I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Retour sur... la ligne de base BEADS : correction et filtrage conjoints de mesures analytiques exploitant positivit´ e et parcimonie X. NING, I. W. SELESNICK Polytechnic School of Engineering, New York University L. DUVAL, A. PIRAYRE IFP Energies nouvelles, Universit´ e Paris-Est 9 octobre 2017 1 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Old peaks cast long shadows 2 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Old peaks cast long shadows 2 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Old peaks cast long shadows 2 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Old peaks cast long shadows 2 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Old peaks cast long shadows 2 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS The quick version ◮ Issue : how to accurately & repeatably quantize peaks? ◮ avoiding separate baseline and noise removal ◮ Question : where is the string behind the bead? ◮ without precise models for: peak, noise, baseline ◮ Answer : use main measurement properties + optimization ◮ sparsity+symmetry, stationarity, smoothness ◮ BEADS: B aseline E stimation A nd D enoising w/ S parsity 3 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Outline I NTRODUCTION F OREWORD O UTLINE * B ACKGROUND BEADS MODEL AND ALGORITHM N OTATIONS C OMPOUND SPARSE DERIVATIVE MODELING M AJORIZE -M INIMIZE TYPE OPTIMIZATION E VALUATION AND RESULTS GC: SIMULATED BASELINE AND G AUSSIAN NOISE GC: SIMULATED P OISSON NOISE GC: REAL DATA GC × GC: REAL DATA O THERS C ONCLUSIONS 4 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Background on background Figure: Image processing: varying illumination ◮ Background affects quantitative evaluation/comparison ◮ In other domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Very rare cases of parametric modeling (piecewise linear, polynomial, spline) 5 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Background on background Figure: Econometrics: trends and seasonality ◮ Background affects quantitative evaluation/comparison ◮ In other domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Very rare cases of parametric modeling (piecewise linear, polynomial, spline) 5 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Background on background Figure: Biomedical: ECG isoelectric line or baseline wander ◮ Background affects quantitative evaluation/comparison ◮ In other domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Very rare cases of parametric modeling (piecewise linear, polynomial, spline) 5 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Background on background Figure: Gas chromatography: baseline ◮ Background affects quantitative evaluation/comparison ◮ In other domains: (instrumental) bias, (seasonal) trend ◮ In analytical chemistry: drift, continuum, wander, baseline ◮ Very rare cases of parametric modeling (piecewise linear, polynomial, spline) 5 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Background on background Analytical chemistry, biological data ◮ Signal separation into three main morphological components 6 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Notations and assumptions Morphological decomposition: y = x + f + w , signals in R N ◮ y : observation (spectrum, analytical data) ◮ x : clean series of peaks (no baseline, no noise) ◮ f : baseline ◮ w : noise Assumption: without peaks, the baseline can be (approx.) recovered from noise-corrupted data by low-pass filtering ◮ ˆ f = L ( y − ˆ x ) : L : low-pass filter; H = I − L : high-pass filter x − ˆ ◮ formulated as � y − ˆ f � 2 x ) � 2 2 = � H ( y − ˆ 2 ◮ Going further with D i : differentiation operators 7 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling An estimate ˆ x can be obtained via: M F ( x ) = 1 � � � 2 � H ( y − x ) � 2 ˆ x = arg min 2 + λ i R i ( D i x ) . x i = 0 8 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling Examples of (smooth) sparsity promoting functions for R i ◮ φ A i = | x | | x | 2 + ǫ ◮ φ B � i = ◮ φ C i = | x | − ǫ log ( | x | + ǫ ) 8 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling Take the positivity of chromatogram peaks into account: F ( x ) = 1 � 2 � H ( y − x ) � 2 ˆ x = arg min 2 x N i − 1 N − 1 M � � � � + λ 0 θ ǫ ( x n ; r ) + λ i φ ([ D i x ] n ) . n = 0 n = 0 i = 1 Start from: � x , x � 0 θ ( x ; r ) = − rx , x < 0 9 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling Take the positivity of chromatogram peaks into account: F ( x ) = 1 � 2 � H ( y − x ) � 2 x = arg min ˆ 2 x N i − 1 N − 1 M � � � � + λ 0 θ ǫ ( x n ; r ) + λ i φ ([ D i x ] n ) . n = 0 n = 0 i = 1 and majorize it The majorizer g(x, v) for the penalty function θ (x; r), r = 3 10 g(x,v) 8 θ r (x) 6 4 (s, θ r (s)) 2 (v, θ r (v)) 0 −5 0 5 x 9 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling Take the positivity of chromatogram peaks into account: F ( x ) = 1 � 2 � H ( y − x ) � 2 x = arg min ˆ 2 x N i − 1 N − 1 M � � � � + λ 0 θ ǫ ( x n ; r ) + λ i φ ([ D i x ] n ) . n = 0 n = 0 i = 1 then smooth it: The smoothed asymmetric penalty function θ ε (x; r), r = 3 10 8 6 4 2 (− ε , f(− ε )) ( ε , f( ε )) 0 −5 0 5 x 9 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Compound sparse derivative modeling Take the positivity of chromatogram peaks into account: F ( x ) = 1 � 2 � H ( y − x ) � 2 ˆ x = arg min 2 x N i − 1 N − 1 M � � � � + λ 0 θ ǫ ( x n ; r ) + λ i φ ([ D i x ] n ) . n = 0 n = 0 i = 1 then majorize it: 4 | v | x 2 + 1 − r 1 + r 2 x + | v | 1 + r | v | > ǫ 4 , g 0 ( x , v ) = 4 ǫ x 2 + 1 − r 1 + r 2 x + ǫ 1 + r 4 , | v | � ǫ. 9 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Overall principle for Majoration-Minimization-Maximization Figure: Courtesy Peng Wang 1 1 https://commons.wikimedia.org/w/index.php?curid=17689902 10 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS BEADS Algorithm (short) Input: y , A , B , λ i , i = 0 , . . . , M b = B T BA − 1 y 1 . 2 . x = y (Initialization) Repeat [ Λ i ] n , n = φ ′ ([ D i x ] n ) 3 . , i = 0 , . . . , M , [ D i x ] n M � λ i D T 4 . M = i Λ i D i i = 0 Q = B T B + A T MA 5 . x = AQ − 1 b 6 . Until converged f = y − x − BA − 1 ( y − x ) 8 . Output: x , f 11 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Evaluation 1 50 50 40 40 30 30 20 20 10 10 0 0 −10 −10 1 2000 1 2000 Time (sample) Time (sample) 50 50 40 40 30 30 20 20 10 10 0 0 −10 −10 1 2000 1 2000 Time (sample) Time (sample) Figure: Simulated chromatograms w/ polynomial+sine baseline 12 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Evaluation 1 with Gaussian noise 13 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Evaluation 2 80 80 60 60 40 40 20 20 0 0 1 2000 1 2000 Time (sample) Time (sample) 50 40 40 30 30 20 20 10 10 0 0 1 2000 1 2000 Time (sample) Time (sample) Figure: Simulated chromatograms w/ limited power spectrum noise 14 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Evaluation 2 with Gaussian noise 15 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Evaluation 3 with Poisson noise Figure: Simulated chromatograms w/ Poisson noise 16 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Results: mono-dimensional chromatography (data 1) Figure: Original, superimposed, clean, noise 17 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Results: two-dimensional chromatography (data 2) Figure: Original data 18 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Results: two-dimensional chromatography (data 2) Figure: 2D background (estimated) 18 / 27
I NTRODUCTION BEADS MODEL AND ALGORITHM E VALUATION AND RESULTS C ONCLUSIONS Results: two-dimensional chromatography (data 2) Figure: Noise (estimated) 18 / 27
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