Discrete Autoregressive Moving Average Research Articles

Discrete ARMA Model Applied for Tumor Growth Inhibition Modeling and LQR-based Chemotherapy Optimization

Mathematical models for tumor growth inhibition (TGI) are an important tool in the battle against cancer allowing preclinical evaluation of potential anti-cancer drugs and treatment schedules. In this article, an autoregressive moving average (ARMA) model for cancer tumor growth is estimated based on laboratory data of TGI in mice and presented. The model was proven capable of describing with accuracy the tumor growth under single-agent chemotherapy. At the same time, an optimal control problem was formulated to identify optimal drug dosages. The linear quadratic regulator (LQR) controller was used with success in optimizing both periodic and intermittent chemotherapy treatment schedules reducing the tumor mass while keeping dosages under acceptable toxicity

Regime-Switching Discrete ARMA Models for Categorical Time Series.

For the modeling of categorical time series, both nominal or ordinal time series, an extension of the basic discrete autoregressive moving-average (ARMA) models is proposed. It uses an observation-driven regime-switching mechanism, leading to the family of RS-DARMA models. After having discussed the stochastic properties of RS-DARMA models in general, we focus on the particular case of the first-order RS-DAR model. This RS-DAR model constitutes a parsimoniously parameterized type of Markov chain, which has an easy-to-interpret data-generating mechanism and may also handle negative forms of serial dependence. Approaches for model fitting are elaborated on, and they are illustrated by two real-data examples: the modeling of a nominal sequence from biology, and of an ordinal time series regarding cloudiness. For future research, one might use the RS-DAR model for constructing parsimonious advanced models, and one might adapt techniques for smoother regime transitions.

Analysis and Modeling of Precipitation Intermittency by Compound Markov‐DARMA Models

AbstractIntermittency in precipitation is analyzed by a new class of stochastic models consisting of the product of two binary stochastic models. The ingoing models in these compound models are confined to a one‐parameter white noise model and either a first‐order or a second‐order autoregressive model, which is described by two or three parameters, respectively. The two resulting second‐order compound models have three and four parameters, respectively. Expressions of these parameters are derived, as well as equations with which to assess the parameters in the ingoing models based on the parameters in the compound models. The compound models are shown to differ considerably from the usual Markov models and discrete autoregressive moving average (DARMA) models. Daily binary precipitation time series from Copenhagen, Denmark, and Alice Springs, Australia, are analyzed by depicting the Markovian transition probabilities in probability trees with time lags of up to 5 days, resulting in 32 branches. Each tree is a kind of signature for the time series considered, but only a few of the branches, typically 3–4, are significant. Furthermore, the significant parts of the trees are rather skewed due to the intermittency of the precipitation process. This skewness makes the application of usual Markov and DARMA models less effective, whereas the new compound models better explain the observations. Based on one of the two new models, explicit but approximate expressions for the cumulative distribution (or the return period) of the durations of wet and dry spells are derived and compared to the observations from Copenhagen.

Generalized Binary Time Series Models

The serial dependence of categorical data is commonly described using Markovian models. Such models are very flexible, but they can suffer from a huge number of parameters if the state space or the model order becomes large. To address the problem of a large number of model parameters, the class of (new) discrete autoregressive moving-average (NDARMA) models has been proposed as a parsimonious alternative to Markov models. However, NDARMA models do not allow any negative model parameters, which might be a severe drawback in practical applications. In particular, this model class cannot capture any negative serial correlation. For the special case of binary data, we propose an extension of the NDARMA model class that allows for negative model parameters, and, hence, autocorrelations leading to the considerably larger and more flexible model class of generalized binary ARMA (gbARMA) processes. We provide stationary conditions, give the stationary solution, and derive stochastic properties of gbARMA processes. For the purely autoregressive case, classical Yule–Walker equations hold that facilitate parameter estimation of gbAR models. Yule–Walker type equations are also derived for gbARMA processes.

Drought probabilities and return period for annual streamflows series

Probabilistic characterization of drought events is extremely important in water resources planning and management. In spite of the large number of studies that have been carried on the subject, the exact derivation of the probabilistic structure of drought characteristics is still an open problem, especially when both duration and accumulated deficit (or intensity) are taken into account. This study focuses on the derivation of the probability distribution of drought episodes considering both drought duration and accumulated deficit (or intensity) as well as of the ensuing return period, when the underlying hydrological variable is autocorrelated. First, the marginal distribution of drought length is investigated, showing that the simple Markov chain, usually adopted to model the sequence of deficits and surpluses, is not adequate when the underlying series exhibits a significant autocorrelation. Following previous studies, a Discrete Autoregressive Moving Average (DARMA) model is proposed to better model drought length. Then, the derivation of the moments of accumulated deficit conditioned on a fixed drought length is pursued. It is shown that the deficit series that is obtained by clipping an autocorrelated stationary series with a constant threshold is not i.i.d., and therefore not stationary. Thus, a fully multivariate approach, based on a truncated multivariate normal model, is applied to derive the moments of accumulated deficit as a function of the distribution of the underlying variable and of the threshold. In order to overcome numerical difficulties related to the integration of the multivariate distributions, empirical approximations are also proposed. Such moments are then used to derive approximate expressions of the bivariate distributions of accumulated deficit (or intensity) and length, based on the assumption that the distribution of accumulated deficit conditioned on a fixed length is beta, and expressions to compute the return period of drought events. The proposed procedure is illustrated by applying it to four streamflow data characterized by different levels of autocorrelation and skewness.

Spatial and temporal analysis of dry spells in Croatia

Systematic statistical analysis of dry day sequences, which are defined according to 0.1, 1, 5 and 10 mm of precipitation-per-day thresholds, is performed on seasonal and yearly basis. The data analysed come from 25 Croatian meteorological stations and cover the period 1961–2000. Climatological features of the mean and maximum dry spell durations, as well as the frequency of long dry spells (>20 days) are discussed. The results affirm the three main climatological regions in Croatia, with the highlands exhibiting shorter dry spells than the mainland, and the coastal region exhibiting longer dry spells. The prevailing positive trend of both mean and maximal durations is detected during winter and spring seasons, while negative trend dominate in autumn for all thresholds. Positive field significant trends of mean dry spell duration with 5 and 10 mm thresholds are found during spring and the same is valid for annual maximum dry spell duration with a 10 mm threshold. It is found that the Discrete Autoregressive Moving Average (DARMA(1,1)) model can be used to estimate the probabilities of dry spells in Croatia that are up to 20–30 days long.

Drought Occurrence Probabilities and Risks of Dependent Hydrologic Processes

The occurrence probabilities, return periods, and risk of drought events are estimated for dependent hydrologic processes. Traditionally, Markovian models have been used for modeling hydrologic processes having short-term time dependence. However, they are inadequate for processes exhibiting longer time dependence. In this paper, low-order discrete autoregressive moving average (DARMA) models are used for modeling the variability of wet and dry years. Specifically, we center our attention on the occurrence of drought events, particularly their duration, by using the concept of runs. The probability distribution of drought occurrence, expected values and variances of first arrival and interarrival times of drought events, and the associated risks are derived. The derived equations and algorithms are verified by Monte Carlo simulation experiments. The applicability of the proposed methods is demonstrated by using annual streamflow data of the South Platte River in Colorado and the Niger River in Africa. It ...

Prediction of Temperature and Humidity in a Naturally Ventilated Pig Building

A mathematical description is given of heat- and moisture-transfer processes in a pig house, for 880 weaned piglets, equipped with natural ventilation, floor heating and local air cooling by means of additional humidification. Dynamic discrete auto-regressive moving average (ARMA) models, describing the air temperature and humidity inside the building and continuous models as transfer functions are derived. Experiments were performed to identify the structure and coefficients of the discrete ARMA models, using recursive prediction error (maximum-likelihood and least-squares) methods. The input and output data for the identification experiments were collected from experiments carried out over three seasons, viz., winter, spring and summer. The structure of the discrete models was established and it was found that the dynamic properties of the heat- and moisture-transfer processes could be described using first-order dynamic models. The model orders were determined by Akaike's test criterion. The identified models were validated by tests to check whether the residuals consisted of white noise. The models were also tested with data sets which were not used in the identification process and their predictions agreed well with measured values. The mean value of the differences between measured data and simulated values for the temperature model was 1·12°C and 1·02 g/kg for the humidity model. Static models describing the air temperature and humidity inside the livestock building in the steady state were also derived. The use of these models offer a quick engineering method for determination of the required supplemental water flow rate for humidifying the indoor air in summer and the required supplemental heat in winter.

DISCRETE TIME ADAPTIVE ACTIVE SUSPENSION FOR A HALF-CAR MODEL

This paper presents a discrete time adaptive active suspension for a multi-degree of freedom (MDOF) half-car model. The study involves formulation of a pitch plane half-car model involving four degrees of freedom and various nonlinear time varying (NTV) parameters. A multi-degree of freedom skyhook reference model has been described in this paper. Discrete auto regressive moving average (DARMA) models for the NTV-MDOF and the reference model have been developed. A modified version of the least squares estimation in which the controller parameters are updated as a matrix rather than as a vector has been derived in this paper. This approach reduces the computational time required for an adaptive controller parameters estimation and enhances the hardware implementation process. Computer simulation results indicate good adaptation to the reference model and maintenance of static equilibrium conditions for large squat and nose diving longitudinal manoeuvres. The responses also indicate an optimal vibration isolation and suspension travel performance by adapting to the reference model.

CHARACTERISTICS OF EXTREME PRECIPITATION1

ABSTRACT: Records of extreme precipitation were investigated using the Discrete Autoregressive Moving Average (DABMA) process, which can explain long persistences of wet and dry spells that exist in daily precipitation data. The results show that the daily precipitation with strong autocorrelation is inclined to be better fit by a Discrete Autoregressive (DAB) model. On the other hand, those data with weak autocorrelations tend to be best fit by a Discrete Moving Average (DMA) model. It can also be concluded that based on the records from extremely wet and dry regions there is no geographic preference regarding the selection of the best model.

Simulation Models of Sequences of Dry and Wet Days

A new statistical model has been developed for the simulation of sequences of dry and wet days. The model is based on the discrete autoregressivemoving average (DARMA) family of stochastic processes, which includes the Markov chain as a particular case. The model building is based on a three‐step procedure consisting of identification, estimation, and model selection. The model identification and parameter estimation are based on the best fit of the autocorrelation function, while the selection of the optimum model is based on the best reproduction of the probability distribution function of the lengths of the runs of dry days and wet days. The model has thus the property of reproducing the persistence of dry spells and wet spells which are important in the evaluation and forecast of droughts and floods. Excellent results were obtained with rainfall data from Indiana, which indicate that the models are useful for scheduling irrigation of crops in the Central United States and possibly elsewhere.

Application of Discrete Autoregressive Moving Average models for estimation of daily runoff

Daily precipitation has been successfully modeled by the Discrete Autoregressive Moving Average (DARMA) family of models. One of these models is used in the estimation of daily streamflows by means of a linear transfer function. The statistical properties of the chosen DARMA precipitation process and the first- and second-order statistics of the relationships between daily precipitation and the corresponding streamflow are used in the determination of the transfer function parameters. The constructed process is applied to a watershed located in Indiana. The model identification, the parameter estimation, and the diagnostic checking are illustrated. This transfer model has the merit that it couples the first- and second-order statistics of the rainfall input and runoff output. The model limitations stem from the stationarity of the DARMA models used for the description of daily rainfall sequences. This requires the division of the year into stationary seasons and introduces minor inaccuracies at the season crossings.

Modeling of Sequences of Wet and Dry Days by Binary Discrete Autoregressive Moving Average Processes

The Binary Discrete Autoregressive Moving Average (B-DARMA) process, which includes the Markov chain as a particular case, is used to describe the wet-dry day sequences that are obtained from daily precipitation time series. A three-step procedure, consisting of identification, estimation and model selection is developed and shown to be effective in model building. The identification step is based on the plot of the autocorrelation function, while the estimation of the parameters is done by fitting the autocorrelation function by a nonlinear least-squares method. The model selection uses the probability distributions of run lengths, which are defined and discussed in this paper. The persistences of wet and dry spells, which are important properties in the study of floods and droughts, are well reproduced through the preservation of the run length properties. The best model is chosen as the one with the run length distribution that has the minimum sum-of-squares error in estimating the actual run length distribution.

Daily Precipitation Modeling by Discrete Autoregressive Moving Average Processes

To account for the statistical dependence among the daily precipitation magnitudes and among the daily precipitation occurrence times, various models from the discrete autoregressive moving average (DARMA) family are constructed for and applied to the daily precipitation data in Indiana. The structure of the DARMA family is specified by the marginal distribution of independent identically distributed (i.i.d.) discrete random variables and by their random linear combination. A fundamental property of the DARMA family is that its correlation structure can be specified independently of its marginal density. The DARMA correlation structure remains invariant under transformation of the marginal density. This property is satisfied by the Indiana daily precipitation data, and it facilitates their modeling by the DARMA family of processes. It is shown that the Markov chain, a popular model for the daily precipitation sequences, is a specified model within DARMA family. The Indiana daily precipitation data is transformed into a multistate discrete precipitation sequence by discretizing the daily precipitation amount into a number of magnitude states. The stochastic process of the daily precipitation sequences is analyzed in terms of the locally stationary seasons defined within the paper. In each season the candidate DARMA models are identified in terms of the covariance function within the season. The parameter estimation of these candidate models is done by fitting their theoretical autocorrelation functions to their empirical counterparts, and the state probabilities are obtained through the multistate mean run lengths. The theoretical run‐length distributions of the candidate models are derived for each of the precipitation magnitude states, including the dry state. The goodness‐of‐fit tests of the candidate models and the best model selection are based on the theoretical multistate run‐length distributions of the daily precipitation data.Supplement (data, tables, calculations, etc.) is available with entire article on microfiche. Order from American Geophysical Union, 2000 Florida Avenue, N.W., Washington, DC 20009. Document W84‐001; $2.50. Payment must accompany order.

A discrete ARMA model for nonlinear system identification

A nonlinear extension of the discrete linear autoregressive moving average (ARMA) model is presented for the identification of nonlinear systems from measurements of input and output signals. This model is linear In the parameters and is shown to be applicable to a broad class of interconnected inear and memoryless nonlinear subsystems. Conditions for identifiability of nonlinear systems and memory requirements for the model are presented. The model contains Volterra and cross product expansions of the measured signals, which in many cases avoids the need for the large number of parameters required in a traditional Volterra expansion. It is also shown that when the model equations are recast in "regular" form, the model parameters can be obtained using multichannel lattice methods based upon the Levinson algorithm.