Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

비만아 부모의 돌봄 경험: q 방법론

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0, 1] s ,w heres may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.

https://web.maths.unsw.edu.au/~josefdick/preprints/DKS2013_Acta_Num_Version.pdf

High-dimensional integration: The quasi-Monte Carlo way

This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.

https://web.maths.unsw.edu.au/~z2265001/preprints/DKS2013_preprint.pdf

Acta Numerica: High dimensional integration - the Quasi-Monte Carlo way

Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced by Dick (Ann Stat 39:1372---1398, 2011) and shown to achieve the optimal rate of convergence of the root mean square error for numerical integration of smooth functions defined on the $$s$$s-dimensional unit cube. The key ingredient there is a digit interlacing function applied to the components of a randomly scrambled digital net whose number of components is $$ds$$ds, where the integer $$d$$d is the so-called interlacing factor. In this paper we replace the randomly scrambled digital nets by randomly scrambled polynomial lattice point sets, which allows us to obtain a better dependence on the dimension while still achieving the optimal rate of convergence. Our results apply to Owen's full scrambling scheme as well as the simplifications studied by Hickernell, Matousek and Owen. We consider weighted function spaces with general weights, whose elements have square integrable partial mixed derivatives of order up to $$\alpha \ge 1$$??1, and derive an upper bound on the variance of the estimator for higher order scrambled polynomial lattice rules. Employing our obtained bound as a quality criterion, we prove that the component-by-component construction can be used to obtain explicit constructions of good polynomial lattice point sets. By first constructing classical polynomial lattice point sets in base $$b$$b and dimension $$ds$$ds, to which we then apply the interlacing scheme of order $$d$$d, we obtain a construction cost of the algorithm of order $$\mathcal {O}(dsmb^m)$$O(dsmbm) operations using $$\mathcal {O}(b^m)$$O(bm) memory in case of product weights, where $$b^m$$bm is the number of points in the polynomial lattice point set.

Construction of Interlaced Scrambled Polynomial Lattice Rules of Arbitrary High Order

Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces

In this paper, we develop a very efficient approach to the Monte Carlo estimation of the expected value of partial perfect information (EVPPI) that measures the average benefit of knowing the value of a subset of uncertain parameters involved in a decision model. The calculation of EVPPI is inherently a nested expectation problem, with an outer expectation with respect to one random variable X and an inner conditional expectation with respect to the other random variable Y. We tackle this problem by using a multilevel Monte Carlo (MLMC) method (Giles in Oper Res 56(3): 607–617, 2008) in which the number of inner samples for Y increases geometrically with level, so that the accuracy of estimating the inner conditional expectation improves and the cost also increases with level. We construct an antithetic MLMC estimator and provide sufficient assumptions on a decision model under which the antithetic property of the estimator is well exploited, and consequently a root-mean-square accuracy of $${\varepsilon }$$
 can be achieved at a cost of $$O({\varepsilon }^{-2})$$
 . Numerical results confirm the considerable computational savings compared to the standard, nested Monte Carlo method for some simple test cases and a more realistic medical application.

Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI

History matching with iterative Latin hypercube samplings and parameterization of reservoir heterogeneity

The expected information gain is an important quality criterion of Bayesian experimental designs, which measures how much the information entropy about uncertain quantity of interest θ is reduced o...

Takashi Goda

Papers

Construction of Interlaced Scrambled Polynomial Lattice Rules of Arbitrary High Order

Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces

Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI

History matching with iterative Latin hypercube samplings and parameterization of reservoir heterogeneity

Multilevel Monte Carlo estimation of expected information gains