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.

\[cov(\theta_m,\theta_k)\geq I(\vec{\theta})^{-1}\]

The Fisher information matrix is defined as

\[I_{mk}(\vec{\theta})= -\langle\left[\sum_{i=1}^{modules} \frac{\partial^2}{\partial\theta_k\partial\theta_m} \ln p(t_i;\vec\theta) \right]\rangle\]

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

\[\vec{r}=\vec{r}_0+\vec{v}\cdot t\]

with

\[\vec{v}=(\cos\Phi\sin\Theta,\sin\Phi\sin\Theta,\cos\Theta)\]

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:

\[\vec{r^\prime}_0=\vec{r}_0+a\cdot\vec{v}=\vec{r}_0-\frac{z\cdot\vec{v}}{\cos\Theta}\]

The coordiantes of the moved vertex are:

\[\begin{split}x^\prime=x-z\cdot\cos\Phi\tan\Theta\\ y^\prime=y-z\cdot\sin\Phi\tan\Theta\\ z^\prime=0\end{split}\]

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:

\[\begin{split}d^2=(\vec{v}\times (\vec{r}_{DOM}-\vec{r^\prime}_0))^2\\ =[(x_{DOM}-x^\prime)\sin\Phi\sin\Theta-(y_{DOM}-y^\prime)\cos\Phi\sin\Theta]^2\\ +[(y_{DOM}-y^\prime)\cos\Theta-(z_{DOM}-z^\prime)\sin\Phi\sin\Theta]^2\\ +(z_{DOM}-z^\prime)\cos\Phi\sin\Theta-(x_{DOM}-x^\prime)\cos\Theta]^2 \\ =[(x_{DOM}-x+z\cdot\cos\Phi\tan\Theta)\sin\Phi\sin\Theta -(y_{DOM}-y+z\cdot\sin\Phi\tan\Theta)\cos\Phi\sin\Theta]^2\\ +\left[(y_{DOM}-y+z\cdot \sin\Phi\tan\Theta)\cos\Theta - z_{DOM}\cdot\sin\Phi\sin\Theta\right]^2\\ +\left[z_{DOM}\cdot\cos\Phi\sin\Theta-(x_{DOM}-x+z\cdot\cos\Phi\tan\Theta)\cos\Theta\right]^2\end{split}\]

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

\[\frac{\partial}{\partial\theta_m} =\frac{\partial}{\partial d}\frac{\partial d}{\partial\theta_m} =\frac{\partial}{\partial d} \frac{\partial d}{\partial d^2} \frac{\partial d^2}{\partial\theta_m} =\frac{1}{2d} \frac{\partial}{\partial d} \frac{\partial d^2}{\partial\theta_m}`\]

the Fisher information matrix can be written as

\[I_{mk}(\vec{\theta})=\sum_{i=1}^{NCh}\underbrace{-\left\langle \left(\frac{\partial^2}{(\partial d_i)^2}\ln p(t_i;d(\vec{\theta}))\right)\right\rangle}_{T(d_i)} \underbrace{\frac{1}{4d_i^2}\frac{\partial d_i^2}{\partial\theta_m}\frac{\partial d_i^2}{\partial\theta_k}}_{D_{mk}(\vec{\theta})}\]

Note that we can write \(T(d_i)\) as:

\[T(d_i) = - \frac{\int_{0}^{\infty} dt' (\frac{\partial ^2}{(\partial d_i)^2} \ln p(t_i; \vec{\theta})) p(t_i; \vec{\theta})}{\int_{0}^{\infty} dt' p(t_i; \vec{\theta})} = \frac{\psi (d/\lambda)}{\lambda^2}\]

where \(\psi\) is the Digamma function: \(\psi(x) = \Gamma'(x)/\Gamma(x)\).

References

  1. 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.

  2. 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.

  3. 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).