FuzzyStatTra (version 1.0)

ADD:

Description

This function calculates the scale measure Average Distance Deviation (ADD) for a matrix of trapezoidal fuzzy numbers F with respect to a fuzzy number U. The employed metric in the calculation can be the 1-norm distance, the mid/spr distance or the \((\varphi,\theta)\)-wabl/ldev/rdev distance. The function first checks if the input matrix F is given in the correct form (tested by checkingTra) and also the input fuzzy number U (tested by checking or checkingTra).

Usage

ADD(F, U, type, a = 1, b = 1, theta = 1/3)

Arguments

F

matrix of dimension n x 4 containing n trapezoidal fuzzy numbers characterized by their four values inf0,inf1,sup1,sup0. The function implicitly checks if the matrix is in the correct form (tested by checkingTra).

U

can be a matrix of dimension 1 x 4 (trapezoidal fuzzy number) or an array of dimension nl x 3 x 1 (general fuzzy number), where nl is the number of considered \(\alpha\)-levels and 3 the number of columns of the array: the first column will be the \(\alpha\)-levels, the second one their infimum values and the third one their supremum values. The function implicitly checks if the fuzzy number U is in the correct form (tested by checking if U is an array and tested by checkingTra if U is a matrix).

type

number 1, 2 or 3: if type==1, the 1-norm distance will be considered in the calculation of the measure ADD. If type==2, the mid/spr distance will be considered. By contrast, if type==3, the \((\varphi,\theta)\)-wabl/ldev/rdev distance will be used.

a

number >0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or in the \((\varphi,\theta)\)-wabl/ldev/rdev distance.

b

number >0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or in the \((\varphi,\theta)\)-wabl/ldev/rdev distance.

theta

number >0, by default theta=1/3. It is the weight of the spread in the mid/spr distance and the weight of the ldev and rdev in the \((\varphi,\theta)\)-wabl/ldev/rdev distance.

Value

The function returns the scale measure ADD, which is a real number.

Details

See examples

References

[1] De la Rosa de Saa, S.; Lubiano M.A.; Sinova, B.; Filzmoser, P.: Robust scale estimators for fuzzy data, Advances in Data Analysis and Classification, pp. 1-28 (2015)

See Also

checkingTra, checking, TransfTra, Rho1Tra, Rho1, DthetaphiTra, Dthetaphi, DwablphiTra, Dwablphi

Examples

# Example 1:
F=SimulCASE1(10)
U=Mean(F)
ADD(F,U,1)

# Example 2:
F=SimulCASE1(100)
U=Median1norm(F)
ADD(F,U,2,2,1,1)

# Example 3:
F=SimulCASE1(100)
U=matrix(c(1,2,3,2),nrow=1)
ADD(F,U,1)

# Example 4:
F=matrix(1:4,nrow=2)
U=matrix(1:4,nrow=1)
ADD(F,U,3,1,1,1)