chapter8_MDS_multmin.Rnw

\chapter{Multiple Minima in  Multidimensional Scaling }
\label{sec:MultMinima}
\chaptermark{Multiple Minima in  Multidimensional Scaling}


The approximate values of the dissimilarity matrix is shown in \autoref{table:diss_mat}.

\begin{table}[ht]
\centering
\ttfamily
\begin{tabular}{r|rrrrrrr}
  \hline
 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 
  \hline
  1 & 0.00 & 1.00 & 1.41 & 1.00 & 0.71 &1.00 & 1.00  \\ 
  2 & 1.00 & 0.00 & 1.00 & 1.41 & 0.71 & 0.00 & 0.01 \\ 
  3 & 1.41 & 1.00 & 0.00 & 1.00 & 0.71 & 1.00 & 1.00  \\ 
  4 & 1.00 & 1.41 & 1.00 & 0.00 & 0.71 & 0.01 & 0.00  \\ 
  5 & 0.71 & 0.71 & 0.71 & 0.71 & 0.00 & 0.71 & 0.71   \\ 
  6 & 1.00 & 0.00 & 1.00 & 0.01 & 0.71 & 0.00 & 1.41  \\ 
  7 & 1.00 & 0.01 & 1.00 & 0.00 & 0.71 & 1.41 & 0.00  \\ 
  
   \hline
\end{tabular}

\caption{The entries of the dissimilarity matrix (rounded to two decimal digits)\label{table:diss_mat}}
\end{table}



<<multMinMDS,eval=FALSE,results='hide',tidy=FALSE,echo=FALSE,error=TRUE,cache=TRUE>>=
verbose=TRUE
root.dir<-Sys.getenv("PROJECT_DIR")
setwd(root.dir)
source("./JOFC_MatchDetect/lib/smacofM.R")
source("./JOFC_MatchDetect/lib/oosIM.R")
source("./JOFC_MatchDetect/lib/multipleMinimaTest_fn.R")

require(ggplot2)

#
# Create point configuration
#

library(ggplot2)
n<-7
X<-matrix(0,n,2)
X[,1]<-c(0,1,1,0,0,0,0.5)#1.01)
X[,2]<-c(0,0,1,1,0,0,0.5)#,0)
X[5,]<-c(1,0)
X[6,]<-c(0,1)
#X[8,]<-c(0,1.0)
in.sample.ind<-c(1:4,7)#,8)
d.X<- dist(X)

epsilon <-1E-3
w <- 0.99

#
# Perturb Dissimilarity matrix
#
i <- 4
j <- 5

ind <- (n*(i-1) - i*(i-1)/2 + j-i)

d.X[ind]<-d.X[ind]-1.4

i<-2
j<-6
ind<-(n*(i-1) - i*(i-1)/2 + j-i)

d.X[ind]<- d.X[ind]-1.4
#X[5,]<-c(0,1)


i<-5
j<-7
ind<-(n*(i-1) - i*(i-1)/2 + j-i)

#d.X[ind]<- d.X[ind]+0.707

i<-5
j<-8
ind<-(n*(i-1) - i*(i-1)/2 + j-i)

#d.X[ind]<- d.X[ind]-1.4

if (verbose) print(d.X)

init.config<-X

w.vals<-c(0.1,seq(0.2,0.4,0.1),seq(0.41,0.54,0.01),seq(0.55,0.7,0.05),seq(0.75,0.85,0.01),seq(0.90,0.95,0.05),0.99)
w.vals.sp<-w.vals[c(1,13,14,15,21,26:31,35,36)]
grid.seq.x<-seq(-0.5,1.5,0.1)
grid.seq.y<-seq(-0.2,1.6,0.1)

x.grid.len <- length(grid.seq.x)
y.grid.len <- length(grid.seq.y)
grid.pts.count<- x.grid.len *y.grid.len 

#
# Define empty arrays for results
#
final.close.to.init.w<-rep(0, length(w.vals))

dir.deriv<-array(0,dim=c(length(w.vals),x.grid.len,length(grid.seq.y),2*n))
hessian.at.pt<-array(list(c(1,1,2,1)),dim=c(length(w.vals),x.grid.len,y.grid.len))
sign.hessian.at.pt<-array(0,dim=c(length(w.vals),x.grid.len,y.grid.len))
stress.at.loc.w <-sign.hessian.at.pt 

min.config.stress.1.w<- rep(0, length(w.vals))
min.config.stress.2.w<- rep(0, length(w.vals))


close.to.init.X5.for.w <- list()
close.to.init.X6.for.w <- list()
far.to.init.X5.for.w   <- list()
far.to.init.X6.for.w   <- list()



final.coords.x.5.w<-array(0,dim=c(length(w.vals),x.grid.len,y.grid.len))
final.coords.y.5.w<-array(0,dim=c(length(w.vals),x.grid.len,y.grid.len))
final.coords.x.6.w<-array(0,dim=c(length(w.vals),x.grid.len,y.grid.len))
final.coords.y.6.w<-array(0,dim=c(length(w.vals),x.grid.len,y.grid.len))
config.points.agg<-data.frame(x=numeric(0),y=numeric(0),pt.name=character(0),w=numeric(0))


for (w.i in 1:length(w.vals)){
  
  stress.at.loc<-array(0,dim=c(x.grid.len,y.grid.len))
  
  #Set up weight vector
  w <- w.vals[w.i]
  W <- matrix(1-w,n,n)
  W[5,2] <- W[2,5]<- w
  W[6,4] <- W[4,6]<- w
  W[5,7] <- W[7,5]<- 1-w
  diag(W)<-0
  
  
  W.oos <- W
  W.oos[in.sample.ind, in.sample.ind]<-0  
  new.index.order<- c(in.sample.ind,c(5,6))
  W.oos<-W.oos[new.index.order,new.index.order]
  
  
  
  in.sample.Bool<-(1:n %in% in.sample.ind)
  X.embed.1.in <- smacofM(as.matrix(d.X)[in.sample.ind,in.sample.ind],
                          ndim    = 2,
                          W       = W[in.sample.ind,in.sample.ind],
                          init    = init.config[in.sample.Bool,]  	,
                          verbose = FALSE,
                          itmax   = 1000,
                          eps     = 1e-6)
  
  X.embed.1.oos <- oosIM(D=as.matrix(d.X),
                         X=X.embed.1.in,
                         init= init.config[!in.sample.Bool,],
                         W=W.oos,
                         verbose = FALSE,
                         itmax   = 1000,
                         eps     = 1e-6,
                         bwOos = TRUE,
                         isWithin = ifelse(in.sample.Bool,1,0) )
  
  row.names(X.embed.1.oos)<-NULL
  row.names(X.embed.1.in)<-NULL
  X.embed.1.norm <- rbind(X.embed.1.in[1:4,] ,X.embed.1.oos,X.embed.1.in[5:(n-2),]) 
  if (verbose) print("Init config ")
  if (verbose) print(X.embed.1.norm)
  stress <- raw.stress.at(X.embed.1.norm)
  if (verbose) print("Init config  stress")
  if (verbose)  print(stress)
  close.to.init.1<-data.frame(x=numeric(),y=numeric())
  close.to.init.2<-data.frame(x=numeric(),y=numeric())
  far.to.init.1<-data.frame(x=numeric(),y=numeric())
  far.to.init.2<-data.frame(x=numeric(),y=numeric())
  min.stress.1<-100
  min.stress.2<-100
  
  
  
  grid.resp<-array(0,dim=c(x.grid.len,y.grid.len))
  grid.coords<-array(0,dim=c(x.grid.len,y.grid.len))
  
  
  mesh.grid.coords<-meshgrid(grid.seq.x,grid.seq.y)
  
  
  final.coords.x.5<-array(0,dim=c(x.grid.len,y.grid.len))
  final.coords.y.5<-array(0,dim=c(x.grid.len,y.grid.len))
  final.coords.x.6<-array(0,dim=c(x.grid.len,y.grid.len))
  final.coords.y.6<-array(0,dim=c(x.grid.len,y.grid.len))
  min.config.1<-matrix(0,n,2)
  min.config.2<-matrix(0,n,2)
  
  for (j in 1:y.grid.len){
    
    for (i in 1:x.grid.len){
      init.config <- X
      
      i.x<-grid.seq.x[i]
      j.y<-grid.seq.y[j]
      #i.x<-mesh.grid.coords$x[i,j] 
      #j.y<-mesh.grid.coords$y[i,j] 
      
      init.config[5,]<-c(i.x,j.y)
      init.config[6,]<-c(1-i.x,1-j.y)
      
      X.embed.2.in<-smacofM(as.matrix(d.X)[in.sample.ind, in.sample.ind],
                            ndim    = 2,
                            W       = W[in.sample.ind,in.sample.ind],
                            init    = init.config[in.sample.Bool,],
                            verbose = FALSE,
                            itmax   = 1000,
                            eps     = 1e-6)
      
      X.embed.2.oos<-oosIM(D = as.matrix(d.X),
                           X = X.embed.2.in,
                           init = init.config[!in.sample.Bool,],
                           
                           verbose =  FALSE,#(i<=(x.grid.len/2)),
                           itmax   = 1000,
                           eps     = 1e-6,
                           W = W.oos,
                           bwOos = TRUE,
                           isWithin = ifelse(in.sample.Bool,1,0))
      
      
      
      
      row.names(X.embed.2.oos)<-NULL
      row.names(X.embed.2.in)<-NULL
      
      X.embed.2.norm<- rbind(X.embed.2.in[1:4,] ,X.embed.2.oos,X.embed.2.in[5:(n-2),])		
      
      hess.mat<- hessian.mat(X.embed.2.norm,n)
      hess.mat[hess.mat==0]<-t(hess.mat)[hess.mat==0]
      
      hessian.at.pt[[w.i,i,j]]<- as.list(hess.mat)
      
      hess.mat<-hess.mat[9:12,9:12]
      hess.eig<- eigen(hess.mat,symmetric=TRUE,only.values=TRUE)
      e.vals<- hess.eig$values
      low.than.thres<- abs(e.vals)<1E-5
      e.vals[low.than.thres] <-  0 #sign(e.vals[low.than.thres])*1E-5 
      
      if (sum(e.vals<0)==0){
        if (sum((e.vals==0)>0)){
          sign.hessian.at.pt[w.i,i,j] <- 2
        } else{
          sign.hessian.at.pt[w.i,i,j] <- 1 #pos definite
        }
      } else if (sum(e.vals>0)==0) {
        if  (sum((e.vals==0)>0)){
          sign.hessian.at.pt[w.i,i,j] <- -2 #neg definite
        } else{
          sign.hessian.at.pt[w.i,i,j] <- -1 #neg definite
          
        }
        
      } else{
        sign.hessian.at.pt[w.i,i,j] <- 0  #neither pos nor neg definite  saddle point
      }
      final.coords.x.5[i,j]<-X.embed.2.norm[5,1]
      final.coords.y.5[i,j]<-X.embed.2.norm[5,2]
      final.coords.x.6[i,j]<-X.embed.2.norm[6,1]
      final.coords.y.6[i,j]<-X.embed.2.norm[6,2]
      #print("X.embed.2.norm")
      #print(X.embed.2.norm)
      stress <- sum(as.dist(W)*((dist(X.embed.2.norm)-d.X)^2))
      stress.unif.wt <- sum(((dist(X.embed.2.norm)-d.X)^2))
      stress.unif.abs <- sum(abs(dist(X.embed.2.norm)-d.X))
      #print(stress)
      #	print(stress.unif.wt)
      #	print(stress.unif.abs)		
      if (((X.embed.2.norm[6,2]>0.5)) & 
            (((X.embed.2.norm[5,1]>X.embed.2.norm[6,1])))) { 
        #& (sum(X.embed.2.norm[6,]<0.3)==1))
        
        
        grid.resp[i,j]<-1
        #	print("First Test Point")
        #	print(X.embed.2.norm[5,])
        if (stress < min.stress.1) {
          min.stress.1<-stress
          if (verbose)	print("Min stress found(real min)")
          if (verbose) print(X.embed.2.norm)
          min.config.1<-X.embed.2.norm
          if (verbose) print(min.stress.1)
          
        }
        close.to.init.1<-rbind(close.to.init.1,X.embed.2.norm[5,])
        close.to.init.2<-rbind(close.to.init.2,X.embed.2.norm[6,])
        
        
      } else{
        grid.resp[i,j]<-0
        #print("First Test Point")
        if (verbose) print(X.embed.2.norm[5,])		
        if (stress < min.stress.2){
          min.stress.2<-stress
          if (verbose) print("Min stress found(second min)")
          if (verbose) print(X.embed.2.norm)
          if (verbose) 	print(min.stress.2)
          min.config.2<-X.embed.2.norm
          
        }
        far.to.init.1<-rbind(far.to.init.1,X.embed.2.norm[5,])
        far.to.init.2<-rbind(far.to.init.2,X.embed.2.norm[6,])
        
        #} else{
        #	grid.resp[i,j] <- NA
      }
      
      if (w.i>9) {
        print("stress")
        print(stress)
        print(as.dist(W))
      }
      stress.at.loc[i,j]<- stress
      
      
      
      
    }
    
  }
  
  stress.at.loc.w[w.i,,] <-stress.at.loc
  final.coords.x.5.w[w.i,,]=final.coords.x.5
  final.coords.x.6.w[w.i,,]=final.coords.x.6
  final.coords.y.5.w[w.i,,]=final.coords.y.5
  final.coords.y.6.w[w.i,,]=final.coords.y.6
  
  #plot(x=grid.seq.x,y=grid.seq.y, col=grid.resp)
  #x.coords = [x1 x2 x3 x4 ... x1 x2 x3 x4 ...
  x.coords <- rep(grid.seq.x,y.grid.len)
  #y.coords = [y1 y1 y1 y1 ... y2 y2 y2 y2 ...
  y.coords <- rep(grid.seq.y,each=x.grid.len)
  #grid.resp<-grid.resp[x.grid.lenA:1,]
  
#  if (w %in% w.vals.sp){
    #if (run.in.linux) {X11()} else {windows()}
#    plot(x.coords, y.coords,
         #plot(unmatrix(mesh.grid.coords$x,byrow=FALSE),
         #     unmatrix(mesh.grid.coords$y,byrow=FALSE),
#         col=ifelse(unmatrix(grid.resp,byrow=FALSE)==1,"red","black"))
#  }
  print("For w.value")
  print(w)
  print("Min stress found(real min)")
  if (verbose) print(min.config.1)
  print(min.stress.1)
  print("Min stress found(second min)")
  if (verbose) print(min.config.2)
  print(min.stress.2)
  min.config.stress.1.w[w.i]<-min.stress.1
  min.config.stress.2.w[w.i]<-min.stress.2
  
  
#   if (!is.vector(close.to.init.1)){
#     config.points <- cbind(close.to.init.1,rep("X_6",grid.pts.count))
#     config.points <- rbind(config.points,cbind(close.to.init.2,rep("X_7",grid.pts.count)) )
#     config.points<- data.frame(x= config.points[,1],y=config.points[,2],pt.name=config.points[,3],w=w.vals[w.i])
#     config.points.agg <- rbind(config.points.agg,config.points)
#   }
#   
  
#   if (!is.vector(far.to.init.1)){
#     config.points <- cbind(far.to.init.1,rep("X_6",grid.pts.count))
#     config.points <- rbind(config.points,cbind(far.to.init.2,rep("X_7",grid.pts.count)) )
#     config.points<- data.frame(x= config.points[,1],y=config.points[,2],pt.name=config.points[,3],w=w.vals[w.i])
#     config.points.agg <- rbind(config.points.agg,config.points)
#   }

  print("Number of final config close to initial config")
  print(dim(close.to.init.1)[1])
  final.close.to.init.w[w.i] <- dim(close.to.init.1)[1]
  
  print("Number of final config far to initial config")
  print(dim(far.to.init.1)[1])
  min.config.stress.1.w[w.i]<- min.stress.1
  min.config.stress.2.w[w.i]<- min.stress.2
  
  if (plot.in.3d){
    
    #if (!w%in%w.vals.sp) rgl.close()
    
  }
  
  if (nrow(far.to.init.1)==0) { far.to.init.1<-NULL}
  if (nrow(far.to.init.2)==0) { far.to.init.2<-NULL}
  if (nrow(close.to.init.1)==0) { close.to.init.1<-NULL}
  if (nrow(close.to.init.2)==0) { close.to.init.2<-NULL}
  
  far.to.init.X5.for.w<- c(far.to.init.X5.for.w, list(far.to.init.1)) 
  far.to.init.X6.for.w<- c(far.to.init.X6.for.w, list(far.to.init.2))
  close.to.init.X5.for.w <- c(close.to.init.X5.for.w, list(close.to.init.1))
  close.to.init.X6.for.w <- c(close.to.init.X6.for.w, list(close.to.init.2)) 
  print("length(far.to.init.X5.for.w)")
  print(length(far.to.init.X5.for.w))  


}
@


<<plot_final_pts_all_w, eval=FALSE, fig.show='asis',results='hide',echo=FALSE,fig.cap="Embedded Point Pairs ($\\hat{X}_6$ and $\\hat{X}_7$) for all initial configurations for different $w$ values">>=
library(ggplot2)
 
  all.final.pts.lf <- data.frame(x=numeric(0),y=numeric(0),pt.name=character(0),close.or.far=character(0),w=numeric(0))
  
  far.to.init.X5.for.w.array <- array(numeric(0),dim=c(0,3))
  pt.name.var <- c("X_6","X_7","X_6","X_7")
  close.or.far.var = c("far","far","close","close")
  data.list<-list(far.to.init.X5.for.w  ,far.to.init.X6.for.w,
                  close.to.init.X5.for.w,close.to.init.X6.for.w)
  
  for (j in 1:4){
  pt.coords<- data.list[[j]]
  pt.name.var.j <- pt.name.var[j]
  close.or.far.var.j <-close.or.far.var[j]
  for (w.i in 1:length(w.vals)){
    final.coords <- pt.coords[[w.i]]
    
   
    if (!is.null(final.coords)){
      colnames(final.coords)<-c("x","y")
      num.pts<-nrow(final.coords)
      new.final.pts.lf <- cbind(final.coords,pt.name=rep(pt.name.var.j,num.pts)
            ,close.or.far=rep(close.or.far.var.j,num.pts),w=rep(w.vals[w.i],num.pts))
      all.final.pts.lf <- rbind(all.final.pts.lf ,new.final.pts.lf)
      
    }
  }
  }
  
  
sub.final.pts.lf.logical<-all.final.pts.lf$w %in% c(0.1,0.5,0.51,0.52,0.8,0.81,0.82,0.84)
sub.final.pts.lf<- all.final.pts.lf[sub.final.pts.lf.logical,]

all.final.pts.lf$w<- as.factor(all.final.pts.lf$w)
sub.final.pts.lf$w<- as.factor(sub.final.pts.lf$w)

levels(sub.final.pts.lf$w)<- paste("w = ",levels(sub.final.pts.lf$w))
g1<-ggplot(sub.final.pts.lf,aes(x=x,y=y,colour=pt.name))+geom_point(alpha = 1/5) +scale_shape(solid = FALSE) + facet_wrap(~w,ncol=2)
print(g1)
@


<<config_w_ani,eval=FALSE, fig.show='hide',echo=FALSE,out.width='6in',fig.cap="Embedded Point Pairs ($\\hat{X}_6$ and $\\hat{X}_7$) for all initial configurations for different $w$ values">>=
for (w.it in w.vals){
sub.final.pts.lf.logical<-all.final.pts.lf$w %in% w.it
sub.final.pts.lf<- all.final.pts.lf[sub.final.pts.lf.logical,]
levels(sub.final.pts.lf$w)<- paste("w=",levels(sub.final.pts.lf$w))

g1<-ggplot(sub.final.pts.lf,aes(x=x,y=y,colour=pt.name))+geom_point(alpha = 1/5)+ 
scale_shape(solid = FALSE) + xlim(-0.1, 1.1)+ylim(-0.1, 1.1) + labs(title=paste("w=", format(w.it,digits=2),collapse = "")) +
  theme(legend.position="bottom")

print(g1)
}
@





<<more_2d_3d_plots,eval=FALSE,results='hide',echo=FALSE>>=


for (w.i in 1:length(w.vals)){
  
  par(pch=1)
  plot(x=close.to.init.1[,1],y=close.to.init.1[,2],col="red",
       xlim=c(min(close.to.init.1[,1],close.to.init.2[,1]),max(close.to.init.1[,1],close.to.init.2[,1]))
       ,ylim=c(min(close.to.init.1[,2],close.to.init.2[,2]),max(close.to.init.1[,2],close.to.init.2[,2])))
  
  par(pch=3)
  points(x=close.to.init.2[,1],y=close.to.init.2[,2],col="blue")
  title(paste("Final config Close to true config- w=",w,collapse=""))
  legend("topright",legend=c(expression(X[6]),expression(X[7])),col=c("red","blue"),pch=c(1,3))
  
  dev.print(paste(results.dir,"/","true-min-w",w,".png",collapse="",sep=""),device=png,width=600,height=600)
  fname<-paste(results.dir,"/","true-min-w",w,".pdf",collapse="",sep="")
  dev.copy2pdf(file=fname)
  dev.off()
  
  #select.x<- sort( sample.int(x.grid.len , 10))
  select.x<-1:x.grid.len 
  #select.y <- sort( sample.int(y.grid.len , 10))
  select.y<-1:y.grid.len 
  #The indexing (select.x,select.y) is mixed because mesh.grid function generates a matrix 
  #whose columns are for x coordinates, while for final.coords, rows are for x coordinates
  
  #}
  
  if(!is.vector(far.to.init.1)){
    #if (run.in.linux) {X11()} else {windows()}
    par(pch=1)
    plot(x=far.to.init.1[,1],y=far.to.init.1[,2],col="red",
         xlim=c(min(grid.seq.x),max(grid.seq.x))
         ,ylim=c(min(grid.seq.y),max(grid.seq.y)))
    par(pch=3)
    points(x=far.to.init.2[,1],y=far.to.init.2[,2],col="blue")
    title(paste("Final config Far to true config- w=",w,collapse=""))
    legend("topright",legend=c(expression(X[6]),expression(X[7])),col=c("red","blue"),pch=c(1,3))
    
    dev.print(paste(results.dir,"/","other-min-w",w,".png",collapse="",sep=""),device=png, width=600,height=600)
    fname<-paste(results.dir,"/","other-min-w",w,".pdf",collapse="",sep="")
    dev.copy2pdf(file=fname)
    
  }
  
  print("Number of final config close to initial config")
  print(dim(close.to.init.1)[1])
  final.close.to.init.w[w.i] <- dim(close.to.init.1)[1]
  
  print("Number of final config far to initial config")
  print(dim(far.to.init.1)[1])
  min.config.stress.1.w[w.i]<- min.stress.1
  min.config.stress.2.w[w.i]<- min.stress.2
  
  if (plot.in.3d){
    
    #if (!w%in%w.vals.sp) rgl.close()
    
  }
  
  if (nrow(far.to.init.1)==0) { far.to.init.1<-NULL}
  if (nrow(far.to.init.2)==0) { far.to.init.2<-NULL}
  if (nrow(close.to.init.1)==0) { close.to.init.1<-NULL}
  if (nrow(close.to.init.2)==0) { close.to.init.2<-NULL}
  
  far.to.init.X5.for.w<- c(far.to.init.X5.for.w, list(far.to.init.1)) 
  far.to.init.X6.for.w<- c(far.to.init.X6.for.w, list(far.to.init.2))
  close.to.init.X5.for.w <- c(close.to.init.X5.for.w, list(close.to.init.1))
  close.to.init.X6.for.w <- c(close.to.init.X6.for.w, list(close.to.init.2)) 
  print("length(far.to.init.X5.for.w)")
  print(length(far.to.init.X5.for.w))
  
  
  
}


min.config.stress.w.table<-rbind(min.config.stress.1.w,min.config.stress.2.w)

colnames(min.config.stress.w.table) <- w.vals
row.names(min.config.stress.w.table) <- c("Local min for real config.","Alternative local min")
#if (plot.in.3d){ rgl.close()}
#graphics.off()



if (create.ani){
  saveGIF(animate.config.w(),"config_w.gif")
  saveGIF(animate.final.loc.stress.w(),"final_loc_vals_w.gif")
  saveMovie(animate.final.loc.stress.w(),movie.name="final_loc_vals_w.mov")
  #saveVideo(animate.final.loc.stress.w(),movie.name="final_loc_vals_w.avi")
  
  open3d(windowRect=c(0,0,480,480))
  play3d(stress.plot3d,duration=65,dev = rgl.cur(),sign.hessian.at.pt=sign.hessian.at.pt,
         x.coords=x.coords,y.coords=y.coords,
         stress.at.loc=stress.at.loc.w,
         grid.seq.x=grid.seq.x,grid.seq.y=grid.seq.y,w.vals=w.vals,rotate.z.angle=pi/3)
  
  movie3d(stress.plot3d,duration=65,dev = rgl.cur(),fps=fps,sign.hessian.at.pt=sign.hessian.at.pt,
          x.coords=x.coords,y.coords=y.coords,
          stress.at.loc=stress.at.loc.w,
          grid.seq.x=grid.seq.x,grid.seq.y=grid.seq.y,w.vals=w.vals, rotate.z.angle=pi/3,
          dir=file.path(getwd(),"./cache/"),type="mov")
  
}

@

\begin{figure}
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[scale=0.41]{true-min-w-0_1.pdf}

\label{fig:Finalconfig-MultMin-w-0_1_a}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[scale=0.41]{other-min-w-0_1.pdf}

\label{fig:Finalconfig-MultMin-w-0_1_b}
\end{minipage}

\caption{Final configurations for different initial configurations, $w=0.1$ }
\label{fig:Finalconfig-MultMin-w-0_1}


\end{figure}




\begin{figure}
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[scale=0.41]{true-min-w0_5}

\label{fig:Finalconfig-MultMin-w-0_5_a}

\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[scale=0.41]{other-min-w0_5.pdf}

\label{fig:Finalconfig-MultMin-w-0_5_b}

\end{minipage}

\caption{Final configurations for different initial configurations, $w=0.5$ }
\label{fig:Finalconfig-MultMin-w-0_5}

\end{figure}

\begin{figure}
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[scale=0.41]{true-min-w0_8.pdf}
\label{fig:Finalconfig-MultMin-w-0_8_a}


\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[scale=0.41]{other-min-w0_8.pdf}
\label{fig:Finalconfig-MultMin-w-0_8_b}


\end{minipage}

\caption{Final configurations for different initial configurations, $w=0.8$ }
\label{fig:Finalconfig-MultMin-w-0_8}

\end{figure}



\begin{figure}
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[scale=0.41]{true-min-w0_81.pdf}


\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[scale=0.41]{other-min-w0_81.pdf}


\end{minipage}

\caption{Final configurations for different initial configurations, $w=0.81$ }
\label{fig:Finalconfig-MultMin-w-0_81}

\end{figure}




\begin{figure}
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[scale=0.41]{true-min-w0_84.pdf}

\label{fig:figure2-1}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[scale=0.41]{other-min-w0_84.pdf}


\end{minipage}

\caption{Final configurations for different initial configurations, $w=0.84$ }
\label{fig:Finalconfig-MultMin-w-0_84}

\end{figure}



% latex table generated in R 2.15.1 by xtable 1.7-0 package
% Sat Jan 05 00:54:38 2013

\begin{landscape}
\begin{table}[ht]
\def\h#1{\multicolumn{1}{p{3em}}{\mbox{}\hskip0pt #1}}
\begin{center}

\begin{tabular}{r|rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr}
\hline
$w$ & 0.1 & 0.2 & 0.3 & 0.4 & 0.41 & 0.42 & 0.43 & 0.44 & 0.45 & 0.46 & 0.47  \\ 
\hline
Local min for real config. & 2.80 & 2.51 & 2.22 & 1.92 & 1.89 & 1.86 & 1.83 & 1.80 & 1.77 & 1.74 & 1.71 \\ 
Alternative local min & 0.39 & 0.76 & 1.10 & 1.40 & 1.43 & 1.46 & 1.48 & 1.51 & 1.53 & 1.56 & 1.58 \\ 
\hline
\end{tabular}


\begin{tabular}{r|rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr}
\hline
$w$ & 0.48 & 0.49 & 0.5 & 0.51 & 0.52 & 0.53 & 0.54 & 0.55 & 0.6 & 0.65 & 0.7 \\ 
\hline
Local min for real config. &  1.68 & 1.65 & 1.62 & 1.59 & 1.56 & 1.53 & 1.50 & 1.47 & 1.32 & 1.17 & 1.01   \\ 
Alternative local min &  1.60 & 1.63 & 1.65 & 1.67 & 1.69 & 1.71 & 1.73 & 1.74 & 1.81 & 1.82 & 1.81  \\ 
\hline
\end{tabular}



\begin{tabular}{r|rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr}
\hline
$w$ & 0.75 & 0.76 & 0.77 & 0.78 & 0.79 & 0.8 & 0.81 & 0.82 & 0.83 & 0.84 & 0.85  \\ 
\hline
Local min for real config. &  0.86 & 0.82 & 0.79 & 0.76 & 0.73 & 0.70 & 0.66 & 0.63 & 0.60 & 0.57 & 0.53  \\ 
Alternative local min &   1.79 & 1.77 & 1.75 & 1.72 & 1.69 & 1.66 & 1.64 & NA & NA & NA & NA \\ 
\hline
\end{tabular}

\end{center}

\caption{Final stress values for the two local minima configurations\label{table:stress_val}}
\end{table}

\end{landscape}





\begin{comment}
%old results

\begin{table}[ht]
\begin{center}
\begin{tabular}{rrrrrr}
\hline
$w$ value & 0.1 & 0.45 & 0.5 & 0.55 & 0.99 \\ 
\hline
Local min for real config. & 2.80 & 1.77 & 1.62 & 1.47 & 0.04 \\ 
Alternative local min & 0.39 & 1.53 & 1.65 & 1.74 & NA \\ 
\hline
\end{tabular}
\end{center}
\label{table:stress_val}
\end{table}
\end{comment}

<<final_stress_plot,echo=FALSE,warning=FALSE,out.width='\\maxwidth',fig.cap='Final stress values vs $w$ for the two true and alternative local minima configurations'>>=
w.cache <- c(
  0.1, 0.2, 0.3, 0.4, 0.41, 0.42, 0.43, 0.44, 0.45, 0.46, 0.47
 , 0.48, 0.49, 0.5, 0.51, 0.52, 0.53, 0.54, 0.55, 0.6, 0.65, 0.7 
 , 0.75, 0.76, 0.77, 0.78, 0.79, 0.8, 0.81, 0.82, 0.83, 0.84, 0.85    
  )
  min.config.stress.1.w.cache <-  c(
    2.80, 2.51, 2.22, 1.92, 1.89, 1.86, 1.83, 1.80, 1.77, 1.74, 1.71,  
    1.68, 1.65, 1.62, 1.59, 1.56, 1.53, 1.50, 1.47, 1.32, 1.17, 1.01,
    0.86, 0.82, 0.79, 0.76, 0.73, 0.70, 0.66, 0.63, 0.60, 0.57, 0.53  
  )
min.config.stress.2.w.cache <- c(
  0.39, 0.76, 1.10, 1.40, 1.43, 1.46, 1.48, 1.51, 1.53, 1.56, 1.58, 
  1.60, 1.63, 1.65, 1.67, 1.69, 1.71, 1.73, 1.74, 1.81, 1.82, 1.81,  
  1.79, 1.77, 1.75, 1.72, 1.69, 1.66, 1.64, NA, NA, NA, NA 
  
)

w.len.c<- length(w.cache)
stress.df<-data.frame (w.ind= 1:w.len.c, true.conf=min.config.stress.1.w.cache,alt.conf=min.config.stress.2.w.cache)
g2 <- ggplot(data=stress.df,aes(w.ind)) + geom_line(aes(y=true.conf,color="True Config"))+
  geom_line(aes(y=alt.conf,color="Alternative Config"))+
  scale_x_discrete("w",labels=w.cache)  + scale_y_continuous("Final Stress Value") +
  theme(text = element_text(size=13),
        axis.text.x = element_text(angle=90,hjust=0.5,vjust=0.5),
           legend.title = element_blank(), #element_text("Config Type")
           legend.position="bottom") 
print(g2)
@