# 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)

```

## Function Members (Methods)

public:
protected:
 void AddToSscp(TMatrixD& sscp, TVectorD& vec) void Classic() void ClearSscp(TMatrixD& sscp) void Correl() void Covar(TMatrixD& sscp, TVectorD& m, TMatrixDSym& cov, TVectorD& sd, Int_t nvec) void CreateOrtSubset(TMatrixD& dat, Int_t* index, Int_t hmerged, Int_t nmerged, TMatrixD& sscp, Double_t* ndist) void CreateSubset(Int_t ntotal, Int_t htotal, Int_t p, Int_t* index, TMatrixD& data, TMatrixD& sscp, Double_t* ndist) Double_t CStep(Int_t ntotal, Int_t htotal, Int_t* index, TMatrixD& data, TMatrixD& sscp, Double_t* ndist) virtual void TObject::DoError(int level, const char* location, const char* fmt, va_list va) const Int_t Exact(Double_t* ndist) Int_t Exact2(TMatrixD& mstockbig, TMatrixD& cstockbig, TMatrixD& hyperplane, Double_t* deti, Int_t nbest, Int_t kgroup, TMatrixD& sscp, Double_t* ndist) Double_t KOrdStat(Int_t ntotal, Double_t* arr, Int_t k, Int_t* work) void TObject::MakeZombie() Int_t Partition(Int_t nmini, Int_t* indsubdat) Int_t RDist(TMatrixD& sscp) void RDraw(Int_t* subdat, Int_t ngroup, Int_t* indsubdat)

## Data Members

private:
 enum TObject::EStatusBits { kCanDelete kMustCleanup kObjInCanvas kIsReferenced kHasUUID kCannotPick kNoContextMenu kInvalidObject }; enum TObject::[unnamed] { kIsOnHeap kNotDeleted kZombie kBitMask kSingleKey kOverwrite kWriteDelete };
protected:
 TMatrixDSym fCorrelation correlation matrix TMatrixDSym fCovariance covariance matrix estimate TMatrixD fData the original data Int_t fExact if there was an exact fit, stores the number of points on a hyperplane Int_t fH algorithm parameter, determining the subsample size TVectorD fHyperplane in case more than fH observations lie on a hyperplane TMatrixDSym fInvcovariance inverse of the covariance matrix TVectorD fMean location estimate (mean values) Int_t fN number of observations Int_t fNvar number of variables TArrayI fOut array of indexes of ouliers, size <0.5*n TVectorD fRd array of robust distances, size n TVectorD fSd array of standard deviations Int_t fVarTemp number of variables already added to the data matrix Int_t fVecTemp number of observations already added to the data matrix

## Function documentation

```this constructor should be used in a univariate case:
first call this constructor, then - the EvaluateUni(..) fucntion
```
TRobustEstimator(Int_t nvectors, Int_t nvariables, Int_t hh = 0)
```constructor
```
```adds a column to the data matrix
it is assumed that the column has size fN
variable fVarTemp keeps the number of columns l
```
```adds a vector to the data matrix
it is supposed that the vector is of size fNvar
```
void Evaluate()
```Finds the estimate of multivariate mean and variance
```
void EvaluateUni(Int_t nvectors, Double_t* data, Double_t& mean, Double_t& sigma, Int_t hh = 0)
```for the univariate case
estimates of location and scatter are returned in mean and sigma parameters
the algorithm works on the same principle as in multivariate case -
it finds a subset of size hh with smallest sigma, and then returns mean and
sigma of this subset
```

```returns the breakdown point of the algorithm
```
Double_t GetChiQuant(Int_t i) const
```returns the chi2 quantiles
```
void GetCovariance(TMatrixDSym& matr)
```returns the covariance matrix
```
void GetCorrelation(TMatrixDSym& matr)
```returns the correlation matrix
```
const TVectorD* GetHyperplane() const
```if the points are on a hyperplane, returns this hyperplane
```
void GetHyperplane(TVectorD& vec)
```if the points are on a hyperplane, returns this hyperplane
```
void GetMean(TVectorD& means)
```return the estimate of the mean
```
void GetRDistances(TVectorD& rdist)
```returns the robust distances (helps to find outliers)
```

```returns the number of outliers
```
```update the sscp matrix with vector vec
```
```clear the sscp matrix, used for covariance and mean calculation
```
void Classic()
```called when h=n. Returns classic covariance matrix
and mean
```
void Covar(TMatrixD& sscp, TVectorD& m, TMatrixDSym& cov, TVectorD& sd, Int_t nvec)
```calculates mean and covariance
```
void Correl()
```transforms covariance matrix into correlation matrix
```
void CreateSubset(Int_t ntotal, Int_t htotal, Int_t p, Int_t* index, TMatrixD& data, TMatrixD& sscp, Double_t* ndist)
```creates a subset of htotal elements from ntotal elements
first, p+1 elements are drawn randomly(without repetitions)
if their covariance matrix is singular, more elements are
added one by one, until their covariance matrix becomes regular
or it becomes clear that htotal observations lie on a hyperplane
If covariance matrix determinant!=0, distances of all ntotal elements
are calculated, using formula d_i=Sqrt((x_i-M)*S_inv*(x_i-M)), where
M is mean and S_inv is the inverse of the covariance matrix
htotal points with smallest distances are included in the returned subset.
```
void CreateOrtSubset(TMatrixD& dat, Int_t* index, Int_t hmerged, Int_t nmerged, TMatrixD& sscp, Double_t* ndist)
```creates a subset of hmerged vectors with smallest orthogonal distances to the hyperplane
hyp[1]*(x1-mean[1])+...+hyp[nvar]*(xnvar-mean[nvar])=0
This function is called in case when less than fH samples lie on a hyperplane.
```
Double_t CStep(Int_t ntotal, Int_t htotal, Int_t* index, TMatrixD& data, TMatrixD& sscp, Double_t* ndist)
```from the input htotal-subset constructs another htotal subset with lower determinant

As proven by Peter J.Rousseeuw and Katrien Van Driessen, if distances for all elements
are calculated, using the formula:d_i=Sqrt((x_i-M)*S_inv*(x_i-M)), where M is the mean
of the input htotal-subset, and S_inv - the inverse of its covariance matrix, then
htotal elements with smallest distances will have covariance matrix with determinant
less or equal to the determinant of the input subset covariance matrix.

determinant for this htotal-subset with smallest distances is returned
```
Int_t Exact(Double_t* ndist)
```for the exact fit situaions
returns number of observations on the hyperplane
```
Int_t Exact2(TMatrixD& mstockbig, TMatrixD& cstockbig, TMatrixD& hyperplane, Double_t* deti, Int_t nbest, Int_t kgroup, TMatrixD& sscp, Double_t* ndist)
```This function is called if determinant of the covariance matrix of a subset=0.

If there are more then fH vectors on a hyperplane,
returns this hyperplane and stops
else stores the hyperplane coordinates in hyperplane matrix
```
Int_t Partition(Int_t nmini, Int_t* indsubdat)
```divides the elements into approximately equal subgroups
number of elements in each subgroup is stored in indsubdat
number of subgroups is returned
```
Int_t RDist(TMatrixD& sscp)
```Calculates robust distances.Then the samples with robust distances
greater than a cutoff value (0.975 quantile of chi2 distribution with
fNvar degrees of freedom, multiplied by a correction factor), are given
weiht=0, and new, reweighted estimates of location and scatter are calculated
The function returns the number of outliers.
```
void RDraw(Int_t* subdat, Int_t ngroup, Int_t* indsubdat)
```Draws ngroup nonoverlapping subdatasets out of a dataset of size n
such that the selected case numbers are uniformly distributed from 1 to n
```
Double_t KOrdStat(Int_t ntotal, Double_t* arr, Int_t k, Int_t* work)
```because I need an Int_t work array
```
virtual ~TRobustEstimator()
`{;}`
void GetCovariance(TMatrixDSym& matr)
void GetCorrelation(TMatrixDSym& matr)

`{return fExact;}`
void GetMean(TVectorD& means)
void GetRDistances(TVectorD& rdist)
Int_t GetNumberObservations() const
`{return fN;}`
Int_t GetNvar() const
`{return fNvar;}`
const TArrayI* GetOuliers() const
`{return &fOut;}`