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Principal Components of Principal Component Analysis
Published in Mark Stamp, Introduction to Machine Learning with Applications in Information Security, 2017
To geometrically illustrate the SVD factorization, we can consider the action of a 2-dimensional shear matrix [8], as illustrated in the top row in Figure 4.7. Shear transformations can be used, for example, to convert a letter in a standard font to its italic or slanted form. The matrix M in Figure 4.7 stretches and rotates the unit disc, yielding an ellipse with the same area. By computing the SVD, we obtain the factorization M = USVT, where, as illustrated in Figure 4.7, the matrices VT and U are rotations, with S having the effect of scaling. Consequently, we see that the SVD factors the linear transformation M in a natural and intuitive way, at least in this particular case. For the purposes of PCA, the matrix U contains the essential information that we’ll need for training and scoring. And we’ll see that the matrices V and S are useful too.
Motorway Impact Attenuation Devices: Past, Present and Future
Published in Norman Jones, Tomasz Wierzbicki, Structural Crashworthiness and Failure, 1993
When the hex-foam material is crushed, the walls of one honeycomb layer shear into the walls of the adjoining honeycomb layer. The polyurethane foam provides both gussetting and stabilization for all the honeycomb cells, as well as additional shearing resistance when the honeycomb walls shear into the foam. Crushing of foam and honeycomb walls also occurs, and adds to the overall crushing force. During crushing, the compressive force level continually increases until full crush is achieved, at approximately 90% compression. Because of the interlocking effect of the honeycomb shear matrix, the rebound after impact is virtually eliminated. In addition, the material provides a certain degree of rate sensitivity during crush.
Strength
Published in Ever J. Barbero, Introduction to Composite Materials Design, 2017
Prediction of matrix failure is important to identify undesirable laminate designs, that is, laminates with fracture controlled by matrix failure. Matrix cracking of the individual laminas of a PMC laminate does not usually lead to overall rupture of the laminate. If load is primarily carried by matrix, such as a [0/90]S $ _{S} $ loaded in shear, matrix cracking leads to collapse. But [0/90]S $ _{S} $ is an inefficient laminate for shear and ±45 $ \pm 45 $ laminas should be added. In general, when the laminate is efficiently designed and it contains at least laminas with all three angles (0, 90, and ±45 $ \pm 45 $ ) matrix cracking does not affect significantly the static strength of the laminate. A common design rule is to use at least 10% of each 0, 90, and ±45 $ \pm 45 $ in any laminate.
A novel framework of multimodal medical image fusion using adaptive NSST and optimized deep learning approach
Published in The Imaging Science Journal, 2023
K. Vanitha, D. Satyanarayana, M. N. GiriPrasad
The matrix representation for anisotropic is given in the form of or , where , is used for scaling transform factor. On the other hand, the shear matrix is signified by or , is used for directional or geometrical factors. Hence, the transform function in NSST is expressed using Equations (2) and (3). Here, the ranges for three factors like , correspondingly and term declares as the basis function of regional space. In Equation (2), and in Equation (3), . Finally, the source image is decomposed into two frequency bands with the aid of the NSLP filter coefficient.
In-situ damage monitoring and numerical characterization of three-point bending and incremental cycle flexural behavior of FMLs
Published in Mechanics of Advanced Materials and Structures, 2023
Lu Yao, Lizhou Mao, Yanchao Wang, Wentao He, Yan Ma, Hang Yu
With the continued bending deformation, the transient stage D has passed, followed by another sudden drop of the flexural load. Compared with stage D, stage E is obviously gone through a more extended period, which can be attributed to the more damage evolution in FMLs-A. As shown in Figure 11, the gradually serious damage morphologies can be observed from E(1) to E(3), including the evident fiber kink-band, the matrix cracking, the fracture of a fiber bundle, the intralaminar delamination between different composite layers, and interlaminar delamination between composite laminates and aluminum layer. The partial enlargement pictures of Figure 11(E1) are also shown in Figure 12. For stage E, the fiber fracture in 0° composite layers and matrix cracking in 90° composite layers are the dominant damage modes, which are caused by the tension stress. It should be noted that the matrix damage during the middle of the specimen includes bending matrix cracking and shear matrix cracking. And the matrix cracking is terminated in the interface between the vertical composite layers, then the interlaminar delamination damage between layers further dissipates the energy from bending deformation [43]. In addition, the aluminum layer has no cracking in the cross-section of FMLs-A, which indicates that the aluminum alloy is still in the plastic deformation and has a certain loading capacity. This also confirms the slightly upward trend of the curve during stage E.
Seismic data denoising using curvelet transforms and fast non-local means
Published in Petroleum Science and Technology, 2022
Siwei Zhao, Ibrar Iqbal, Xiaokang Yin, Tianyu Zhang, Mingkun Jia, Meng Chen
Defining as the one-dimensional low-pass window inner product, where and introducing a set of equally spaced slopes then we can get: where, is a shear matrix, the angle is not uniformly distributed, but its slope is uniformly distributed, is the window function in the spectral domain,