[R] Simplify formula for heterogeneity

(Ted Harding) Ted.Harding at nessie.mcc.ac.uk
Thu May 26 18:05:17 CEST 2005

```On 26-May-05 Stefaan Lhermitte wrote:
> Dear R-ians,
>
> I'm looking for a computational simplified formula to calculate a
> measure for heterogeneity (let's say H ):
>
> H = sqrt [ (Si (Sj (Xi - Xj)² ) ) /n ]
>
> where:
> sqrt = square root
> Si = summation over i  (= 0 to n)
> Sj = summation over j (= 0 to n)
> Xi = element of X with index i
> Xj = element of X with index j

If I have understood your formula correctly (and you are
applying it to a vector X of length n) then it seems that

sqrt[(Si(n*(Xi - Xbar)^2) + Sj(n*(Xj - Xbar)^2))/n]

= sqrt[2*(n-1)var(X)] = sd(X)*sqrt(2*(n-1))

(where Xbar is the mean of the values in X).

So I don't see what the special point of H is anyway.
But at least this simplifies it1

Best wishes,
Ted.

> I can simplify the formula to:
>
> H = sqrt [ ( 2 * n * Si (Xi) - 2 Si (Sj ( Xi * Xj)) ) / n]
>
> Unfortunately this formula stays difficult in iterative programming,
> because I have to keep every element of X to calculate H.
>
> I know a computional simplified formula exists for the standard
> deviation (sd) that is much easier in iterative programming.
> Therefore I wondered I anybody knew about analog simplifications to
> simplify H:
>
> sd = sqrt [ ( Si (Xi - mean(X) )² ) /n  ]  -> simplified computation ->
> sqrt [ (n * Si( X² ) - ( Si( X ) )² )/ n² ]
>
> This simplied formula is much easier in iterative programming, since I
> don't have to keep every element of X.
> E.g.: I have a vector X[1:10]  and I already have caculated Si(
> X[1:10]²
> ) (I will call this A) and Si( X ) (I will call this B).
> When X gets extendend by 1 element (eg. X[11]) it easy fairly simple to
> calculate sd(X[1:11]) without having to reuse the elements of X[1:10].
> I just have to calculate:
>
> sd = sqrt [ (n * (A + X[11]²) - (A + X[11]²)² ) / n² ]
>
> This is failry easy in an iterative process, since before we continue
> with the next step we set:
> A = (A + X[11]²)
> B = (B + X[11])
>
> Can anybody help me to do something comparable for H? Any other help to
> calculate H easily in an iterative process is also welcome!
>
>
> Kind regards,
> Stef
>
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help