The hardware and bandwidth for this mirror is donated by dogado GmbH, the Webhosting and Full Service-Cloud Provider. Check out our Wordpress Tutorial.
If you wish to report a bug, or if you are interested in having us mirror your free-software or open-source project, please feel free to contact us at mirror[@]dogado.de.
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 0.6PJM_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 0.7PJM_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 1PJM_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 1PJM_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 1PJM_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 specnb mass
## 1 w1y A + w1y C + w1y P 1 0.4
## 2 w1n A + w1n C + w1n P 2 0.6PJM_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 specnb mass
## 1 w1y A + w1y C 1 0.4
## 2 w1n A + w1n C + w1n P 2 0.6PJM_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 specnb mass
## 1 A + C 1 0.4
## 2 frame 2 0.6PJM_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 0.7PJM_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 specnb mass
## 1 C 1 0.12
## 2 A + C 2 0.28
## 3 C + P 3 0.18
## 4 frame 4 0.42PJM_example_DSC_Multivalued_Map.R
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.