The routine may be called by the names g03ecf or nagf_mv_cluster_hier.
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 : ${d}_{i.jk}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}({d}_{ij},{d}_{ik})$.
2.Complete link or furthest neighbour : ${d}_{i.jk}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}({d}_{ij},{d}_{ik})$.
3.Group average : ${d}_{i.jk}=\frac{{n}_{j}}{{n}_{j}+{n}_{k}}{d}_{ij}+\frac{{n}_{k}}{{n}_{j}+{n}_{k}}{d}_{ik}$.
If the clusters are numbered $1,2,\dots ,n$ then, for convenience, if clusters $j$ and $k$, $j<k$, 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.
${\mathbf{method}}=1$
Single link.
${\mathbf{method}}=2$
Complete link.
${\mathbf{method}}=3$
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: $\mathbf{d}\left({\mathbf{n}}\times ({\mathbf{n}}-1)/2\right)$ – 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., ${\mathbf{d}}\left((i-1)(i-2)/2+j\right)$, $i>j$ must contain ${d}_{ij}$.
On exit: is overwritten.
Constraint:
${\mathbf{d}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n(n-1)/2$.
On exit: ${\mathbf{ilc}}\left(\mathit{l}\right)$ contains the number, $j$, of the cluster merged with cluster $k$ (see iuc), $j<k$, at step $\mathit{l}$, for $\mathit{l}=1,2,\dots ,n-1$.
On exit: ${\mathbf{iuc}}\left(\mathit{l}\right)$ contains the number, $k$, of the cluster merged with cluster $j$, $j<k$, 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 $k$, $j<k$, merged at step $\mathit{l}$, for $\mathit{l}=1,2,\dots ,n-1$.
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.
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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 $\mathrm{-1}$, 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, ${\mathbf{method}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{method}}=1$, $2$, $3$, $4$, $5$ or $6$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 2$.
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.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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 ${d}_{kl}$, ($i<k$) or ($i=k$), and $j<l$, 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.
9Further Comments
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.