kquantiles {QuClu}

Quantile-based clustering algorithm

Description

This function allows to run the $k$-quantile clustering algorithm, allowing for different constraints: common theta and unscaled variables (CU), common theta and scaled variables (CS), variable-wise theta and unscaled variables (VU) and the variable-wise theta and scaled variables (VS).

Usage

kquantiles(
  data,
  k = 2,
  method = "VS",
  eps = 1e-08,
  it.max = 100,
  B = 30,
  lambda = NULL
)

Arguments

data

A numeric vector, matrix, or data frame of observations. Categorical variables are not allowed. If a matrix or data frame, rows correspond to observations and columns correspond to variables.

k

The number of clusters. The default is k=2.

method

The chosen constrained method. The options are: CU (Common theta and Unscaled variables), CS (Common theta and Scaled variables), VU (Variable-wise theta and Unscaled variables), VS (Variable-wise theta and Scaled variables).The default is the unconstrained method, VS.

eps

The relative convergence tolerances for objective function. The default is set to 1e-8.

it.max

A number that gives integer limits on the number of the algorithm iterations. By default, it is set to 100.

B

The number of times the initialization step is repeated; the default is 30.

lambda

The initial value for lambda_j, the variable scaling parameters, for models CS and VS. By default, lambdas are set to be equal to 1.

Details

Algorithm CU: Common theta and Unscaled variables. A common value of theta for all the variables is assumed. Algorithm CS: Common theta and Scaled variables via lambda_j. A common value of theta is taken but variables are scaled through lambda_j. Algorithm VU: Variable-wise theta_j and Unscaled variables. A different theta for every single variable is estimated to better accomodate different degree of skeweness in the data. Algorithm VS: Variable-wise theta_j and Scaled variables via lambda_j. A different theta for every single variable is estimated to better accomodate different degree of skeweness in the data and variables are scaled through lambda_j.

Value

A list containing the following elements:

method

The chosen parameterization.

k

The number of clusters.

cl

A vector whose [i]th entry is classification of observation i in the test data.

qq

A matrix whose [h,j]th entry is the theta-quantile of variable j in cluster h.

theta

A vector whose [j]th entry is the percentile theta for variable j.

Vseq

The values of the objective function V at each step of the algorithm.

V

The final value of the objective function V.

lambda

A vector containing the scaling factor for each variable.

References

Hennig, C., Viroli, C., Anderlucci, L. (2019) "Quantile-based clustering" Electronic Journal of Statistics, 13 (2) 4849-4883 <doi:10.1214/19-EJS1640>

Examples

out <- kquantiles(iris[,-5],k=3,method="VS")
out$theta
out$qq

table(out$cl)