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

Wavelet analysis applications

Gennady OsoskovLIT JINR, Dubna

Semeon LebedevGSI, Darmstadt and LIT JINR, Dubna

Wavelet analysis applications CBM Collaboration meeting, 26.02.2008 2

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


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 hastwo steps procedure:1. Approximate the spectrum pedestal and subtract it from the spectrum


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 We have tested those algorithms, their efficiency depends on the particular efficiency depends on the particular spectrum peculiarity, but they are feasible. spectrum peculiarity, but they are feasible. So So the main focus should be on the step 2.the main focus should be on the step 2.


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 Cψ < ∞ 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 knownwavelet G2 is named “the Mexican hat”


Recall to wavelet introduction (cont)

Applicatios for extracting special features of mixed and contaminated signal

G2 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 bythresholding of scales, to be eliminatedor extracted, and then by making the inverse transform


Continuous or discrete waveletsContinuous 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 symmetricAn 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.


Peak parameter estimating by gaussian wavelets

When a signal is bell-shaped one, itcan 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 WG2(a,b;x0,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 x0 and σ, can be estimated directly in the G2 domain, we do not need the inverse transform!

Then it can be derived analytically that its wavelet transformation looksas the corresponding wavelet. For instance, for G2(x)

one has

Considering WG2 as a function of the dilation b we obtain its maximum

and then solving

the equation we obtain .

)()(

),(22

02

2

322

2

5

2ba

xbG

a

AagbaWG

2

322

2

5

2

)(

),(max

a

AabaWGb

0)(max

a

ab 5max a


Step 2. Peak parameters estimating in G2 wavelet domain

5

ˆˆ

a

2AI

How it works:How it works:after stage 1 we have a noisy spectrumafter stage 1 we have a noisy spectrum

It is transformed by GIt is transformed by G2 2 into wavelet domain, into wavelet domain,

where we look for the wavelet surface maximum where we look for the wavelet surface maximum

2

322

2

5 )ˆˆ(

ˆˆ

maxˆ

a

a

WA

bmax amax and then fit this surface by the analytical formula for WG2(a,b;x0,σ)g starting fit from x0=bmax

and . 5max

0

a

ax ˆ,ˆ0

Eventually, we should find the maximum of this fitted surface and use its coordinates as

estimations . From them we can

obtain and . Integral 2AI


Comparative accuracy study

Compare result with the LSFCompare result with the LSF

original peakoriginal peak reconstructed by waveletsreconstructed by wavelets reconstructed by LSFreconstructed 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: Two methods were compared: 1.1. nonlinear least square fit by a gaussian,nonlinear least square fit by a gaussian,

2.2. GG22 wavelet approach wavelet approach

Example of the peak restoration.Noise dispersion = signal amplitude


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


Comparative accuracy study III

2. At each point of the spectrum with amplitude A gaussian noise is added with σ =2*A.

mAσ

mA

σ

Summary of results for various signal distortions is shown in this plotwhere errors of reconstructed parameters marked in red for LSF method and in blue for wavelets approachWavelets advantage is doubtless!

Cut applied


CBM spectra. First results 1.

Λc invariant mass spectrum (by courtesy of Iou.Vassilev)

and its G2 spectrum

more andmore detailed

Wavelet method results:A=15.0σ =0.0116mean=2.2840Iw=0.435

PDG m=2.285

Igauss=0.365 (19% less)


CBM spectra. First results 2.

Low-mass dileptons (muon channel) ω. Gauss fit of reco signalM=0.7785σ =0.0125A=1.8166Ig=0.0569

ω. WaveletsM=0.7700σ =0.0143A=1.8430Iw=0.0598

- ω– wavelet spectrum

ω.

ω-meson

φ-meson

Even φ- and mesons have been visiblein the wavelet space, so we could extract their parameters.

Thanks to Ana Kiseleva


Summary and outlook• Algorithms and programs have been developed for estimating resonance

peak parameters in invariant mass spectra on the basis of G2 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 G4 wavelets

• Make a comparison of G2 and G4 wavelet applications

• Develop discrete wavelet algorithms and corresponding programs for resonance peak detection and background elimination

• Commit, eventually, wavelet-oriented software into SVN

Documents

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