Zsolt Talata

Tuning mechanical performance of poly(ethylene glycol) and agarose interpenetrating network hydrogels for cartilage tissue engineering

Hydrogels are attractive for tissue engineering applications due to their incredible versatility, but they can be limited in cartilage tissue engineering applications due to inadequate mechanical performance. In an effort to address this limitation, our team previously reported the drastic improvement in the mechanical performance of interpenetrating networks (IPNs) of poly(ethylene glycol) diacrylate (PEG-DA) and agarose relative to pure PEG-DA and agarose networks. The goal of the current study was specifically to determine the relative importance of PEG-DA concentration, agarose concentration, and PEG-DA molecular weight in controlling mechanical performance, swelling characteristics, and network parameters. IPNs consistently had compressive and shear moduli greater than the additive sum of either single network when compared to pure PEG-DA gels with a similar PEG-DA content. IPNs withstood a maximum stress of up to 4.0 MPa in unconfined compression, with increased PEG-DA molecular weight being the greatest contributing factor to improved failure properties. However, aside from failure properties, PEG-DA concentration was the most influential factor for the large majority of properties. Increasing the agarose and PEG-DA concentrations as well as the PEG-DA molecular weight of agarose/PEG-DA IPNs and pure PEG-DA gels improved moduli and maximum stresses by as much as an order of magnitude or greater compared to pure PEG-DA gels in our previous studies. Although the viability of encapsulated chondrocytes was not significantly affected by IPN formulation, glycosaminoglycan (GAG) content was significantly influenced, with a 12-fold increase over a three-week period in gels with a lower PEG-DA concentration. These results suggest that mechanical performance of IPNs may be tuned with partial but not complete independence from biological performance of encapsulated cells.

Consistent estimation of the basic neighborhood of Markov random fields

For Markov random fields on Z d with finite state space, we address the statistical estimation of the basic neighborhood, the smallest region that determines the conditional distribution at a site on the condition that the values at all other sites are given. A modification of the Bayesian Information Criterion, replacing likelihood by pseudo-likelihood, is proved to provide strongly consistent estimation from observing a realization of the field on increasing finite regions: the estimated basic neighborhood equals the true one eventually almost surely, not assuming any prior bound on the size of the latter. Stationarity of the Markov field is not required, and phase transition does not affect the results.

Context tree estimation for not necessarily finite memory processes, via BIC and MDL

The concept of context tree, usually defined for finite memory processes, is extended to arbitrary stationary ergodic processes (with finite alphabet). These context trees are not necessarily complete, and may be of infinite depth. The familiar BIC and MDL principles are shown to provide strongly consistent estimators of the context tree, via optimization of a criterion for hypothetical context trees of finite depth, allowed to grow with the sample size n as o(log n). Algorithms are provided to compute these estimators in O(n) time, and to compute them on-line for all i les n in o(nlog n) time

On Rate of Convergence of Statistical Estimation of Stationary Ergodic Processes

Stationary ergodic processes with finite alphabets are approximated by finite memory processes based on an n-length realization of the process. Under the assumptions of summable continuity rate and non-nullness, a rate of convergence in d̅-distance is obtained, with explicit constants. Asymptotically, as n → ∞, the result is near the optimum.

MODEL SELECTION VIA INFORMATION CRITERIA

SummaryThis is a survey of the information criterion approach to model selection problems. New results about context tree estimation and the estimation of the basic neighborhood of Markov random fields are also mentioned.

Unrestricted BIC context tree estimation for not necessarily finite memory processes

Context trees of arbitrary stationary ergodic processes with finite alphabets are considered. Such a process is not necessarily a Markov chain, so the context tree may be of infinite depth. Calculated from a sample of size n, the Bayesian information criterion (BIC) is shown to provide a strongly consistent estimator of the context tree of the process, via minimization over hypothetical context trees, without any restriction on the hypothetical context trees. Strong consistency means that the estimated context tree recovers the true one up to any fixed level K, eventually almost surely as n tends to infinity. This generalizes the previous results, where either the context trees were assumed to be of finite depth or the depth of the hypothetical context trees was bounded by o(log n). Moreover, under some conditions on the process it is also shown that the level K above can grow with n at a specific rate determined by the distribution of the process; thus the BIC estimator can recover the true context tree to larger and larger depths.

BIC Context Tree Estimation for Stationary Ergodic Processes

Context trees of arbitrary stationary ergodic processes with finite alphabets are considered. Such a process is not necessarily a Markov chain, so the context tree may be of infinite depth. Calculated from a sample of size n, the Bayesian information criterion (BIC) is shown to provide a strongly consistent estimator of the context tree of the process, via minimization over hypothetical context trees, without any restriction on the hypothetical context trees. Strong consistency means that the estimated context tree recovers the true one up to a level K, eventually almost surely as n tends to infinity. Under some conditions on the process, it is shown that the recovery level K can grow with n at a specific rate determined by the distribution of the process; thus, the BIC estimator can recover the true context tree to larger and larger depths. The results include for the special case of K being an arbitrary constant that the strong consistency is satisfied without any assumption on the stationary ergodic process, which itself improves the existing results, where either the true context tree was assumed to be of finite depth or the depth of the hypothetical context trees was bounded by o(log n).

Divergence Rates of Markov Order Estimators and Their Application to Statistical Estimation of Stationary Ergodic Processes

Stationary ergodic processes with finite alphabets are estimated by finite memory processes from a sample, an n-length realization of the process, where the memory depth of the estimator process is also estimated from the sample using penalized maximum likelihood (PML). Under some assumptions on the continuity rate and the assumption of non-nullness, a rate of convergence in ¯ d-distance is obtained, with explicit constants. The result requires an analysis of the divergence of PML Markov order estimators for not necessarily finite memory processes. This divergence problem is investigated in more generality for three information criteria: the Bayesian information criterion with generalized penalty term yielding the PML, and the normalized maximum likelihood and the Krichevsky–Trofimov code lengths. Lower and upper bounds on the estimated order are obtained. The notion of consistent Markov order estimation is generalized for infinite memory processes using the concept of oracle order estimates, and generalized consistency of the PML Markov order estimator is presented.

Divergence of information-criterion based Markov order estimators for infinite memory processes

For finite-alphabet stationary ergodic processes with infinite memory, Markov order estimators that optimize an information criterion over the candidate orders based on a sample of size n are investigated. Three familiar information criteria are considered: the Bayesian information criterion (BIC) with generalized penalty term yielding the penalized maximum likelihood (PML), and the normalized maximum likelihood (NML) and the Krichevsky-Trofimov (KT) code lengths. A bound on the probability that the estimated order is greater than some order is obtained under the assumption that the process is weakly non-null and α-summable. This gives an O(log n) upper bound on the estimated order eventually almost surely as n → +∞. Moreover, a bound on the probability that the estimated order is less than some order is obtained if the decay of the continuity rate of the weakly non-null process is in some exponential range. This implies that then the estimated order attains the O(log n) divergence rate eventually almost surely as n → +∞.

On the rate of approximation in finite-alphabet longest increasing subsequence problems

The rate of convergence of the distribution of the length of the longest increasing subsequence, toward the maximal eigenvalue of certain matrix ensembles, is investigated. For finite-alphabet uniform and nonuniform i.i.d. sources, a rate of logn/ p n is obtained. The uniform binary case is further explored, and an improved 1/ p n rate obtained.