# Two_Node_Process

#### 2020-07-06

library(BayesMassBal)

The function BMB is used with a two node process and simulated data.

The constraints around these process nodes are:

\begin{align} y_1 &= y_2 +y_4\\ y_2 &= y_3 +y_5 \end{align}

Therefore the matrix of constraints, C is:

C <- matrix(c(1,-1,0,-1,0,0,1,-1,0,-1), nrow = 2, ncol = 5, byrow = TRUE)
C
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1   -1    0   -1    0
#> [2,]    0    1   -1    0   -1

The constrainProcess function in the BayesMassBal package is used to generate an X matrix based on C that will later be used with the BMB function.

X <- constrainProcess(C = C)
X
#>      [,1] [,2] [,3]
#> [1,]    1    1    1
#> [2,]    1    0    1
#> [3,]    1    0    0
#> [4,]    0    1    0
#> [5,]    0    0    1

The previously simulated data is loaded from a .csv file using the importObservations() function. The local location of the the file imported below can be found by typing system.file("extdata", "twonode_example.csv",package = "BayesMassBal"). View the document in Excel to see how your data should be formatted for import. Note: it is not required that the entries into the *.csv file are separated by ";".

y <- importObservations(file = system.file("extdata", "twonode_example.csv",
package = "BayesMassBal"),
header = TRUE, csv.params = list(sep = ";"))

Then, the BMB function is used to generate the distribution of constrained masses from the data with cov.structure = "indep".

indep.samples <- BMB(X = X, y = y, cov.structure = "indep", BTE = c(100,3000,1), lml = TRUE, verb = 0)

The output of BMB is a BayesMassBal object. Special instructions are designated when feeding a BayesMassBal object to the plot() function. Adding the argument layout = "dens" and indicating the mass balanced flow rate for CuFeS2 at $$y_3$$ should be plotted using a list supplied to sample.params, the desired distribution can be plotted with its 95% Highest Posterior Density Interval.


plot(indep.samples,sample.params = list(ybal = list(CuFeS2 = 3)),
layout = "dens",hdi.params = c(1,0.95))

It is also possible to generate trace plots to inspect convergence of the Gibbs sampler. Here are trace plots for $$\beta$$

plot(indep.samples,sample.params = list(beta = list(CuFeS2 = 1:3, gangue = 1:3)),layout = "trace",hdi.params = c(1,0.95))

The model with independent variances may not be the best fitting model. Models specifying covariance between sample locations for a single component, and covariance between components at a single location are fit.

component.samples <- BMB(X = X, y = y, cov.structure = "component", BTE = c(100,3000,1), lml = TRUE, verb = 0)
location.samples <- BMB(X = X, y = y, cov.structure = "location", BTE = c(100,3000,1), lml = TRUE, verb = 0)

Computing $$\log(\mathrm{Bayes Factor})$$ for $$BF = p(y|\texttt{indep})/p(y|\texttt{component})$$:

indep.samples$lml - component.samples$lml
#> [1] -138.7161

Then comparing $$p(y|\texttt{component})$$ to $$p(y|\texttt{location})$$

component.samples$lml - location.samples$lml
#> [1] 0.8340885

Shows there is little difference between the models where cov.structure = "location" and cov.structure = "component", but both of these models better explain the data than cov.structure = "indep".

The main effect of a variable independent of the process can be calculated by supplying a function, fn that takes the arguments of mass balanced flow rates ybal, and the random independent and uniformly distributed variables x. Information can be gained on the main effect of a particular element of x, xj, on fn using the mainEff function. Output from mainEff includes information on the distribution of $$E_x\lbrack f(x,y_{\mathrm{bal}})|x_j \rbrack$$.

fn_example <- function(X,ybal){
cu.frac <- 63.546/183.5
feed.mass <- ybal$CuFeS2[1] + ybal$gangue[1]
# Concentrate mass per ton feed
con.mass <- (ybal$CuFeS2[3] + ybal$gangue[3])/feed.mass
# Copper mass per ton feed
cu.mass <- (ybal$CuFeS2[3]*cu.frac)/feed.mass gam <- c(-1,-1/feed.mass,cu.mass,-con.mass,-cu.mass,-con.mass) f <- X %*% gam return(f) } rangex <- matrix(c(4.00 ,6.25,1125,1875,3880,9080,20,60,96,208,20.0,62.5), ncol = 6, nrow = 2) mE_example <- mainEff(indep.samples, fn = "fn_example",rangex = rangex,xj = 3, N = 25, res = 25) A plot of the output can be made. To get lines that are better connected, change increase N in the mainEff function. m.sens<- mE_example$fn.out[2,]
hpd.sens <- mE_example$fn.out[c(1,3),] row.names(hpd.sens) <- c("upper", "lower") g.plot <- mE_example$g/2000

y.lim <- range(hpd.sens)

lzero.bound <- apply(hpd.sens,1,function(X){which(X <= 0)})
lzero.mean <- which(m.sens <= 0)

main.grid <- pretty(g.plot)
minor.grid <- pretty(g.plot,25)
minor.grid <- minor.grid[-which(minor.grid %in% main.grid)]

y.main <- pretty(hpd.sens)

par(mar = c(4.2,4,1,1))
plot(g.plot,m.sens, type = "n", xlim = range(g.plot), ylim = y.lim, ylab = "Net Revenue ($/ton Feed)", xlab= "Cu Price ($/lb)")

abline(v = main.grid, lty = 6, col = "grey", lwd = 1)
abline(v = minor.grid, lty =3, col = "grey", lwd = 0.75)

abline(h = 0, col = "red", lwd = 1, lty = 6)

lines(g.plot[lzero.mean],m.sens[lzero.mean],col = "red", lwd =2)
lines(g.plot[-lzero.mean[-length(lzero.mean)]],m.sens[-lzero.mean[-length(lzero.mean)]],col = "darkgreen", lwd =2)

lines(g.plot[lzero.bound$lower],hpd.sens[2,][lzero.bound$lower], lty = 5, lwd = 2, col = "red")
lines(g.plot[-lzero.bound$lower],hpd.sens[2,][-lzero.bound$lower], lty = 5, lwd = 2, col = "darkgreen")

lines(g.plot[lzero.bound$upper],hpd.sens[1,][lzero.bound$upper], lty = 5, lwd = 2, col = "red")
lines(g.plot[-lzero.bound$upper],hpd.sens[1,][-lzero.bound$upper], lty = 5, lwd = 2, col= "darkgreen")

legend("topleft", legend = c("Expected Main Effect", "95% Bounds", "Net Revenue < $0", "Net Revenue >$0"), col = c("black","black","red", "darkgreen"), lty = c(1,6,1,1), lwd = c(2,2,2,2), bg = "white")


par(opar)