@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="BBiometrics 25(1)",url="https://doi.org/10.2307/2528688",bibkey="doi:10.2307/2528688") @Alias(value={"ward","MISSQ","de.lmu.ifi.dbs.elki.algorithm.clustering.hierarchical.WardLinkageMethod"}) @Priority(value=101) public class WardLinkage extends java.lang.Object implements Linkage
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)
| Modifier and Type | Class and Description |
|---|---|
static class |
WardLinkage.Parameterizer
Class parameterizer.
|
| Modifier and Type | Field and Description |
|---|---|
static WardLinkage |
STATIC
Static instance of class.
|
| Constructor and Description |
|---|
WardLinkage()
Deprecated.
use the static instance
STATIC instead. |
| Modifier and Type | Method and Description |
|---|---|
double |
combine(int sizex,
double dx,
int sizey,
double dy,
int sizej,
double dxy)
Compute combined linkage for two clusters.
|
double |
initial(double d,
boolean issquare)
Initialization of the distance matrix.
|
double |
restore(double d,
boolean issquare)
Restore a distance to the original scale.
|
public static final WardLinkage STATIC
@Deprecated public WardLinkage()
STATIC instead.public double initial(double d,
boolean issquare)
Linkagepublic double restore(double d,
boolean issquare)
Linkagepublic double combine(int sizex,
double dx,
int sizey,
double dy,
int sizej,
double dxy)
Linkagecombine in interface Linkagesizex - Size of first cluster x before mergingdx - Distance of cluster x to j before mergingsizey - Size of second cluster y before mergingdy - Distance of cluster y to j before mergingsizej - Size of candidate cluster jdxy - Distance between clusters x and y before mergingCopyright © 2019 ELKI Development Team. License information.