Partition Of Interval Research Articles

Nonparametric Bayesian Estimation of a HHlder Continuous Diffusion Coefficient

We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a Holder-continuous diffusion coefficient with smoothness parameter 0<λ≤1. Our approach is straightforward to implement, as the posterior distributions turn out to be inverse Gamma again, and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets.

A spectra partition algorithm based on spectral clustering for interval variable selection

Variable selection is recognized as a way to build a robust model and improve its prediction performance in NIR spectral analysis. Interval variable selection as one of two main strategies of variable selection, is believed to have a good spectra interpretation. Many interval variable selection algorithms have been proposed and developed. Among them dividing the spectra into intervals manually is normally needed, and generally the equal-width partition is adopted, which is arbitrary and subjective. Clustering is known for self-discovering the inner structure of data. In this work, a new method called spectral clustering-based interval partition (SCIP), is purposed as an alternative to the equal-width partition. It partitions the spectra by using spectral clustering based on the correlation between variables. When SCIP is adopted, compared with the equal-width partition under the same number of partitions, the commonly used interval variable selection methods, such as forward interval partial least regression (fiPLS), backward interval partial least regression (biPLS), synergy interval partial least regression (siPLS), genetic algorithm-interval partial least regression (GA-iPLS), and interval combination optimization (ICO) can supply more reasonable wavelength intervals and contribute better prediction performances in four near infrared datasets.

Stability analysis and L1-gain characterization for impulsive positive systems with time-varying delay

This paper is concerned with the issues of stability analysis and L1-gain characterization for impulsive positive systems with time-varying delay. To obtain less conservative results, the impulse interval partitioning idea is employed to construct an impulse-time-dependent discretized copositive Lyapunov-Krasovskii function. For the noiseless case, a delay-dependent criterion is derived to ensure the globally asymptotic stability of the concerned systems under a ranged dwell-time constraint. When external disturbances are taken into account, the proposed copositive Lyapunov–Krasovskii function is further employed to the L1-gain analysis for the concerned system. The differential evolution algorithm is proved to be an effective alternative to solve the associated non-linear optimization problems, e.g., calculating the optimal L1-gain and the allowable upper (lower) bound for the dwell-time. Three examples are provided to validate the effectiveness of the theoretical results, and some comparative studies with the Razumikhin’s method are also given to illustrate the superiority of proposed approach.

Diffusions on a space of interval partitions: construction from marked Lévy processes

Consider a spectrally positive Stable$(1\!+\!\alpha )$ process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning “sizes” varying during the lifetime. As for Crump–Mode–Jagers processes (with “characteristics”), we consider for each level the collection of individuals alive. We arrange their “sizes” at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable$(1\!+\!\alpha )$ process, this yields new theorems of Ray–Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson–Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.

Ordinal Regression Modeling of Mother to Infant HIV Transmission in Nyeri County, Kenya

The transmission rates of HIV from a HIV-positive mother to her child during pregnancy, delivery or breastfeeding remains of much concern. Various governments and non-governmental organizations have aimed at coming up with policies aimed at minimizing the transmissions. For this to be achievable, there is a need for sound statistical procedures in the analysis of the mother to infant HIV transmission data. The study gives an application of the ordinal regression to the modeling of such data, a case of Nyeri County-Kenya. The logistic regression has been described as the best methodology for modeling binary response variables. However it does not provide a best fit for an ordered categorical variable with more than two categories. This calls for the extensions of the logistic regression which can be used when modeling such kind of variables, such an extension is the ordinal regression methodology. This study proposes the use of the ordinal regression methodology with probit and logit link functions to model infant feeding, arv regimen, maternal cell count and maternal viral load effect on mother to infant HIV transmission in Nyeri county, Kenya a case of Karatina sub-county referral hospital. An aspect of the ordinal link models, which can be useful for this implementation is particularly emphasized as it is in their interpretation that the classes of the dependent variable can be considered from the partition of the variation interval of an underlying continuous random variable. Data to be used shall be secondary data collected from Karatina sub-county referral hospital. Inference on parameters and model diagnostics is also provided.

A Social Beetle Swarm Algorithm Based on Grey Target Decision-Making for a Multiobjective Distribution Network Reconfiguration Considering Partition of Time Intervals

With the increased distributed generation (DG) and the combination of residential, commercial, and industrial loads connected to the distribution networks, it is more difficult to ensure the safe and economic operation of the distribution networks because of the great volatility and randomness of DG and complex loads. In this paper, with the aim of minimizing network loss, load balance, and maximum voltage deviation, a multiobjective reconfiguration model of the distribution network is established under the condition of satisfying network constraints. Moreover, a new social beetle swarm optimization algorithm (SBSO) considering two social behaviors is adopted to solve the complex problem according to the characteristics of the distribution network reconfiguration (DNRC). Based on the SBSO algorithm, grey target decision-making (GTDM) strategy is used to choose the best beetle in the process of solving the multiobjective problem. Additionally, The grey relation projection (GRP) method is used to divide the time period of DNRC according to the change of DG and loads in a day, in order to reduce the number of switching operations. Finally, the effectiveness of the proposed multiobjective model and algorithm are verified on the standard IEEE-33 system and IEEE-69 system.

Characterising random partitions by random colouring

Let $(X_1,X_2,...)$ be a random partition of the unit interval $[0,1]$, i.e. $X_i\geq0$ and $\sum_{i\geq1} X_i=1$, and let $(\varepsilon_1,\varepsilon_2,...)$ be i.i.d. Bernoulli random variables of parameter $p \in (0,1)$. The Bernoulli convolution of the partition is the random variable $Z =\sum_{i\geq1} \varepsilon_i X_i$. The question addressed in this article is: Knowing the distribution of $Z$ for some fixed $p\in(0,1)$, what can we infer about the random partition? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter $p$ is not equal to $1/2$.

Diffusions on a space of interval partitions: construction from Bertoin’s ${\tt BES}_{0}(d)$, $d\in (0,1)$

In 1990, Bertoin constructed a measure-valued Markov process in the framework of a Bessel process of dimension between 0 and 1. In the present paper, we represent this process in a space of interval partitions. We show that this is a member of a class of interval partition diffusions introduced recently and independently by Forman, Pal, Rizzolo and Winkel using a completely different construction from spectrally positive stable Lévy processes with index between 1 and 2 and with jumps marked by squared Bessel excursions of a corresponding dimension between $-2$ and 0.

Exchangeable hierarchies and mass-structure of weighted real trees

Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal {T},d,r,p)$, where $(\mathcal {T},d)$ is a tree-like metric space, $r\in \mathcal {T}$ is a distinguished root, and $p$ is a probability measure on this space. Intuitively, these trees have a combinatorial “underlying branching structure” implied by their topology but otherwise independent of the metric $d$. We explore various ways of making this rigorous, using the weight $p$ to do so without losing the fractal complexity possible in continuum trees. We introduce a notion of mass-structural equivalence and show that two rooted, weighted $\mathbb {R}$-trees are equivalent in this sense if and only if the discrete hierarchies derived by i.i.d. sampling from their weights, in a manner analogous to Kingman’s paintbox, have the same distribution. We introduce a family of trees, called “interval partition trees” that serve as representatives of mass-structure equivalence classes, and which naturally represent the laws of the aforementioned hierarchies.

A high order time discretization of the solution of the non-linear filtering problem

The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the m-power of the mesh of the partition for arbitrary $$m\in {\mathbb {N}}$$ . The result paves the way for constructing high order numerical approximation for the solution of the filtering problem.

Unifying approach to score based statistical inference in physical sciences

In this contribution the statistical inference based on score functions is developed with the aim of future utilization throughout different fields of physics, for example in detector collision data processing or neutrino prongs matching. New score functions between theoretical and empirical probability measures are defined and the corresponding minimum score estimators are presented. We find that consistency of different estimators in various score functions leads to the well-known consistency in commonly used statistical distances or disparity measures between probability distributions. Conditions under which a specific score function pass to ˚–divergence are formulated. Conversely, each ˚–divergence is a score function. Furthermore, the minimization of arbitrary divergence score function leads to the classical histogram density estimator and thus can be used to alternative interpretation of histogram based calculations in (high energy) physics. The Kolmogorov-Smirnov testing statistics can be achieved through absolute score function under the class of mutually complement interval partitioning of the real line. It means that the most popular statistical methods, such as histogram estimation and Kolmogorov goodness of fit testing used in physics, can be covered by one unifying score based statistical approach. Also, these methods were previously successfully applied to data sets originated from the particular material elasticity testing (nondestructive defectoscopy) within Preisach-Mayergoyz space modeling.

Efficient discretisation of stochastic differential equations

The aim of this study is to find a generic method for generating a path of the solution of a given stochastic differential equation which is more efficient than the standard Euler–Maruyama scheme with Gaussian increments. First we characterize the asymptotic distribution of pathwise error in the Euler–Maruyama scheme with a general partition of time interval and then, show that the error is reduced by a factor when using a partition associated with the hitting times of sphere for the driving d-dimensional Brownian motion. This reduction ratio is the best possible in a symmetric class of partitions. Next we show that a reduction which is close to the best possible is achieved by using the hitting time of a moving sphere that is easier to implement.

Adaptive symbolic transfer entropy and its applications in modeling for complex industrial systems.

Directed coupling between variables is the foundation of studying the dynamical behavior of complex systems. We propose an adaptive symbolic transfer entropy (ASTE) method based on the principle of equal probability division. First, the adaptive kernel density method is used to obtain an accurate probability density function for an observation series. Second, the complete phase space of the system can be obtained by using the multivariable phase space reconstruction method. This provides common parameters for symbolizing a time series, including delay time and embedding dimension. Third, an optimization strategy is used to select the appropriate symbolic parameters of a time series, such as the symbol set and partition intervals, which can be used to convert the time series to a symbol sequence. Then the transfer entropy between the symbolic sequences can be carried out. Finally, the proposed method is analyzed and validated using the chaotic Lorenz system and typical complex industrial systems. The results show that the ASTE method is superior to the existing transfer entropy and symbolic transfer entropy methods in terms of measurement accuracy and noise resistance, and it can be applied to the network modeling and performance safety analysis of complex industrial systems.

Modeling of Hybrid Energy Harvesting Communication Systems

Energy harvesting is considered as a prominent technology, particularly for low-power wireless nodes. In this paper, we propose the use of hybrid energy harvesting (HEH) utilizing multiple type energy sources and present the modeling of HEH communication systems based on their probabilistic natures. According to our approach, received energy levels of an HEH system for possible combinations of energy arrivals are characterized by using Gaussian mixture models, which are used to determine harvested energy levels. The range of harvested energy is partitioned, and probabilities of partitioning intervals are used to form a finite-state Markov energy channel (FSMEC) model as an energy channel. Similar to the energy arrival, we also include the probabilistic energy consumption of wireless node in this model, depending on multiple application services, by means of the FSMEC model. Thus, we develop an integrated Markov energy model for HEH communication systems corresponding to the energy harvesting and energy consumption profiles. To evaluate the performance of an HEH communication system, we derive the expressions of energy outage, energy shortage, and service loss probabilities analytically. In numerical studies, the derived expressions are verified by matching simulation results, and it is shown that the performance improves significantly with energy source diversity.

Fuzzy time series forecasting based on proportions of intervals and particle swarm optimization techniques

In this paper, we propose a new fuzzy time series (FTS) forecasting method based on the proportions of intervals and particle swarm optimization (PSO) techniques. First, it uses PSO techniques to obtain the optimal partitions of intervals in the universe of discourse (UOD). Then, each historical testing datum (HTD) is transformed into one of the obtained optimal intervals to construct logical relationships (LRs). Then, based on the current states of the constructed LRs, it constructs logical relationship groups (LRGs). Finally, it performs the forecasting based on the constructed LRGs and the proportions of intervals. The proposed method outperforms the existing methods for forecasting the enrollments of the University of Alabama (UA), the time series data of “killed in car road accidents in Belgium” and the “spot gold in Turkey”.

Heart Rate Variability as a Prognostic Factor for Cancer Survival - A Systematic Review

This paper addresses the problems of robust stability analysis and L1-gain controller synthesis for uncertain impulsive positive systems. First, by employing the idea of impulse interval partitioning, an impulse-time-dependent discretized copositive Lyapunov function is proposed to analyze the robust stability and L1-gain performance of the considered system without control inputs, and several stability conditions and L1-gain criteria are respectively derived. Subsequently, a sufficient condition is formulated for the existence of state-feedback controllers with which not only the positivity and robust uniform asymptotic stability of the resulting closed-loop system are guaranteed, but also a prescribed L1-gain performance is satisfied simultaneously. Furthermore, to make the controller synthesis problem numerically tractable, we propose an iterative convex optimization algorithm to compute the desired controller parameters. Finally, three numerical examples and two realistic examples are presented to show the effectiveness and applicability of the proposed methodology.

CRSO approach for microgrid power dispatching

The optimal dispatch of microgrids is greatly challenged by various uncertainties, introduced by dynamic power fluctuations in loads and variations in renewable energy sources. To depress the over-conservativeness of traditional robust optimisation, this study puts forward a two-stage coupled robust and stochastic optimal (CRSO) dispatching model with mixed integer recourse for the islanded hybrid AC/DC microgrid. The CRSO model comprehensively considers the uncertainties in source–load power and bidirectional converter overhauling, and the stochastic source–load uncertainty sets are constructed by interval partitioning methods. The CRSO model optimises the binary states of the diesel generator and the bidirectional converter in the first stage, and the second stage obtains the flexible operating states and power of each unit. The daily worst-case operation cost is minimised by the nested column-and-constraint generation algorithm. Numerical simulations have validated the rationality and superiority of the developed dispatching model and its solution method.

Robust stability analysis and controller synthesis for uncertain impulsive positive systems under [formula omitted]-gain performance

This paper addresses the problems of robust stability analysis and L1-gain controller synthesis for uncertain impulsive positive systems. First, by employing the idea of impulse interval partitioning, an impulse-time-dependent discretized copositive Lyapunov function is proposed to analyze the robust stability and L1-gain performance of the considered system without control inputs, and several stability conditions and L1-gain criteria are respectively derived. Subsequently, a sufficient condition is formulated for the existence of state-feedback controllers with which not only the positivity and robust uniform asymptotic stability of the resulting closed-loop system are guaranteed, but also a prescribed L1-gain performance is satisfied simultaneously. Furthermore, to make the controller synthesis problem numerically tractable, we propose an iterative convex optimization algorithm to compute the desired controller parameters. Finally, three numerical examples and two realistic examples are presented to show the effectiveness and applicability of the proposed methodology.

Fuzzy time series for real-time flood forecasting

This study applied fuzzy time series (FTS) analysis to develop a real-time flood forecasting model to forecast typhoon flood discharges. Two crucial factors that influence the performance of FTS are the partition of intervals of the variable and the defuzzification method. This study examined the effects of various interval lengths and two defuzzification methods, the centroid and the midpoint methods, on the model performance. Criteria of model completeness and consistency principle were used to determine the effective interval length, and analytic results showed that the midpoint method outperforms the centroid method. Two structures of forecasting models were proposed to make multiple-hour-ahead flood forecasts. Validation results from typhoon flood events in the Wu River in Taiwan showed that the proposed FTS model, which is novel in hydrologic forecasting, can effectively forecast flood discharges.

Saxena-Easo Fuzzy Time Series on Indonesia’s Inflation Rate Forecasting

Saxena-Easo Fuzzy Time Series (FTS) is a softcomputing method for forecasting using fuzzy concept. It doesn’t need any assumption like conventional forecasting method. Generally it’s focused on three important steps like percentage change as the universe of discourse, interval partition, and defuzzification. In this research, this method is applied to Indonesia’s inflation rate data. The aim of this research is to forecast Indonesia’s inflation rate in 2017 by using input from Autoregressive Integrated Moving Average (ARIMA) process, Saxena-Easo FTS, and actual data from 1970-2016. ARIMA is focused on four steps like identifying, parameter estimation, diagnostic checking, and forecasting. The result for Indonesia’s inflation rate forecasting in 2017 is about 5.9182 using Saxena-Easo FTS. Root Mean Square Error (RMSE) is also computed to compare the accuracy rate from each method between Saxena-Easo FTS and ARIMA. RMSE from Saxena-Easo FTS is about 0.9743 while ARIMA is about 6.3046. Keywords: saxena-easo fuzzy time series, ARIMA, inflation rate, RMSE.