• java.lang.Object
• All Implemented Interfaces:
GeometricLinkage, Linkage

@Reference(authors="J. H. Ward Jr.",title="Hierarchical grouping to optimize an objective function",booktitle="Journal of the American statistical association 58.301",url="https://doi.org/10.1080/01621459.1963.10500845",bibkey="doi:10.1080/01621459.1963.10500845") @Reference(authors="D. Wishart",title="256. Note: An Algorithm for Hierarchical Classifications",booktitle="Biometrics 25(1)",url="https://doi.org/10.2307/2528688",bibkey="doi:10.2307/2528688")
@Alias({"ward","MISSQ"})
@Priority(101)
extends java.lang.Object
implements GeometricLinkage
Ward's method clustering method.

This criterion minimizes the increase of squared errors, and should be used with squared Euclidean distance. Usually, ELKI will try to automatically square distances when you combine this with Euclidean distance. For performance reasons, the direct use of squared distances is preferable!

The distance of two clusters in this method is: $d_{\text{Ward}}(A,B):=\text{SSE}(A\cup B)-\text{SSE}(A)-\text{SSE}(B)$ where the sum of squared errors is defined as: $\text{SSE}(X):=\sum\nolimits_{x\in X} (x-\mu_X)^2 \qquad \text{with } \mu_X=\tfrac{1}{|X|}\sum\nolimits_{x\in X} X$ This objective can be rewritten to $d_{\text{Ward}}(A,B):=\tfrac{|A|\cdot|B|}{|A|+|B|} ||\mu_A-\mu_B||^2 = \tfrac{1}{1/|A|+1/|B|} ||\mu_A-\mu_B||^2$

For Lance-Williams, we can then obtain the following recursive definition: $d_{\text{Ward}}(A\cup B,C)=\tfrac{|A|+|C|}{|A|+|B|+|C|} d(A,C) + \tfrac{|B|+|C|}{|A|+|B|+|C|} d(B,C) - \tfrac{|C|}{|A|+|B|+|C|} d(A,B)$

These transformations rely on properties of the L2-norm, so they cannot be used with arbitrary metrics, unless they are equivalent to the L2-norm in some transformed space.

Because the resulting distances are squared, when used with a non-squared distance, ELKI implementations will apply the square root before returning the final result. This is statistically somewhat questionable, but usually yields more interpretable distances that — roughly — correspond to the increase in standard deviation. With ELKI, you can get both behavior: Either choose squared Euclidean distance, or regular Euclidean distance.

This method is also referred to as "minimize increase of sum of squares" (MISSQ) by Podani.

Reference:

J. H. Ward Jr.
Hierarchical grouping to optimize an objective function
Journal of the American statistical association 58.301

The formulation using Lance-Williams equations is due to:

D. Wishart
256. Note: An Algorithm for Hierarchical Classifications
Biometrics 25(1)

Since:
0.6.0
Author:
Erich Schubert
• ### Nested Class Summary

Nested Classes
Modifier and Type Class Description
static class  WardLinkage.Par
Class parameterizer.
• ### Field Summary

Fields
Modifier and Type Field Description
static WardLinkage STATIC
Static instance of class.
• ### Constructor Summary

Constructors
Constructor Description
WardLinkage()
Deprecated.
use the static instance STATIC instead.
• ### Method Summary

All Methods
Modifier and Type Method Description
double combine​(int sizex, double dx, int sizey, double dy, int sizej, double dxy)
Compute combined linkage for two clusters.
double distance​(double[] x, int sizex, double[] y, int sizey)
Distance of two aggregated clusters.
double initial​(double d, boolean issquare)
Initialization of the distance matrix.
double[] merge​(double[] x, int sizex, double[] y, int sizey)
Merge the aggregated vectors.
double restore​(double d, boolean issquare)
Restore a distance to the original scale.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Field Detail

• #### STATIC

public static final WardLinkage STATIC
Static instance of class.
• ### Constructor Detail

@Deprecated
public WardLinkage()
Deprecated.
use the static instance STATIC instead.
Constructor.
• ### Method Detail

• #### initial

public double initial​(double d,
boolean issquare)
Description copied from interface: Linkage
Initialization of the distance matrix.
Specified by:
initial in interface Linkage
Parameters:
d - Distance
issquare - Flag to indicate the input values are already squared
Returns:
Initial value
• #### restore

public double restore​(double d,
boolean issquare)
Description copied from interface: Linkage
Restore a distance to the original scale.
Specified by:
restore in interface Linkage
Parameters:
d - Distance
issquare - Flag to indicate the input values were already squared
Returns:
Initial value
• #### combine

public double combine​(int sizex,
double dx,
int sizey,
double dy,
int sizej,
double dxy)
Description copied from interface: Linkage
Compute combined linkage for two clusters.
Specified by:
combine in interface Linkage
Parameters:
sizex - Size of first cluster x before merging
dx - Distance of cluster x to j before merging
sizey - Size of second cluster y before merging
dy - Distance of cluster y to j before merging
sizej - Size of candidate cluster j
dxy - Distance between clusters x and y before merging
Returns:
Combined distance
• #### merge

public double[] merge​(double[] x,
int sizex,
double[] y,
int sizey)
Description copied from interface: GeometricLinkage
Merge the aggregated vectors.
Specified by:
merge in interface GeometricLinkage
Parameters:
x - Center of the first cluster
sizex - Weight of the first cluster
y - Center of the second cluster
sizey - Weight of the second cluster
Returns:
Combined vector
• #### distance

public double distance​(double[] x,
int sizex,
double[] y,
int sizey)
Description copied from interface: GeometricLinkage
Distance of two aggregated clusters.
Specified by:
distance in interface GeometricLinkage
Parameters:
x - Center of the first cluster
sizex - Weight of the first cluster
y - Center of the second cluster
sizey - Weight of the second cluster
Returns:
Distance