Wavelet analysis applications Gennady Ososkov LIT JINR, Dubna Semeon Lebedev GSI, Darmstadt and LIT JINR, Dubna

Wavelet analysis applications CBM Collaboration meeting, Outline Formulating the peak finding problem Background estimation and elimination Wavelet features to be applied New idea: work in the wavelet domain A comparative accuracy study First results of CBM data handling Summary and outlook

Wavelet analysis applications CBM Collaboration meeting, Resonance peak identifying from invariant mass spectra 2. Detect a resonance peak in question and estimate its parameters Assuming a spectrum as a composition of background, peaks and statistical disturbances (noise), one has two steps procedure: 1. Approximate the spectrum pedestal and subtract it from the spectrum

Wavelet analysis applications CBM Collaboration meeting, Step 1. Background estimation There are many well elaborated algorithms, 1.Background simulation by Monte Carlo as either event mixing or like sign technique. However, it supposes, one has the adequate knowledge about background processes. Since is not always the case, an arbitrariness appears which leads to a stray background and/or worsen the accuracy. 2.Approximate by a polynomial of the 4th order 3.2. Sensitive Nonlinear Iterative Peak (SNIP) clipping algorithm is avialavle in ROOT with simultaneous smoothing, when signal values are to be recalculated as where p=1,2,… is iteration number. 4. Wavelet filtering on the basis of the orthogonal discrete wavelets We have tested those algorithms, their efficiency depends on the particular spectrum peculiarity, but they are feasible. So the main focus should be on the step 2. We have tested those algorithms, their efficiency depends on the particular spectrum peculiarity, but they are feasible. So the main focus should be on the step 2.

Wavelet analysis applications CBM Collaboration meeting, Recall to wavelet introduction One-dimensional wavelet transform (WT) of the signal f(x) has 2D form where the function  is the wavelet, b is a displacement (shift), and a is a scale. Condition C ψ < ∞ guarantees the existence of  and the wavelet inverse transform. Due to freedom in  choice, many different wavelets were invented. The family of continuous wavelets is presented here by Gaussian wavelets, which are generated by derivatives of Gaussian function Two of them, we use, are and Most known wavelet G 2 is named “the Mexican hat”

Wavelet analysis applications CBM Collaboration meeting, Recall to wavelet introduction (cont) Applicatios for extracting special features of mixed and contaminated signal G 2 wavelet spectrum of this signal Filtering results. Noise is removed and high frequency part perfectly localized An example of the signal with a localized high frequency part and considerable contamination then wavelet filtering is applied Filtering works in the wavelet domain by thresholding of scales, to be eliminated or extracted, and then by making the inverse transform

Wavelet analysis applications CBM Collaboration meeting, Continuous or discrete wavelets Continuous wavelets are remarkably resistant to noise (robust), but because of their non- orthogonality one obtains non-admissible signal distortions after inverse transform. Besides, real signals to be analysed by computer are always discrete. So orthogonal discrete wavelets look preferable. The discrete wavelet transform (DWT) was built by Mallat as multi-resolution analysis. It consists in representing a given data as a signal decomposition into basis functions φ and ψ, which must be compact. Various types of discrete wavelets One of Daubechie’s wavelets Coiflet – most symmetric An example of Daub2 spectrum The discrete wavelets are a good tool for background eliminating and peak detecting. However the main problem of wavelet applications was the absence of corresponding C++ software in any of available frameworks. So we had to build it ourselve.

Wavelet analysis applications CBM Collaboration meeting, Peak parameter estimating by gaussian wavelets When a signal is bell-shaped one, it can be approximated by a gaussian Thus, we can work directly in the wavelet domain instead of time/space domain and use this analytical formula for W G2 (a,b;x 0,σ)g surface in order to fit it to the surface, obtained for a real invariant mass spectrum. The most remarkable point is: since the fitting parameters x 0 and σ, can be estimated directly in the G 2 domain, we do not need the inverse transform! Then it can be derived analytically that its wavelet transformation looks as the corresponding wavelet. For instance, for G 2 (x) one has Considering W G2 as a function of the dilation b we obtain its maximum and then solving the equation we obtain.

Wavelet analysis applications CBM Collaboration meeting, Step 2. Peak parameters estimating in G 2 wavelet domain How it works: after stage 1 we have a noisy spectrum It is transformed by G 2 into wavelet domain, where we look for the wavelet surface maximum b max a max and then fit this surface by the analytical formula for W G2 (a,b;x 0,σ)g starting fit from x 0 =b max and. Eventually, we should find the maximum of this fitted surface and use its coordinates as estimations. From them we can obtain and. Integral

Wavelet analysis applications CBM Collaboration meeting, Comparative accuracy study Compare result with the LSF original peak original peak reconstructed by wavelets reconstructed by wavelets reconstructed by LSF reconstructed by LSF The accuracy test has been done on several samples of 500 simulated spectrum of invariant mass, consisting of small gaussian peak at the point 0.5 and the white noise with various s/n ratio Two methods were compared: 1.nonlinear least square fit by a gaussian, 2.G 2 wavelet approach Example of the peak restoration. Noise dispersion = signal amplitude

Wavelet analysis applications CBM Collaboration meeting, Comparative accuracy study II We present two examples utmost from their noisiness point of view: 1. At each point of the spectrum with amplitude A gaussian noise is added with σ =0.2*A Results of estimating signal parameters A, σ, mean by two methods are shown on the histograms below. Histogramed values: Δ=(MC-Rec)/MC - First three histograms – wavelet approach - Second three histograms – least square estimations

Wavelet analysis applications CBM Collaboration meeting, Comparative accuracy study III 2. At each point of the spectrum with amplitude A gaussian noise is added with σ =2*A. m A σ m A σ Summary of results for various signal distortions is shown in this plot where errors of reconstructed parameters marked in red for LSF method and in blue for wavelets approach Wavelets advantage is doubtless! Cut applied

Wavelet analysis applications CBM Collaboration meeting, CBM spectra. First results 1. Λc invariant mass spectrum (by courtesy of Iou.Vassilev) and its G 2 spectrum more and more detailed Wavelet method results: A=15.0 σ = mean= I w =0.435 PDG m=2.285 I gauss =0.365 (19% less)

Wavelet analysis applications CBM Collaboration meeting, CBM spectra. First results 2. Low-mass dileptons (muon channel) ω. Gauss fit of reco signal M= σ = A= I g = ω. Wavelets M= σ = A= I w = ω– wavelet spectrum ω.ω. ω-meson φ-meson Even φ- and mesons have been visible in the wavelet space, so we could extract their parameters. Thanks to Ana Kiseleva

Wavelet analysis applications CBM Collaboration meeting, Summary and outlook Algorithms and programs have been developed for estimating resonance peak parameters in invariant mass spectra on the basis of G 2 continuous wavelets Accuracy study has been performed, which shows significant advantages of the wavelet approach in comparison with LSF First attempts of the wavelet applications to CBM open charm and meson data are very promising What to do – a lot! Tuning of running software in close contacts with physicists interested in peak finding business Extend this software by including ready algorithms for applying G 4 wavelets Make a comparison of G 2 and G 4 wavelet applications Develop discrete wavelet algorithms and corresponding programs for resonance peak detection and background elimination Commit, eventually, wavelet-oriented software into SVN