Spectral methods for growth curve clustering

Abstract

The growth curve clustering problem is analyzed and its connection with the spectral relaxation method is described. For a given set of growth curves and similarity function, a similarity matrix is defined, from which the corresponding similarity graph is constructed. It is shown that a nearly optimal growth curve partition can be obtained from the eigendecomposition of a specific matrix associated with a similarity graph. The results are illustrated and analyzed on the set of synthetically generated growth curves. One real-world problem is also given.

Appendix A: Adjusted rand index

Let S be the set of n data items, $S=\{x_1,x_2,\ldots ,x_n\}$, and let $U=\{U_1,U_2,\ldots ,U_k\}$ and $V=\{V_1,V_2,\ldots ,V_r\}$ be two partitions of S. The measure of similarity between partitions U and V is based on counting the pairs of data items that are in the same/different partition sets in U and V. Each pair $(x_i,x_j)$ of data items is classified into one of four groups based on their comembership in U and V, which results in the pair-counts derived using the contingency table. The contingency table is a $k\times r$ matrix of all possible overlaps between each pair of clusters U and V, where its ijth element shows the intersection of cluster $U_i$ and $V_j$, that is, $n_{ij}=|U_i\cap V_j|$.

	$V_1$	$V_2$	$\ldots$	$V_r$	Marginal sums
$U_1$	$n_{11}$	$n_{12}$	$\ldots$	$n_{1r}$	$n_{1.}$
$U_2$	$n_{21}$	$n_{22}$	$\ldots$	$n_{2r}$	$n_{2.}$
$\vdots$	$\vdots$	$\vdots$	$\ddots$	$\vdots$	$\vdots$
$U_k$	$n_{k1}$	$n_{k2}$	$\ldots$	$n_{kr}$	$n_{k.}$
Marginal sums	$n_{.1}$	$n_{.2}$	$\ldots$	$n_{.r}$	n

In the case of disjoint clusters we have $n_{i.}=|U_i|$ and $n_{.j}=|V_j|$. Let us define the following numbers:

• a—the number of pairs of elements in S that are in the same set in U and in the same set in V;

• b—the number of pairs of elements in S that are in different sets in U and in different sets in V;

• c—the number of pairs of elements in S that are in the same set in U and in different sets in V;

• d—the number of pairs of elements in S that are in different sets in U and in the same set in V.

The Rand index is a measure of similarity between U and V defined as

$$\begin{aligned} RI=\frac{a+b}{a+b+c+d}=\frac{a+b}{{n\atopwithdelims ()2}}. \end{aligned}$$

The numbers a, b, c, d can be easily obtained by using simple combinatorial methods.

The Rand index has a fixed range of [0, 1]. A problem with this type of similarity measure is that the expected value of this index of two random partitions does not take a constant value (say zero). The adjusted Rand index ARI is proposed in Hubert and Arabie (1985) assuming that the contingency table is constructed randomly when the size of the clusters in U and V is fixed. The ARI is calculated based on an upper bound 1 on the RI and its expected value

$$\begin{aligned} ARI=\frac{\sum _{i=1}^k\sum _{j=1}^r{n_{ij}\atopwithdelims ()2}-\sum _{i=1}^k{n_{i.}\atopwithdelims ()2}\sum _{j=1}^r{n_{.j}\atopwithdelims ()2}/{n\atopwithdelims ()2}}{\frac{1}{2}\left[ \sum _{i=1}^k{n_{i.}\atopwithdelims ()2}+\sum _{j=1}^r{n_{.j}\atopwithdelims ()2}\right] -\sum _{i=1}^k{n_{i.}\atopwithdelims ()2}\sum _{j=1}^r{n_{.j}\atopwithdelims ()2}/{n\atopwithdelims ()2}}. \end{aligned}$$

It is the normalized difference of the Rand Index and its expected value under the null hypothesis (Wagner and Wagner 2007).