Ergodic control of partially observed Markov chains

The ergodic or long-run average cost control problem for a partially observed finite-state Markov chain is studied via the associated fully observed separated control problem for the nonlinear filter. Dynamic programming equations for the latter are derived, leading to existence and characterization of optimal stationary policies.

An Explicit Guidance Scheme For A Low-Thrust Terminal Rendezvous

An explicit near-optimal guidance scheme is developed for a terminal rendezvous of a spacecraft with a passive target in circular orbit around the earth. The thrust angle versus time profile for the continuous-thrust, constant-acceleration maneuver is derived, based on the assumption that the components of inertial acceleration due to relative position and velocity are negligible on account of the close proximity between the two spacecraft. The control law is obtained as a ''bilinear tangent law'' and an analytic solution to the state differential equations is obtained by expanding a portion of the integrand as an infinite series in time. A differential corrector method is proposed, to obtain real-time updates to the guidance parameters at regular time intervals. Simulation of the guidance scheme is carried out using the Clohessy-Wiltshire equations of relative motion as well as the inverse-square two-body equations of motion. Results for typical examples are presented.

Solution of a System of Generalized Abel Integral Equations Using Fractional Calculus

A method is presented for obtaining useful closed form solution of a system of generalized Abel integral equations by using the ideas of fractional integral operators and their applications. This system appears in solving certain mixed boundary value problems arising in the classical theory of elasticity.

Precipitation In Small Systems--II. Mean Field Equations More Effective Than Population Balance

Part I (Manjunath et al., 1994, Chem. Engng Sci. 49, 1451-1463) of this paper showed that the random particle numbers and size distributions in precipitation processes in very small drops obtained by stochastic simulation techniques deviate substantially from the predictions of conventional population balance. The foregoing problem is considered in this paper in terms of a mean field approximation obtained by applying a first-order closure to an unclosed set of mean field equations presented in Part I. The mean field approximation consists of two mutually coupled partial differential equations featuring (i) the probability distribution for residual supersaturation and (ii) the mean number density of particles for each size and supersaturation from which all average properties and fluctuations can be calculated. The mean field equations have been solved by finite difference methods for (i) crystallization and (ii) precipitation of a metal hydroxide both occurring in a single drop of specified initial supersaturation. The results for the average number of particles, average residual supersaturation, the average size distribution, and fluctuations about the average values have been compared with those obtained by stochastic simulation techniques and by population balance. This comparison shows that the mean field predictions are substantially superior to those of population balance as judged by the close proximity of results from the former to those from stochastic simulations. The agreement is excellent for broad initial supersaturations at short times but deteriorates progressively at larger times. For steep initial supersaturation distributions, predictions of the mean field theory are not satisfactory thus calling for higher-order approximations. The merit of the mean field approximation over stochastic simulation lies in its potential to reduce expensive computation times involved in simulation. More effective computational techniques could not only enhance this advantage of the mean field approximation but also make it possible to use higher-order approximations eliminating the constraints under which the stochastic dynamics of the process can be predicted accurately.

Fatigue Laws Via Functional Equations

In routine industrial design, fatigue life estimation is largely based on S-N curves and ad hoc cycle counting algorithms used with Miner's rule for predicting life under complex loading. However, there are well known deficiencies of the conventional approach. Of the many cumulative damage rules that have been proposed, Manson's Double Linear Damage Rule (DLDR) has been the most successful. Here we follow up, through comparisons with experimental data from many sources, on a new approach to empirical fatigue life estimation (A Constructive Empirical Theory for Metal Fatigue Under Block Cyclic Loading', Proceedings of the Royal Society A, in press). The basic modeling approach is first described: it depends on enforcing mathematical consistency between predictions of simple empirical models that include indeterminate functional forms, and published fatigue data from handbooks. This consistency is enforced through setting up and (with luck) solving a functional equation with three independent variables and six unknown functions. The model, after eliminating or identifying various parameters, retains three fitted parameters; for the experimental data available, one of these may be set to zero. On comparison against data from several different sources, with two fitted parameters, we find that our model works about as well as the DLDR and much better than Miner's rule. We finally discuss some ways in which the model might be used, beyond the scope of the DLDR.

Solution of singular integral equations with logarithmic and Cauchy kernels

A direct method of solution is presented for singular integral equations of the first kind, involving the combination of a logarithmic and a Cauchy type singularity. Two typical cages are considered, in one of which the range of integration is a Single finite interval and, in the other, the range of integration is a union of disjoint finite intervals. More such general equations associated with a finite number (greater than two) of finite, disjoint, intervals can also be handled by the technique employed here.

A linear programming approach for estimating the structure of a sparse linear genetic network from transcript profiling data

Background: A genetic network can be represented as a directed graph in which a node corresponds to a gene and a directed edge specifies the direction of influence of one gene on another. The reconstruction of such networks from transcript profiling data remains an important yet challenging endeavor. A transcript profile specifies the abundances of many genes in a biological sample of interest. Prevailing strategies for learning the structure of a genetic network from high-dimensional transcript profiling data assume sparsity and linearity. Many methods consider relatively small directed graphs, inferring graphs with up to a few hundred nodes. This work examines large undirected graphs representations of genetic networks, graphs with many thousands of nodes where an undirected edge between two nodes does not indicate the direction of influence, and the problem of estimating the structure of such a sparse linear genetic network (SLGN) from transcript profiling data. Results: The structure learning task is cast as a sparse linear regression problem which is then posed as a LASSO (l1-constrained fitting) problem and solved finally by formulating a Linear Program (LP). A bound on the Generalization Error of this approach is given in terms of the Leave-One-Out Error. The accuracy and utility of LP-SLGNs is assessed quantitatively and qualitatively using simulated and real data. The Dialogue for Reverse Engineering Assessments and Methods (DREAM) initiative provides gold standard data sets and evaluation metrics that enable and facilitate the comparison of algorithms for deducing the structure of networks. The structures of LP-SLGNs estimated from the INSILICO1, INSILICO2 and INSILICO3 simulated DREAM2 data sets are comparable to those proposed by the first and/or second ranked teams in the DREAM2 competition. The structures of LP-SLGNs estimated from two published Saccharomyces cerevisae cell cycle transcript profiling data sets capture known regulatory associations. In each S. cerevisiae LP-SLGN, the number of nodes with a particular degree follows an approximate power law suggesting that its degree distributions is similar to that observed in real-world networks. Inspection of these LP-SLGNs suggests biological hypotheses amenable to experimental verification. Conclusion: A statistically robust and computationally efficient LP-based method for estimating the topology of a large sparse undirected graph from high-dimensional data yields representations of genetic networks that are biologically plausible and useful abstractions of the structures of real genetic networks. Analysis of the statistical and topological properties of learned LP-SLGNs may have practical value; for example, genes with high random walk betweenness, a measure of the centrality of a node in a graph, are good candidates for intervention studies and hence integrated computational – experimental investigations designed to infer more realistic and sophisticated probabilistic directed graphical model representations of genetic networks. The LP-based solutions of the sparse linear regression problem described here may provide a method for learning the structure of transcription factor networks from transcript profiling and transcription factor binding motif data.

Optimal barrier subdivision for Kramers' escape rate

We examine the effect of subdividing the potential barrier along the reaction coordinate on Kramers' escape rate for a model potential. Using the known supersymmetric potential approach, we show the existence of an optimal number of subdivisions that maximizes the rate.

On the nature of power flow equations in security analysis

The power system network is assumed to be in steady-state even during low frequency transients. However, depending on generator dynamics, and toad and control characteristics, the system model and the nature of power flow equations can vary The nature of power flow equations describing the system during a contingency is investigated in detail. It is shown that under some mild assumptions on load-voltage characteristics, the power flow equations can be decoupled in an exact manner. When the generator dynamics are considered, the solutions for the load voltages are exact if load nodes are not directly connected to each other

Deriving optimal fishing effort for managing Australia's Moreton Bay multispecies trawl fishery with aggregated effort data

Deriving an estimate of optimal fishing effort or even an approximate estimate is very valuable for managing fisheries with multiple target species. The most challenging task associated with this is allocating effort to individual species when only the total effort is recorded. Spatial information on the distribution of each species within a fishery can be used to justify the allocations, but often such information is not available. To determine the long-term overall effort required to achieve maximum sustainable yield (MSY) and maximum economic yield (MEY), we consider three methods for allocating effort: (i) optimal allocation, which optimally allocates effort among target species; (ii) fixed proportions, which chooses proportions based on past catch data; and (iii) economic allocation, which splits effort based on the expected catch value of each species. Determining the overall fishing effort required to achieve these management objectives is a maximizing problem subject to constraints due to economic and social considerations. We illustrated the approaches using a case study of the Moreton Bay Prawn Trawl Fishery in Queensland (Australia). The results were consistent across the three methods. Importantly, our analysis demonstrated the optimal total effort was very sensitive to daily fishing costs-the effort ranged from 9500-11 500 to 6000-7000, 4000 and 2500 boat-days, using daily cost estimates of $0, $500, $750, and $950, respectively. The zero daily cost corresponds to the MSY, while a daily cost of $750 most closely represents the actual present fishing cost. Given the recent debate on which costs should be factored into the analyses for deriving MEY, our findings highlight the importance of including an appropriate cost function for practical management advice. The approaches developed here could be applied to other multispecies fisheries where only aggregated fishing effort data are recorded, as the literature on this type of modelling is sparse.

Load estimation with uncertainties from opportunistic sampling data: A semiparametric approach

We consider estimating the total load from frequent flow data but less frequent concentration data. There are numerous load estimation methods available, some of which are captured in various online tools. However, most estimators are subject to large biases statistically, and their associated uncertainties are often not reported. This makes interpretation difficult and the estimation of trends or determination of optimal sampling regimes impossible to assess. In this paper, we first propose two indices for measuring the extent of sampling bias, and then provide steps for obtaining reliable load estimates that minimizes the biases and makes use of informative predictive variables. The key step to this approach is in the development of an appropriate predictive model for concentration. This is achieved using a generalized rating-curve approach with additional predictors that capture unique features in the flow data, such as the concept of the first flush, the location of the event on the hydrograph (e.g. rise or fall) and the discounted flow. The latter may be thought of as a measure of constituent exhaustion occurring during flood events. Forming this additional information can significantly improve the predictability of concentration, and ultimately the precision with which the pollutant load is estimated. We also provide a measure of the standard error of the load estimate which incorporates model, spatial and/or temporal errors. This method also has the capacity to incorporate measurement error incurred through the sampling of flow. We illustrate this approach for two rivers delivering to the Great Barrier Reef, Queensland, Australia. One is a data set from the Burdekin River, and consists of the total suspended sediment (TSS) and nitrogen oxide (NO(x)) and gauged flow for 1997. The other dataset is from the Tully River, for the period of July 2000 to June 2008. For NO(x) Burdekin, the new estimates are very similar to the ratio estimates even when there is no relationship between the concentration and the flow. However, for the Tully dataset, by incorporating the additional predictive variables namely the discounted flow and flow phases (rising or recessing), we substantially improved the model fit, and thus the certainty with which the load is estimated.

Nonparametric rank regression for analyzing water quality concentration data with multiple detection limits

Environmental data usually include measurements, such as water quality data, which fall below detection limits, because of limitations of the instruments or of certain analytical methods used. The fact that some responses are not detected needs to be properly taken into account in statistical analysis of such data. However, it is well-known that it is challenging to analyze a data set with detection limits, and we often have to rely on the traditional parametric methods or simple imputation methods. Distributional assumptions can lead to biased inference and justification of distributions is often not possible when the data are correlated and there is a large proportion of data below detection limits. The extent of bias is usually unknown. To draw valid conclusions and hence provide useful advice for environmental management authorities, it is essential to develop and apply an appropriate statistical methodology. This paper proposes rank-based procedures for analyzing non-normally distributed data collected at different sites over a period of time in the presence of multiple detection limits. To take account of temporal correlations within each site, we propose an optimal linear combination of estimating functions and apply the induced smoothing method to reduce the computational burden. Finally, we apply the proposed method to the water quality data collected at Susquehanna River Basin in United States of America, which dearly demonstrates the advantages of the rank regression models.

Efficient designs for sampling and subsampling in fisheries research based on ranked sets

Sampling strategies are developed based on the idea of ranked set sampling (RSS) to increase efficiency and therefore to reduce the cost of sampling in fishery research. The RSS incorporates information on concomitant variables that are correlated with the variable of interest in the selection of samples. For example, estimating a monitoring survey abundance index would be more efficient if the sampling sites were selected based on the information from previous surveys or catch rates of the fishery. We use two practical fishery examples to demonstrate the approach: site selection for a fishery-independent monitoring survey in the Australian northern prawn fishery (NPF) and fish age prediction by simple linear regression modelling a short-lived tropical clupeoid. The relative efficiencies of the new designs were derived analytically and compared with the traditional simple random sampling (SRS). Optimal sampling schemes were measured by different optimality criteria. For the NPF monitoring survey, the efficiency in terms of variance or mean squared errors of the estimated mean abundance index ranged from 114 to 199% compared with the SRS. In the case of a fish ageing study for Tenualosa ilisha in Bangladesh, the efficiency of age prediction from fish body weight reached 140%.

Optimal sign tests for data from ranked set samples

This paper considers the one-sample sign test for data obtained from general ranked set sampling when the number of observations for each rank are not necessarily the same, and proposes a weighted sign test because observations with different ranks are not identically distributed. The optimal weight for each observation is distribution free and only depends on its associated rank. It is shown analytically that (1) the weighted version always improves the Pitman efficiency for all distributions; and (2) the optimal design is to select the median from each ranked set.

A maximum likelihood approach for estimating growth from tag–recapture data

The Fabens method is commonly used to estimate growth parameters k and l infinity in the von Bertalanffy model from tag-recapture data. However, the Fabens method of estimation has an inherent bias when individual growth is variable. This paper presents an asymptotically unbiassed method using a maximum likelihood approach that takes account of individual variability in both maximum length and age-at-tagging. It is assumed that each individual's growth follows a von Bertalanffy curve with its own maximum length and age-at-tagging. The parameter k is assumed to be a constant to ensure that the mean growth follows a von Bertalanffy curve and to avoid overparameterization. Our method also makes more efficient use nf thp measurements at tno and recapture and includes diagnostic techniques for checking distributional assumptions. The method is reasonably robust and performs better than the Fabens method when individual growth differs from the von Bertalanffy relationship. When measurement error is negligible, the estimation involves maximizing the profile likelihood of one parameter only. The method is applied to tag-recapture data for the grooved tiger prawn (Penaeus semisulcatus) from the Gulf of Carpentaria, Australia.