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Kinematic data
Published in Paul Grimshaw, Michael Cole, Adrian Burden, Neil Fowler, Instant Notes in Sport and Exercise Biomechanics, 2019
Numerical differentiation is a mathematical process that quantifies the change in one variable with respect to another. In the case of sport and exercise biomechanics, these variables are often displacement, velocity and acceleration with respect to time. The biomechanical study of human motion requires an understanding of the precise relationship between the changes in these variables. As outlined elsewhere in this text, it is shown that the average velocity of any moving object is given by the change in position (displacement) divided by the time over which the change takes place. If position is represented by the letter p and time by the letter t, the average velocity (v) between time 1 and time 2 may be determined using: v=p2−p1t2−t1orΔpΔt
Machine Learning - A gentle introduction
Published in Nailong Zhang, A Tour of Data Science, 2020
It is worth distinguishing numerical differentiation, symbolic differentiation and automatic differentiation. Symbolic differentiation30 aims at finding the derivative of a given formula with respect to a specified variable. It takes a symbolic formula as input and also returns a symbolic formula. For example, by using symbolic differentiation we could simplify ∂(x2 + y2)/∂x to 2x. Numerical differentiation is often used when the expression/formula of the function is unknown (or in a black box, for example, a computer program). Compared with symbolic differentiation, numerical differentiation and automatic differentiation try to evaluate the derivative of a function at a fixed point. For example, we may want to know the value of ∂f(x, y)/∂x when x = x0; y = y0. The most basic formula for numerical differentiation is limh→0(f(x + h)) − f(x))/h. In implementation, this limit could be approximated by setting h to a very small value, for example 1e − 6. Automatic differentiation also aims at finding the numerical values of derivatives, in a more efficient approach based on chain rule. However, unlike the numerical differentiation, automatic differentiation requires the exact function formula/structure. Also, automatic differentiation does not need the approximation for the limit which is done in numerical differentiation. At the lowest level, it evaluates the derivatives based on symbolic rules. Thus automatic differentiation is partly symbolic and partly numerical [2].
Basics of Numerical Calculation and Series
Published in Caio Lima Firme, Quantum Mechanics, 2022
The numerical differentiation is the mathematical procedure to estimate the derivative of a determined function. f′(x)=limh→0f(x+h)−f(x)h
Robust design optimisation of adaptive cruise controller considering uncertainties of vehicle parameters and occupants
Published in Vehicle System Dynamics, 2020
Hansu Kim, Tae Hee Lee, Yuho Song, Kunsoo Huh
The Taylor series expansion method involves the approximation of performance function through Taylor series expansion as follows [22,23]: where is mean of random variable , is the first derivative of the performance function, and is the second derivative of the performance function. Performance functions are approximated to their first derivative term owing to the weak nonlinearity of these functions. Statistical moments, such as mean and variance of the performance functions, can be calculated using the approximated performance functions, as follows [24]: where is the number of random variables, is the mean of the performance functions, is the variance of performance functions, and is the standard deviation of the i-th random variable. The derivative term of performance functions is evaluated using the central difference method, which is a numerical differentiation.
Computational graph-based framework for integrating econometric models and machine learning algorithms in emerging data-driven analytical environments
Published in Transportmetrica A: Transport Science, 2022
Taehooie Kim, Xuesong (Simon) Zhou, Ram M. Pendyala
In the area of discrete choice modeling, maximum likelihood estimation (MLE) is one of the fundamentally important estimation methods. By computing the first order (gradient) and second order (curvature) derivatives of the likelihood function, MLE furnishes values of parameters by maximizing the likelihood function through the use of the Hessian matrix. The derivatives are computed by three approaches: manual/analytical, finite difference, and automatic differentiation (AD) (Bartholomew-Biggs et al. 2000). Due to the difficulty of embedding/coding highly nonlinear forms in complicated functions, manual differentiation could be used for some very small cases. The numerical differentiation aims to approximate derivatives through the finite differencing, but the solution quality is greatly affected by the potential truncation and round-off errors associated with different finite difference formulas (Wright and Nocedal 1999). On the other hand, the automatic differentiation (AD) technique utilizes the chain rule-based principle and intermediate variables to evaluate complex derivatives analytically (Wright and Nocedal 1999; Griewank and Walther 2008). Specifically, in the new generation of low-level computational graph libraries such as Tensorflow and PyTorch, the computing architecture can enable modelers to represent the analytical optimization model through a graph of simple elementary operations (i.e. addition, subtraction, multiplication, and division) and elementary functions (e.g. natural logarithm), and further execute a sequential and complex structure of computations easily. In new domain-specific languages (DSLs) for convex optimization such as CVXPY, progress has been made recently to convert standard convex optimization to detailed CG representations with low-level solver interfaces (Agrawal et al. 2018). It should be noted that AD might still encounter the difficulty of computing piecewise rational functions, especially when estimating gradients of non-smooth composite functions (Beck and Fischer 1994; Nocedal and Wright 2006).
A case study of density functional theory and domain-based local pair natural orbital coupled cluster for vibrational effects on EPR hyperfine coupling constants: vibrational perturbation theory versus ab initio molecular dynamics
Published in Molecular Physics, 2020
Alexander A. Auer, Van Anh Tran, Bikramjit Sharma, Georgi L. Stoychev, Dominik Marx, Frank Neese
As we have adopted the latter approach in the following, we will reiterate the basic expressions of the VPT2 framework used to compute vibrational averages (see Equation 1) [47–49]. The vibrational average of any given property can be evaluated by inclusion of two correction terms that appear in the same order of perturbation theory and are added to the property computed at the equilibrium geometry, . The first correction term depends on the derivative of the property and the expectation value of a normal mode displacement with associated harmonic frequency , which requires the evaluation of elements of the cubic FF φ (second term in Equation 1 and Equation 2). The second correction term consists of the second derivative of the property and the expectation value of the square normal mode displacement which only requires the harmonic FF (Equation 3). Together, these two terms are denoted as the zero-point vibrational correction (ZPVC) of property A. The necessary property derivatives are typically evaluated by numerical differentiation using central finite differences. Note that in the well-established framework of perturbative treatments of vibrational corrections, these ZPVCs corresponding to the quantum vibrational ground state contribute the majority of the effects with respect to , while additional contributions due to the thermal population of excited vibrational quantum states and classical rigid-body rotation (collectively denoted as the ‘temperature correction’ (TC)) are found to be typically one order of magnitude smaller at room temperature [50,51]. Hence, the majority of vibrational effects should be captured by the ZPVCs from VPT2 considering only the quantum-mechanical ground vibrational state formally corresponding to 0 K.