The imputation of missingness in cyclic and non-cyclic Electromyography(EMG) signaling data

Abstract

Multidimensional datasets in healthcare and life sciences often reflect temporal variations, but are often incomplete, complicating the analysis, and reducing statistical accuracy. To address missing data, imputation techniques are widely used, with machine learning algorithms like random forest and k-nearest neighbors and nonparametric methods such as spline and linear interpolation among the common approaches. This study examines electromyography data, a time-series biomedical data set, by evaluating 11 imputation methods in four datasets. We introduce four approaches, normal ratio, weighted normal ratio, expectation maximization, and Gibbs sampling, and assess each for accuracy and computational efficiency. Two scenarios were simulated: unaltered and down-sampled data, each with scattered and intermittent missingness. The comparative assessment emphasizes the notable precision of the expectation maximization method, with the random forest emerging as a robust alternative. Moreover, the normal ratio and weighted normal ratio methods demonstrate computational efficiency akin to mean and median imputation while improving accuracy. We also address cyclic data, a critical factor for improving accuracy. Using Fourier transformation, spline, and autoregressive models, we propose pattern-based and sinusoidal-based approaches to improve imputation. Results indicate that pattern-based improves accuracy, while sinusoidal-based offers efficiency, particularly for k-nearest neighbors.

Keywords

• Cyclic electromyography data
• missing imputation methods
• periodic patterns

References

1. [1] Z. C. Lipton, D. C. Kale and R. Wetzel, Modeling missing data in clinical time series with RNNs, Mach. Learn. Healthc. 56(56), 253–270, 2016.
2. [2] C. Yozgatligil, S. Aslan, C. Iyigun and I. Batmaz, Comparison of missing value imputation methods in time series: the case of Turkish meteorological data, Theor. Appl. Climatol. 112, 143–167, 2013.
3. [3] M. E. Magnello, Karl Pearson and the establishment of mathematical statistics, Int. Stat. Rev. 77(1), 3–29, 2009.
4. [4] D. Millett, Hans Berger: From psychic energy to the EEG, Perspect. Biol. Med. 44(4), 522–542, 2001.
5. [5] R. A. Fisher, The use of multiple measurements in taxonomic problems, Ann. Eugen. 7(2), 179–188, 1936.
6. [6] A. H. Herring and J. G. Ibrahim, Likelihood-based methods for missing covariates in the Cox proportional hazards model, J. Am. Stat. Assoc. 96(453), 292–302, 2001.
7. [7] D. B. Chaffin, Surface electromyography frequency analysis as a diagnostic tool, J. Occup. Med. 11(3), 109–115, 1969.
8. [8] T. R. Ten Have, Statistical and design issues in geriatric psychiatric research, in: Annu. Rev. Gerontol. Geriatr. 19, 37–52, Springer, 1999.

9. [9] M. T. Tan, G.-L. Tian and K. W. Ng, Bayesian missing data problems: EM, data augmentation and noniterative computation, Chapman & Hall/CRC, 2009.
10. [10] J. L. Schafer, Analysis of incomplete multivariate data, Chapman & Hall/CRC, 1997.
11. [11] R. J. A. Little and D. B. Rubin, Statistical Analysis with Missing Data, Wiley, Chichester, 1987.
12. [12] P. D. Allison, Missing Data, Sage, Thousand Oaks, CA, 2001.
13. [13] J. L. Schafer and J. W. Graham, Missing data: our view of the state of the art, Psychol. Methods 7, 147–177, 2002.
14. [14] O. Kulkarni and R. Chandra, Bayes-CATSI: A variational Bayesian deep learning framework for medical time series data imputation, Expert Syst. Appl. 2024.
15. [15] P. Bansal, P. Deshpande and S. Sarawagi, Missing Value Imputation on Multidimensional Time Series, arXiv preprint arXiv:2103.01600, 2021.
16. [16] J. Liu, J. Yao, J. Liu, Z. Wang and L. Huang, Missing data imputation and classification of small sample missing time series data based on gradient penalized adversarial multi-task learning, Appl. Intell. 54(3), 2528–2550, 2024.
17. [17] J. Wang, W. Du, W. Cao, K. Zhang, W. Wang, Y. Liang and Q. Wen, Deep learning for multivariate time series imputation: A survey, arXiv preprint arXiv:2402.04059, 2024.
18. [18] J. Park, J. Müller, B. Arora, B. Faybishenko, G. Pastorello, C. Varadharajan and D. Agarwal, Long-term missing value imputation for time series data using deep neural networks, Neural Comput. Appl. 35(12), 9071–9091, 2023.
19. [19] N. I. Fisher, Statistical Analysis of Circular Data, Cambridge Univ. Press, 1995.
20. [20] S. R. Jammalamadaka and A. SenGupta, Topics in Circular Statistics, World Sci., 2001.
21. [21] A. Pewsey, M. Neuhauser and G. D. Ruxton, Circular Statistics in R, Oxford Univ. Press, 2013.
22. [22] E. Batschelet, Circular Statistics in Biology, Academic Press, 1981.
23. [23] T. Abe and A. Pewsey, Sine-skewed circular distributions, Stat. Pap. 52(3), 683–707, 2011.
24. [24] M. J. Downes and K. L. Mengersen, Bayesian estimation of the concentration parameter of the von Mises distribution, Stat. Comput. 20(4), 433–444, 2010.
25. [25] S. Azimi and G. Tamazian, Missing Value Imputation of Circular Data Using Neural Networks, J. Appl. Stat. 48(6), 1086–1104, 2021.
26. [26] M. Chierici et al., Deep learning for circular data, Int. J. Comput. Intell. Syst. 13(1), 723–739, 2020.
27. [27] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Pearson, 2009.
28. [28] P. Bloomfield, Fourier Analysis of Time Series: An Introduction, Wiley, 2000.
29. [29] P. S. Addison, The Illustrated Wavelet Transform Handbook, CRC Press, 2017.
30. [30] R. J. Hyndman and G. Athanasopoulos, Forecasting: Principles and Practice, 2nd ed., OTexts, 2018.
31. [31] G. E. P. Box, G. M. Jenkins, G. C. Reinsel and G. M. Ljung, Time Series Analysis: Forecasting and Control, Wiley, 2015.
32. [32] P. J. Brockwell and R. A. Davis, Introduction to Time Series and Forecasting, Springer, 2016.
33. [33] D. U. Silverthorn, Human Physiology: An Integrated Approach, 6th ed., Pearson Press, 2012.
34. [34] A. Belouchrani and M. G. Amin, Blind source separation based on time-frequency signal representations, IEEE Trans. Signal Process. 46(11), 2888–2897, 1998.
35. [35] X. Hu, A. Song, J.Wang, H. Zeng and W.Wei, Finger movement recognition via highdensity electromyography of intrinsic and extrinsic hand muscles, Sci. Data 9(1), 373, 2022.
36. [36] V. Novak, K. Hu, L. Desrochers, P. Novak, L. Caplan, L. Lipsitz and M. Selim, Cerebral flow velocities during daily activities depend on blood pressure in patients with chronic ischemic infarctions, Stroke, 2010.
37. [37] A. L. Goldberger, L. A. N. Amaral, L. Glass, J. M. Hausdorff, P. Ch. Ivanov, R. G. Mark and H. E. Stanley, PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals, Circulation 101(23), e215– e220, 2000.
38. [38] F. Di Nardo, C. Morbidoni and S. Fioretti, Surface electromyographic signals collected during long-lasting ground walking of young able-bodied subjects, PhysioNet, version 1.0.0, 2022.
39. [39] F. Di Nardo, C. Morbidoni, A. Cucchiarelli and S. Fioretti, Influence of EMG-signal processing and experimental set-up on prediction of gait events by neural network, Biomed. Signal Process. Control 63, 102232, 2021.
40. [40] M. X. Cohen, MATLAB for Brain and Cognitive Scientists, MIT Press, 2017.
41. [41] M. Atzori, A. Gijsberts, I. Kuzborskij, S. Elsig, A. G. M. Hager, O. Deriaz and B. Caputo, Characterization of a benchmark database for myoelectric movement classification, IEEE Trans. Neural Syst. Rehabil. Eng. 23(1), 73–83, 2014.
42. [42] M. N. Norazian, Y. A. Shukri, R. N. Azam and A. M. M. Al Bakri, Estimation of missing values in air pollution data using single imputation techniques, ScienceAsia 34(3), 341–345, 2008.
43. [43] N. M. Noor, M. M. Al Bakri Abdullah, A. S. Yahaya and N. A. Ramli, Comparison of linear interpolation method and mean method to replace the missing values in environmental data set, Mater. Sci. Forum 803, 278–281, 2015.
44. [44] S. Thangaraj, V. T. Goh and T. T. V. Yap, Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU), F1000Research 11, 246, 2023.
45. [45] K. A. Bekiashev and V. V. Serebriakov, World Meteorological Organization (WMO), in: Int. Mar. Organ.: Essays Struct. Act., pp. 540–552, Springer, 1981.
46. [46] R. P. De Silva, N. D. K. Dayawansa and M. D. Ratnasiri, A comparison of methods used in estimating missing rainfall data, J. Agric. Sci. 3(2), 2007.
47. [47] W. Sanusi, W. Z. Wan Zin, U. Mulbar, M. Danial and S. Side, Comparison of the methods to estimate missing values in monthly precipitation data, Int. J. Adv. Sci. Eng. Inf. Technol. 7(6), 2168–2174, 2017.
48. [48] K. C. Young, A three-way model for interpolating for monthly precipitation values, Mon. Weather Rev. 120(11), 2561–2569, 1992.
49. [49] D. E. Frossard, I. O. Nunes and R. A. Krohling, An approach to dealing with missing values in heterogeneous data using k-nearest neighbors, arXiv preprint arXiv:1608.04037, 2016.
50. [50] F. Tang and H. Ishwaran, Random forest missing data algorithms, Stat. Anal. Data Min. ASA Data Sci. J. 10(6), 363–377, 2017.
51. [51] R. J. Little and D. B. Rubin, Statistical Analysis with Missing Data, 2nd ed., John Wiley & Sons, 2019.
52. [52] A. Phinyomark, F. Quaine, Y. Laurillau, S. Thongpanja, C. Limsakul and P. Phukpattaranont, EMG amplitude estimators based on probability distribution for musclecomputer interface, Fluct. Noise Lett. 12(03), 1350016, 2013.
53. [53] R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, Prentice Hall, 2002.
54. [54] A. Joshi, Y. Van de Peer and T. Michoel, Analysis of a Gibbs sampler method for model-based clustering of gene expression data, Bioinformatics 24(2), 176–183, 2007.
55. [55] A. Gelman, J. B. Carlin, H. S. Stern and D. B. Rubin, Bayesian Data Analysis, Chapman & Hall/CRC, 1995.
56. [56] J. W. Cooley, P. A. Lewis and P. D. Welch, The fast Fourier transform and its applications, IEEE Trans. Educ. 12(1), 27–34, 1969.
57. [57] A. A. Ligun, Approximation of periodic functions by splines of minimal defect, Ukr. Math. J. 32(3), 259–261, 1980.
58. [58] A. D. Roy and S. Karmakar, Time-varying auto-regressive models for count time series, Electron. J. Stat. 15(1), 2905–2938, 2021.
59. [59] A. Özmen, Y. Ylmaz and G.-W.Weber, Natural gas consumption forecast with MARS and CMARS models for residential users, Energy Econ. 70, 357–381, 2018.
60. [60] O. Kaynar, I. Yilmaz and F. Demirkoparan, Forecasting of natural gas consumption with neural network and neuro-fuzzy system, Energy Educ. Sci. Technol. Part A: Energy Sci. Res. 26(2), 221–238, 2011.
61. [61] E. Kürüm, G.-W. Weber and C. yigün, Early warning on stock market bubbles via methods of optimization, clustering and inverse problems, Ann. Oper. Res. 260(1–2), 293–320, 2018.
62. [62] S. Kuter, Completing the machine learning saga in fractional snow cover estimation from MODIS Terra reflectance data: Random forests versus support vector regression, Remote Sens. Environ. 255, 112294, 2021.
63. [63] Ö. Yaar and Ö. Y. Diner, On the applied mathematics of discrete tomography, J. Comput. Technol. 9(5), 14–32, 2004.
64. [64] S. Özogür et al., Real-Time SVM Classification of EMG Signals for Detecting Finger Movements in Prosthetic and Robotic Arm Control, Front. Neurosci. 16, 1001266, 2022.
65. [65] P. Taylan, F. Yerlikaya-Özkurt, B. B. Uçak and G.-W.Weber, A new outlier detection method based on convex optimization: Application to diagnosis of Parkinsons disease, J. Appl. Stat. 48(13–15), 2421–2440, 2020.
66. [66] A. Özmen, A. Çevik and G.-W. Weber, Voxel-MARS: A method for early detection of Alzheimer’s disease by classification of structural brain MRI, Ann. Oper. Res. 253(1– 2), 627–646, 2017.
67. [67] Ö. N. Onak, Y. S. Dorusöz and G.-W. Weber, Evaluation of multivariate adaptive non-parametric reduced-order model for solving the inverse electrocardiography problem: A simulation study, Med. Biol. Eng. Comput. 57(5), 967–993, 2019.
68. [68] F. Oriani, S. Stisen, M. C. Demirel and G. Mariethoz, Missing data imputation for multisite rainfall networks: a comparison between geostatistical interpolation and pattern-based estimation on different terrain types, J. Hydrometeorol. 21(10), 2325– 2341, 2020.
69. [69] T. Schneider, Analysis of incomplete climate data: Estimation of mean values and covariance matrices and imputation of missing values, J. Climate 14(5), 853–871, 2001.
70. [70] M. P. Tingley and P. Huybers, A Bayesian algorithm for reconstructing climate anomalies in space and time. Part I: Development and applications to paleoclimate reconstruction problems, J. Climate 23(10), 2759–2781, 2010.
71. [71] T. Chai and R. R. Draxler, Root mean square error (RMSE) or mean absolute error (MAE)?Arguments against avoiding RMSE in the literature, Geosci. Model Dev., 7(3) 1247–1250, 2014.
72. [72] R. J. Hyndman and A. B. Koehler, Another look at measures of forecast accuracy, Int. J. Forecast. 22(4), 679–688, 2006.
73. [73] C. J. Willmott and K. Matsuura, Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance, Climate Res. 30(1), 79–82, 2005.