One-sided Error Wiki
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because it lacks inline citations. Please help to improve this article by introducing more precise citations. (August 2011) (Learn how and when to remove this template message) In computing, a Monte Carlo algorithm is a wikipedia monte carlo simulation randomized algorithm whose running time is deterministic, but whose output may be incorrect with
Las Vegas Algorithm Example
a certain (typically small) probability. The name refers to the grand casino in the Principality of Monaco at Monte Carlo, which
Las Vegas Algorithm Explanation
is well-known around the world as an icon of gambling. The related class of Las Vegas algorithms are also randomized, but in a different way: they take an amount of time that varies randomly, but
Monte Carlo Algorithm Example
always produce the correct answer. A Monte Carlo algorithm can be converted into a Las Vegas algorithm whenever there exists a procedure to verify that the output produced by the algorithm is indeed correct. If so, then the resulting Las Vegas algorithm is merely to repeatedly run the Monte Carlo algorithm until one of the runs produces an output that can be verified to be correct. Contents 1 One-sided vs two-sided atlantic city algorithm error 2 Amplification 3 Complexity classes 4 Applications in computational number theory 5 See also 6 References One-sided vs two-sided error[edit] Whereas the answer returned by a deterministic algorithm is always expected to be correct, this is not the case for Monte Carlo algorithms. For decision problems, these algorithms are generally classified as either false-biased or true-biased. A false-biased Monte Carlo algorithm is always correct when it returns false; a true-biased algorithm is always correct when it returns true. While this describes algorithms with one-sided errors, others might have no bias; these are said to have two-sided errors. The answer they provide (either true or false) will be incorrect, or correct, with some bounded probability. For instance, the Solovay–Strassen primality test is used to determine whether a given number is a prime number. It always answers true for prime number inputs; for composite inputs, it answers false with probability at least ½ and true with probability at most ½. Thus, false answers from the algorithm are certain to be correct, whereas the true answers remain uncertain; this is said to be a ½-correct false-biased algorithm. Amplification[edit] For a Monte Carlo algorithm with one-sided errors, the failure probability can be reduced (and the success probability amplified) by running the algorithm
linear model Generalized linear model Discrete choice Logistic regression Multinomial logit Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson Multilevel model Fixed effects Random effects Mixed model Nonlinear regression Nonparametric monte carlo algorithm code Semiparametric Robust Quantile Isotonic Principal components Least angle Local Segmented Errors-in-variables Estimation las vegas algorithm ppt Least squares Ordinary least squares Linear (math) Partial Total Generalized Weighted Non-linear Non-negative Iteratively reweighted Ridge regression Least absolute wiki chevy monte carlo deviations Bayesian Bayesian multivariate Background Regression model validation Mean and predicted response Errors and residuals Goodness of fit Studentized residual Gauss–Markov theorem Statistics portal v t e Least absolute deviations (LAD), https://en.wikipedia.org/wiki/Monte_Carlo_algorithm also known as least absolute errors (LAE), least absolute value (LAV), least absolute residual (LAR), sum of absolute deviations, or the L1 norm condition, is a statistical optimality criterion and the statistical optimization technique that relies on it. Similar to the popular least squares technique, it attempts to find a function which closely approximates a set of data. In the simple case https://en.wikipedia.org/wiki/Least_absolute_deviations of a set of (x,y) data, the approximation function is a simple "trend line" in two-dimensional Cartesian coordinates. The method minimizes the sum of absolute errors (SAE) (the sum of the absolute values of the vertical "residuals" between points generated by the function and corresponding points in the data). The least absolute deviations estimate also arises as the maximum likelihood estimate if the errors have a Laplace distribution. Contents 1 Formulation of the problem 2 Contrasting least squares with least absolute deviations 3 Other properties 4 Variations, extensions, specializations 5 Solving methods 5.1 Solving using linear programming 6 See also 7 References 8 External links Formulation of the problem[edit] Suppose that the data set consists of the points (xi, yi) with i = 1, 2, ..., n. We want to find a function f such that f ( x i ) ≈ y i . {\displaystyle f(x_{i})\approx y_{i}.} To attain this goal, we suppose that the function f is of a particular form containing some parameters which need to be determined. For instance, the simplest form would be linear: f(x) = bx + c, whe
that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0 x e − t 2 d t . {\displaystyle {\begin − https://en.wikipedia.org/wiki/Error_function 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac https://en.wikipedia.org/wiki/RL_(complexity) − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1 monte carlo (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / las vegas algorithm 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 8 (x)} is real when x is real. When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w ( z ) = e − z 2 erfc ( − i z ) = erfcx ( − i z ) . {\displaystyle w(z)=e^{-z^ 6}\operatorname 5 (-iz)=\operatorname 4 (-iz).} Contents 1 The name "error function" 2 Properties 2.1 Taylor series 2.2 Derivative and integral 2.3 Bürmann series 2.4 Inverse functions 2.5 Asymptotic expansion 2.6 Continued fraction expansion 2.7 Integral of error function
theory problems solvable in logarithmic space and polynomial time with probabilistic Turing machines with one-sided error. It is named in analogy with RP, which is similar but has no logarithmic space restriction. The probabilistic Turing machines in the definition of RL never accept incorrectly but are allowed to reject incorrectly less than 1/3 of the time; this is called one-sided error. The constant 1/3 is arbitrary; any x with 0 < x < 1 would suffice. This error can be made 2−p(x) times smaller for any polynomial p(x) without using more than polynomial time or logarithmic space by running the algorithm repeatedly. Sometimes the name RL is reserved for the class of problems solvable by logarithmic-space probabilistic machines in unbounded time. However, this class can be shown to be equal to NL using a probabilistic counter, and so is usually referred to as NL instead; this also shows that RL is contained in NL. RL is contained in BPL, which is similar but allows two-sided error (incorrect accepts). RL contains L, the problems solvable by deterministic Turing machines in log space, since its definition is just more general. Noam Nisan showed in 1992 the weak derandomization result that RL is contained in SC,[3] the class of problems solvable in polynomial time and polylogarithmic space on a deterministic Turing machine; in other words, given polylogarithmic space, a deterministic machine can simulate logarithmic space probabilistic algorithms. It is believed that RL is equal to L, that is, that polynomial-time logspace computation can be completely derandomized; major evidence for this was presented by Reingold et al. in 2005.[4] A proof of this is the holy grail of the efforts in the field of unconditional derandomization of complexity classes. A major step forward was Omer Reingold's proof that SL is equal to L. References[edit] ^ Complexity Zoo: RL ^ A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3):559–578. 1989. ^ Nisan, Noam (1992), "RL ⊆ SC", Proceedings of the 24th ACM Symposium on Theory of computing (STOC '92), Victoria, British Columbia, Canada, pp.619–623, doi:10.1145/129712.129772. ^ O. Reingold and L. Trevisan and S. Vadhan. Pseudorandom walks in biregular graphs and the RL vs. L problem, ECCCTR05-022, 2004. v t e Importa