kmeans clustering
performs kmeans
clustering to partition the observations of the nbyp data
matrix idx
= kmeans(X
,k
)X
into k
clusters, and
returns an nby1 vector (idx
)
containing cluster indices of each observation. Rows of X
correspond
to points and columns correspond to variables.
By default, kmeans
uses the squared Euclidean
distance measure and the kmeans++
algorithm for cluster center initialization.
returns
the cluster indices with additional options specified by one or more idx
= kmeans(X
,k
,Name,Value
)Name,Value
pair
arguments.
For example, specify the cosine distance, the number of times to repeat the clustering using new initial values, or to use parallel computing.
Cluster data using kmeans clustering, then plot the cluster regions.
Load Fisher's iris data set. Use the petal lengths and widths as predictors.
load fisheriris X = meas(:,3:4); figure; plot(X(:,1),X(:,2),'k*','MarkerSize',5); title 'Fisher''s Iris Data'; xlabel 'Petal Lengths (cm)'; ylabel 'Petal Widths (cm)';
The larger cluster seems to be split into a lower variance region and a higher variance region. This might indicate that the larger cluster is two, overlapping clusters.
Cluster the data. Specify k = 3 clusters.
rng(1); % For reproducibility
[idx,C] = kmeans(X,3);
kmeans
uses the kmeans++ algorithm for centroid initialization and squared Euclidean distance by default. It is good practice to search for lower, local minima by setting the 'Replicates'
namevalue pair argument.
idx
is a vector of predicted cluster indices corrresponding to the observations in X
. C
is a 3by2 matrix containing the final centroid locations.
Use kmeans
to compute the distance from each centroid to points on a grid. To do this, pass the centroids (C
) and points on a grid to kmeans
, and implement one iteration of the algorithm.
x1 = min(X(:,1)):0.01:max(X(:,1)); x2 = min(X(:,2)):0.01:max(X(:,2)); [x1G,x2G] = meshgrid(x1,x2); XGrid = [x1G(:),x2G(:)]; % Defines a fine grid on the plot idx2Region = kmeans(XGrid,3,'MaxIter',1,'Start',C); % Assigns each node in the grid to the closest centroid
Warning: Failed to converge in 1 iterations.
kmeans
displays a warning stating that the algorithm did not converge, which you should expect since the software only implemented one iteration.
Plot the cluster regions.
figure; gscatter(XGrid(:,1),XGrid(:,2),idx2Region,... [0,0.75,0.75;0.75,0,0.75;0.75,0.75,0],'..'); hold on; plot(X(:,1),X(:,2),'k*','MarkerSize',5); title 'Fisher''s Iris Data'; xlabel 'Petal Lengths (cm)'; ylabel 'Petal Widths (cm)'; legend('Region 1','Region 2','Region 3','Data','Location','SouthEast'); hold off;
Randomly generate the sample data.
rng default; % For reproducibility X = [randn(100,2)*0.75+ones(100,2); randn(100,2)*0.5ones(100,2)]; figure; plot(X(:,1),X(:,2),'.'); title 'Randomly Generated Data';
There appears to be two clusters in the data.
Partition the data into two clusters, and choose the best arrangement out of five intializations. Display the final output.
opts = statset('Display','final'); [idx,C] = kmeans(X,2,'Distance','cityblock',... 'Replicates',5,'Options',opts);
Replicate 1, 3 iterations, total sum of distances = 201.533. Replicate 2, 5 iterations, total sum of distances = 201.533. Replicate 3, 3 iterations, total sum of distances = 201.533. Replicate 4, 3 iterations, total sum of distances = 201.533. Replicate 5, 2 iterations, total sum of distances = 201.533. Best total sum of distances = 201.533
By default, the software initializes the replicates separatly using kmeans++.
Plot the clusters and the cluster centroids.
figure; plot(X(idx==1,1),X(idx==1,2),'r.','MarkerSize',12) hold on plot(X(idx==2,1),X(idx==2,2),'b.','MarkerSize',12) plot(C(:,1),C(:,2),'kx',... 'MarkerSize',15,'LineWidth',3) legend('Cluster 1','Cluster 2','Centroids',... 'Location','NW') title 'Cluster Assignments and Centroids' hold off
You can determine how well separated the clusters are by passing idx
to silhouette
.
Clustering large data sets might take time,
particularly if you use online updates (set by default). If you have
a Parallel Computing Toolbox™ license and you invoke a pool of workers,
then kmeans
runs each clustering task (or replicate)
in parallel. Therefore, if Replicates
> 1, then
the parallel computing decreases time to convergence.
Randomly generate a large data set from a Gaussian mixture model.
Mu = bsxfun(@times,ones(20,30),(1:20)'); % Gaussian mixture mean rn30 = randn(30,30); Sigma = rn30'*rn30; % Symmetric and positivedefinite covariance Mdl = gmdistribution(Mu,Sigma); rng(1); % For reproducibility X = random(Mdl,10000);
Mdl
is a 30dimensional gmdistribution
model
with 20 components. X
is a 10000
by30
matrix
of data generated from Mdl
.
Invoke a parallel pool of workers. Specify options for parallel computing.
pool = parpool; % Invokes workers stream = RandStream('mlfg6331_64'); % Random number stream options = statset('UseParallel',1,'UseSubstreams',1,... 'Streams',stream);
Starting parallel pool (parpool) using the 'local' profile ... connected to 4 workers.
The input argument 'mlfg6331_64'
of RandStream
specifies
to use the multiplicative lagged Fibonacci generator algorithm. options
is
a structure array containing fields that specify options for controlling
estimation.
The Command Window indicates that four workers are available. The number of workers might vary on your system.
Cluster the data using kmeans clustering. Specify that there are k = 20 clusters in the data and increase the number of iterations. Typically, the objective function contains local minima. Specify 10 replicates to help find a lower, local minimum.
tic; % Start stopwatch timer [idx,C,sumd,D] = kmeans(X,20,'Options',options,'MaxIter',10000,... 'Display','final','Replicates',10); toc % Terminate stopwatch timer
Replicate 7, 44 iterations, total sum of distances = 7.55218e+06. Replicate 4, 95 iterations, total sum of distances = 7.53848e+06. Replicate 2, 104 iterations, total sum of distances = 7.54232e+06. Replicate 6, 80 iterations, total sum of distances = 7.54237e+06. Replicate 8, 111 iterations, total sum of distances = 7.54445e+06. Replicate 1, 52 iterations, total sum of distances = 7.55817e+06. Replicate 5, 70 iterations, total sum of distances = 7.55278e+06. Replicate 3, 94 iterations, total sum of distances = 7.54858e+06. Replicate 10, 56 iterations, total sum of distances = 7.54547e+06. Replicate 9, 83 iterations, total sum of distances = 7.53701e+06. Best total sum of distances = 7.53701e+06 Elapsed time is 3.239232 seconds.
The Command Window displays the number of iterations and the
terminal objective function value for each replicate. The output arguments
contain the results of replicate 9
because it has
the lowest total sum of distances.
X
— Datanumeric matrixData, specified as a numeric matrix. The rows of X
correspond
to observations, and the columns correspond to variables.
If X
is a numeric vector, then kmeans
treats
it as an nby1 data matrix, regardless of its
orientation.
Data Types: single
 double
k
— Number of clusterspositive integerNumber of clusters in the data, specified as a positive integer.
Data Types: single
 double
Specify optional commaseparated pairs of Name,Value
arguments.
Name
is the argument
name and Value
is the corresponding
value. Name
must appear
inside single quotes (' '
).
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN
.
'Distance','cosine','Replicates',10,'Options',statset('UseParallel',1)
specifies
the cosine distance, 10
replicate clusters at different
starting values, and to use parallel computing.'Display'
— Level of output to display'off'
(default)  'final'
 'iter'
Level of output to display in the Command Window, specified
as the commaseparated pair consisting of 'Display'
and
a string. Available options are:
'final'
— Displays results
of the final iteration
'iter'
— Displays results
of each iteration
'off'
— Displays nothing
Example: 'Display','final'
Data Types: char
'Distance'
— Distance measure'sqeuclidean'
(default)  'cityblock'
 'cosine'
 'correlation'
 'hamming'
Distance measure, in p
dimensional space,
used for minimization, specified as the commaseparated pair consisting
of 'Distance'
and a string.
kmeans
computes centroid clusters differently
for the different, supported distance measures. This table summarizes
the available distance measures. In the formulae, x is
an observation (that is, a row of X
) and c is
a centroid (a row vector).
Distance Measure  Description  Formula 

'sqeuclidean'  Squared Euclidean distance (default). Each centroid is the mean of the points in that cluster.  $$d(x,c)=(xc)(xc{)}^{\prime}$$ 
'cityblock'  Sum of absolute differences, i.e., the L1 distance. Each centroid is the componentwise median of the points in that cluster.  $$d(x,c)={\displaystyle \sum _{j=1}^{p}\left{x}_{j}{c}_{j}\right}$$ 
'cosine'  One minus the cosine of the included angle between points (treated as vectors). Each centroid is the mean of the points in that cluster, after normalizing those points to unit Euclidean length.  $$d(x,c)=1\frac{xc\prime}{\sqrt{\left(x{x}^{\prime}\right)\left(cc\prime \right)}}$$ 
'correlation'  One minus the sample correlation between points (treated as sequences of values). Each centroid is the componentwise mean of the points in that cluster, after centering and normalizing those points to zero mean and unit standard deviation.  $$d(x,c)=1\frac{\left(x\overrightarrow{\overline{x}}\right){\left(c\overrightarrow{\overline{c}}\right)}^{\prime}}{\sqrt{\left(x\overrightarrow{\overline{x}}\right){\left(x\overrightarrow{\overline{x}}\right)}^{\prime}}\sqrt{\left(c\overrightarrow{\overline{c}}\right){\left(c\overrightarrow{\overline{c}}\right)}^{\prime}}},$$ where

'hamming'  This measure is only suitable for binary data. It is the proportion of bits that differ. Each centroid is the componentwise median of points in that cluster.  $$d(x,y)=\frac{1}{p}{\displaystyle \sum}_{j=1}^{p}I\left\{{x}_{j}\ne {y}_{j}\right\},$$ where I is the indicator function. 
Example: 'Distance','cityblock'
Data Types: char
'EmptyAction'
— Action to take if cluster loses all member observations'singleton'
(default)  'error'
 'drop'
Action to take if a cluster loses all its member observations,
specified as the commaseparated pair consisting of 'EmptyAction'
and
a string. This table summarizes the available options.
Value  Description 

'error'  Treat an empty cluster as an error. 
'drop'  Remove any clusters that become empty. 
'singleton'  Create a new cluster consisting of the one point furthest from its centroid (default). 
Example: 'EmptyAction','error'
Data Types: char
'MaxIter'
— Maximum number of iterations100
(default)  positive integerMaximum number of iterations, specified as the commaseparated
pair consisting of 'MaxIter'
and a positive integer.
Example: 'MaxIter',1000
Data Types: double
 single
'OnlinePhase'
— Online update flag'off'
(default)  'on'
Online update flag, specified as the commaseparated pair consisting
of 'OnlinePhase'
and 'off'
or 'on'
.
If OnlinePhase
is on
,
then kmeans
performs an online update phase in
addition to a batch update phase. The online phase can be time consuming
for large data sets, but guarantees a solution that is a local minimum
of the distance criterion. In other words, the software finds a partition
of the data in which moving any single point to a different cluster
increases the total sum of distances.
Example: 'OnlinePhase','on'
Data Types: char
'Options'
— Options for controlling iterative algorithm for minimizing fitting criteria[]
(default)  structure array returned by statset
Options for controlling the iterative algorithm for minimizing
the fitting criteria, specified as the commaseparated pair consisting
of 'Options'
and a structure array returned by statset
. These options require Parallel
Computing Toolbox™.
This table summarizes the available options.
Option  Description 

'Streams'  A RandStream object
or cell array of such objects. If you do not specify Streams , kmeans uses
the default stream or streams. If you specify Streams ,
use a single object except when:
In that case, use a cell array the same size
as the parallel pool. If a parallel pool is not open, then 
'UseParallel' 

'UseSubstreams'  Set to true to compute in parallel in a
reproducible fashion. Default is false . To compute
reproducibly, set Streams to a type allowing substreams: 'mlfg6331_64' or 'mrg32k3a' . 
To ensure more predictable
results, use parpool
and explicitly
create a parallel pool before invoking kmeans
and
setting 'Options',statset('UseParallel',1)
.
Example: 'Options',statset('UseParallel',1)
Data Types: struct
'Replicates'
— Number of times to repeat clustering using new initial cluster centroid positions1
(default)  positive integerNumber of times to repeat clustering using new initial cluster
centroid positions, specified as the commaseparated pair consisting
of 'Replicates'
and an integer. kmeans
returns
the solution with the lowest sumd
.
You can set 'Replicates'
implicitly by supplying
a 3D array as the value for the 'Start'
namevalue
pair argument.
Example: 'Replicates',5
Data Types: double
 single
'Start'
— Method for choosing initial cluster centroid positions'plus'
(default)  'cluster'
 'sample'
 'uniform'
 numeric matrix  numeric arrayMethod for choosing initial cluster centroid positions (or seeds),
specified as the commaseparated pair consisting of 'Start'
and
a string, a numeric matrix, or a numeric array. This table summarizes
the available options for choosing seeds.
Value  Description 

'cluster'  Perform a preliminary clustering phase on a random 10% subsample
of X . This preliminary phase is itself initialized
using 'sample' . 
'plus' (default)  Select k seeds by implementing the kmeans++
algorithm for cluster center initialization. 
'sample'  Select k observations from X at
random. 
'uniform'  Select k points uniformly at random from
the range of X . Not valid with the Hamming distance. 
numeric matrix  k byp matrix of centroid
starting locations. The rows of Start correspond
to seeds. The software infers k from the first
dimension of Start , so you can pass in [] for k . 
numeric array  k bypr array
of centroid starting locations. The rows of each page correspond to
seeds. The third dimension invokes replication of the clustering routine.
Page j contains the set of seeds for replicate j.
The software infers the number of replicates (specified by the 'Replicates' namevalue
pair argument) from the size of the third dimension. 
Example: 'Start','sample'
Data Types: char
 double
 single
Note:
The software treats 
idx
— Cluster indicesnumeric column vectorCluster indices, returned as a numeric column vector. idx
has
as many rows as X
, and each row indicates the cluster
assignment of the corresponding observation.
C
— Cluster centroid locationsnumeric matrixCluster centroid locations, returned as a numeric matrix. C
is
a k
byp matrix, where row j is
the centroid of cluster j.
sumd
— Withincluster sums of pointtocentroid distancesnumeric column vectorWithincluster sums of pointtocentroid distances, returned
as a numeric column vector. sumd
is a k
by1
vector, where element j is the sum of pointtocentroid
distances within cluster j.
D
— Distances from each point to every centroidnumeric matrixDistances from each point to every centroid, returned as a numeric
matrix. D
is an nbyk
matrix,
where element (j,m) is the distance
from observation j to centroid m.
kmeans clustering, or Lloyd's algorithm [2], is an iterative, datapartitioning algorithm that assigns n observations to exactly one of k clusters defined by centroids, where k is chosen before the algorithm starts.
The algorithm proceeds as follows:
Choose k initial cluster centers
(centroid). For example, choose k observations
at random (by using 'Start','sample'
) or use the kmeans
++ algorithm for cluster center initialization (the default).
Compute pointtoclustercentroid distances of all observations to each centroid.
There are two ways to proceed (specified by OnlinePhase
):
Batch update — Assign each observation to the cluster with the closest centroid.
Online update — Individually assign observations to a different centroid if the reassignment decreases the sum of the withincluster, sumofsquares pointtoclustercentroid distances.
For more details, see Algorithms.
Compute the average of the observations in each cluster to obtain k new centroid locations.
Repeat steps 2 through 4 until cluster assignments do not change, or the maximum number of iterations is reached.
The kmeans++ algorithm uses an heuristic to find centroid seeds for kmeans clustering. According to Arthur and Vassilvitskii [1], kmeans++ improves the running time of Lloyd's algorithm, and the quality of the final solution.
The kmeans++ algorithm chooses seeds as follows, assuming the number of clusters is k.
Select an observation uniformly at random from the data set, X. The chosen observation is the first centroid, and is denoted c_{1}.
Compute distances from each observation to c_{1}. Denote the distance between c_{j} and the observation m as $$d\left({x}_{m},{c}_{j}\right)$$.
Select the next centroid, c_{2} at random from X with probability
$$\frac{{d}^{2}\left({x}_{m},{c}_{1}\right)}{{\displaystyle \sum}_{j=1}^{n}{d}^{2}\left({x}_{j},{c}_{1}\right)}.$$
To choose center j:
Compute the distances from each observation to each centroid, and assign each observation to its closest centroid.
For m = 1,...,n and p = 1,...,j – 1, select centroid j at random from X with probability
$$\frac{{d}^{2}\left({x}_{m},{c}_{p}\right)}{{\displaystyle \sum}_{\{h;{x}_{h}\in {C}_{p}\}}^{}{d}^{2}\left({x}_{h},{c}_{p}\right)},$$
where C_{p} is the set of all observations closest to centroid c_{p} and x_{m} belongs to C_{p}.
That is, select each subsequent center with a probability proportional to the distance from itself to the closest center that you already chose.
Repeat step 4 until k centroids are chosen.
Arthur and Vassilvitskii [1] demonstrate, using a simulation study for several cluster orientations, that kmeans++ achieves faster convergence to a lower sum of withincluster, sumofsquares pointtoclustercentroid distances than Lloyd's algorithm.
kmeans
uses a twophase iterative
algorithm to minimize the sum of pointtocentroid distances, summed
over all k
clusters.
This first phase uses batch updates, where each iteration consists of reassigning points to their nearest cluster centroid, all at once, followed by recalculation of cluster centroids. This phase occasionally does not converge to solution that is a local minimum. That is, a partition of the data where moving any single point to a different cluster increases the total sum of distances. This is more likely for small data sets. The batch phase is fast, but potentially only approximates a solution as a starting point for the second phase.
This second phase uses online updates, where points are individually reassigned if doing so reduces the sum of distances, and cluster centroids are recomputed after each reassignment. Each iteration during this phase consists of one pass though all the points. This phase converges to a local minimum, although there might be other local minima with lower total sum of distances. In general, finding the global minimum is solved by an exhaustive choice of starting points, but using several replicates with random starting points typically results in a solution that is a global minimum.
If Replicates
= r >
1 and Start
is plus
(the default),
then the software selects r possibly different
sets of seeds according to the kmeans++
algorithm.
If you enable the UseParallel
option
in Options
and Replicates
>
1, then each worker selects seeds and clusters in parallel.
[1] Arthur, David, and Sergi Vassilvitskii. "Kmeans++: The Advantages of Careful Seeding." SODA ‘07: Proceedings of the Eighteenth Annual ACMSIAM Symposium on Discrete Algorithms. 2007, pp. 1027–1035.
[2] Lloyd, Stuart P. "Least Squares Quantization in PCM." IEEE Transactions on Information Theory. Vol. 28, 1982, pp. 129–137.
[3] Seber, G. A. F. Multivariate Observations. Hoboken, NJ: John Wiley & Sons, Inc., 1984.
[4] Spath, H. Cluster Dissection and Analysis: Theory, FORTRAN Programs, Examples. Translated by J. Goldschmidt. New York: Halsted Press, 1985.
clusterdata
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