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Robustness of Multiway Methods in Relation to Homoscedastic and Hetroscedastic Noise

Robustness of Multiway Methods in Relation to Homoscedastic and Hetroscedastic Noise. T. Khayamian Department of Chemistry , Isfahan University of Technology, Isfahan 84154, Iran. Outline. Introduction

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Robustness of Multiway Methods in Relation to Homoscedastic and Hetroscedastic Noise

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  1. Robustness of Multiway Methods in Relation to Homoscedastic and Hetroscedastic Noise T. Khayamian Department of Chemistry , Isfahan University of Technology, Isfahan 84154, Iran

  2. Outline • Introduction • Prediction of component concentrations in Claus data using PARAFAC and N-PLS multi-way methods • original data • (original + noise) data • denoised data (using wavelet as a denoising method) - Homoscedastic noise (level independent method) - Hetroscedastic noise (level dependent and minimum description length) • Conclusions

  3. Noise definition • Noise is any component of a signal which impedes observation, detection or utilization of the information that the signal is carrying. • Noise is measured by its standard deviation or peak to peak fluctuation

  4. Different types of noise Homoscedastic Noise Hetroscedastic

  5. Homoscedastic and Hetroscedastic Noise • Homoscedastic noise: Noise is independent of variable, sample and signal with the normal distribution and a constant variance. • Hetroscedastic Noise: Noise is dependent on the variable, sample and signal. (Noise from different variables or samples can be correlated) 2 2 R 2 1 1 2

  6. Least squares method Homoscedastic noise : ij is constant, uniform and independent of the signal, variables and samples Hetroscedastic noise : ij is dependent on signal, variables or samples

  7. Hetroscedastic noise in Univariate and Multivariate Calibration Methods • Zeroth order calibration weighted linear regression • First order calibration weighted principle component analysis • Second order calibration Positive matrix factorization Maximum likelihood PARAFAC

  8. Claus data “fluorescence Spectroscopy” C. A. Andesson and R. Bro. The N-way Toolbox for MATAB Chemom. Intell. Lab. Sys. 2000, 52 (1), 1- 4 http://www. models . kvl. dk

  9. Fluorescence excitation and emission spectrum of five samples

  10. • • • • • • Score (a2) Score (a1) Score (a3) • • • • • • • • • Concentration analyte 2 Concentration analyte 1 Concentration analyte 3 PARAFAC: four samples were used for modeling c1 c2 c3 61 b2 b3 b1 = X 4 + + a2 a3 a1 201 Claus data

  11. Calculation of Score for a New Sample Un = 61 201 Z = kr (B, C) Un = reshape (Un, 12261, 1) Score Un = pinv(Z) * Un

  12. Relative Errors of Predicted Concentrations for Samples 4 & 5 (without adding noise)

  13. Generating of Noise Matrix 61 unfolding 5 4 201×61 Claus data + Noise 201 (Claus data) 5 Noise 201×61 Homoscedastic nois: Standard deviation of noise = 2%, 5%, 10% of the maximum value in the claus data Hetroscedastic noise : N = N(0,1) .* 1/10 X Element by element was multiplied by one-tenth of the claus data

  14. Homoscedastic and Hetroscedastic noise were added to original data Hetroscedastic noise (10%) Homoscedastic noise (10%)

  15. The effect of adding Homoscedastic noise Reshape of Sample One Sample one with adding Homoscedastic noise

  16. The effect of adding Hetroscedastic noise Sample one with adding Hetroscedastic noise Reshape of Sample One

  17. Wavelet can be used as a powerful tool for signal denoising Wavelet Denoising : • Wavelet decomposition of the signal • Selecting the threshold • Applying the threshold to the wavelet coefficients • Inverse transformation to the native domain

  18. Thresholding methods : • Global thresholding • Level dependent thresholding • Data dependent thresholding • Cycle – spin thresholding • Wavelet packet thresholding

  19. Universal threshold : N = length of data array Xi = detail part of coefficient

  20. Prediction of Analyte Concentrations for Samples 4 & 5using PARAFAC

  21. Comparison of Sum of the Square of Residuals (Homoscedastic noise - PARAFAC) model 1 : sample 1, 2 , 3, 4 / model 2 : sample 1, 2, 3, 5 Each number × 105

  22. Comparison of explained variation (Homoscedastic noise - PARAFAC)

  23. Relative Errors of Predicted Concentrations for Sample 4 ( Homoscedastic noise – PARAFAC )

  24. Relative Errors of Predicted Concentrations for Sample 5 ( Homoscedastic noise - PARAFAC )

  25. Comparison of Sum of the Square of Residuals (Hetroscedastic noise) (Each number * 105) wavelet denoising (level dependent method)

  26. Comparison of explained variation (Hetroscedastic noise - PARAFAC) wavelet denoising (level dependent method)

  27. Relative Errors of Predicted Concentrations for sample 4 ( Hetroscedastic noise - PARAFAC) wavelet denoising (level dependent method)

  28. Relative Errors of Predicted Concentrations for Sample 5 ( Hetroscedastic noise - PARAFAC) wavelet denoising (level dependent method)

  29. Minimum Description Length • The MDL is an approach to simultaneous noise suppression and signal compression. • It is free from any parameter setting such as threshold selection, which can be particularly useful for real data where the noise level is difficult to estimate. m = filter type lm = the number of major coefficients retained γj,k = the vector of wavelet coefficients of transformed type m γj,k = the vector of the contractedwavelet coefficients ml

  30. Signal Denoising with MDL method

  31. Comparison of Sum of the Square of Residuals (Hetroscedastic noise - PARAFAC) Wavelet Denoising (MDL) Each number × 105

  32. Comparison of explained variation (Hetroscedastic noise - PARAFAC) Wavelet Denoising (MDL)

  33. Relative Errors of Predicted Concentrations for sample 4 ( Hetroscedastic noise - PARAFAC) Wavelet Denoising (MDL)

  34. Relative Errors of Predicted Concentrations for Sample 5 ( Hetroscedastic noise - PARAFAC ) Wavelet Denoising (MDL)

  35. Prediction of Analyte Concentrations for Samples 4 & 5using N-PLS

  36. Relative Errors of Predicted Concentrations for Sample 4 ( Homoscedastic noise – NPLS model) X-block > 99 Y-block > 99

  37. Relative Errors of Predicted Concentrations for Sample 5 ( Homoscedastic noise – NPLS model ) X-block > 99 Y-block > 99

  38. Relative Errors of Predicted Concentrations for Sample 4 ( Hetroscedastic noise – NPLS model) Wavelet Denoising (MDL) X-block > 99 Y-block > 99

  39. Comparison of Sum of the Square of Residuals (Hetroscedastic noise - PARAFAC) Each number × 105

  40. Relative Errors of Predicted Concentrations for Sample 4 ( Hetroscedastic noise – NPLS model)

  41. Relative Errors of Predicted Concentrations for Sample 5 ( Hetroscedastic noise - NPLS model )

  42. Relative Errors of Predicted Concentrations for Sample 5 ( Hetroscedastic noise – NPLS model ) Wavelet Denoising (MDL)

  43. Comparison of explained variation (Homoscedastic noise – NPLS model)

  44. Comparison of explained variation (Hetroscedastic noise – NPLS model)

  45. Comparison of explained variation (Hetroscedastic noise – NPLS model) Wavelet Denoising (MDL)

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