Approximate Maximum Likelihood Method Research Articles

Estimating local interaction from spatiotemporal forest data, and Monte Carlo bias correction

We point out a general problem in fitting continuous time spatially explicit models to a temporal sequence of spatial data observed at discrete times. To illustrate the problem, we examined the continuous time Markov model for forest gap dynamics. A forest is assumed to be apportioned into discrete cells (or sites) arranged in a regular square lattice. Each site is characterized as either a gap or a non-gap site according to the vegetation height of trees. The model incorporates the influence of neighboring sites on transition rate: transition rate from a non-gap to a gap site increases linearly with the number of neighbors that are currently in the gap state, and vice versa. We fitted the model to the spatiotemporal data of canopy height observed at the permanent plot in Barro Colorado Island (BCI). When we used the approximate maximum likelihood method to estimate the parameters of the model, the estimated transition rates included a large bias—in particular, the strength of interaction between nearby sites was underestimated. This bias originated from the assumption that each transition between two observation times is independent. The interaction between sites at local scale creates a long chain of transitions within a single census interval, which violates the independence of each transition. We show that a computer-intensive method, called Monte Carlo bias correction (MCBC), is very effective in removing the bias included in the estimate. The global and local gap densities measuring spatial aggregation of gap sites were computed from simulated and real gap dynamics to assess the model. When the approximate likelihood estimates were applied to the model, the predicted local gap density was clearly lower than the observed one. The use of MCBC estimates, suggesting a strong interaction between sites, improved this discrepancy.

On possibilities for estimating continuous-time ARMA parameters

The problem of estimating the parameters in continuous-time ARMA processes from discrete-time data is considered. Three different approaches, based on the prediction error method, the instrumental variable method and an approximate maximum likelihood method, respectively, arc studied. All three techniques provide reliable solutions to the estimation problem. A general discussion of the inherent difficulties of the problem is given together with an extensive numerical study.

Bayesian inference strategies for the prediction of genetic merit using threshold models with an application to calving ease scores in Italian Piemontese cattle

SummaryFirst parity calving difficulty scores from Italian Piemontese cattle were analysed using a threshold mixed effects model. The model included the fixed effects of age of dam and sex of calf and their interaction and the random effects of sire, maternal grandsire, and herd‐year‐season. Covariances between sire and maternal grandsire effects were modelled using a numerator relationship matrix based on male ancestors. Field data consisted of 23 953 records collected between 1989 and 1998 from 4741 herd‐year‐seasons. Variance and covariance components were estimated using two alternative approximate marginal maximum likelihood (MML) methods, one based on expectation‐maximization (EM) and the other based on Laplacian integration. Inferences were compared to those based on three separate runs or sequences of Markov Chain Monte Carlo (MCMC) sampling in order to assess the validity of approximate MML estimates derived from data with similar size and design structure. Point estimates of direct heritability were 0.24, 0.25 and 0.26 for EM, Laplacian and MCMC (posterior mean), respectively, whereas corresponding maternal heritability estimates were 0.10, 0.11 and 0.12, respectively. The covariance between additive direct and maternal effects was found to be not different from zero based on MCMC‐derived confidence sets. The conventional joint modal estimates of sire effects and associated standard errors based on MML estimates of variance and covariance components differed little from the respective posterior means and standard deviations derived from MCMC. Therefore, there may be little need to pursue computation‐intensive MCMC methods for inference on genetic parameters and genetic merits using conventional threshold sire and maternal grandsire models for large datasets on calving ease.

Further Long Memory Properties of Inflationary Shocks

Several previous studies have found fractionally integrated, or long memory behavior, in the conditional mean of inflation. This paper notes that extremely similar phenomena are also apparent in the squared and absolute values of residuals from fractionally filtered inflation series. Hence, the inflation process appears to have a dual long memory feature in both its first and its second conditional moments. We suggest a parametric model of long memory in both the conditional mean and the conditional variance. Some Monte Carlo evidence is presented that supports estimation of the model by approximate maximum likelihood methods. We then report estimated models for the inflation series for several different industrialized countries, including the United States. For nearly all of the countries in our study, there is strong evidence of statistically significant long memory parameters in both the conditional mean and the variance. We note some of the implications for modeling inflation.

Further Long Memory Properties of Inflationary Shocks

Several previous studies have found fractionally integrated, or long memory behavior, in the conditional mean of inflation. This paper notes that extremely similar phenomena are also apparent in the squared and absolute values of residuals from fractionally filtered inflation series. Hence, the inflation process appears to have a dual long memory feature in both its first and its second conditional moments. We suggest a parametric model of long memory in both the conditional mean and the conditional variance. Some Monte Carlo evidence is presented that supports estimation of the model by approximate maximum likelihood methods. We then report estimated models for the inflation series for several different industrialized countries, including the United States. For nearly all of the countries in our study, there is strong evidence of statistically significant long memory parameters in both the conditional mean and the variance. We note some of the implications for modeling inflation.

A general empirical model of protein evolution derived from multiple protein families using a maximum-likelihood approach.

Phylogenetic inference from amino acid sequence data uses mainly empirical models of amino acid replacement and is therefore dependent on those models. Two of the more widely used models, the Dayhoff and JTT models, are estimated using similar methods that can utilize large numbers of sequences from many unrelated protein families but are somewhat unsatisfactory because they rely on assumptions that may lead to systematic error and discard a large amount of the information within the sequences. The alternative method of maximum-likelihood estimation may utilize the information in the sequence data more efficiently and suffers from no systematic error, but it has previously been applicable to relatively few sequences related by a single phylogenetic tree. Here, we combine the best attributes of these two methods using an approximate maximum-likelihood method. We implemented this approach to estimate a new model of amino acid replacement from a database of globular protein sequences comprising 3,905 amino acid sequences split into 182 protein families. While the new model has an overall structure similar to those of other commonly used models, there are significant differences. The new model outperforms the Dayhoff and JTT models with respect to maximum-likelihood values for a large majority of the protein families in our database. This suggests that it provides a better overall fit to the evolutionary process in globular proteins and may lead to more accurate phylogenetic tree estimates. Potentially, this matrix, and the methods used to generate it, may also be useful in other areas of research, such as biological sequence database searching, sequence alignment, and protein structure prediction, for which an accurate description of amino acid replacement is required.

Joint space-time channel parameterestimation for DS-CDMA system in multipathRayleigh fading channels

An approximate maximum likelihood method for joint estimation of time delays and directions-of-arrival for direct-sequence code-division multiple-access systems over multipath Rayleigh fading channels is presented. To reduce the computational complexity, the multi-dimensional maximisation problem is transformed into two simple one-dimensional maximisation problems using an iterative scheme. To illustrate the performance, simulation results are compared with Cramer-Rao bounds.

Estimation of seismic-wave parameters and signal detection using maximum-likelihood methods

A broad-band maximum likelihood method is presented for the detection of and parameter extraction from seismic events using wideband data recorded by an array of seismic stations. The statistical characteristics of finite Fourier-transformed data motivate the use of approximate maximum-likelihood (ML) methods which allow simultaneous detection and wave-parameter estimation. The detection strategy based on the likelihood ratio indicates the presence of a seismic event and resolves different phases of seismic events arriving within a time interval of interest. The corresponding slowness vectors of the phases are simultaneously estimated by optimization of the likelihood function over parameters of interest. The potential of the wideband ML method is demonstrated on GERESS data and compared to conventional f– k analysis showing advantages of the former in detection and resolution.

Further results and insights on subspace based sinusoidal frequency estimation

Subspace-based methods for parameter identification have received considerable attention in the literature. Starting with a scalar-valued process, it is well known that subspace-based identification of sinusoidal frequencies is possible if the scalar valued data is windowed to form a low-rank vector-valued process. MUSIC and ESPRIT-like estimators have, for some time, been applied to this vector model. In addition, a statistically attractive Markov-like procedure for this class of methods has been proposed. Herein, the Markov-like procedure is reinvestigated. Several results regarding rank, performance, and structure are given in a compact manner. The large sample equivalence with the approximate maximum likelihood method by Stoica et al. (1988) is also established.

Hyperspectral imagery: clutter adaptation in anomaly detection

Hyperspectral sensors are passive sensors that simultaneously record images for hundreds of contiguous and narrowly spaced regions of the electromagnetic spectrum. Each image corresponds to the same ground scene, thus creating a cube of images that contain both spatial and spectral information about the objects and backgrounds in the scene. In this paper, we present an adaptive anomaly detector designed assuming that the background clutter in the hyperspectral imagery is a three-dimensional Gauss-Markov random field. This model leads to an efficient and effective algorithm for discriminating man-made objects (the anomalies) in real hyperspectral imagery. The major focus of the paper is on the adaptive stage of the detector, i.e., the estimation of the Gauss-Markov random field parameters. We develop three methods: maximum-likelihood; least squares; and approximate maximum-likelihood. We study these approaches along three directions: estimation error performance, computational cost, and detection performance. In terms of estimation error, we derive the Cramer-Rao bounds and carry out Monte Carlo simulation studies that show that the three estimation procedures have similar performance when the fields are highly correlated, as is often the case with real hyperspectral imagery. The approximate maximum-likelihood method has a clear advantage from the computational point of view. Finally, we test extensively with real hyperspectral imagery the adaptive anomaly detector incorporating either the least squares or the approximate maximum-likelihood estimators. Its performance compares very favorably with that of the RX algorithm.

Persistence in International Inflation Rates

We test for fractional dynamics in inflation rates based on the consumer price index (CPI) for 27 countries and inflation rates based on the wholesale price index (WPI) for 22 countries. The fractional differencing parameter is estimated using semiparametric and approximate maximum likelihood methods. Significant evidence of fractional dynamics with long‐memory features is found in both CPI‐ and WPI‐based inflation rates for industrial as well as developing countries. Implications of the findings are considered, and sources of long memory are hypothesized.

Persistence in International Inflation Rates

We test for fractional dynamics in inflation rates based on the consumer price index (CPI) for 27 countries and inflation rates based on the wholesale price index (WPI) for 22 countries. The fractional differencing parameter is estimated using semiparametric and approximate maximum likelihood methods. Significant evidence of fractional dynamics with long-memory features is found in both CPI- and WPI-based inflation rates for industrial as well as developing countries. Implications of the findings are considered, and sources of long memory are hypothesized.

A population pharmacokinetic-pharmacodynamic analysis of repeated measures time-to-event pharmacodynamic responses: the antiemetic effect of ondansetron.

This paper presents and illustrates methodology for specifying, estimating, and evaluating a predictive model for repeated measures time-to-event responses. The illustrative example specifies a model of the antiemetic effect vs. concentration relationship for the 5-HT3 antagonist ondansetron in the human ipecac model for emesis. A key part of this model is a time-dependent log hazard function for emesis that is increased by ipecac administration and decreased by ondansetron concentration. The model is fit using an approximate maximum likelihood method. The data consist of the time free of emeses and, for those individuals with emetic episodes, the time(s) of the episode(s). Model evaluation is accomplished using residual plots adapted to time-to-event data and a "posterior predictive check" wherein observed data statistics are compared to those obtained from data simulated from the fitted model. The ondansetron concentration required to obtain a 50% reduction in the hazard of emesis is estimated to be 1.4 +/- 0.2 ng/ml, and the rate constant for elimination of ipecac-induced hazard is 1.5 +/- 0.2 hr-1.

MAXIMUM LIKELIHOOD INFERENCE OF SPATIAL COVARIANCE PARAMETERS OF SOIL PROPERTIES

The statistical description of the spatial variability of soil properties is a preliminary step to the application of other quantitative techniques, for example, optimal spatial interpolation by kriging. The spatial correlation between the experimental data is characterized by the covariance function that is usually inferred from the experimental measurements. An appealing method for performing this inference is maximum likelihood estimation. This paper describes the results of the application of maximum likelihood estimation to the inference of the spatial covariance of infiltration measurements for a small catchment. Because there are a large number of experimental data, the approximate maximum likelihood method has been used to overcome the computational burden imposed by the calculation of the complete likelihood. As will be shown, this method, used in conjunction with classical graphical variogram analysis (method of moments), provides more objective inference of covariance parameters.

Comparison between some approximate maximum-likelihood methods for quantitative trait locus detection in progeny test designs

The power and efficiency of parameter estimation of four approximate maximum-likelihood segregation-analysis methods for QTL detection were numerically compared using Monte Carlo simulation. The approximations were designed to avoid the long computation required by exact maximum-likelihood segregation analysis for populations composed of large, independent half-sib families, as found in forest-tree and animal-breeding programs. The methods were compared both when information from a marker closely linked to the QTL was available and when it was not. Three of the approximations were from the literature: the Modal-Estimation method initially developed by Le Roy et al., an approximate Regressive Model from Demenais and Bonney, and the Within-Sire method used by Boichard et al. The fourth method was derived from this Within-Sire method by ignoring between-male-parent information and segregation within families due to the alleles inherited from the female parents. The relative advantages of the criteria are compared for various hypotheses concerning the characteristics of the QTL and the size of the population. No one method was clearly superior over all situations studied. The fourth, and simplest, method, however, performed sufficiently well when marker data were available, particularly in terms of power, for it to provide a tool for rapid preliminary screening of data from QTL mapping studies.

Random Component Threshold Models

Three threshold models, in which the response variable is related to regression variables with fixed coefficients as well as random components, are fitted to ordinal data. Estimation procedures are based on the use of penalized likelihood methods to provide the initial computations for both approximate maximum likelihood and residual maximum likelihood methods. Procedures are applied to a problem arising in the testing of dairy products.

Maximum Likelihood Estimation of the Differencing Parameter for Invertible Short and Long Memory Autoregressive Integrated Moving Average Models

SUMMARY In practical applications of Box-Jenkins autoregressive integrated moving average (ARIMA) models, the number of times that the observed time series must be differenced to achieve approximate stationarity is usually determined by careful, but mostly informal, analysis of the differenced series. For many time series, some differencing seems appropriate, but taking the first or the second difference may be too strong. As an alternative, Hosking, and Granger and Joyeux proposed the use of fractional differences. For -1/2 < d < 1/2, the resulting fractional ARIMA processes are stationary. For 0 < d < 1/2, the correlations are not summable. The parameter d can be estimated, for instance by maximum likelihood. Unfortunately, estimation methods known so far have been restricted to the stationary range -1/2 < d < 1/2. In this paper, we show how any real d > -1/2 can be estimated by an approximate maximum likelihood method. We thus obtain a unified approach to fitting traditional Box-Jenkins ARIMA processes as well as stationary and non-stationary fractional ARIMA processes. A confidence interval for d can be given. Tests, such as for unit roots in the autoregressive parameter or for stationarity, follow immediately. The resulting confidence intervals for the ARMA parameters take into account the additional uncertainty due to estimation of d. A simple algorithm for calculating the estimate of d and the ARMA parameters is given. Simulations and two data examples illustrate the results.

ESTIMATION OF THE LONG‐MEMORY PARAMETER, BASED ON A MULTIVARIATE CENTRAL LIMIT THEOREM

Abstract. Long memory is known to occur in many fields of statistical application. Stationary processes whose correlations decay asymptotically like ‖k‖2H‐2, where k is the lag and Hε (0.5, 1), provide useful parsimonious models with long memory. The parameter H characterizes the long‐memory features of the data. For long time series, maximum likelihood estimation of H can be costly in terms of CPU time. In this paper, we show that, for disjoint stretches of the data, estimates of H and other parameters that characterize the dependence structure are asymptotically independent. Averaging these estimates leads to a fast and efficient approximate maximum likelihood method.

A simple noniterative estimator for moving average models

SUMMARY We examine a simple noniterative estimator for the parameters of a general moving average process. This non-maximum-likelihood estimator derives moving average model parameters directly from the coefficients of an approximating autoregressive model. The estimator is evaluated through asymptotic expansions and by simulation, and is also compared with maximum likelihood and the related estimator of Durbin (1959). The comparison with maximum likelihood by simulation suggests a variety of circumstances in which the simpler estimator may be appropriate despite the advantage of maximum likelihood for properly-specified low-order models. The difficulties involved in estimating a stationary autoregressive-moving average (ARMA) process with a nontrivial moving average part are well known. They include problems arising from nonlinearity, problems associated with nonuniqueness of the density function, and difficulties in deriving exact finite-sample properties. In addition, computational considerations which may be minor difficulties for a single estimation can become prominent in Monte Carlo experiments used, for example, to investigate finite-sample performance. In this paper we advance a simple estimator which may alleviate some of these difficulties, and we offer a comparison of this estimator with maximum likelihood. Most algorithms for the estimation of moving-average models are based on nonlinear least squares, or on numerical optimization of the exact or approximate likelihood function for the sample. Important contributions to the latter class include those of Osborn (1977), Godolphin (1977) and Ansley (1979). Box & Jenkins (1976) describe approximate maximum likelihood methods; see Fuller (1976, Ch. 8) for a review of nonlinear least squares methods. Hannan & Rissanen (1982) and Koreisha & Pukkila (1990) discuss estimation using a sequence of long autoregressions. These methods may suffer from a number of drawbacks. In general they require iteration, implying the potential for slow convergence of even nonconvergence. Results may be sensitive to starting values. Some methods are prone to generate noninvertible estimates or, where constrained to invertibility, estimates clustered near the boundary of the invertibility region (Ansley & Newbold, 1980). Finally, these algorithms tend to be computationally burdensome. The new, non-maximum-likelihood estimator is related to the estimator proposed by Durbin (1959). Like Durbin's, it involves approximating a moving average process by an autoregressive model, and using the pattern of autoregressive coefficients to deduce estimates of the parameters of the underlying process. The resulting estimator is simple and

Long Memory in Foreign-Exchange Rates

Using the Geweke–Porter-Hudak test, we find evidence of long memory in exchange-rate data. This implies that the empirical evidence of unit roots in exchange rates may not be robust to long-memory alternatives. Fractionally integrated autoregressive moving average (ARFIMA) models are estimated by both the time-domain exact maximum likelihood (ML) method and the frequency-domain approximate ML method. Impulse-response functions and forecasts based on these estimated ARFIMA models are evaluated to gain insight into the long-memory characteristics of exchange rates. Some tentative explanations of the long memory found in the exchange rates are discussed.