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Monte Carlo Methods

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Mathematische Geodäsie/Mathematical Geodesy

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Abstract

Monte Carlo methods deal with generating random variates from probability density functions in order to estimate unknown parameters or general functions of unknown parameters and to compute their expected values, variances and covariances. One generally works with the multivariate normal distribution due to the central limit theorem. However, if random variables with the normal distribution and random variables with a different distribution are combined, the normal distribution is not valid anymore. The Monte Carlo method is then needed to get the expected values, variances and covariances for the random variables with distributions different from the normal distribution.

The error propagation by the Monte Carlo method is discussed and methods for generating random variates from the multivariate normal distribution and from the multivariate uniform distribution. The Monte Carlo integration is presented leading to the sampling-importance-resampling (SIR) algorithm. Markov Chain Monte Carlo methods provide by the Metropolis algorithm and the Gibbs sampler additional ways of generating random variates. A special topic is the Gibbs sampler for computing and propagating large covariance matrices. This task arises when the geopotential is determined from satellite observations. The example of the minimal detectable outlier shows, how the Monte Carlo method is used to determine the power of a hypothesis test.

Zusammenfassung

Monte-Carlo-Methoden arbeiten mit Zufallszahlen aus Verteilungsfunktionen, um unbekannte Parameter oder allgemeine Funktionen unbekannter Parameter zu schätzen, und um ihre Erwartungswerte, Varianzen und Kovarianzen zu berechnen. Im allgemeinen nutzt man wegen des zentralen Grenzwertsatzes die multivariate Normalverteilung. Wenn jedoch Zufallsvariable mit der Normalverteilung mit Zufallsvariablen unterschiedlicher Verteilungen kombiniert werden, gilt die Normalverteilung nicht mehr. Die Monte-Carlo-Methode wird dann benötigt, die Erwartungswerte, Varianzen und Kovarianzen der Zufallsvariablen mit Verteilungen zu erhalten, die sich von der Normalverteilung unterscheiden.

Die Fehlerfortpflanzung durch die Monte-Carlo-Methode wird diskutiert und Methoden für die Generierung von Zufallswerten aus der multivariaten Nor- malverteilung und aus der multivariaten Gleichverteilung. Die Monte-Carlo-Integration der wesentlichen Stichprobe führt auf den SIR (sampling-importance-resampling) Algorithmus. Monte-Carlo-Methoden mit Markoff-Ketten verschaffen durch den Metropolis-Algorithmus und das Gibbs-Verfahren weitere Methoden, Zufallswerte zu generieren. Als besondere Aufgabe wird das Gibbs-Verfahren zur Berechnung und Propagation großer Kovarianzmatrizen behandelt. Dieses Problem tritt auf, wenn das Schwerefeld der Erde aus Satel- litenbeobachtungen bestimmt wird. Das Beispiel der minimal aufzudecken- den Ausreißer zeigt, wie die Monte-Carlo-Methode benutzt wird, um die Trennschärfe eines Hypothesentests zu bestimmen.

This chapter is part of the series Handbuch der Geodäsie, volume “Mathematische Geodäsie/ Mathematical Geodesy”, edited by Willi Freeden, Kaiserslautern.

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Acknowledgments

The author is indebted to Willi Freeden for his invitation to this contribution for HbMG and to Jan Martin Brockmann for his valuable comments.

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  1. Institute for Geodesy and Geoinformation, Theoretical Geodesy Group, University of Bonn, Bonn, Germany

    Karl-Rudolf Koch

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  1. Karl-Rudolf Koch
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Correspondence to Karl-Rudolf Koch .

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  1. Geomathematics Group, Mathematics Department, University of Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

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Koch, KR. (2020). Monte Carlo Methods. In: Freeden, W. (eds) Mathematische Geodäsie/Mathematical Geodesy. Springer Reference Naturwissenschaften . Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55854-6_100

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