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Matrix Calculus for Machine Learning
Published in Richard M. Golden, Statistical Machine Learning, 2020
Let Ω⊆Rn be an open set whose elements are column vectors. The Jacobian matrix of a vector-valued function f:Ω→Rm is a matrix-valued function df/dx:Ω→Rm×n whose element in the ith row and jth column is the partial derivative of the ith element of f with respect to the jth element of the n-dimensional vector variable x.
Some Thoughts on Mathematical Models For Aircraft Accidents Simulation
Published in Hans M. Soekkha, Aviation Safety, 2020
where the Jacobian matrix J describes the rate of change of the output vector with respect to the control vector. Assuming that the problem to be solved involves a vehicle with n controls flying a manoeuvre defined by n parameters then for the Kth estimate at the Cth time point, the Jacobian is an n×n matrix, the entries of which, ji,j(tc)K are evaluated by differentiating each of the elements of the output error vector yd(tC+1)K with respect to each of the elements of the control vector dt(tc)k. The expression for determining a Jacobian elements is: ji,j(tC)K=∂yej(tC+1)K∂dj(tC)K
Constrained Optimization and Applications
Published in James A. Momoh, Electric Power System Applications of Optimization, 2017
The Jacobian matrix is the array of all first-order partial derivatives of a vector-valued function such that: J(x)=∇(f(x))=[∂fi∂xj]n×n,wherei∈{1,n}andi∈{1,n}
Computed Torque Control Method for Two-Axis Planar Serial Robotic Arm
Published in IETE Journal of Research, 2023
G. Perumalsamy, Deepak Kumar, Joel Jose, S. Joseph Winston, S. Murugan
The inverse position kinematics determines incremental joint angle for the given change in end effector position. The incremental joint angle, needed to achieve the incremental end effector movement, is calculated by Equation (3) where represents steam generator robotic arm linear displacement and denotes incremental angular displacement of revolute joints of the robotic arm for the corresponding change in linear displacement. Where The Jacobian matrix transforms joint velocity to end effector velocity. The Jacobian matrix is configuration dependent [5].
Wind Speed Predictability Accuracy with Height Using LiDAR Based Measurements and Artificial Neural Networks
Published in Applied Artificial Intelligence, 2021
M. Mohandes, S. Rehman, H. Nuha, M.S. Islam, F.H. Schulze
where and are the values of the weights at and iterations, correspondingly. The Jacobian matrix J contains first derivatives of model output with respect to the optimizing parameters. Measured and predicted output values are denoted byand, respectively. Increasing the damping parameterdecreases the step size, and vice versa. Therefore, if a step is unacceptable, should be increased for a smaller step. If a step is accepted, is decreased in order to proceed more quickly in the correct descent direction, speeding up the convergence.
Evolutionary game analysis of pedestrian-autonomous vehicle interactions at unsignalized road sections: a policy intervention perspective
Published in Transportation Letters, 2022
Rong Rui, Xusheng Yao, Shunqiang Ye, Shoufeng Ma
In the construction of an evolutionary game model, the Jacobian matrix and the evolutionary stable strategy (ESS) are commonly adopted mathematical forms. The Jacobian matrix of a vector-valued function is the matrix of first-order partial derivatives. Equilibrium points can be derived by solving the Jacobian matrix and first-order partial derivatives. The ESS is a strategy adopted by a population in the adaptation process, and it cannot be replaced by an alternative strategy (Roca, Cuesta, and Sánchez 2009). It is an important terminology in evolutionary biology and game theory. Specifically, an ESS refers to an evolutionarily stable equilibrium. In most simple games, the ESS and Nash equilibrium can be considered equivalent.