Constant Transition Probabilities Research Articles

Spectral quantization of discrete random walks on half-line and orthogonal polynomials on the unit circle

We define quantization scheme for discrete-time random walks on the half-line consistent with Szegedy’s quantization of finite Markov chains. Motivated by the Karlin and McGregor description of discrete-time random walks in terms of polynomials orthogonal with respect to a measure with support in the segment [-1,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1,1]$$\\end{document}, we represent the unitary evolution operator of the quantum walk in terms of orthogonal polynomials on the unit circle. We find the relation between transition probabilities of the random walk with the Verblunsky coefficients of the corresponding polynomials of the quantum walk. We show that the both polynomials systems and their measures are connected by the classical Szegő map. Our scheme can be applied to arbitrary Karlin and McGregor random walks and generalizes the so-called Cantero–Grünbaum–Moral–Velázquez method. We illustrate our approach on example of random walks related to the Jacobi polynomials. Then we study quantization of random walks with constant transition probabilities where the corresponding polynomials on the unit circle have two-periodic real Verblunsky coefficients. We present geometric construction of the spectrum of such polynomials (in the general complex case) which generalizes the known construction for the Geronimus polynomials. In the Appendix, we present the explicit form, in terms of Chebyshev polynomials of the second kind, of polynomials orthogonal on the unit circle and polynomials orthogonal on the real line with coefficients of arbitrary period.

Mortality (Dis)Similarities in the Context of Demographic Aging: The Countries of Ex-Yugoslavia

Generally, mortality declines at all ages, with varying intensities, as a result of heterogeneous factors affecting human lifespan. This paper considers switching regression estimation for six ex-Yugoslavian countries with the specification of a time-varying transition probability model of crude death rate. A two-state Markov switching means VAR estimates are used in which the mean growth rate of crude death rate is subject to regime switching, where the errors follow a constant transition probability. The UN data were employed consisting of the crude mortality rate series containing the log difference of yearly crude mortality rate in the six countries of ex-Yugoslavia for 1990–2021. The results show that the estimates of coefficients on the intercept in the mean equation both differ from zero in Bosnia & Herzegovina, Serbia, Montenegro, and Slovenia and are with opposite and statistically significant signs only for Montenegro and Slovenia. The transition probability summaries show a higher probability of remaining in the first high regime state for Macedonia, Serbia, and Bosnia & Herzegovina (0.96, 0.94, and 0.87, respectively). The higher probability of remaining in the second low-medium regime state was found in Slovenia, Croatia, and Montenegro (0.96, 0.87, and 0.56, respectively). The appropriate expected durations in the first regime are approximately 26.41, 16.19, and 7.78 years for Macedonia, Serbia, and Bosnia & Herzegovina, and 28.30, 7.80, and 2.28 years were the corresponding expected durations in the second regime for Slovenia, Croatia, and Montenegro.

Birth-death chains on a spider: Spectral analysis and reflecting-absorbing factorization

We consider discrete-time birth-death chains on a spider, i.e. a graph consisting of N discrete half lines on the plane that are joined at the origin. This process can be identified with a discrete-time quasi-birth-death process on the state space N0×{1,2,…,N}, represented by a block tridiagonal transition probability matrix. We prove that we can analyze this process by using spectral methods and obtain the n-step transition probabilities in terms of a weight matrix and the corresponding matrix-valued orthogonal polynomials (the so-called Karlin-McGregor formula). We also study under what conditions we can get a reflecting-absorbing factorization of the birth-death chain on a spider which can be seen as a stochastic UL block factorization of the transition probability matrix of the quasi-birth-death process. With this factorization we can perform a discrete Darboux transformation and get new families of “almost” birth-death chains on a spider. The spectral matrix associated with the Darboux transformation will be a Geronimus transformation of the original spectral matrix. Finally, we apply our results to the random walk on a spider, i.e. with constant transition probabilities.

Asymptotic properties of the maximum likelihood estimator in regime-switching models with time-varying transition probabilities

Summary Time-varying transition probability (TVTP) regime-switching models extend the constant regime transition probability in Markov-switching models to include information from observations. We show consistency and asymptotic normality of the maximum likelihood estimator (MLE) in general TVTP regime-switching models where the conditional distribution of $Y_t$ depends on lagged regimes. Consistency of the MLE is also shown under misspecification. The assumptions are verified in regime-switching autoregressive models with widely applied TVTP specifications. A simulation study examines the finite-sample distributions of the MLE and compares the asymptotic variance estimates constructed from the Hessian matrix and the outer product of the score. The simulation results favour the latter. As an empirical example, we compare three leading economic indicators in terms of describing U.S. industrial production.

Model parameters influencing the cost-effectiveness of sacubitril/valsartan in heart failure: evidence from a systematic literature review

ObjectivesTo summarize cost-effectiveness (CE) evidence of sacubitril/valsartan for the treatment of heart failure (HF) patients with reduced ejection fraction (HFrEF). The impact of different modeling approaches and parameters on the CE results is also described.MethodsWe conducted a systematic literature review using multiple databases: Embase®; MEDLINE®; MEDLINE®-In Process; NIHR CRD database including DARE, NHS EED, and HTA databases; and the Cost Effectiveness Analysis registry. We also reviewed HTA countries’ websites to identify CE reports of sacubitril/valsartan, published up to 25-July-2021. Articles published in English as full-texts, conference-abstracts, or HTA reports were included.ResultsWe included 44 CE models [39 from 37 publications (22 full-texts; 15 conference-abstracts) and 5 HTAs; Europe, n = 20; North and South Americas, n = 14; Asia and Australia, n = 10]. Most models adopted a Markov structure with constant transition probabilities of events (n = 27) or a mix of Markov and regression-based models (n = 16), with variations in structural assumptions and chosen parameters. Study authors concluded sacubitril/valsartan to be a cost-effective therapy in 37/41 models in chronic HFrEF patients and 2/3 models in hospitalized patients stabilized after an acute decompensation for HF. CE models showing sacubitril/valsartan not to be a cost-effective treatment generally modeled a shorter time horizon. Effect of sacubitril/valsartan on cardiovascular and all-cause mortality, cost, duration of effect and time horizon was the main model drivers.ConclusionsMost evidence indicated sacubitril/valsartan is cost-effective in HFrEF. The use of a lifetime horizon is recommended in future models as HF is a chronic disease. Data on the CE of sacubitril/valsartan in the inpatient setting were limited and further research is warranted.

HEDOS—a computational tool to assess radiation dose to circulating blood cells during external beam radiotherapy based on whole-body blood flow simulations

We have developed a time-dependent computational framework, hematological dose (HEDOS), to estimate dose to circulating blood cells from radiation therapy treatment fields for any treatment site. Two independent dynamic models were implemented in HEDOS: one describing the spatiotemporal distribution of blood particles (BPs) in organs and the second describing the time-dependent radiation field delivery. A whole-body blood flow network based on blood volumes and flow rates from ICRP Publication 89 was simulated to produce the spatiotemporal distribution of BPs in organs across the entire body using a discrete-time Markov process. Constant or time-varying transition probabilities were applied and their impact on transition time was investigated. The impact of treatment time and anatomical site were investigated using imaging data and dose distributions from a liver cancer and a brain cancer patient. The simulations revealed different dose levels to the circulating blood for brain irradiation compared to liver irradiation even for similar field sizes due to the different blood flow properties of the two organs. The volume of blood receiving any dose (V >0 Gy) after a single radiation fraction increases from 1.2% for a 1 s delivery time to 20.9% for 120 s delivery time for the brain cancer treatment, and from 10% (1 s) to 48.7% (120 s) for a liver cancer treatment. Other measures of the low-dose bath to the circulating blood such as the dose to small volumes of blood (D 2%) decreases with longer delivery time. Furthermore, we demonstrate that the blood dose-volume histogram is highly sensitive to changes in the treatment time, indicating that dynamic modeling of blood flow and radiation fields is necessary to evaluate dose to circulating blood cells for the assessment of radiation-induced lymphopenia. HEDOS is publicly available and allows for the estimation of patient-specific dose to circulating blood cells based on organ DVHs, thus enabling the study of the impact of different treatment plans, dose rates, and fractionation schemes.

Absorbing–reflecting factorizations for birth–death chains on the integers and their Darboux transformations

We consider a new way of factorizing the transition probability matrix of a discrete-time birth–death chain on the integers by means of an absorbing and a reflecting birth–death chain to the state 0 and viceversa. First we will consider reflecting–absorbing factorizations of birth–death chains on the integers. We give conditions on the two free parameters such that each of the factors is a stochastic matrix. By inverting the order of the factors (also known as a Darboux transformation) we get new families of “almost” birth–death chains on the integers with the only difference that we have new probabilities going from the state 1 to the state −1 and viceversa. On the other hand an absorbing–reflecting factorization of birth–death chains on the integers is only possible if both factors are split into two separated birth–death chains at the state 0. Therefore it makes more sense to consider absorbing–reflecting factorizations of “almost” birth–death chains with extra transitions between the states 1 and −1 and with some conditions. This factorization is now unique and by inverting the order of the factors we get a birth–death chain on the integers. In both cases we identify the spectral matrices associated with the Darboux transformation, the first one being a Geronimus transformation and the second one a Christoffel transformation of the original spectral matrix. We also study the probabilistic implications of both transformations. Finally, we apply our results to examples of chains with constant transition probabilities.

Dynamic volatility modelling of Bitcoin using time-varying transition probability Markov-switching GARCH model

Bitcoin (BTC), as the dominant cryptocurrency, has attracted tremendous attention lately due to its excessive volatility. This paper proposes the time-varying transition probability Markov-switching GARCH (TV-MSGARCH) models incorporated with BTC daily trading volume and daily Google searches singly and jointly as exogenous variables to model the volatility dynamics of BTC return series. Extensive comparisons are carried out to evaluate the modelling performances of the proposed model with the benchmark models such as GARCH, GJRGARCH, threshold GARCH, constant transition probability MSGARCH and MSGJRGARCH. Results reveal that the TV-MSGARCH models with skewed and fat-tailed distribution predominate other models for the in-sample model fitting based on Akaike information criterion and other benchmark criteria. Furthermore, it is found that the TV-MSGARCH model with BTC daily trading volume and student-t error distribution offers the best out-of-sample forecast evaluated based on the mean square error loss function using Hansen’s model confidence set. Filardo’s weighted transition probabilities are also computed and the results show the existence of time-varying effect on transition probabilities. Lastly, different levels of long and short positions of value-at-risk and the expected shortfall forecasts based on MSGARCH, MSGJRGARCH and TV-MSGARCH models are also examined.

Emergence of power law distributions for odd and even lifetimes.

Avalanche lifetime distributions have been related to first-return random walk processes. In this sense, the theory for random walks can be employed to understand, for instance, the origin of power law distributions in self-organized criticality. In this work, we study first-return probability distributions, f^{(n)}, for discrete random walks with constant one-step transition probabilities. Explicit expressions are given in terms of _{2}F_{1} hypergeometric functions, allowing us to study the different behaviors of f^{(n)} for odd and even values of n. We show that the first-return probabilities have a power law behavior with exponent -3/2 only when the random walk is unbiased. In any other case, it presents an exponential decay.

The spectral matrices associated with the stochastic Darboux transformations of random walks on the integers

We consider UL and LU stochastic factorizations of the transition probability matrix of a random walk on the integers, which is a doubly infinite tridiagonal stochastic Jacobi matrix. We give conditions on the free parameter of both factorizations in terms of certain continued fractions such that this stochastic factorization is always possible. By inverting the order of the factors (also known as a Darboux transformation) we get new families of random walks on the integers. We identify the spectral matrices associated with these Darboux transformations (in both cases) which are basically conjugations by a matrix polynomial of degree one of a Geronimus transformation of the original spectral matrix. Finally, we apply our results to the random walk with constant transition probabilities with or without an attractive or repulsive force.

Separation of Residential Space Cooling Usage From Smart Meter Data

For demand response (DR) programs that focus on the energy consumption of heating, ventilation, and air conditioning (HVAC), the knowledge on HVAC usage of individual customers is of great value for DR program implementation. This paper presents a novel methodology to extract space cooling usage from smart meter data. A multisequence, non-homogeneous Factorial Hidden Markov Model (MN-FHMM) is proposed to disaggregate the whole-house energy consumption into an HVAC component and a baseload (i.e., the sum of non-temperature-sensitive loads) component. Rather than holding a constant transition probability over the whole process, a time-varying transition probability model is developed to characterize the dynamic nature of the evolution of users' energy consumption. We also discuss how to use the disaggregation results to estimate the HVAC related DR potential for individual customers, which can further inform DR programs to target customers more cost-effectively. Data experiments on ground truth data validate both the accuracy and the robustness of the proposed model as well as the effectiveness of the targeting strategy based on the disaggregation results.

Dynamic Volatility Modelling of Bitcoin Using Time-Varying Transition Probability Markov-Switching GARCH Model

Bitcoin (BTC), as the dominant cryptocurrency, has attracted tremendous attention lately due to its excessive volatility. This paper proposes the time-varying transition probability Markov-switching GARCH (TV-MSGARCH) models incorporated BTC logarithmic daily trading volume or Google daily searches as exogenous factors to model the volatility dynamics of BTC return series. Extensive comparisons were carried out to evaluate the modelling and forecasting performances of TV-MSGARCH models with the benchmark models such as GARCH, GJRGARCH, threshold GARCH and constant transition probability MSGARCH. Results reveal that the TV-MSGARCH models predominate other models for the in-sample model fitting based on log-likelihood function and other benchmark criteria. Furthermore, the TV-MSGARCH model with Google daily searches offers the best out-of-sample forecast evaluated based on quasi-likelihood loss function and also using Hansen’s model confidence set. In addition, the Filardo’s weighted transition probabilities is computed and the results show the existence of time-varying effect on transition probabilities. Lastly, new approach for computing different levels of long and short positions of value-at-risk forecasts based on TV-MSGARCH models are also provided and tested.

Robert S. Kirsner Gary Delhougne Richard J. Searle

The purpose of this economic evaluation was to determine the cost-effectiveness of single-use negative pressure wound therapy (sNPWT) compared with traditional NPWT (tNPWT) for the treatment of VLUs and DFUs in the United States. A Markov decision-analytic model was used to compare the incremental cost and ulcer weeks avoided for a time horizon of 12 and 26 weeks using lower extremity ulcer closure rates from a published randomized controlled trial (N = 161) that compared sNPWT with tNPWT. Treatment costs were extracted from a retrospective cost-minimization study of sNPWT and tNPWT from the payer perspective using US national 2016 Medicare claims data inflated to 2018 costs and multiplied by 7 to estimate the weekly costs of treatment for sNPWT and tNPWT. Two (2) arms of the model, tNPWT and sNPWT, were calculated separately for a combination of both VLU and DFU ulcer types. In this model, a hypothetical cohort of patients began in the open ulcer health state, and at the end of each weekly cycle a proportion of the cohort moved into the closed ulcer health state according to a constant transition probability. The costs over the defined timescale were summed to give a total cost of treatment for each arm of the model, and then the difference between the arms was calculated. Effectiveness was calculated by noting the incidence of healing at 12 and 26 weeks and the total number of open ulcer weeks; the incremental effectiveness was calculated as sNPWT effectiveness minus tNPWT effectiveness. Data were extracted to Excel spreadsheets and subjected to one-way sensitivity, scenario (where patients with unhealed ulcers were changed to standard care at 4 or 12 weeks), probabilistic, and threshold analyses. sNPWT was found to provide an expected cost saving of $7756 per patient and an expected reduction of 1.67 open ulcer weeks per patient over 12 weeks and a cost reduction of $15 749 and 5.31 open ulcer weeks over 26 weeks. Probabilistic analysis at 26 weeks showed 99.8% of the simulations resulted in sNPWT dominating tNPWT. Scenario analyses showed that sNPWT remained dominant over tNPWT (cost reductions over 26 weeks of $2536 and $7976 per patient, respectively). Using sNPWT for VLUs and DFUs is likely to be more cost-effective than tNPWT from the US payer perspective and may provide an opportunity for policymakers to reduce the economic burden of lower extremity ulcers.

Forecasting realised volatility: a Markov switching approach with time‐varying transition probabilities

AbstractThis paper introduces a markov‐switching heterogeneous autoregressive (MS‐HAR) model with time‐varying transition probabilities (TVTP) for the realised volatility of Shanghai securities composite index returns. Its various extensions have been obtained by including negative returns outside trading hours in addition to the leverage effects and trading volume. The findings show asymmetries in the impact of explanatory variables on the realised volatility. Moreover, the out‐of‐sample results show that the benchmark MS‐HAR with TVTP model and its extensions consistently outperform the simple HAR model, MS‐HAR model with constant transition probabilities (CTP) and their extensions. These results are robust to alternative realised measurements, and have economic implications.

Stochastic LU factorizations, Darboux transformations and urn models

Abstract We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

A class of iterated function systems with adapted piecewise constant transition probabilities: Asymptotic stability and Hausdorff dimension of the invariant measure

A class of iterated function systems (IFS) with non-overlapping or just-touching contractions on closed real intervals and adapted piecewise constant transition probabilities are studied. We give criteria for the existence and the uniqueness of an invariant probability measure for the IFSs and for the asymptotic stability of the system in terms of bounds of transition probabilities. The proofs are mainly based on the symbolic system associated with the contractions, an extended alphabet and usual theorems for Markov chains. Additionally, in case there exists a unique invariant measure, we obtain its Hausdorff dimension as the ratio of the entropy over the Lyapunov exponent. This result extends the formula, established in the literature for continuous transition probabilities, to the case considered here of piecewise constant probabilities. The main idea of this proof consists in finding good lower and upper bounds for the invariant measure.

Revisiting the Relationship between Military Expenditure and Economic Growth in Pakistan

This study aims to examine relationship of military expenditure and economic growth in different phases of military regimes in the context of Pakistan. This study uses two-state Markov switching models with Constant Transition Probability (CTP) and Time Varying Transition Probabilities (TVTP) for the time period: 1973-2014. This investigation analyses two sorts of relations between military expenditures and economic development through fixed transition probability Markov exchanging models. To begin with, there is negative connection between GDP growth and military expenditures during a high variance state (i.e. having low economic growth). Second, there is positive relation between both variables, during low variance state (i.e. having higher economic growth) which is also supported by idea of Keynesian income multiplier. Another, empirical test of time varying transition probability model was used to capture the switch through indicator variable. Results of the study suggest that chances of switching are increased from low to high economic growth. The chances of switching increase from lower to higher economic growth period (or high variance period) if non-military expenditure increases. The study concludes that military expenditure and economic growth are state dependent. If conditions of economy are stable then increase of expenditure results in positive outcomes, otherwise, it affects negatively. Empirical findings suggest that military spending should be planned in accordance to the economic performance of the country.

An Alternative to Fixed Transition Probabilities for the Projection of Interprovincial Migration in Canada

Internal migration plays a critical role in subnational population projections. The multiregional model is often seen as a gold standard, for its capacity to project several interconnected regions simultaneously and coherently. However, undesirable effects may occur when assumptions of constant transition probabilities are used. This paper investigates these limits, explores a few solutions provided in the literature and describes the alternative methodology used by Statistics Canada in its most recent edition of population projections for the Canadian provinces and territories. Among other things, the new method is shown to improve the consistency between internal migration assumptions and results and to facilitate the projection of the uncertainty associated with this component.

Dynamic role of the intramolecular hydrogen bonding in nonadiabatic chemistry revealed in the UV photodissociation reactions of 2-fluorothiophenol and 2-chlorothiophenol.

The dynamic interplay between the intramolecular hydrogen bonding and intramolecular vibrational redistribution is found to be critical in nonadiabatic reaction dynamics. Herein, it has been demonstrated that the molecular planarity, directed by the intramolecular hydrogen bonding, plays an important role in the nonadiabatic passage of the reactive flux at the conical intersection in the photodissociation reactions of 2-fluorothiophenol and 2-chlorothiophenol. As the internal energy increases in the excited state, the intramolecular hydrogen bonding of 2-fluorothiophenol loosens. The floppiness brought into the molecular structure then modifies the dynamic path of the reactive flux, leading to the diminishment of the nonadiabatic transition probability at the conical intersection. On the contrary, for 2-chlorothiophenol having the relatively stronger intramolecular hydrogen bonding, the reactive flux seems to retain the molecular planarity even with the increase of the internal energy as manifested by the constant nonadiabatic transition probability over the wide range of the S1 internal energy. The effect of the intramolecular hydrogen bonding on the molecular structure and its relation to the nonadiabatic dynamics along the tunneling path has been experimentally demonstrated.

An Implementation of Markov Regime Switching Model with Time Varying Transition Probabilities in Matlab

Abstract This memo explains how to use the MATLAB code for estimating a Markov Regime Switching Model with time varying transition probabilities. The code is developed by Zhuanxin Ding based on the original code by Marcelo Perlin for estimating a Markov Regime Switching Model with constant transition probability matrix. 1. Introduction The Matlab code developed here (MS_Regress_tvtp) is based on the original Matlab code by Marcelo Perlin on Markov Regime Switching Model for constant transition probability matrix (MS_Regress). In order to understand the code here, please go through the original code and explanation by Marcelo Perlin. Perlin’s code is available at https://sites.google.com/site/marceloperlin/matlab-code/ms_regress---a-package-for-markov-regime-switching-models-in-matlab An explanation of Perlin’s code can be found at SSRN: http://ssrn.com/abstract=1714016 or http://dx.doi.org/10.2139/ssrn.1714016 The code structure, including subroutines and functions are almost the same. I attached a _tvtp for subroutine and function names used in this code. For example, Perlin’s original code has a subroutine named MS_Regress_Fit.m, my corresponding subroutine will be named MS_Regress_Fit_tvtp.m. I would suggest the user build an m_files_tvtp directory in your computer to store the Matlab code here.