ogi: Objective General Index
In OGI: Objective General Index

Description Usage Arguments Details Value References Examples

ogi(X) returns the objective general index (OGI) of the covariance matrix S of X.

1 2	ogi(X, se = FALSE, force = FALSE, se.loop = 1000, nu = rep(1, ncol(X)), center = TRUE, mar = FALSE)

`X`	Numeric or ordered matrix.
`se`	Logical: if se=TRUE, it additionally computes `w.se` and `v.se` by bootstrap. Default: FALSE.
`force`	Logical: if force=FALSE, `S` should be strictly positive definite. Default: FALSE.
`se.loop`	Iteration number in bootstrap for computation of standard error.
`nu`	Numeric vector of subjective importance. It determines the importance of each column of `X`.
`center`	Logical: if center=TRUE, `ogi(X)$Z` is centered. Default:TRUE.
`mar`	Logical: if mar=TRUE, each of ordered categorical variates of `X` (if exists) is marginally converted into a numeric vector in advance by the univariate OGI quantification. If mar=FALSE, the simultaneous OGI quantification is applied. Default:FALSE.

Consider a data matrix of n individuals with p variates. The objective general index (OGI) is a general index that combines the p variates into a univariate index in order to rank the n individuals. The OGI is always positively correlated with each of the variates. For more details, see the references.

`value`	The objective general index (OGI).
`X`	The input matrix `X`.
`scaled`	The product of `Z %*% diag(weight)`, where `Z` and `weight` are as follows.
`Z`	Numerical matrix converted from `X`. If center = TRUE, it is centered.
`weight`	The output of `cov2weight(S, nu=nu, force=force)`, where `S` is the covariance matrix of `X`.
`rel.weight`	The product of `weight * sqrt(diag(S))`, where `S` is the covariance matrix of `X`.
`biu`	The bi-unit canonical form of the covariance matrix of `X`.
`idx`	Numeric vector. If `X` has ordered categorical variates, `idx` has (number of levels) -1 number of indexes.
`w.se`	If requested, `w.se` is numeric vector of the standard error of `weight`. It is calculated by bootstrap.
`v.se`	If requested, `v.se` is numeric vector of the standard error of `value`. It is calculated by bootstrap.

Sei, T. (2016). An objective general index for multivariate ordered data, Journal of Multivariate Analysis, 147, 247-264. http://www.sciencedirect.com/science/article/pii/S0047259X16000269

CT = matrix(c(
2,1,1,0,0,
8,3,3,0,0,
0,2,1,1,1,
0,0,0,1,1,
0,0,0,0,1), 5, 5, byrow=TRUE)
X = matrix(0, 0, 2)
for(i in 1:5){
  for(j in 1:5){
    if(CT[i,j]>0){
      X = rbind(X, matrix(c(6-i,6-j), CT[i,j], 2, byrow=TRUE))
    }
  }
}
X0 = X
X = as.data.frame(X0)
X[,1] = factor(X0[,1], ordered=TRUE)
X[,2] = factor(X0[,2], ordered=TRUE)
ogiX = ogi(X)
par(pty="s", cex=1.7, mar=c(4.5,3,1,1))
plot(ogiX$scaled, xlim=c(-3,3), ylim=c(-3,3), xlab="Geometry", ylab="Probability")
for(t in 1:nrow(ogiX$scaled)){
  xy = ogiX$scaled[t,]
  g = rep(sum(xy)/2, 2)
  segments(xy[1], xy[2], g[1], g[2], lty=2)
}
arrows(-3, -3, 3, 3)
text(2.5, 2, "OGI/2")
ogiX


f = ordered(1:10)
f[sample(1:10, 20, replace=TRUE)]
Y = ogi(f)$value
plot((1:10)/(10+1), Y, type="b")
xs = (1:1000)/1001
points(xs, qnorm(xs), type="l", col="red")


X = USJudgeRatings
ogiX = ogi(X)
nameX = ordered(names(X), names(X))
plot(nameX, ogiX$weight, las=3, cex.axis=0.8, ylim=c(0,1.2), ylab="weight")