Two_Node_Process

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

Constraints can also be imported from a .csv file. The path to a file, included in the BayesMassBal package, for this process can be found and constraints can be imported by specifying the location for the file argument for constrainProcess as shown below:

constraint_file_location <- system.file("extdata", "twonode_constraints.csv",package = "BayesMassBal")
X <- constrainProcess(file = constraint_file_location)

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))

A quantitative diagnostics for convergence and autocorrelation are available as part of the output from BMB:

indep.samples$diagnostics
#> $beta
#> $beta$CuFeS2
#>   index         cd      ess
#> 1     1 -0.6730579 2709.133
#> 2     2  2.5661023 2631.620
#> 3     3 -0.6703322 2248.571
#> 
#> $beta$gangue
#>   index         cd      ess
#> 1     1 -0.8579396 2553.158
#> 2     2  0.5161251 2900.000
#> 3     3 -0.6090662 1989.773
#> 
#> 
#> $Sig
#> $Sig[[1]]
#>   index         cd      ess
#> 1     1  0.2065817 2900.000
#> 2     2  2.0302338 2900.000
#> 3     3 -1.1139300 2680.417
#> 4     4  0.1070069 1780.752
#> 5     5  0.8672792 1288.647
#> 
#> $Sig[[2]]
#>   index        cd      ess
#> 1     1 -0.273248 3103.046
#> 2     2 -1.263225 2506.492
#> 3     3  1.352281 1712.572
#> 4     4 -2.418153 2724.410
#> 5     5  1.088375 2365.488

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] -126.7283

component.samples$lml - location.samples$lml
#> [1] 0.7817279

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".

We can view a summary of the favored model by passing a BayesMassBal object to the summary function. While not done in this case, the summary table can be saved by passing the desired name of a *.csv file to the export argument.

summary(component.samples, export = NA)
#> Mass Flow Rates:
#> 
#> CuFeS2:
#> --------------------
#>  Sampling Location Expected Value     95% LB     95% UB
#>                  1      1.1976820 1.14864886 1.24366309
#>                  2      1.1724315 1.12335717 1.21938312
#>                  3      1.1095675 1.05916172 1.16087921
#>                  4      0.0252505 0.01670985 0.03423872
#>                  5      0.0628640 0.05352902 0.07428994
#> 
#> gangue:
#> --------------------
#>  Sampling Location Expected Value     95% LB      95% UB
#>                  1    100.2170669 95.2270032 105.1671009
#>                  2      6.6722496  5.8571286   7.5356527
#>                  3      0.2608555  0.2261483   0.2941051
#>                  4     93.5448172 88.5328497  97.8662773
#>                  5      6.4113941  5.6036402   7.2517879
#> 
#> Total:
#> --------------------
#>  Sampling Location Expected Value    95% LB     95% UB
#>                  1     101.414749 96.407736 106.350292
#>                  2       7.844681  7.037650   8.736746
#>                  3       1.370423  1.309481   1.431934
#>                  4      93.570068 88.554228  97.885080
#>                  5       6.474258  5.659293   7.307979
#> 
#> 
#> log-marginal likelihood:
#> -66.0614765716421

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)

opar <- par(no.readonly =TRUE) 
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")

Two_Node_Process

Scott Koermer

2022-06-17