DBindexL2dist: Functional Davies–Bouldin (fDB) index and L2 (or with...

In sysbioTurin/connector: Fitting and clustering of longitudinal data

Functional Davies–Bouldin (fDB) index and L2 (or with derivatives) distances

Description

L2dist.curve2mu calculates the L2 (or with derivatives) distance between all the spline estimated for each sample with respect to the corresponding cluster center curve. L2dist.mu2mu calculates the L2 distance between the cluster centers curves. L2dist.mu20 calculates the L2 distance between the cluster centers curves and zero. Sclust.coeff calculates the S coefficents for each cluster. Rclust.coeff calculates the R coefficents for each cluster.

Usage

L2dist.curve2mu(clust, fcm.curve, database, fcm.fit = NULL, deriv = 0, q)

L2dist.mu2mu(clust, fcm.curve, database, fcm.fit = NULL, deriv = 0, q)

L2dist.mu20(clust, fcm.curve, database, fcm.fit = NULL, deriv = 0, q)

L2dist.curve20(clust, fcm.curve, database, fcm.fit = NULL, deriv = 0, q)

Sclust.coeff(clust, fcm.curve, database, fcm.fit = NULL, deriv = 0, q)

Rclust.coeff(clust, fcm.curve, database, fcm.fit = NULL, deriv = 0, q)

Arguments

clust	The vector reporting the cluster membership for each sample.
fcm.curve	The list obtained applying the function fclust.curvepred to the fitfclust output, the FCM algorithm, more details in Sugar and James.
database	...
fcm.fit	The fitfclust output, the FCM algorithm, more details in Sugar and James.
deriv	The derivative values (0, 1 or 2).

Value

The following indexes are calculated for obtaining a cluster separation measure:

• T: the total tightness representing the dispersion measure given by

  T = \sum_{k=1}^G \sum_{i=1}^n D_0(\hat{g}_i, \bar{g}^k);

  (see L2dist.curve2mu)

• S_k:

  S_k = \sqrt{\frac{1}{G_k} \sum_{i=1}^{G_k} D_q^2(\hat{g}_i, \bar{g}^k);}

  with G_k the number of curves in the k-th cluster; (see Sclust.coeff)

• M_hk: the distance between centroids (mean-curves) of h-th and k-th cluster

  M_{hk} = D_q(\bar{g}^h, \bar{g}^k);

  (seeL2dist.mu2mu)

• R_hk: a measure of how good the clustering is,

  R_{hk} = \frac{S_h + S_k}{M_{hk}};

  (seeRclust.coeff )

• fDB_q: functional Davies-Bouldin index, the cluster separation measure

  fDB_q = \frac{1}{G} \sum_{k=1}^G \max_{h \neq k} { \frac{S_h + S_k}{M_{hk}} }.

  (seeRclust.coeff)

Where the proximities measures choosen is defined as follow

D_q(f,g) = \sqrt( \integral | f^{(q)}(s)-g^{(q)}(s) |^2 ds ), d=0,1,2

whit f and g are two curves and f^(q) and g^(q) are their q-th derivatives. Note that for q=0, the equation becomes the distance induced by the classical L^2-norm.

@references Gareth M. James and Catherine A. Sugar, (2003). Clustering for Sparsely Sampled Functional Data. Journal of the American Statistical Association.

Author(s)

Cordero Francesca, Pernice Simone, Sirovich Roberta

sysbioTurin/connector documentation built on April 9, 2024, 12:10 p.m.