K. Taji, Takahito Miyake, Hiroki Tamura · 2003
We propose error backpropagation using the absolute error function as an objective function. The error backpropagation is the most popular learning algorithm for multi-layered neural networks. In the error backpropagation, the square error function is usually used as the objective function. But a square function has a drawback in which it is enormously large if the data set includes a few anomalous data, which may be observational errors. On the other hand, the absolute error function is less affected by such data. But since the absolute error function is not differentiable, the standard backpropagation's way cannot be applied directly by using an absolute error function as an objective function. Therefore, we first introduce differentiable approximate functions for the absolute value function. The purpose of introducing an approximate function is to construct a differentiable error function which is close to the absolute error function. Then we propose an error backpropagation algorithm minimizing a differentiable approximate error function. Some computational experiments indicate that the proposed method is practically efficient. In particular, it is observed that the method is more robust and learns faster than the backpropagation with the square error function when teacher signals include some incorrect data.
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Dieter Jaeger, Ranu Jung · Encyclopedia of Computational Neuroscience · 2015
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Vassili F. Pastushenko · Journal of Modern Applied Statistical Methods · 2009
Email: The empirical distribution function (ecdf) is unbiased in the usual sense, but shows certain order bias. Pyke suggested discrete ecdf using expectations of order statistics. Piecewise constant optimal ecdf saves 200%/N of sample size N. Results are compared with linear interpolation for U(0, 1), which require up to sixfold shorter samples at the same accuracy.
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Ruze Zhang, Xuan Guang · 2023
In this paper, we put forward the model of zero-error distributed function compression system of two binary memoryless sources X and Y as depicted in Fig. 1. In the model, there are two encoders En1 and En2 and one decoder De, connected by two channels with capacity constraints C<inf>1</inf> and C<inf>2</inf>, respectively. The encoder En1 can observe X or (X, Y), and the encoder En2 can observe Y or (X, Y). Here, we use two switches s<inf>1</inf> and s<inf>2</inf> open or closed (taking values 0 or 1) to represent whether En1 can observe Y and En2 can observe X, respectively. The decoder De is required to compress the binary arithmetic sum f(X, Y) = X + Y with zero error by using the system multiple times. We use (s<inf>1</inf>s<inf>2</inf>; C<inf>1</inf>, C<inf>2</inf>; f) to denote the model. The compression capacity is defined as the maximum average number of times that the function f can be compressed with zero error for one use of the system, which measures the efficiency for using the system. In the paper, the compression capacities for all the four models are fully characterized. Amongst them, the characterization of the compression capacity for (01; C<inf>1</inf>, C<inf>2</inf>; f) is very difficult. Toward this end, we develop a novel graph coloring approach in the converse part and the proof is highly nontrivial. Furthermore, we apply the compression capacity for (01; C<inf>1</inf>, C<inf>2</inf>; f) to the open problem in network function computation that whether the best known upper bound by Guang et al. on computing capacity is in general tight. This upper bound is always tight for all previously considered network function computation problems whose computing capacities are known. By considering equivalent network function computation models of (01; C<inf>1</inf>, C<inf>2</inf>; f), we give the answer that in general the upper bound of Guang et al. is not tight.
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S. Salon, M.V.K. Chari, Lale T. Ergene et al. · 2023
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Leo F. Epstein · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 1955
reasonably well the elastic scattering angular distributions. One of these sets appears to be more accurate than the other in predicting the inelastic cross sections. For Al, one set of parameters was used, for which both the elastic and inelastic predictions agree very well with the experirnental data. The data presented will be incorporated, as appropriate, into multi- and few-group computer programs for reactor analysis. (auth)
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Annie Cuyt, Vigdis Brevik Petersen, Brigitte Verdonk et al. · 2008
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D. Kavitha, K. Dhanalakshmi, K. Anitha · Mathematics and Statistics · 2023
Normalised Error function has been coined and analyzed in 2018 [13].The concept of normalised error function discussed in [13], motivated us to find the new results of Toeplitz determinant for the subclasses of analytic univalent functions concurrent with error function. By seeing the history of error function in Geometric functions theory, Ramachandran et. al [13] derived the coefficient estimates followed by the Fekete-Szeg¨o problem for the normalised subclasses of starlike and convex functions associated with error function. Finding coefficient estimates is one of the most provoking concepts in geometric function theory. In current scenario scientists are concentrating on special functions which are connected with univalent functions. Based on these concepts, the present paper deals with supremum and infimum of Toeplitz determinant for starlike and convex in terms of error function with convolution product using the concept of subordination. Also, we derive the sharp bounds for probability distribution associated with error starlike and error convex functions.
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Xianping Tu, Xianqing Lei · 2015
According to definitions, geometric characteristics and corresponding error evaluation functions of straightness error, flatness error, roundness error, cylindricity error and sphericity error, the concept of evaluation reference of form error is put forward by analyzing the minimum zone evaluation method of form error. The evaluation reference corresponding to form error is obtained by studying the function of all kinds of form error. The relationship between the evaluation references and the all kinds of form error are studied. The unified formula of the form error is gained and the unification of the form error is realized by summing up the relationships. The foundation of researching the unified shape error evaluation algorithm is laid by unified mathematical description.
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Sang-Hoon Oh · IEEE Transactions on Neural Networks · 1997
This letter proposes a modified error function to improve the error backpropagation (EBP) algorithm of multilayer perceptrons (MLPs) which suffers from slow learning speed. To accelerate the learning speed of the EBP algorithm, the proposed method reduces the probability that output nodes are near the wrong extreme value of sigmoid activation function. This is acquired through a strong error signal for the incorrectly saturated output node and a weak error signal for the correctly saturated output node. The weak error signal for the correctly saturated output node, also, prevents overspecialization of learning for training patterns. The effectiveness of the proposed method is demonstrated in a handwritten digit recognition task.
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Ibrahim Tunji Awolere, Ukeje Emelike · Asia Pacific Journal of Mathematics · 2019
. In this investigation we introduce and study a subclass of harmonic univalent functions associated with error function and salagean operator using convolutional approach. Coefficient estimate, distortion bounds, growth theorem, extreme points and convex combination were established for a new class T H,n (). Consequently, our results are new approaches to those corresponding to previously known results.
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Alan Jeffrey · Elsevier eBooks · 2004
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james Robertson · Bulletin of Miscellaneous Information (Royal Gardens Kew)
You can bind and error handler using a JQuery selector, which can be a little simpler than using the stock Javascript bind() function
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František Čuřík · Časopis pro pěstování matematiky a fysiky · 1913
typu a téhož řádu.Užitá symbolika má zde opět týž význam jako v odstavci třetím při definici matic prvního typu.
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Hessein Ali · 2014
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J. A. C. Weideman · SIAM Journal on Numerical Analysis · 1994
Rational expansions for computing the complex error function $w(z) = e^{ - z^2 } {\text{erfc}}( - iz)$ are presented. These expansions have the following attractive properties: (1) they can be evaluated using a polynomial evaluation routine such as Homer’s method, (2) the polynomial coefficients can be computed once and for all by a single Fast Fourier Transform (FFT), and (3) high accuracy is achieved uniformly in the complex plane with only a small number of terms. Comparisons reveal that in some parts of the complex plane certain competitors may be more efficient. However, the difference in efficiency is never great, and the new algorithms are simpler than existing ones: a complete program takes eight lines of Matlab code.
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Wang, Pengzhao · DR-NTU (Nanyang Technological University) · 2023
The Gaussian error function is a non-fundamental function that is commonly used in probability theory and statistics. In this paper, we present a new method for approximating the error function. The expression consists of two parts: one part is a function that decreases at a rate similar to the error function, while the other part is a polynomial function with several parameters that assists the approximation in approaching the exact error function. Two criteria, absolute error and relative error, are utilized to analyze the error of the approximation, as they have distinct impacts. Absolute error aims to minimize the error at the start of the interval because the error function declines exponentially, causing the error at the right end of the interval to be much larger. Conversely, all errors throughout the argument region have the same importance when computing the relative error. As a result, approximations based on relative error exhibit more exceptional performance as the variable's magnitude increases. By considering these two types of errors, we can merge them together, with the first half utilizing an approximation determined by absolute error and the second half using relative error. Moreover, the number of terms in the polynomial function also affects the performance of the approximations. After comparing the errors, we select the third-order function. The novel approximation we presented surpasses previous approximations in terms of performance. The results of the calculations indicate that this new method has a tenfold reduction in error and requires only half the computation time compared to Karagiannidis' approximation. Consequently, the proposed approximation in this paper represents a remarkable advancement.
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Alex A. Villanueva (679351), Shashank Priya (299920) · Figshare · 2015
<p>Error (E) for different number of segments found by the error analysis method. The data has been fitted with power curves with equations with a coefficient of determination of for the optimized data.</p>
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Tzeng Jengnan, Horng-Shing Lu Henry, Li Wen-Hsiung · Figshare · 2011
The solid line refers to the case where the points in each group are chosen randomly, while the dash line refers to the case where the points in each group are chosen from neighbors. In all cases, is fixed to 200.<b>Copyright information:</b>Taken from "Multidimensional scaling for large genomic data sets"http://www.biomedcentral.com/1471-2105/9/179BMC Bioinformatics 2008;9():179-179.Published online 4 Apr 2008PMCID:PMC2375126.
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Krishnanand Verma · Applied Mathematics Letters · 1989
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