Use The Coiflets and Daubechies Wavelet Transform To Reduce Data Noise For a Simple Experiment

Section: Research Paper
Issue
Vol. 19 No. 2 (2022): Volume 19 Issue 2
Published
Jun 25, 2025
Pages
91-103

Abstract

In this research, a simple experiment in the field of agriculture was studied, in terms of the effect of out-of-control noise as a result of several reasons, including the effect of environmental conditions on the observations of agricultural experiments, through the use of Discrete Wavelet transformation, specifically (The Coiflets transform of wavelength 1 to 2 and the Daubechies transform of wavelength 2 To 3) based on two levels of transform (J-4) and (J-5), and applying the hard threshold rules, soft and non-negative, and comparing the wavelet transformation methods using real data for an experiment with a size of 26 observations. The application was carried out through a program in the language of MATLAB. The researcher concluded that using the wavelet transform with the Suggested threshold reduced the noise of observations through the comparison criteria.

References

  1. ,
  2. , observation value of experiment , Experiment data after transformation
  3. : Abbreviation for the name of the researcher Ronald Coifman
  4. : Wavelet rank
  5. Ali, Taha, Husean & Mawlod , Kurdestan ,Ibrahem (2010)." Addressing the problem of contamination and variance heterogeneity in a complete random design using a small wave filter",Iraqi Journal of Statistical Science,No.18,Issue.10.
  6. . Nason, G. P. (Ed.). (2008). Wavelet methods in statistics with R. New York, NY: Springer New York.
  7. . Taher, M. M., & Ridha, S. M. (2022). The suggested threshold to reduce data noise for A factorial experiment. International Journal of Nonlinear Analysis and Applications, 13(1), 3861-3872.
  8. . Tammireddy, P. R., & Tammu, R. (2014). Image reconstruction using wavelet Transform with extended fractional Fourier transform.
  9. . Tang, H., Liu, Z. L., Chen, L., & Chen, Z. Y. (2013). Wavelet image denoising based the new threshold function. In Applied Mechanics and Materials (Vol. 347, pp. 2231-2235). Trans Tech Publications Ltd.
  10. . Zaeni, A., Kasnalestari, T., & Khayam, U. (2018, October). Application of wavelet Transformation symlet type and coiflet type for partial discharge signals denoising. In 2018 5th International Conference on Electric Vehicular Technology (ICEVT) (pp. 78- 82). IEEE.
  11. . Zhang, Y., Zhou, H., Dong, Y., & Wang, L. (2021). Restraining EMI of Displacement Sensors Based on Wavelet Fuzzy Threshold Denoising. In Signal and Information Processing, Networking, and Computers (pp. 543-551). Springer, Singapore.
  12. . Bruce, A. G., & Gao, H. Y. (1995, September). WaveShrink: Shrinkage functions and thresholds. In wavelet applications in signal and image processing III (Vol. 2569, pp. 270-281). SPIE.
  13. . Dehda, B., & Melkemi, K. (2017). Image denoising using new wavelet thresholding , Function. Journal of Applied Mathematics and Computational Mechanics, 16(2), 55-65.
  14. . Gao, H. Y. (1997). Wavelet Shrinkage Denoising Using the Non-Negative Garrote," Mathsoft. Inc. Tech. Rep.
  15. . Genay, R., Seluk, F., & Whitcher, B. J. (2001). An introduction to wavelets and other Filtering methods in finance and economics. Elsevier.
  16. . Han, G., & Xu, Z. (2016). Electrocardiogram signal denoising based on a new improved, Wavelet thresholding. Review of Scientific Instruments, 87(8), 084303.
  17. . He, C., Xing, J. C., & Yang, Q. L. (2014). Optimal wavelet basis selection for wavelet denoising of structural vibration signal. In Applied Mechanics and Materials (Vol. 578, pp. 1059-1063). Trans Tech Publications Ltd.
  18. . In, F., & Kim, S. (2013). An introduction to wavelet theory in finance: a wavelet multiscale Approach. World Scientific.
  19. . Montgomery, D. C. (2020). Design and analysis of experiments. John Wiley & sons.
  20. *The Scheme was prepared by the researcher
  21. +
  22. 23
  23. 24
  24. 26
  25. 27
  26. 28
  27. 29
  28. 30
  29. 31
  30. 32
  31. 33
  32. 34
  33. 35
  34. -
  35. -When applying The Discrete wavelet transform at the level (j-4), it led to a decline in the value of MSe and CV for all cases, but at the level (j-5), there was an increase and decrease in the values of the Criteria MSe and CV,in this experiment.
  36. Introduction
  37. Conclusion
  38. 12
  39. 21
  40. 25
  41. 57
  42. 68
  43. 78
  44. 93
  45. 1
  46. 8
  47. 9
  48. 3
  49. 02
  50. 03
  51. 45
  52. 3
  53. 1
  54. 9
  55. 41
  56. 2
  57. 61
  58. 1
  59. 8
  60. 4
  61. 83
  62. 4
  63. 1
  64. 01
  65. 11
  66. 22
  67. 9
  68. 1
  69. -Obtaining the best results when applying the hard threshold rule with the Universal and suggested threshold according to standards.
  70. Discrete Wavelet transform (DWT)
  71. 5
  72. 4
  73. -When processing the observations noise (wheat crop 26) and the level (j-4), the suggested threshold gave better results than the Universal Threshold based on the criteria MSe, CV, MSe(w), SNR. and The best results were obtained for the first-order Coiflet wavelet transformation filter with the hard threshold rule, the second-order Daubechies wavelet transformation filter with the non-negative threshold rule, and second-order Coiflet wavelet transformation filter with the non-negative threshold rule,in this experiment.
  74. The Transfer levels Multi Resolution Analysis
  75. 2
  76. 6
  77. 5
  78. 4
  79. -When processing the observations noise (wheat crop 26) and the level (j-5), the suggested threshold gave better results than the Universal Threshold based on the criteria MSe, CV, MSe(w), SNR. and The best results were obtained for the first-order Coiflet wavelet transformation filter with the hard threshold rule, the second-order Daubechies wavelet transformation filter with the hard threshold rule, and the second-order Coiflet wavelet transformation filter with the hard threshold rule,in this experiment.
  80. Stages of Discrete Wavelet Transform Using Orthogonality
  81. 1. Representing the observations of a randomized complete block design (RCBD) with a vectorThat contains all the observations of the experiment, which the following mathematical model represents
  82. 2.Applying Coiflets transform and the wavelet Daubechies on observation the Experiment, we will have wavelet coefficients that can be represented Using an orthogonal matrix , and multiplying it by the observations vector, the Wavelet Coiflets and Daubechies coefficients vector of the following form is Obtained.
  83. 3. We note that the formula (6) can be obtained through which the values of the original Data Depending on the orthogonality condition of the Discrete wavelet coefficients.
  84. 4.Threshold rules (hard, soft, non-negative) are applied with threshold limit Universal Threshold and suggested threshold.
  85. 82
  86. 3
  87. 1
  88. 9
  89. 2
  90. 9
  91. Daubechies Wavelet
  92. 43
  93. 55
  94. 64
  95. 94
  96. 8
  97. 8
  98. 5
  99. Coiflets Wavelet
  100. 23
  101. 31
  102. 44
  103. 49
  104. 79
  105. 83
  106. 92
  107. 8
  108. Threshold Rules
  109. 05
  110. 1. Hard Threshold Rule
  111. 17
  112. 2. Soft Threshold Rule
  113. 3 Non-Negative Garrote Threshold Rule
  114. 5
  115. 4
  116. 1
  117. Evaluation Criteria
  118. 03
  119. 08
  120. 1.The mean square error for design is defined by the following formula(Montgomery, 2020):
  121. 19
  122. 2. The coefficient of variation(cv)is defined by the following formula: (Montgomery, 2020):
  123. 3. The mean of squares for original and transform observation is defined by the following Formula (He et al.,2014):
  124. 4.The Signal-to-noise ratio (SNR) is defined by the following formula(He et al.,2014):
  125. 90
  126. The practical side
  127. 1 Results
  128. 11
  129. 36
  130. 75
  131. 2
  132. 9
  133. 4
  134. According to the complete random blocks design (CRBD) of four blocks, each block contained 16 experimental units. Then the characteristics of the field yield were taken, which are (number of branches, plant height cm, dry weight of g, number of spikes / m2, number of seeds/spike, weight of 1000 grains of/ g, Grain yield / m), where the trait was studied: grain yield / m2.
  135. And It is considered one of the standard techniques for processing observational noise (Tang et al., 2013) (Han & Xu, 2016).
  136. and The following diagram shows the steps of the wavelet transform, with the comparison criteria
  137. Application Non N rule with Suggested
  138. Application Non N rule with Universal
  139. Application Hard rule with Suggested
  140. Application Hard rule with Universal
  141. Application Soft rulewith Suggested
  142. Application Soft rulewith Universal
  143. Approximation
  144. Approximation coefficients
  145. Best Vector
  146. By applying equations (10-19) on experiment observations, The results are shown in table (1), Which represents a summary of wavelet transformation when the level of analysis is (J-4)
  147. By applying equations (10-19) on experiment observations, The results are shown in table (2), Which represents a summary of wavelet transformation when the level of analysis is (J-5)
  148. CA
  149. CAA
  150. CAAA
  151. CAAAA
  152. CAAAD
  153. CAAD
  154. CAD
  155. Calculation of the Suggested Threshold value
  156. Calculation of the Universal Threshold value
  157. CD
  158. Coif N
  159. Coiflets Wavelet
  160. considered wavelet haar of one the members of this family of wavelets and is symbolized by the symbol db1 or The wavelet is called Daubechies the first order Because built from the function of the father (father wave), the function of the mother (mother wave), as follows (Tammireddy & Tammu, 2014).
  161. CV
  162. Data
  163. Data filter coefficients
  164. Daubchies Wavelet
  165. Db: Abbreviation for the name of the researcher Daubechies
  166. DbN
  167. Detail coefficients
  168. Details
  169. DWT1
  170. DWT2
  171. DWT3
  172. DWT4
  173. Equation (7) represents the Universal Threshold (UT) value (Genay et al.,2001)(Zhang et al., 2021), while equation (8) represents the Suggested Threshold (ST) (Taher & Ridha, 2022).
  174. Evaluation Criteria's
  175. father wavelet Coiflets
  176. father wavelet Daubechies
  177. Figure (1): High-pass filter and low-pass filter for x vector
  178. Figure (5): Stages of The Wavelet Transform in this research
  179. Figure Next represents a Daubechies Wavelet of several lengths
  180. Figure(2):The wavelet transform levels for size N = 32
  181. Figure(3): the wave function and The scale function of a wave Daubechies
  182. Figure(4): the wave function and The scale function of a wave Coiflets
  183. For the purpose of comparing the results between the Coiflets Wavelet Transformation and the Daubechies Wavelet Transformation, several criteria were applied,
  184. From the formula (9), we get a vector coefficient of a discrete wavelet which can be represented in the following form.
  185. he Second component is called Approximate, which includes low frequencies (noise) or (anomalous) values according to the nature of the study and its application, and it can be calculated from the father wavelet by the following formula: (In, F., & Kim, S 2013)
  186. High pass filter
  187. Homogeneity of variance
  188. In general, the discrete wavelet transform is used with data that contain discrete variables and have discrete outputs.
  189. Input Data
  190. It is also built from the father function (father wavelet) and mother function (mother wavelet), in the following formula (Tammireddy & Tammu, 2014)
  191. It is also the base of wavelet Shrink introduced by Gao
  192. It is another type of threshold rule and can be written as (Bruce & Gao , 1995).
  193. Level 1
  194. Level 2
  195. Level 3
  196. Level 4
  197. Low passfilter
  198. Mother wavelet Coiflets
  199. Mother wavelet Daubechies
  200. MSe
  201. MSe ,CV. MSe(w),SNR
  202. MSe(w)
  203. N: Wavelet rank
  204. Normal
  205. observation value (j) from treatment (i),The general arithmetic mean,The effect of the i-treatment for this observation,The amount of random error , number of treatments number of blocks
  206. One of the types of threshold rules, it is applied in discrete wavelet transform and takes the following form (Dehda & Melkemi, 2017),( Zaeni et al.,2018).
  207. Orthogonal matrix
  208. P.v=0.02
  209. P.v=0.08
  210. P.v=0.09
  211. P.v=0.1
  212. P.v=0.13
  213. P.v=0.14
  214. P.v=0.17
  215. P.v=0.82
  216. P.v=0.85
  217. P.v=0.86
  218. P.v=0.88
  219. P.v=0.89
  220. P.v=0.91
  221. P.v=0.92
  222. P.v=0.93
  223. P.v=0.94
  224. P.v=0.95
  225. P.v=0.96
  226. P.v=0.97
  227. P.v=0.98
  228. P.v=0.99
  229. P.v>0.15
  230. Pv>0.05
  231. References
  232. Represents a high-pass filter (In & Kim, 2013)
  233. Several types of Wavelets exist through the above offer About Discrete Wavelet transformation and orthogonality. In this research, we will address: Daubechies Wavelet, Coiflets Wavelet
  234. SNR
  235. start
  236. Table (1) represents a summary of the wavelet transform results for all cases when the transform level is (J-4), where led to decline in the MSe value of the used design and a significant improvement in the CV criterion. In addition to obtaining low values for the standard .with an increase in the SNR criterion value, Especially for ( XD2h3st , XD2n2st , XD3n2st , XC1h2st , XC2h3st , XC2n2st )
  237. Table (1): The best results of wavelet transformation when the
  238. Table (2): The best results of wavelet transformation when the
  239. Table(2) represents a summary of the wavelet transform results for all cases when the transform level is (J-5), where led to decline in the MSe value of the used design and a significant improvement in the CV criterion. In addition to obtaining low values for the standard .with an increase in the SNR criterion value, Especially for( XD2n2th, XD3h3th , XC1h2th, XC2h2th).
  240. Test
  241. The Discrete wave transform is one of the most Transfers used in the wavelet due to its multiple applications in various practical fields and its theoretical uses in various sciences. The researcher will give a comprehensive idea of this transformation and focus on its use in designing experiments Through the application of a simple experiment.
  242. The field experiment on wheat cultivation was conducted in one of the stations of the National Program for the Development of Wheat Cultivation in Iraq, and sixteen wheat varieties were included.(al sds (12) A1, al sahel (1)A2 , al sds (1)A3 , Egypt (1)A4 .Egypt (2) A5, Sakha (93)A6, al Geza (11)A7, al Geza (168)A8, Apaa (99)A9, Italia(1)A10, Italia (2)A11, Caronia A12, gold kernels A13, aom al rabee A14, Smitto A15, Waha A16).
  243. The following figure shows the division of data into two components
  244. The name came in relation to the researcher Ingrid Daubechies (Tammireddy & Tammu, 2014). It has made a boom in the wavelet theory, as it is generated from a group of intermittent wavelets. The most crucial characteristic of this family of wavelets is their smoothness. It is abbreviated as follows.
  245. The next stage is how to extract information from vector , where the information extracted from vector It is called (detail), which can be obtained from different locations and levels, and in general, the word "detail" means "degree of difference" or "variance" in the observations of the vector. This information is calculated based on the following two equations (Nason,2008).
  246. The next step is calculating the elaboration and measurement coefficients for the other levels (Nason,2008).
  247. The principle of The work of this algorithm is to create filters for smoothing and heterogeneity from the wavelet coefficients, and these filters are used frequently to obtain data for all scales, meaning that the wavelet transform splits the data into two components, the first component is called detail, which includes high frequencies and can be calculated from the mother wavelet by the following formula ( In & Kim, 2013)
  248. The vector x contains all the observations of the experiment. One of the critical conditions in the wavelet transformation is The size of the observations fulfills the following condition.
  249. The wavelet coefficients vector
  250. The wavelet transform has been discussed in addressing pollution and heterogeneity (Ali & Mawlood,2010)for the complete randomized design using wavelet filters and some types of threshold rules. In addition, a threshold was Suggested to reduce the observations noise for a factorial experiment by (Taher&Sabah,2022) compared with the Universal Threshold. In this research, the application of different levels of analysis, through the use of the Daubechies transform of wavelengths from 2 to 3 and the transformation of Coiflets of wavelengths from 1 to 2 with hard, soft, and non-negative threshold rules, and comparison of the results.
  251. The work of the discrete wavelet transform depends on the Mallat pyramidal algorithm, which is an efficient algorithm proposed by the researcher Mallat (1989) (Nason, 2008) to calculate the wavelet coefficients for a set of data containing noise
  252. They are discrete wavelets designed by Ingrid Daubechies, at the request of Ronald Coifman (Tammireddy & Tammu, 2014).
  253. This family of wavelets is characterized by the presence of a relationship that relates the length of the filter with its rank
  254. This family of wavelets is considered orthogonal and is close to symmetry, as it connects the mother and father function through the high-pass filter and the low-pass filter by obtaining the vanishing torque, unlike each filter separately.
  255. This process is called the multiscale transform algorithm, Through the simple experiment applied, we will have the following levels { (j) , (j-1) , (j-2) , (j-3) , (j-4) , (j-5)}.
  256. This threshold is characterized by the small samples. It is less sensitive to observation than the hard threshold, especially in small fluctuations, and it is less biased than the soft threshold and can be written as follows (Gao,1997).
  257. Three types of threshold rules will be applied in this research, namely:
  258. Through the above formulas (10) and (11), we note that if the coefficients of the wavelet are less than the threshold value, it goes to zero, but in the case of being greater than the threshold value, it preserves its value. Using a Shrinking wavelet based on a soft-threshold rule tends to bias because all large coefficients Shrink towards zero.
  259. Through the formula (9) where the coefficients whose values are greater and equal to the threshold do not change, and the coefficients whose values are less than the threshold are replaced by the value zero
  260. Through the observations vector, the levels of analysis for this experiment will be (Nason, 2008).
  261. To clarify what was mentioned above, we take the following figure, which shows the discrete wavelet transformation coefficients for the data for four levels .
  262. Transform levels are determined from the design observation, and through the application side, we will have a simple experiment containing sixteen treatments and four blocks represented by the following vector.
  263. transformation level is (j-4) for The first experiment
  264. transformation level is (j-5) for The first experiment
  265. Vector observation
  266. wavelets
  267. We will give a brief overview of the key concepts of multiscale analysis before attempting formal definitions of wavelets and the wavelet transform and How we extract multiscale "information" from the vector y . We identify the "detail" in the sequence at various scales and places as the essential information.
  268. Where total sum of squares , treatments Sum of squares , blocks Sum of squares
  269. Where Coiflets scaling function coefficients
  270. Where Coiflets wavelet function coefficients
  271. Where Daubechies scaling function coefficients
  272. Where Daubechies wavelet function coefficients
  273. Where XD2h3st :Represents the second-order Daubechies wavelet transformation filter with the hard rule and suggested threshold,And so on for the rest of the vectors
  274. where's
  275. Wheres:,
  276. Whereas
  277. Whereas Represents a low-pass filter, (In & Kim, 2013).and it is related to through
  278. Whereas, Represents the first component of the transform, which is the detail coefficients computed from the rate of the difference of the data at each measurement and is symbolized by CD. As for Represents the second component of the transform, which is the approximation coefficients and represents the rate of The measurement is symbolized by CA.
  279. X
  280. X[n]
  281. XC1h2st
  282. XC1h2th
  283. XC1h3st
  284. XC1n2st
  285. XC1n2th
  286. XC1n3st
  287. XC1n3th
  288. XC1s2st
  289. XC1s2th
  290. XC2h2th
  291. XC2h3st
  292. XC2h3th
  293. XC2n2st
  294. XC2n2th
  295. XC2n3st
  296. XC2n3th
  297. XC2s2st
  298. XC2s2th
  299. XD2h2th
  300. XD2h3st
  301. XD2h3th
  302. XD2h4st
  303. XD2n2st
  304. XD2n2th
  305. XD2n3st
  306. XD2n3th
  307. XD2s2st
  308. XD2s2th
  309. XD2s3st
  310. XD3h2th
  311. XD3h3st
  312. XD3h3th
  313. XD3n2st
  314. XD3n2th
  315. XD3n3st
  316. XD3n3th
  317. XD3s2st
  318. XD3s2th

Authors

Mahmood M Taher

Department of Informatics & Statistic, College of Computer & Mathematical Science, University of Mosul, Mosul, Iraq

Sabah Manfi Redha

Department of Statistics, College of Administration And Economics , Baghdad University, Iraq.

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M Taher, M., محمود, Manfi Redha, S., & صباح. (2025). Use The Coiflets and Daubechies Wavelet Transform To Reduce Data Noise For a Simple Experiment. IRAQI JOURNAL OF STATISTICAL SCIENCES, 19(2), 91–103. Retrieved from https://rjps.uomosul.edu.iq/index.php/stats/article/view/20898