Excitation energy dependent raman signatures of ABA- and ABC-stacked few-layer graphene

The dependence of the Raman spectrum on the excitation energy has been investigated for ABA-and ABC- stacked few-layer graphene in order to establish the fingerprint of the stacking order and the number of layers, which affect the transport and optical properties of few-layer graphene. Five different excitation sources with energies of 1.96, 2.33, 2.41, 2.54 and 2.81â €...eV were used. The position and the line shape of the Raman 2D, G*, N, M, and other combination modes show dependence on the excitation energy as well as the stacking order and the thickness. One can unambiguously determine the stacking order and the thickness by comparing the 2D band spectra measured with 2 different excitation energies or by carefully comparing weaker combination Raman modes such as N, M, or LOLA modes. The criteria for unambiguous determination of the stacking order and the number of layers up to 5 layers are established.

Asymptotic behaviour of approximate Bayesian estimators

Although approximate Bayesian computation (ABC) has become a popular technique for performing parameter estimation when the likelihood functions are analytically intractable there has not as yet been a complete investigation of the theoretical properties of the resulting estimators. In this paper we give a theoretical analysis of the asymptotic properties of ABC based parameter estimators for hidden Markov models and show that ABC based estimators satisfy asymptotically biased versions of the standard results in the statistical literature.

Parameter Estimation for Hidden Markov Models with Intractable Likelihoods

Approximate Bayesian computation (ABC) is a popular technique for analysing data for complex models where the likelihood function is intractable. It involves using simulation from the model to approximate the likelihood, with this approximate likelihood then being used to construct an approximate posterior. In this paper, we consider methods that estimate the parameters by maximizing the approximate likelihood used in ABC. We give a theoretical analysis of the asymptotic properties of the resulting estimator. In particular, we derive results analogous to those of consistency and asymptotic normality for standard maximum likelihood estimation. We also discuss how sequential Monte Carlo methods provide a natural method for implementing our likelihood-based ABC procedures.

Filtering via approximate Bayesian computation

Approximate Bayesian computation (ABC) has become a popular technique to facilitate Bayesian inference from complex models. In this article we present an ABC approximation designed to perform biased filtering for a Hidden Markov Model when the likelihood function is intractable. We use a sequential Monte Carlo (SMC) algorithm to both fit and sample from our ABC approximation of the target probability density. This approach is shown to, empirically, be more accurate w.r.t.~the original filter than competing methods. The theoretical bias of our method is investigated; it is shown that the bias goes to zero at the expense of increased computational effort. Our approach is illustrated on a constrained sequential lasso for portfolio allocation to 15 constituents of the FTSE 100 share index.

Approximate Bayesian Computation for Smoothing

We consider a method for approximate inference in hidden Markov models (HMMs). The method circumvents the need to evaluate conditional densities of observations given the hidden states. It may be considered an instance of Approximate Bayesian Computation (ABC) and it involves the introduction of auxiliary variables valued in the same space as the observations. The quality of the approximation may be controlled to arbitrary precision through a parameter ε > 0. We provide theoretical results which quantify, in terms of ε, the ABC error in approximation of expectations of additive functionals with respect to the smoothing distributions. Under regularity assumptions, this error is, where n is the number of time steps over which smoothing is performed. For numerical implementation, we adopt the forward-only sequential Monte Carlo (SMC) scheme of [14] and quantify the combined error from the ABC and SMC approximations. This forms some of the first quantitative results for ABC methods which jointly treat the ABC and simulation errors, with a finite number of data and simulated samples. © Taylor & Francis Group, LLC.