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Classification of sonic wave Chapter 9 271

Soft voting can lead to different decision as compared with hard voting. Soft
voting entails computing a weighed sum of the predicted probabilities of all
models for each class. Hard voting uses predicted class labels for majority rule
voting, which is a simple majority vote for accuracy. For example, a voting clas-
sifier ensembles three classifiers. When the individual classifiers 1, 2, and 3 pre-
dict Class Y, Class N, and Class N, respectively, hard voting will lead to Class N
as the ensemble decision because 2/3 classifiers predict the Class N. When the
individual classifiers 1, 2, and 3 predict Class Y with probability of 90%, 45%,
and 36%, soft voting will lead to final prediction of Class Y because the average
probability is (90 + 45 + 36)/3 ¼ 57%. Soft voting can improve on hard voting
because it takes into account more information; it uses each classifier’s uncer-
tainty in the final decision. We use hard voting in this study.

5 Results for the classification-based noninvasive
characterization of static mechanical discontinuities
in materials

5.1 Characterization of material containing static discontinuities
of various dispersions around the primary orientation

5.1.1 Background
In this section, nine classifiers (discussed in Section 4.1) process compressional
wavefront travel times to categorize materials containing discontinuities in
terms of the dispersion around the primary orientation. The von Mises distribu-
tion is used to generate 100 discontinuities with various dispersions around a
primary orientation. The von Mises probability density function for the orien-
tation of discontinuity can be expressed as [19]
e kcos x μÞ
ð
fxj μ,kÞ ¼ (9.8)
ð
2πI 0 kðÞ
where x is the orientation of discontinuity, μ is the mode of the orientations of
the discontinuities, k is the concentration (inverse of dispersion) of the orienta-
tions around the mode, and I 0 is the modified Bessel function of order 0. The
distribution is clustered around μ and the dispersion around the mode is
expressed as 1/k. The dispersion 1/k controls the deviation of the orientations
around the mode μ. Mode and dispersion (1/k) are statistical parameters of
the von Mises distribution analogous to mean and standard deviation of Gauss-
ian distribution. von Mises distribution is an approximation to the wrapped nor-
mal distribution resulting from the “wrapping” of a normal distribution around a
unit circle. Three types of material containing discontinuities are generated with
concentration parameter (k) of 0, 5, and 1000 (Fig. 9.22).
Fig. 9.22 illustrates the materials containing discontinuities (upper row) and
the corresponding distribution of dispersion around the primary orientation of