Abstract

A major problem in applying the FCM method for clustering microarray data is the choice of the fuzziness parameter m. We show that the commonly used value m=2 is not appropriate for some datasets, and that optimal values for m vary widely from one dataset to another. We propose an empirical method, based on the distribution of distances between genes in a given dataset, to determine an adequate value for . By setting threshold levels for the membership values, genes which are the most tigthly associated to a given cluster can be selected. Using a yeast cell cycle dataset as an example, we show that this selection increases the overall biological significance of the genes within the cluster.

Datasets used

Serum dataset : serum.txt
can be downloaded also from : http://www.sciencemag.org/feature/data/984559.shl
the entire dataset is available at : http://genome-www.stanford.edu/serum

Yeast dataset : yeast.txt (normalized data used)
The entire dataset can be downloaded from : all dataset or from http://genomics.stanford.edu
see also http://arep.med.harvard.edu/network_discovery

Human cancer dataset : hc728g.txt
The complete dataset can be downloaded from : discover.nci.nih.gov/nature2000/

Iris dataset : iris.txt
can be downloaded also from : ftp://ftp.ics.uci.edu/pub/machine-learning-databases/

Synthetic dataset 1 : y3c.txt

Synthetic dataset 2 : y14c.txt

Color figures

figure 1 :

Influence of the fuzziness parameter m on the distribution of membership values.
Boxplot representations of sorted membership values from FCM clustering are shown.
For fixed values of m, the K membership values of each gene were sorted
in decreasing order. For a point in each plot, horizontal segments are 99 centile, third quartile,
median, first quartile and first centile values respectively ; isolated segments represent outliers.
a) distribution of membership values when m is fixed to 2. Note that, in the case
of the yeast and cancer datasets, all membership values for all genes were equal to 1/K.
b) distribution of membership values when $m$ is equal to the computed upper bound value m_ub.
c) effect of varying m in the case of the serum data.

figure 2 :

Boxplot of sorted membership values for real and randomized datasets (see figure 1 for
description of the representation). Only the first 15 significant values are shown for the
cancer dataset. Randomized datasets was generated as follows. To the first gene in the list of
the dataset, we associated an expression value selected randomly from the N values of the
experiment j. To the second gene in the list, we associated an expression value selected
radomly from the remaining (N-1) values of experiment j. We repeated this process until
we associated the remaining expression value to the last gene in the list. The overall randomized
dataset was formed by doing these operations for the p experiments.

figure 3 :

Scatterplot of the two highest membership values of all genes in the datasets.
Vertical lines correspond to median value of the highest membership value.
(a) serum dataset, (b) yeast dataset, (c) cancer dataset.

figure 4 :


Cancer data, threshold-based selection of genes and expression profile representation.
We use the value of the threshold (0.67) to divide the data set into two groups :
genes with a U_max greater than 0.67 (red in part (a)) and the genes which
have a U_max lower than 0.67 (magenta in part (a)). Finally, we replace the
normalized data of each gene by color codes in which red stands for the highest
expression value while green is used for the lower value. The panel (c) of this figure
shows a clean separation of clusters for genes having a membership value greater
than 0.67. In contrast, expression profile of the genes in magenta (part (a))
shows a more fuzzy pattern (the panel (b) of the figure).

figure 5 :

Boxplot of silhouettes values of genes in clusters. For each gene a silhouette value is computed,
see text. When this value is lower than zero, the corresponding gene is poorly classified.
(a) serum data set, top : no selection, bottom : gene selection with a threshold equal to 0.87 ;
(b) yeast data set, top : no selection, bottom : gene selection with a threshold equal to 0.80 ;
(c) cancer data set, top : no selection, bottom : gene selection with a threshold equal to 0.67.

Supplementary material

For the same values of the fuzziness parameter m, we computed the sample mean, standard deviation and the coefficient of variation of the derivative distances Y_m. We also computed the ratio of the coefficient of variation.
The results obtained are summarized in tables the following two tables

Synthetic dataset 1
We varied the fuzziness parameter m and we performed the same computations as for the iris dataset. The results obtained are summarized in the following table

Synthetic dataset 2
We varied the fuzzy parameter m and computed U_{min} and U_{max} and the sample statistics of distances Y_m (see the following table)

Serum dataset
The sample statistics of the normalized Y_m are summarized in the following table

Yeast dataset
Two different selections of genes are made from the 6200 ORF in the yeast dataset. The first selection contains 2945 genes while the second contains 1159 genes. The sample statistics of the normalized Y_m of these datasets are summarized in the following tables

Direct computation of the upper bound value for m
The flowchart in the following figure summarizes the computation steps used to define the upper bound value of the fuzziness parameter m.

figure 6 :

Computation flowchart for determining the upper bound value of the fuzziness
parameter m. The algorithm stops when the absolute value of the error between
computed cv and 0.03p is lower than e. It also stops when the number of
iterations is greater than maxIter. Default values of e and maxIter are 0.001
and 500 respectively, and can be ajusted by the user. The maximum value of
m, m_2=1000 can also be changed by the user.

Matlab functions