ADD function

Average Distance Deviation of a trapezoidal fuzzy sample with respect to a fuzzy number

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).

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.

Details

See examples

Returns

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

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)

Author(s)

Asun Lubiano lubiano@uniovi.es, Sara de la Rosa de Saa rosasara@uniovi.es

Note

In case you find (almost surely existing) bugs or have recommendations for improving the functions comments are welcome to the above mentioned mail addresses.

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)