# NAG Library Routine Document

## 1Purpose

g03ecf performs hierarchical cluster analysis.

## 2Specification

Fortran Interface
 Subroutine g03ecf ( n, d, ilc, iuc, cd, iord, dord, iwk,
 Integer, Intent (In) :: method, n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ilc(n-1), iuc(n-1), iord(n), iwk(2*n) Real (Kind=nag_wp), Intent (Inout) :: d(n*(n-1)/2) Real (Kind=nag_wp), Intent (Out) :: cd(n-1), dord(n)
#include nagmk26.h
 void g03ecf_ (const Integer *method, const Integer *n, double d[], Integer ilc[], Integer iuc[], double cd[], Integer iord[], double dord[], Integer iwk[], Integer *ifail)

## 3Description

Given a distance or dissimilarity matrix for $n$ objects (see g03eaf), cluster analysis aims to group the $n$ objects into a number of more or less homogeneous groups or clusters. With agglomerative clustering methods, a hierarchical tree is produced by starting with $n$ clusters, each with a single object and then at each of $n-1$ stages, merging two clusters to form a larger cluster, until all objects are in a single cluster. This process may be represented by a dendrogram (see g03ehf).
At each stage, the clusters that are nearest are merged, methods differ as to how the distances between the new cluster and other clusters are computed. For three clusters $i$, $j$ and $k$ let ${n}_{i}$, ${n}_{j}$ and ${n}_{k}$ be the number of objects in each cluster and let ${d}_{ij}$, ${d}_{ik}$ and ${d}_{jk}$ be the distances between the clusters. Let clusters $j$ and $k$ be merged to give cluster $jk$, then the distance from cluster $i$ to cluster $jk$, ${d}_{i.jk}$ can be computed in the following ways.
 1 Single link or nearest neighbour : ${d}_{i.jk}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({d}_{ij},{d}_{ik}\right)$. 2 Complete link or furthest neighbour : ${d}_{i.jk}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({d}_{ij},{d}_{ik}\right)$. 3 Group average : ${d}_{i.jk}=\frac{{n}_{j}}{{n}_{j}+{n}_{k}}{d}_{ij}+\frac{{n}_{k}}{{n}_{j}+{n}_{k}}{d}_{ik}$. 4 Centroid : ${d}_{i.jk}=\frac{{n}_{j}}{{n}_{j}+{n}_{k}}{d}_{ij}+\frac{{n}_{k}}{{n}_{j}+{n}_{k}}{d}_{ik}-\frac{{n}_{j}{n}_{k}}{{\left({n}_{j}+{n}_{k}\right)}^{2}}{d}_{jk}$. 5 Median : ${d}_{i.jk}=\frac{1}{2}{d}_{ij}+\frac{1}{2}{d}_{ik}-\frac{1}{4}{d}_{jk}$. 6 Minimum variance : ${d}_{i.jk}=\left\{\left({n}_{i}+{n}_{j}\right){d}_{ij}+\left({n}_{i}+{n}_{k}\right){d}_{ik}-{n}_{i}{d}_{jk}\right\}/\left({n}_{i}+{n}_{j}+{n}_{k}\right)$.
For further details see Everitt (1974) or Krzanowski (1990).
If the clusters are numbered $1,2,\dots ,n$ then, for convenience, if clusters $j$ and $k$, $j, merge then the new cluster will be referred to as cluster $j$. Information on the clustering history is given by the values of $j$, $k$ and ${d}_{jk}$ for each of the $n-1$ clustering steps. In order to produce a dendrogram, the ordering of the objects such that the clusters that merge are adjacent is required. This ordering is computed so that the first element is $1$. The associated distances with this ordering are also computed.

## 4References

Everitt B S (1974) Cluster Analysis Heinemann
Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press

## 5Arguments

1:     $\mathbf{method}$ – IntegerInput
On entry: indicates which clustering method is used.
${\mathbf{method}}=1$
${\mathbf{method}}=2$
${\mathbf{method}}=3$
Group average.
${\mathbf{method}}=4$
Centroid.
${\mathbf{method}}=5$
Median.
${\mathbf{method}}=6$
Minimum variance.
Constraint: ${\mathbf{method}}=1$, $2$, $3$, $4$, $5$ or $6$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of objects.
Constraint: ${\mathbf{n}}\ge 2$.
3:     $\mathbf{d}\left({\mathbf{n}}×\left({\mathbf{n}}-1\right)/2\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the strictly lower triangle of the distance matrix. $D$ must be stored packed by rows, i.e., ${\mathbf{d}}\left(\left(i-1\right)\left(i-2\right)/2+j\right)$, $i>j$ must contain ${d}_{ij}$.
On exit: is overwritten.
Constraint: ${\mathbf{d}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n\left(n-1\right)/2$.
4:     $\mathbf{ilc}\left({\mathbf{n}}-1\right)$ – Integer arrayOutput
On exit: ${\mathbf{ilc}}\left(\mathit{l}\right)$ contains the number, $j$, of the cluster merged with cluster $k$ (see iuc), $j, at step $\mathit{l}$, for $\mathit{l}=1,2,\dots ,n-1$.
5:     $\mathbf{iuc}\left({\mathbf{n}}-1\right)$ – Integer arrayOutput
On exit: ${\mathbf{iuc}}\left(\mathit{l}\right)$ contains the number, $k$, of the cluster merged with cluster $j$, $j, at step $\mathit{l}$, for $\mathit{l}=1,2,\dots ,n-1$.
6:     $\mathbf{cd}\left({\mathbf{n}}-1\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{cd}}\left(\mathit{l}\right)$ contains the distance ${d}_{jk}$, between clusters $j$ and $k$, $j, merged at step $\mathit{l}$, for $\mathit{l}=1,2,\dots ,n-1$.
7:     $\mathbf{iord}\left({\mathbf{n}}\right)$ – Integer arrayOutput
On exit: the objects in dendrogram order.
8:     $\mathbf{dord}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the clustering distances corresponding to the order in iord. ${\mathbf{dord}}\left(\mathit{l}\right)$ contains the distance at which cluster ${\mathbf{iord}}\left(\mathit{l}\right)$ and ${\mathbf{iord}}\left(\mathit{l}+1\right)$ merge, for $\mathit{l}=1,2,\dots ,n-1$. ${\mathbf{dord}}\left(n\right)$ contains the maximum distance.
9:     $\mathbf{iwk}\left(2×{\mathbf{n}}\right)$ – Integer arrayWorkspace
10:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{method}}\ne 1$, $2$, $3$, $4$, $5$ or $6$, or ${\mathbf{n}}<2$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{d}}\left(i\right)<0.0$ for some $i=1,2,\dots ,n\left(n-1\right)/2$.
${\mathbf{ifail}}=3$
A true dendrogram cannot be formed because the distances at which clusters have merged are not increasing for all steps, i.e., ${\mathbf{cd}}\left(l\right)<{\mathbf{cd}}\left(l-1\right)$ for some $l=2,3,\dots ,n-1$. This can occur for the median and centroid methods.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

For ${\mathbf{method}}\ge 3$ slight rounding errors may occur in the calculations of the updated distances. These would not normally significantly affect the results, however there may be an effect if distances are (almost) equal.
If at a stage, two distances ${d}_{ij}$ and ${d}_{kl}$, ($i) or ($i=k$), and $j, are equal then clusters $k$ and $l$ will be merged rather than clusters $i$ and $j$. For single link clustering this choice will only affect the order of the objects in the dendrogram. However, for other methods the choice of $kl$ rather than $ij$ may affect the shape of the dendrogram. If either of the distances ${d}_{ij}$ and ${d}_{kl}$ is affected by rounding errors then their equality, and hence the dendrogram, may be affected.

## 8Parallelism and Performance

g03ecf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The dendrogram may be formed using g03ehf. Groupings based on the clusters formed at a given distance can be computed using g03ejf.

## 10Example

Data consisting of three variables on five objects are read in. Euclidean squared distances based on two variables are computed using g03eaf, the objects are clustered using g03ecf and the dendrogram computed using g03ehf. The dendrogram is then printed.

### 10.1Program Text

Program Text (g03ecfe.f90)

### 10.2Program Data

Program Data (g03ecfe.d)

### 10.3Program Results

Program Results (g03ecfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017