is symmetric chi-squared distance "A" metric?

Question

Is symmetric chi squared distance $$\int \frac{(p-q)^2}{pq}\mbox{d}\mu(x)$$ a metric?

I am searching web since long time ago but I couldnt find anything. It is positive and is zero whenever $p=q$ a.e. and symmetric but I dont know if it also satisfies the triangle inequality.

Please let me know if you know if it is metric or not.

Thank you very much.

Answer 1

So I saw a discrete version of this described here:
https://www.researchgate.net/post/What_is_chi-squared_distance_I_need_help_with_the_source_code

The denominator is not the same, in yours it is the product while in the reference it is the difference.

Here is my first bit of code:

##   Libraries
library(dplyr)

##   Parameters

samp_size <- 2  #how many "bins" in our "histogram"
num_loops <- 1e5 #how many times to run this

##   Subroutines

#my distance function
ascad <- function(x,y){
  sum( (1/(x+y))*(x-y )^2 )
}

##   Main Program

#predeclare data frame
df <- as.data.frame(matrix(data = 0, nrow = samp_size, ncol=3))
names(df) <- c("x", "y", "xpy")

#predeclare output store
tests <- as.data.frame(matrix(data = F, nrow = num_loops, ncol=4))
names(tests) <- c("nonnegative", 
                 "identity", 
                 "symmetry",
                 "triangle")

#main loop
for(i in 1:num_loops){

  #draw random positive values
  val1 <- (1/runif(n = samp_size,min = 0, max = 1))-1
  val2 <- (1/runif(n = samp_size,min = 0, max = 1))-1
  val3 <- (1/runif(n = samp_size,min = 0, max = 1))-1

  #normalize, and compute 
  df$x <- val1/sum(val1)
  df$y <- val2/sum(val2)
  df$y3 <- (val1 + val2)/(sum(val1)+sum(val2))

  #compute additve symmetric chi-squared distance
  d1 <- ascad(df$x, df$y)   #(x vs y)
  d2 <- ascad(df$y, df$xpy) #(y vs x+y)
  d3 <- ascad(df$x, df$xpy) #(x vs x+y)
  d4 <- ascad(df$y, df$x)   #(y vs x)

  #non-negativity test
  if(d3 >=0){
    tests[i,1] <- TRUE
  } 

  #symmetry
  if(d1 == d4){
    tests[i,3] <- TRUE
  }

  #triangle
  if(d1 <= d2 + d3){
    tests[i,4] <- TRUE
  }

  #recontrive it to test identity
  if(sample(1:5,1)==1){
    val2 <- val1
  } else {
    val1 <- val2
  }

  d1 <- ascad(val1, val2)   #(x vs y)

  #identity test
  if( sum(val1 == val2)==samp_size & d1==0){
    tests[i,2] <- TRUE
  }

}

#make summary of tests
summary(tests)

It gives the following result:

> summary(tests)
 nonnegative    identity       symmetry       triangle      
 Mode:logical   Mode:logical   Mode:logical   Mode:logical  
 TRUE:100000    TRUE:100000    TRUE:100000    TRUE:100000     

What I interpret it to say is:

• given a hundred thousand trials
• with identity testing by clamping either x=y or y=x
• and the default result is FALSE

None of the tests came out false.

This SUGGESTS that it is in fact a distance metric, though restricting the input domain to positive might make it a degenerate metric of some sort. If in the summary there were not 1e5 in all cases, then there would have been failure to pass the test, a valid counter-example, and it would not qualify as a metric of some sort.

You could

• increase the number of elements in the vector, but I think that even 1 should generalize well.
• test more uniformly or non-randomly, although I feel like inverting 'runif' was clever. If you try every value it may take longer but be more exhaustive
• make a version of this that runs through a symbolic algebra engine like sympy or yacas.
• test it with a difference for a denominator

This is, however, a possibly non-terrible start toward a solid answer.