Tuning the hierarchical procedure searching for the key where there is light S.Frasca – Potsdam, December 2006

The hierarchical procedure to detect periodic sources (rough scheme)  Data set division: the data are divided in a small number of sets (2, 3,...)  Coherent step: for each set the data are divided in chunks of the length T coh and for each chunk a coherent analysis is done  Incoherent step: from the results of this first step, an incoherent step follows (Hough or Radon transform), that takes all the data of the set. This step is normally the most computationally heavy. Candidate source of level 1 are produced.  Coincidence: the candidates from 2 or more sets are searched for coincidence within the parameters (frequency, spin-down, sky position), obtaining candidate sources of level 2  Refining the analysis: this candidates are “followed up” with a new more refined coherent step followed by a new incoherent step

Coherent step: limit on T coh to use just a periodogram as the first coherent step (as those produced by the SFTs)

“Input” parameters for a hierarchical procedure  Resources: The data: observation time T obs (normally fragmented and with varying antenna(s) sensitivity)The data: observation time T obs (normally fragmented and with varying antenna(s) sensitivity) Available computing powerAvailable computing power Maximum number of candidates we can manageMaximum number of candidates we can manage  Target choice Sky areaSky area Frequency rangeFrequency range Spin-down interval(s)Spin-down interval(s)  Procedure tuning First coherent step time T cohFirst coherent step time T coh Number of sets of data for separate hierarchical analyses in order to do candidate coincidenceNumber of sets of data for separate hierarchical analyses in order to do candidate coincidence

“Output” parameters for a hierarchical procedure  Sensitivity  Needed computing power (to do the analysis in a “reasonable” time)  Probability of “success”

Now we discuss only a simple case of hierarchical procedure, neglecting the data set division problem and considering the case of using the Hough transform for the incoherent step. The coherent time is taken variable (smaller or larger of T 0 ). Some parameters, like the sensitivity, will be normalized to the case of using T coh =T 0.

Number of points in parameter space for the incoherent step Number of frequency bins Freq. bins in the Doppler band at the mid freq. of the band Sky points Spin-down points Total number of points

Basic equations: needed computing power Number of operations for the incoherent step Needed computing power (to do the job in ½ of T obs )

Basic equations: sensitivity Optimal detection nominal sensitivity Hierarchical method nominal sensitivity Optimal detection (whole obs. time) nominal sensitivity: Hierarchical method nominal sensitivity Nominal sensitivity vs CP To double the “nominal” sensitivity, we need 4048 times more CP

1 billion candidate sensitivity reduction  To reduce the number of candidates to a manageable number (e.g. 1 billion) we must put a threshold on the Hough map. This reduces the sensitivity, respect to the “nominal” (SNR=1) by a factor given in figure: This farther reduces the dependence of the sensitivity on CP.

1G Sensitivity vs CP The sensitivity is normalized to the case of using T coh =T 0 In red there is the cost of only the incoherent step, sperimposed in blue there is the entire cost. We see that to gain a factor 2 in sensitivity, we need to increase the CP of a factor of more than

Source distribution Reasonably the distribution of the amplitude of the sources ( at the detector ), in the “threshold” range, can be believed as a power law of exponent m (for example m=2). So the probability to have a source over a certain threshold is and So a gain in sensitivity of the detection algorithm of the order of 2, that we paid times more in computing power, gives us only a factor 2 in “success” probability.

So what ?  It is better to enhance the sensitivity with longer T coh or enlarge the possible targets investigating more spin-down and more sky ?