chisq {DstarM}

Calculates the distance between two probability densities.

Description

Calculates the distance between two probability densities.

Usage

chisq(tt, a, b)

battacharyya(tt, a, b)

hellinger(tt, a, b)

Arguments

tt

the time grid on which the densities are evaluated.

a

a vector with values of the first density.

b

a vector with values of the second density.

Value

The distance between densities a and b.

Examples

# Lets simulate a bunch of parameters and compare the three distance measures.

tt = seq(0, 5, .001)
parsMatV = cbind(.8, seq(0, 5, .5), .5, .5, .5) # differ only in drift speed
parsMatA = cbind(seq(.5, 2, .15), 2, .5, .5, .5)# differ only in boundary
# calculate densities for all these parameters
dV = apply(parsMatV, 1, function(x, tt) Voss.density(tt, x, boundary = 'upper'), tt = tt)
dA = apply(parsMatA, 1, function(x, tt) Voss.density(tt, x, boundary = 'upper'), tt = tt)
# make plots of the densities
matplot(tt, dA, xlim = c(0, .6), main = 'Densities with different Boundary',
        col = rainbow(ncol(dA)),type = 'l', lty = 1, las = 1, bty = 'n',
        xlab = 'Time', ylab = 'Density')
legend('topright', lty = 1, bty = 'n', col = rainbow(ncol(dA)),
       legend = paste('a = ', parsMatA[, 1]))
matplot(tt, dV, xlim = c(0, .6), main = 'Densities with different Drift Speed',
        col = rainbow(ncol(dV)), type = 'l', lty = 1, las = 1, bty = 'n',
        xlab = 'Time', ylab = 'Density')
legend('topright', lty = 1, bty = 'n', col = rainbow(ncol(dV)),
       legend = paste('v = ',parsMatV[, 2]))
# empty matrices for data storage
distMatV = matrix(NA, nrow = ncol(dV) - 1, ncol = 3,
                  dimnames = list(NULL, c('Chisq', 'Bhattacharyya', 'Hellinger')))
distMatA = matrix(NA, nrow = ncol(dA) - 1, ncol = 3,
                  dimnames = list(NULL, c('Chisq', 'Bhattacharyya', 'Hellinger')))
# calculate distances between densities in column i and i + 1.
# this is done using three different distance measures
for (i in 1:(ncol(dA) - 1)) {
  distMatV[i, ] = c(chisq(tt, dV[, i], dV[, i + 1]),
                    battacharyya(tt, dV[, i], dV[, i + 1]),
                    hellinger(tt, dV[, i], dV[, i + 1]))
  distMatA[i, ] = c(chisq(tt, dA[, i], dA[, i + 1]),
                    battacharyya(tt, dA[, i], dA[, i + 1]),
                    hellinger(tt, dA[, i], dA[, i + 1]))
}
# The three distance measures correlate highly for differences in Boundary
cor(distMatA)
# The battacharyya distance measures does not correlate with the others
# when calculating differences in drift speed
cor(distMatV)