Likelihood functions for state space models with diffuse initial conditions

State space models with non‐stationary processes and/or fixed regression effects require a state vector with diffuse initial conditions. Different likelihood functions can be adopted for the estimation of parameters in time‐series models with diffuse initial conditions. In this article, we consider profile, diffuse and marginal likelihood functions. The marginal likelihood function is defined as the likelihood function of a transformation of the data vector. The transformation is not unique. The diffuse likelihood is a marginal likelihood for a data transformation that may depend on parameters. Therefore, the diffuse likelihood cannot be used generally for parameter estimation. The marginal likelihood function is based on an orthonormal data transformation that does not depend on parameters. Here we develop a marginal likelihood function for state space models that can be evaluated by the Kalman filter. The so‐called diffuse Kalman filter is designed for computing the diffuse likelihood function. We show that a minor modification of the diffuse Kalman filter is needed for the evaluation of our marginal likelihood function. Diffuse and marginal likelihood functions have better small sample properties compared with the profile likelihood function for the estimation of parameters in linear time series models. The results in our article confirm the earlier findings and show that the diffuse likelihood function is not appropriate for a range of state space model specifications.     

Penalised Complexity Priors for Stationary Autoregressive Processes

The autoregressive (AR) process of order p(AR(p)) is a central model in time series analysis. A Bayesian approach requires the user to define a prior distribution for the coefficients of the AR(p) model. Although it is easy to write down some prior, it is not at all obvious how to understand and interpret the prior distribution, to ensure that it behaves according to the users' prior knowledge. In this article, we approach this problem using the recently developed ideas of penalised complexity (PC) priors. These prior have important properties like robustness and invariance to reparameterisations, as well as a clear interpretation. A PC prior is computed based on specific principles, where model component complexity is penalised in terms of deviation from simple base model formulations. In the AR(1) case, we discuss two natural base model choices, corresponding to either independence in time or no change in time. The latter case is illustrated in a survival model with possible time‐dependent frailty. For higher‐order processes, we propose a sequential approach, where the base model for AR(p) is the corresponding AR(p?1) model expressed using the partial autocorrelations. The properties of the new prior distribution are compared with the reference prior in a simulation study.     

An approximate likelihood function for panel data with a mixed ARMA(p, q) remainder disturbance model

Abstract. An approximate likelihood function for panel data with an autoregressive moving‐average (ARMA)(p, q) model remainder disturbance is presented and Whittle's approximate maximum likelihood estimator (MLE) is used to yield an asymptotic estimator. Although an asymptotic approach, the power test is quite successful for estimating and testing. In this approach, we do not need to calculate the transformation matrix in exact form. Through the Riemann sum approach, we can construct a simple approximate concentrated likelihood function. In addition, the model is also extended to the restricted maximum likelihood (REML) function, in which the package of Gilmour, Thompson and Cullis [Biometrics (1995) Vol. 51, pp. 1440–1450] is applied without difficulty. In the case study, we implement the model on the characteristic line for the investment analysis of Taiwanese computer motherboard makers.     

Empirical likelihood in long‐memory time series models

This article studies the empirical likelihood method for long‐memory time series models. By virtue of the Whittle likelihood, one obtains a score function that can be viewed as an estimating equation of the parameters of a fractional integrated autoregressive moving average (ARFIMA) model. This score function is used to obtain an empirical likelihood ratio which is shown to be asymptotically chi‐square distributed. Confidence regions for the parameters are constructed based on the asymptotic distribution of the empirical likelihood ratio. Bartlett correction and finite sample properties of the empirical likelihood confidence regions are examined.     

The analysis and design of binary vapour-liquid equilibrium experiments. Part I: Parameter estimation and consistency tests

When estimating the unknown parameters in semi-empirical relationships from vapour-liquid equilibrium experiments, it has been common practice to minimize the sum of squares of some arbitrary function of one or more of the measured variables (x, y, P, T) without regard to the statistical consequences of such a procedure. It is shown that because these arbitrary procedures do not fully account for the errors in all the variables in the functional relationship they can lead to poor parameter estimates when compared to a statistically sound procedure based on maximum likelihood. The analysis of some binary data sets and a simulation study based on one of them are used to demonstrate the kinds of errors that arise. Use of the maximum likelihood method of estimation also aids one in checking the adequacy of the model used, and in testing the consistency of the experimental data. This is accomplished by comparing the deviations between predicted and measured responses with their computed confidence limits.     

TESTING CHANGE-POINTS IN THE EXPLOSIVE GAUSSIAN AUTOREGRESSIVE PROCESSES

Abstract. A functional limit theorem with a particular function class and topology is derived for non-ergodic type time series. This limit theorem allows us to study the asymptotic law of the associated likelihood ratio test (LRT) statistic for testing the presence of a change in the covariance parameter in the explosive Gaussian autoregressive model. We show that the level of the LRT cannot be approximated without introducing appropriate normalization. The limit law of a particular weighted likelihood ratio test is examined through a simulation study and is compared with the well-known Kolmogorov distribution obtained in the stationary case; we conclude that for practical applications when the root is really close to unity one can use the same thresholds as in the stationary case. This procedure is applied to the study of three real time series known to be non-stationary.     

Intermolecular Interactions in Crystal Structures of Imatinib-Containing Compounds

Imatinib, one of the most used therapeutic agents to treat leukemia, is an inhibitor that specifically blocks the activity of tyrosine kinases. The molecule of imatinib is flexible and contains several functional groups able to take part in H-bonding and hydrophobic interactions. Analysis of molecular conformations for this drug was carried out using density functional theory calculations of rotation potentials along single bonds and by analyzing crystal structures of imatinib-containing compounds taken from the Cambridge Structural Database and the Protein Data Bank. Rotation along the N-C bond in the region of the amide group was found to be the reason for two relatively stable molecular conformations, an extended and a folded one. The role of various types of intermolecular interactions in stabilization of the particular molecular conformation was studied in terms of (i) the likelihood of H-bond formation, and (ii) their contribution to the Voronoi molecular surface. It is shown that experimentally observed hydrogen bonds are in accord with the likelihood of their formation. The number of H-bonds in ligand-receptor complexes surpasses that in imatinib salts due to the large number of donors and acceptors of H-bonding within the binding pocket of tyrosine kinases. Contribution of hydrophilic intermolecular interactions to the Voronoi molecular surface is similar for both conformations, while π...π stacking is more typical for the folded conformation of imatinib.     

EFFICIENT METHOD OF MOMENTS ESTIMATORS FOR INTEGER TIME SERIES MODELS

The parameters of integer autoregressive models with Poisson, or negative binomial innovations can be estimated by maximum likelihood where the prediction error decomposition, together with convolution methods, is used to write down the likelihood function. When a moving average component is introduced this is not the case. To address this problem an efficient method of moment estimator is proposed where the estimated standard errors for the parameters are obtained using subsampling methods. The small sample properties of the estimator are investigated using Monte Carlo methods, while the approach is demonstrated using two well‐known examples from the time series literature.     

FAST LINEAR ESTIMATION METHODS FOR VECTOR AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. Three linear methods for estimating parameter values of vector auto-regressive moving-average (VARMA) models which are in general at least an order of magnitude faster than maximum likelihood estimation are developed in this paper. Simulation results for different model structures with varying numbers of component series and observations suggest that the accuracy of these procedures is in most cases comparable with maximum likelihood estimation. Procedures for estimating parameter standard error are also discussed and used for identification of nonzero elements in the VARMA polynomial structures. These methods can also be used to establish the order of the VARMA structure. We note, however, that the primary purpose of these estimates is to generate initial estimates for the nonzero parameters in order to reduce subsequent computational time of more efficient estimation procedures such as exact maximum likelihood.     

Recursive Prediction and Likelihood Evaluation for Periodic ARMA Models

This paper explores recursive prediction and likelihood evaluation techniques for periodic autoregressive moving-average (PARMA) time series models. The innovations algorithm is used to develop a simple recursive scheme for computing one-step-ahead predictors and their mean squared errors. The asymptotic form of this recursion is explored. The prediction results are then used to develop an efficient (and exact) PARMA likelihood evaluation algorithm for Gaussian series. We then show how a multivariate autoregressive moving average (ARMA) likelihood can be evaluated by writing the multivariate ARMA model in PARMA form. Explicit calculations for PARMA(1, 1) models and periodic autoregressions are included.     

Marginal Estimation of Parameter Driven Binomial Time Series Models

This article develops asymptotic theory for estimation of parameters in regression models for binomial response time series where serial dependence is present through a latent process. Use of generalized linear model estimating equations leads to asymptotically biased estimates of regression coefficients for binomial responses. An alternative is to use marginal likelihood, in which the variance of the latent process but not the serial dependence is accounted for. In practice, this is equivalent to using generalized linear mixed model estimation procedures treating the observations as independent with a random effect on the intercept term in the regression model. We prove that this method leads to consistent and asymptotically normal estimates even if there is an autocorrelated latent process. Simulations suggest that the use of marginal likelihood can lead to generalized linear model estimates result. This problem reduces rapidly with increasing number of binomial trials at each time point, but for binary data, the chance of it can remain over 45% even in very long time series. We provide a combination of theoretical and heuristic explanations for this phenomenon in terms of the properties of the regression component of the model, and these can be used to guide application of the method in practice.     

Spatio-temporal point processes, partial likelihood, foot and mouth disease

Spatio-temporal point process data arise in many fields of application. An intuitively natural way to specify a model for a spatio-temporal point process is through its conditional intensity at location x and time t, given the history of the process up to time t. Often, this results in an analytically intractable likelihood. Likelihood-based inference then relies on Monte Carlo methods which are computationally intensive and require careful tuning to each application. A partial likelihood alternative is proposed, which is computationally straightforward and can be applied routinely. The method is applied to data from the 2001 foot and mouth epidemic in the UK, using a previously published model for the spatio-temporal spread of the disease.     

NON-LINEAR TIME SERIES MODELLING AND DISTRIBUTIONAL FLEXIBILITY

Abstract. Most of the existing work in non-linear time series analysis has concentrated on generating flexible functional models by specifying non-linear specifications for the mean of a particular process, without much, if any, attention given to the distributional properties of the model. However, as Martin ( J. Time Ser. Anal. 13 (1992), 79–94) has shown, greater flexibility in perhaps a more natural way can be achieved by consideration of distributions from the generalized exponential class. This paper represents an extension of the earlier work of Martin by introducing a flexible class of non-linear time series models which can capture a wide range of empirical behaviour such as skewed, fat-tailed and even multimodal distributions. This class of models is referred to as generalized exponential non-linear time series. A maximum likelihood algorithm is given for estimating the parameters of the model and the framework is applied to estimating the distribution of the movements of the exchange rate.     

Some Results on Cointegration with Random Coefficients in the Error Correction Form: Estimation and Testing

Abstract.  This study considers a time-series model with random coefficients in cointegration. The estimation problem can be solved by maximizing the log-likelihood. Asymptotic distributions of the least squares and maximum likelihood estimates are considered. The randomness of the cointegration vector is checked by a score-based test approach. The test statistic converges asymptotically to a functional of Brownian processes. An empirical application to two cointegrated series, federal fund rate and 90-day treasury bill rate is considered.     

Integer‐Valued Autoregressive Models With Survival Probability Driven By A Stochastic Recurrence Equation

This paper proposes a new class of integer‐valued autoregressive models with a dynamic survival probability. The peculiarity of this class of models lies in the specification of the survival probability through a stochastic recurrence equation. The proposed models can effectively capture changing dependence over time and enhance both the in‐sample and out‐of‐sample performance of integer‐valued autoregressive models. This point is illustrated through an empirical application to a real‐time series of crime reports. Additionally, this paper discusses the reliability of likelihood‐based inference for the class of models. In particular, this study proves the consistency of the maximum likelihood estimator and a plug‐in estimator for the conditional probability mass function in a misspecified model setting.     

A selectivity method for modelling the kinetics of pentene-2 hydrogenation

A maximum likelihood ratio was used as a statistical criterion for discriminating among twenty-two kinetic models posited to describe the hydrogenation of mixed pentenes (cis- and trans-2-pentenes) with a Raney nickel catalyst. Selectivity ratio, the ratio of reaction rates of two different species, was used and found to be a suitable extension of conventional techniques for identifying rate controlling steps owing in this case to its reproducibility. Selectivity ratio models were of the Langmuir-Hinshelwood form. Hydrogenation was performed isothermally at 20°C in benzene at five constant pressures corresponding to partial pressures of hydrogen of 22.5, 42.3, 62.3, 112.3 and 122.1 psia. Discrimination was attempted at each pressure, but found acceptable only at the three intermediate pressures. At each of these pressures, a model of similar functional form was found to represent the selectivity data; however, this model appears to be empirical. The likelihood ratio approach appears to be a powerful one for discriminating among a large number of models even with only a moderate number (? 20) of experimental observations.     

Latent class and finite mixture models for multilevel data sets

An extension of latent class (LC) and finite mixture models is described for the analysis of hierarchical data sets. As is typical in multilevel analysis, the dependence between lower-level units within higher-level units is dealt with by assuming that certain model parameters differ randomly across higher-level observations. One of the special cases is an LC model in which group-level differences in the logit of belonging to a particular LC are captured with continuous random effects. Other variants are obtained by including random effects in the model for the response variables rather than for the LCs. The variant that receives most attention in this article is an LC model with discrete random effects: higher-level units are clustered based on the likelihood of their members belonging to the various LCs. This yields a model with mixture distributions at two levels, namely at the group and the subject level. This model is illustrated with three rather different empirical examples. The appendix describes an adapted version of the expectation-maximization algorithm that can be used for maximum likelihood estimation, as well as providing setups for estimating the multilevel LC model with generally available software.     

Prediction of protein residue contacts with a PDB-derived likelihood matrix

Proteins with similar folds often display common patterns ofresidue variability. A widely discussed question is how thesepatterns can be identified and deconvoluted to predict proteinstructure. In this respect, correlated mutation analysis (CMA)has shown considerable promise. CMA compares multiple membersof a protein family and detects residues that remain constantor mutate in tandem. Often this behavior points to structuralor functional interdependence between residues. CMA has beenused to predict pairs of amino acids that are distant in theprimary sequence but likely to form close contacts in the nativethree-dimensional structure. Until now these methods have usedevolutionary or biophysical models to score the fit betweenresidues. We wished to test whether empirical methods, derivedfrom known protein structures, would provide useful predictivepower for CMA. We analyzed 672 known protein structures, derivedcontact likelihood scores for all possible amino acid pairs,and used these scores to predict contacts. We then tested themethod on 118 different protein families for which structureshave been solved to atomic resolution. The mean performancewas almost seven times better than random prediction. Used inconcert with secondary structure prediction, the new CMA methodcould supply restraints for predicting still undetermined structures.     

AFI技术原理及应用

AFI为AVO流体反演（AVO Fluid Inversion）的简称。所有AVO属性都存在很大的＂不确定性＂,即存在一个大范围的岩性和流体组合,不同组合可以产生类似的AVO响应。AVO分析预测的油气存在的可能性有多大,这正是AFI技术要解决的问题,该技术能够提供烃类流体定量的可能性分布图。本文首先介绍了AFI技术的原理及实现过程,然后以在X地区的应用为例进行了说明,应用中取得了良好效果。     

Expert elicitation on ultrafine particles: likelihood of health effects and causal pathways

Background

Exposure to fine ambient particulate matter (PM) has consistently been associated with increased morbidity and mortality. The relationship between exposure to ultrafine particles (UFP) and health effects is less firmly established. If UFP cause health effects independently from coarser fractions, this could affect health impact assessment of air pollution, which would possibly lead to alternative policy options to be considered to reduce the disease burden of PM. Therefore, we organized an expert elicitation workshop to assess the evidence for a causal relationship between exposure to UFP and health endpoints.

Methods

An expert elicitation on the health effects of ambient ultrafine particle exposure was carried out, focusing on: 1) the likelihood of causal relationships with key health endpoints, and 2) the likelihood of potential causal pathways for cardiac events. Based on a systematic peer-nomination procedure, fourteen European experts (epidemiologists, toxicologists and clinicians) were selected, of whom twelve attended. They were provided with a briefing book containing key literature. After a group discussion, individual expert judgments in the form of ratings of the likelihood of causal relationships and pathways were obtained using a confidence scheme adapted from the one used by the Intergovernmental Panel on Climate Change.

Results

The likelihood of an independent causal relationship between increased short-term UFP exposure and increased all-cause mortality, hospital admissions for cardiovascular and respiratory diseases, aggravation of asthma symptoms and lung function decrements was rated medium to high by most experts. The likelihood for long-term UFP exposure to be causally related to all cause mortality, cardiovascular and respiratory morbidity and lung cancer was rated slightly lower, mostly medium. The experts rated the likelihood of each of the six identified possible causal pathways separately. Out of these six, the highest likelihood was rated for the pathway involving respiratory inflammation and subsequent thrombotic effects.

Conclusion

The overall medium to high likelihood rating of causality of health effects of UFP exposure and the high likelihood rating of at least one of the proposed causal mechanisms explaining associations between UFP and cardiac events, stresses the importance of considering UFP in future health impact assessments of (transport-related) air pollution, and the need for further research on UFP exposure and health effects.