Variable-order Markov Chains Research Articles

Auditory learning of recurrent tone sequences is present in the newborn's brain

The seemingly effortless ability of our auditory system to rapidly detect new events in a dynamic environment is crucial for survival. Whether the underlying brain processes are innate is unknown. To answer this question, electroencephalography was recorded while regularly patterned (REG) versus random (RAND) tone sequences were presented to sleeping neonates. Regular relative to random sequences elicited differential neural responses after only a single repetition of the pattern indicating the existence of an innate capacity of the auditory system to detect auditory sequential regularities. We show that the newborn auditory system accumulates evidence only somewhat longer than the minimum amount determined by the ideal Bayesian observer model (the prediction from a variable-order Markov chain model) before detecting a repeating pattern. Thus, newborns can quickly form representations for regular features of the sound input, preparing the way for learning the contingencies of the environment.

A dynamic “predict, then optimize” preventive maintenance approach using operational intervention data

We investigate whether historical machine failures and maintenance records may be used to derive future machine failure estimates and, in turn, prescribe advancements of scheduled preventive maintenance interventions. We model the problem using a sequential predict, then optimize approach. In our prescriptive optimization model, we use a finite horizon Markov decision process with a variable order Markov chain, in which the chain length varies depending on the time since the last preventive maintenance action was performed. The model therefore captures the dependency of a machine’s failures on both recent failures as well as preventive maintenance actions, via our prediction model. We validate our model using an original equipment manufacturer data set and obtain policies that prescribe when to deviate from the planned periodic maintenance schedule. To improve our predictions for machine failure behavior with limited to no past data, we pool our data set over different machine classes by means of a Poisson generalized linear model. We find that our policies can supplement and improve on those currently applied by 5%, on average.

A New Phenotyping Pipeline Reveals Three Types of Lateral Roots and a Random Branching Pattern in Two Cereals.

Recent progress in root phenotyping has focused mainly on increasing throughput for genetic studies, while identifying root developmental patterns has been comparatively underexplored. We introduce a new phenotyping pipeline for producing high-quality spatiotemporal root system development data and identifying developmental patterns within these data. The SmartRoot image-analysis system and temporal and spatial statistical models were applied to two cereals, pearl millet (Pennisetum glaucum) and maize (Zea mays). Semi-Markov switching linear models were used to cluster lateral roots based on their growth rate profiles. These models revealed three types of lateral roots with similar characteristics in both species. The first type corresponds to fast and accelerating roots, the second to rapidly arrested roots, and the third to an intermediate type where roots cease elongation after a few days. These types of lateral roots were retrieved in different proportions in a maize mutant affected in auxin signaling, while the first most vigorous type was absent in maize plants exposed to severe shading. Moreover, the classification of growth rate profiles was mirrored by a ranking of anatomical traits in pearl millet. Potential dependencies in the succession of lateral root types along the primary root were then analyzed using variable-order Markov chains. The lateral root type was not influenced by the shootward neighbor root type or by the distance from this root. This random branching pattern of primary roots was remarkably conserved, despite the high variability of root systems in both species. Our phenotyping pipeline opens the door to exploring the genetic variability of lateral root developmental patterns.

The impact of long-term water stress on tree architecture and production is related to changes in transitions between vegetative and reproductive growth in the 'Granny Smith' apple cultivar.

Water stress (WS) generates a number of physiological and morphological responses in plants that depend on the intensity and duration of stress as well as the plant species and development stage. In perennial plants, WS may affect plant development through cumulative effects that modify plant functions, architecture and production over time. Plant architecture depends on the fate of the terminal and axillary buds that can give rise, in the particular case of apple, to reproductive or vegetative growth units (GUs) of different lengths. In this study, the impact of long-term WS (7 years) on the fate of terminal and axillary buds was investigated in relation to flowering occurrence and production pattern (biennial vs regular) in the 'Granny Smith' cultivar. It was observed that WS decreased the total number of GUs per branch, regardless of their type. Conversely, WS did not modify the timing of the two successive developmental phases characterized by the production of long and medium GUs and an alternation of floral GUs over time, respectively. The analysis of GU successions over time using a variable-order Markov chain that included both the effects of the predecessor and water treatment revealed that WS reduced the transition towards long and medium GUs and increased the transition toward floral, short and dead GUs. WS also slightly increased the proportion of axillary floral GUs. The higher relative frequency of floral GUs compared with vegetative ones reduced the tendency to biennial bearing under WS. The accelerated ontogenetic trend observed under WS suggests lower vegetative growth that could, in turn, be beneficial to floral induction and fruit set.

Pattern identification and characterization reveal permutations of organs as a key genetically controlled property of post-meristematic phyllotaxis

In vascular plants, the arrangement of organs around the stem generates geometric patterns called phyllotaxis. In the model plant, Arabidopsis thaliana, as in the majority of species, single organs are initiated successively at a divergence angle from the previous organ close to the canonical angle of 137.5°, producing a Fibonacci spiral. Given that little is known about the robustness of these geometric arrangements, we undertook to characterize phyllotaxis by measuring divergence angles between organs along the stems of wild-type and specific mutant plants with obvious defects in phyllotaxis. Sequences of measured divergence angles exhibit segments of non-canonical angles in both genotypes, albeit to a far greater extent in the mutant. We thus designed a pipeline of methods for analyzing these perturbations. The latent structure models used in this pipeline combine a non-observable model representing perturbation patterns (either a variable-order Markov chain or a combinatorial model) with von Mises distributions representing divergence angle uncertainty. We show that the segments of non-canonical angles in both wild-type and mutant plants can be explained by permutations in the order of insertion along the stem of two or three consecutive organs. The number of successive organs between two permutations reveals specific patterns that depend on the nature of the preceding permutation (2- or 3-permutation). We also highlight significant individual deviations from 137.5° in the level of baseline segments and a marked relationship between permutation of organs and defects in the elongation of the internodes between these organs. These results demonstrate that permutations are an intrinsic property of spiral phyllotaxis and that their occurrence is genetically regulated.

Deciphering the ontogeny of a sympodial tree

This paper addresses the identification and characterization of developmental patterns in the whole structure of a sympodial species, the apple tree. Dedicated stochastic models (hidden variable-order Markov chains) were used to (i) categorise growth units (GUs) on the basis of their morphological characteristics (number of nodes and presence/absence of flowering) and position along axes, (ii) analyse dependencies between successive GUs and (iii) identify repeated patterns in GU sequences. Two successive phases, referred to as “adolescent” and “adult”, were identified in two apple tree cultivars. In the adolescent phase, “very” long monocyclic GUs were followed by long polycyclic GUs, whereas in the adult phase medium GUs were preferentially followed by short GUs. Flowering GUs constituted a preferential pathway between vegetative GUs of decreasing vigour (long, medium and short) and generated patterns that were interpreted with respect to fruiting regularity. The proposed modelling gave a global and quantitative picture of the two-scale structuring of apple tree ontogeny: a coarse scale corresponding to the succession of the previously mentioned phases and a fine scale corresponding to the alternation between flowering and vegetative GUs. This led us to propose a synthetic scheme of apple tree ontogeny that combines growth phases, polycyclism and flowering, and which could be transposed to other sympodial trees.

Bayesian analysis of variable-order, reversible Markov chains

We define a conjugate prior for the reversible Markov chain of order $r$. The prior arises from a partially exchangeable reinforced random walk, in the same way that the Beta distribution arises from the exchangeable Poly\'{a} urn. An extension to variable-order Markov chains is also derived. We show the utility of this prior in testing the order and estimating the parameters of a reversible Markov model.

Variable context Markov chains for HIV protease cleavage site prediction

Deciphering the knowledge of HIV protease specificity and developing computational tools for detecting its cleavage sites in protein polypeptide chain are very desirable for designing efficient and specific chemical inhibitors to prevent acquired immunodeficiency syndrome. In this study, we developed a generative model based on a generalization of variable order Markov chains (VOMC) for peptide sequences and adapted the model for prediction of their cleavability by certain proteases. The new method, called variable context Markov chains (VCMC), attempts to identify the context equivalence based on the evolutionary similarities between individual amino acids. It was applied for HIV-1 protease cleavage site prediction problem and shown to outperform existing methods in terms of prediction accuracy on a common dataset. In general, the method is a promising tool for prediction of cleavage sites of all proteases and encouraged to be used for any kind of peptide classification problem as well.

A fast haplotype inference method for large population genotype data

With the rapid progress of genotyping techniques, many large-scale, genome-wide disease studies are now under way. One of the challenges of large disease-association studies is developing a fast and accurate computing method for haplotype inference from genotype data. In this paper, a new computing method for population-based haplotype inference problem is proposed. The designed method does not assume haplotype blocks in the population and allows each individual haplotype to have its own structure, and thus is able to accommodate recombination and obtain higher adaptivity to the genotype data, specifically in the case of long marker maps. This method develops a dynamic programming algorithm, which is theoretically guaranteed to find exact maximum likelihood solutions of the variable order Markov chain model for haplotype inference problem within linear running time. Hence, it is fast and, as a result, practicable for large genotype datasets. Through extensive computational experiments on large-scale real genotype data, the proposed method is shown to be fast and efficient.

A Variable-Order Markov-Chain-Based Model for Rayleigh Fading and Rake Receiver

The reliability of usual Markov chain models for Rayleigh fading depends strongly on the channel variation regime. In this letter, a flat Rayleigh fading channel and RAKE receiver output model for all fading variation speeds is proposed. The complex Gaussian fading and the RAKE output processes are bandlimited. A variable-order Markov chain provides the desirable discrete process with suitable correlation properties at a frequency that respects the sampling theorem. An interpolator provides samples at the transmission rate. This model is available for all fading variation speeds, and the Markov chain is built only once. High soft-output reliability with important simulation time reduction is obtained.

A proof of the stability of highly negative ions in the absence of the Pauli principle: 554 (P8,T8) Rafael Benguria, Departamento de Fisica, Universidad de Chile, Casilla 5487, Santiago, CHILE; Elliott H. Lieb, Departments of Mathematics and Physics, Princeton University, P.O.B. 708, Princeton, NJ 08544 USA

In vascular plants, the arrangement of organs around the stem generates geometric patterns called phyllotaxis. In the model plant, Arabidopsis thaliana, as in the majority of species, single organs are initiated successively at a divergence angle from the previous organ close to the canonical angle of 137.5°, producing a Fibonacci spiral. Given that little is known about the robustness of these geometric arrangements, we undertook to characterize phyllotaxis by measuring divergence angles between organs along the stems of wild-type and specific mutant plants with obvious defects in phyllotaxis. Sequences of measured divergence angles exhibit segments of non-canonical angles in both genotypes, albeit to a far greater extent in the mutant. We thus designed a pipeline of methods for analyzing these perturbations. The latent structure models used in this pipeline combine a non-observable model representing perturbation patterns (either a variable-order Markov chain or a combinatorial model) with von Mises distributions representing divergence angle uncertainty. We show that the segments of non-canonical angles in both wild-type and mutant plants can be explained by permutations in the order of insertion along the stem of two or three consecutive organs. The number of successive organs between two permutations reveals specific patterns that depend on the nature of the preceding permutation (2- or 3-permutation). We also highlight significant individual deviations from 137.5° in the level of baseline segments and a marked relationship between permutation of organs and defects in the elongation of the internodes between these organs. These results demonstrate that permutations are an intrinsic property of spiral phyllotaxis and that their occurrence is genetically regulated.