The Best Method of Noise Filtering Yuri Kalambet, Sergey Maltsev, Ampersand Ltd., Moscow, Russia; Yuri Kozmin, Shemyakin Institute of Bioorganic Chemistry, Moscow, Russia
[email protected]
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History: Adaptive peak approximation
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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 halfheight of the peak. Count obtained is an estimate of the slope width. 3
Properties of adaptive peak approximation • • • •
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 4
Improvement 1: Non-central approximation
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Confidence interval estimate CY t n(1/p2 ) S u* where 1 (Y Xβˆ ) (Y Xβˆ ) 2 u x ( X X) x* * * S n p 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; x* {1, x* ,..., x*p } βˆ ( X X) 1 X Y t m - Student’s coefficient for confidence probability (1-δ) and m degrees of freedom x* - position at which smoothed (approximated) value is estimated. 7
Approximation using confidence intervals G1
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Algorithm of simple Confidence filter approximation • Evaluate points and confidence intervals for new (shifted) window
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Algorithm of simple Confidence filter approximation • 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.
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Algorithm of simple Confidence filter approximation • 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).
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Bonus #1: Correct handling of baseline steps and array boundaries mv Original SG ASG 2000
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dotted – raw data; thick line – Confidence Filter; thin line – Savitzky-Golay filter 12
Confidence filter algorithm improvement: Adaptive gap of the polynomial • 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)
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Confidence interval estimate CY t n(1/p2 ) S u* where 1 (Y Xβˆ ) (Y Xβˆ ) 2 u x ( X X) x* * * S n p 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; x* {1, x* ,..., x*p } βˆ ( X X) 1 X Y t m - Student’s coefficient for confidence probability (1-δ) and m degrees of freedom x* - position at which smoothed (approximated) value is estimated. 14
t(df) for confidence probability 0.975
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Confidence interval profiles for different slits (degree = 3) 1
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Confidence Interval profiles, 31 points, 0…5 degrees
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σ evaluation problems: • 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 18
Handling σ estimate
CY t
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S 2 2 , Formula (a ) CY 2 2 S , Formula ( b )
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Noise Filtering: How it works 1 23
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Noise Filtering: How it works 2
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Is pump pulsation a noise or a signal?
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Conclusions: • 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
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Thank you!
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