class TRobustEstimator: public TObject

  TRobustEstimator

 Minimum Covariance Determinant Estimator - a Fast Algorithm
 invented by Peter J.Rousseeuw and Katrien Van Dreissen
 "A Fast Algorithm for the Minimum covariance Determinant Estimator"
 Technometrics, August 1999, Vol.41, NO.3

 What are robust estimators?
 "An important property of an estimator is its robustness. An estimator
 is called robust if it is insensitive to measurements that deviate
 from the expected behaviour. There are 2 ways to treat such deviating
 measurements: one may either try to recongize them and then remove
 them from the data sample; or one may leave them in the sample, taking
 care that they do not influence the estimate unduly. In both cases robust
 estimators are needed...Robust procedures compensate for systematic errors
 as much as possible, and indicate any situation in which a danger of not being
 able to operate reliably is detected."
 R.Fruhwirth, M.Regler, R.K.Bock, H.Grote, D.Notz
 "Data Analysis Techniques for High-Energy Physics", 2nd edition

 What does this algorithm do?
 It computes a highly robust estimator of multivariate location and scatter.
 Then, it takes those estimates to compute robust distances of all the
 data vectors. Those with large robust distances are considered outliers.
 Robust distances can then be plotted for better visualization of the data.

 How does this algorithm do it?
 The MCD objective is to find h observations(out of n) whose classical
 covariance matrix has the lowest determinant. The MCD estimator of location
 is then the average of those h points and the MCD estimate of scatter
 is their covariance matrix. The minimum(and default) h = (n+nvariables+1)/2
 so the algorithm is effective when less than (n+nvar+1)/2 variables are outliers.
 The algorithm also allows for exact fit situations - that is, when h or more
 observations lie on a hyperplane. Then the algorithm still yields the MCD location T
 and scatter matrix S, the latter being singular as it should be. From (T,S) the
 program then computes the equation of the hyperplane.

 How can this algorithm be used?
 In any case, when contamination of data is suspected, that might influence
 the classical estimates.
 Also, robust estimation of location and scatter is a tool to robustify
 other multivariate techniques such as, for example, principal-component analysis
 and discriminant analysis.




 Technical details of the algorithm:
 0.The default h = (n+nvariables+1)/2, but the user may choose any interger h with
   (n+nvariables+1)/2<=h<=n. The program then reports the MCD's breakdown value
   (n-h+1)/n. If you are sure that the dataset contains less than 25% contamination
   which is usually the case, a good compromise between breakdown value and
  efficiency is obtained by putting h=[.75*n].
 1.If h=n,the MCD location estimate is the average of the whole dataset, and
   the MCD scatter estimate is its covariance matrix. Report this and stop
 2.If nvariables=1 (univariate data), compute the MCD estimate by the exact
   algorithm of Rousseeuw and Leroy (1987, pp.171-172) in O(nlogn)time and stop
 3.From here on, h<n and nvariables>=2.
   3a.If n is small:
    - repeat (say) 500 times:
    -- construct an initial h-subset, starting from a random (nvar+1)-subset
    -- carry out 2 C-steps (described in the comments of CStep funtion)
    - for the 10 results with lowest det(S):
    -- carry out C-steps until convergence
    - report the solution (T, S) with the lowest det(S)
   3b.If n is larger (say, n>600), then
    - construct up to 5 disjoint random subsets of size nsub (say, nsub=300)
    - inside each subset repeat 500/5 times:
    -- construct an initial subset of size hsub=[nsub*h/n]
    -- carry out 2 C-steps
    -- keep the best 10 results (Tsub, Ssub)
    - pool the subsets, yielding the merged set (say, of size nmerged=1500)
    - in the merged set, repeat for each of the 50 solutions (Tsub, Ssub)
    -- carry out 2 C-steps
    -- keep the 10 best results
    - in the full dataset, repeat for those best results:
    -- take several C-steps, using n and h
    -- report the best final result (T, S)
 4.To obtain consistency when the data comes from a multivariate normal
   distribution, covariance matrix is multiplied by a correction factor
 5.Robust distances for all elements, using the final (T, S) are calculated
   Then the very final mean and covariance estimates are calculated only for
   values, whose robust distances are less than a cutoff value (0.975 quantile
   of chi2 distribution with nvariables degrees of freedom)