PJM_example_DSC_Simplified

In dst: Using the Theory of Belief Functions

#devtools::load_all(".") # only used in place of dst when testing with R-devel
library(dst) 
knitr::opts_chunk$set(echo = TRUE)

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)

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)

We also need three placeholder SSMs, DSMs. On $SSM_A:{A\text{ is T},A\text{ is F}}$, we define vacuous $DSM_A:\mathcal{P}(SSM_A)\rightarrow[0,1]$ where $DSM_A({A\text{ is T},A\text{ is F}})=1$ and $DSM_A(X)=0$ for all other $X=\emptyset,{A\text{ is T}},{A\text{ is F}}$.

tt_SSMA <- matrix(c(1,1), nrow = 1, ncol = 2, byrow = TRUE)
m_DSMA <- matrix(c(1), nrow = 1, ncol = 1)
cnames_SSMA <- c("Ay", "An") 
varnames_SSMA <- "A"
idvar_SSMA <- 3
DSMA <- bca(tt_SSMA, m_DSMA, cnames = cnames_SSMA, idvar = idvar_SSMA, varnames = varnames_SSMA)
bcaPrint(DSMA)

Similarly, on $SSM_C:{C\text{ is T},C\text{ is F}}$, we define vacuous $DSM_C:\mathcal{P}(SSM_C)\rightarrow[0,1]$ where $DSM_C({C\text{ is T},C\text{ is F}})=1$ and $DSM_C(X)=0$ for all other $X=\emptyset,{C\text{ is T}},{C\text{ is F}}$.

tt_SSMC <- matrix(c(1,1), nrow = 1, ncol = 2, byrow = TRUE)
m_DSMC <- matrix(c(1), nrow = 1, ncol = 1)
cnames_SSMC <- c("Cy", "Cn") 
varnames_SSMC <- "C"
idvar_SSMC <- 4
DSMC <- bca(tt_SSMC, m_DSMC, cnames = cnames_SSMC, idvar = idvar_SSMC, varnames = varnames_SSMC)
bcaPrint(DSMC)

Similarly, on $SSM_P:{P\text{ is T},P\text{ is F}}$, we define vacuous $DSM_P:\mathcal{P}(SSM_P)\rightarrow[0,1]$ where $DSM_P({P\text{ is T},P\text{ is F}})=1$ and $DSM_P(X)=0$ for all other $X=\emptyset,{P\text{ is T}},{P\text{ is F}}$.

tt_SSMP <- matrix(c(1,1), nrow = 1, ncol = 2, byrow = TRUE)
m_DSMP <- matrix(c(1), nrow = 1, ncol = 1)
cnames_SSMP <- c("Py", "Pn") 
varnames_SSMP <- "P"
idvar_SSMP <- 5
DSMP <- bca(tt_SSMP, m_DSMP, cnames = cnames_SSMP, idvar = idvar_SSMP, varnames = varnames_SSMP)
bcaPrint(DSMP)

$SSM_{R_1}$ is on the product space of $W_1 \times A \times C \times P$. $DSM_{R_1}: \mathcal{P}(SSM_{R_2}) \rightarrow [0,1]$. When w1 is true, one of A, C are true, which has \binom{2}{2} + \binom{2}{1} = 1 + 2 = 3 cases; when w1 is false, everything can be true, which has $\binom{3}{3} + \binom{3}{2} + \binom{3}{1} = 1 + 3 + 3 = 7$ cases. So $DSM_{R_1}(X)=1$ if $X$ is the subset of all these cases and $0$ otherwise.

tt_SSMR_1 <- matrix(c(1,0,1,0,0,1,0,1,
                     1,0,0,1,1,0,0,1,

                     0,1,1,0,0,1,0,1,
                     0,1,0,1,1,0,0,1,
                     0,1,0,1,0,1,1,0,

                     1,1,1,1,1,1,1,1), nrow = 2 + 3 + 1, ncol = 8, byrow = TRUE, dimnames = list(NULL, c("w1y","w1n","Ay","An","Cy","Cn","Py","Pn")))
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(1,3,4,5,2,2,2,2), nrow = 4, ncol = 2)
varnames_SSMR_1 <- c("w1", "A", "C", "P")
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)

$SSM_{R_2}$ is on the product space of $W_2 \times A \times C \times P$. $DSM_{R_2}: \mathcal{P}(SSM_{R_2}) \rightarrow [0,1]$. When w2 is true, one of C, P are true, which has 2 + 1 = 3 cases; when w1 is false, everything can be true, which has $\binom{3}{3} + \binom{3}{2} + \binom{3}{1} = 1 + 3 + 3 = 7$ cases. So $DSM_{R_2}(X)=1$ if $X$ is the subset of all these cases and $0$ otherwise.

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

                     0,1,1,0,0,1,0,1,
                     0,1,0,1,1,0,0,1,
                     0,1,0,1,0,1,1,0,

                     1,1,1,1,1,1,1,1), nrow = 2 + 3 + 1, ncol = 8, byrow = TRUE, dimnames = list(NULL, c("w2y","w2n","Ay","An","Cy","Cn","Py","Pn")))
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(2,3,4,5,2,2,2,2), nrow = 4, ncol = 2)
varnames_SSMR_2 <- c("w2", "A", "C", "P")
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)

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

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

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

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

Then, down-project $DSM1$ to $SSM_A\times SSM_C\times SSM_P$ to get $DSM1_{dproj_{SSM_A\times SSM_C\times SSM_P}}$ where $DSM1_{dproj_{SSM_A\times SSM_C\times SSM_P}}({\text{one of A,C is T}})=\sum_{X|{SSM{W_1}} \in SSM_{W_1}}DSM1(X)=0.4$ and $DSM1_{dproj_{SSM_A\times SSM_C\times SSM_P}}(SSM_A\times SSM_C\times SSM_P\backslash{\text{all of }A,C,P\text{ are F}})=\sum_{X|{SSM{W_1}} \in SSM_{W_1}}DSM1(X)=0.6$ and $DSM1_{dproj_{SSM_A\times SSM_C\times SSM_P}}(X)=0$ for all other $X$.

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

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 $SSM_A\times SSM_C\times SSM_P$ to get $DSM2_{dproj_{SSM_A\times SSM_C\times SSM_P}}$.

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

Now we can combine $DSM1_{dproj_{SSM_A\times SSM_C\times SSM_P}}$ and $DSM2_{dproj_{SSM_A\times SSM_C\times SSM_P}}$ on $SSM_A\times SSM_C\times SSM_P$ to get $DSM3$ where $DSM3({\text{A is F and C is T and P is F}})=0.12$ and $DSM3({\text{(A is T or C is T) and P is F}})=0.12$ and $DSM3({\text{A is F and (C is T or P is T)}})=0.28$ and $DSM3({\text{One of A,C,P is T}})=0.42$.

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

Now, we can marginalize $DSM3$ to $C$ to get $DSM3_{dproj_{SSM_C}}$ where $DSM3_{dproj_{SSM_C}}({\text{C is T}})=\sum_{X|{SSM_A\times SSM_P}\in SSM_A\times SSM_P}DSM3(X)=0.12$ and $DSM3{dproj_{SSM_C}}({\text{C is F}})=\sum_{X|{SSM_A\times SSM_P}\in SSM_A\times SSM_P}DSM3(X)=0$ and $DSM3{dproj_{SSM_C}}(X)=0$ for all others. The (p,q,r) triplet on $SSM_C$ is then $(0.12,0,0.88)$.

DSM3_dprojSSMC <- elim(elim(DSM3, 3), 5)
bcaPrint(DSM3_dprojSSMC)

dst documentation built on Sept. 11, 2024, 7:05 p.m.