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M-Dimensional Unified (r,s)-J-Divergences

The three different ways of defining $M-$dimensional unified $(r,s)-J-$divergence measures are given by

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P_1,P_2,...,P_......P_M), & r\neq 1,\ \ s=1 \\ J(P_1,P_2,...,P_M), &r=1,\ \ s=1\end{array}\right.$$ (5.18)

$\alpha=1,2$ and 3, where for $\alpha =1$, we have

$^1J^s_r(P_1,P_2,...,P_M)$

$$=(1-2^{1-s})^{-1}\Big\{\Big[ \sum_{i=1}^n{\Big(\sum_{{j,k=1}_......}}^M{\lambda_j\lambda_k}}\Big)\Big]^{s-1\over r-1}-1\Big\},\r\neq 1,\ s\neq 1$$ $$^1J^s_1(P_1,P_2,...,P_M)=(1-2^{1-s})^{-1}\Big\{2^{(s-1)J(P_1,P_2,...,P_M)}-1\Big\},\ s\neq 1$$ and  $$^1 J^1_r(P_1,P_2,...,P_M)=(r-1)^{-1} log_2\Big[\sum_{i=1}^n{ \B......_{ji}p^{1-r}_{ki}}/ \sum_{{j,k=1}_{j\neqk}}^M{\lambda_j \lambda_k}\Big)}\Big]$$ For $\alpha=2$, we have

$$^2 {\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P_1,P_2,...,P_M)= ......da_j \lambda_k} {\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P_j\vert\vert P_k).$$ (5.19)

For $\alpha =3$, we have

$^3 J^s_r(P_1,P_2,...,P_M)$

$$= (1-2^{1-s})^{-1}\Big\{ \Big[\sum_{i=1}^n{\Big(\sum_{j=1}^M{\l......^{\lambda_k}_{ki}}\Big)^{1-r}} \Big]^{s-1\overr-1}-1\Big\},\ r\neq 1, s\neq 1$$ $$^3J^s_1(P_1,P_2,...,P_M)=(1-2^{1-s})^{-1}\Big\{2^{(s-1)J(P_1,P_2,...,P_M)}-1\Big\}, s\neq 1$$ $$^3J^1_r(P_1,P_2,...,P_M)=(r-1)^{-1}\,\,log_2\Big\{\sum_{i=1}^......ji}}\Big)\Big(\prod_{k=1}^M{p^{\lambda_k}_{ki}}\Big)^{1-r}}\Big\}, \ r\neq1$$ In particular, we have  $$^1 {\ensuremath{\boldsymbol{\mathscr{J}}}}^s_s(P_1,P_2,...,P_M)=......P_1,P_2,...,P_M)}= {\ensuremath{\boldsymbol{\mathscr{J}}}}^s_s(P_1,P_2,...,P_M)$$ where $${\ensuremath{\boldsymbol{\mathscr{J}}}}^s_s(P_1,P_2,...,P_M)=\lef......P_1,P_2,...,P_M), & s\neq 1 \\ J(P_1,P_2,...,P_M), & s=1\end{array}\right.$$ with

$J^s_s(P_1,P_2,...,P_M)=$

$$(1-2^{1-s})^{-1}\Big\{ \Big[\sum_{i=1}^n{\Big(\sum_{{j,k=1}_{j\....../ \sum_{{j,k=1}_{j\neq k}}^M{\lambda_j\lambda_k}\Big)}\Big]-1\Big\},\ s\neq 1$$

For two dimensional case, i.e., when M=2,$\lambda_1=\lambda_2={1\over 2}$, $P_1=P$ and $P_2=Q$, we have

$2\, ^1J^s_r(P_1,P_2)=\ ^1J^s_r(P\vert\vert Q)$

$$= 2(1-2^{1-s})^{-1}\Big\{\Big[\sum_{i=1}^n{\Big({p^r_iq^{1-r}_i+p^{1-r}_iq^r_i\over2}\Big)} \Big]^{s-1\over r-1}-1\Big\},$$ (5.20)

$4\, ^2J^s_r(P_1,P_2)=\, ^2J^s_r(P\vert\vert Q)$

$$= (1-2^{1-s})^{-1}\Big\{\Big(\sum_{i=1}^n{p^r_iq^{1-r}_i}\Big)^{s-1\over r-1}+\Big(\sum_{i=1}^n{q^r_ip^{1-r}_i}\Big)^{s-1\over r-1}-2\Big\},$$ (5.21)

and

$^3J^s_r(P_1,P_2)=\ ^3I^s_r(P\vert\vert Q)$

$$= (1-2^{1-s})^{-1}\Big\{ \Big[ \sum_{i=1}^n{ \Big( {p^{1+r\over2......+ p^{1+r\over 2}_i q^{1-r\over 2}_i\over 2}\Big)}\Big]^{s-1\over r-1}-1\Big\},$$ (5.22)

for all $r\neq 1, \ s\neq 1$.