Pairwise Markov Models Research Articles

Pairwise Markov Models and Hybrid Segmentation Approach

The article studies segmentation problem (also known as classification problem) with pairwise Markov models (PMMs). A PMM is a process where the observation process and underlying state sequence form a two-dimensional Markov chain, it is a natural generalization of a hidden Markov model. To demonstrate the richness of the class of PMMs, we examine closer a few examples of rather different types of PMMs: a model for two related Markov chains, a model that allows to model an inhomogeneous Markov chain as a conditional marginal process of a homogeneous PMM, and a semi-Markov model. The segmentation problem assumes that one of the marginal processes is observed and the other one is not, the problem is to estimate the unobserved state path given the observations. The standard state path estimators often used are the so-called Viterbi path (a sequence with maximum state path probability given the observations) or the pointwise maximum a posteriori (PMAP) path (a sequence that maximizes the conditional state probability for given observations pointwise). Both these estimators have their limitations, therefore we derive formulas for calculating the so-called hybrid path estimators which interpolate between the PMAP and Viterbi path. We apply the introduced algorithms to the studied models in order to demonstrate the properties of different segmentation methods, and to illustrate large variation in behaviour of different segmentation methods in different PMMs. The studied examples show that a segmentation method should always be chosen with care by taking into account the purpose of modelling and the particular model of interest.

Local Viterbi property in decoding

Abstract The article studies the decoding problem (also known as the classification or the segmentation problem) with pairwise Markov models (PMMs). A PMM is a process where the observation process and the underlying state sequence form a two-dimensional Markov chain, a natural generalization of hidden Markov model. The standard solutions to the decoding problem are the so-called Viterbi path—a sequence with maximum state path probability given the observations—or the pointwise maximum a posteriori (PMAP) path that maximizes the expected number of correctly classified entries. When the goal is to simultaneously maximize both criterions—conditional probability (corresponding to Viterbi path) and pointwise conditional probability (corresponding to PMAP path)—then they are combined into one single criterion via the regularization parameter $C$. The main objective of the article is to study the behaviour of the solution—called the hybrid path—as $C$ grows. Increasing $C$ increases the conditional probability of the hybrid path and when $C$ is big enough then every hybrid path is a Viterbi path. We show that hybrid paths also approach the Viterbi path locally: we define $m$-locally Viterbi paths and show that the hybrid path is $m$-locally Viterbi whenever $C$ is big enough. This all might lead to an impression that when $C$ is relatively big then any hybrid path that is not yet Viterbi differs from the Viterbi path by a few single entries only. We argue that this intuition is wrong, because when unique and $m$-locally Viterbi, then different hybrid paths differ by at least $m$ entries. Thus, when $C$ increases then the different hybrid paths tend to differ from each other by larger and larger intervals. Hence the hybrid paths might offer a variety of rather different solutions to the decoding problem.

Exponential forgetting of smoothing distributions for pairwise Markov models

We consider a bivariate Markov chain Z={Zk}k≥1={(Xk,Yk)}k≥1 taking values on product space Z=X×Y, where X is possibly uncountable space and Y={1,…,|Y|} is a finite state-space. The purpose of the paper is to find sufficient conditions that guarantee the exponential convergence of smoothing, filtering and predictive probabilities: supn≥t‖P(Yt:∞∈⋅|Xl:n)−P(Yt:∞∈⋅|Xs:n)‖TV≤Ksαt,a.s. Here t≥s≥l≥1, Ks is σ(Xs:∞)-measurable finite random variable and α∈(0,1) is fixed. In the second part of the paper, we establish two-sided versions of the above-mentioned convergence. We show that the desired convergences hold under fairly general conditions. A special case of above-mentioned very general model is popular hidden Markov model (HMM). We prove that in HMM-case, our assumptions are more general than all similar mixing-type of conditions encountered in practice, yet relatively easy to verify.

State estimation in pairwise Markov models with improved robustness using unbiased FIR filtering

We propose a novel estimation procedure for linear time-varying pairwise Markov models (PMM), that is robust to system parameter uncertainties occurring in real-world applications. In order to cope with mismodeling errors and ignorance of noise/initial state statistics, we solve a finite-horizon state estimation problem. The resulting unbiased finite impulse response filter for PMMs (PMM-UFIR) is first derived in batch form and then converted to a recursive Kalman-like form for the sake of complexity reduction. Closed forms for the error covariance matrix of the state estimate are also provided for analytical performance assessment.Numerical results illustrate the effectiveness of the proposed estimation method over Gaussian processes, by showing that the PMM-UFIR is nearly as accurate as (resp. more robust than) optimal filtering under perfect (resp. uncertain) system parameters after tuning the horizon size.

SVD-based state and parameter estimation approach for generalized Kalman filtering with application to GARCH-in-Mean estimation

In this paper, a gradient-based adaptive filtering technique for unknown state and parameter estimation is proposed for some extensions of classical state-space models: (i) linear time-invariant multiple-input, multiple-output (LTI MIMO) systems, and (ii) linear pairwise Markov models (PMMs) with the related pairwise Kalman filter (PKF). The solution to robust adaptive filtering problem is grounded in the singular value decomposition (SVD) recently adopted for the classical Kalman filter. In contrast to our previous result, the new methods are designed for the case of thin matrices (pre-arrays) involved in the filtering recursions and sensitivities calculation. Additionally, the paper discusses a time-varying variance models’ calibration by the proposed innovative adaptive filters. In particular, the state-space approach for estimating GARCH-in-Mean(1,1) is explored.

Assessing the segmentation performance of pairwise and triplet Markov models

The hidden Markov models (HMMs) are state-space models widely applied in time series analysis. Well-known Bayesian state estimation methods designed for HMMs, such as the Baum–Welch algorithm and the Viterbi algorithm, allow state estimation with a complexity linear in the sample size. We consider recent extensions of HMMs, specifically the pairwise Markov models (PMMs) and the triplet Markov models (TMMs), in which the Baum–Welch algorithm also has a complexity linear in the sample size. However, the state process is not necessarily Markovian in PMMs and TMMs, which offers a considerable flexibility of modeling. This study explores potential performance gains achievable if PMMs and TMMs are used to describe the state-space system rather than HMMs. This is done through extensive comparative Monte-Carlo experiments among HMMs, PMMs and TMMs in the case of discrete state space models. A simple comparative example of the use of PMMs and HMMs to predict market direction is also given. These experiments confirm the interest of PMMs and TMMs in the time series modeling: specifically, the classification rate can be improved by nearly fifty percent. These findings mean that PMMs and TMMs may be more suitable than classic HMMs for real-world applications.

The UD-based Approach for Designing Pairwise Kalman Filtering Algorithms

Over the past few years we have observed a growing interest in the class of Pairwise Markov Models (PMMs). In this paper, we explore the Pairwise Kalman Filter (PKF) intended for the hidden state estimation in linear PMMs in the presence of Gaussian noises. Previous works produced the robust square-root PKF algorithm for improving a numerical stability of the estimator with respect to roundoff errors. The square-root approach is, definitely, the most popular technique for designing numerically stable filter implementations. However, the use of the modified Cholesky decomposition of corresponding error covariance matrix together with the modified weighted Gram-Schmidt orthogonalization at each iteration step of the Kalman filter was shown to improve the estimation accuracy with the reduced computational cost compared to the square-root methods. Here, we propose the UD-based approach for designing the PKF implementations. As for the classical Kalman filter, we may anticipate a high accuracy of the new UD-based PKF implementation with the reduced CPU cost. The methodology is derived in terms of covariance quantities.

Dempster–Shafer Fusion of Evidential Pairwise Markov Chains

Hidden Markov models have been extended in many directions, leading to pairwise Markov models, triplet Markov models, or discriminative random fields, all of which have been successfully applied in many fields covering signal and image processing. The Dempster–Shafer theory of evidence has also shown its interest in a wide range of situations involving reasoning under uncertainty and/or information fusion. There are, however, only few works dealing with both of these modeling tools simultaneously. The aim of this paper, which falls under this category of works, is to propose a general evidential Markov model offering wide modeling and processing possibilities regarding information imprecision, sensor unreliability, and data fusion. The main interest of the proposed model relies in the possibility of achieving, easily, the Dempster–Shafer fusion without destroying the Markovianity.

Integration of Architectural and Cytologic Driven Image Algorithms for Prostate Adenocarcinoma Identification

Introduction: The advent of digital slides offers new opportunities within the practice of pathology such as the use of image analysis techniques to facilitate computer aided diagnosis (CAD) solutions. Use of CAD holds promise to enable new levels of decision support and allow for additional layers of quality assurance and consistency in rendered diagnoses. However, the development and testing of prostate cancer CAD solutions requires a ground truth map of the cancer to enable the generation of receiver operator characteristic (ROC) curves. This requires a pathologist to annotate, or paint, each of the malignant glands in prostate cancer with an image editor software - a time consuming and exhaustive process.Recently, two CAD algorithms have been described: probabilistic pairwise Markov models (PPMM) and spatially-invariant vector quantization (SIVQ). Briefly, SIVQ operates as a highly sensitive and specific pattern matching algorithm, making it optimal for the identification of any epithelial morphology, whereas PPMM operates as a highly sensitive detector of malignant perturbations in glandular lumenal architecture.Methods: By recapitulating algorithmically how a pathologist reviews prostate tissue sections, we created an algorithmic cascade of PPMM and SIVQ algorithms as previously described by Doyle el al. [1] where PPMM identifies the glands with abnormal lumenal architecture, and this area is then screened by SIVQ to identify the epithelium.Results: The performance of this algorithm cascade was assessed qualitatively (with the use of heatmaps) and quantitatively (with the use of ROC curves) and demonstrates greater performance in the identification of malignant prostatic epithelium.Conclusion: This ability to semi-autonomously paint nearly all the malignant epithelium of prostate cancer has immediate applications to future prostate cancer CAD development as a validated ground truth generator. In addition, such an approach has potential applications as a pre-screening/quality assurance tool.

Integration of architectural and cytologic driven image algorithms for prostate adenocarcinoma identification.

Introduction: The advent of digital slides offers new opportunities within the practice of pathology such as the use of image analysis techniques to facilitate computer aided diagnosis (CAD) solutions. Use of CAD holds promise to enable new levels of decision support and allow for additional layers of quality assurance and consistency in rendered diagnoses. However, the development and testing of prostate cancer CAD solutions requires a ground truth map of the cancer to enable the generation of receiver operator characteristic (ROC) curves. This requires a pathologist to annotate, or paint, each of the malignant glands in prostate cancer with an image editor software - a time consuming and exhaustive process.Recently, two CAD algorithms have been described: probabilistic pairwise Markov models (PPMM) and spatially-invariant vector quantization (SIVQ). Briefly, SIVQ operates as a highly sensitive and specific pattern matching algorithm, making it optimal for the identification of any epithelial morphology, whereas PPMM operates as a highly sensitive detector of malignant perturbations in glandular lumenal architecture. Methods: By recapitulating algorithmically how a pathologist reviews prostate tissue sections, we created an algorithmic cascade of PPMM and SIVQ algorithms as previously described by Doyle el al. [1] where PPMM identifies the glands with abnormal lumenal architecture, and this area is then screened by SIVQ to identify the epithelium. Results: The performance of this algorithm cascade was assessed qualitatively (with the use of heatmaps) and quantitatively (with the use of ROC curves) and demonstrates greater performance in the identification of malignant prostatic epithelium. Conclusion: This ability to semi-autonomously paint nearly all the malignant epithelium of prostate cancer has immediate applications to future prostate cancer CAD development as a validated ground truth generator. In addition, such an approach has potential applications as a pre-screening/quality assurance tool.

High-throughput detection of prostate cancer in histological sections using probabilistic pairwise Markov models

In this paper we present a high-throughput system for detecting regions of carcinoma of the prostate (CaP) in HSs from radical prostatectomies (RPs) using probabilistic pairwise Markov models (PPMMs), a novel type of Markov random field (MRF). At diagnostic resolution a digitized HS can contain 80Kx70K pixels - far too many for current automated Gleason grading algorithms to process. However, grading can be separated into two distinct steps: (1) detecting cancerous regions and (2) then grading these regions. The detection step does not require diagnostic resolution and can be performed much more quickly. Thus, we introduce a CaP detection system capable of analyzing an entire digitized whole-mount HS (2x1.75cm(2)) in under three minutes (on a desktop computer) while achieving a CaP detection sensitivity and specificity of 0.87 and 0.90, respectively. We obtain this high-throughput by tailoring the system to analyze the HSs at low resolution (8microm per pixel). This motivates the following algorithm: (Step 1) glands are segmented, (Step 2) the segmented glands are classified as malignant or benign, and (Step 3) the malignant glands are consolidated into continuous regions. The classification of individual glands leverages two features: gland size and the tendency for proximate glands to share the same class. The latter feature describes a spatial dependency which we model using a Markov prior. Typically, Markov priors are expressed as the product of potential functions. Unfortunately, potential functions are mathematical abstractions, and constructing priors through their selection becomes an ad hoc procedure, resulting in simplistic models such as the Potts. Addressing this problem, we introduce PPMMs which formulate priors in terms of probability density functions, allowing the creation of more sophisticated models. To demonstrate the efficacy of our CaP detection system and assess the advantages of using a PPMM prior instead of the Potts, we alternately incorporate both priors into our algorithm and rigorously evaluate system performance, extracting statistics from over 6000 simulations run across 40 RP specimens. Perhaps the most indicative result is as follows: at a CaP sensitivity of 0.87 the accompanying false positive rates of the system when alternately employing the PPMM and Potts priors are 0.10 and 0.20, respectively.