PJM_example_DSC_Multivalued_Map

Peiyuan Zhu

2024-06-01

Now we code the PJM (using ACP here) example in DS-ECP.

On $SSM_{W_1}:\{w_1\text{ is T},w_1\text{ is F}\}$, we define $DSM_{W_1}:\mathcal{P}(SSM_{W_1})\rightarrow[0,1]$ where $DSM_{W_1}(\{w_1\text{ is T}\})=0.4$ and $DSM_{W_1}(\{w_1\text{ is F}\})=0.6$ and $DSM_{W_2}(X)=0$ for all other $X=\emptyset,\{w_2\text{ is T},w_2\text{ is F}\}$.

tt_SSMw1 <- matrix(c(1,0,0,1,1,1), nrow = 3, ncol = 2, byrow = TRUE)
m_DSMw1 <- matrix(c(0.4,0.6,0), nrow = 3, ncol = 1)
cnames_SSMw1 <- c("w1y", "w1n") 
varnames_SSMw1 <- "w1"
idvar_SSMw1 <- 1
DSMw1 <- bca(tt_SSMw1, m_DSMw1, cnames = cnames_SSMw1, idvar = idvar_SSMw1, varnames = varnames_SSMw1)
bcaPrint(DSMw1)

##   DSMw1 specnb mass
## 1   w1y      1  0.4
## 2   w1n      2  0.6

Similarly, on $SSM_{W_2}:\{w_2\text{ is T},w_2\text{ is F}\}$, we define $DSM_{W_2}(\mathcal{P})SSM_{W_2}\rightarrow[0,1]$ where $DSM_{W_2}(\{w_2\text{ is T}\})=0.3$ and $DSM_{W_2}(\{w_2\text{ is F}\})=0.7$ and $DSM_{W_2}(X)=0$ for all other $X=\emptyset,\{w_2\text{ is T},w_2\text{ is F}\}$.

tt_SSMw2 <- matrix(c(1,0,0,1,1,1), nrow = 3, ncol = 2, byrow = TRUE)
m_DSMw2 <- matrix(c(0.3,0.7,0), nrow = 3, ncol = 1)
cnames_SSMw2 <- c("w2y", "w2n") 
varnames_SSMw2 <- "w2"
idvar_SSMw2 <- 2
DSMw2 <- bca(tt_SSMw2, m_DSMw2, cnames = cnames_SSMw2, idvar = idvar_SSMw2, varnames = varnames_SSMw2)
bcaPrint(DSMw2)

##   DSMw2 specnb mass
## 1   w2y      1  0.3
## 2   w2n      2  0.7

We also need three placeholder $SSM_{ACP}$, $DSMs_{ACP}$ on $\{A,C,P\}$.

tt_SSMacp <- matrix(c(1,1,1), nrow = 1, ncol = 3, byrow = TRUE)
m_DSMacp <- matrix(c(1), nrow = 1, ncol = 1)
cnames_SSMacp <- c("A", "C", "P") 
varnames_SSMacp <- "ACP"
idvar_SSMacp <- 3
DSMacp <- bca(tt_SSMacp, m_DSMacp, cnames = cnames_SSMacp, idvar = idvar_SSMacp, varnames = varnames_SSMacp)
bcaPrint(DSMacp)

##   DSMacp specnb mass
## 1  frame      1    1

On $SSM_{R1}:W1\times\{A,C,P\}$, we define multivalued mapping $DSM_{R1}:\mathcal{P}(SSM_{R1})\rightarrow[0,1]$ where $DSM_{R1}(\{(w1y,A),(w1y,C)\})=0.3$ and $DSM_{R1}(\{(w1n,A),(w1n,C),(w1n,P)\})=0.7$ and $DSM_{R1}(X)=0$ for all other $X$.

tt_SSMR_1 <- matrix(c(1,0,0,1,0,
                     1,0,1,0,0,
                     
                     0,1,1,0,0,
                     0,1,0,1,0,
                     0,1,0,0,1,
                     
                     1,1,1,1,1), nrow = 2 + 3 + 1, ncol = 2 + 3, byrow = TRUE, dimnames = list(NULL, c("w1y","w1n","A","C","P")))
spec_DSMR_1 <- matrix(c(1,1,1,1,1,2,1,1,1,1,1,0), nrow = 2 + 3 + 1, ncol = 2)
infovar_SSMR_1 <- matrix(c(1,3,2,3), nrow = 2, ncol = 2)
varnames_SSMR_1 <- c("w1", "ACP")
relnb_SSMR_1 <- 1
DSMR_1 <- bcaRel(tt_SSMR_1, spec_DSMR_1, infovar_SSMR_1, varnames_SSMR_1, relnb_SSMR_1)
bcaPrint(DSMR_1)

##                                  DSMR_1 specnb mass
## 1 w1y A + w1y C + w1n A + w1n C + w1n P      1    1

Similarly, we define multivalued mapping $SSM_{R2}$ and $DSMR2$.

tt_SSMR_2 <- matrix(c(1,0,0,1,0,
                     1,0,0,0,1,

                     0,1,1,0,0,
                     0,1,0,1,0,
                     0,1,0,0,1,
                     
                     1,1,1,1,1), nrow = 2 + 3 + 1, ncol = 2 + 3, byrow = TRUE, dimnames = list(NULL, c("w2y","w2n","A","C","P")))
spec_DSMR_2 <- matrix(c(1,1,1,1,1,2,1,1,1,1,1,0), nrow = 2 + 3 + 1, ncol = 2)
infovar_SSMR_2 <- matrix(c(2,3,2,3), nrow = 2, ncol = 2)
varnames_SSMR_2 <- c("w2", "ACP")
relnb_SSMR_2 <- 2
DSMR_2 <- bcaRel(tt_SSMR_2, spec_DSMR_2, infovar_SSMR_2, varnames_SSMR_2, relnb_SSMR_2)
bcaPrint(DSMR_2)

##                                  DSMR_2 specnb mass
## 1 w2y C + w2y P + w2n A + w2n C + w2n P      1    1

Now we apply Dempster-Shafer calculus. First, we up-project $DSM_{W_1}$ onto $SSM_{R_1}$ to get $DSM1_{uproj_{SSM_{R_1}}}=(\{w_1\text{ is T}\}\times SSM_{ACP})=0.4$ and $DSM1_{uproj_{SSM_{R_2}}}(\{w_1\text{ is F}\}\times SSM_{ACP})=0.6$ and $DSM1_{uproj_{SSM_{R_1}}}(X)=0$ for all other $X$.

DSMw1_uproj <- extmin(DSMw1,DSMR_1)
bcaPrint(DSMw1_uproj)

##             DSMw1_uproj specnb mass
## 1 w1y A + w1y C + w1y P      1  0.4
## 2 w1n A + w1n C + w1n P      2  0.6

Combining $DSM_{W_1}$ with $DSM_{R_1}$ to get $DSM1$ where $DSM1(\{w_1\text{ is T}\}\times\{A,C\})=0.4$ and $DSM1(\{w_1\text{ is F}\}\times(\{A,C,P\}))=0.6$ and $DSM1(X)=0$ for all other $X$.

DSM1 <- dsrwon(DSMw1_uproj,DSMR_1)
bcaPrint(DSM1)

##                    DSM1 specnb mass
## 1         w1y A + w1y C      1  0.4
## 2 w1n A + w1n C + w1n P      2  0.6

Then, down-project $DSM1$ to $SSM_{ACP}$ to get $DSM1_{dproj_{SSM_{ACP}}}$ where $DSM1_{dproj_{SSM_{ACP}}}(\{A,C\})=\sum_{X|_{SSM_{W_1}} \in SSM_{W_1}}DSM1(X)=0.4$ and $DSM1_{dproj_{SSM_{ACP}}}(\{A,C,P\})=\sum_{X|_{SSM_{W_1}} \in SSM_{W_1}}DSM1(X)=0.6$ and $DSM1_{dproj_{SSM_{ACP}}}(X)=0$ for all other $X$.

DSM1_dproj <- elim(DSM1,1)
bcaPrint(DSM1_dproj)

##   DSM1_dproj specnb mass
## 1      A + C      1  0.4
## 2      frame      2  0.6

Similarly, we up-project $DSM_{W_2}$ onto $SSM_{R_2}$ to get $DSM2_{uproj_{SSM_{R_2}}}$. Combining $DSM_{W_2}$ with $DSM_{R_2}$ to get $DSM2$. Then, down-project $DSM2$ to $SSM_{ACP}$ to get $DSM2_{dproj_{SSM_{ACP}}}$.

DSMw2_uproj <- extmin(DSMw2,DSMR_2)
DSM2 <- dsrwon(DSMw2_uproj,DSMR_2)
DSM2_dproj <- elim(DSM2,2)
bcaPrint(DSM2_dproj)

##   DSM2_dproj specnb mass
## 1      C + P      1  0.3
## 2      frame      2  0.7

Now we can combine $DSM1_{dproj_{SSM_{ACP}}}$ and $DSM2_{dproj_{SSM_{ACP}}}$ on $SSM_{ACP}$ to get $DSM3$ where $DSM3(\{C\})=0.12$ and $DSM3(\{A,C\})=0.12$ and $DSM3(\{T,P\})=0.28$ and $DSM3(\{A,C,P\})=0.42$.

DSM3 <- dsrwon(DSM1_dproj,DSM2_dproj)
bcaPrint(DSM3)

##    DSM3 specnb mass
## 1     C      1 0.12
## 2 A + C      2 0.28
## 3 C + P      3 0.18
## 4 frame      4 0.42