2.1.0 User Documentation for Apache MADlib
k-Means Clustering

Clustering refers to the problem of partitioning a set of objects according to some problem-dependent measure of similarity. In the k-means variant, given $$n$$ points $$x_1, \dots, x_n \in \mathbb R^d$$, the goal is to position $$k$$ centroids $$c_1, \dots, c_k \in \mathbb R^d$$ so that the sum of distances between each point and its closest centroid is minimized. Each centroid represents a cluster that consists of all points to which this centroid is closest.

This module can compute clusters given the number of centroids k as an input, using a variety of seeding methods. It can also automatically select the best k value from a range of suggested k values, using the simplified silhouette method or the elbow method.

Clustering Function

The k-means algorithm can be invoked in different ways, depending on the source of the initial set of centroids:

• Uniform random centroid seeding method.
kmeans_random( rel_source,
expr_point,
k,
fn_dist,
agg_centroid,
max_num_iterations,
min_frac_reassigned
)

• k-means++ centroid seeding method [1]. This method can speed up convergence by seeding centroids spread out over the whole range of the input points, while at the same time not being too susceptible to outliers.
kmeanspp( rel_source,
expr_point,
k,
fn_dist,
agg_centroid,
max_num_iterations,
min_frac_reassigned,
seeding_sample_ratio
)

• Supply an initial centroid set in a relation identified by the rel_initial_centroids argument, for the case where initial centroids are stored in a table.
kmeans( rel_source,
expr_point,
rel_initial_centroids,
expr_centroid,
fn_dist,
agg_centroid,
max_num_iterations,
min_frac_reassigned
)

• Provide an initial centroid set as an array expression in the initial_centroids argument.
kmeans( rel_source,
expr_point,
initial_centroids,
fn_dist,
agg_centroid,
max_num_iterations,
min_frac_reassigned
)


Arguments

rel_source

TEXT. The name of the table containing the input data points. Data points and predefined centroids (if used) are expected to be stored row-wise, in a column of type SVEC (or any type convertible to SVEC, like FLOAT[] or INTEGER[]). Data points with non-finite values (NULL, NaN, infinity) in any component are skipped during analysis.

expr_point

TEXT. The name of the column with point coordinates or an array expression.

k

INTEGER. The number of centroids to calculate.

fn_dist (optional)

TEXT, default: 'squared_dist_norm2'. The name of the function to use to calculate the distance from a data point vector to a centroid vector. The following distance functions can be used (computation of barycenter/mean in parentheses):

• dist_norm1: 1-norm/Manhattan (element-wise median). MADlib does not provide a median aggregate function for performance reasons.
• dist_norm2: 2-norm/Euclidean (element-wise mean)
• squared_dist_norm2: squared Euclidean distance (element-wise mean)
• dist_angle: angle (element-wise mean of normalized points)
• dist_tanimoto: tanimoto (element-wise mean of normalized points [5])
• user defined function with signature DOUBLE PRECISION[] x, DOUBLE PRECISION[] y -> DOUBLE PRECISION

agg_centroid (optional)

TEXT, default: 'avg'. The name of the aggregate function used to determine centroids. The following aggregate functions can be used:

max_num_iterations (optional)

INTEGER, default: 20. The maximum number of iterations to perform.

min_frac_reassigned (optional)

DOUBLE PRECISION, default: 0.001. The minimum fraction of centroids reassigned to continue iterating. When fewer than this fraction of centroids are reassigned in an iteration, the calculation completes.

seeding_sample_ratio (optional)

DOUBLE PRECISION, default: 1.0. The proportion of subsample of original dataset to use for kmeans++ centroid seeding method. Kmeans++ scans through the data sequentially 'k' times and can be too slow for big datasets. When 'seeding_sample_ratio' is greater than 0 (thresholded to be maximum value of 1.0), the seeding is run on a uniform random subsample of the data. Note: the final K-means algorithm is run on the complete dataset. This parameter only builds a subsample for the seeding and is only available for kmeans++.

rel_initial_centroids

TEXT. Table or view containing the set of initial centroids.

expr_centroid

TEXT. The name of the column (or the array expression) in the rel_initial_centroids table or view that contains the centroid coordinates.

initial_centroids
DOUBLE PRECISION[][]. Array expression with the initial centroid coordinates.

Output
The output of the k-means module is a composite type with the following columns:

centroids DOUBLE PRECISION[][]. The final centroid positions. DOUBLE PRECISION[]. The value of the objective function per cluster. DOUBLE PRECISION. The value of the objective function. DOUBLE PRECISION. The fraction of points reassigned in the last iteration. INTEGER. The total number of iterations executed.

Auto Clustering Function

The optimal number of centroids can be determined automatically, from a set of candidate values that you provide, for random seeding or k-means++ seeding. The simplified silhouette method or the elbow method are used to pick the best k value.

• Uniform random centroid seeding method.
kmeans_random_auto( rel_source,
output_table,
expr_point,
k,
fn_dist,
agg_centroid,
max_num_iterations,
min_frac_reassigned,
k_selection_algorithm
)

• k-means++ centroid seeding method.
kmeanspp_auto( rel_source,
output_table,
expr_point,
k,
fn_dist,
agg_centroid,
max_num_iterations,
min_frac_reassigned,
seeding_sample_ratio,
k_selection_algorithm
)


Arguments
The arguments for auto k selection are the same as described above, with the following additions:

output_table

TEXT. Name of the output table containing results for each k value. Details of the output table are shown below. A summary table called <output_table>_summary will also be created for the best k value as per the selection algorithm.

k

INTEGER[]. Array of k values to test. Does not need to be contiguous but all elements must be >1 and cannot repeat within the array. Parameter 'k_selection_algorithm' determines the evaluation method.

k_selection_algorithm (optional)

TEXT, default: 'silhouette'. Method to evaluate optimal number of centroids k. Current approaches supported: 'silhouette', 'elbow' or 'both'. The text can be any subset of the strings; for e.g., 'silh' will use the silhouette method. Note that for large data sets, the silhouette computation can be expensive.

Note
Note that the auto k-means algorithms require the NumPy python package to be installed on the system since the elbow method uses the NumPy gradient function [2]. For Greenplum clusters, installing NumPy only on the host machine of the master segment will be sufficient. The suggested installation method is: pip install numpy –user

Output Tables
Two output tables are created for auto k-means. The first is called 'output_table' and contains one row per k value:

k INTEGER. Number of centroids. DOUBLE PRECISION[][]. The final centroid positions. DOUBLE PRECISION[]. The value of the objective function per cluster. DOUBLE PRECISION. The value of the objective function. DOUBLE PRECISION. The fraction of points reassigned in the last iteration. INTEGER. The total number of iterations executed. INTEGER. Number of centroids as per the specified 'k_selection_algorithm'. If 'both' is specified, the best k value will correspond to the silhouette method. DOUBLE PRECISION. The value of the simplified silhouette score for the k value, if computed. DOUBLE PRECISION. The value of the elbow score for the k value, if computed. TEXT. Algorithm used to select the best k (either 'silhouette' or 'elbow')

A summary table named <output_table>_summary is also created, which has the same output as the 'output_table' above but only contains one row for best k value as per the selection algorithm. If 'both' is specified for 'k_selection_algorithm' the best k value returned will correspond to the silhouette method.

Cluster Assignment

After clustering, the cluster assignment for each data point can be computed with the help of the closest_column() utility function:

closest_column( m,
x,
dist
)


Arguments

m

DOUBLE PRECISION[][]. Learned centroids from the training function.

x

DOUBLE PRECISION[]. Data points.

dist (optional)
TEXT, default: 'squared_dist_norm2'. The name of the function to use to calculate the distance from a data point vector to a centroid vector. See the fn_dist argument of the k-means training function for more details on distance functions.

Output
The output is a composite type with the following columns:

column_id INTEGER. Cluster assignment (zero-based, i.e., 0,1,2...). DOUBLE PRECISION. Distance to the cluster centroid.

Simple Silhouette

A common method to assess the quality of the clustering is the silhouette coefficient, a simplified version of which is implemented in MADlib [3]. There are two silhouette functions: average score across all data points, and a per-point score. The average silhouette function has the following syntax:

simple_silhouette( rel_source,
expr_point,
centroids,
fn_dist
)


The per-point silhouette function has two formats. The first takes an array of centroids:

simple_silhouette_points(rel_source,
output_table,
pid,
expr_point,
centroids,
fn_dist
)


The second takes a centroids column from a table:

simple_silhouette_points(rel_source,
output_table,
pid,
expr_point,
centroids_table,
centroids_col,
fn_dist
)


Arguments

rel_source

TEXT. The name of the table containing the input data points.

expr_point

TEXT. The name of the column with point coordinates or an array expression.

centroids

DOUBLE PRECISION[][]. An expression evaluating to an array of centroids.

fn_dist (optional)

TEXT, default: 'squared_dist_norm2'. The name of the function to use to calculate the distance from a data point vector to a centroid vector. See the fn_dist argument of the k-means training function for more details on distance functions.

output_table

TEXT. Name of the output table to contain sihouette score for each point.

pid

TEXT. Column containing point ID in the table 'rel_source' containing the data points.

centroids_table

TEXT. Name of the table containing the centroids.

centroids_col

TEXT. Name of the column in the table 'centroids_table' containing the centroids.

Output
For the average function, a single value for simple silhouette score is returned. For the per-point function, the output table contains the following columns:

pid INTEGER. Point ID. INTEGER. The cluster that the point is assigned to. INTEGER. The next closest cluster to the one that the point is assigned to. DOUBLE PRECISION. Simplified silhouette score for the point.

Examples
Note
Your results may not be exactly the same as below due to the nature random nature of the k-means algorithm. Also, remember to be consistent in the distance functions that you use in the k-means, silhouette and helper functions.

#### Clustering for Single k Value

1. Create input data:
DROP TABLE IF EXISTS km_sample;
CREATE TABLE km_sample(pid int, points double precision[]);
INSERT INTO km_sample VALUES
(1,  '{14.23, 1.71, 2.43, 15.6, 127, 2.8, 3.0600, 0.2800, 2.29, 5.64, 1.04, 3.92, 1065}'),
(2,  '{13.2, 1.78, 2.14, 11.2, 1, 2.65, 2.76, 0.26, 1.28, 4.38, 1.05, 3.49, 1050}'),
(3,  '{13.16, 2.36,  2.67, 18.6, 101, 2.8,  3.24, 0.3, 2.81, 5.6799, 1.03, 3.17, 1185}'),
(4,  '{14.37, 1.95, 2.5, 16.8, 113, 3.85, 3.49, 0.24, 2.18, 7.8, 0.86, 3.45, 1480}'),
(5,  '{13.24, 2.59, 2.87, 21, 118, 2.8, 2.69, 0.39, 1.82, 4.32, 1.04, 2.93, 735}'),
(6,  '{14.2, 1.76, 2.45, 15.2, 112, 3.27, 3.39, 0.34, 1.97, 6.75, 1.05, 2.85, 1450}'),
(7,  '{14.39, 1.87, 2.45, 14.6, 96, 2.5, 2.52, 0.3, 1.98, 5.25, 1.02, 3.58, 1290}'),
(8,  '{14.06, 2.15, 2.61, 17.6, 121, 2.6, 2.51, 0.31, 1.25, 5.05, 1.06, 3.58, 1295}'),
(9,  '{14.83, 1.64, 2.17, 14, 97, 2.8, 2.98, 0.29, 1.98, 5.2, 1.08, 2.85, 1045}'),
(10, '{13.86, 1.35, 2.27, 16, 98, 2.98, 3.15, 0.22, 1.8500, 7.2199, 1.01, 3.55, 1045}');

2. Run k-means clustering using kmeans++ for centroid seeding. Use squared Euclidean distance which is a commonly used distance function.
DROP TABLE IF EXISTS km_result;
CREATE TABLE km_result AS
SELECT * FROM madlib.kmeanspp( 'km_sample',   -- Table of source data
'points',      -- Column containing point co-ordinates
2,             -- Number of centroids to calculate
20,            -- Number of iterations
0.001          -- Fraction of centroids reassigned to keep iterating
);
\x on;
SELECT * FROM km_result;

-[ RECORD 1 ]----+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
centroids        | {{14.255,1.9325,2.5025,16.05,110.5,3.055,2.9775,0.2975,1.845,6.2125,0.9975,3.365,1378.75},{13.7533333333333,1.905,2.425,16.0666666666667,90.3333333333333,2.805,2.98,0.29,2.005,5.40663333333333,1.04166666666667,3.31833333333333,1020.83333333333}}
cluster_variance | {30561.74805,122999.110416013}
objective_fn     | 153560.858466013
frac_reassigned  | 0
num_iterations   | 2

3. Simplified silhouette coefficient. First find average for whole data set. Make sure to use the same distance function as k-means above.
SELECT * FROM madlib.simple_silhouette( 'km_sample',          -- Input points table
'points',             -- Column containing points
(SELECT centroids FROM km_result),  -- Centroids
);

-[ RECORD 1 ]-----+------------------
simple_silhouette | 0.868174608939623

Now calculate simplified silhouette coefficient for each point in the data set:
DROP TABLE IF EXISTS km_points_silh;
SELECT * FROM madlib.simple_silhouette_points( 'km_sample',          -- Input points table
'km_points_silh',      -- Output table
'pid',                 -- Point ID column in input table
'points',              -- Points column in input table
'km_result',           -- Centroids table
'centroids',           -- Column in centroids table containing centroids
);
\x off
SELECT * FROM km_points_silh ORDER BY pid;

 pid | centroid_id | neighbor_centroid_id |       silh
-----+-------------+----------------------+-------------------
1 |           1 |                    0 | 0.966608819821713
2 |           1 |                    0 | 0.926251125077039
3 |           1 |                    0 |  0.28073008848306
4 |           0 |                    1 | 0.951447083910869
5 |           1 |                    0 |  0.80098239014753
6 |           0 |                    1 | 0.972487557020722
7 |           0 |                    1 |  0.88838568723116
8 |           0 |                    1 | 0.906334719972002
9 |           1 |                    0 | 0.994315140692314
10 |           1 |                    0 |  0.99420347703982
(10 rows)

4. Find the cluster assignment for each point. Use the closest_column() function to map each point to the cluster that it belongs to.
DROP TABLE IF EXISTS point_cluster_map;
CREATE TABLE point_cluster_map AS
FROM km_sample as data, km_result;
ALTER TABLE point_cluster_map RENAME column_id to cluster_id; -- change column name
SELECT * FROM point_cluster_map ORDER BY pid;

 pid |                               points                               | cluster_id |     distance
-----+--------------------------------------------------------------------+------------+------------------
1 | {14.23,1.71,2.43,15.6,127,2.8,3.06,0.28,2.29,5.64,1.04,3.92,1065}  |          1 | 3296.12614333444
2 | {13.2,1.78,2.14,11.2,1,2.65,2.76,0.26,1.28,4.38,1.05,3.49,1050}    |          1 | 8856.90882600111
3 | {13.16,2.36,2.67,18.6,101,2.8,3.24,0.3,2.81,5.6799,1.03,3.17,1185} |          1 | 27072.3216580044
4 | {14.37,1.95,2.5,16.8,113,3.85,3.49,0.24,2.18,7.8,0.86,3.45,1480}   |          0 |    10261.9450375
5 | {13.24,2.59,2.87,21,118,2.8,2.69,0.39,1.82,4.32,1.04,2.93,735}     |          1 | 82492.8673553345
6 | {14.2,1.76,2.45,15.2,112,3.27,3.39,0.34,1.97,6.75,1.05,2.85,1450}  |          0 |     5080.3612375
7 | {14.39,1.87,2.45,14.6,96,2.5,2.52,0.3,1.98,5.25,1.02,3.58,1290}    |          0 |     8090.4485875
8 | {14.06,2.15,2.61,17.6,121,2.6,2.51,0.31,1.25,5.05,1.06,3.58,1295}  |          0 |     7128.9931875
9 | {14.83,1.64,2.17,14,97,2.8,2.98,0.29,1.98,5.2,1.08,2.85,1045}      |          1 | 634.301947334443
10 | {13.86,1.35,2.27,16,98,2.98,3.15,0.22,1.85,7.2199,1.01,3.55,1045}  |          1 | 646.584486004443
(10 rows)

5. Display cluster ID. There are two steps to get the cluster id associated with the centroid coordinates, if you need it. First unnest the cluster centroids 2-D array to get a set of 1-D centroid arrays:
DROP TABLE IF EXISTS km_centroids_unnest;
-- Run unnest function
CREATE TABLE km_centroids_unnest AS
FROM km_result;
SELECT * FROM km_centroids_unnest ORDER BY 1;

 unnest_row_id |                                                                       unnest_result
---------------+------------------------------------------------------------------------------------------------------------------------------------------------------------
1 | {14.255,1.9325,2.5025,16.05,110.5,3.055,2.9775,0.2975,1.845,6.2125,0.9975,3.365,1378.75}
2 | {13.7533333333333,1.905,2.425,16.0666666666667,90.3333333333333,2.805,2.98,0.29,2.005,5.40663333333333,1.04166666666667,3.31833333333333,1020.83333333333}
(2 rows)

Note that the ID column returned by array_unnest_2d_to_1d() is just a row ID and not the cluster ID assigned by k-means. The second step to get the cluster_id is:
SELECT cent.*,  (madlib.closest_column(centroids, unnest_result, 'madlib.squared_dist_norm2')).column_id as cluster_id
FROM km_centroids_unnest as cent, km_result
ORDER BY cent.unnest_row_id;

 unnest_row_id |                                                                       unnest_result                                                                        | cluster_id
---------------+------------------------------------------------------------------------------------------------------------------------------------------------------------+------------
1 | {14.255,1.9325,2.5025,16.05,110.5,3.055,2.9775,0.2975,1.845,6.2125,0.9975,3.365,1378.75}                                                                   |          0
2 | {13.7533333333333,1.905,2.425,16.0666666666667,90.3333333333333,2.805,2.98,0.29,2.005,5.40663333333333,1.04166666666667,3.31833333333333,1020.83333333333} |          1
(2 rows)

6. Run the same example as above, but using array input. Create the input table:
DROP TABLE IF EXISTS km_arrayin CASCADE;
CREATE TABLE km_arrayin(pid int,
p1 float,
p2 float,
p3 float,
p4 float,
p5 float,
p6 float,
p7 float,
p8 float,
p9 float,
p10 float,
p11 float,
p12 float,
p13 float);
INSERT INTO km_arrayin VALUES
(1,  14.23, 1.71, 2.43, 15.6, 127, 2.8, 3.0600, 0.2800, 2.29, 5.64, 1.04, 3.92, 1065),
(2,  13.2, 1.78, 2.14, 11.2, 1, 2.65, 2.76, 0.26, 1.28, 4.38, 1.05, 3.49, 1050),
(3,  13.16, 2.36,  2.67, 18.6, 101, 2.8,  3.24, 0.3, 2.81, 5.6799, 1.03, 3.17, 1185),
(4,  14.37, 1.95, 2.5, 16.8, 113, 3.85, 3.49, 0.24, 2.18, 7.8, 0.86, 3.45, 1480),
(5,  13.24, 2.59, 2.87, 21, 118, 2.8, 2.69, 0.39, 1.82, 4.32, 1.04, 2.93, 735),
(6,  14.2, 1.76, 2.45, 15.2, 112, 3.27, 3.39, 0.34, 1.97, 6.75, 1.05, 2.85, 1450),
(7,  14.39, 1.87, 2.45, 14.6, 96, 2.5, 2.52, 0.3, 1.98, 5.25, 1.02, 3.58, 1290),
(8,  14.06, 2.15, 2.61, 17.6, 121, 2.6, 2.51, 0.31, 1.25, 5.05, 1.06, 3.58, 1295),
(9,  14.83, 1.64, 2.17, 14, 97, 2.8, 2.98, 0.29, 1.98, 5.2, 1.08, 2.85, 1045),
(10, 13.86, 1.35, 2.27, 16, 98, 2.98, 3.15, 0.22, 1.8500, 7.2199, 1.01, 3.55, 1045);

Now find the cluster assignment for each point using random seeding:
DROP TABLE IF EXISTS km_result_array;
-- Run kmeans algorithm
CREATE TABLE km_result_array AS
SELECT * FROM madlib.kmeans_random('km_arrayin',                 -- Table of source data
'ARRAY[p1, p2, p3, p4, p5, p6,   -- Points
p7, p8, p9, p10, p11, p12, p13]',
2,                               -- Number of centroids to calculate
20,                              -- Number of iterations
0.001);                          -- Fraction of centroids reassigned to keep iterating
-- Get point assignment
ARRAY[p1, p2, p3, p4, p5, p6,
p7, p8, p9, p10, p11, p12, p13],
FROM km_arrayin as data, km_result_array
ORDER BY data.pid;

This produces the result in column format:
 pid |  p1   |  p2  |  p3  |  p4  | p5  |  p6  |  p7  |  p8  |  p9  |  p10   | p11  | p12  | p13  | cluster_id
-----+-------+------+------+------+-----+------+------+------+------+--------+------+------+------+------------
1 | 14.23 | 1.71 | 2.43 | 15.6 | 127 |  2.8 | 3.06 | 0.28 | 2.29 |   5.64 | 1.04 | 3.92 | 1065 |          0
2 |  13.2 | 1.78 | 2.14 | 11.2 |   1 | 2.65 | 2.76 | 0.26 | 1.28 |   4.38 | 1.05 | 3.49 | 1050 |          0
3 | 13.16 | 2.36 | 2.67 | 18.6 | 101 |  2.8 | 3.24 |  0.3 | 2.81 | 5.6799 | 1.03 | 3.17 | 1185 |          1
4 | 14.37 | 1.95 |  2.5 | 16.8 | 113 | 3.85 | 3.49 | 0.24 | 2.18 |    7.8 | 0.86 | 3.45 | 1480 |          1
5 | 13.24 | 2.59 | 2.87 |   21 | 118 |  2.8 | 2.69 | 0.39 | 1.82 |   4.32 | 1.04 | 2.93 |  735 |          0
6 |  14.2 | 1.76 | 2.45 | 15.2 | 112 | 3.27 | 3.39 | 0.34 | 1.97 |   6.75 | 1.05 | 2.85 | 1450 |          1
7 | 14.39 | 1.87 | 2.45 | 14.6 |  96 |  2.5 | 2.52 |  0.3 | 1.98 |   5.25 | 1.02 | 3.58 | 1290 |          1
8 | 14.06 | 2.15 | 2.61 | 17.6 | 121 |  2.6 | 2.51 | 0.31 | 1.25 |   5.05 | 1.06 | 3.58 | 1295 |          1
9 | 14.83 | 1.64 | 2.17 |   14 |  97 |  2.8 | 2.98 | 0.29 | 1.98 |    5.2 | 1.08 | 2.85 | 1045 |          0
10 | 13.86 | 1.35 | 2.27 |   16 |  98 | 2.98 | 3.15 | 0.22 | 1.85 | 7.2199 | 1.01 | 3.55 | 1045 |          0
(10 rows)


#### Auto Clustering for Range of k Values

1. Auto k selection. Now let's run k-means random for a variety of k values and compare using simple silhouette and elbow methods.
DROP TABLE IF EXISTS km_result_auto, km_result_auto_summary;
'km_sample',                   -- points table
'km_result_auto',              -- output table
'points',                      -- column name in point table
ARRAY[2,3,4,5,6],              -- k values to try
20,                            -- max iterations
0.001,                         -- minimum fraction of centroids reassigned to continue iterating
'both'                         -- k selection algorithm  (simple silhouette and elbow)
);
\x on
SELECT * FROM km_result_auto_summary;

-[ RECORD 1 ]-------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
k                   | 6
centroids           | {{14.23,1.71,2.43,15.6,127,2.8,3.06,0.28,2.29,5.64,1.04,3.92,1065},{13.24,2.59,2.87,21,118,2.8,2.69,0.39,1.82,4.32,1.04,2.93,735},{14.285,1.855,2.475,16,112.5,3.56,3.44,0.29,2.075,7.275,0.955,3.15,1465},{13.87,2.12666666666667,2.57666666666667,16.9333333333333,106,2.63333333333333,2.75666666666667,0.303333333333333,2.01333333333333,5.32663333333333,1.03666666666667,3.44333333333333,1256.66666666667},{14.345,1.495,2.22,15,97.5,2.89,3.065,0.255,1.915,6.20995,1.045,3.2,1045},{13.2,1.78,2.14,11.2,1,2.65,2.76,0.26,1.28,4.38,1.05,3.49,1050}}
cluster_variance    | {0,0,452.7633,8078.22646267333,5.346498005,0}
objective_fn        | 8536.33626067833
frac_reassigned     | 0
num_iterations      | 3
silhouette          | 0.954496117675027
elbow               | -50296.3677976633
selection_algorithm | silhouette

The best selection above is made by the silhouette algorithm by default. Note that the elbow method may select a different k value as the best. To see results for all k values:
SELECT * FROM km_result_auto ORDER BY k;

-[ RECORD 1 ]----+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
k                | 2
centroids        | {{14.036,2.018,2.536,16.56,108.6,3.004,3.03,0.298,2.038,6.10598,1.004,3.326,1340},{13.872,1.814,2.376,15.56,88.2,2.806,2.928,0.288,1.844,5.35198,1.044,3.348,988}}
cluster_variance | {60672.638245208,90512.324426408}
objective_fn     | 151184.962671616
frac_reassigned  | 0
num_iterations   | 3
silhouette       | 0.872087020146542
elbow            | -1056.25028126836
-[ RECORD 2 ]----+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
k                | 3
centroids        | {{13.87,2.12666666666667,2.57666666666667,16.9333333333333,106,2.63333333333333,2.75666666666667,0.303333333333333,2.01333333333333,5.32663333333333,1.03666666666667,3.44333333333333,1256.66666666667},{14.285,1.855,2.475,16,112.5,3.56,3.44,0.29,2.075,7.275,0.955,3.15,1465},{13.872,1.814,2.376,15.56,88.2,2.806,2.928,0.288,1.844,5.35198,1.044,3.348,988}}
cluster_variance | {8078.22646267333,452.7633,90512.324426408}
objective_fn     | 99043.3141890814
frac_reassigned  | 0
num_iterations   | 2
silhouette       | 0.895849874221733
elbow            | 20549.7753189989
-[ RECORD 3 ]----+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
k                | 4
centroids        | {{14.02,1.765,2.385,16.05,105.75,2.845,3.1075,0.2725,2.2325,5.93495,1.04,3.3725,1085},{13.24,2.59,2.87,21,118,2.8,2.69,0.39,1.82,4.32,1.04,2.93,735},{14.255,1.9325,2.5025,16.05,110.5,3.055,2.9775,0.2975,1.845,6.2125,0.9975,3.365,1378.75},{13.2,1.78,2.14,11.2,1,2.65,2.76,0.26,1.28,4.38,1.05,3.49,1050}}
cluster_variance | {14227.41709401,0,30561.74805,0}
objective_fn     | 44789.16514401
frac_reassigned  | 0
num_iterations   | 3
silhouette       | 0.877839150000438
elbow            | 17535.7421610686
-[ RECORD 4 ]----+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
k                | 5
centroids        | {{14.285,1.855,2.475,16,112.5,3.56,3.44,0.29,2.075,7.275,0.955,3.15,1465},{14.225,2.01,2.53,16.1,108.5,2.55,2.515,0.305,1.615,5.15,1.04,3.58,1292.5},{13.2,1.78,2.14,11.2,1,2.65,2.76,0.26,1.28,4.38,1.05,3.49,1050},{14.04,1.8225,2.435,16.65,110,2.845,2.97,0.295,1.985,5.594975,1.0425,3.3125,972.5},{13.16,2.36,2.67,18.6,101,2.8,3.24,0.3,2.81,5.6799,1.03,3.17,1185}}
cluster_variance | {452.7633,329.8988,0,76176.4564000075,0}
objective_fn     | 76959.1185000075
frac_reassigned  | 0
num_iterations   | 2
silhouette       | 0.771207558995578
elbow            | -28690.3421973961
-[ RECORD 5 ]----+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
k                | 6
centroids        | {{14.23,1.71,2.43,15.6,127,2.8,3.06,0.28,2.29,5.64,1.04,3.92,1065},{13.24,2.59,2.87,21,118,2.8,2.69,0.39,1.82,4.32,1.04,2.93,735},{14.285,1.855,2.475,16,112.5,3.56,3.44,0.29,2.075,7.275,0.955,3.15,1465},{13.87,2.12666666666667,2.57666666666667,16.9333333333333,106,2.63333333333333,2.75666666666667,0.303333333333333,2.01333333333333,5.32663333333333,1.03666666666667,3.44333333333333,1256.66666666667},{14.345,1.495,2.22,15,97.5,2.89,3.065,0.255,1.915,6.20995,1.045,3.2,1045},{13.2,1.78,2.14,11.2,1,2.65,2.76,0.26,1.28,4.38,1.05,3.49,1050}}
cluster_variance | {0,0,452.7633,8078.22646267333,5.346498005,0}
objective_fn     | 8536.33626067833
frac_reassigned  | 0
num_iterations   | 3
silhouette       | 0.954496117675027
elbow            | -50296.3677976633

2. Simplified silhouette per point. Let's say we want the simplified silhouette coefficient for each point in the data set, for the case where k=3:
DROP TABLE IF EXISTS km_points_silh;
SELECT * FROM madlib.simple_silhouette_points( 'km_sample',          -- Input points table
'km_points_silh',     -- Output table
'pid',                -- Point ID column in input table
'points',             -- Points column in input table
(SELECT centroids FROM km_result_auto WHERE k=3), -- centroids array
);
\x off
SELECT * FROM km_points_silh ORDER BY pid;

 pid | centroid_id | neighbor_centroid_id |       silh
-----+-------------+----------------------+-------------------
1 |           2 |                    0 | 0.800019825058391
2 |           2 |                    0 | 0.786712987562913
3 |           0 |                    2 | 0.867496080386644
4 |           1 |                    0 | 0.995466498952947
5 |           2 |                    0 | 0.761551610381542
6 |           1 |                    0 | 0.993950286967157
7 |           0 |                    1 | 0.960621637528528
8 |           0 |                    1 | 0.941493577405087
9 |           2 |                    0 | 0.925822020308802
10 |           2 |                    0 |  0.92536421766532
(10 rows)


Technical Background

k-means Algorithm
Formally, we wish to minimize the following objective function:

$(c_1, \dots, c_k) \mapsto \sum_{i=1}^n \min_{j=1}^k \operatorname{dist}(x_i, c_j)$

In the most common case, $$\operatorname{dist}$$ is the square of the Euclidean distance, though other distance measures can be used.

This problem is computationally difficult (NP-hard), yet the local-search heuristic proposed by Lloyd [4] performs reasonably well in practice. In fact, it is so ubiquitous today that it is often referred to as the standard algorithm or even just the k-means algorithm. It works as follows:

1. Seed the $$k$$ centroids, meaning specify their initial positions (see below).
2. Repeat until convergence:
1. Assign each point to its closest centroid.
2. Move each centroid to a position that minimizes the sum of distances in this cluster.
3. Convergence is achieved when no points change their assignments during step 2a.

Since the objective function decreases in every step, this algorithm is guaranteed to converge to a local optimum.

The quality of k-means is highly influenced by the choice of the seeding. In MADlib we support uniform-at-random sampling, kmeans++ as well as the ability for the user to enter manual seeding based on other methods.

k-means++ seeding [1] starts with a single centroid chosen randomly among the input points. It then iteratively chooses new centroids from the input points until there are a total of k centroids. The probability for picking a particular point is proportional to its minimum distance to any existing centroid. Intuitively, k-means++ favors seedings where centroids are spread out over the whole range of the input points, while at the same time not being too susceptible to outliers.

Silhouette

To evaluate the validity of clustering with different k values, the objective function is not ideal because it decreases as k value increases, so it has a bias toward selecting the largest k as the best result [6]. Therefore we use other internal measures to evaluate cluster validity.

One such measure is silhouette score. The original formulation [7] computes the average distance of a data point to all the other data points in its own cluster, and to all the data points in the neighbouring cluster nearest to the data point. This is expensive for a large number of points since it requires the full pairwise distance matrix over all data.

In the simplified silhouette [3] which is used in MADlib, the distance of a data point to a cluster is represented with the distance to the cluster centroid instead of the average distance to all (other) data points in the cluster. This has the advantage of being much faster to compute, and can be shown to be comparable in performance to the full silhouette [8].

Elbow Method
The elbow method considers the percentage of variance explained as a function of number of clusters. The idea is not to add another cluster if it doesn't model the data better. Graphically it means identifying the "elbow" in the curve of sum of squared errors vs. number of clusters (k). This was possibly originally suggested in [9]. To locate the elbow, we use the 2nd derivative of the objective function using the NumPy gradient function [2].

Literature

[1] David Arthur, Sergei Vassilvitskii: k-means++: the advantages of careful seeding, Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'07), pp. 1027-1035.

[3] E. R. Hruschka, L. N. C. Silva, R. J. G. B. Campello: Clustering Gene-Expression Data: A Hybrid Approach that Iterates Between k-Means and Evolutionary Search. In: Studies in Computational Intelligence - Hybrid Evolutionary Algorithms. pp. 313-335. Springer. 2007.

[4] Lloyd, Stuart: Least squares quantization in PCM. Technical Note, Bell Laboratories. Published much later in: IEEE Transactions on Information Theory 28(2), pp. 128-137. 1982.

[5] Leisch, Friedrich: A Toolbox for K-Centroids Cluster Analysis. In: Computational Statistics and Data Analysis, 51(2). pp. 526-544. 2006.

[6] Jain, A.K.: Data clustering: 50 years beyond k-means. Pattern recognition letters 31(8), 651–666 (2010)

[7] Peter J. Rousseeuw (1987). “Silhouettes: a Graphical Aid to the Interpretation and Validation of Cluster Analysis”. Computational and Applied Mathematics 20: 53-65

[8] Wang F., Franco-Penya HH., Kelleher J.D., Pugh J., Ross R. (2017) An Analysis of the Application of Simplified Silhouette to the Evaluation of k-means Clustering Validity. In: Perner P. (eds) Machine Learning and Data Mining in Pattern Recognition. MLDM 2017. Lecture Notes in Computer Science, vol 10358. Springer, Cham

[9] Robert L. Thorndike (December 1953). "Who Belongs in the Family?". Psychometrika. 18 (4): 267–276. doi:10.1007/BF02289263.

Related Topics

File kmeans.sql_in documenting the k-Means SQL functions

Sparse Vectors

closest_column()

array_unnest_2d_to_1d()