Stochastic template placement algorithm for gravitational wave data analysis

This paper presents an algorithm for constructing matched-filter template banks in an arbitrary parameter space. The method places templates at random, then removes those which are “too close” together. The properties and optimality of stochastic template banks generated in this manner are investigated for some simple models. The effectiveness of these template banks for gravitational wave searches for binary inspiral waveforms is also examined. The properties of a stochastic template bank are then compared to the deterministically placed template banks that are currently used in gravitational wave data analysis.

Figure 1

The theoretical upper and lower bounds on normalized thickness of a stochastic template bank [21] and the normalized thickness of known lattice algorithms as a function of dimension as defined by Eqs. (2.7, 2.8). Also, the obtained thickness of stochastic banks at dimensions less than 5.

Figure 2

A histogram of the distribution of distances from a template to the nearest template, in units of the closest possible spacing $\Delta$, for a simple three-dimensional example. The distances are clustered close to the minimum possible spacing $\Delta$, showing that the covering locations found by the stochastic template placement method are close to the positions found by packing spheres of radius $\Delta /2$.

Figure 3

The relationship between the covering fraction and the thickness of the bank in two, three, and four dimensions. This is also compared to what one would expect in the case of the random template bank [20] as well as the case where no templates overlap each other.

Figure 4

The computational cost (number of distance calculations) depends on the covering fraction.

Figure 5

As Fig. 3 but setting the value of $N_{\mathrm{lower}\mathrm{\text{−}}\mathrm{bound}}$ to be 50. By comparing the two figures, one can see how boundary effects manifest themselves both by decreasing the initial slope $df/dn$ and by requiring a much larger number of templates than the estimate.

Figure 6

The distribution of trial points chosen uniformly in polar coordinates (left-hand panel) and the points that remain as templates after the application of the stochastic placement algorithm (right-hand panel).

Figure 7

Trial points chosen uniformly in the $(\theta ,\phi )$ coordinates (left-hand panel) and the templates on the surface of a sphere of unit area that remain after the application of the stochastic template placement algorithm (middle panel). Also, the distribution of these templates in $\theta$-coordinate (right-hand panel), where the solid line shows the expected distribution.

Figure 8

The distribution of templates placed by the hexagonal lattice algorithm in $({\tau }_0,{\tau }_3)$ coordinates (left-hand panel) and the same templates in $(m_1,m_2)$ coordinates (right-hand panel). Clearly, the distribution is highly skewed in the latter coordinates.

Figure 9

The number of templates as a function of the number of random trial points is shown when the trial points are assigned uniformly in $({\tau }_0,{\tau }_3)$ coordinates as well as uniformly in $(m_1,m_2)$ coordinates.

Figure 10

The distribution of stochastically generated templates in $({\tau }_0,{\tau }_3)$ coordinates (left-hand panel) and in $(m_1,m_2)$ coordinates (right-hand panel).

Figure 11

The signals which had an overlap larger than 0.95 (gray crosses) as well as signals with an overlap less than 0.95 (black circles) in $(m_1,m_2)$ coordinates (left-hand panel) and $({\tau }_0,{\tau }_3)$ coordinates (middle panel). Also, a histogram of the overlaps of all injections with the stochastic template bank (right-hand panel).