)) :=
nearestCenterHelper(point, centers, 2, distance(point, centers[1]), 1).second;
nearestCenter(point, centers(1)) :=
nearestCenterHelper(point, centers, 2, distance(point, centers[1]), 1).first;
nearestCenterHelper(point, centers(1), counter, minDistance, minIndex) :=
let
d := distance(point, centers[counter]);
in
(first : minIndex, second : minDistance) when counter > size(centers) else
nearestCenterHelper(point, centers, counter + 1, d, counter) when minDistance > d else
nearestCenterHelper(point, centers, counter + 1, minDistance, minIndex);
// ------------ KPP -------------- kpp(points(1), k, RG) :=
let
randomValues := getRandomSequence(RG, k).Value;
centers := initialCenters(points, k, randomValues / (RG.RandomMax - 1.0),
[points[randomValues[1] mod size(points)]]);
in
nearestCenter(points, centers);
initialCenters(points(1), k, randoms(1), centers(1)) :=
let
distances := nearestDistance(points, centers);
randomSum := randoms[size(centers) + 1] * sum(distances);
newCenter := points[findNewCenter(randomSum, distances, 1)];
in
centers when size(centers) = k else
initialCenters(points, k, randoms, centers++[newCenter]);
findNewCenter(s, distances(1), counter) :=
let
new_s := s - distances[counter];
in
counter when new_s <= 0 else
findNewCenter(new_s, distances, counter + 1);
// ------------ K Means -------------- kMeans(points(1), groups(1), k) :=
let
newCenters := clusterAverage(points, groups, k);
newGroups := nearestCenter(points, newCenters);
threshold := size(points)/1024;
// Calculate the number of changes between iterations
changes[i] := 1 when groups[i] /= newGroups[i] else 0;
in
(first : newGroups, second : newCenters) when sum(changes) < threshold else
kMeans(points, newGroups, k);
clusterAverage(points(1), groups(1), k) :=
clusterAverageHelper(points, groups, 1, duplicate((x:0.0, y:0.0), k), duplicate(0, k));
clusterAverageHelper(points(1), groups(1), counter, averages(1), sizes(1)) :=
let
group := groups[counter];
result[i] := (x : averages[i].x / sizes[i], y : averages[i].y / sizes[i]);
in
result when counter > size(points) else
clusterAverageHelper(points, groups, counter + 1,
setElementAt(averages, group,
(x : averages[group].x + points[counter].x,
y : averages[group].y + points[counter].y)),
setElementAt(sizes, group, sizes[group] + 1));
// ------------ Generate Points -------------- gen2DPoints(count, radius, RG) :=
let
randA := getRandomSequence(RG, count);
randR := getRandomSequence(randA.Generator, count);
angles := 2*pi*(randA.Value / (RG.RandomMax - 1.0));
radiuses := radius * (randR.Value / (RG.RandomMax - 1.0));
points[i] := (x: radiuses[i] * cos(angles[i]), y : radiuses[i] * sin(angles[i]));
in
(first : points, second : randR.Generator);
// ------------ Visualize -------------- printEPS(points(1),groups(1),centers(1),k,maxVal) :=
let
scale := min(W / (maxVal * 2), H / (maxVal * 2));
printedGroups := printGroup(points, groups, centers, k, 0.0, scale, 1 ... k);
in
"%!-PS-Adobe-3.0\n%%BoundingBox: -5 -5 " ++ toString(W + 10) ++ " " ++
toString(H + 10) ++
"\n/l {rlineto} def /m {rmoveto} def\n" ++
"/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def\n" ++
"/s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath " ++
" gsave 1 setgray fill grestore gsave 3 setlinewidth" ++
" 1 setgray stroke grestore 0 setgray stroke }def\n" ++
join(printedGroups) ++
"\n%%EOF";
printGroup(points(1), groups(1), centers(1), k, maxVal, scale, group) :=
let
printedPoints[i] :=
toString((points[i].x - maxVal) * scale + W/2) ++ " " ++
toString((points[i].y - maxVal) * scale + H/2) ++ " c\n"
when groups[i] = group;
colors := toString((3 * group mod k) / (k * 1.0)) ++ " " ++
toString((7 * (group - 1) mod k) / (k * 1.0)) ++ " " ++
toString((9 * (group - 1) mod k) / (k * 1.0)) ++
" setrgbcolor\n";
printedCenters := "\n0 setgray " ++
toString((centers[group].x - maxVal) * scale + W/2) ++ " " ++
toString((centers[group].y - maxVal) * scale + H/2) ++ " s\n";
in
colors ++ join(printedPoints) ++ printedCenters;
// Take number of points, K and seed for random data as command line inputs main(args(2)) :=
let
n := stringToInt(args[1]) when size(args) >= 1 else 1000;
k := stringToInt(args[2]) when size(args) >= 2 else 7;
seed := stringToInt(args[3]) when size(args) >= 3 else 13;
points := gen2DPoints(n, 10.0, seedRandom(seed));
initialGroups := kpp(points.first, k, points.second);
result := kMeans(points.first, initialGroups, k);
in
printEPS(points.first, result.first, result.second,k,10.0);
</lang>
Tcl
<lang tcl>package require Tcl 8.5 package require math::constants math::constants::constants pi proc tcl::mathfunc::randf m {expr {$m * rand()}}
proc genXY {count radius} {
global pi
for {set i 0} {$i < $count} {incr i} {
set ang [expr {randf(2 * $pi)}] set r [expr {randf($radius)}] lappend pt [list [expr {$r*cos($ang)}] [expr {$r*sin($ang)}] -1]
} return $pt
} proc dist2 {a b} {
lassign $a ax ay
lassign $b bx by
return [expr {($ax-$bx)**2 + ($ay-$by)**2}]
}
proc nearest {pt cent {d2var ""}} {
set minD 1e30
set minI [lindex $pt 2]
set i -1
foreach c $cent {
incr i set d [dist2 $c $pt] if {$minD > $d} { set minD $d set minI $i }
}
if {$d2var ne ""} {
upvar 1 $d2var d2 set d2 $minD
} return $minI
}
proc kpp {ptsVar centVar numClusters} {
upvar 1 $ptsVar pts $centVar cent
set idx [expr {int([llength $pts] * rand())}]
set cent [list [lindex $pts $idx]]
for {set nCent 1} {$nCent < $numClusters} {incr nCent} {
set sum 0 set d {} foreach p $pts { nearest $p $cent dd set sum [expr {$sum + $dd}] lappend d $dd } set sum [expr {randf($sum)}] foreach p $pts dj $d { set sum [expr {$sum - $dj}] if {$sum <= 0} { lappend cent $p break } }
}
set i -1
foreach p $pts {
lset pts [incr i] 2 [nearest $p $cent]
}
}
proc lloyd {ptsVar numClusters} {
upvar 1 $ptsVar pts
kpp pts cent $numClusters
while 1 {
# Find centroids for round set groupCounts [lrepeat [llength $cent] 0] foreach p $pts { lassign $p cx cy group lset groupCounts $group [expr {[lindex $groupCounts $group] + 1}] lset cent $group 0 [expr {[lindex $cent $group 0] + $cx}] lset cent $group 1 [expr {[lindex $cent $group 1] + $cy}] } set i -1 foreach groupn $groupCounts { incr i lset cent $i 0 [expr {[lindex $cent $i 0] / $groupn}] lset cent $i 1 [expr {[lindex $cent $i 1] / $groupn}] }
set changed 0 set i -1 foreach p $pts { incr i set minI [nearest $p $cent] if {$minI != [lindex $p 2]} { incr changed lset pts $i 2 $minI } } if {$changed < ([llength $pts] >> 10)} break
}
set i -1
foreach c $cent {
lset cent [incr i] 2 $i
} return $cent
}</lang> Demonstration/visualization code:
<lang tcl>package require Tk image create photo disp -width 400 -height 400 pack [label .l -image disp] update proc plot {x y color} {
disp put $color -to [expr {int(200+19.9*$x)}] [expr {int(200+19.9*$y)}]
} apply {{} {
set POINTS [genXY 100000 10]
set CENTROIDS [lloyd POINTS 11]
foreach c $CENTROIDS {
lappend colors [list [list [format "#%02x%02x%02x" \ [expr {64+int(128*rand())}] [expr {64+int(128*rand())}] \ [expr {64+int(128*rand())}]]]]
}
foreach pt $POINTS {
lassign $pt px py group plot $px $py [lindex $colors $group]
}
foreach c $CENTROIDS {
lassign $c cx cy group plot $cx $cy black
}
}}</lang>
Wren
<lang ecmascript>import "random" for Random import "/dynamic" for Struct import "/fmt" for Fmt
var Point = Struct.create("Point", ["x", "y", "group"])
var r = Random.new() var hugeVal = Num.infinity
var RAND_MAX = Num.maxSafeInteger var PTS = 100000 var K = 11 var W = 400 var H = 400
var rand = Fn.new { r.int(RAND_MAX) } var randf = Fn.new { |m| m * rand.call() / (RAND_MAX - 1) }
var genXY = Fn.new { |count, radius|
var pts = List.filled(count, null)
/* note: this is not a uniform 2-d distribution */
for (i in 0...count) {
pts[i] = Point.new(0, 0, 0)
var ang = randf.call(2 * Num.pi)
var r = randf.call(radius)
pts[i].x = r * ang.cos
pts[i].y = r * ang.sin
}
return pts
}
var dist2 = Fn.new { |a, b|
var x = a.x - b.x var y = a.y - b.y return x * x + y * y
}
var nearest = Fn.new { |pt, cent, nCluster|
var minD = hugeVal
var minI = pt.group
for (i in 0...nCluster) {
var d = dist2.call(cent[i], pt)
if (minD > d) {
minD = d
minI = i
}
}
return [minI, minD]
}
var copyPoint = Fn.new { |pt| Point.new(pt.x, pt.y, pt.group) }
var kpp = Fn.new { |pts, len, cent|
var nCent = cent.count
var d = List.filled(len, 0)
cent[0] = copyPoint.call(pts[rand.call() % len])
for (nCluster in 1...nCent) {
var sum = 0
for (j in 0...len) {
d[j] = nearest.call(pts[j], cent, nCluster)[1]
sum = sum + d[j]
}
sum = randf.call(sum)
for (j in 0...len) {
sum = sum - d[j]
if (sum > 0) continue
cent[nCluster] = copyPoint.call(pts[j])
break
}
}
for (j in 0...len) pts[j].group = nearest.call(pts[j], cent, nCent)[0]
}
var lloyd = Fn.new { |pts, len, nCluster|
var cent = List.filled(nCluster, null)
for (i in 0...nCluster) cent[i] = Point.new(0, 0, 0)
kpp.call(pts, len, cent)
while(true) {
/* group element for centroids are used as counters */
for (i in 0...nCluster) {
cent[i].x = 0
cent[i].y = 0
cent[i].group = 0
}
for (j in 0...len) {
var p = pts[j]
var c = cent[p.group]
c.group = c.group + 1
c.x = c.x + p.x
c.y = c.y + p.y
}
for (i in 0...nCluster) {
var c = cent[i]
c.x = c.x / c.group
c.y = c.y / c.group
}
var changed = 0
/* find closest centroid of each point */
for (j in 0...len) {
var p = pts[j]
var minI = nearest.call(p, cent, nCluster)[0]
if (minI != p.group) {
changed = changed + 1
p.group = minI
}
}
/* stop when 99.9% of points are good */
if (changed <= (len >> 10)) break
}
for (i in 0...nCluster) cent[i].group = i
return cent
}
var printEps = Fn.new { |pts, len, cent, nCluster|
var colors = List.filled(nCluster * 3, 0)
for (i in 0...nCluster) {
colors[3 * i + 0] = (3 * (i + 1) % 11) / 11
colors[3 * i + 1] = (7 * i % 11) / 11
colors[3 * i + 2] = (9 * i % 11) / 11
}
var minX = hugeVal
var minY = hugeVal
var maxX = -hugeVal
var maxY = -hugeVal
for (j in 0...len) {
var p = pts[j]
if (maxX < p.x) maxX = p.x
if (minX > p.x) minX = p.x
if (maxY < p.y) maxY = p.y
if (minY > p.y) minY = p.y
}
var scale = (W / (maxX - minX)).min(H / (maxY - minY))
var cx = (maxX + minX) / 2
var cy = (maxY + minY) / 2
System.print("\%!PS-Adobe-3.0\n\%\%BoundingBox: -5 -5 %(W + 10) %(H + 10)")
System.print("/l {rlineto} def /m {rmoveto} def")
System.print("/c { .25 sub exch .25 sub exch .5 0 360 arc fill } def")
System.write("/s { moveto -2 0 m 2 2 l 2 -2 l -2 -2 l closepath ")
System.write(" gsave 1 setgray fill grestore gsave 3 setlinewidth")
System.print(" 1 setgray stroke grestore 0 setgray stroke }def")
var f1 = "$g $g $g setrgbcolor"
var f2 = "$.3f $.3f c"
var f3 = "\n0 setgray $g $g s"
for (i in 0...nCluster) {
var c = cent[i]
Fmt.print(f1, colors[3 * i], colors[3 * i + 1], colors[3 * i + 2])
for (j in 0...len) {
var p = pts[j]
if (p.group != i) continue
Fmt.print(f2, (p.x - cx) * scale + W / 2, (p.y - cy) * scale + H / 2)
}
Fmt.print(f3, (c.x - cx) * scale + W / 2, (c.y - cy) * scale + H / 2)
}
System.write("\n\%\%EOF")
}
var v = genXY.call(PTS, 10) var c = lloyd.call(v, PTS, K) printEps.call(v, PTS, c, K)</lang>
XPL0
Like C, simplicity and clarity was chosen over extra credit. Also, the dataset is global, and the arrays are separate instead of being packed into two arguments and passed into the KMeans procedure. Hopefully the animated display, showing the convergence of the clusters, compensates somewhat for these sins. Alas, image uploads appears to be broken.
<lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations
def N = 30000; \number of points def K = 6; \number of clusters int Px(N), Py(N), Pc(N), \coordinates of points and their cluster
Cx(K), Cy(K); \coordinates of centroid of cluster
func Centroid; \Find new centroids of points grouped with current centroids
int Change, Cx0(K), Cy0(K), C, Count, I;
[Change:= false;
for C:= 0 to K-1 do \for each centroid...
[Cx0(C):= Cx(C); Cy0(C):= Cy(C); \save current centroid
Cx(C):= 0; Cx(C):= 0; Count:= 0;\find new centroid
for I:= 0 to N-1 do \for all points
if Pc(I) = C then \ grouped with current centroid...
[Cx(C):= Cx(C) + Px(I);
Cy(C):= Cy(C) + Py(I);
Count:= Count+1;
];
Cx(C):= Cx(C)/Count; Cy(C):= Cy(C)/Count;
if Cx(C)#Cx0(C) or Cy(C)#Cy0(C) then Change:= true;
];
return Change; ];
proc Voronoi; \Group points with their nearest centroid
int D2, MinD2, I, C; \distance squared, minimum distance squared
[for I:= 0 to N-1 do \for each point...
[MinD2:= -1>>1; \find closest centroid
for C:= 0 to K-1 do
[D2:= sq(Px(I)-Cx(C)) + sq(Py(I)-Cy(C));
if D2 < MinD2 then
[MinD2:= D2; Pc(I):= C]; \update closest centroid
];
];
];
proc KMeans; \Group points into K clusters
int Change, I;
repeat Voronoi;
Change:= Centroid;
SetVid($101); \show result on 640x480x8 screen
for I:= 0 to N-1 do Point(Px(I), Py(I), Pc(I)+1);
for I:= 0 to K-1 do Point(Cx(I), Cy(I), \bright white\ $F);
until Change = false;
proc Random(X, Y); \Return random X,Y biased for polar coordinates
int X, Y;
real A, D;
[D:= float(Ran(240)); \distance: 0..239
A:= float(Ran(314159*2)) / 10000.0; \angle: 0..2pi
X(0):= fix(D*Cos(A)) + 320; \rectangular coords centered on screen
Y(0):= fix(D*Sin(A)) + 240;
];
int I;
[for I:= 0 to N-1 do Random(@Px(I), @Py(I)); \random set of points
for I:= 0 to K-1 do Random(@Cx(I), @Cy(I)); \random set of cluster centroids
KMeans; I:= ChIn(1); \wait for keystroke SetVid($03); \restore normal text screen ]</lang>