# PJM_example_DSC_Multivalued_Map

#### 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

PJM_example_DSC_Multivalued_Map.R

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

PJM_example_DSC_Multivalued_Map.R

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

PJM_example_DSC_Multivalued_Map.R

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

PJM_example_DSC_Multivalued_Map.R

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

PJM_example_DSC_Multivalued_Map.R

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

PJM_example_DSC_Multivalued_Map.R

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

PJM_example_DSC_Multivalued_Map.R

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

PJM_example_DSC_Multivalued_Map.R

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

PJM_example_DSC_Multivalued_Map.R

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

PJM_example_DSC_Multivalued_Map.R