Cramer-Rao¶
Maintainer: Chujie Chen
Introduction¶
CramerRao is a module to calculate an estimator for the angular resolution of a reconstructed track. As it is an analytical algorithm that does not need a minimization process, it is much faster (factor ~200) than Paraboloid.
Cramer-Rao relation¶
The Cramer-Rao relation states that the inverted Fisher information matrix \(I^{-1}\) provides a lower bound on the variance of the estimator of a parameter.
The Fisher information matrix is defined as
where \(\vec{\theta}\) represents the track parameters, i.e direction and vertex of the track. \(p\) is the propability density function. In our case \(p(t_i;\vec\theta)\) is the Pandel function.
Technical details of the implementation¶
If calculating the covariance matrix for a track in IceCube, it is important to take into account, that only four parameters are independent, as explained in the following: a track can be described by
with
Due to the averaging, the CR-inequation doesn’t include any time information. That means that the vertex is not uniquely defined: it can be placed anywhere on the track. This leads to the fact that the information matrix is over-determined, thus not invertible. Therefore the number of parameters must be reduced by giving a unique definition of the vertex. We fix the the vertex position at z=0, i.e. the intersection of the track with the x-y plane:
The coordiantes of the moved vertex are:
With this definition, only four independent parameters remain: the intersection x’, y’, \(Phi\) and \(Theta\). The distance between the track and a DOM (r_DOM = x_DOM,y_DOM,z_DOM) can be calculated with:
The disadvantage of this choice of coordiates is that it cannot describe tracks that are parallel to the x-y plane. These tracks cannot be processed by the cramer-rao module. Fortunately it is very rare, that a track has a zenith of exactly 90 degree.
The searched parameters \(\vec{\theta}\) are connected to the propability density \(p(t_i,d(\theta))\). With the chain rule
the Fisher information matrix can be written as
Note that we can write \(T(d_i)\) as:
where \(\psi\) is the Digamma function: \(\psi(x) = \Gamma'(x)/\Gamma(x)\).
References¶
An approach to estimate the optimal resolution was introduced by Marek Kowalski: “Applying information theory to cascade reconstruction”, talk presented at IceCube Collaboration meeting, 6.10.2006, Zeuthen.
The application on muon tracks by was introduced by Lutz Köepke, Jan Lünemann, Heinz-Georg Sander: “Muon track reconstruction resolution (note in progress)”, circulated in the muon group in March 2008.
This documentation is mainly translated from chapter 4.4 of Jan Lünemanns thesis: “Suche nach Dunkler Materie in Galaxien und Galaxienhaufen mit dem Neutrinoteleskop IceCube” (2013).