Future work

Given the signicant computational cost attached to the implementation of the WMC, the future work related to WMC should be highly focused on the computational optimisation of the algorithm. In particular, ecient ways how to construct and produce samples from the desired wavelet of interest in real time is a top priority. The same could be said about the computation of wavelet coecients, required to construct sampling distributions for wavelet resolution levels. If these two goals could be achieved, WMC algorithm has real chances of becoming one of the more popular sampling algorithms in the scientic community.

Bibliography

Akay, M. (1995), `Wavelets in biomedical engineering', Annals of biomedical Engineering 23(5), 531542.

Almeida, A. (2005), `Wavelet bases in generalized Besov spaces', Journal of Mathematical Analysis and Applications 304(1), 198211.

Beskos, A., Roberts, G. & Stuart, A. (2009), `Optimal scalings for local Metropolis - Hastings chains on nonproduct targets in high dimensions', The Annals of Applied Probability 19(3), 863898.

Besov, O. (1959), On a family of function spaces. embedding theorems and extensions, in `Doklady Akademii Nauk SSSR', Vol. 126, pp. 11631165.

Besov, O. V. (1961), `Investigation of a class of function spaces in connection with imbedding and extension theorems', Trudy Matematicheskogo Instituta Imeni VA Steklova 60, 4281.

Bijaoui, A., Slezak, E., Rue, F. & Lega, E. (1996), `Wavelets and the study of the distant universe', Proceedings of the IEEE 84(4), 670679.

Camussi, R. & Guj, G. (1997), `Orthonormal wavelet decomposition of turbulent ows: intermittency and coherent structures', Journal of Fluid Mechanics 348, 177199.

Celeux, G., Hurn, M. & Robert, C. P. (2000), `Computational and inferential diculties with mixture posterior distributions', Journal of the American Statistical Association 95(451), 957970.

Cohen, A., Dahmen, W. & Devore, R. (2001), `Adaptive wavelet methods for elliptic operator equations: Convergence rates', Mathematics of Computation 70(233), 2775.

Cooley, J. W. & Tukey, J. W. (1965), `An algorithm for the machine calculation of complex Fourier series', Mathematics of Computation 19, 297301.

Daubechies, I. (1988), `Orthonormal bases of compactly supported wavelets', Communications on Pure and Applied Mathematics 41(7), 909996.

Daubechies, I., Grossmann, A. & Meyer, Y. (1986), `Painless nonorthogonal expansions', Journal of Mathematical Physics 27(5), 12711283.

de Valpine, P. (2008), `Improved estimation of normalizing constants from Markov chain Monte Carlo output', Journal of Computational and Graphical Statistics 17(2), 333351.

Devroye, L. (1986), Non-Uniform Random Variate Generation, Springer-Verlag. Donoho, D. L. (1995), `De-noising by soft-thresholding', IEEE Transactions on

Information Theory 41(3), 613627.

Donoho, D. L. & Johnstone, J. M. (1994), `Ideal spatial adaptation by wavelet shrinkage', Biometrika 81(3), 425455.

Farge, M. (1992), `Wavelet transforms and their applications to turbulence', Annual review of uid mechanics 24(1), 395458.

Frazier, M., Frazier, M. W., Jawerth, B. & Weiss, G. L. (1991), Littlewood-Paley theory and the study of function spaces, number 79, American Mathematical Soc.

Gabor, D. (1946), `Theory of Communication', Journal of the Institution of Electrical Engineers 93(26), 429457.

Gallegati, M. (2012), `A wavelet-based approach to test for nancial market contagion', Computational Statistics & Data Analysis 56(11), 34913497.

Geman, S. & Geman, D. (1984), `Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images', IEEE Transactions on Pattern Analysis and Machine Intelligence 6(6), 721741.

Gilks, W. R. (2017), `Notes on Wavelet Monte Carlo', Personal communications. Gilks, W. R., Best, N. G. & Tan, K. K. C. (1995), `Adaptive rejection Metropolis

sampling within Gibbs sampling', Journal of the Royal Statistical Society. Series C (Applied Statistics) 44(4), 455472.

Gilks, W. R. & Wild, P. (1992), `Adaptive rejection sampling for Gibbs sampling', Journal of the Royal Statistical Society. Series C (Applied Statistics) 41(2), 337 348.

Gilks, W., Richardson, S. & Spiegelhalter, D. (1995), Markov Chain Monte Carlo in Practice, Chapman & Hall/CRC Interdisciplinary Statistics, Taylor & Francis. Grossmann, A. & Morlet, J. (1982), `Decomposition of Hardy functions into square integrable wavelets of constant shape', SIAM Journal on Mathematical Analysis 15(4), 723736.

Haar, A. (1910), `Zur theorie der orthogonalen funktionensysteme', Mathematische Annalen 69(3), 331371.

Hastings, W. K. (1970), `Monte Carlo sampling methods using Markov chains and their applications', Biometrika 57(1), 97109.

Kahn, H. & Harris, T. E. (1951), `Estimation of particle transmission by random sampling', National Bureau of Standards Applied Mathematics Series 12, 2730. Kartsonaki, C. (2016), `Survival analysis', Diagnostic Histopathology 22(7), 263270. Kinderman, A. J. & Monahan, J. F. (1977), `Computer generation of random variables using the ratio of uniform deviates', ACM Transactions on Mathematical Software 3(3), 257260.

Kyriazis, G. (2003), `Decomposition systems for function spaces', Studia Mathematica 2(157), 133169.

Kyriazis, G. & Petrushev, P. (2002), `New bases for Triebel-Lizorkin and Besov spaces', Transactions of the American Mathematical Society 354(2), 749776. Lan, S., Streets, J. & Shahbaba, B. (2014), Wormhole Hamiltonian Monte Carlo, in

`Proceedings of the Twenty-Eighth AAAI Conference on Articial Intelligence', AAAI'14, AAAI Press, pp. 19531959.

Mallat, S. (2008), A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way, 3rd edn, Academic Press.

Mallat, S. G. (1989a), `Multiresolution approximations and wavelet orthonormal bases of L2

(R)', Transactions of the American Mathematical Society 315(1), 69 87.

Mallat, S. G. (1989b), `A theory for multiresolution signal decomposition: The wavelet representation', IEEE Transactions on Pattern Analysis and Machine Intelligence 11(7), 674693.

Mark, G. & Ben, C. (2011), `Riemann manifold Langevin and Hamiltonian Monte Carlo methods', Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73(2), 123214.

Meng, X.-L. & Wong, W. H. (1996), `Simulating ratios of normalizing constants via a simple identity: A theoretical exploration', Statistica Sinica 6(4), 831860. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller,

E. (1953), `Equation of state calculations by fast computing machines', JCP 21, 10871092.

Meyer, Y. (1986-1987a), `Ondelettes et fonctions splines', Séminaire Équations aux dérivées partielles (Polytechnique) pp. 118.

Meyer, Y. (1986-1987b), `Ondelettes et fonctions splines', Séminaire Équations aux dérivées partielles (Polytechnique) pp. 118.

Neal, R. M. (1993), Probabilistic inference using Markov Chain Monte Carlo methods, Technical report, CRG-TR-93-1, Dept. of Computer Science, University of Toronto.

Neal, R. M. (2005), Estimating ratios of normalizing constants using linked importance sampling, Technical report, Department of Statistics and Department of Computer Science, University of Toronto.

R Core Team (2013), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria.

URL: http://www.R-project.org/

Rao, R. & Bopardikar, A. (1998), Wavelet Transforms: Introduction to Theory and Applications, Addison-Wesley.

Roberts, G. O., Gelman, A. & Gilks, W. R. (1997), `Weak convergence and optimal scaling of random walk Metropolis algorithms', The Annals of Applied Probability 7(1), 110120.