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About 10,000 papers for “Error function”

Error function description and test optimization results.

Timothy Rumbell (7419269), James Kozloski (7419272) · Figshare · 2019

<p>(A) The traditional transfer function from normalized feature value to error value. The error value is the number of experimentally observed standard deviations from the experimentally observed mean for the model generated feature value. This method targets one ‘best’ feature value at the observed mean (function minimum). (B) Soft-thresholded error, where a number of standard deviations (in this case 2) is permitted to give equivalent error by reducing error to zero within that region. This approach considers any model within an observed range as equally plausible. (C) The decision making value when ranks are equal in the NSDE algorithm is crowdedness in feature space. Models with lower crowdedness (e.g. in the blue box) are preferred over models with higher crowdedness (e.g. in the red box). This promotes diversity of model activity and fills the full range of experimentally observed activity. (D) Sigmoidal transfer functions converting parameter values (x-axis) to feature values (y-axis) for the test optimizations. Solid lines show transfer functions for parameter 1 into feature 1; the dotted line shows transfer function for parameter 2 into feature 2 in one optimization (orange). In the other two optimizations feature 2 is calculated from a sigmoidal transfer function on either (<i>P</i>1 − <i>P</i>2) × 2 (green), or (<i>P</i>1 + <i>P</i>2)/2 (purple). (E) Test optimization result. Models shown in feature space are approximately uniformly distributed regardless of location of transfer function in parameter space. Curves at top and right in (E) and (F) are based on histograms with 100 bins. (F) Test optimization result. All model parameter sets plotted in the 2D parameter space. Distributions are Gaussian, with mean and standard deviation dependent on center and slope of the sigmoidal transfer function shown in (D).</p>

Cited by 0Open access

Rational Chebyshev approximations for the error function

William J. Cody · Mathematics of Computation · 1969

This note presents nearly-best rational approximations for the functions erf <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and erfc <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with maximal relative errors ranging down to between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="6 times 10 Superscript negative 19"> <mml:semantics> <mml:mrow> <mml:mn>6</mml:mn> <mml:mo>×</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>19</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">6 \times {10^{ - 19}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 times 10 Superscript negative 20"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>×</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>20</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">3 \times {10^{ - 20}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Cited by 261Open access

Production Function Estimation and Capital Measurement Error

Allan Collard‐Wexler, Jan De Loecker · National Bureau of Economic Research · 2016

We look into the impact of measurement error in capital on the estimation of production functions. We introduce an identification scheme and an estimation procedure that jointly deals with measurement error in capital and the standard simultaneity bias due to unobserved productivity shocks. We use lagged investment to instrument for potentially mis-measured capital stock, while conditioning on the part of productivity that is persistent. Our estimation routine nests standard approaches in the literature, such as Ackerberg, It requires no additional data as investment is usually collected with other producer level data. We show through Monte-Carlo experiments that a 40 percent measurement error in capital yields capital coefficients that are biased downward by a factor of two. We illustrate our approach using data for three distinct economies: China, India and Chile; which experienced radically different processes of capital and productivity dynamics. We find capital coefficients that are typically two times larger than those using standard approaches that only control for simultaneity.

Cited by 28Open access

Table of complementary error function Q(x)

Samuel O. Agbo, Matthew N. O. Sadiku · 2017

Written specifically for a one-semester course, this textbook introduces the physical and engineering principles of communication systems using an accessible, yet mathematically rigorous, approach. Beginning with valuable background material on signals and systems, and random processes, the text then guides students through the core topics, including amplitude modulation, pulse modulation, and noise. Key terms and formulae are highlighted throughout to help students identify essential points easily. Worked examples, practice problems, and review questions reinforce concepts and enable students to develop confidence in solving problems on their own. To help visualize the concepts discussed, MATLAB-based exercises and examples are provided throughout, supported by an introductory appendix for students who are new to MATLAB. Each chapter ends with a practical applications section, showing students how concepts are used in real-life communication scenarios and devices. Figures from the book and a solutions manual, password-protected for instructors, are available online.

Cited by 0

Robust Learning Algorithm with LTS Error Function

Andrzej Rusiecki · IGI Global eBooks · 2009

Feedforward neural networks (FFNs) are often considered as universal tools and find their applications in areas such as function approximation, pattern recognition, or signal and image processing. One of the main advantages of using FFNs is that they usually do not require, in the learning process, exact mathematical knowledge about input-output dependencies. In other words, they may be regarded as model-free approximators (Hornik, 1989). They learn by minimizing some kind of an error function to fit training data as close as possible. Such learning scheme doesn’t take into account a quality of the training data, so its performance depends strongly on the fact whether the assumption, that the data are reliable and trustable, is hold. This is why when the data are corrupted by the large noise, or when outliers and gross errors appear, the network builds a model that can be very inaccurate. In most real-world cases the assumption that errors are normal and iid, simply doesn’t hold. The data obtained from the environment are very often affected by noise of unknown form or outliers, suspected to be gross errors. The quantity of outliers in routine data ranges from 1 to 10% (Hampel, 1986). They usually appear in data sets during obtaining the information and pre-processing them when, for instance, measurement errors, long-tailed noise, or results of human mistakes may occur. Intuitively we can define an outlier as an observation that significantly deviates from the bulk of data. Nevertheless, this definition doesn’t help in classifying an outlier as a gross error or a meaningful and important observation. To deal with the problem of outliers a separate branch of statistics, called robust statistics (Hampel, 1986, Huber, 1981), was developed. Robust statistical methods are designed to act well when the true underlying model deviates from the assumed parametric model. Ideally, they should be efficient and reliable for the observations that are very close to the assumed model and simultaneously for the observations containing larger deviations and outliers. The other way is to detect and remove outliers before the beginning of the model building process. Such methods are more universal but they do not take into account the specific type of modeling philosophy (e.g. modeling by the FFNs). In this article we propose new robust FFNs learning algorithm based on the least trimmed squares estimator.

Cited by 1

Continued Fraction Evaluation of the Error Function

A. S. Al-Fhaid · Journal of King Abdulaziz University-Science · 2006

Abstract. In this paper, continued fraction expansion of the error function is developed. An efficient and simple computational al-gorithm based on this expansion is also developed using top-down evaluation procedure. Numerical results of the algorithm are in full agreement at least to fifteen digits accuracy with that of the standard tables.

Cited by 1

Univariate error function based neural network approximation

George A. Anastassiou · arXiv (Cornell University) · 2014

Here we research the univariate quantitative approximation of real and complex valued continuous functions on a compact interval or all the real line by quasi-interpolation, Baskakov type and quadrature type neural network operators. We perform also the related fractional approximation. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its high order derivative or fractional derivatives. Our operators are defined by using a density function induced by the error function. The approximations are pointwise and with respect to the uniform norm. The related feed-forward neural networks are with one hidden layer.

Cited by 0Open access

Error as a function of trial number for experiments 1–4 (± standard error).

Mor Ben-Tov (304500), Shelly Levy-Tzedek (304501), Amir Karniel (304502) · Figshare · 2015

&lt;p&gt;Each shape represents a different condition of the task: circles – discrete small, triangles – discrete big, rectangles – rhythmic small, diamonds – rhythmic big. In all transitions, from discrete movements to rhythmic ones (panels A and B) and from rhythmic movements to discrete ones (panels C and D), there was a significant increase in error.&lt;/p&gt;

Cited by 0Open access

Error as a function of trial number for experiments 5–8 (±standard error).

Mor Ben-Tov (304500), Shelly Levy-Tzedek (304501), Amir Karniel (304502) · Figshare · 2015

&lt;p&gt;Each shape represents a different condition of the task: circles – discrete small, triangles – discrete big, rectangles – rhythmic small, diamonds – rhythmic big. In the transition from small movements to big movements (panels A and C, first transition), there is a significant increase in error. In the transition from big movements to small movements (panels B and D, first transition), there is no significant change in error. In the transition from discrete movements to rhythmic ones (panels A and B, second transition) and from rhythmic movements to discrete ones (panels C and D, second transition), there is a significant increase in error.&lt;/p&gt;

Cited by 0Open access

A close approximation related to the error function

Roger G. Hart · Mathematics of Computation · 1966

The quantity fi(p) Ti(p) must be nonnegative, for otherwise we could choose (xi pi) so small that (5) could not hold. (We note here fni(x) > 0 for x E D by hypothesis.) A similar consideration in the case where pi > xi shows that fi(p)Ti(p) < 0. Hence fi(p) = T1(p). In the same manner one can show that ft(p) = Ti(p), i = 2, *.. , n. Thus Q*(x) and T(x) are identical. The idea for this note occurred to the author after hearing a lecture by Prof. Ranko Bojanic [1] on best one sided approximation in the case of functions of one variable.

Cited by 31Open access

Non‐Markovian Cost Function for Quantum Error Mitigation

Doyeol Ahn, Byeongyong Park · Advanced Quantum Technologies · 2024

Abstract In near‐term quantum computers like noisy intermediate‐scale quantum (NISQ) devices, reducing the impact of errors and decoherence is critical for practical implementation. Existing studies have primarily focused on Markovian noise sources; however, understanding the relationship between quantum error mitigation (QEM) and non‐Markovian noise sources is essential, as these effects are practically unavoidable in most solid‐state devices used for quantum computing. Here, a non‐Markovian model of quantum state evolution and a QEM cost function of controlled‐NOT (CNOT) gate operation are presented for NISQ devices interacting with an environment characterized by simple harmonic oscillators as a noise source. Using the projection operator formalism and both advanced and retarded propagators in time, the reduced‐density operator is derived for output quantum states in a time‐convolutionless form by solving the quantum Liouville equation. Output quantum state fluctuations are analyzed for identity and CNOT gate operations in two‐qubit operations across various input states and compare these results with experimental data from ion‐trap and superconducting quantum computing systems to estimate the key parameters of the QEM cost functions. These findings demonstrate that the QEM cost function increases as the coupling strength between the quantum system and its environment intensifies. This study highlights the significance of non‐Markovian models for understanding quantum state evolution and the practical implications of the QEM cost function in assessing experimental results from NISQ devices.

Cited by 3Open access

Calibration error as loss function in binary classification

Fragomeni G., · 2024

Calibration, as an evaluation criterion for classification models, is often not given as much consideration as discrimination, possibly because it is harder to measure. Nevertheless, it is a crucial measure in biomedical data analysis, because a probability estimate by a model should reflect the relative frequency with which it occurs. Poor calibration can lead to suboptimal decisions; by ensuring a close correspondence between predicted probabilities and actual chance of outcome, medical professionals can make these decisions in the most informed manner possible. We consider the special case of binary classification, and calibration measures for this task. We show how the multi-class calibration measure expected calibration error can be adapted to the two-class case, and used directly as a loss function in training neural network models. We also consider the alternative two-class calibration measure of the Hosmer-Lemeshow test statistic, and demonstrate empirically how calibration measures, both stand-alone and in combination with cross-entropy error, can serve as loss functions for classifying two sample data sets from the biomedical domain. Our experiments demonstrate that explicitly optimizing for calibration loss results in models that are well calibrated without losing their ability to discriminate between two classes.

Cited by 0Open access

Error-minimizing dead zone for basis function networks

M. Heiss · IEEE Transactions on Neural Networks · 1996

The incorporation of dead zones in the error signal of basis function networks avoids the networks' overtraining and guarantees the convergence of the normalized least mean square (LMS) algorithm and related algorithms. A new so-called error-minimizing dead zone is presented providing the least a posteriori error out of the set of all convergence assuring dead zones. A general convergence proof is developed for LMS algorithms with dead zones, and the error-minimizing dead zone is derived from the resulting convergence condition. The performance is compared with the performance of classical dead zones.

Cited by 27

Average error function for 13 n-heptane models

Niemeyer, Kyle · Figshare · 2016

Average error function for 13 n-heptane models based on available ignition delay data. Presented by Kyle Niemeyer, "An autoignition performance comparison of chemical kinetics models for n-heptane", Spring 2016 Meeting of the Western States Section of the Combustion Institute, Seattle, WA, USA. 21–22 March 2016. http://www.slideshare.net/kyleniemeyer/an-autoignition-performance-comparison-of-chemical-kinetics-models-for-nheptane

Cited by 0Open access