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## calculates the kernel maximum mean discrepancy for samples from two distributions
## author: alexandros karatzoglou
setGeneric("kmmd",function(x,...) standardGeneric("kmmd"))
setMethod("kmmd", signature(x = "matrix"),
function(x, y, kernel="rbfdot",kpar="automatic", alpha = 0.05, asymptotic = FALSE, replace = TRUE, ntimes = 150, frac = 1, ...)
{
x <- as.matrix(x)
y <- as.matrix(y)
res <- new("kmmd")
if(is.character(kernel)){
kernel <- match.arg(kernel,c("rbfdot","polydot","tanhdot","vanilladot","laplacedot","besseldot","anovadot","splinedot","matrix"))
if(kernel == "matrix")
if(dim(x)[1]==dim(x)[2])
return(kmmd(x= as.kernelMatrix(x), y = y, Kxy = as.kernelMatrix(x)%*%y, alpha = 0.05, asymptotic = FALSE, replace = TRUE, ntimes = 100, frac = 1, ...))
else
stop(" kernel matrix not square!")
if(is.character(kpar))
if((kernel == "tanhdot" || kernel == "vanilladot" || kernel == "polydot"|| kernel == "besseldot" || kernel== "anovadot"|| kernel=="splinedot") && kpar=="automatic" )
{
cat (" Setting default kernel parameters ","\n")
kpar <- list()
}
}
if (!is.function(kernel))
if (!is.list(kpar)&&is.character(kpar)&&(is(kernel, "laplacedot")|| kernel=="rbfdot")){
kp <- match.arg(kpar,"automatic")
if(kp=="automatic")
kpar <- list(sigma=sigest(rbind(x,y),scaled=FALSE)[2])
cat("Using automatic sigma estimation (sigest) for RBF or laplace kernel","\n")
}
if(!is(kernel,"kernel"))
{
if(is(kernel,"function")) kernel <- deparse(substitute(kernel))
kernel <- do.call(kernel, kpar)
}
if(!is(kernel,"kernel")) stop("kernel must inherit from class `kernel'")
m <- dim(x)[1]
n <- dim(y)[1]
N <- max(m,n)
M <- min(m,n)
Kxx <- kernelMatrix(kernel,x)
Kyy <- kernelMatrix(kernel,y)
Kxy <- kernelMatrix(kernel,x,y)
resmmd <- .submmd(Kxx, Kyy, Kxy, alpha)
H0(res) <- (resmmd$mmd1 > resmmd$D1)
Radbound(res) <- resmmd$D1
Asymbound(res) <- 0
mmdstats(res)[1] <- resmmd$mmd1
mmdstats(res)[2] <- resmmd$mmd3
if(asymptotic){
boundA <- .submmd3bound(Kxx, Kyy, Kxy, alpha, frac, ntimes, replace)
AsympH0(res) <- (resmmd$mmd3 > boundA)
Asymbound(res) <- boundA
}
kernelf(res) <- kernel
return(res)
})
setMethod("kmmd",signature(x="list"),
function(x, y, kernel="stringdot",kpar=list(type="spectrum",length=4), alpha = 0.05, asymptotic = FALSE, replace = TRUE, ntimes = 150, frac = 1, ...)
{
if(!is(kernel,"kernel"))
{
if(is(kernel,"function")) kernel <- deparse(substitute(kernel))
kernel <- do.call(kernel, kpar)
}
if(!is(kernel,"kernel")) stop("kernel must inherit from class `kernel'")
Kxx <- kernelMatrix(kernel,x)
Kyy <- kernelMatrix(kernel,y)
Kxy <- kernelMatrix(kernel,x,y)
ret <- kmmd(x=Kxx,y = Kyy,Kxy=Kxy, alpha=alpha, asymptotic= asymptotic, replace = replace, ntimes = ntimes, frac= frac)
kernelf(ret) <- kernel
return(ret)
})
setMethod("kmmd",signature(x="kernelMatrix"), function (x, y, Kxy, alpha = 0.05, asymptotic = FALSE, replace = TRUE, ntimes = 100, frac = 1, ...)
{
res <- new("kmmd")
resmmd <- .submmd(x, y, Kxy, alpha)
H0(res) <- (resmmd$mmd1 > resmmd$D1)
Radbound(res) <- resmmd$D1
Asymbound(res) <- 0
mmdstats(res)[1] <- resmmd$mmd1
mmdstats(res)[2] <- resmmd$mmd3
if(asymptotic){
boundA <- .submmd3bound(x, y, Kxy, alpha, frac, ntimes, replace)
AsympH0(res) <- (resmmd$mmd1 > boundA)
Asymbound(res) <- boundA
}
kernelf(res) <- " Kernel matrix used as input."
return(res)
})
.submmd <- function(Kxx,Kyy, Kxy, alpha)
{
m <- dim(Kxx)[1]
n <- dim(Kyy)[1]
N <- max(m,n)
M <- min(m,n)
sumKxx <- sum(Kxx)
if(m!=n)
sumKxxM <- sum(Kxx[1:M,1:M])
else
sumKxxM <- sumKxx
dgxx <- diag(Kxx)
sumKxxnd <- sumKxx - sum(dgxx)
R <- max(dgxx)
RM <- max(dgxx[1:M])
hu <- colSums(Kxx[1:M,1:M]) - dgxx[1:M]
sumKyy <- sum(Kyy)
if(m!=n)
sumKyyM <- sum(Kyy[1:M,1:M])
else
sumKyyM <- sumKyy
dgyy <- diag(Kyy)
sumKyynd <- sum(Kyy) - sum(dgyy)
R <- max(R,dgyy)
RM <- max(RM,dgyy[1:M]) # RM instead of R in original
hu <- hu + colSums(Kyy[1:M,1:M]) - dgyy[1:M]
sumKxy <- sum(Kxy)
if (m!=n)
sumKxyM <- sum(Kxy[1:M,1:M])
else
sumKxyM <- sumKxy
dg <- diag(Kxy) # up to M only
hu <- hu - colSums(Kxy[1:M,1:M]) - colSums(t(Kxy[1:M,1:M])) + 2*dg # one sided sum
mmd1 <- sqrt(max(0,sumKxx/(m*m) + sumKyy/(n*n) - 2/m/n* sumKxy))
mmd3 <- sum(hu)/M/(M-1)
D1 <- 2*sqrt(RM/M)+sqrt(log(1/alpha)*4*RM/M)
return(list(mmd1=mmd1,mmd3=mmd3,D1=D1))
}
.submmd3bound <- function(Kxx,Kyy, Kxy, alpha, frac, ntimes, replace)
{
## implements the bootstrapping approach to the MMD3 bound by shuffling
## the kernel matrix
## frac : fraction of data used for bootstrap
## ntimes : how many times MMD is to be evaluated
m <- dim(Kxx)[1]
n <- dim(Kyy)[1]
M <- min(m,n)
N <- max(m,n)
poslabels <- 1:m
neglabels <- (m+1):(m+n)
## bootstrap
bootmmd3 <- rep(0,ntimes)
for (i in 1:ntimes)
{
nsamples <- ceiling(frac*min(m,n))
xinds <- sample(1:m,nsamples,replace=replace)
yinds <- sample(1:n,nsamples,replace=replace)
newlab <- c(poslabels[xinds],neglabels[yinds])
samplenew <- sample(newlab, length(newlab), replace=FALSE)
xinds <- samplenew[1:nsamples]
yinds <- samplenew[(nsamples+1):length(samplenew)]
newm <- length(xinds)
newn <- length(yinds)
newM <- min(newm,newn)
##get new kernel matrices (without concat to big matrix to save memory)
xind1 <- xinds[xinds<=m]
xind2 <- xinds[xinds>m]- m
yind1 <- yinds[yinds<=m]
yind2 <- yinds[yinds>m]-m
##Kxx (this should be implemented with kernelMult for memory efficiency)
nKxx <- rbind(cbind(Kxx[xind1,xind1],Kxy[xind1,xind2]), cbind(t(Kxy[xind1,xind2]),Kyy[xind2,xind2]))
dgxx <- diag(nKxx)
hu <- colSums(nKxx[1:newM,1:newM]) - dgxx[1:newM] # one sided sum
rm(nKxx)
#Kyy
nKyy <- rbind(cbind(Kxx[yind1,yind1],Kxy[yind1,yind2]), cbind(t(Kxy[yind1,yind2]), Kyy[yind2,yind2]))
dgyy <- diag(nKyy)
hu <- hu + colSums(nKyy[1:newM,1:newM]) - dgyy[1:newM]
rm(nKyy)
## Kxy
nKxy <- rbind(cbind(Kxx[yind1,xind1],Kxy[yind1,xind2]), cbind(t(Kxy[xind1,yind2]),Kyy[yind2,xind2]))
dg <- diag(nKxy)
hu <- hu - colSums(nKxy[1:newM,1:newM]) - colSums(t(nKxy[1:newM,1:newM])) + 2*dg
rm(nKxy)
## now calculate mmd3
bootmmd3[i] <- sum(hu)/newM/(newM-1)
}
bootmmd3 <- sort(bootmmd3, decreasing=TRUE);
aind <- floor(alpha*ntimes) ## better less than too much (-> floor);
## take threshold in between aind and the next smaller value:
bound <- sum(bootmmd3[c(aind,aind+1)])/2;
return(bound)
}
setMethod("show","kmmd",
function(object){
cat("Kernel Maximum Mean Discrepancy object of class \"kmmd\"","\n","\n")
show(kernelf(object))
if(is.logical(object@H0)){
cat("\n")
cat("\n","H0 Hypothesis rejected : ", paste(H0(object)))
cat("\n","Rademacher bound : ", paste(Radbound(object)))
}
cat("\n")
if(Asymbound(object)!=0){
cat("\n","H0 Hypothesis rejected (based on Asymptotic bound): ", paste(AsympH0(object)))
cat("\n","Asymptotic bound : ", paste(Asymbound(object)))
}
cat("\n","1st and 3rd order MMD Statistics : ", paste( mmdstats(object)))
cat("\n")
})
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