1. The Best Method of Noise Filtering Yuri Kalambet, Sergey Maltsev, Ampersand Ltd., Moscow, Russia; Yuri Kozmin, Shemyakin Institute of Bioorganic Chemistry, Moscow, Russia kalambet@ampersand.ru 1

2. History: Adaptive peak approximation 2

3. Rough slope width estimate Evaluate baseline using default gap (minimum peak width Integration parameter) Evaluate peak height using default gap Count all points from peak apex to slope end with height bigger than half- height of the peak. Count obtained is an estimate of the slope width. 3

4. Properties of adaptive peak approximation 4 Good noise suppression at each slope Minimal peak shape disturbances All peak parameters are resistant to oversampling Baseline approximation may be poor – either noisy (small gap) or disturbed (large gap). No approximation outside of peaks Does not improve formal signal/noise ratio Baseline position is one of the most important sources of error

5. Improvement 1: Non-central approximation 5

6. Confidence intervals 6

7. Confidence interval estimate 7 where n - number of data points used for polynomial approximation (gap of the filter); p - power of the polynomial; X - matrix of x power values on independent axis (time); Y - vector of detector response values; - Student’s coefficient for confidence probability (1-δ) and m degrees of freedom x * - position at which smoothed (approximated) value is estimated.

8. Approximation using confidence intervals 8 * x 2 G confidence interval 1 G

9. Algorithm of simple Confidence filter approximation 9 Evaluate points and confidence intervals for new (shifted) window

10. Algorithm of simple Confidence filter approximation 10 Evaluate points and confidence intervals for new (shifted) window Compare new confidence interval with that for previously evaluated point. If the new one is smaller than previous, replace approximated point and its confidence interval.

11. Algorithm of simple Confidence filter approximation 11 Evaluate points and confidence intervals for new (shifted) window Compare new confidence interval with that for previously evaluated point. If the new one is smaller than previous, replace approximated point and its confidence interval. Computational complexity of Confidence filter is comparable to that of simple convolution, (e.g. Savitzky-Golay) and linearly depends on the product gap∙ (degree of the polynomial).

12. Bonus #1: Correct handling of baseline steps and array boundaries 12 dotted – raw data; thick line – Confidence Filter; thin line – Savitzky-Golay filter

13. Confidence filter algorithm improvement: Adaptive gap of the polynomial 13 Repeat confidential filter algorithm for approximations with different windows (gaps) Computational complexity: degree∙gap∙(gap-1)/2 Logarithmic step: next gap is k times smaller, than previous, e.g. gap2 = gap1/k, k>1; Computational complexity: degree∙gap∙k/(k-1)

14. Confidence interval estimate 14 where n - number of data points used for polynomial approximation (gap of the filter); p - power of the polynomial; X - matrix of x power values on independent axis (time); Y - vector of detector response values; - Student’s coefficient for confidence probability (1-δ) and m degrees of freedom x * - position at which smoothed (approximated) value is estimated.

15. t(df) for confidence probability 0.975 15

16. Confidence interval profiles for different slits (degree = 3) 16

17. Confidence Interval profiles, 31 points, 0…5 degrees 17

18. σ evaluation problems: 18 Small gaps: accidental perfect fit Large gaps: treating small peaks as a noise due to large number of degrees of freedom Is pump pulsation a noise or a signal? Small gaps: confidence interval depends on confidence level σ evaluation solutions: Evaluate in advance using the whole data array Use the estimate for evaluation of confidence intervals

19. Handling σ estimate 19

20. 20 Noise Filtering: How it works 1

21. 21 Noise Filtering: How it works 2

22. Automatic selection of degree and gap of approximating polynomial 22

23. Is pump pulsation a noise or a signal? 23

24. Conclusions: 24 Confidence filter introduces a measure of approximation quality Confidence filter helps to select the best set of functions that approximate the data set Confidence filter is metrologically the best noise filtering method and can be used in the fight with legal metrology Patent pending

25. Thank you! 25