func TestDoPairPointCentroidJobs(t *testing.T) {
r := 4
c := 2
dataPoints := matrix.Zeros(r, c)
// centroidSqDist := matrix.Zeros(r, c)
centroids := matrix.Zeros(r, c)
done := make(chan int)
jobs := make(chan PairPointCentroidJob, r)
results := make(chan PairPointCentroidResult, minimum(1024, r))
var md ManhattanDist
go addPairPointCentroidJobs(jobs, dataPoints, centroids, md, results)
for i := 0; i < r; i++ {
go doPairPointCentroidJobs(done, jobs)
}
j := 0
for ; j < r; j++ {
<-done
}
if j != r {
t.Errorf("doPairPointToCentroidJobs jobs processed=%d. Should be %d", j, r)
}
}
// Kmeansp returns means and distance squared of the coordinates for each
// centroid using parallel computation.
//
// Input values
//
// datapoints - a kX2 matrix of R^2 coordinates
//
// centroids - a kX2 matrix of R^2 coordinates for centroids.
//
// measurer - anythng that implements the matutil.VectorMeasurer interface to
// calculate the distance between a centroid and datapoint. (e.g., Euclidian
// distance)
//
// Return values
//
// centroidMean - a kX2 matrix where the row number corresponds to the same
// row in the centroid matrix and the two columns are the means of the
// coordinates for that cluster. i.e., the best centroids that could
// be determined.
//
// ____ ______
// | 12.29 32.94 | <-- The mean of coordinates for centroid 0
// | 4.6 29.22 | <-- The mean of coordinates for centroid 1
// |_____ ______|
//
//
// centroidSqErr - a kX2 matrix where the first column contains a number
// indicating the centroid and the second column contains the minimum
// distance between centroid and point squared. (i.e., the squared error)
//
// ____ _______
// | 0 38.01 | <-- Centroid 0, squared error for the coordinates in row 0 of datapoints
// | 1 23 .21| <-- Centroid 1, squared error for the coordinates in row 1 of datapoints
// | 0 14.12 | <-- Centroid 0, squared error for the coordinates in row 2 of datapoints
// _____ _______
//func Kmeansp(datapoints, centroids *matrix.DenseMatrix, measurer matutil.VectorMeasurer) (centroidMean,
func Kmeansp(datapoints *matrix.DenseMatrix, k int, cc CentroidChooser, measurer matutil.VectorMeasurer) (centroidMean,
centroidSqErr *matrix.DenseMatrix, err error) {
//k, _ := centroids.GetSize()
fp, _ := os.Create("/var/tmp/km.log")
w := io.Writer(fp)
log.SetOutput(w)
centroids := cc.ChooseCentroids(datapoints, k)
numRows, numCols := datapoints.GetSize()
centroidSqErr = matrix.Zeros(numRows, numCols)
centroidMean = matrix.Zeros(k, numCols)
jobs := make(chan PairPointCentroidJob, numworkers)
results := make(chan PairPointCentroidResult, minimum(1024, numRows))
done := make(chan int, numworkers)
go addPairPointCentroidJobs(jobs, datapoints, centroidSqErr, centroids, measurer, results)
for i := 0; i < numworkers; i++ {
go doPairPointCentroidJobs(done, jobs)
}
go awaitPairPointCentroidCompletion(done, results)
processPairPointToCentroidResults(centroidSqErr, results) // This blocks so that all the results can be processed
// Now that you have each data point grouped with a centroid, iterate
// through the centroidSqErr martix and for each centroid retrieve the
// original coordinates from datapoints and place the results in
// pointsInCuster.
for c := 0; c < k; c++ {
// c is the index that identifies the current centroid.
// d is the index that identifies a row in centroidSqErr and datapoints.
// Select all the rows in centroidSqErr whose first col value == c.
// Get the corresponding row vector from datapoints and place it in pointsInCluster.
matches, err := centroidSqErr.FiltColMap(float64(c), float64(c), 0) //rows with c in column 0.
if err != nil {
return centroidMean, centroidSqErr, nil
}
// It is possible that some centroids will not have any points, so there
// may not be any matches in the first column of centroidSqErr.
if len(matches) == 0 {
continue
}
pointsInCluster := matrix.Zeros(len(matches), 2)
for d, rownum := range matches {
pointsInCluster.Set(d, 0, datapoints.Get(int(rownum), 0))
pointsInCluster.Set(d, 1, datapoints.Get(int(rownum), 1))
}
// pointsInCluster now contains all the data points for the current
// centroid. Take the mean of each of the 2 cols in pointsInCluster.
means := pointsInCluster.MeanCols()
centroidMean.Set(c, 0, means.Get(0, 0))
centroidMean.Set(c, 1, means.Get(0, 1))
}
return
}
func TestAssessClusters(t *testing.T) {
r, c := DATAPOINTS.GetSize()
CentPointDist := matrix.Zeros(r, c)
done := make(chan int)
jobs := make(chan PairPointCentroidJob, r)
results := make(chan PairPointCentroidResult, minimum(1024, r))
var md ManhattanDist
go addPairPointCentroidJobs(jobs, DATAPOINTS, CENTROIDS, md, results)
for i := 0; i < r; i++ {
go doPairPointCentroidJobs(done, jobs)
}
go awaitPairPointCentroidCompletion(done, results)
clusterChanged := assessClusters(CentPointDist, results)
if clusterChanged != true {
t.Errorf("TestAssessClusters: clusterChanged=%b and should be true.", clusterChanged)
}
if CentPointDist.Get(9, 0) != 0 || CentPointDist.Get(10, 0) != 1 {
t.Errorf("TestAssessClusters: rows 9 and 10 should have 0 and 1 in column 0, but received %v", CentPointDist)
}
}
func TestComputeCentroid(t *testing.T) {
empty := matrix.Zeros(0, 0)
_, err := ComputeCentroid(empty)
if err == nil {
t.Errorf("Did not raise error on empty matrix")
}
twoByTwo := matrix.Ones(2, 2)
centr, err := ComputeCentroid(twoByTwo)
if err != nil {
t.Errorf("Could not compute centroid, err=%v", err)
}
expected := matrix.MakeDenseMatrix([]float64{1.0, 1.0}, 1, 2)
if !matrix.Equals(centr, expected) {
t.Errorf("Incorrect centroid: was %v, should have been %v", expected, centr)
}
twoByTwo.Set(0, 0, 3.0)
expected.Set(0, 0, 2.0)
centr, err = ComputeCentroid(twoByTwo)
if err != nil {
t.Errorf("Could not compute centroid, err=%v", err)
}
if !matrix.Equals(centr, expected) {
t.Errorf("Incorrect centroid: was %v, should have been %v", expected, centr)
}
}
// chooseCentroids picks random centroids based on the min and max values in the matrix
// and return a k by m matrix of the centroids.
func (c randCentroids) ChooseCentroids(mat *matrix.DenseMatrix, k int) *matrix.DenseMatrix {
_, cols := mat.GetSize()
centroids := matrix.Zeros(k, cols)
for colnum := 0; colnum < cols; colnum++ {
r := mat.ColSlice(colnum)
minj := float64(0)
// min value from column
for _, val := range r {
minj = math.Min(minj, val)
}
// max value from column
maxj := float64(0)
for _, val := range r {
maxj = math.Max(maxj, val)
}
// create a slice of random centroids
// based on maxj + minJ * random num to stay in range
for h := 0; h < k; h++ {
randInRange := ((maxj - minj) * rand.Float64()) + minj
centroids.Set(h, colnum, randInRange)
}
}
return centroids
}
// modelMean calculates the mean between all points in a model and a centroid.
func modelMean(points, centroid *matrix.DenseMatrix) *matrix.DenseMatrix {
prows, pcols := points.GetSize()
pdist := matrix.Zeros(prows, pcols)
for i := 0; i < prows; i++ {
diff := matrix.Difference(centroid, points.GetRowVector(i))
pdist.SetRowVector(diff, i)
}
return pdist.MeanCols()
}
func TestAddPairPointToCentroidJob(t *testing.T) {
r := 4
c := 2
jobs := make(chan PairPointCentroidJob, r)
results := make(chan PairPointCentroidResult, minimum(1024, r))
dataPoints := matrix.Zeros(r, c)
// centroidSqDist := matrix.Zeros(r, c)
centroids := matrix.Zeros(r, c)
var ed EuclidDist
go addPairPointCentroidJobs(jobs, dataPoints, centroids, ed, results)
i := 0
for ; i < r; i++ {
<-jobs
}
if i != r {
t.Errorf("addPairPointToCentroidJobs number of jobs=%d. Should be %d", i, r)
}
}
func TestProcessPairPointToCentroidResults(t *testing.T) {
r := 4
c := 2
dataPoints := matrix.Zeros(r, c)
centroidSqDist := matrix.Zeros(r, c)
centroids := matrix.Zeros(r, c)
done := make(chan int)
jobs := make(chan PairPointCentroidJob, r)
results := make(chan PairPointCentroidResult, minimum(1024, r))
var md matutil.ManhattanDist
go addPairPointCentroidJobs(jobs, dataPoints, centroids, centroidSqDist, md, results)
for i := 0; i < r; i++ {
go doPairPointCentroidJobs(done, jobs)
}
go awaitPairPointCentroidCompletion(done, results)
//TODO check deterministic results of centroidDistSq
processPairPointToCentroidResults(centroidSqDist, results)
}
// Needs comments
func (c EllipseCentroids) ChooseCentroids(mat *matrix.DenseMatrix, k int) *matrix.DenseMatrix {
_, cols := mat.GetSize()
var xmin, xmax, ymin, ymax = matutil.GetBoundaries(mat)
x0, y0 := xmin+(xmax-xmin)/2.0, ymin+(ymax-ymin)/2.0
centroids := matrix.Zeros(k, cols)
rx, ry := xmax-x0, ymax-y0
thetaInit := rand.Float64() * math.Pi
for i := 0; i < k; i++ {
centroids.Set(i, 0, rx*c.frac*math.Cos(thetaInit+float64(i)*math.Pi/float64(k)))
centroids.Set(i, 1, ry*c.frac*math.Sin(thetaInit+float64(i)*math.Pi/float64(k)))
}
return centroids
}
// EllipseCentroids lays out the initial centroids evenly along an elipse inscribed and centered within the boundaries of the dataset.
// It is only defined for M=2
// * Frac: This must be a float between 0 and 1. It determines the scale of the inscribing ellipse relative to the dataset,
// so Frac==1.0 produces an ellipse that spans the entire dataset, while Frac==0.5 produces an ellipse spanning half the dataset.
func (c EllipseCentroids) ChooseCentroids(mat *matrix.DenseMatrix, k int) *matrix.DenseMatrix {
_, cols := mat.GetSize()
// TODO Cache boundaries call for each matrix so that it is not called on each bisect
var xmin, xmax, ymin, ymax = boundaries(mat)
x0, y0 := xmin+(xmax-xmin)/2.0, ymin+(ymax-ymin)/2.0
centroids := matrix.Zeros(k, cols)
rx, ry := xmax-x0, ymax-y0
thetaInit := rand.Float64() * math.Pi
for i := 0; i < k; i++ {
centroids.Set(i, 0, rx*c.Frac*math.Cos(thetaInit+float64(i)*2.0*math.Pi/float64(k)))
centroids.Set(i, 1, ry*c.Frac*math.Sin(thetaInit+float64(i)*2.0*math.Pi/float64(k)))
}
return centroids
}
// DataCentroids picks k distinct points from the dataset. If k is > points in
// the matrix then k is set to the number of points.
func (c DataCentroids) ChooseCentroids(mat *matrix.DenseMatrix, k int) *matrix.DenseMatrix {
// first set up a map to keep track of which data points have already been chosen so we don't dupe
rows, cols := mat.GetSize()
centroids := matrix.Zeros(k, cols)
if k > rows {
k = rows
}
chosenIdxs := make(map[int]bool, k)
for len(chosenIdxs) < k {
index := rand.Intn(rows)
chosenIdxs[index] = true
}
i := 0
for idx, _ := range chosenIdxs {
centroids.SetRowVector(mat.GetRowVector(idx).Copy(), i)
i += 1
}
return centroids
}
func makeCentPointDist(datapoints, centroids *matrix.DenseMatrix) *matrix.DenseMatrix {
r, c := datapoints.GetSize()
CentPointDist := matrix.Zeros(r, c)
done := make(chan int)
jobs := make(chan PairPointCentroidJob, r)
results := make(chan PairPointCentroidResult, minimum(1024, r))
var ed EuclidDist
go addPairPointCentroidJobs(jobs, datapoints, centroids, ed, results)
for i := 0; i < r; i++ {
go doPairPointCentroidJobs(done, jobs)
}
go awaitPairPointCentroidCompletion(done, results)
clusterChanged := assessClusters(CentPointDist, results)
if clusterChanged == true || clusterChanged == false {
}
return CentPointDist
}
// Load loads a tab delimited text file of floats into a matrix.
func Load(fname, sep string) (*matrix.DenseMatrix, error) {
z := matrix.Zeros(1, 1)
fp, err := os.Open(fname)
if err != nil {
return z, err
}
defer fp.Close()
data := make([]float64, 0)
cols := 0
r := bufio.NewReader(fp)
linenum := 0
eof := false
for !eof {
var line string
var buf []byte
buf, _, err := r.ReadLine()
line = string(buf)
if err == io.EOF {
err = nil
eof = true
break
} else if err != nil {
return z, errors.New(fmt.Sprintf("goxmean.Load: reading linenum %d: %v", linenum, err))
}
l1 := strings.TrimRight(line, "\n")
l := strings.Split(l1, sep)
// If each line does not have the same number of columns then error
if linenum == 0 {
cols = len(l)
}
if len(l) != cols {
return z, errors.New(fmt.Sprintf("Load(): linenum %d has %d columns. It should have %d columns.", linenum, len(line), cols))
}
if len(l) < 2 {
return z, errors.New(fmt.Sprintf("Load(): linenum %d has only %d elements", linenum, len(line)))
}
linenum++
// Convert the strings to float64 and build up the slice t by appending.
t := make([]float64, 0)
for _, v := range l {
v = strings.TrimSpace(v)
f, err := strconv.ParseFloat(v, 64)
if err != nil {
return z, errors.New(fmt.Sprintf("goxmeans.Load: cannot convert value %s to float64.", v))
}
t = append(t, f)
}
data = append(data, t...)
}
mat := matrix.MakeDenseMatrix(data, linenum, cols)
//fmt.Println(time.Now())n // flag for debugging
return mat, nil
}
// kmeans partitions datapoints into K clusters. This results in a partitioning of
// the data space into Voronoi cells. The problem is NP-hard so here we attempt
// to parallelize or make concurrent as many processes as possible to reduce the
// running time.
//
// 1. Place K points into the space represented by the objects that are being clustered.
// These points represent initial group centroids.
//
// 2. Assign each object to the group that has the closest centroid.
//
// 3. When all objects have been assigned, recalculate the positions of the K centroids
// by calculating the mean of all cooridnates in a cluster and making that
// the new centroid.
//
// 4. Repeat Steps 2 and 3 until the centroids no longer move.
//
// centroids is K x M matrix that cotains the coordinates for the centroids.
// The centroids are indexed by the 0 based rows of this matrix.
// ____ _________
// | 12.29 32.94 ... | <-- The coordinates for centroid 0
// | 4.6 29.22 ... | <-- The coordinates for centroid 1
// |_____ __________|
//
//
// CentPointDist is ax R x M matrix. The rows have a 1:1 relationship to
// the rows in datapoints. Column 0 contains the row number in centroids
// that corresponds to the centroid for the datapoint in row i of this matrix.
// Column 1 contains (x_i - mu(i))^2.
// ____ _______
// | 3 38.01 | <-- Centroid 3, squared error for the coordinates in row 0 of datapoints
// | 1 23 .21| <-- Centroid 1, squared error for the coordinates in row 1 of datapoints
// | 0 14.12 | <-- Centroid 0, squared error for the coordinates in row 2 of datapoints
// _____ _______
//
func kmeans(datapoints, centroids *matrix.DenseMatrix, measurer VectorMeasurer) Model {
/* datapoints CentPoinDist centroids
________________
____ ____ __|__ ______ | ____ ___________
| ... | | ... | V | ... |
| 3.0 5.1| <-- row i --> | 3 32.12 | row 3 | 3 38.1, ... |
|____ ___| |____ ______| |___ __________ |
*/
R, M := datapoints.GetSize()
CentPointDist := matrix.Zeros(R, 2)
k, _ := centroids.GetSize()
clusterChanged := true
var clusters []cluster
for clusterChanged == true {
clusterChanged = false
clusters = make([]cluster, 0)
jobs := make(chan PairPointCentroidJob, 1024)
results := make(chan PairPointCentroidResult, 1024)
done := make(chan int, 1024)
// Pair each point with its closest centroid.
go addPairPointCentroidJobs(jobs, datapoints, centroids, measurer, results)
for i := 0; i < numworkers; i++ {
go doPairPointCentroidJobs(done, jobs)
}
go awaitPairPointCentroidCompletion(done, results)
clusterChanged = assessClusters(CentPointDist, results) // This blocks so that all the results can be processed
// You have each data point grouped with a centroid,
for idx, cent := 0, 0; cent < k; cent++ {
// Select all the rows in CentPointDist whose first col value == cent.
// Get the corresponding row vector from datapoints and place it in pointsInCluster.
r, _ := CentPointDist.GetSize()
matches := make([]int, 0)
for i := 0; i < r; i++ {
v := CentPointDist.Get(i, 0)
if v == float64(cent) {
matches = append(matches, i)
}
}
// It is possible that some centroids may have zero points, so there
// may not be any matches.
if len(matches) == 0 {
continue
}
pointsInCluster := matrix.Zeros(len(matches), M)
i := 0
for _, rownum := range matches {
pointsInCluster.Set(i, 0, datapoints.Get(int(rownum), 0))
pointsInCluster.Set(i, 1, datapoints.Get(int(rownum), 1))
i++
}
// pointsInCluster now contains all the data points for the current
// centroid. The mean of the coordinates for this cluster becomes
// the new centroid for this cluster.
mean := pointsInCluster.MeanCols()
centroids.SetRowVector(mean, cent)
clust := cluster{pointsInCluster, mean, 0}
clust.Variance = variance(clust, measurer)
clusters = append(clusters, clust)
idx++
}
}
modelbic := calcbic(R, M, clusters)
model := Model{modelbic, clusters}
return model
}
// Load loads a tab delimited text file of floats into a slice.
// Assume last column is the target.
// For now, we limit ourselves to two columns
func Load(fname string) (*matrix.DenseMatrix, error) {
datamatrix := matrix.Zeros(1, 1)
data := make([]float64, 2048)
idx := 0
fp, err := os.Open(fname)
if err != nil {
return datamatrix, err
}
defer fp.Close()
r := bufio.NewReader(fp)
linenum := 1
eof := false
for !eof {
var line string
var buf []byte
// line, err := r.ReadString('\n')
buf, _, err := r.ReadLine()
line = string(buf)
// fmt.Printf("linenum=%d buf=%v line=%v\n",linenum,buf, line)
if err == io.EOF {
err = nil
eof = true
break
} else if err != nil {
return datamatrix, errors.New(fmt.Sprintf("means.Load: reading linenum %d: %v", linenum, err))
}
linenum++
l1 := strings.TrimRight(line, "\n")
l := strings.Split(l1, "\t")
if len(l) < 2 {
return datamatrix, errors.New(fmt.Sprintf("means.Load: linenum %d has only %d elements", linenum, len(line)))
}
// for now assume 2 dimensions only
f0, err := Atof64(string(l[0]))
if err != nil {
return datamatrix, errors.New(fmt.Sprintf("means.Load: cannot convert f0 %s to float64.", l[0]))
}
f1, err := Atof64(string(l[1]))
if err != nil {
return datamatrix, errors.New(fmt.Sprintf("means.Load: cannot convert f1 %s to float64.", l[1]))
}
if linenum >= len(data) {
data = append(data, f0, f1)
} else {
data[idx] = f0
idx++
data[idx] = f1
idx++
}
}
numcols := 2
datamatrix = matrix.MakeDenseMatrix(data, linenum-1, numcols)
return datamatrix, nil
}
// Kmeansbi bisects a given cluster and determines which centroids give the lowest error.
// Take the points in a cluster
// While the number of cluster < k
// for every cluster
// measure total error
// cacl kmeansp with k=2 on a given cluster
// measure total error after kmeansp split
// choose the cluster split with the lowest SSE
// commit the chosen split
//
// N.B. We are using SSE until the BIC is completed.
func Kmeansbi(datapoints *matrix.DenseMatrix, k int, cc CentroidChooser, measurer matutil.VectorMeasurer) (matCentroidlist, clusterAssignment *matrix.DenseMatrix, err error) {
numRows, numCols := datapoints.GetSize()
clusterAssignment = matrix.Zeros(numRows, numCols)
matCentroidlist = matrix.Zeros(k, numCols)
centroid0 := datapoints.MeanCols()
centroidlist := []*matrix.DenseMatrix{centroid0}
// Initially create one cluster.
for j := 0; j < numRows; j++ {
point := datapoints.GetRowVector(j)
distJ, err := measurer.CalcDist(centroid0, point)
if err != nil {
return matCentroidlist, clusterAssignment, errors.New(fmt.Sprintf("Kmeansbi: CalcDist returned err=%v", err))
}
clusterAssignment.Set(j, 1, math.Pow(distJ, 2))
}
var bestClusterAssignment, bestNewCentroids *matrix.DenseMatrix
var bestCentroidToSplit int
// Find the best centroid configuration.
for len(centroidlist) < k {
lowestSSE := math.Inf(1)
// Split cluster
for i, _ := range centroidlist {
// Get the points in this cluster
pointsCurCluster, err := clusterAssignment.FiltCol(float64(i), float64(i), 0)
if err != nil {
return matCentroidlist, clusterAssignment, err
}
centroids, splitClusterAssignment, err := Kmeansp(pointsCurCluster, 2, cc, measurer)
if err != nil {
return matCentroidlist, clusterAssignment, err
}
/* centroids is a 2X2 matrix of the best centroids found by kmeans
splitClustAssignment is a mX2 matrix where col0 is either 0 or 1 and refers to the rows in centroids
where col1 cotains the squared error between a centroid and a point. The rows here correspond to
the rows in ptsInCurrCluster. For example, if row 2 contains [1, 7.999] this means that centroid 1
has been paired with the point in row 2 of splitClustAssignment and that the squared error (distance
between centroid and point) is 7.999.
*/
// Calculate the sum of squared errors for each centroid.
// This give a statistcal measurement of how good
// the clustering is for this cluster.
sseSplit := splitClusterAssignment.SumCol(1)
// Calculate the SSE for the original cluster
sqerr, err := clusterAssignment.FiltCol(float64(0), math.Inf(1), 0)
if err != nil {
return matCentroidlist, clusterAssignment, err
}
sseNotSplit := sqerr.SumCol(1)
// TODO: Pre-BCI is this the best way to evaluate?
if sseSplit+sseNotSplit < lowestSSE {
bestCentroidToSplit = 1
bestNewCentroids = matrix.MakeDenseCopy(centroids)
bestClusterAssignment = matrix.MakeDenseCopy(splitClusterAssignment)
}
}
// Applying the split overwrites the existing cluster assginments for the
// cluster you have decided to split. Kmeansp() returned two clusters
// labeled 0 and 1. Change these cluster numbers to the cluster number
// you are splitting and the next cluster to be added.
m, err := bestClusterAssignment.FiltColMap(1, 1, 0)
if err != nil {
return matCentroidlist, clusterAssignment, err
}
for i, _ := range m {
bestClusterAssignment.Set(i, 0, float64(len(centroidlist)))
}
n, err := bestClusterAssignment.FiltColMap(0, 0, 0)
if err != nil {
return matCentroidlist, clusterAssignment, err
}
for i, _ := range n {
bestClusterAssignment.Set(i, 1, float64(bestCentroidToSplit))
}
fmt.Printf("Best centroid to split %f\n", bestCentroidToSplit)
r, _ := bestClusterAssignment.GetSize()
fmt.Printf("The length of best cluster assesment is %f\n", r)
// Replace a centroid with the two best centroids from the split.
centroidlist[bestCentroidToSplit] = bestNewCentroids.GetRowVector(0)
centroidlist = append(centroidlist, bestNewCentroids.GetRowVector(1))
// Reassign new clusters and SSE
rows, _ := clusterAssignment.GetSize()
for i, j := 0, 0; i < rows; i++ {
if clusterAssignment.Get(i, 0) == float64(bestCentroidToSplit) {
clusterAssignment.Set(i, 0, bestClusterAssignment.Get(j, 0))
clusterAssignment.Set(i, 1, bestClusterAssignment.Get(j, 1))
j++
}
}
// make centroidlist into a matrix
s := make([][]float64, len(centroidlist))
for i, mat := range centroidlist {
s[i][0] = mat.Get(0, 0)
s[i][1] = mat.Get(0, 1)
}
matCentroidlist = matrix.MakeDenseMatrixStacked(s)
}
return matCentroidlist, clusterAssignment, nil
}
