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)� ) ) /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 your H reduces to 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) )� ) /n ] -> simplified computation -> > sqrt [ (n * Si( X� ) - ( Si( X ) )� )/ n� ] > > 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]� > ) (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]�) - (A + X[11]�)� ) / n� ] > > This is failry easy in an iterative process, since before we continue > with the next step we set: > A = (A + X[11]�) > 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! > > Thanx in advance! > > Kind regards, > Stef > > ______________________________________________ > [email protected] mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide! > http://www.R-project.org/posting-guide.html -------------------------------------------------------------------- E-Mail: (Ted Harding) <[EMAIL PROTECTED]> Fax-to-email: +44 (0)870 094 0861 Date: 26-May-05 Time: 17:05:13 ------------------------------ XFMail ------------------------------ ______________________________________________ [email protected] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
