Non-homogeneous Markov Model Research Articles

Projection of future rainfall for the North China Plain using two statistical downscaling models and its hydrological implications

This study projected the future rainfall (2046–2065 and 2081–2100) for the North China Plain (NCP) using two stochastic statistical downscaling models, the non-homogeneous hidden Markov model and the generalized linear model for daily climate time series, conditioned by the large-scale atmospheric predictors from six general circulation models for three emission scenarios (A1B, A2 and B1). The results indicated that the annual total rainfall, the extreme daily rainfall and the maximum length of consecutive wet/dry days would decline, while the number of annual rainfall days would slightly increase (correspondingly rainfall intensity would decrease) in the NCP, in comparison with the base period (1961–2010). Moreover, the summer monsoon rainfall, which accounted for 50–75 % of the total annual rainfalls in NCP, was projected to decrease in the latter half of twenty-first century. The spatial patterns of change showed generally north–south gradients with relatively larger magnitude decrease in the northern NCP and less decrease (or even slightly increase) in the southern NCP. This could result in decline of the annual runoff by −5.5 % (A1B), −3.3 % (A2) and −4.1 % (B1) for 2046–2065 and −5.3 % (A1B), −4.6 % (A2) and −1.9 % (B1) decrease for 2081–2100. These rainfall changes, combined with the warming temperature, could lead to drier catchment soil profiles and further reduce runoff potential, would hence provide valuable references for the water availability and related climate change adaption in the NCP.

A Sequential Monte Carlo Approach for MLE in a Plant Growth Model

Parametric identification of plant growth models formalized as discrete dynamical systems is a challenging problem due to specific data acquisition (system observation is generally done with destructive measurements), non-linear dynamics, model uncertainties and high-dimensional parameter space. In this study, we present a novel idea of modeling plant growth in the framework of non-homogeneous hidden Markov models (Cappé, Moulines, and Rydén 2005), for a certain class of plants with known organogenesis (structural development). Unknown parameters of the models are estimated via a stochastic variant of a generalized EM (Expectation-Maximization) algorithm and approximate confidence intervals are given via parametric bootstrap. The complexity of the model makes both the E-step (expectation step) and the M-step (maximization step) non-explicit. For this reason, the E-step is approximated via a sequential Monte Carlo procedure (sequential importance sampling with resampling) and the M-step is separated into two steps (Conditional-Maximization), where before applying a numerical maximization procedure (quasi-Newton type), a large subset of unknown parameters is updated explicitly conditioned on the other subset. A simulation study and a case-study with real data from the sugar beet are considered and a model comparison is performed based on these data. Appendices are available online.

Scenario development for estimating potential climate change impacts on crop production in the North China Plain

ABSTRACTIt is important to investigate potential changes in temperature, precipitation and solar radiation for assessing the impacts of future climate change on agricultural production for specific regions. In this study, climate scenarios of precipitation, temperature and solar radiation for the North China Plain (NCP) were constructed in terms of stochastic daily weather sequences. A nonhomogeneous hidden Markov model (NHMM) was used to downscale daily precipitation projections at 32 stations during winter wheat and summer maize growing seasons for a baseline (1966–2005) and a 21st century (2080–2099) A1B scenario, using selected general circulation models (GCMs). A climatological seasonal cycle of regional‐averaged daily reanalysis precipitation was used as input to the down‐scaling for the baseline simulation; this input was then scaled by the precipitation changes from GCM projections to generate down‐scaled stochastic simulations of precipitation in the 21st century. Temperature was generated using a weakly stationary generating process, conditional on precipitation occurrence, with 21st century additive changes taken from the GCMs at the regional scale. Three hypotheses about changes in solar radiation (−20%, 0% and 20%) were made considering the large uncertainty in its future change. The down‐scaled simulations exhibit station increases in the mean daily rainfall of 13.9–69.7% in the scenarios driven by the GCM with the projected largest and multi‐model mean precipitation increase for the wheat season, with changes of 0.4–29.9% for the maize season. In the scenario driven by the GCM with the largest projected precipitation decrease, the simulated rainfall decreases at all stations, with changes ranging from −24.6 to −0.1% for the wheat and maize seasons, respectively. Temperature increases by about 3.7 °C for the wheat season and 3.6 °C for the maize season.

Nowcasting influenza epidemics using non‐homogeneous hidden Markov models

Timeliness of a public health surveillance system is one of its most important characteristics. The process of predicting the present situation using available incomplete information from surveillance systems has received the term nowcasting and has high public health interest. Generally in Europe, general practitioners' sentinel networks support the epidemiological surveillance of influenza activity, and each week's epidemiological bulletins are usually issued between Wednesday and Friday of the following week. In this work, we have developed a non-homogeneous hidden Markov model (HMM) that, on a weekly basis, uses as covariates an early observation of influenza-like illness (ILI) incidence rate and the number of ILI cases tested positive to nowcast the current week ILI rate and the probability that the influenza activity is in an epidemic state. We use Bayesian inference to find estimates of the model parameters and nowcasted quantities. The results obtained with data provided by the Portuguese influenza surveillance system show the additional value of using a non-homogeneous HMM instead of a homogeneous one. The use of a non-homogeneous HMM improves the surveillance system timeliness in 2 weeks.

Daily rainfall projections from general circulation models with a downscaling nonhomogeneous hidden Markov model (NHMM) for south‐eastern Australia

AbstractAn ensemble of stochastic daily rainfall projections has been generated for 30 stations across south‐eastern Australia using the downscaling nonhomogeneous hidden Markov model, which was driven by atmospheric predictors from four climate models for three IPCC emissions scenarios (A1B, A2, and B1) and for two periods (2046–2065 and 2081–2100). The results indicate that the annual rainfall is projected to decrease for both periods for all scenarios and climate models, with the exception of a few scenarios of no statistically significant changes. However, there is a seasonal difference: two downscaled GCMs consistently project a decline of summer rainfall, and two an increase. In contrast, all four downscaled GCMs show a decrease of winter rainfall. Because winter rainfall accounts for two‐thirds of the annual rainfall and produces the majority of streamflow for this region, this decrease in winter rainfall would cause additional water availability concerns in the southern Murray–Darling basin, given that water shortage is already a critical problem in the region. In addition, the annual maximum daily rainfall is projected to intensify in the future, particularly by the end of the 21st century; the maximum length of consecutive dry days is projected to increase, and correspondingly, the maximum length of consecutive wet days is projected to decrease. These changes in daily sequencing, combined with fewer events of reduced amount, could lead to drier catchment soil profiles and further reduce runoff potential and, hence, also have streamflow and water availability implications. Copyright © 2012 John Wiley & Sons, Ltd.

Downscaling daily precipitation over the Yellow River source region in China: a comparison of three statistical downscaling methods

Three statistical downscaling methods are compared with regard to their ability to downscale summer (June–September) daily precipitation at a network of 14 stations over the Yellow River source region from the NCEP/NCAR reanalysis data with the aim of constructing high-resolution regional precipitation scenarios for impact studies. The methods used are the Statistical Downscaling Model (SDSM), the Generalized LInear Model for daily CLIMate (GLIMCLIM), and the non-homogeneous Hidden Markov Model (NHMM). The methods are compared in terms of several statistics including spatial dependence, wet- and dry spell length distributions and inter-annual variability. In comparison with other two models, NHMM shows better performance in reproducing the spatial correlation structure, inter-annual variability and magnitude of the observed precipitation. However, it shows difficulty in reproducing observed wet- and dry spell length distributions at some stations. SDSM and GLIMCLIM showed better performance in reproducing the temporal dependence than NHMM. These models are also applied to derive future scenarios for six precipitation indices for the period 2046–2065 using the predictors from two global climate models (GCMs; CGCM3 and ECHAM5) under the IPCC SRES A2, A1B and B1scenarios. There is a strong consensus among two GCMs, three downscaling methods and three emission scenarios in the precipitation change signal. Under the future climate scenarios considered, all parts of the study region would experience increases in rainfall totals and extremes that are statistically significant at most stations. The magnitude of the projected changes is more intense for the SDSM than for other two models, which indicates that climate projection based on results from only one downscaling method should be interpreted with caution. The increase in the magnitude of rainfall totals and extremes is also accompanied by an increase in their inter-annual variability.

A comparison of three multi-site statistical downscaling models for daily rainfall in the North China Plain

Three statistical downscaling methods (conditional resampling statistical downscaling model: CR-SDSM, the generalised linear model for daily climate time series: GLIMCLIM, and the non-homogeneous hidden Markov model: NHMM) for multi-site daily rainfall were evaluated and compared in the North China Plain (NCP). The comparison focused on a range of statistics important for hydrological studies including rainfall amount, extreme rainfall, intra-annual variability, and spatial coherency. The results showed that no single model performed well over all statistics/timescales, suggesting that the user should chose appropriate methods after assessing their advantages and limitations when applying downscaling methods for particular purposes. Specifically, the CR-SDSM provided relatively robust results for annual/monthly statistics and extreme characteristics, but exhibited weakness for some daily statistics, such as daily rainfall amount, dry-spell length, and annual wet/dry days. GLIMCLIM performed well for annual dry/wet days, dry/wet spell length, and spatial coherency, but slightly overestimated the daily rainfall. Additionally, NHMM performed better for daily rainfall and annual wet/dry days, but slightly underestimated dry/wet spell length and overestimated the daily extremes. The results of this study could be applied when investigating climate change impact on hydrology and water availability for the NCP, which suffers from intense water shortages due to climate change and human activities in recent years.

Downscaling of Precipitation over Vancouver Island using a Synoptic Typing Approach

A statistical downscaling technique is employed to link atmospheric circulation produced by an ensemble of global climate model (GCM) simulations over the twenty-first century to precipitation recorded at weather stations on Vancouver Island. Relationships between the different spatial scales are established with synoptic typing, coupled with non-homogeneous Markov models to simulate precipitation intensity and occurrence. Types are generated from daily precipitation observations spanning 1971 to 2000. Atmospheric predictors used to influence the Markov models are derived from two versions of GCM output: averages of GCM grid cells selected by correlation maps of circulation and precipitation data and an approach involving common Empirical Orthogonal Functions (EOFs) calculated from GCM output over the northeast Pacific Ocean. Projections for 2081 to 2100 made using averaged grid cells find that winter (November–February) precipitation anomalies produce modestly positive values, with gains of 7.5% in average precipitation, typical increases of 9.0% rising to 20% in the case of high-intensity precipitation, and little spatial dependence. In contrast, average and high-intensity summer precipitation (June–September) decline negligibly at most island weather stations with the exception of those in the southwestern sections, which experience reductions of 15% relative to 1971 to 2000. Projections made using common EOFs display a strong spatial dependence. Future winter precipitation is expected to increase only on the west coast of the island by 11%, on average, while the southeastern coast will experience decreases of 5% to 10%. The same pattern repeats in summer, though with negligible increases on the west coast and declines of 12% to 16% on the southeastern coast. The reliability of this novel EOF method remains to be confirmed definitively, however. In both seasons precipitation occurrence decreases slightly at all stations with declines in the total days with measurable precipitation ranging from 2% to 8%. RÉSUMÉ [Traduit par la rédaction] Nous employons une technique statistique de réduction d’échelle pour lier la circulation atmosphérique produite par un ensemble de simulations du GCM (Global Climate Model) durant le XXIe siècle aux précipitations enregistrées à des stations météorologiques sur l’île de Vancouver. Les relations entre les différentes échelles spatiales sont établies au moyen d'un typage synoptique couplé avec des modèles markoviens non homogènes pour simuler l'intensité et la fréquence des précipitations. Les types sont générés à partir des observations quotidiennes de précipitations au cours de la période 1971–2000. Les prédicteurs atmosphériques utilisés pour influencer les modèles markoviens sont dérivés de deux versions de sorties du GCM : les moyennes de mailles du GCM sélectionnées par tables de corrélation des données de circulation et de précipitations et une approche fondée sur les fonctions orthogonales empiriques (EOF) communes calculées d'après la sortie du GCM pour le nord-est du Pacifique. Les projections pour la période 2081–2100 basées sur des moyennes de mailles montrent que les anomalies de précipitations hivernales (novembre–février) produisent de faibles valeurs positives, avec des gains de 7.5% dans les précipitations moyennes, des accroissements caractéristiques de 9.0% augmentant à 20% dans le cas des précipitations de forte intensité, et peu de dépendance spatiale. En revanche, les précipitations estivales (juin–septembre) moyennes et de forte intensité diminuent de façon négligeable à la plupart des stations météorologiques de l’île, à l'exception de celles situées dans secteur sud-ouest qui subissent une réduction de 15% par rapport à 1971–2000. Les projections faites à l'aide des fonctions orthogonales empiriques communes exhibent une forte dépendance spatiale. Les précipitations hivernales futures devraient augmenter seulement sur la côte ouest de l’île de 11% en moyenne alors que la côte sud-est connaîtra des diminutions de 5 à 10%. La même configuration se répète en été, bien qu'avec des accroissements négligeables sur la côte ouest et des diminutions de 12 à 16% sur la côte sud-est. La fiabilité de cette nouvelle méthode EOF reste toutefois à établir. Dans les deux saisons, la fréquence des précipitations diminue légèrement à toutes les stations, les diminutions du nombre total de jours avec précipitations mesurables variant entre 2 et 8%.

Integrating Prior Knowledge in Multiple Testing under Dependence with Applications to Detecting Differential DNA Methylation

DNA methylation has emerged as an important hallmark of epigenetics. Numerous platforms including tiling arrays and next generation sequencing, and experimental protocols are available for profiling DNA methylation. Similar to other tiling array data, DNA methylation data shares the characteristics of inherent correlation structure among nearby probes. However, unlike gene expression or protein DNA binding data, the varying CpG density which gives rise to CpG island, shore and shelf definition provides exogenous information in detecting differential methylation. This article aims to introduce a robust testing and probe ranking procedure based on a nonhomogeneous hidden Markov model that incorporates the above-mentioned features for detecting differential methylation. We revisit the seminal work of Sun and Cai (2009, Journal of the Royal Statistical Society: Series B (Statistical Methodology)71, 393-424) and propose modeling the nonnull using a nonparametric symmetric distribution in two-sided hypothesis testing. We show that this model improves probe ranking and is robust to model misspecification based on extensive simulation studies. We further illustrate that our proposed framework achieves good operating characteristics as compared to commonly used methods in real DNA methylation data that aims to detect differential methylation sites.

Optimal replacement policy for safety-related multi-component multi-state systems

This paper investigates replacement scheduling for non-repairable safety-related systems (SRS) with multiple components and states. The aim is to determine the cost-minimizing time for replacing SRS while meeting the required safety. Traditionally, such scheduling decisions are made without considering the interaction between the SRS and the production system under protection, the interaction being essential to formulate the expected cost to be minimized. In this paper, the SRS is represented by a non-homogeneous continuous time Markov model, and its state distribution is evaluated with the aid of the universal generating function. Moreover, a structure function of SRS with recursive property is developed to evaluate the state distribution efficiently. These methods form the basis to derive an explicit expression of the expected system cost per unit time, and to determine the optimal time to replace the SRS. The proposed methodology is demonstrated through an illustrative example.

Linking atmospheric circulation to daily precipitation patterns over the territory of Bulgaria

An eight states non-homogeneous hidden Markov model is developed for linking daily precipitation amounts data at a network of 32 stations broadly covering the territory of Bulgaria to large-scale atmospheric patterns. At each site a 40-year record 1960–2000 of daily October–March precipitation amounts is modeled. The atmospheric data consists of daily sea-level pressure, geopotential height at 500 and 850 hPa, air temperature at 850 hPa and relative humidity at 700 and 850 hPa on a 2.5 × 2.5 grid based on NCEP-NCAR reanalysis data set covering the Europe-Atlantic sector 30W–60E, 20N–70N for the same period. The first 30 years data are used for model fitting purposes while the remaining 10 years are used for model evaluation. Detailed model validation is carried out on various aspects. The proposed model reproduces well the rainfall statistics for the observed and reserved data whereas the identified states are found to be physically interpretable in terms of regional climatology.

Asymptotic normality of M-estimators in nonhomogeneous hidden Markov models

Results on asymptotic normality for the maximum likelihood estimate in hidden Markov models are extended in two directions. The stationarity assumption is relaxed, which allows for a covariate process influencing the hidden Markov process. Furthermore, a class of estimating equations is considered instead of the maximum likelihood estimate. The basic ingredients are mixing properties of the process and a general central limit theorem for weakly dependent variables.

Asymptotic normality of M-estimators in nonhomogeneous hidden Markov models

Results on asymptotic normality for the maximum likelihood estimate in hidden Markov models are extended in two directions. The stationarity assumption is relaxed, which allows for a covariate process influencing the hidden Markov process. Furthermore, a class of estimating equations is considered instead of the maximum likelihood estimate. The basic ingredients are mixing properties of the process and a general central limit theorem for weakly dependent variables.

Statistical downscaling of rainfall data using sparse variable selection methods

In many statistical downscaling methods, atmospheric variables are chosen by using a combination of expert knowledge with empirical measures such as correlations and partial correlations. In this short communication, we describe the use of a fast, sparse variable selection method, known as RaVE, for selecting atmospheric predictors, and illustrate its use on rainfall occurrence at stations in South Australia. We show that RaVE generates parsimonious models that are both sensible and interpretable, and whose results compare favourably to those obtained by a non-homogeneous hidden Markov model ( Hughes et al., 1999).

Flexible Nonhomogeneous Markov Models for Panel Observed Data

Methods for fitting nonhomogeneous Markov models to panel-observed data using direct numerical solution to the Kolmogorov Forward equations are developed. Nonhomogeneous Markov models occur most commonly when baseline transition intensities depend on calendar time, but may also occur with deterministic time-dependent covariates such as age. We propose transition intensities based on B-splines as a smooth alternative to piecewise constant intensities and also as a generalization of time transformation models. An expansion of the system of differential equations allows first derivatives of the likelihood to be obtained, which can be used in a Fisher scoring algorithm for maximum likelihood estimation. The method is evaluated through a small simulation study and demonstrated on data relating to the development of cardiac allograft vasculopathy in posttransplantation patients.

Identity-by-descent filtering of exome sequence data for disease-gene identification in autosomal recessive disorders.

Motivation: Next-generation sequencing and exome-capture technologies are currently revolutionizing the way geneticists screen for disease-causing mutations in rare Mendelian disorders. However, the identification of causal mutations is challenging due to the sheer number of variants that are identified in individual exomes. Although databases such as dbSNP or HapMap can be used to reduce the plethora of candidate genes by filtering out common variants, the remaining set of genes still remains on the order of dozens.Results: Our algorithm uses a non-homogeneous hidden Markov model that employs local recombination rates to identify chromosomal regions that are identical by descent (IBD = 2) in children of consanguineous or non-consanguineous parents solely based on genotype data of siblings derived from high-throughput sequencing platforms. Using simulated and real exome sequence data, we show that our algorithm is able to reduce the search space for the causative disease gene to a fifth or a tenth of the entire exome.Availability: An R script and an accompanying tutorial are available at http://compbio.charite.de/index.php/ibd2.html.Contact: peter.robinson@charite.de

Evaluation of two statistical downscaling models for daily precipitation over an arid basin in China

AbstractTwo statistical downscaling (SD) models, the nonhomogeneous hidden Markov model (NHMM) and the statistical down‐scaling model (SDSM), which have been widely applied and proved skillful in terms of downscaling precipitation, were evaluated based on observed daily precipitation over the Tarim River basin, an arid basin located in China. The evaluated metrics included residual functions, correlation analyses, probability density functions (PDFs) and distributions. Overall, both models exhibited stability with little model performance difference between the calibration and validation periods. There was little difference for model performance on dry‐spell length (dsl) and wet‐spell length (wsl) between NHMM and SDSM. NHMM showed skill in simulating wet‐day precipitation amount (wpa), whilst SDSM performed relatively poorly on extreme values of wpa, especially for dry stations with annual precipitation lower than 200 mm. Both NHMM and SDSM captured the spatial distribution characteristics of precipitation for most of the stations, as 11 months had an at‐station correlation coefficient being greater than 0.9 and 0.8 for NHMM and SDSM in calibration period. NHMM showed slightly better model performance than SDSM on simulating monthly precipitation, as the former was able to model precipitation well for all months, whereas the later was well only for certain months. SDSM was able to capture the inter‐site correlation characteristics of observed series, whilst the NHMM multi‐site simulation over estimated inter‐site correlation. Both NHMM and SDSM had less skill downscaling annual series because of stochastic components in precipitation amounts modelling. Copyright © 2011 Royal Meteorological Society

Two Stationary Nonhomogeneous Markov Models of Nucleotide Sequence Evolution

The general Markov model (GMM) of nucleotide substitution does not assume the evolutionary process to be stationary, reversible, or homogeneous. The GMM can be simplified by assuming the evolutionary process to be stationary. A stationary GMM is appropriate for analyses of phylogenetic data sets that are compositionally homogeneous; a data set is considered to be compositionally homogeneous if a statistical test does not detect significant differences in the marginal distributions of the sequences. Though the general time-reversible (GTR) model assumes stationarity, it also assumes reversibility and homogeneity. We propose two new stationary and nonhomogeneous models--one constrains the GMM to be reversible, whereas the other does not. The two models, coupled with the GTR model, comprise a set of nested models that can be used to test the assumptions of reversibility and homogeneity for stationary processes. The two models are extended to incorporate invariable sites and used to analyze a seven-taxon hominoid data set that displays compositional homogeneity. We show that within the class of stationary models, a nonhomogeneous model fits the hominoid data better than the GTR model. We note that if one considers a wider set of models that are not constrained to be stationary, then an even better fit can be obtained for the hominoid data. However, the methods for reducing model complexity from an extremely large set of nonstationary models are yet to be developed.

Dynamic Allocation of Pharmaceutical Detailing and Sampling for Long-Term Profitability

The U.S. pharmaceutical industry spent upwards of $18 billion on marketing drugs in 2005; detailing and drug sampling activities accounted for the bulk of this spending. To stay competitive, pharmaceutical managers need to maximize the return on these marketing investments by determining which physicians to target as well as when and how to target them. In this paper, we present a two-stage approach for dynamically allocating detailing and sampling activities across physicians to maximize long-run profitability. In the first stage, we estimate a hierarchical Bayesian, nonhomogeneous hidden Markov model to assess the short- and long-term effects of pharmaceutical marketing activities. The model captures physicians' heterogeneity and dynamics in prescription behavior. In the second stage, we formulate a partially observable Markov decision process that integrates over the posterior distribution of the hidden Markov model parameters to derive a dynamic marketing resource allocation policy across physicians. We apply the proposed approach in the context of a new drug introduction by a major pharmaceutical firm. We identify three prescription-behavior states, a high degree of physicians' dynamics, and substantial long-term effects for detailing and sampling. We find that detailing is most effective as an acquisition tool, whereas sampling is most effective as a retention tool. The optimization results suggest that the firm could increase its profits substantially while decreasing its marketing spending. Our suggested framework provides important implications for dynamically managing customers and maximizing long-run profitability.

Optimal Replacement Policy for Multi-State System Under Imperfect Maintenance

A multi-state system (MSS) has more than two discrete states corresponding to different performance rates. Usually, MSS is viewed as in a failure state once its performance rate falls below user demand, and maintenance is carried out immediately. Generally, the repaired system cannot be regarded as good as new, and oftentimes the system restoration is stochastic. We introduce an optimal replacement policy for MSSs, called policy N. Under this policy, a MSS is replaced whenever its failure number reaches N . We assess the dynamic element state probabilities of each aging multi-state element (MSE) using a stochastic process model which is identified as a non-homogeneous continuous time Markov model (NHCTMM), and we evaluate the state distribution of the entire MSS via the combination of the stochastic process, and the universal generating function (UGF). To quantify the quality of imperfect maintenance, a quasi-renewal process is used to describe the stochastic behavior of each individual MSE after repair. Moreover, we derive an explicit expression of the long-run expected profit per unit time, and determine the optimal failure number N* to replace the entire system. The proposed models are demonstrated via an illustrative case, followed by some comparative studies.