Triplet Markov Chain Research Articles

Filtering in Triplet Markov Chain Model in the Presence of Non-Gaussian Noise with Application to Target Tracking

In hidden Markov chain (HMC) models, widely used for target tracking, the process noise and measurement noise are in general assumed to be independent and Gaussian for mathematical simplicity. However, the independence and Gaussian assumptions do not always hold in practice. For instance, in a typical radar tracking application, the measurement noise is correlated over time as the sampling frequency of a radar is generally much higher than the bandwidth of the measurement noise. In addition, target maneuvers and measurement outliers imply that the process noise and measurement noise are non-Gaussian. To solve this problem, we resort to triplet Markov chain (TMC) models to describe stochastic systems with correlated noise and derive a new filter under the maximum correntropy criterion to deal with non-Gaussian noise. By stacking the state vector, measurement vector, and auxiliary vector into a triplet state vector, the TMC model can capture the complete dynamics of stochastic systems, which may be subjected to potential parameter uncertainty, non-stationarity, or error sources. Correntropy is used to measure the similarity of two random variables; unlike the commonly used minimum mean square error criterion, which uses only second-order statistics, correntropy uses second-order and higher-order information, and is more suitable for systems in the presence of non-Gaussian noise, particularly some heavy-tailed noise disturbances. Furthermore, to reduce the influence of round-off errors, a square-root implementation of the new filter is provided using QR decomposition. Instead of the full covariance matrices, corresponding Cholesky factors are recursively calculated in the square-root filtering algorithm. This is more numerically stable for ill-conditioned problems compared to the conventional filter. Finally, the effectiveness of the proposed algorithms is illustrated via three numerical examples.

A non-stationary NDVI time series modelling using triplet Markov chain

Nowadays, vegetation monitoring using remotely sensed data is an important far-reaching real-world issue. The main purpose of this study is to build a triplet Markov chain (TMC) to model and analyse vegetation dynamics on large-scales using non-stationary normalised difference vegetation index (NDVI) time series. TMC is a generalisation of hidden Markov models (HMMs), which have been widely used to represent satellite time series images but which they proved to be inefficient for non-stationary data. The TMC model proposed in this paper overcomes this limit by adding an auxiliary process which allows modelling non-stationarity. In order to assess the performance of the proposed model, experimentation is carried out using moderate resolution imaging spectroradiometer (MODIS) NDVI time series of the north-western region of Tunisia. The TMC model is compared to standard HMM and seasonal auto regressive integrated moving average model (SARIMA) and proved to achieve the best performance with an overall accuracy prediction rate of 92.8% and a kappa coefficient of 0.885.

A non-stationary NDVI time series modelling using triplet Markov chain

Nowadays, vegetation monitoring using remotely sensed data is an important far-reaching real-world issue. The main purpose of this study is to build a triplet Markov chain (TMC) to model and analyse vegetation dynamics on large-scales using non-stationary normalised difference vegetation index (NDVI) time series. TMC is a generalisation of hidden Markov models (HMMs), which have been widely used to represent satellite time series images but which they proved to be inefficient for non-stationary data. The TMC model proposed in this paper overcomes this limit by adding an auxiliary process which allows modelling non-stationarity. In order to assess the performance of the proposed model, experimentation is carried out using moderate resolution imaging spectroradiometer (MODIS) NDVI time series of the north-western region of Tunisia. The TMC model is compared to standard HMM and seasonal auto regressive integrated moving average model (SARIMA) and proved to achieve the best performance with an overall accuracy prediction rate of 92.8% and a kappa coefficient of 0.885.

Phasic Triplet Markov Chains.

Hidden Markov chains have been shown to be inadequate for data modeling under some complex conditions. In this work, we address the problem of statistical modeling of phenomena involving two heterogeneous system states. Such phenomena may arise in biology or communications, among other fields. Namely, we consider that a sequence of meaningful words is to be searched within a whole observation that also contains arbitrary one-by-one symbols. Moreover, a word may be interrupted at some site to be carried on later. Applying plain hidden Markov chains to such data, while ignoring their specificity, yields unsatisfactory results. The Phasic triplet Markov chain, proposed in this paper, overcomes this difficulty by means of an auxiliary underlying process in accordance with the triplet Markov chains theory. Related Bayesian restoration techniques and parameters estimation procedures according to the new model are then described. Finally, to assess the performance of the proposed model against the conventional hidden Markov chain model, experiments are conducted on synthetic and real data.

Unsupervised Segmentation of Random Discrete Data Hidden With Switching Noise Distributions

Hidden Markov models are very robust and have been widely used in a wide range of application fields; however, they can prove some limitations for data restoration under some complex situations. These latter include cases when the data to be recovered are nonstationary. The recent triplet Markov models have overcome such difficulty thanks to their rich formalism, that allows considering more complex data structures while keeping the computational complexity of the different algorithms linear to the data size. In this letter, we propose a new triplet Markov chain that allows the unsupervised restoration of random discrete data hidden with switching noise distributions. We also provide genuine parameters estimation and MPM restoration algorithms. The new model is validated through experiments conducted on synthetic data and on real images, whose results show its interest with respect to the standard hidden Markov chain.

Unsupervised segmentation of hidden semi-Markov non-stationary chains

The Bayesian segmentation using Hidden Markov Chains (HMC) is widely used in various domains such as speech recognition, acoustics, biosciences, climatology, text recognition, automatic translation and image processing. On the one hand, hidden semi-Markov chains (HSMC), which extend HMC, have turned out to be of interest in many situations and have improved HMC-based results. On the other hand, the case of non-stationary data can pose an important problem in real-life situations, especially when the model parameters have to be estimated. The aim of this paper is to consider these two extensions simultaneously: we propose using a particular triplet Markov chain (TMC) to deal with non-stationary hidden semi-Markov chains. In addition, we consider a recent particular HSMC having the same computation complexity as the classical HMC. We propose a related parameter estimation method and the resulting unsupervised Bayesian segmentation is validated through experiments; in particular, a real radar image segmentations are provided.

Unsupervised segmentation of triplet Markov chains hidden with long-memory noise

The hidden Markov chain (HMC) model is a couple of random sequences ( X, Y), in which X is an unobservable Markov chain, and Y is its observable noisy version. Classically, the distribution p( y| x) is simple enough to ensure the Markovianity of p( x| y), that enables one to use different Bayesian restoration techniques. HMC model has recently been extended to “pairwise Markov chain” (PMC) model, in which one directly assumes the Markovianity of the pair Z=( X,Y), and which still enables one to recover X from Y. Finally, PMC has been extended to “triplet Markov chain” (TMC) model, which is obtained by adding a third chain U and considering the Markovianity of the triplet T=( X, U, Y). When U is not too complex, X can still be recovered from Y. Then U can model different situations, like non-stationarity or semi-Markovianity of ( X, Y). Otherwise, PMC and TMC have been extended to pairwise “partially” Markov chains (PPMC) and triplet “partially” Markov chains (TPMC), respectively. In a PPMC Z=( X, Y) the distribution p( x| y) is a Markov distribution, but p( y| x) may not be and, similarly, in a TPMC T=( X, U, Y) the distribution p( x, u| y) is a Markov distribution, but p( y| x, u) may not be. However, both PPMC and TPMC can enable one to recover X from Y, and TPMC include different long-memory noises. The aim of this paper is to show how a particular Gaussian TPMC can be used to segment a discrete signal hidden with long-memory noise. An original parameter estimation method, based on “Iterative Conditional Estimation” (ICE) principle, is proposed and some experiments concerned with unsupervised segmentation are provided. The particular unsupervised segmentation method used in experiments can also be seen as identification of different stationarities in fractional Brownian noise, which is widely used in different problems in telecommunications, economics, finance, or hydrology.