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Review of Basic Laws and Equations
Published in Pradip Majumdar, Computational Methods for Heat and Mass Transfer, 2005
A norm is a single-valued quantity that represents the size of vectors or/and matrices. For example, for a vector {x}, the Euclidean norm is defined as ∥x∥e=∑i=1nxi2=x12+x22+⋯+xn2
Approximation
Published in Phillip A. Regalia, Adaptive IIR Filtering in Signal Processing and Control, 2018
The Frobenius (or Hilbert-Schmidt) norm, denoted ║ · ║F, is defined as the square root of the sum of the squares of each entry, i.e., ‖P‖F=(∑k,l=1∞pk,l2)1/2
Localization
Published in Prabhakar S. Naidu, Distributed Sensor Arrays Localization, 2017
where ʌ is the error matrix in the system matrix, (A), and ‖⋅‖F stands for the Frobenius norm of a matrix. The Frobenius norm is defined as a square root of the sum of the mod square of all matrix elements. We minimally perturb the system matrix A and data vector b while the constraint in Equation 4.75 is satisfied. Let us express this constraint in a slightly different form,
Efficient learning algorithm for sparse subsequence pattern-based classification and applications to comparative animal trajectory data analysis
Published in Advanced Robotics, 2019
Takuto Sakuma, Kazuya Nishi, Kaoru Kishimoto, Kazuya Nakagawa, Masayuki Karasuyama, Yuta Umezu, Shinsuke Kajioka, Shuhei J. Yamazaki, Koutarou D. Kimura, Sakiko Matsumoto, Ken Yoda, Matasaburo Fukutomi, Hisashi Shidara, Hiroto Ogawa, Ichiro Takeuchi
We use the following notation in the rest of the paper. For any natural number n, we define . For an n-dimensional vector and a set , represents a subvector of whose elements are indexed by . The indicator function is written as ; i.e. if z is true, and otherwise. The L1 norm of a vector is written as . A sequence (an ordered list of discrete symbols) with length T is represented as .
Multilevel weather detection based on images: a machine learning approach with histogram of oriented gradient and local binary pattern-based features
Published in Journal of Intelligent Transportation Systems, 2021
Md Nasim Khan, Anik Das, Mohamed M. Ahmed, Shaun S. Wulff
Once the histogram of gradient was obtained, a block normalization technique with a block size of 2 × 2 cells was applied to each image, as illustrated in Figure 2c. The normalization was necessary for eliminating any potential effect of lighting variations (Dalal & Triggs, 2005). Since a 2 × 2 block contained four histograms with five bins in each histogram, concentrating the block produced a vector of size 20 × 1. A normalization factor was then calculated for the vector containing all histograms of a given block using “L2 norm”. The “L2 norm” is the square root of the sum of the squared vector values and can be described using Equation 4 (Strang, 2009; Suard et al., 2006).
High-order methods for the simulation of unsteady counterflow flames subject to stochastic forcing of large amplitude
Published in Combustion Theory and Modelling, 2023
Errors are shown in Figure 3 for T (temperature) and U (axial mass flux) as 2-norm and infinity norm of the error. Recall that the infinity norm is defined as the maximum absolute value of the vector elements. The data in Figure 3 show that the asymptotic orders of convergence are close to the theoretical ones.