MAXIMUM‐LIKELIHOOD ESTIMATION OF SEISMIC IMPULSE RESPONSES*

Felsenstein, J. (Department of Genetics SK-50, University of Washington, Seattle, Washington 98195). 1973. Maximum likelihood and minimum-steps methods for estimating evolutionary trees from data on discrete characters. Syst. Zool. 22:240-249.The general maximum likelihood approach to the statistical estimation of phylogenies is outlined, for data in which there are a number of discrete states for each character. The details of the maximum likelihood method will depend on the details of the probabilistic model of evolution assumed. There are a very large number of possible models of evolution. For a few of the simpler models, the calculation of the likelihood of an evolutionary tree is outlined. For these models, the maximum likelihood tree will be the same as the parsimonious (or minimum-steps) tree if the probability of change during the evolution of the group is assumed a priori to be very small. However, most sets of data require too many assumed state changes per character to be compatible with this assumption. Farris (1973) has argued that maximum likelihood and parsimony methods are identical under a much less restrictive set of assumptions. It is argued that the present methods are preferable to his, and a counterexample to his argument is presented. An algorithm which enables rapid calculation of the likelihood of a phylogeny is described. [Evolutionary trees: maximum likelihood.] The first systematic attempt to apply standard statistical inference procedures to the estimation of evolutionary trees was the work of Edwards and Cavalli-Sforza (1964; see also Cavalli-Sforza and Edwards, 1967). At about the same time, the parsimony' or minimum evolutionary steps method of Camin and Sokal (1965) gave a great impetus to the development of welldefined procedures for obtaining evolutionary trees. Edwards and Cavalli-Sforza concerned themselves with data from continuous variables such as gene frequencies and quantitative characters. The CaminSokal approach, on the other hand, was developed for characters which are recorded as a series of discrete states. Although some taxonomists have declared that the problem of guessing phylogenies should be viewed as a problem of statistical inference (Farris, 1967, 1968; Throckmorton, 1968), until recently there have been no attempts to explore the relationship between the statistical inference and minimum-steps approaches. Recently, Farris (1973) has presented a detailed argument that, under certain reasonable assumptions, the maximum-likelihood method of statistical inference appropriate to discrete-character data is precisely the parsimony method of Camin and Sokal. In this paper, I will examine the application of maximum likelihood methods to discrete characters, and will show that parsimony methods are not maximum likelihood methods under the assumptions made by Farris. They are maximum likelihood methods under considerably more restrictive assumptions about evolution. METHODS OF MAXIMUM LIKELIHOOD Suppose that we want to estimate the evolutionary tree, T, which is to be specified by the topological form of the tree and the times of branching. We are given a set of data, D, and a model of evolution, M, which incorporates not only the evolutionary processes, but also the processes of sampling by which we obtained the data. This model will usually be probabilistic, involving random events such as changes of the environment, occurrence of favorable

Many clinical or prevention studies involve missing or censored outcomes. Maximum likelihood (ML) methods provide a conceptually straightforward approach to estimation when the outcome is partially missing. Methods of implementing ML methods range from the simple to the complex, depending on the type of data and the missing-data mechanism. Simple ML methods for ignorable missing-data mechanisms (when data are missing at random) include complete-case analysis, complete-case analysis with covariate adjustment, survival analysis with covariate adjustment, and analysis via propensity-to-be-missing scores. More complex ML methods for ignorable missing-data mechanisms include the analysis of longitudinal dropouts via a marginal model for continuous data or a conditional model for categorical data. A moderately complex ML method for categorical data with a saturated model and either ignorable or nonignorable missing-data mechanisms is a perfect fit analysis, an algebraic method involving closed-form estimates and variances. A complex and flexible ML method with categorical data and either ignorable or nonignorable missing-data mechanisms is the method of composite linear models, a matrix method requiring specialized software. Except for the method of composite linear models, which can involve challenging matrix specifications, the implementation of these ML methods ranges in difficulty from easy to moderate.

Relative Efficiencies of the Maximum Likelihood, Maximum Parsimony, and Neighbor-Joining Methods for Estimating Protein Phylogeny

Calculation of the uncertainty in complication probability for various dose–response models, applied to the parotid gland

A Poisson ridge regression estimator

An eigenvector (EV) method for the determination of directional spectra from heave, pitch and roll buoy data is presented. Both a direct and an iterative form (based on an algorithm by Pawka) of this data-adaptive procedure are developed. The direct form outperforms the Longuet-Higgins et al. method, the cosine spread model and the maximum likelihood (ML) method for both simulated and real data. In the iterative form, the iterative EV method is superior to the other methods tested, including the iterative ML method for unimodal peaks when the noise-to-signal ratio is greater than 0.2 and for bimodal peaks at all noise levels. With real data, the direct EV method produced errors within an 80% confidence zone of the data cross-spectral matrix within the stopping criteria of Lawson and Long. Errors of the iterative EV and iterative ML methods were lower than for both the direct EV and ML results and were about the same magnitude when compared to each other. The iterative EV method, however, produced...

In an urban road network, travel times are not uniquely determined by the traffic states due to stochastic properties of traffic flow, stochastic arrivals and departures at intersections and traffic signal control. As a result, for a given traffic state, a range of travel times (delays) is found. This can be represented by a distribution of travel times (delays). Calibrating a model for the travel time only for the expectation value gives a large ‘noise’ such that the model will have little value for the prediction purpose. In this paper, the delay distribution function as derived from the analytical model under different circumstances is introduced. The overflow queue distribution which is the parameter in the delay distribution function is estimated based on traffic measurements, e.g., the measured delays, flows and cycle time. The Least Squares (LS) and Maximum Likelihood (ML) Estimation are used to perform the parameter estimation in the delay distribution. The Genetic Algorithm (GA) is applied to find the optimal solution for the objective functions in terms of minimizing square error and maximizing the likelihood function. Based on the estimated model parameters, the delay distribution is reconstructed. The estimated delay distribution is compared with that obtained from VISSIM simulation. Results show that both ML and LS estimation methods perform well in the undersaturated condition. While in the oversaturated condition, the ML method performs considerably better than the LS method.

Three-parameter Weibull probability distribution function is used to represent the time-to-failure data. For parameter estimation, the errors-in-variables, maximum likelihood, and least squares methods are compared. The results obtained from five data sets show that maximum likelihood method gives the most reliable parameter estimates which are close to those obtained by errors-in-variables approach. The least squares method gave the poorest results in most cases. For parameter estimation, the Luus-Jaakola direct search optimization procedure yielded the optimal parameter values in negligible computation time, taking less than a second of computation time on a Pentium4/2.66MHz personal computer for 50 data points.

The objective of this paper was to predict a model for describing stand structure of tree heights (h) and diameters at breast height (dbh). The research material consisted of data collected from 64 stands of Norway spruce (Picea abies Karst.) and 91 stands of Scots pine (Pinus sylvestris L.) located in southern Finland. Both stand types contained birch (Betula pendula Roth and B. pubescent Ehrh.) admixtures. The traditional univariate approach (Model I) of using the dbh distribution (Johnson’s SB) together with a height curve (Näslund’s function) was compared against the bivariate approaches, Johnson’s SBB distribution (Model II) and Model Ie. In Model Ie within-dbh-class h-variation was included by transforming a normally distributed homogenous error of linearized Näslund’s function to concern real heights. Basal-area-weighted distributions were estimated using the maximum likelihood (ML) method. Species-specific prediction models were derived using linear regression analysis. The models were compared with Kolmogorov-Smirnov tests for marginal distributions, accuracy of stand variables and the dbh-h relationship of individual trees. The differences in the stand characteristics between the models were marginal. Model I gave a slightly better fit for spruce, but Model II was better for pine stands. The univariate Model I resulted in clearly too narrow marginal h-distribution for pine. It is recommended applying of a constrained ML method for reasonable dbh-h relationship instead of using a pure ML method when fitting the SBB model.

Recent work has developed maximum likelihood (ML) methods for track-to-track data association and fusion in a multisensor, i.e., more than two sensor, environment. In order to conserve bandwidth, only the state estimates and corresponding covariance matrices are shared amongst the nodes. The fusion engine uses this track information to determine which tracks associate to the same target and then computes a fused track to improve the accuracy of the state estimates. The simplest class of ML methods assumes that the track errors from different sensors are uncorrelated. The more computationally demanding ML methods incorporate the cross-correlations that are due to the common process noise in the kinematic model of the target. In order to account for track correlations in practice, the cross-covariance matrices must be approximated from the single sensor covariance matrices. This paper introduces new methods to approximate the cross-covariance matrices, and these approximations lead to a third class of association and estimation methods. The paper then uses simulations to assess the performance of the different association and estimation techniques. The simulations include results when the sensor tracks are produced by either a Kalman filter or an interacting multiple model (IMM) filter.

Deconvolution techniques provide efficient implementations for super-resolution imaging for forward-looking scanning radar. However, deconvolution is normally an ill-posed problem, and the solution is extremely sensitive to noise. From a statistical perspective, maximum likelihood (ML) methods are able to condition the ill-posed problem into a well-posed one. Nevertheless, traditional ML methods only consider the amplitude of the echo and by ignoring the phase that do not adequately model the radar imaging system. In this letter, an I/Q-channel modeling ML method is proposed for forward-looking scanning radar. First, the probability model of the echo is deduced by jointly considering noise in the I and Q channels. Then, a probability density function of the received data is deduced and used to formulate the likelihood function. Finally, the targets can be precisely estimated by maximizing this likelihood function. The results of simulations and experiments are provided to illustrate the effectiveness of the proposed method.

Accurate determination of the effective reproduction number (Rt) is a very important strategy in the epidemiology of contagious diseases, including coronavirus disease 2019 (COVID-19). This study compares different methods of estimating the Rt of susceptible population to identify the most accurate method for estimating Rt. A secondary study. The value of Rt was estimated using attack rate (AR), exponential growth (EG), maximum likelihood (ML), time-dependent (TD), and sequential Bayesian (SB) methods, for Iran, the United States, the United Kingdom, India, and Brazil from June to October 2021. In order to accurately compare these methods, a simulation study was designed using forty scenarios. The lowest mean square error (MSE) was observed for TD and ML methods, with 15 and 12 cases, respectively. Therefore, considering the estimated values of Rt based on the TD method, it was found that Rt values in the United Kingdom (1.33; 95% CI: 1.14-1.52) and the United States (1.25; 95% CI: 1.12-1.38) substantially have been more than those in other countries, such as Iran (1.07; 95% CI: 0.95-1.19), India (0.99; 95% CI: 0.89-1.08), and Brazil (0.98; 95% CI: 0.84-1.14) from June to October 2021. The important result of this study is that TD and ML methods lead to a more accurate estimation of Rt of population than other methods. Therefore, in order to monitor and determine the epidemic situation and have a more accurate prediction of the incidence rate, as well as control COVID-19 and similar diseases, the use of these two methods is suggested to more accurately estimate Rt.

We propose a super-resolution resolution algorithm on the basis of maximum likelihood (ML) method and edge-orient diffusion. By using Hammerseley-Clifford theorem, an image field assumed to be a Markov random field is Gibbs distributed. An edge-orient diffusion function is introduced and employed in the Gibbs prior. According to Bayesian theorem, the solution to the maximum likelihood function is equal to that to maximum a posterior function. Therefore we incorporate ML with a prior distributed function. Experimental results illustrate that our method has a powerful super-resolution restoration performance. Compared with traditional ML method, our approach can not only obtain super-resolution images, but also eliminate noise artifacts effectively without smoothing edges.

The problem of distributed parameter estimation from binary quantized observations is studied when the unquantized observations are corrupted by combined multiplicative and additive Gaussian noise. These results are applicable to sensor networks where the sensors observe a parameter in combined additive and nonadditive noise and to a case where dispersed receivers are employed with analog communication over fading channels where the receivers employ binary quantization before noise-free digital communications to a fusion center. We first discuss the case in which all the quantizers use an identical threshold. The parameter identifiability condition is given, and, surprisingly, it is shown that unless the common threshold is chosen properly and the parameter lies in an open interval, the parameter will not generally be identifiable, in contrast to the additive noise case. The best achievable mean square error (MSE) performance is characterized by deriving the corresponding Cramer-Rao Lower Bound (CRLB). A closed-form expression describing the corresponding maximum likelihood (ML) estimator is presented. The stability of the performance of the ML estimators is improved when a nonidentical threshold strategy is utilized to estimate the unknown parameter. The thresholds are designed by maximizing the minimum asymptotic relative efficiency (ARE) between quantized and unquantized ML estimators. Although the ML estimation problem is nonconvex, it is shown that one can relax the optimization to make it convex. The solution to the relaxed problem is used as an initial solution in a gradient algorithm to solve the original problem. Next, the case where both the variances of the additive noise and multiplicative noise are unknown is studied. The corresponding CRLB is obtained, and the ML estimation problem is transformed to a convex optimization, which can be solved efficiently. Finally, numerical simulations are performed to substantiate the theoretical analysis.

Analysis of snow load probabilistic models and calculation of reference snow pressure using maximum likelihood method for multiple cities in Liaoning Province, China