Joint Distribution Of Time Research Articles

Beam-pointing drift prediction in pulsed lasers by a probabilistic learning approach.

In laser systems, it is well known that beam pointing is shifted due to many un-modeled factors, such as vibrations from the hardware platform and air disturbance. In addition, beam-pointing shift also varies with laser sources as well as time, rendering the modeling of shifting errors difficult. While a few works have addressed the problem of predicting shift dynamics, several challenges still remain. Specifically, a generic approach that can be easily applied to different laser systems is highly desired. In contrast to physical modeling approaches, we aim to predict beam-pointing drift using a well-established probabilistic learning approach, i.e., the Gaussian mixture model. By exploiting sampled datapoints (collected from the laser system) comprising time and corresponding shifting errors, the joint distribution of time and shifting error can be estimated. Subsequently, Gaussian mixture regression is employed to predict the shifting error at any query time. The proposed learning scheme is verified in a pulsed laser system (1064nm, Nd:YAG, 100Hz), showing that the drift prediction approach achieves remarkable performances.

A maximum likelihood estimator for left-truncated lifetimes based on probabilistic prior information about time of occurrence

ABSTRACTIn forestry, many processes of interest are binary and they can be modeled using lifetime analysis. However, available data are often incomplete, being interval- and right-censored as well as left-truncated, which may lead to biased parameter estimates. While censoring can be easily considered in lifetime analysis, left truncation is more complicated when individual age at selection is unknown. In this study, we designed and tested a maximum likelihood estimator that deals with left truncation by taking advantage of prior knowledge about the time when the individuals enter the experiment. Whenever a model is available for predicting the time of selection, the distribution of the delayed entries can be obtained using Bayes' theorem. It is then possible to marginalize the likelihood function over the distribution of the delayed entries in the experiment to assess the joint distribution of time of selection and time to event. This estimator was tested with continuous and discrete Gompertz-distributed lifetimes. It was then compared with two other estimators: a standard one in which left truncation was not considered and a second estimator that implemented an analytical correction. Our new estimator yielded unbiased parameter estimates with empirical coverage of confidence intervals close to their nominal value. The standard estimator leaded to an overestimation of the long-term probability of survival.

Joint modeling of time to recurrence and cancer stage at recurrence in oncology trials

ABSTRACTThis research was motivated by a clinical trial with bladder cancer patients who went through a surgery and were followed up for cancer recurrence. One of the main objectives of the trial was to evaluate the time to cancer recurrence in patients in control and experimental groups. At the time of recurrence, the disease stage was also evaluated. Because the stage of cancer at recurrence significantly impacts future treatment and patient prognosis of survival, analyzing the time to cancer recurrence and the stage at recurrence jointly provides more clinically relevant information than analyzing the time to recurrence alone. In this paper, we propose a stochastic model for the joint distribution of time to recurrence and cancer stage that (1) accounts for the recurrence caused by cancer cells surviving a treatment or a surgery and for the recurrence caused by spontaneous carcinogenesis, and (2) incorporates parameters that have biological meaning. To estimate the parameters, we use the maximum-likelihood method combined with the EM algorithm. To demonstrate the performance of our modeling, we evaluate the data from a clinical trial in patients with bladder cancer. We also use simulations to assess the sensitivity of the method.

A Mixture Model for the Regression Analysis of Competing Risks Data

SUMMARY A parametric mixture model provides a regression framework for analysing failure-time data that are subject to censoring and multiple modes of failure. The regression context allows us to adjust for concomitant variables and to assess their effects on the joint distribution of time and type of failure. The mixing parameters correspond to the marginal probabilities of the various failure types and are modelled as logistic functions of the covariates. The hazard rate for each conditional distribution of time to failure, given type of failure, is modelled as the product of a piece-wise exponential function of time and a log-linear function of the covariates. An EM algorithm facilitates the maximum likelihood analysis and illuminates the contributions of the censored observations. The methods are illustrated with data from a heart transplant study and are compared with a cause-specific hazard analysis. The proposed mixture model can also be used to analyse multivariate failure-time data.

Expected Critical Path Lengths in PERT Networks

A method of obtaining a fairly good lower approximation to the expected duration time of a project whose individual job times are discrete random variables is described. It is assumed that jobs that immediately precede any given job may have a joint distribution of times, with independence between immediate predecessors of different jobs. The method provides an estimate that is usually much better, and never worse, than the one obtained by replacing each random job time by its expected value. An application to expected minimal path lengths in networks whose arc lengths are random variables is discussed.