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 0.6Similarly, 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 0.7We 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 1On \(SSM_{R1}:W1\times\{A,C,P\}\), 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 1Similarly, 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 1Now 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 specnb mass
## 1 w1y A + w1y C + w1y P 1 0.4
## 2 w1n A + w1n C + w1n P 2 0.6Combining \(DSM_{W_1}\) 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 specnb mass
## 1 w1y A + w1y C 1 0.4
## 2 w1n A + w1n C + w1n P 2 0.6Then, 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 specnb mass
## 1 A + C 1 0.4
## 2 frame 2 0.6Similarly, 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 0.7Now 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 specnb mass
## 1 C 1 0.12
## 2 A + C 2 0.28
## 3 C + P 3 0.18
## 4 frame 4 0.42