Modeling Background Noise for Denoising in Chemical Spectroscopy - PowerPoint PPT Presentation

Modeling Background Noise for Denoising in Chemical Spectroscopy Modeling Background Noise for Denoising in Chemical Spectroscopy Problem Formulation An Algorithm for Denoising Modelling the Noise Estimating Coefficients Segmentation Tikhonov Regularization June 29, 2009 Numerical Results Conclusions and Future Work Richard Barnard Department of Mathematics Louisiana State University

Modeling Talk Outline Background Noise for Denoising in Chemical Spectroscopy Problem Formulation Problem Formulation An Algorithm for Denoising An Algorithm for Denoising Modelling the Noise Estimating Modelling the Noise Coefficients Segmentation Estimating Coefficients Tikhonov Regularization Segmentation Numerical Results Tikhonov Regularization Conclusions and Future Work Numerical Results Conclusions and Future Work

Modeling MALDI-TOF Mass Spectrometer Background Noise for Denoising in Chemical Spectroscopy Problem Formulation We will consider data sets obtained via M atrix A ssisted An Algorithm for L aser Desorption / I onization T ime O f F light Mass Denoising Spectrometer. Modelling the Noise Estimating Coefficients ◮ Analyte sample is placed in a matrix solution. Segmentation Tikhonov Regularization Numerical Results Conclusions and Future Work

Modeling MALDI-TOF Mass Spectrometer Background Noise for Denoising in Chemical Spectroscopy Problem Formulation We will consider data sets obtained via M atrix A ssisted An Algorithm for L aser Desorption / I onization T ime O f F light Mass Denoising Spectrometer. Modelling the Noise Estimating Coefficients ◮ Analyte sample is placed in a matrix solution. Segmentation Tikhonov Regularization ◮ Pulsed laser fired at mixture, ionizing analyte. Numerical Results Conclusions and Future Work

Modeling MALDI-TOF Mass Spectrometer Background Noise for Denoising in Chemical Spectroscopy Problem Formulation We will consider data sets obtained via M atrix A ssisted An Algorithm for L aser Desorption / I onization T ime O f F light Mass Denoising Spectrometer. Modelling the Noise Estimating Coefficients ◮ Analyte sample is placed in a matrix solution. Segmentation Tikhonov Regularization ◮ Pulsed laser fired at mixture, ionizing analyte. Numerical Results ◮ Analyte ions travel along a path of known length, Conclusions and Future Work striking a detector.

Modeling MALDI-TOF Mass Spectrometer Background Noise for Denoising in Chemical Spectroscopy Problem Formulation We will consider data sets obtained via M atrix A ssisted An Algorithm for L aser Desorption / I onization T ime O f F light Mass Denoising Spectrometer. Modelling the Noise Estimating Coefficients ◮ Analyte sample is placed in a matrix solution. Segmentation Tikhonov Regularization ◮ Pulsed laser fired at mixture, ionizing analyte. Numerical Results ◮ Analyte ions travel along a path of known length, Conclusions and Future Work striking a detector. ◮ Time of flight can be used to determine mass to charge ratio.

Modeling Mass Spectrum Background Noise for Denoising in Resulting data is a set of 50,000-100,000 data pairs Chemical Spectroscopy (time/mass-to-charge ratio and intensity). Our spectra will be from SRM 2881, a polystyrene, obtained from NIST. Problem Noise from various sources can lead to uncertainty (see Formulation Guttman, Flynn, Wallace, and Kearsley 2009). An Algorithm for Denoising Modelling the Noise Estimating Coefficients Segmentation Tikhonov Regularization Numerical Results Conclusions and Future Work Figure: Analyte(red) and corresponding background(blue), low noise

Modeling Mass Spectrum Background Noise for Denoising in Resulting data is a set of 50,000-100,000 data pairs Chemical Spectroscopy (time/mass-to-charge ratio and intensity). Our spectra will be from SRM 2881, a polystyrene, obtained from NIST. Problem Noise from various sources can lead to uncertainty (see Formulation Guttman, Flynn, Wallace, and Kearsley 2009). An Algorithm for Denoising Modelling the Noise Estimating Coefficients Segmentation Tikhonov Regularization Numerical Results Conclusions and Future Work Figure: Analyte(red) and corresponding background(blue), with noise

Modeling Mass Spectrum Background Noise for Denoising in Resulting data is a set of 50,000-100,000 data pairs Chemical Spectroscopy (time/mass-to-charge ratio and intensity). Our spectra will be from SRM 2881, a polystyrene, obtained from NIST. Problem Noise from various sources can lead to uncertainty (see Formulation Guttman, Flynn, Wallace, and Kearsley 2009). An Algorithm for Denoising Modelling the Noise Estimating Coefficients Segmentation Tikhonov Regularization Numerical Results Conclusions and Future Work Figure: Analyte(red) and corresponding background(blue),with noise

Modeling Overview Background Noise for Denoising in Chemical Spectroscopy Problem Formulation An Algorithm for Denoising ◮ Fit background spectrum to stochastic differential Modelling the Noise Estimating model Coefficients Segmentation Tikhonov Regularization Numerical Results Conclusions and Future Work

Modeling Overview Background Noise for Denoising in Chemical Spectroscopy Problem Formulation An Algorithm for Denoising ◮ Fit background spectrum to stochastic differential Modelling the Noise Estimating model Coefficients Segmentation Tikhonov ◮ Determine the mean and variance of noise Regularization Numerical Results Conclusions and Future Work

Modeling Overview Background Noise for Denoising in Chemical Spectroscopy Problem Formulation An Algorithm for Denoising ◮ Fit background spectrum to stochastic differential Modelling the Noise Estimating model Coefficients Segmentation Tikhonov ◮ Determine the mean and variance of noise Regularization Numerical Results ◮ Segment spectrum Conclusions and Future Work

Modeling Overview Background Noise for Denoising in Chemical Spectroscopy Problem Formulation An Algorithm for Denoising ◮ Fit background spectrum to stochastic differential Modelling the Noise Estimating model Coefficients Segmentation Tikhonov ◮ Determine the mean and variance of noise Regularization Numerical Results ◮ Segment spectrum Conclusions and ◮ Use Tikhonov regularization on each segment Future Work

Modeling Background Model Background Noise for Denoising in Chemical Spectroscopy Problem Formulation An Algorithm for Denoising We fit the analyte-free spectrum to a Stochastic Differential Modelling the Noise Estimating Equation with time dependent coefficients Coefficients Segmentation Tikhonov Regularization dX t = ( a 0 ( t ) + a 1 ( t ) X t ) dt + b 0 ( t ) X t ( t ) dW t Numerical Results Conclusions and { W t } is a Wiener Process, W t − W s ∼ N (0 , t − s ) , s < t Future Work

Modeling Discretization Background Noise for Denoising in Chemical Spectroscopy Problem Given the background data { X ( i ) } at discrete points, we use Formulation Euler-Maruyama discretization: An Algorithm for Denoising Modelling the Noise ∆ X ( i ) = ( a 0 ( i ) + a 1 ( i ) X ( i ) δ + b 0 ( i ) X ( i )∆ W i (1) Estimating Coefficients Segmentation Tikhonov Regularization Numerical Results Conclusions and Future Work

Modeling Discretization Background Noise for Denoising in Chemical Spectroscopy Problem Given the background data { X ( i ) } at discrete points, we use Formulation Euler-Maruyama discretization: An Algorithm for Denoising Modelling the Noise ∆ X ( i ) = ( a 0 ( i ) + a 1 ( i ) X ( i ) δ + b 0 ( i ) X ( i )∆ W i (1) Estimating Coefficients Segmentation Tikhonov Regularization Given a window size for regression h , we use the Numerical Results Epanechnikov Kernel Conclusions and Future Work K h ( z ) = 3 4 h (1 − z 2 ) for z ∈ ( − 1 , 0) and K h ≡ 0 off ( − 1 , 0) .

Modeling Estimating a 0 , a 1 Background Noise for Denoising in Chemical Spectroscopy Problem Formulation An Algorithm for In order to estimate a 0 , a 1 at each i , we look to minimize Denoising Modelling the Noise Estimating Coefficients N ( X ( j + 1) − X ( j ) − a 0 ( i ) − a 1 ( i ) X ( j )) 2 K h ( δ ( j − i ) Segmentation � min ) . Tikhonov Regularization δ h a 0 , a 1 j =1 Numerical Results (2) Conclusions and Future Work

Modeling Estimating a 0 , a 1 Background Noise for Denoising in Chemical Spectroscopy Problem In order to estimate a 0 , a 1 at each i , we look to minimize Formulation An Algorithm for Denoising N ( X ( j + 1) − X ( j ) − a 0 ( i ) − a 1 ( i ) X ( j )) 2 K h ( δ ( j − i ) Modelling the Noise � min ) . Estimating δ Coefficients h a 0 , a 1 Segmentation j =1 Tikhonov (2) Regularization Numerical Results For Y ( j ) = X ( j + 1) − X ( j ) , τ ij = δ ( j − i ) h Conclusions and � Y ( j ) K h ( τ ij ) − δ a 1 ( i ) K h ( τ ij ) Future Work ˜ a 0 ( i ) = δ K h ( τ ij )