Evaluation of Effectiveness of Wavelet Based Denoising Schemes Using ANN and SVM for Bearing Condition Classification

Table 3

Time and frequency domain features extracted for the study.

Sl. number	Feature
(1)	𝑇1=𝑛^𝑛−1∑𝑖=1𝑥𝑖
(2)	𝑇2=𝑛^𝑛−1∑𝑖=1𝑥^2𝑖^1/2
(3)	𝑇3=(𝑛−1)^𝑛−1∑𝑖=1𝑥𝑖−𝑇1^21/2
(4)	𝑇4=12Max𝑥𝑖−Min𝑥𝑖
(5)	𝑇5=𝑛^𝑛−1∑𝑖=1𝑥𝑖^2
(6)	𝑇6=𝑛^𝑛−1∑𝑖=1𝑥𝑖
(7)	𝑇7=(𝑛−1)^𝑛−1∑𝑖=1𝑥𝑖−𝑇1𝑇3^3
(8)	𝑇8=(𝑛−1)^𝑛−1∑𝑖=1𝑥𝑖−𝑇1𝑇3^4
(9)	𝑇9=𝑇4𝑇2
(10)	𝑇10=𝑇4𝑇5
(11)	𝑇11=𝑇2𝑇6
(12)	𝑇12=𝑇4𝑇6
(13)	𝑇13=𝑇3𝑇6
(14)	𝑇14=log𝑇3^𝑛−1∑𝑖=1log𝑥𝑖+1
(15)	𝑇15=𝑇5𝑇3
(16)	𝑇16=−∑^𝑛𝑖=1log𝑇3√2𝜋^−1−𝑥exp^𝑖−𝑇1^22𝑇^23
(17)	𝑇17=−∑^𝑛𝑖=1log𝛽𝜂^−𝛽𝑥𝑖^𝛽−1𝑥exp^𝑖𝜂^𝛽,
	where 𝛽 is the shape factor and 𝜂 is the scale factor
(18)	𝐹1=𝑁^𝑁−1∑𝑘=1𝑆(𝑘)
(19)	𝐹2=(𝑁−1)^𝑁−1∑𝑘=1𝑆(𝑘)−𝐹1^2
(20)	𝐹3=𝑁^−1𝐹2^𝑁−1.5∑𝑘=1𝑆(𝑘)−𝐹1^3
(21)	𝐹4=𝑁^−1𝐹2^𝑁−2∑𝑘=1𝑆(𝑘)−𝐹1^4
(22)	𝐹5=∑^𝑁𝑘=1𝑓𝑘𝑆(𝑘)∑^𝑁𝑘=1𝑆(𝑘)^−1
(23)	𝐹6=𝑁^𝑁−1∑𝑘=1𝑓𝑘−𝐹5^2𝑆(𝑘)^1/2
(24)	𝐹7=∑^𝑁𝑘=1𝑓^2𝑘𝑆(𝑘)∑^𝑁𝑘=1𝑆(𝑘)^−11/2
(25)	𝐹8=∑^𝑁𝑘=1𝑓^4𝑘𝑆(𝑘)∑^𝑁𝑘=1𝑓^2𝑘𝑆(𝑘)^−11/2
(26)	𝐹9=∑^𝑁𝑘=1𝑓^2𝑘𝑆(𝑘)∑^𝑁𝑘=1𝑆(𝑘)∑^𝑁𝑘=1𝑓^4𝑘𝑆(𝑘)^−1/2
(27)	𝐹10=𝐹6𝐹5
(28)	𝐹11=𝑁^−1𝐹6^𝑁−3∑𝑘=1𝑓𝑘−𝐹5^3𝑆(𝑘)
(29)	𝐹12=𝑁^−1𝐹6^𝑁−4∑𝑘=1𝑓𝑘−𝐹5^4𝑆(𝑘)
(30)	𝐹13=𝑁^−1𝐹6^𝑁−0.5∑𝑘=1𝑓𝑘−𝐹5^1/2𝑆(𝑘)

*Features 1 to 17 (𝑇₁ to 𝑇₁₇) are statistical features extracted from data in time domain and 18 to 30 (𝐹₁ to 𝐹₁₃) are features extracted from data in the frequency domain. 𝑛 is the number of data points in the time domain signal, 𝑥_𝑖 is the acceleration amplitude of 𝑖th data point in the time domain signal, 𝑁 is the number of lines in the frequency spectrum, 𝑆(𝑘) is the amplitude of the 𝑘th line in the frequency spectrum, 𝑓_𝑘 is the frequency value of the 𝑘th line in the frequency spectrum [18, 19].