BurstMon Basics

     BurstMon performs Monte-Carlo simulation of detection process to determine the amplitude at which signal of particular form (waveform) may be detected with 50% probability. To achieve this goal BurstMon makes software injections of signals to be detected into signal from interferometer channel. The form of injected signal is defined by user and supplied to BurstMon as ASCII file of simple format. To obtain reasonable accuracy in calculation of the detection efficiency the injection must be repeated many times for the same sample of signal from interferometer. BurstMon takes data of one stride length (option -t, 60 sec by default) from interferometer channel and performs 500 injections to achieve 3-4%  accuracy. The total number of injections is number of simultaneous injections (-nsim, 10 by default) multiplied by number of injection cycles (-ncycle, 10 by default) by number of measurements with different amplitudes (-nmes, 5 by default), nsim×ncycle×nmes=500.

     Input data from interferometer are subjected to whitening with the use of binary tree wavelets as bandpass filters with specified frequency bandwidth (option  -fw, whitening frequency resolution, 16 Hz by default).  For detection BurstMon uses wavelet expansion of signal with the same wavelets as for whitening but with higher time resolution which implies lower frequency resolution (option -f, 64 Hz by default). To speed up calculations the waveforms are injected in wavelet domain. This allows the wavelet transform of signal from interferometer to be computed once per time stride.  Wavelet transforms of each waveform are computed once at BurstMon start up. For each injection cycle waveforms are subjected to amplitude scaling,  whitening and application of few steps of wavelet inverse transform to match decomposition level of signal from interferometer. Wavelet expansion of input signal is combined with waveform expansions randomly shifted in time. Detection process  includes application of percentile threshold (defined by option -p, 1% by default) to the wavelet amplitudes and further cluster analysis to reveal loudest events on time-frequency plane. Final step of detection process consists of comparison of average time for each cluster with times of injections. The clusters with average time within time gate (option -gate, 20 msec by default) around injection time are counted as found injections. The same procedure is applied to signal from interferometer without injections. This gives number of clusters which randomly coincide in time with injections. This number is subtracted from the number of detected injections to obtain true number of detected injections. Detection efficiency is a ratio of number of found injections to total number of injections. To obtain the efficiency with reasonable accuracy 100 injections (nsim×ncycle=100, by default) with the same amplitude are performed.

    To obtain amplitude estimation at 50% detection efficiency the amplitudes of injections are choosen as close as possible to this point. The approximate amplitude of injection at first step is calculated using noise estimate returned by whitening procedure applied to data from ineterferometer channel. The amplitudes for further steps may be estimated using efficiency obtained at previous steps. We assume that efficiency eff dependency on amplitude h  is described by "sigmoid" function eff(h)=1/(1+(h/h₅₀)^α),  where h₅₀ is amplitude at which efficiency reaches 50% and α is unknown exponent. From actual fits of this dependency we found α to be in the range -2..-10. To choose next amplitude of injection we use α= -5. This gives estimated h₅₀=h / (1/eff -1)^(1/α)=h (1/eff-1)^(0.2), where h and eff are amplitude of injection at previous step and corresponding efficiency. This iteration is repeated at next steps using newly obtained efficiency. Having 3 or more efficiency measurements it is possible to apply more advanced technique to obtain estimation of h₅₀. For new variables y=ln(1/eff-1) and x=log(h) relation is: y=α (x-x₀), where x₀=log(h₅₀) and  50% efficiency corresponds to y=0. Using Least Square fit for linear function y=y(x) it is possible to calculate more accurate estimation of h₅₀. By default BurstMon uses 5 (option -nmes) points for final Least Square fit.

    The amplitude h₅₀ estimated by BurstMon depends on both noise level in interferometer channel and details of cluster analysis. User may set preffered values of thresholds for cluster analysis. Currently there are two: pixel fraction taken into analysis (option -p,  1% by default) and minimal cluster size (option -mcs, 1 by default), which means that clusters of this size and less will not be taken into analysis.