Model-free methods are widely used for the processing of brain fMRI data collected under natural stimulations, sleep, or rest. Among them is the popular fuzzy

There are many contexts where model-based methods are inadequate to map brain function, including for instance tasks that cannot be fully controlled (e.g., sleep, learning, natural stimulation, continuous rest;

Several data-driven methods have previously been used in fMRI (

It is critical for any reliable clustering method to be able to determine whether: (i) the data contains any structure and (ii) the segregated clusters are ‘true’ representations of the data (_{opt}). To do that, previous studies have introduced many measures, called cluster validity (CV) indices, to estimate _{opt} in an unsupervised manner (for a review see _{opt} clusters).

A few studies have previously investigated the effectiveness of CV indices in the context of fMRI data clustering (e.g.,

Our clustering method was based on the popular fuzzy _{i} of _{j}, that represents its characteristic timecourse (prototype). The resemblance between each voxel _{j} is assessed by the distance _{ij} between _{i} and _{j}. The degree of membership _{ij} is calculated for each voxel _{ij} for each cluster

In brief, the standard FCM algorithm iteratively minimises the following objective function _{m}:

Degrees of membership

Optimal clustering depends on the choice of the similarity _{opt}, as detailed below.

Here I used a modified version of the hyperbolic correlation distance proposed previously by

Where _{ij} is the Pearson correlation coefficient between _{i} and _{j}.

Here, a modified version of

This new formula uses the square root function, a monotonically increasing function over

A good and robust clustering should yield compact and well-separated clusters. This is assumed to be the case when the number of clusters reaches an optimal value _{opt}. The exact _{opt} value is however unknown in fMRI data. Previous reports have suggested that _{opt} can be found within the interval [2, _{opt} can only be estimated empirically. Typically, FCM is repeated several times with different _{opt}, and that criterion is typically defined as a trade-off between compactness and separation.

Before introducing the different CV indices used here, it might be helpful to define the core measures of compactness and separation using unified mathematical notations. These measures can be seen as building blocks that can be combined into different CV indices. Ultimately, the definition of those measures would help appreciate the inherent links (or similarity) between previously suggested CV indices, before introducing the rationale of the new CV index.

Two core quantities, noted _{m,j} and _{m,j}, were defined as following:

The measures _{1,j} and _{2,j} represent the fuzzy cardinality and the fuzzy partition of cluster _{m,j} denotes the fuzzy variation of cluster _{1,j} as a measure of fuzzy variation (e.g.,

Those core quantities can then be combined into different forms to give away different measures of fuzzy compactness (cohesiveness) for a given _{m,1} (_{m,m} (

Likewise, the fuzzy separation (isolation) between clusters was previously estimated with several fuzzy separations quantities called _{m} (

Interestingly, the ratio

Furthermore, different measures of between-centroid distance have been proposed, including the minimum distance _{dmin} (e.g., _{dmax} (e.g., _{dmin,j} between a cluster

These measures, based on the distance between estimated centroids, can be seen as alternative separation measures. They can be handy when the clustering is showing redundant clusters.

This section introduces two new measures of separation and discrimination between voxels by combining different measures of fuzzy cardinality and variation (cf. _{intra}) dissimilarity coefficient and an inter-cluster (_{inter}) dissimilarity coefficient:

Small _{intra} values would indicate that, across all clusters, voxels that are close to a given cluster are well-isolated from voxels that are far from that cluster, whereas high _{inter} values indicate well-discriminated voxels (i.e., small fuzzy overlap between clusters). Our initial tests with noisy simulated data showed the need to define new separation measures that are robust to noise and can handle

There are many CV indices in the literature (probably more than 50 indices), hence it is beyond the scope of this study to test all of them. In a preliminary analysis (results not shown here), about 20 selected CV indices were first tested on several simulated datasets (as defined in

From this preliminary analysis, seven CV indices (out of twenty) were selected according to the following four criteria: CV indices should (i) combine both measures of separation and compactness; (ii) not suffer from monotonic dependency with the number of expected clusters; (iii) not necessitate the categorisation or the binarisation of

(1)- The Rezaee-Lelieveldt-Reider index _{RLR} (

The constant _{X} is the variance of the whole data set. The best _{RLR} with respect to the number of clusters _{RLR}, the constant _{max}) as suggested previously (

CV index | Proposed by | Range | Value at _{opt} |
---|---|---|---|

_{RLR} |
Rezaee-Lelieveldt-Reider index (1998). A modified version was used here ( |
[0, +∞[ | Minimal |

_{ZLE} |
Zahid-Limouri-Essaid index (1999) | ]−∞, +∞[ | Maximal |

_{GV} |
Geva index (2000) | [0, +∞[ | Maximal |

_{KP} |
Kim-Park index (2001) | [0, +∞[ | Minimal |

_{PBM} |
Pakhira-Bandyopadhyay-Maulik index (2004) | [0, +∞[ | Maximal |

_{WY} |
Wu-Yang index (2005) | [ − |
Maximal |

_{BWS} |
Bouguessa-Wang-Sun index (2006) | [0, +∞[ | Maximal |

_{new} |
A new CV index | [0, +∞[ | Maximal |

(2)- The Zahid-Limouri-Essaid index _{ZLE} (

The constant _{max} (note that in the original paper of Zahid et al., _{ZLE} with respect to _{ZLE} index has previously been used for fMRI analysis (

Note that the ratio

(3)- Among several CV indices suggested by Geva and colleagues (_{GV} was selected here to measure the ratio of the between-cluster scatter matrix to the within-cluster scatter matrix (

The normalisation with the number of clusters _{GV} when

(4)- The Kim-Park index, noted _{KP} (

The best _{KP} with respect to the number of clusters

(5)- The Pakhira-Bandyopadhyay-Maulik index _{PBM} (

With _{PBM} with respect to the number of clusters

(6)- The Wu-Yang index _{WY} (

This index compared the fuzzy partition of each cluster to its exponential separation, with −_{WY} < _{WY} is maximal at _{opt}.

(7)- The Bouguessa-Wang-Sun index _{BWS} index (

This index _{BWS} should be maximised with respect to

(8)- Our new _{new}, combined different measures of compactness and separation as following:

The best _{new}. The rationale behind incorporating those specific compactness and separation measures (_{inter}, _{m}, _{intra}, _{1}) in the definition of _{new} is illustrated below with simulated (noisy) datasets.

Twenty-two simulated datasets were generated as following. First, a fixed number _{j} times, with _{opt}, and

Specifically, the following 22 datasets were generated: (i) a single-cluster dataset (noted 1-cluster; e.g., the ‘null’ case, see _{opt} = 1; _{opt} near to _{opt} varying from 2 to 11 and low noise level (_{opt} = 3); (iv) ten datasets with a known number of clusters _{opt} varying from 2 to 11 and high noise level (_{opt} = 3).

All simulated datasets were clustered by FCM with _{min} = 2 and _{max} = 19. All analyses were carried out with homemade Matlab-based scripts (MathWorks, Natick, MA, USA).

Real data consisted of single subject fMRI data with a block paradigm design (freely available at: _{min} = 2 and _{max} = 39. To identify relevant FCM cluster(s) with activated auditory regions, the centroids (prototype) _{j} (

To appreciate the distribution of brain regions’ sizes in each _{ij} in

This dataset was also analysed with SPM12 software package (Wellcome Trust Centre for Neuroimaging, London UK;

The degree of fuzziness _{ij} takes the value 0 (voxel _{ij} is near to 1/

More specifically, there are two issues to be considered when selecting

Here,

One important issue during the clustering of fMRI datasets is the selection of the relevant

Depending on the initialisation of the degrees of membership

FCM on the one-cluster (A) and the

Clustering the 1-cluster dataset (_{opt} = 1) showed how compactness and separation measures behave when data cannot be clustered any further. In this context of high redundancy, it is expected to observe: (i) high similar or identical centroids _{m,1} and separation _{m} showed monotonic dependency with ^{m−1} and ^{1−m} respectively, suggesting that the product _{m,1}⋅_{m} remained constant (independent from _{dmin}, _{dmax}, _{intra} and _{inter}) were independent from

Clustering the _{opt} towards _{1} and _{intra} and _{inter}.

What emerged from above is that _{m}, _{intra}, _{inter}, and _{1} behaved differently on 1-cluster and _{new} index (as defined in

The different measures of compactness and separation are shown in _{1} decreased in the interval _{m,1}, _{intra}) and separation (e.g., _{inter}, S, _{m}). When the data became noisy, some measures were less sensitive to the structure of the data (i.e., the presence of seven clusters). As illustrated in _{m,1} showed comparable behaviour as in the clustering of the _{dmin}, quantities _{intra}, _{m} and _{1} were more robust to noise and showed high discriminability with an optimal value around the expected number of classes (

The behaviour of different measures of compactness and separation during FCM of the 7-cluster dataset with low (A,

_{ZLE}, _{GV}, _{PBM}, _{WY}, _{BWS}, _{new}; minimum value for _{RLR} and _{KP}). Note that the new index _{new} is highly discriminative in pointing to the optimal _{BWS} and _{new}, failed to indicate the optimal _{opt} (e.g., _{opt} > 9), only the new index _{new} identified the true number of clusters, albeit with lower discriminability (e.g., compare

Plots of the CV indices for the simulated 3-cluster (A), 7-cluster (B) and 11-cluster (C) datasets with low noise level (_{opt} = 3 in a, _{opt} = 7 in b and _{opt} = 11 in c). See full definition of these indices in the ‘Methods’.

Plots of the CV indices for the simulated 3-cluster (A), 7-cluster (B) and 11-cluster (C) datasets with high noise levels (

An ad hoc analysis was conducted to monitor the behaviour of _{new} over different degrees of fuzziness _{new} correctly identified the true number of clusters _{opt} in almost all simulated datasets for _{opt} > 9) where _{new} failed to identify _{opt} when _{new} underestimated _{opt} at higher

As expected, the number of iterations for the convergence of the FCM algorithm varied across the 10 different initialisations. However, for a given _{m} (_{m} value compared to the other nine initialisations).

_{inter}, and _{dmin} showed an interesting pattern when _{dmax}, _{m}). More specifically, the fuzzy separation measures _{m} showed optimal values for higher numbers of expected clusters at

(A) Different measures of compactness and separation and (B) the different CV indices. The number of clusters varied from 2 and 39.

_{RLR} and _{KP}) showed optimal values for low _{ZLE}, _{BWS}, _{GV}, and _{PBM}) showed optimal values at an intermediate number of expected clusters (i.e., maximal fuzzy separation). Interestingly, the new index _{new} went through different phases (i.e., different plateaus), depending on the weight of fuzzy separation and compactness (a change of behaviour visible at _{new} reached its maximum value at

The results of the morphological granulometry at

(A) Regions’ sizes (in number of voxels) for each crisp _{new}), there was less than 4 single-voxel regions per cluster on average. Total number of voxels ^{3}.

_{new} (

Each obtained cluster (a 3D image) of each

SPM results illustrated with the function ‘montage’ of SPM12, with axial slices varying between MNI-z = −16 mm to MNI-z = +36 mm. (A) Results at a very liberal threshold of

Using both simulated and real fMRI data, this study explored the usefulness of CV indices in identifying the best _{new}) was introduced here and it showed relatively good robustness when clustering noisy data with high number of classes. Our study also highlighted the importance of analysing different measures of separation and compactness in order to get a better understating of the complex structure of the data.

The typical low signal-to-noise ratio in fMRI might be the most challenging issue that can hinder the success of clustering techniques. Here, simulated data were based on Gaussian-like noise distributions, and the success of different CV indices depended on the level of noise in the data. Our findings are in line with previous studies that compared several CV indices on different simulated datasets and found that CV indices may fail to indicate the true number of clusters in noisy data that have high number of classes (

Perhaps more importantly, the results stressed the importance of reading the behaviour of different separation and compactness measures, defined here as building blocks of CV indices, in order to depict an accurate description of the fMRI data (cf. _{opt} must be unique, and look instead for complementary explanations of the data at different _{opt} values. For instance, using fuzzy clustering on resting-state fMRI data, Lee and colleagues (

Previous work suggested that, when CV indices fail to agree on the true number of clusters for high-dimensional datasets, a combination of different indices into a single index should be considered (

The existence of different plausible explanations (_{new} went through three different phases: (i) low values for _{m} and _{intra}; however, when c increased the _{inter}). Given the expected small proportion of task-related activations in the auditory cortex, a segregation of relevant auditory voxels was only achieved with _{new} index, can provide a richer representation of the clustering results so that users can select the most useful

Other methodological issues warrant further investigations. For instance, it might be interesting to test these CV indices with other varieties of FCM algorithms that incorporated spatial constraints during the minimisation of the objective function _{m} (e.g.,

Although FCM can provide useful data-driven explanations, deciding which clustering method is best for fMRI data remains an open question (

Unsupervised FCM with different CV indices is a useful tool for analysing model-free fMRI datasets, an alternative to the widely used independent component analysis methods. It is recommended to combine different CV indices in order to draw a complete picture of the structure of the data. The assumption here is that different CV indices may point to different optimal

The authors declare there are no competing interests.

The following information was supplied regarding data availability:

The fMRI dataset used in this paper is freely available on SPM’s website (