13 Pattern Recognition and Reconstruction
593
A different kind of re-weighted least-squares estimator is proposed in [132]. The
weights are computed according to Tukey’s bi-square function [128]:
w i =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
1 −
r 2
i /σ 2
i
c 2
2
, |r i | ≤ cσ i ,
0,
otherwise,
where r 2
i is the squared residual of track i with respect to the vertex, σ 2
i is its
variance, and c is again the robustness constant. The estimator is now equivalent to a
redescending M-estimator, and consequently less sensitive to outliers than Huber’s
M-estimator.
The combination of a redescending M-estimator with the concept of deterministic annealing [133] leads to the adaptive method of vertex fitting [113, 117, 134,
135]. The concept of the adaptive vertex fit is derived from the Deterministic
Annealing Filter [64] (see Sect. 13.1.3.5). The weights are computed according to
w i =
exp
−χ 2
i /2T
exp
−χ 2
i /2T
+ exp
−χ 2
cut /2T
,
where χ 2
i is the χ 2 -contribution of track i, χ 2
cut is a cutoff value, and T is a
temperature parameter. The computation of the redescending M-estimator can be
interpreted as an EM (expectation–maximization) algorithm [136, 137]. Alternatively it can be viewed as the minimization of the energy function of an elastic
arm algorithm [18, 19]. If annealing is employed, the iteration starts at high T . The
temperature is then gradually decreased. At low T the weights approach either zero
or one. The final weights can be used for classification of the tracks as inliers or
outliers. A comparison of the adaptive method with other robust estimators can be
found in [138]. The adaptive estimator has been extended to a multi-vertex estimator
fitting several vertices simultaneously, including competition of all vertices for all
tracks [139].
Iterated re-weighted least-squares estimators require a good starting point, in
order to ensure convergence to the global minimum and to minimize the number of
iterations required. In many cases a standard least-squares estimate is sufficient. In
the presence of a large number of outliers also the starting point should be estimated
robustly, preferably by an estimator with a high breakdown point [129]. Several such
initial estimators have been proposed and studied in [117].
The M-estimators and the adaptive estimator presented above do not presuppose
an explicit outlier model. If it is possible to describe the outliers by a Gaussian
mixture model, estimation of the vertex can be carried out by the Gaussian-sum
filter [140].
Several of the estimators described here are implemented in RAVE, a detectorindependent toolkit for reconstruction of interaction vertices [120, 121].
593
A different kind of re-weighted least-squares estimator is proposed in [132]. The
weights are computed according to Tukey’s bi-square function [128]:
w i =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
1 −
r 2
i /σ 2
i
c 2
2
, |r i | ≤ cσ i ,
0,
otherwise,
where r 2
i is the squared residual of track i with respect to the vertex, σ 2
i is its
variance, and c is again the robustness constant. The estimator is now equivalent to a
redescending M-estimator, and consequently less sensitive to outliers than Huber’s
M-estimator.
The combination of a redescending M-estimator with the concept of deterministic annealing [133] leads to the adaptive method of vertex fitting [113, 117, 134,
135]. The concept of the adaptive vertex fit is derived from the Deterministic
Annealing Filter [64] (see Sect. 13.1.3.5). The weights are computed according to
w i =
exp
−χ 2
i /2T
exp
−χ 2
i /2T
+ exp
−χ 2
cut /2T
,
where χ 2
i is the χ 2 -contribution of track i, χ 2
cut is a cutoff value, and T is a
temperature parameter. The computation of the redescending M-estimator can be
interpreted as an EM (expectation–maximization) algorithm [136, 137]. Alternatively it can be viewed as the minimization of the energy function of an elastic
arm algorithm [18, 19]. If annealing is employed, the iteration starts at high T . The
temperature is then gradually decreased. At low T the weights approach either zero
or one. The final weights can be used for classification of the tracks as inliers or
outliers. A comparison of the adaptive method with other robust estimators can be
found in [138]. The adaptive estimator has been extended to a multi-vertex estimator
fitting several vertices simultaneously, including competition of all vertices for all
tracks [139].
Iterated re-weighted least-squares estimators require a good starting point, in
order to ensure convergence to the global minimum and to minimize the number of
iterations required. In many cases a standard least-squares estimate is sufficient. In
the presence of a large number of outliers also the starting point should be estimated
robustly, preferably by an estimator with a high breakdown point [129]. Several such
initial estimators have been proposed and studied in [117].
The M-estimators and the adaptive estimator presented above do not presuppose
an explicit outlier model. If it is possible to describe the outliers by a Gaussian
mixture model, estimation of the vertex can be carried out by the Gaussian-sum
filter [140].
Several of the estimators described here are implemented in RAVE, a detectorindependent toolkit for reconstruction of interaction vertices [120, 121].