Chain Rule Of Calculus Research Articles

Modified Backpropagation Algorithm with Multiplicative Calculus in Neural Networks

Backpropagation is one of the most widely used algorithms for training feedforward deep neural networks. The algorithm requires a differentiable activation function and it performs computations of the gradient proceeding backwards through the feedforward deep neural network from the last layer through to the first layer. In order to calculate the gradient at a specific layer, the gradients of all layers are combined via the chain rule of calculus. One of the biggest disadvantages of the backpropagation is that it requires a large amount of training time. To overcome this issue, this paper proposes a modified backpropagation algorithm with multiplicative calculus. Multiplicative calculus provides an alternative to the classical calculus and it defines new kinds of derivative and integral forms in multiplicative form rather than addition and subtraction forms. The performance analyzes are discussed in various case studies and the results are given comparatively with classical backpropagation algorithm. It is found that the proposed modified backpropagation algorithm converges in less time to the solution and thus provides fast training in the given case studies. It is also shown that the proposed algorithm avoids the local minima problem.

A general method for N-order integral-form Gauss's variational equations under impulsive control

In this paper, an innovative derivatives-based method is proposed to derive any higher-order integral-form Gauss's Variational Equations (GVEs) of the orbital elements under impulsive control. In this new method, the Taylor series expansion is applied to express the variation of each orbital element as a sum of terms that are calculated from the values of the derivatives of orbital elements. The required N-order derivatives of orbital elements can be successively computed from the (N-1)-order differential-form GVEs by using the chain rule of calculus. Using this method, the final expressions of the second-order and third-order integral-form GVEs are provided in detail. To testify the accuracy of these new results, the Monte Carlo simulations are performed randomly and the histogram data are analyzed. It is demonstrated that the proposed higher-order GVEs are more accurate than the traditional lower-order ones. The method and the resultant higher-order GVEs can be potentially used on spacecraft impulsive orbital maneuver or formation flying.

Risk-Index of Colorectal Cancer to Triage for Screening

Colorectal cancer (CRC) is the unregulated growth of one or more adenomatous polyps (most frequently adenocarcinomas) occurring in the colon and/or the rectum. Of all new cancer cases in the US, 8.0% are colorectal. Colorectal cancer claims 8.4% of all cancer-deaths and thus has above-average mortality vs. other cancers. To detect CRC at early stages, the United States Preventative Services Task Force (USPSTF) recommends screening by colonoscopy or sigmoidoscopy for ages 50-75. However, the high false positive rates of these procedures lead to a lot of unnecessary screening. The goal of this study is to develop an artificial neural network (ANN) for colorectal cancer risk prediction usable for triaging people for screening based on their personal health data. The data to train and validate the ANN is the set of 1997-2016 responses (excepting 2004) to the National Health Interview Survey (NHIS) personal health questionnaire, in which colon and/or rectal cancer occurring 4 years or less from the survey date is counted as one instance of CRC. Respondents with 1 or more null entries were discarded. Using 70% of the NHIS data, training uses gradient descent in parameter space to adjust the parameters of a softmax-function and minimize a logistic cost function (regression). The extent to which the parameters are adjusted is calculated using the chain rule of calculus (backpropagation). The validation tests the regression-fitted function upon the remaining 30% of the data. As a binary test, sensitivity (TPR), specificity (SPC), and positive/negative predictive values (PPV/NPV) are calculated and compared with the USPSTF. As a trinary test, three levels of risk-stratification are defined by requiring that no more than 1% of CRC/non-CRC cases be misclassified as low/high-risk. If a person of medium risk having/not having CRC is counted as half of a false negative/positive (e.g., annual/biennial screenings for those of high/medium risk), the trinary test of risk-stratification also has an associated TPR, SPC, PPV, and NPV. As a binary test, the ANN has sensitivity of 0.7, specificity of 0.7, PPV of 0.09, and NPV of 0., all (except for NPV) exceeding the USPSTF guidelines and independent of tumorous advancement. As a trinary test, lowered SPC and PPV are exchanged for higher TPR and NPV.Abstract MO_30_2748; Table 1Performance of USPSTF (entire data set) compared to the ANN (validation-portion of data set, only).CRC, # screenedCRC, # not screenedNo CRC, # screenedNo CRC, # not screenedANN (bi.)517255106,864235,029ANN (tri.)497145142,603.5207,004USPSTF762490232,545297,065TPRSPCPPVNPVANN (bi.)0.6700.6870.009450.997ANN (tri.)0.7740.4080.003470.9993USPSTF0.6090.4390.003270.998 Open table in a new tab CRC risk calculated by ANN from personal health data is noninvasive, insensitive to tumorous advancement, & outperforms USPSTF screening guidelines as a binary and trinary test. In addition, the ANN offers the prospect of CRC risk assessment in real time and on the world map.

How can Students Generalize the Chain Rule? The Roles of Abduction in Mathematical Modeling

The purpose of this study is to design a modeling task to facilitate students’ inquiries into the chain rule in calculus and to analyze the results after implementation of the task. Researchers have pointed out that students may encounter difficulties in understanding the chain rule due to the formal approaches used to teach and learn this concept. In this study, we take a modeling approach to the teaching and learning of the chain rule by facilitating the generalization of students’ models and modeling activities. We assumed abductive reasoning to be one of the key factors which can support the generalization of students’ models and modeling activities. We believe that analogical reasoning and diagrammatic reasoning are key factors in fostering students’ use of abduction. As a result, we determined that the students’ models and modeling activities were generalized to the chain rule by their use of abductive reasoning, and the students found the chain rule to be a generalized rule for describing changes of various quantities.

From human activity to conceptual understanding of a mathematical concept

Abstract:This article reports on a study which investigated first year university engineering students’ construction of the definition of the concept of the chain rule in differential calculus at a University of Technology in South Africa. An APOS (Action-Process-Objects-Schema) approach was used to explore conceptual understanding displayed by students in learning the chain rule in calculus. Structured worksheets based on instruction designed to induce construction of conceptual understanding of the chain rule were used. A number of students used the straight form technique in differentiating complicated tasks while very few used either the link and Leibniz form techniques. In this manner differentiation of each function within the composite function was accomplished. Students either operated in the Inter- or Trans stages of the Triad. It was found that even students who had inadequate understanding of composition of functions, performed well in the application of the chain rule.Keywords: Calculus, chain rule, APOS, Genetic decomposition

ADIC: an extensible automatic differentiation tool for ANSI‐C

In scientific computing, we often require the derivatives ∂f/∂x of a function f expressed as a program with respect to some input parameter(s) x, say. Automatic Differentiation (AD) techniques augment the program with derivative computation by applying the chain rule of calculus to elementary operations in an automated fashion. This article introduces ADIC (Automatic Differentiation of C), a new AD tool for ANSI-C programs. ADIC is currently the only tool for ANSI-C that employs a source-to-source program transformation approach; that is, it takes a C code and produces a new C code that computes the original results as well as the derivatives. We first present ADIC ‘by example’ to illustrate the functionality and ease of use of ADIC and then describe in detail the architecture of ADIC. ADIC incorporates a modular design that provides a foundation for both rapid prototyping of better AD algorithms and their sharing across AD tools for different languages. A component architecture called AIF (Automatic Differentiation Intermediate Form) separates core AD concepts from their language-specific implementation and allows the development of generic AD modules that can be reused directly in other AIF-based AD tools. The language-specific ADIC front-end and back-end canonicalize C programs to make them fit for semantic augmentation and manage, for example, the association of a program variable with its derivative object. We also report on applications of ADIC to a semiconductor device simulator, 3-D CFD grid generator, vehicle simulator, and neural network code. © 1997 John Wiley & Sons, Ltd.