The revised algorithms of fuzzy variance and an application to portfolio selection

Abstract

Fuzzy statistics have been developing for decades and though many contributions have gone into the expansion of theorems, most practitioners in the field of finance who usually use statistical methods actually seldom apply fuzzy set theory. One of the more likely reasons is that many operation rules of fuzzy statistics are still in progress. Among them, the statistical measures of mean, variance, and standard deviation of fuzzy numbers are the most practically used in descriptive and inferential statistics. Although they have been investigated before, previous studies on fuzzy variance and standard deviation are either defective or too rough to follow. This work therefore step-by-step develops their definitions, computational algorithms, propositions, and proofs. The deficiency of fuzzy variance is mended by substituting the requisite equality-constraint operation for standard fuzzy arithmetic. The derivation of membership functions completely depicts the shapes of the fuzzy measures and is not just an approximation. Finally a numerical example of portfolio selection illustrates the calculation process and their use.

Appendix

The Appendix presents the proofs of Propositions 1–3.

Proof of Proposition 1

From Definition 1, $A_{L} ,\;B_{L} ,\;A_{U} ,$ and $B_{U}$ are constant numbers. In Eq. 12 both $f_{{\tilde{\mu }_{{\tilde{X}}} }}^{L} (r)$ and $f_{{\tilde{\mu }_{{\tilde{X}}} }}^{R} (r)$ are linear functions of r. Because $\tilde{\mu }_{{\tilde{X}}}$ has linear left and right membership functions, it is a triangular fuzzy number. □

Proof of Proposition 2

• (1) $\because \;{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }}^{2} ,\;{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }}^{2} ,$ 0, and $\max \left( {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }}^{2} ,{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }}^{2} } \right)$ are zero or positive, (see Eq. 13)

$\therefore \;{}^{\alpha }\underline{{\sigma_{{\tilde{X}}}^{2} }}$ and ${}^{\alpha }\overline{{\sigma_{{\tilde{X}}}^{2} }}$ are non-negative. (Eq. 14)

• (2) If $\tilde{\sigma }_{{\tilde{X}}}^{2}$ is a fuzzy number, then it is to prove: ${}^{0}\underline{{\sigma_{{\tilde{X}}}^{2} }} \le {}^{1}\underline{{\sigma_{{\tilde{X}}}^{2} }} \le {}^{1}\overline{{\sigma_{{\tilde{X}}}^{2} }} \le {}^{0}\overline{{\sigma_{{\tilde{X}}}^{2} }} .$

$$ \because \;{}^{0}\underline{{\sigma_{{\tilde{X}}}^{2} }} = I_{L} ,\;{}^{1}\underline{{\sigma_{{\tilde{X}}}^{2} }} = G_{L} + H_{L} + I_{L} ,\;{}^{1}\overline{{\sigma_{{\tilde{X}}}^{2} }} = G_{U} + H_{U} + I_{U} ,\;{}^{0}\overline{{\sigma_{{\tilde{X}}}^{2} }} = I_{U} , $$

$\therefore$ This is to prove: $I_{L} \le G_{L} + H_{L} + I_{L} \le G_{U} + H_{U} + I_{U} \le I_{U} .$

$$ \because \;I_{L} = \frac{1}{N}\sum\limits_{j = 1}^{N} {\left\{ {\begin{array}{*{20}c} {(c_{j} - \mu_{a} )^{2} ,} & {{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} < 0,} \\ {(a_{j} - \mu_{c} )^{2} ,} & {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} > 0,} \\ {0,} & {0 \in [{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} ,{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} ],} \\ \end{array} } \right.} \;G_{L} + H_{L} + I_{L} = \frac{1}{N}\sum\limits_{j = 1}^{N} {\left\{ {\begin{array}{*{20}c} {(b_{j} - \mu_{b} )^{2} ,} & {{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} < 0,} \\ {(b_{j} - \mu_{b} )^{2} ,} & {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} > 0,} \\ {0,} & {0 \in \left[ {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} ,{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} } \right],} \\ \end{array} } \right.} $$

$$ G_{U} + H_{U} + I_{U} = \frac{1}{N}\sum\limits_{j = 1}^{N} {\left\{ {\begin{array}{*{20}c} {(b_{j} - \mu_{b} )^{2} ,} & {{}^{\alpha }\overline{{D_{{\tilde{X}_{{_{j} }} }} }} < 0,} \\ {(b_{j} - \mu_{b} )^{2} ,} & {{}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} > 0,} \\ {(b_{j} - \mu_{b} )^{2} ,} & {0 \in \left[ {{}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} ,{}^{\alpha }\overline{{D_{{\tilde{X}_{j} }} }} } \right],\left| {{}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} } \right| \le \left| {{}^{\alpha }\overline{{D_{{\tilde{X}_{j} }} }} } \right|,} \\ {(b_{j} - \mu_{b} )^{2} ,} & {0 \in \left[ {{}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} ,{}^{\alpha }\overline{{D_{{\tilde{X}_{j} }} }} } \right],\left| {{}^{\alpha }\underline{{D_{{\tilde{X}_{{_{j} }} }} }} } \right| \ge \left| {{}^{\alpha }\overline{{D_{{\tilde{X}_{{_{j} }} }} }} } \right|,} \\ \end{array} } \right.} \;I_{U} = \frac{1}{N}\sum\limits_{j = 1}^{N} {\left\{ {\begin{array}{*{20}c} {(a_{j} - \mu_{c} )^{2} ,} & {{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} < 0,} \\ {(c_{j} - \mu_{a} )^{2} ,} & {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} > 0,} \\ {(c_{j} - \mu_{a} )^{2} ,} & {0 \in \left[ {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} ,{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} } \right],\left| {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} } \right| \le \left| {{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} } \right|,} \\ {(a_{j} - \mu_{c} )^{2} ,} & {0 \in \left[ {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} ,{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} } \right],\left| {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} } \right| \ge \left| {{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} } \right|.} \\ \end{array} } \right.} $$

$\therefore$ This is to prove that:

• 1. $(c_{j} - \mu_{a} )^{2} \le (b_{j} - \mu_{b} )^{2} \le (b_{j} - \mu_{b} )^{2} \le (a_{j} - \mu_{c} )^{2},$ when ${}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} < 0;$

• 2. $(a_{j} - \mu_{c} )^{2} \le (b_{j} - \mu_{b} )^{2} \le (b_{j} - \mu_{b} )^{2} \le (c_{j} - \mu_{a} )^{2},$ when ${}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} > 0;$

• 3. $0 \le 0 \le (b_{j} - \mu_{b} )^{2} \le (c_{j} - \mu_{a} )^{2},$ when $0 \in \left[ {{}^{\alpha }\underline{{D_{{\tilde{X}_{{_{j} }} }} }} ,{}^{\alpha }\overline{{D_{{\tilde{X}_{j} }} }} } \right],\left| {{}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} } \right| \le \left| {{}^{\alpha }\overline{{D_{{\tilde{X}_{j} }} }} } \right|;$

• 4. $0 \le 0 \le (b_{j} - \mu_{b} )^{2} \le (a_{j} - \mu_{c} )^{2},$ when $0 \in \left[ {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} ,{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} } \right],\left| {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }} } \right| \ge \left| {{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} } \right|.$

Refer to Eq. 13 $\Rightarrow \;{}^{0}\underline{{D_{{\tilde{x}_{j} }} }} = a_{j} - \mu_{c} \le {}^{1}\underline{{D_{{\tilde{x}_{j} }} }} = b_{j} - \mu_{b} \le {}^{1}\overline{{D_{{\tilde{x}_{j} }} }} = b_{j} - \mu_{b} \le {}^{0}\overline{{D_{{\tilde{x}_{j} }} }} = c_{j} - \mu_{a}$

If ${}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }} < 0 \Rightarrow 0 < {}^{0}\overline{{D_{{\tilde{x}_{j} }} }}^{2} \le {}^{1}\overline{{D_{{\tilde{x}_{j} }} }}^{2} \le {}^{1}\underline{{D_{{\tilde{x}_{j} }} }}^{2} \le {}^{0}\underline{{D_{{\tilde{x}_{j} }} }}^{2}$

$\Rightarrow \;0 < (c_{j} - \mu_{a} )^{2} \le (b_{j} - \mu_{b} )^{2} \le (b_{j} - \mu_{b} )^{2} \le (a_{j} - \mu_{c} )^{2}$ {1. is proven}.

If ${}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} > 0 \Rightarrow 0 < {}^{0}\underline{{D_{{\tilde{x}_{j} }} }}^{2} \le {}^{1}\underline{{D_{{\tilde{x}_{j} }} }}^{2} \le {}^{1}\overline{{D_{{\tilde{x}_{j} }} }}^{2} \le {}^{0}\overline{{D_{{\tilde{x}_{j} }} }}^{2}$

$\Rightarrow \;0 < (a_{j} - \mu_{c} )^{2} \le (b_{j} - \mu_{b} )^{2} \le (b_{j} - \mu_{b} )^{2} \le (c_{j} - \mu_{a} )^{2}$ {2. is proven}.

If $0 \in \left[ {{}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} ,{}^{\alpha }\overline{{D_{{\tilde{X}_{j} }} }} } \right],\left| {{}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} } \right| \le \left| {{}^{\alpha }\overline{{D_{{\tilde{X}_{j} }} }} } \right| \Rightarrow {}^{\alpha }\tilde{D}_{{\tilde{x}_{j} }}^{2} = \left[ {0,\max \left( {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }}^{2} ,{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }}^{2} } \right)} \right] = \left[ {0,{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }}^{2} } \right]$

$\Rightarrow 0 \le {}^{1}\overline{{D_{{\tilde{x}_{j} }} }}^{2} \le {}^{0}\overline{{D_{{\tilde{x}_{j} }} }}^{2} \Rightarrow 0 \le (b_{j} - \mu_{b} )^{2} \le (c_{j} - \mu_{a} )^{2}$ {3. is proven}.

If $0 \in \left[ {{}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} ,{}^{\alpha }\overline{{D_{{\tilde{X}_{j} }} }} } \right],\left| {{}^{\alpha }\underline{{D_{{\tilde{X}_{j} }} }} } \right| \ge \left| {{}^{\alpha }\overline{{D_{{\tilde{X}_{j} }} }} } \right| \Rightarrow {}^{\alpha }\tilde{D}_{{\tilde{x}_{j} }}^{2} = \left[ {0,\max \left( {{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }}^{2} ,{}^{\alpha }\overline{{D_{{\tilde{x}_{j} }} }}^{2} } \right)} \right] = \left[ {0,{}^{\alpha }\underline{{D_{{\tilde{x}_{j} }} }}^{2} } \right]$

$\Rightarrow 0 \le {}^{1}\underline{{D_{{\tilde{x}_{j} }} }}^{2} \le {}^{0}\underline{{D_{{\tilde{x}_{j} }} }}^{2} \Rightarrow 0 \le (b_{j} - \mu_{b} )^{2} \le (a_{j} - \mu_{c} )^{2}$ {4. is proven}.

• (3) $\because \;f_{{\tilde{\sigma }_{{\tilde{X}}}^{2} }}$ are non-linear. (see Eq. 16) $\therefore \;\tilde{\sigma }_{{\tilde{X}}}^{2}$ is not in a triangular shape. □

Proof of Proposition 3

Since ${}^{\alpha }\tilde{\sigma }_{{\tilde{X}}} = \sqrt {{}^{\alpha }\tilde{\sigma }_{{\tilde{X}}}^{2} } ,\;{}^{\alpha }\tilde{\sigma }_{{\tilde{X}}}$ complies with the properties of ${}^{\alpha }\tilde{\sigma }_{{\tilde{X}}}^{2}$ as in Proposition 2. □