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Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z )
Error_function
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
Optimization algorithm for artificial neural networks
loss function or "cost function" For classification, this is usually cross-entropy (XC, log loss), while for regression it is usually squared error loss
Backpropagation
Probability distribution
Robert (August 7, 2003). "The Q-function". Connexions. Barak, Ohad (April 6, 2006). "Q Function and Error Function" (PDF). Tel Aviv University. Archived
Normal_distribution
Complex complementary error function
The Faddeeva function or Kramp function is a scaled complex complementary error function, w ( z ) := e − z 2 erfc ( − i z ) = erfcx ( − i z ) = e
Faddeeva_function
Statistics function
distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an important function in applied
Q-function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
example is the error function, which is related to the cumulative distribution function of a normal distribution; another is the arctan function, which is
Sigmoid_function
Mathematical function
lack elementary antiderivatives; the integral of the Gaussian function is the error function: ∫ exp ( − x 2 ) d x = π 2 erf x + C . {\displaystyle \int
Gaussian_function
Method for estimating new data within known data points
at the point xk. The following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by
Interpolation
Reliable digital data delivery methods on unreliable channels
applications in computer science and telecommunications, error detection and correction (EDAC) or error control are techniques that enable reliable delivery
Error detection and correction
Error_detection_and_correction
Measure of the error of an estimator
squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk function, corresponding to
Mean_squared_error
Effect of variables' uncertainties on the uncertainty of a function based on them
combination of variables in the function. The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error Δx. Uncertainties can
Propagation_of_uncertainty
Error function: An integral important for normal random variables. Fresnel integral: related to the error function; used in optics. Dawson function:
List of mathematical functions
List_of_mathematical_functions
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Function that is holomorphic on the whole complex plane
well as derivatives and integrals of entire functions such as the error function. If an entire function f ( z ) {\displaystyle f(z)} has a root at
Entire_function
Mathematical function
_{0}^{\infty }e^{-t^{2}/4}\,\sin(xt)\,dt.} It is closely related to the error function erf, as D + ( x ) = π 2 e − x 2 erfi ( x ) = − i π 2 e − x 2 erf
Dawson_function
Theory of getting acceptably close inexact mathematical calculations
is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What
Approximation_theory
Graphical representations of the variability of data
are statistically significant. Error bars can also suggest goodness of fit of a given function, i.e., how well the function describes the data. Scientific
Error_bar
Integral of the Gaussian function, equal to sqrt(π)
statistical mechanics, to find its partition function. Although no elementary function exists for the error function, as can be proven by the Risch algorithm
Gaussian_integral
Continuous probability distribution
{5}{4}}\right)}}} The Kaniadakis κ-Error function (or κ-Error function) is a one-parameter generalization of the ordinary error function defined as: erf κ ( x
Kaniadakis Gaussian distribution
Kaniadakis_Gaussian_distribution
Internet error message
death – Fatal system error screen Funky caching – Generation, display and storage of dynamic content Link rot – URLs ceasing to function List of HTTP status
HTTP_404
Extension of the factorial function
integrals thereof, such as the error function. There are many interrelations between these functions and the gamma function; notably, the factor π {\displaystyle
Gamma_function
{\text{ for }}a>0,\ a\neq 1} In the following formulas, erf is the error function and Ei is the exponential integral. ∫ e c x ln x d x = 1 c ( e c x
List of integrals of exponential functions
List_of_integrals_of_exponential_functions
Computer programming concept
(VTA) and substantia nigra (SNc) appear to mimic the error function in the algorithm. The error function reports back the difference between the estimated
Temporal_difference_learning
Measure of complexity of real-valued functions
{\displaystyle h} represents a binary classifier, the error function is a 0–1 loss function, i.e. the error function f h {\displaystyle f_{h}} returns 0 if h {\displaystyle
Rademacher_complexity
}}'={\hat {H}}\mathbf {{\hat {x}}_{i}} } that minimize the reprojection error function given by ∑ i d ( x i , x i ^ ) 2 + d ( x i ′ , x i ^ ′ ) 2 {\displaystyle
Reprojection_error
Association of one output to each input
= 1. Another common example is the error function. More generally, many functions, including most special functions, can be defined as solutions of differential
Function_(mathematics)
Criterion for integration in terms of elementary functions
example of such a function is e − x 2 , {\displaystyle e^{-x^{2}},} whose antiderivative is (with a multiplier of a constant) the error function, familiar in
Liouville's theorem (differential algebra)
Liouville's_theorem_(differential_algebra)
Types of special mathematical functions
\operatorname {erf} } is the error function, and erfc {\displaystyle \operatorname {erfc} } is the complementary error function, erfc ( x ) = 1 − erf
Incomplete_gamma_function
Special function defined by an integral
functions F(x) and G(x) are transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function
Fresnel_integral
C standard library header file
operations are a group of functions in the standard library of the C programming language implementing basic mathematical functions. Different C standards
C_mathematical_functions
Artificial neural network node function
In artificial neural networks, the activation function of a node is a function that calculates the output of the node based on its individual inputs and
Activation_function
Measure of algorithm accuracy
estimates of the generalization error throughout the learning process. In a learning problem, the goal is to develop a function f n ( x → ) {\displaystyle
Generalization_error
Elliptic partial differential equation
Furthermore, the error function approaches 1 extremely quickly as its argument increases; in practice, for r > 3.4σ the relative error is smaller than
Poisson's_equation
Special mathematical function defined as sin(x)/x
"Sampling by incomplete cosine expansion of the sinc function: Application to the Voigt/complex error function". Appl. Math. Comput. 258: 425–435. arXiv:1407
Sinc_function
Probability density function
{\left|\alpha \right|}{\sqrt {2}}}\right)\right)} , with the error function erf. The parameters of the function (that are usually determined by a fit) are: N {\displaystyle
Crystal_Ball_function
Computer programming error
Compilation error or compile error refers to a state when a compiler fails to compile a piece of computer program source code, either due to errors in the
Compilation_error
Statistical error measure
In statistics, mean absolute error (MAE) is a measure of errors between paired observations expressing the same phenomenon. Examples of Y versus X include
Mean_absolute_error
Type of activation function
function of the standard normal distribution and erf ( z ) {\displaystyle \operatorname {erf} (z)} is the error function. This activation function is
Rectified_linear_unit
Scheme for controlling errors in data over noisy communication channels
groups of two to eight bits). Formally, an error-correcting code is given by its (injective) encoding function f : Σ ∗ ↪ Δ ∗ {\displaystyle f:\Sigma ^{*}\hookrightarrow
Error_correction_code
Quantization error in numerical analysis
computational physics, and simulation, discretization error is the error resulting from the fact that a function of a continuous variable is represented in the
Discretization_error
Concepts from statistical hypothesis testing
Type I error, or a false positive, is the incorrect rejection of a true null hypothesis in statistical hypothesis testing. A type II error, or a false
Type_I_and_type_II_errors
Numerical method for solving physical or engineering problems
PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual
Finite_element_method
Tasks in machine learning
the error function using data which is independent of that used for training. Various networks are trained by minimization of an appropriate error function
Training, validation, and test data sets
Training,_validation,_and_test_data_sets
Elementary functions and their finitely iterated integrals
of well-known functions which are Liouvillian but not elementary are the nonelementary antiderivatives, for example: The error function, e r f ( x ) =
Liouvillian_function
Analytic function that does not satisfy a polynomial equation
logarithm and inverse trigonometric functions. All special functions such as the gamma, error, bessel, and Riemann zeta functions are transcendental. Equations
Transcendental_function
Series of functions in mathematics
compute the zeta function by using a large enough value of N, for instance N > | s | {\displaystyle N>\vert s\vert } . Error function π x e x 2 e r f
Asymptotic_expansion
Functions such that f(–x) equals f(x) or –f(x)
sine sinh , {\displaystyle \sinh ,} The error function erf . {\displaystyle \operatorname {erf} .} If a function is both even and odd, it is equal to 0
Even_and_odd_functions
Mathematical approximation of a function
gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit
Taylor_series
Set of rules defining correctly structured programs
name of a function or variable with a lower-case letter. Example: var a = 5; console.log(a); // 5 console.log(A); // throws a ReferenceError: A is not
JavaScript_syntax
Error that stops the operating system
Windows, a fatal system error can be deliberately caused from a kernel-mode driver with either the KeBugCheck or KeBugCheckEx function. However, this should
Fatal_system_error
Loss function used in robust regression
the Huber loss is a loss function used in robust regression, that is less sensitive to outliers in data than the squared error loss. A variant for classification
Huber_loss
Probability distribution
dz\right).} Substituting the definition of the Faddeeva function (scaled complex error function) yields for the indefinite integral: 1 π ∫ w ( z ) d z
Voigt_profile
Probability distribution
}}\right),} where erf is the error function, a standard function in many mathematical software packages. The quantile function (or inverse CDF) is written:
Half-normal_distribution
Statistics concept
In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element
Errors_and_residuals
Statistical function that converts a probability to a standard normal score
environments where numerical implementations of the inverse error function are available, the probit function may be obtained as probit ( p ) = 2 erf − 1 (
Probit
Reinforcement learning method
taking action a {\displaystyle a} in state s {\displaystyle s} . An error function E ( o , p ) {\displaystyle E(o,p)} that compares the actual outcome
Error-driven_learning
Measure of prediction accuracy of a forecast
possible (see section below). Mean absolute percentage error is commonly used as a loss function for regression problems and in model evaluation, because
Mean absolute percentage error
Mean_absolute_percentage_error
Type of mathematical function
logarithmic integral (Li or li) and Fresnel integrals (S and C) the error function, e r f ( x ) = 2 π ∫ 0 x e − t 2 d t {\displaystyle \mathrm {erf}
Elementary_function
Error-detecting code for detecting data changes
that generates it is occasionally used as a hash function. CRCs are based on the theory of cyclic error-correcting codes. The use of systematic cyclic codes
Cyclic_redundancy_check
JavaScript test framework running on Node.js
for completion. describe('Foo', function(){ describe('#bar()', function() { it('should work without error', function(done) { var foo = new Foo(128); foo
Mocha_(JavaScript_framework)
equation. erf – error function. erfc – complementary error function. erfcx – scaled complementary error function. erfi – imaginary error function. etr – exponent
List of mathematical abbreviations
List_of_mathematical_abbreviations
Study of Galois symmetry groups of differential fields
function, the error function erf x {\displaystyle \operatorname {erf} x} , familiar in statistics. The indefinite integrals of elementary functions
Differential_Galois_theory
Mathematical function
{\displaystyle \alpha ={\tfrac {1}{2}},1,2} we find: (Section 2 of ) Error function: E 1 2 ( z ) = exp ( z 2 ) erfc ( − z ) . {\displaystyle E_{\frac
Mittag-Leffler_function
Study of kind and quantity of error
both forward error analysis and backward error analysis. Forward error analysis involves the analysis of a function z ′ = f ′ ( a 0 , a 1 , … , a n ) {\displaystyle
Error_analysis_(mathematics)
Solution of a confluent hypergeometric equation
integer, this equals 2−aθ−a(x/2) where θ is a Bessel polynomial. The error function can be expressed as e r f ( x ) = 2 π ∫ 0 x e − t 2 d t = 2 x π 1
Confluent hypergeometric function
Confluent_hypergeometric_function
Mathematical formula involving a given set of operations
functions such as the error function or gamma function to be basic. It is possible to solve the quintic equation if general hypergeometric functions are
Closed-form_expression
Serial communications protocol
is no error during those steps, the server's responding function code will also be the function code sent from the client. If there is any error during
Modbus
Methods of calculating definite integrals
behavior of the approximation error as a function of the number of integrand evaluations. A method that yields a small error for a small number of evaluations
Numerical_integration
Description of limiting behavior of a function
increasingly larger, the approximating function becomes increasingly more accurate, leading to an increasingly smaller relative error. This asymptotic relationship
Asymptotic_analysis
Probability distribution
x-\mu }{\sigma {\sqrt {2}}}}\right)} where erfc is the complementary error function. If X ∼ N ( μ , Σ ) {\displaystyle {\boldsymbol {X}}\sim {\mathcal {N}}({\boldsymbol
Log-normal_distribution
Integrals not expressible in closed-form from elementary functions
{1}{\ln x}}} (logarithmic integral) e − x 2 {\displaystyle e^{-x^{2}}} (error function, Gaussian integral) sin ( x 2 ) {\displaystyle \sin(x^{2})} and cos
Nonelementary_integral
Adaptive boosting based classification algorithm
feature of the choice of exponential error function is that the error of the final additive model is the product of the error of each stage, that is, e ∑ m −
AdaBoost
Probability distribution
instance of the hypergeometric function. For information on its inverse cumulative distribution function, see quantile function § Student's t-distribution
Student's_t-distribution
Numerical integration method
accuracy. For functions that are not in C2, the error bound given above is not applicable. Still, error bounds for such rough functions can be derived
Trapezoidal_rule
Measure of variation in statistics
{erf} } is the error function. The proportion that is less than or equal to a number, x, is given by the cumulative distribution function: Proportion ≤
Standard_deviation
Control loop feedback mechanism
below the setpoint. The integral function of the controller tends to compensate for error by introducing another error in the positive direction. This
PID_controller
Process of mapping a continuous set to a countable set
value (such as round-off error) is referred to as quantization error, noise or distortion. A device or algorithmic function that performs quantization
Quantization (signal processing)
Quantization_(signal_processing)
Data used to detect errors in other data
is especially true of cryptographic hash functions, which may be used to detect many data corruption errors and verify overall data integrity; if the
Checksum
Probability distribution
\operatorname {erfc} (z)} is the complementary error function, and Φ ( x ) {\displaystyle \Phi (x)} is the Laplace function (CDF of the standard normal distribution)
Lévy_distribution
error and NaN warning). Just as with the Python version, the squared-error (i.e., x 2 {\displaystyle x^{2}} ) returned by the "peirce_dev" function must
Peirce's_criterion
Limit of the tangent line at a point that tends to infinity
− x 2 ) , {\displaystyle x\mapsto \exp(-x^{2}),} the error function, and the logistic function. When a linear asymptote is not parallel to the x- or
Asymptote
Header file for C programs
library function can alter the value stored before return, whether or not they detect errors. Most functions indicate that they detected an error by returning
Errno.h
Describes the sum of independent normal and exponential random variables
^{2}-x}{{\sqrt {2}}\sigma }}\right),} where erfc is the complementary error function defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t .
Exponentially modified Gaussian distribution
Exponentially_modified_Gaussian_distribution
Probability distribution
{\displaystyle \operatorname {erfi} (z)} is the imaginary error function. The moment generating function is given by M ( t ) = 1 + σ t e 1 2 σ 2 t 2 π 2 [ erf
Rayleigh_distribution
Report of stack frames during program execution
line 13, in c error() NameError: name 'error' is not defined The stack trace shows where the error occurs, namely in the c function. It also shows that
Stack_trace
Fatal system error screen
(BSoD) – or blue screen error, blue screen, fatal error, bugcheck, and officially known as a stop error – is a critical error screen displayed in many
Blue_screen_of_death
Measure of inequality of a statistical distribution
}{2}}\right)} where erf {\displaystyle \operatorname {erf} } is the error function ( since G = 2 Φ ( σ 2 ) − 1 {\displaystyle G=2\Phi \left({\frac {\sigma
Gini_coefficient
Statistical learning theory
increasing real-valued function g : [ 0 , ∞ ) → R {\displaystyle g\colon [0,\infty )\to \mathbb {R} } , and an arbitrary error function E : ( X × R 2 ) n →
Representer_theorem
Recurrent neural network architecture
In 2006 Graves, Fernandez, Gomez, and Schmidhuber introduced a new error function for LSTM: Connectionist Temporal Classification (CTC) for simultaneous
Long_short-term_memory
Type of reservoir computer
synapses that connect the hidden neurons to output neurons. Thus, the error function is quadratic with respect to the parameter vector and can be differentiated
Echo_state_network
Mathematical functions having established names and notations
cosine ( cos {\displaystyle \cos } ), exponential function ( exp {\displaystyle \exp } ), and error function ( erf {\displaystyle \operatorname {erf} } or
Special_functions
Bug in a program that causes incorrect operation, but not termination
mathematical calculation. This example function in C to calculate the average of two numbers contains a logic error. It is missing parentheses in the calculation
Logic_error
Type of statistics
function. Tukey's biweight (also known as bisquare) function behaves in a similar way to the squared error function at first, but for larger errors,
Robust_statistics
On the distribution of prime numbers in arithmetic progressions
where φ {\displaystyle \varphi } is Euler's totient function. If we then define the error function E ( x ; q ) = max gcd ( a , q ) = 1 | π ( x ; q , a
Elliott–Halberstam_conjecture
Keywords provided by a programming language
structure(simpleError("mySpecialError message"),class=c("specialError","error","condition")) ) } ,specialError=function(e){ cat("catches errors of class specialError\n")
Exception_handling_syntax
Signal processing design process
also subject to the weighting function W {\displaystyle W} before the error function is computed. Once the error function is established, the optimal filter
Filter_design
Browser-based graphing calculator
sufficiently high iterations. Other functions like trigonometric and other transcendental functions, as well as the error function, factorial, statistical operations
Desmos
Oscillatory error in Fourier series
differentiable periodic function around a jump discontinuity. The N {\textstyle N} th partial Fourier series of the function (formed by summing the N
Gibbs_phenomenon
Function's sensitivity to argument change
measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently
Condition_number
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