A Characteristic Polynomial for the Transition Probability Matrix of Correlated Random Walks on a Graph

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph.

4525 Estimates of Dose Response using the Dixon Up-and-Down Method and BOIN study designs

OBJECTIVES/GOALS: The Dixon up-and-down method (U/D), original developed for testing explosives, is especially common in anesthesia research studies. The objective of this research is to compare the performance of the U/D method for obtaining and analyzing sensitivity data with that of the Bayesian Optimal Interval (BOIN) method. METHODS/STUDY POPULATION: A simulation study will compare the performance of the U/D method with the BOIN design. The two study designs offer alternative decision-making algorithms with respect to the dose at which the next experimental unit is treated. These alternative decisions may impact the precision of point estimates of the mean and standard deviation of the effective dose to elicit a response. Transition probability matrices are developed, and maximum likelihood estimates of the unknown parameters assessed for accuracy. For simulation, the effective dose is assumed to be randomly distributed with a known mean and standard deviation. Fixed dose levels are defined, and decisions for what level the next experimental unit should be treated at are defined by the Dixon up-and-down method and the BOIN design. For the U/D method, the stimulus is increased by one level in the absence of a response or decreased if a response occurs from an initial stimulus. A target toxicity probability of 0.50 is used to define the dose escalation or de-escalation rules for the application of the BOIN design. RESULTS/ANTICIPATED RESULTS: A feature of both methods is that the consecutive observations are concentrated about the mean value of the effective dose. However, the BOIN design tends to be more concentrated between these two dose levels. In the presence of severe adverse events, the BOIN design can choose to eliminate doses that are too toxic whereas the U/D design cannot eliminate any dose levels. Transition probability matrices are defined and parameters for the distribution of the effective dose are estimated using maximum likelihood estimation. Mean squared errors for the estimated mean and standard deviations compare the two study designs. DISCUSSION/SIGNIFICANCE OF IMPACT: The BOIN design offers an alternative method for decision-making compared with the U/D method. The BOIN design tends to concentrate dose levels about the mean more than the U/D. This may provide better estimates of the mean and standard deviation of the effective dose for eliciting a response in some circumstances.

Characterising regime behaviour in the stably stratified nocturnal boundary layer on the basis of stationary Markov chains

Recent research has demonstrated that hidden Markov model (HMM) analysis is an effective tool to classify regimes of the stratified nocturnal boundary layer (SBL) at different tower sites. Here we analyse if SBL regime statistics (the occurrence of regime transitions, subsequent transitions after the first, and very persistent nights) in observations match theoretical calculations obtained from a stationary Markov chain with the goal of developing the foundations of novel Markov-chain-based boundary layer schemes which capture the effects of SBL regime dynamics. The regime statistics of a stationary Markov chain using the best estimate transition probabilities from the HMM analyses generally overestimate occurrence probabilities of regime transitions, resulting in an underestimation of persistent nights. Across the locations considered, sensitivity analyses of transition probability matrices in the HMM and the stationary Markov chain reveal that regimes are generally required to be more persistent in the stationary Markov chain in order to simulate observations accurately. A range of transition probability matrices allowing for a relatively accurate description of the occurrence of at least one transition within a night, multiple transitions, and the mean event durations is identified. The occurrence of very persistent nights (nights without regime transitions) is found to depend highly on the season. Therefore, for better representations of very persistent nights a nonstationary Markov chain linked to external drivers is likely appropriate. The observed transition probability maximum between one and two hours after a previous transition cannot be accounted for by two-state Markov processes (stationary or not). The use of these results in the development of SBL turbulence parameterisations is discussed.

Deriving the transition probability matrix using computational mechanics

Purpose The combined effects of several complex phenomena cause the deterioration of elements in steel hydraulic structures (SHSs) within the US lock system: corrosion, cracking and fatigue, impact and overloads. Predicting the future condition state of these structures by the use of current condition state inspection data can be achieved through the probabilistic chain deterioration model. The purpose of this study is to derive the transition probability matrix using final elements modeling of a miter gate. Design/methodology/approach If predicted accurately, this information would yield benefits in determining the need for rehabilitation or replacement of SHS. However, because of the complexity and difficulties on obtaining sufficient inspection data, there is a lack of available condition states needed to formulate proper transition probability matrices for each deterioration case. Findings This study focuses on using a three-dimensional explicit finite element analysis (FEM) of a miter gate that has been fully validated with experimental data to derive the transition probability matrix when the loss of flexural capacity in a corroded member is simulated. Practical implications New methodology using computational mechanics to derive the transition probability matrices of navigation steel structures has been presented. Originality/value The difficulty of deriving the transition probability matrix to perform a Markovian analysis increases when limited amount of inspection data is available. The used state of practice FEM to derive the transition probability matrix is not just necessary but also essential when the need for proper maintenance is required but limited amount of the condition of the structural system is unknown.

The ENSO Transition Probabilities

ENSO is investigated here by considering it as a transition from different states. Transition probability matrices can be defined to describe the evolution of ENSO in this way. Sea surface temperature anomalies are classified into four categories, or states, and the probability to move from one state to another has been calculated for both observations and a simulation from a GCM. This could be useful for understanding and diagnosing general circulation models elucidating the mechanisms that govern ENSO in models. Furthermore, these matrices have been used to define a predictability index of ENSO based on the entropy concept introduced by Shannon. The index correctly identifies the emergence of the spring predictability barrier and the seasonal variations of the transition probabilities. The transition probability matrices could also be used to formulate a basic prediction model for ENSO that was tested here on a case study.