If you are interested in measuring the separability of clusters, you can use the silhouette approach, which gives a measure of how similar that point is to points in its own cluster compared to points in other clusters, and ranges from -1 to +1. A further explanation about silhouette is found in "Kaufman L., and P. J. Rousseeuw. Finding Groups in Data: An Introduction to Cluster Analysis. Hoboken, NJ: John Wiley & Sons, Inc., 1990."

On the other hand, if you are interested in comparing the selected features instead of  the classes, a simple approach that works well is the Fisher's Discriminant Ratio (FDR). The FDR gives a measure for each feature revealing the most relevant features in the classification point of view.

Support Vector Machines(SVM) classifiers

May be PBM Index. You might need to do experiments to see the which one best suits the data. 

You could refer this page if you are interested.

http://cran.r-project.org/web/packages/clusterCrit/vignettes/clusterCrit.pdf

I think what by use a discriminant functions is better methods.

Thank you Anthony for your recommendations

I do not understand the answer of Hernando

Talking about clustering and Dunn's Index we are in the realm of unsupervised classification. These techniques help exploring a data set with respect to the presence of internal heterogeneities, which can be measured with different metrics. Euclidean is one of this and indices for testing clustering (like the Davies-Bouldin Index) are based on this measure. Ohter metrics may require different criteria. The Identification of groups using Kohonen Maps and the U-Matrix, for instance,  cannot be so easily handled. Often unsupervised classification is said to be user-defined and application driven. So I'm not afraid to evaluate a-posteri whether a clustering makes sense from the view-point of a physical interpretation. Sounds a bit subjective and must be well discussed, but using only formal criteria may drive you off the road.

Linear discriminant analysis (LDA) and the related Fisher's linear discriminant are methods used in statistics, pattern recognition and machine learning to find a linear combination of features which characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification.

Hi George,

Let me deal with your question from another aspect. In my opinion you can not study separability  of your classes independent from your classifiers because this is exactly the point that make distinction between classifiers and demonstrate their abilities. Let me bring you an example, for SVM you can use a kernel to increase the dimension of your data (map your data to a higher order space) but make it more separable. In contrary, some time you need to decrease the dimension of your data to make in more separable such as PCA. Therefore you can not generally talk about separability without considering specific condition(e.g. keep the dimension of data, do not map the data,...). By the way, please pay attention even if you determine your condition you may miss many chances for better classifications. In addition each classifier has specific criteria to make decision about your data, such as covariance, distance ,... so should make decision that which criteria are the most important ones for you and talk about separability in that area.

Actually there are several methods and there is not a specific method for a particular task. The results may depend on the type of data clustering method, clustering parameters, etc. A first step in the way  to get the appropriate response is to outline the clusters by the method of 'silhoutte' (Rousseeuw, 1987), to have a graphical view of the clusters.

There are various things needs to be considered while clustering it depends on the you need to implement a bast method in terms of time for datasets. if you have large datasets then you have to pay in terms of quality for the clusters and if you have small datasets, then you have other options for the clustering which takes more time for running. In all SVM is the better approach.

Dunn's index works fine with well separated data (clusters) only. The PBM index is able to resolve the cluster overlapping to a significant extent. You can try features of this index. However, if your problem is only to measure between class separability, you may use different distance measures as are used in the literature.

Have you tried mahalanobis distance?

Whilst you are wanting to keep the measure independent of the classifier, the classification method is important when defining separability. If your classifier is probabilistic, should you not be looking at the difference between probability distributions (e.g. Bhattacharyya distance, Kullback–Leibler divergence, ...)? Arguably, SVM, NN, are not based on probability distributions (depending on how you define them) and so you may want to use a more geometric measure (and is your feature space Euclidean or otherwise?). Also, are you interested in the distance between centroids or the separability between classes? Feature selection always feels like such a subjective beast, what about just chucking the whole lot at the classifier ;-) ? Any help?

If you are interested in measuring the separability of clusters, you can use the silhouette approach, which gives a measure of how similar that point is to points in its own cluster compared to points in other clusters, and ranges from -1 to +1. A further explanation about silhouette is found in "Kaufman L., and P. J. Rousseeuw. Finding Groups in Data: An Introduction to Cluster Analysis. Hoboken, NJ: John Wiley & Sons, Inc., 1990."

On the other hand, if you are interested in comparing the selected features instead of  the classes, a simple approach that works well is the Fisher's Discriminant Ratio (FDR). The FDR gives a measure for each feature revealing the most relevant features in the classification point of view.

I agree with Danton using an Linear Diskriminat Analysis (LDA) and computing the FDR and Rayleigh coefficient is a good possibility to generate a good measure. We used this approach for a multi-cluster analysis for electrial drives.

Bayer, Christian; Bator, Martyna; Enge-Rosenblatt, Olaf; Mönks, Uwe; Dicks, Alexander; Lohweg, Volker: Sensorless Drive Diagnosis Using Automated Feature Extraction, Significance Ranking and Reduction. 18th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA), Cagliari, Italy, Sep 2013.

Hello.

This our article will be helpful for you, I think.

http://www.sciencedirect.com/science/article/pii/S1532046414001373

It is good for cases of nonlinear discrimination, when ANN for further classification is used.

hi...

you can use scatter matrices, divergence , bhattacharya divergence.

For more details, See below book, Chapter-4,

An introduction to pattern recognition A MATLAB approach,

By: Theodoridis, Sergios