Normalised Error Function
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Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered error function table in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined by (1) Note that some authors (e.g., Whittaker and Watson inverse error function 1990, p.341) define without the leading factor of . Erf is implemented in the Wolfram Language as Erf[z]. A two-argument form giving is also implemented as Erf[z0, z1]. Erf satisfies the identities (2) (3) (4) where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. For , (5) where is the incomplete gamma function.
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Erf can also be defined as a Maclaurin series (6) (7) (OEIS A007680). Similarly, (8) (OEIS A103979 and A103980). For , may be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990). For , (11) (12) Using integration by parts gives (13) (14) (15) (16) so (17) and continuing the procedure gives the asymptotic series (18) (19) (20) (OEIS A001147 and A000079). Erf has the values (21) (22) It is an odd function (23) and satisfies (24) Erf may be expressed in terms of a confluent hypergeometric function of the first kind as (25) (26) Its derivative is (27) where is a Hermite polynomial. The first derivative is (28) and the integral is (29) Min Max Re Im Erf can also be extended to the complex plane, as illustrated above. A simple integral involving erf that Wolfram Language cannot do is given by (30) (M.R.D'Orsogna, pers. comm., May 9, 2004). More complicated integrals include (31) (M.R.D'Orsogna, pers. comm., Dec.15, 2005). Erf has the continued fraction (32) (33) (Wall 1948, p.357), first stated by Laplace in 1805 and Lege
to reliable sources. Unsourced material may be challenged and removed. (July 2012) (Learn how and when to remove this template message) In statistics and applications of statistics, normalization can have a range error function python of meanings.[1] In the simplest cases, normalization of ratings means adjusting values erf(1) measured on different scales to a notionally common scale, often prior to averaging. In more complicated cases, normalization may
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refer to more sophisticated adjustments where the intention is to bring the entire probability distributions of adjusted values into alignment. In the case of normalization of scores in educational assessment, there may http://mathworld.wolfram.com/Erf.html be an intention to align distributions to a normal distribution. A different approach to normalization of probability distributions is quantile normalization, where the quantiles of the different measures are brought into alignment. In another usage in statistics, normalization refers to the creation of shifted and scaled versions of statistics, where the intention is that these normalized values allow the comparison of corresponding normalized https://en.wikipedia.org/wiki/Normalization_(statistics) values for different datasets in a way that eliminates the effects of certain gross influences, as in an anomaly time series. Some types of normalization involve only a rescaling, to arrive at values relative to some size variable. In terms of levels of measurement, such ratios only make sense for ratio measurements (where ratios of measurements are meaningful), not interval measurements (where only distances are meaningful, but not ratios). In theoretical statistics, parametric normalization can often lead to pivotal quantities – functions whose sampling distribution does not depend on the parameters – and to ancillary statistics – pivotal quantities that can be computed from observations, without knowing parameters. Contents 1 Examples 2 Other types 3 See also 4 References Examples[edit] There are various normalizations in statistics – nondimensional ratios of errors, residuals, means and standard deviations, which are hence scale invariant – some of which may be summarized as follows. Note that in terms of levels of measurement, these ratios only make sense for ratio measurements (where ratios of measurements are meaningful), not interval measurements (where only distances are meaningful, but not ratios). See also Category:Statistical
For other uses, see Bell curve (disambiguation). Normal distribution Probability density function The red curve is the standard normal distribution Cumulative distribution function Notation N ( μ , σ 2 ) {\displaystyle {\mathcal σ 4}(\mu ,\,\sigma https://en.wikipedia.org/wiki/Normal_distribution ^ σ 3)} Parameters μ ∈ R — mean (location) σ2 > 0 http://www.fresco.org.uk/minuit/cern/node31.html — variance (squared scale) Support x ∈ R PDF 1 2 σ 2 π e − ( x − μ ) 2 2 σ 2 {\displaystyle {\frac σ 0{\sqrt − 9\pi }}}\,e^{-{\frac {(x-\mu )^ − 8} − 7}}}} CDF 1 2 [ 1 + erf ( x − μ σ 2 ) error function ] {\displaystyle {\frac − 2 − 1}\left[1+\operatorname − 0 \left({\frac 9{\sigma {\sqrt 8}}}\right)\right]} Quantile μ + σ 2 erf − 1 ( 2 F − 1 ) {\displaystyle \mu +\sigma {\sqrt 2}\operatorname 1 ^{-1}(2F-1)} Mean μ Median μ Mode μ Variance σ 2 {\displaystyle \sigma ^ − 8\,} Skewness 0 Ex. kurtosis 0 Entropy 1 2 ln ( 2 σ normalised error function 2 π e ) {\displaystyle {\tfrac − 6 − 5}\ln(2\sigma ^ − 4\pi \,e\,)} MGF exp { μ t + 1 2 σ 2 t 2 } {\displaystyle \exp\{\mu t+{\frac − 0 σ 9}\sigma ^ σ 8t^ σ 7\}} CF exp { i μ t − 1 2 σ 2 t 2 } {\displaystyle \exp\ σ 2 σ 1}\sigma ^ σ 0t^ μ 9\}} Fisher information ( 1 / σ 2 0 0 1 / ( 2 σ 4 ) ) {\displaystyle {\begin μ 41/\sigma ^ μ 3&0\\0&1/(2\sigma ^ μ 2)\end μ 1}} In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.[1][2] The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of random variables is sufficiently large. Physical quantities that are expected to
provide for full generality in the user-defined function value, the user is allowed to define a normalization factor known internally as UP and defined by the Minuit user on an ERROR DEF command card. The default value is one. The Minuit error on a parameter is defined as the change of parameter which would produce a change of the function value equal to UP. This is the most general way to define the error, although in statistics it is more usual to define it in terms of the second derivative of the function - with respect to the parameter in question. In the simplest linear case (when the function is exactly parabolic at the minimum), the value UP=1.0 corresponds to defining the error as the inverse of the second derivative at the minimum. The fact that Minuit defines the error in terms of a function change does not mean that it always calculates such a function change. Indeed it sometimes (HESSE) calculates the second derivative matrix and inverts it, assuming a parabolic behaviour. This distinction is discussed in section 7.2. The purpose of defining errors by function changes is threefold: 1. to preserve its meaning in the non-parabolic case (see section 7.2); 2. to allow generality when the user-defined function is not a chi- square or likelihood, but has some other origin; 3. to allow calculation not only of ``one-standard deviation'' errors, but also two or more standard deviations, or more general 'confidence regions', especially in the multiparameter case (see section 7.3). Chi-square normalization If the user's function value F is supposed to be a chisquare, it must of course be properly normalized. That is, the ``weights'' must in fact correspond to the one-standard-deviation errors on the observations. The most general expression for the chi-square is of the form (see [5], p.163): = (xi - yi(a))Vij(xj - yj(a)) where x is the vector of observations, y(a) is the vector of fitted values (or theoretical expressions for them) containing the variable fit parameters a, and V is the inverse of the error matrix of the observations x, also known as the covariance matrix of the observations. Fortunately, in most real cases the observations x are statistically independent of each other (e.g., the contents of the bins of a histogram, or measurements of points on a trajectory), so the matrix V is diagonal only. The expression for then simplifies to the more familiar form: = where e2 is the inverse of the diagonal element of V, the square of the error on the corresponding observation x. In the case where the x are integer numbers of events in an unweighted histogram, for example, the e2 are just equa