Fractional Differential Whittaker Smoother

doi:10.12068/j.issn.1005-3026.2021.11.002

Abstract

Abstract: At present， the Whittaker Smoother(WS)algorithm is widely used. The core of the algorithm lies in the use of integer-order differentiation to express roughness. However， the integer-order differential representation is single and not flexible enough to truly reflect the roughness of the signal. On the contrary， the fractional differential expression is flexible and can better describe the roughness of the real signal. Therefore， the fractional differentiation is used to improve the WS algorithm and make it more flexible and effective. As two fractional differential calculation methods， Riemann-Liouville(RL)and Grümwald-Letnikov(GL)are adopted to implement the fractional WS algorithm.Furthermore， the automatic parameter selection of the fractional WS algorithm is realized by mathematical derivation. The experimental results of the nuclear magnetic resonance spectrum with sharp peaks show that the fractional-order WS algorithm can extract more real information; additionally， the experimental results of Marzipan infrared spectra show that the precision of spectral quantitative analysis is higher compared with the original integer-order WS algorithm.

Key words: fractional differention; spectral preprocessing; smoothing and denoising; quantitative analysis; partial least squares

CLC Number:

O433.4

SHAN Peng， HE Nian， LI Zhi-gang， WU Zhui. Fractional Differential Whittaker Smoother[J]. Journal of Northeastern University(Natural Science), 2021, 42(11): 1527-1532.

References

[1]Wilde A S，Hanghey S A，Galvin-king P，et al.The feasibility of applying NIR and FT-IR fingerprinting to detect adulteration in black pepper［J］.Food Control，2019，100:1-7.
[2]Viderman D，Abdildin Y G.Near-infrared spectroscopy in neurocritical care:a review of recent updates［J］.World Neurosurgery，2021，151(5):23-28.
[3]Fan S，Zhong Q，Fauhl-Hassek C，et al.Classification of Chinese wine varieties using 1H NMR spectroscopy combined with multivariate statistical analysis［J］.Food Control，2017，88:113-122.
[4]Chen T H，Zhao S，Shao S Q，et al.Non-invasive diagnosis methods of coronary disease based on wavelet denoising and sound analyzing［J］.Saudi Journal of Biological Sciences，2017，24(3):526-536.
[5]赵肖宇，方一鸣，王志刚，等.EEMD自适应去噪在拉曼光谱中的应用［J］.光谱学与光谱分析，2013，33(12):3255-3258.(Zhao Xiao-yu，Fang Yi-ming，Wang Zhi-gang，et al.Application of EEMD adaptive denoising in Raman spectroscopy［J］.Spectroscopy and Spectral Analysis，2013，33(12):3255-3258.)
[6]Savitzky A，Golay M J E.Smoothing and differentiation of data by simplified least squares procedures［J］.Analytical Chemistry，1964，36(8):1627-1639.
[7]Eilers P H.A perfect smoother［J］.Analytical Chemistry，2003，75:3631-3636.
[8]谢天婷，张路，王飞，等.双频驱动下分数阶过阻尼马达在空间对称势中的定向输运［J］.物理学报，2014，63(23):105-113.(Xie Tian-ting，Zhang Lu，Wang Fei，et al.The directional transport of a fractional overdamped motor in a space symmetric potential under dual-frequency drive［J］.Acta Physica Science，2014，63(23):105-113.)
[9]Oldham K B.Fractional differential equations in electrochemistry［J］.Advances in Engineering Software，2010，41(1):9-12.
[10]Smith A Z，Kartci A，Brancˇík L.Fractional-order lossy transmission line with skin effect using NILT method［C］//2017 40th International Conference on TSP.Barcelona: IEEE，2017:730-734.
[11]Wu S L，Zhou T.Parareal algorithms with local time-integrators for time fractional differential equations［J］.Journal of Computational Physics，2018(358):135-149.
[12]Allagui A，Freeborn T J，Elwakil A S，et al.Review of fractional-order electrical characterization of supercapacitors［J］.Journal of Power Sources，2018，400(1):457-467.
[13]Toman J，Brown S D.Peak resolution by semiderivative voltammetry［J］.Analytical Chemistry，1981，53(9):1497-1504.
[14]卢小泉，刘宏德，张敏，等.分数导数结合傅里叶最小二乘拟合处理含噪音的重迭信号［J］.分析化学，2003(2):143-147.(Lu Xiao-quan，Liu Hong-de，Zhang Min，et al.Fractional derivative combined with Fourier least squares fitting to deal with overlapping signals with noise［J］.Chinese Journal of Analytical Chemistry，2003(2):143-147.)
[15]Kharintsev S，Salakhov M K.A simple method to extract spectral parameters using fractional derivative spectrometry［J］.Spectrochimica Acta Part A: Molecular & Biomolecular Spectroscopy，2004，60(8/9):2125-2133.
[16]Mocak J，Janiga I，Rievaj M，et al.The use of fractional differentiation or integration for signal improvement ［J］.Measurement Science Review，2007，7(5):39-42.
[17]李远禄，于盛林，郑罡.基于分数阶微分的重叠峰分辨方法［J］.中国科学(B辑:化学)，2007，4:361-368.(Li Yuan-lu，Yu Sheng-lin，Zheng Gang.Overlapping peak resolution method based on fractional differential［J］.Science in China(Series B:Chemistry)，2007，4:361-368.)
[18]Li Y L，Tang H Q，Chen H X.Fractional-order derivative spectroscopy for resolving simulated overlapped Lorenztian peaks［J］.Chemometrics & Intelligent Laboratory Systems，2011，107(1):83-89.
[19]徐继刚，冯新泸，管亮，等.分数阶微分在红外光谱数据预处理中的应用［J］.化工自动化及仪表，2012，39(3):347-351.(Xu Ji-gang，Feng Xin-lu，Guan Liang，et al.Fractional differential application in reprocessing infrared spectral data［J］.Control and Instruments in Chemical Industry，2012，39(3):347-351.)
[20]张东，塔西甫拉提·特依拜，张飞，等.分数阶微分在盐渍土高光谱数据预处理中的应用［J］.农业工程学报，2014，30(24):151-160.(Zhang Dong，Tashpolat·Tiyip，Zhang Fei，et al.Application of fractional differential in preprocessing hyperspectral data of saline soil［J］.Transactions of the Chinese Society of Agricultural Engineering，2014，30(24):151-160.)
[21]田安红，熊黑钢，赵俊三，等.分数阶微分对盐渍土野外光谱预处理精度提升的机理分析［J］.光谱学与光谱分析，2019，39(8):2495-2500.(Tian An-hong，Xiong Hei-gang，Zhao Jun-san，et al.Mechanism of improvement for pretreatment accuracy of field spectra of saline soil using fractional differention algorithm［J］.Spectroscopy and Spectral Analysis，2019，39(8):2495-2500.)
[22]Hastie T，Tibshirani R.Generalized additive models［M］.London:Chapman and Hall，1990:42-49.
[23]Kilicman A，Zhour Z.Kronecker operational matrices for fractional calculus and some applications［J］.Applied Mathematics and Computation，2007(187):250-265.
[24]Zhang L，Li G，Sun M，et al.Kennard-Stone combined with least square support vector machine method for noncontact discriminating human blood species［J］.Infrared Physics and Technology，2017(86):116-119.
[25]Zheng J H，Song Z H，Ge Z Q.Probabilistic learning of partial least squares regression model:theory and industrial applications［J］.Chemometrics and Intelligent Laboratory Systems，2016，158(5):80-90.