This work concerns Lotka–Volterra models that are formulated using stochastic differential equations with regime-switching.Distinct from the existing formulations,the Markov chain that models random environments is u...This work concerns Lotka–Volterra models that are formulated using stochastic differential equations with regime-switching.Distinct from the existing formulations,the Markov chain that models random environments is unobservable.For such partially observed systems,we use Wonham’s filter to estimate the Markov chain from the observable evolution of the population,and convert the original system to a completely observable one.We then show that the positive solution of our model does not explode in finite time with probability 1.Several properties including stochastic boundedness,finite moments,sample path continuity and large-time asymptotic behaviour are also obtained.Moreover,stochastic permanence,extinction and feedback controls are also investigated.展开更多
We prove some transportation inequalities for hidden Markov chains, generalize the results proved by Kontorovich and Ramanan in two directions and give some applications to log-likelihood functions and hypothesis test...We prove some transportation inequalities for hidden Markov chains, generalize the results proved by Kontorovich and Ramanan in two directions and give some applications to log-likelihood functions and hypothesis testing.展开更多
The stage of a tumor is sometimes hard to predict, especially early in its development. The size and complexity of its observations are the major problems that lead to false diagnoses. Even experienced doctors can mak...The stage of a tumor is sometimes hard to predict, especially early in its development. The size and complexity of its observations are the major problems that lead to false diagnoses. Even experienced doctors can make a mistake in causing terrible consequences for the patient. We propose a mathematical tool for the diagnosis of breast cancer. The aim is to help specialists in making a decision on the likelihood of a patient’s condition knowing the series of observations available. This may increase the patient’s chances of recovery. With a multivariate observational hidden Markov model, we describe the evolution of the disease by taking the geometric properties of the tumor as observable variables. The latent variable corresponds to the type of tumor: malignant or benign. The analysis of the covariance matrix makes it possible to delineate the zones of occurrence for each group belonging to a type of tumors. It is therefore possible to summarize the properties that characterize each of the tumor categories using the parameters of the model. These parameters highlight the differences between the types of tumors.展开更多
基金This work was supported in part by the National Science Foundation under DMS-1207667.
文摘This work concerns Lotka–Volterra models that are formulated using stochastic differential equations with regime-switching.Distinct from the existing formulations,the Markov chain that models random environments is unobservable.For such partially observed systems,we use Wonham’s filter to estimate the Markov chain from the observable evolution of the population,and convert the original system to a completely observable one.We then show that the positive solution of our model does not explode in finite time with probability 1.Several properties including stochastic boundedness,finite moments,sample path continuity and large-time asymptotic behaviour are also obtained.Moreover,stochastic permanence,extinction and feedback controls are also investigated.
基金supported by the Youth Innovation Foundation of Zhongnan University of Economics and Law from the Fundamental Research Funds for the Central Universities of China (Grant No. 2009004/31540911202)
文摘We prove some transportation inequalities for hidden Markov chains, generalize the results proved by Kontorovich and Ramanan in two directions and give some applications to log-likelihood functions and hypothesis testing.
文摘The stage of a tumor is sometimes hard to predict, especially early in its development. The size and complexity of its observations are the major problems that lead to false diagnoses. Even experienced doctors can make a mistake in causing terrible consequences for the patient. We propose a mathematical tool for the diagnosis of breast cancer. The aim is to help specialists in making a decision on the likelihood of a patient’s condition knowing the series of observations available. This may increase the patient’s chances of recovery. With a multivariate observational hidden Markov model, we describe the evolution of the disease by taking the geometric properties of the tumor as observable variables. The latent variable corresponds to the type of tumor: malignant or benign. The analysis of the covariance matrix makes it possible to delineate the zones of occurrence for each group belonging to a type of tumors. It is therefore possible to summarize the properties that characterize each of the tumor categories using the parameters of the model. These parameters highlight the differences between the types of tumors.
