Кіріспе
Сандық әдіс
Numerical technique
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
Математикада Монте-Карло интеграциясы – кездейсоқ сандарды қолдана отырып сандық интеграция жасау әдісі. Бұл нақты интегралды сандық түрде есептейтін Монте-Карло әдісінің бір түрі. Басқа алгоритмдер көбінесе интегралдың мәнін тұрақты торда есептейді, ал Монте-Карло интегралдың мәнін есептеу үшін нүктелерді кездейсоқ түрде таңдайды. Бұл әдіс жоғары өлшемді интегралдар үшін ерекше пайдалы. N біртекті үлгіні ескере отырып, I-дің жуық мәнін табуға болады:
Numerical technique
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
Бұл үлкен сандар заңының нәтижесінде болады, ол келесіні қамтамасыз етеді:
Numerical technique
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
I-ді QN арқылы бағалау кезінде, QN-нің қателіктері дисперсияның бөгде бағалауын қолдана отырып, үлгі дисперсиясымен бағалануы мүмкін, нәтижесінде:
Numerical technique
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
Егер тізбек шектелген болса, бұл дисперсия асимптотикалық түрде 1/N-ге дейін төмендейді. Осылайша, QN қателігінің бағалауы:
Numerical technique
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
Бұл орташа мәннің стандартты қатесіне көбейтілген шама, және бұл нәтиже интегралдың өлшемдерінің санына байланысты емес. Бұл Монте-Карло интеграциясының детерминистік әдістерге қарағандағы артықшылығы, себебі детерминистік әдістер көбінесе өлшемге экспоненциалды түрде тәуелді болады. Детерминистік әдістерден айырмашылығы, қателік бағалауы нақты қателік шегі емес екенін ескеру маңызды; кездейсоқ үлгі алу интегралдың қателікті төмендетуіне әкелуі мүмкін барлық маңызды ерекшеліктерін ашпауы мүмкін. Наive Монте-Карло әдісі қарапайым мысалдар үшін қолданылса да, детерминистік алгоритмдерге қарағанда жақсарту тек проблемаға қатысты үлгі алу үлестірімдерін қолданатын алгоритмдермен ғана мүмкін болады. Тиісті үлгі алу үлестірімін қолдану арқылы жоғары өлшемді интегралдардың көпшілігі локалданған екенін және интегралға тек кішігірім кеңістіктер елеулі үлес қосатынын пайдалануға болады. Монте-Карло әдебиетінің үлкен бөлігі қателік бағалауларын жақсарту стратегияларын әзірлеуге арналған. Атап айтқанда, қабатталған үлгі алу (аумақты субдомендерге бөлу) және маңыздылық үлгі алу (біртекті емес үлестірімдерден үлгі алу) – осындай әдістердің екі мысалы.
Numerical technique
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher dimensional integrals. given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as This is standard error of the mean multiplied with This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher dimensional integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub domains—and importance sampling—sampling from non uniform distributions—are two examples of such techniques.
Рекурсивті қабатталған сынама алу
Рекурсивті қабатталған үлгі алу – бір өлшемді бейімделген квадратуралардың көп өлшемді интегралдарға жасалған жалпылауы. Әрбір рекурсиялық қадамда интеграл және қателік қарапайым Монте-Карло алгоритмі арқылы бағаланады. Егер қателік бағалауы қажетті дәлдіктен артық болса, интеграциялық көлем кіші көлемдерге бөлінеді және процедура кіші көлемдерге рекурсивті түрде қолданылады. Көп өлшемділік үшін әдеттегі "екіге бөлу" стратегиясы тиімсіз, себебі кіші көлемдердің саны тым жылдам өсіп, бақылау қиынға түседі. Оның орнына, қай өлшем бойынша бөлу ең тиімді екені бағаланып, көлем тек сол өлшем бойынша бөлінеді. Қабатталған үлгі алу алгоритмі үлгілеу нүктелерін функцияның дисперсиясы ең жоғары аймақтарда шоғырландырады, осылайша жалпы дисперсияны азайтып, үлгілеуді тиімдірек етеді, бұл суретте көрсетілгендей. Көп таралған MISER процедурасы да ұқсас алгоритмді қолданады.
Recursive stratified sampling is a generalization of one dimensional adaptive quadratures to multi dimensional integrals. On each recursion step the integral and the error are estimated using a plain Monte Carlo algorithm. If the error estimate is larger than the required accuracy the integration volume is divided into sub volumes and the procedure is recursively applied to sub volumes. The ordinary 'dividing by two' strategy does not work for multi dimensions as the number of sub volumes grows far too quickly to keep track. Instead one estimates along which dimension a subdivision should bring the most dividends and only subdivides the volume along this dimension. The stratified sampling algorithm concentrates the sampling points in the regions where the variance of the function is largest thus reducing the grand variance and making the sampling more effective, as shown on the illustration. The popular MISER routine implements a similar algorithm.