Rのmvtnormパッケージの関数 pmvnorm() の引数 lower,upper の意味について

*R言語のmvtnormパッケージの多変数累積密度関数pmvnorm()を使用して、２次元の正規分布関数の任意の範囲の累積確率密度（確率）を取得したいのですが、この関数の引数lower,upper（ベクトルで指定）が何を意味しているのか分かりません。この引数の意味を教えて頂きたいですm(__)m

関数pmvnorm()のヘルプ
http://math.furman.edu/~dcs/courses/math47/R/library/mvtnorm/html/pmvnorm.html

以下実装中コードです。

library( MASS )     # MASS package（ピマ・インディアンPima のデータを使用）
library( mvtnorm )  # 多変量正規分布を扱う

# set options
options( digits=5 ) # 表示桁数

# Pima data expand on memory
data( Pima.tr )
data( Pima.te )

# copy data from Pima data
lstPimaTrain <- list(
  numNoDiabetes = 0,       # 糖尿病未発症人数（0で初期化）
  numDiabetes = 0,         # 糖尿病発症人数（0で初期化）
  glu = Pima.tr$glu, 
  bmi = Pima.tr$bmi, 
  bResult = rep(FALSE, length(Pima.tr$glu)) # 糖尿病か否か？（FALSE:糖尿病でない、    TRUE:糖尿病）
)

# Pima.tr$type のデータ（Yes,No）を符号化[encoding]
for ( i in 1:length(Pima.tr$type) ) 
{
  if(Pima.tr$type[i] == "Yes" )
  {
    lstPimaTrain$numDiabetes <- (lstPimaTrain$numDiabetes + 1)
    lstPimaTrain$bResult[i] <- TRUE
  }
  else if( Pima.tr$type[i] == "No" )
  {
    lstPimaTrain$numNoDiabetes <- (lstPimaTrain$numNoDiabetes + 1)
    lstPimaTrain$bResult[i] <- FALSE
  }
  else{
    # Do Nothing
  }
}

# sort Pima data
lstPimaTrain$glu <- lstPimaTrain$glu[ order(lstPimaTrain$bResult) ]
lstPimaTrain$bmi <- lstPimaTrain$bmi[ order(lstPimaTrain$bResult) ]
lstPimaTrain$bResult <- lstPimaTrain$bResult[ order(lstPimaTrain$bResult) ]

# split data to class C1 and C2
dfPimaTrain_C1 <- data.frame(
  glu = lstPimaTrain$glu[1:lstPimaTrain$numNoDiabetes], 
  bmi = lstPimaTrain$bmi[1:lstPimaTrain$numNoDiabetes],
  bResult = FALSE             # 糖尿病か否か？（FALSE:糖尿病でない、TRUE:糖尿病）
)
dfPimaTrain_C2 <- data.frame(
  glu = lstPimaTrain$glu[(lstPimaTrain$numNoDiabetes+1):    (lstPimaTrain$numNoDiabetes+lstPimaTrain$numDiabetes)], 
  bmi = lstPimaTrain$bmi[(lstPimaTrain$numNoDiabetes+1):(lstPimaTrain$numNoDiabetes+lstPimaTrain$numDiabetes)],
  bResult = TRUE             # 糖尿病か否か？（FALSE:糖尿病でない、TRUE:糖尿病）
)

# sort class C1,C2 data
dfPimaTrain_C1$glu <- dfPimaTrain_C1$glu[ order(dfPimaTrain_C1$glu) ]
dfPimaTrain_C1$bmi <- dfPimaTrain_C1$bmi[ order(dfPimaTrain_C1$glu) ]
dfPimaTrain_C2$glu <- dfPimaTrain_C2$glu[ order(dfPimaTrain_C2$glu) ]
dfPimaTrain_C2$bmi <- dfPimaTrain_C2$bmi[ order(dfPimaTrain_C2$glu) ]

# release memory
rm(Pima.tr) 
rm(Pima.te)

#-------------------
# set class C1 data
#-------------------
datU1_C1 <- mean( dfPimaTrain_C1$glu ) # クラス１（糖尿病発症なし）の変数１（glu）の平均値
datU2_C1 <- mean( dfPimaTrain_C1$bmi ) # クラス１（糖尿病発症有り）の変数２（BMI）の平均値
datU_C1 <- matrix( c(datU1_C1,datU2_C1), nrow = 2, ncol = 1)      # クラス１（糖尿病発症なし）の平均ベクトル

matS_C1 <- matrix( c(0,0,0,0), nrow = 2, ncol = 2 ) # クラス１（糖尿病発症なし）の共分散行列
matS_C1[1,1] <- sqrt( var( dfPimaTrain_C1$glu, dfPimaTrain_C1$glu ) )
matS_C1[1,2] <- sqrt( var( dfPimaTrain_C1$glu, dfPimaTrain_C1$bmi ) )
matS_C1[2,1] <- sqrt( var( dfPimaTrain_C1$bmi, dfPimaTrain_C1$glu ) )
matS_C1[2,2] <- sqrt( var( dfPimaTrain_C1$bmi, dfPimaTrain_C1$bmi ) )

#-------------------
# set class C2 data
#-------------------
datU1_C2 <- mean( dfPimaTrain_C2$glu ) # クラス２（糖尿病発症なし）の変数１（glu）の平均値
datU2_C2 <- mean( dfPimaTrain_C2$bmi ) # クラス２（糖尿病発症有り）の変数２（BMI）の平均値
datU_C2 <- matrix( c(datU1_C2,datU2_C2), nrow = 2, ncol = 1)      # クラス１（糖尿病発症なし）の平均ベクトル

matS_C2 <- matrix( c(0,0,0,0), nrow = 2, ncol = 2 ) # クラス２（糖尿病発症なし）の共分散行列
matS_C2[1,1] <- sqrt( var( dfPimaTrain_C2$glu, dfPimaTrain_C2$glu ) )
matS_C2[1,2] <- sqrt( var( dfPimaTrain_C2$glu, dfPimaTrain_C2$bmi ) )
matS_C2[2,1] <- sqrt( var( dfPimaTrain_C2$bmi, dfPimaTrain_C2$glu ) )
matS_C2[2,2] <- sqrt( var( dfPimaTrain_C2$bmi, dfPimaTrain_C2$bmi ) )


##################################
# secondary idification function
##################################
Dim2IdificationFunc <- function( x1, x2, u_C1, u_C2, matS1, matS2, P_C1=0.5, P_C2=0.5 )
{
  datX <- matrix( 0, nrow = 2, ncol = 1  )
  datX[1,] <- x1
  datX[2,] <- x2
  matW <- matrix( c(0,0,0,0), nrow = 2, ncol = 2 )
  matW <- ( solve(matS1) - solve(matS2) )
  vct <- ( t(u_C2)%*%solve(matS2) - t(u_C1)%*%solve(matS1) )
  r <- ( t(u_C1)%*%solve(matS1)%*%u_C1 - t(u_C2)%*%solve(matS2)%*%u_C2 + log(     det(matS1)/det(matS2) ) - 2*log(P_C1/P_C2) )

  z0 <- ( t(datX)%*%matW%*%datX )
  z1 <- ( 2*vct%*%datX )
  z <- ( z0 + z1 + r )
  return(z)
}

###############################
# liner idification function
###############################
Dim1IdificationFunc <- function( x1, x2, u_C1, u_C2, matS1, matS2, P_C1=0.5, P_C2=0.5 )
{
  datX <- matrix( 0, nrow = 2, ncol = 1  )
  datX[1,] <- x1
  datX[2,] <- x2

  matSp <- matrix( c(0,0,0,0), nrow = 2, ncol = 2 )
  matSp <- P_C1*matS1 + P_C2*matS2
  vct <- ( t(u_C2)%*%solve(matSp) - t(u_C1)%*%solve(matSp) )
  r <- ( t(u_C1)%*%solve(matSp)%*%u_C1 - t(u_C2)%*%solve(matSp)%*%u_C2 + log( det(matSp)/det(matSp) ) - 2*log(P_C1/P_C2) )

  z <- ( 2*vct%*%datX + r )
  return(z)
}

#------------------------------
# set functions values
#-------------------------------
datX1 <- seq( from=0, to=200, by=5 )  # glu 軸の値ベクトル
datX2 <- seq( from=0, to=50,  by=1 )  # BMI 軸の値ベクトル
datX1 <- as.matrix( datX1 )
datX2 <- as.matrix( datX2 )
matZDim2 <- matrix( 0, nrow=length(datX1), ncol=length(datX2) )
matZDim1 <- matrix( 0, nrow=length(datX1), ncol=length(datX2) )

# for loop matZDim2[i,j], matZDim1[i,j]
for( j in 1:length(datX2) )
{
  for( i in 1:length(datX1) )
  {
    matZDim2[i,j] <- Dim2IdificationFunc( 
      x1 = datX1[i], x2 = datX2[j],
      u_C1 = datU_C1, u_C2 = datU_C2, matS1 = matS_C1, matS2 = matS_C2 
    )

    matZDim1[i,j] <- Dim1IdificationFunc( 
      x1 = datX1[i], x2 = datX2[j],
      u_C1 = datU_C1, u_C2 = datU_C2, matS1 = matS_C1, matS2 = matS_C2 , 
      P_C1 = lstPimaTrain$numNoDiabetes/(lstPimaTrain$numNoDiabetes+lstPimaTrain$numDiabetes),
      P_C2 = lstPimaTrain$numNoDiabetes/(lstPimaTrain$numDiabetes+lstPimaTrain$numDiabetes)
    )
  }
}


############################
# ROC Curve                #
############################
#-------------------------------------------------
# get error ratio from Normal distribution (2dim)
#-------------------------------------------------
datLow = seq( from = 0.0, to = 200, by = 1.0 ) # idefication-boundary <lower>
datUp  = seq( from = 0.0, to = 200, by = 1.0 ) # idefication-boundary <upper>
dfROC2 <- data.frame(
  sigma1 = matrix( 0, nrow = 10, ncol = 1 ),
  sigma2 = matrix( 0, nrow = 10, ncol = 1 ),
  sigma = matrix( 0, nrow = 10, ncol = 1 )
)

# pmvnormの動作確認
dfROC2$sigma1[1] <- pmvnorm( lower = -Inf, upper = +Inf, mean = as.vector(datU_C1), sigma = matS_C1 )
dfROC2$sigma1[2] <- pmvnorm( lower = -Inf, upper = datU1_C1, mean = as.vector(datU_C1), sigma = matS_C1 )
dfROC2$sigma1[3] <- pmvnorm( lower = -Inf, upper = datU1_C1+10, mean = as.vector(datU_C1), sigma = matS_C1 )
dfROC2$sigma1[4] <- pmvnorm( lower = datU1_C1, upper = +Inf, mean = as.vector(datU_C1), sigma = matS_C1 )

dfROC2$sigma2[1] <- pmvnorm( lower = -Inf, upper = c(0,0), mean = as.vector(datU_C2), sigma = matS_C2 )
dfROC2$sigma2[2] <- pmvnorm( lower = -Inf, upper = c(50,10), mean = as.vector(datU_C2), sigma = matS_C2 )
dfROC2$sigma2[3] <- pmvnorm( lower = -Inf, upper = c(100,30), mean = as.vector(datU_C2), sigma = matS_C2 )
dfROC2$sigma2[4] <- pmvnorm( lower = -Inf, upper = c(150,40), mean = as.vector(datU_C2), sigma = matS_C2 )


dfROC2$sigma <- dfROC2$sigma1 + dfROC2$sigma2

print( dfROC2 )

############################
# set graphics parameters  #
############################
# 軸に関してのデータリスト
lstAxis <- list(                        
  xMin = 0.0, xMax = 1.0,  # x軸の最小値、最大値
  yMin = 0.0, yMax = 1.0,   # y軸の最小値、最大値
  zMin = 0.0, zMax = 1.0,   # z軸の最小値、最大値
  xlim = range( c(0.0, 1.0) ), 
  ylim = range( c(0.0, 1.0) ), 
  zlim = range( c(0.0, 1.0) ),
  mainTitle1 = "ROC曲線（２次元正規分布）\n２次識別関数[secondary idification function]", # 図のメインタイトル（図の上）
  mainTitle2 = "ROC曲線（２次元正規分布）\n線形識別関数[liner idification function]",     # 図のメインタイトル（図の上）
  mainTitle3 = "ROC曲線（２次元正規分布）\n",     # 図のメインタイトル（図の上）
  subTitle1  = "２次識別関数[dim2 idification function]",    # 図のサブタイトル（図の下）
  subTitle2  = "線形識別関数[liner idification function]",    # 図のサブタイトル（図の下）
  subTitle3  = "２次識別関数+線形識別関数",    # 図のサブタイトル（図の下）
  xlab      = "偽陽性率 [false positive rate]",         # x軸の名前
  ylab      = "真陽性率 [true positive rate]",         # y軸の名前
  zlab      = "z"            # z軸の名前
)
lstAxis$xlim = range( c(lstAxis$xMin, lstAxis$xMax) )
lstAxis$ylim = range( c(lstAxis$yMin, lstAxis$yMax) )
lstAxis$zlim = range( c(lstAxis$zMin, lstAxis$zMax) )

#########################
# Draw figure           #
#########################
# plot ROC Curve
plot(
  dfROC2$sigma1, dfROC2$sigma2,
  main = lstAxis$mainTitle1,
  xlab = lstAxis$xlab, ylab = lstAxis$ylab,
  xlim = lstAxis$xlim, ylim = lstAxis$ylim,
  type = "p"      
)
grid()  # 図にグリッド線を追加強調太字テキスト