scholarly journals Fractional Calculus as a Simple Tool for Modeling and Analysis of Long Memory Process in Industry

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 511 ◽  
Author(s):  
Ivo Petráš ◽  
Ján Terpák
Keyword(s):  
Time Series ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Long Memory ◽  
Real Data ◽  
Data Set ◽  
Modeling And Analysis ◽  
Long Memory Process ◽  
Real Process ◽  
Definition Of

This paper deals with the application of the fractional calculus as a tool for mathematical modeling and analysis of real processes, so called fractional-order processes. It is well-known that most real industrial processes are fractional-order ones. The main purpose of the article is to demonstrate a simple and effective method for the treatment of the output of fractional processes in the form of time series. The proposed method is based on fractional-order differentiation/integration using the Grünwald–Letnikov definition of the fractional-order operators. With this simple approach, we observe important properties in the time series and make decisions in real process control. Finally, an illustrative example for a real data set from a steelmaking process is presented.

The bouncing ball and the Grünwald-Letnikov definition of fractional derivative

2021 ◽  
Vol 24 (4) ◽  
pp. 1003-1014
Author(s):  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Restitution Coefficient ◽  
Classical Physics ◽  
Bouncing Ball ◽  
Mechanical Experiment ◽  
Fractional Models ◽  
Definition Of

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

scholarly journals Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive?

2020 ◽  
Vol 4 (3) ◽  
pp. 40
Author(s):  
Jocelyn Sabatier
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Initial Conditions ◽  
Fractional Integration ◽  
Current Trend ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Definition Of ◽  
A Current

In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations.

Bayesian Structural Time Series

2020 ◽  
Vol 12 (1) ◽  
pp. 54-61
Author(s):  
Abdullah M. Almarashi ◽  
Khushnoor Khan
Keyword(s):  
Time Series ◽  
Stock Prices ◽  
Predictive Accuracy ◽  
Local Level ◽  
Real Life ◽  
Secondary Data ◽  
Real Data ◽  
Data Set ◽  
Structural Time Series ◽  
Structural Time

The current study focused on modeling times series using Bayesian Structural Time Series technique (BSTS) on a univariate data-set. Real-life secondary data from stock prices for flying cement covering a period of one year was used for analysis. Statistical results were based on simulation procedures using Kalman filter and Monte Carlo Markov Chain (MCMC). Though the current study involved stock prices data, the same approach can be applied to complex engineering process involving lead times. Results from the current study were compared with classical Autoregressive Integrated Moving Average (ARIMA) technique. For working out the Bayesian posterior sampling distributions BSTS package run with R software was used. Four BSTS models were used on a real data set to demonstrate the working of BSTS technique. The predictive accuracy for competing models was assessed using Forecasts plots and Mean Absolute Percent Error (MAPE). An easyto-follow approach was adopted so that both academicians and practitioners can easily replicate the mechanism. Findings from the study revealed that, for short-term forecasting, both ARIMA and BSTS are equally good but for long term forecasting, BSTS with local level is the most plausible option.

Fractional Order Modeling and Analysis of Buck-Boost Converter

2015 ◽  
Vol 789-790 ◽  
pp. 842-848
Author(s):  
Li Feng Yi ◽  
Kai Ru Zhang ◽  
Jun Liu
Keyword(s):  
Mathematical Model ◽  
Theoretical Analysis ◽  
Fractional Calculus ◽  
Simulation Model ◽  
Fractional Order ◽  
Theoretical Foundation ◽  
Boost Converter ◽  
Modeling And Analysis ◽  
Simulation Results ◽  
Conduction Mode

Considered the theoretical foundation of fractional order, the fractional mathematical model of the Buck-Boost converter in continuous conduction mode operation is built and analyzed in theory. Based on the improved Oustaloup fractional calculus for filter algorithm, the simulation model is framed by using the Matlab/Simulink software. And the simulation results are compared with that of integer order. It proves the correctness of the fractional order mathematical model and the theoretical analysis.

scholarly journals An update on multivariate return periods in hydrology

2016 ◽  
Vol 373 ◽  
pp. 175-178 ◽  
Author(s):  
Benedikt Gräler ◽  
Andrea Petroselli ◽  
Salvatore Grimaldi ◽  
Bernard De Baets ◽  
Niko Verhoest
Keyword(s):  
Time Series ◽  
Return Period ◽  
Simulated Data ◽  
Time Span ◽  
Return Periods ◽  
Data Set ◽  
Univariate Case ◽  
Definition Of ◽  
Selection Of

Abstract. Many hydrological studies are devoted to the identification of events that are expected to occur on average within a certain time span. While this topic is well established in the univariate case, recent advances focus on a multivariate characterization of events based on copulas. Following a previous study, we show how the definition of the survival Kendall return period fits into the set of multivariate return periods.Moreover, we preliminary investigate the ability of the multivariate return period definitions to select maximal events from a time series. Starting from a rich simulated data set, we show how similar the selection of events from a data set is. It can be deduced from the study and theoretically underpinned that the strength of correlation in the sample influences the differences between the selection of maximal events.

scholarly journals Confidence Regions for Parameters in Stationary Time Series Models With Gaussian Noise

2022 ◽  
Vol 9 ◽  
Author(s):  
Xiuzhen Zhang ◽  
Riquan Zhang ◽  
Zhiping Lu
Keyword(s):  
Time Series ◽  
Empirical Likelihood ◽  
Long Memory ◽  
Score Function ◽  
Real Data ◽  
Confidence Regions ◽  
Stationary Time Series ◽  
Time Series Models ◽  
Finite Sample ◽  
Memory Time

This article develops two new empirical likelihood methods for long-memory time series models based on adjusted empirical likelihood and mean empirical likelihood. By application of Whittle likelihood, one obtains a score function that can be viewed as the estimating equation of the parameters of the long-memory time series model. An empirical likelihood ratio is obtained which is shown to be asymptotically chi-square distributed. It can be used to construct confidence regions. By adding pseudo samples, we simultaneously eliminate the non-definition of the original empirical likelihood and enhance the coverage probability. Finite sample properties of the empirical likelihood confidence regions are explored through Monte Carlo simulation, and some real data applications are carried out.

Stochastic Modeling of Time Series with Application to Local Damage Detection in Rotating Machinery

2013 ◽  
Vol 569-570 ◽  
pp. 441-448 ◽  
Author(s):  
Jakub Obuchowski ◽  
Agnieszka Wylomanska ◽  
Radoslaw Zimroz
Keyword(s):  
Time Series ◽  
Good Condition ◽  
Real Data ◽  
Statistical Properties ◽  
Normal Operation ◽  
Local Damage ◽  
Data Set ◽  
Time Frequency ◽  
Frequency Map ◽  
Vibration Time

Raw vibration signals measured on the machine housing in industrial conditions are complex and can be modeled as an additive mixture of several processes (with different statistical properties) related to normal operation of machine, damage related to one (or more) of its part, some noise, etc. In the case of local damage in rotating machines, contribution of informative process related to damage is hidden in the raw signal so its detection is difficult. In this paper we propose to use the statistical modeling of vibration time series to identify these components. Building the model of raw signal may be ineffective. It is proposed to decompose signal into set of narrowband sub-signals using time-frequency representation. Next, it is proposed to model each sub-signal in the given frequency range and classify all signals using their statistical properties. We have used several parameters (called selectors because they will be used for selection of sub-signals from time-frequency map for further processing) for analysis of sub-signals. They have base in statistics theory and can be useful for example in testing of normality of data set (vibration time series from machine in good condition is close to Gaussian, damaged not). Results of such modeling will be used in the sub-signals classification procedure but also in defects detection. We illustrate effectiveness of novel technique using real data from heavy machinery system.

On the Mittag–Leffler Stability of Impulsive Fractional Solow-Type Models

2017 ◽  
Vol 18 (5) ◽  
pp. 315-325 ◽  
Author(s):  
Ivanka M. Stamova ◽  
Gani Tr. Stamov
Keyword(s):  
Fractional Calculus ◽  
Dynamic Behavior ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Mathematical Finance ◽  
Financial Systems ◽  
Modeling And Analysis ◽  
Modeling Approach ◽  
Suitable Tool ◽  
Impulsive Perturbations

AbstractIn this article, we introduce fractional-order Solow-type models as a new tool for modeling and analysis in mathematical finance. Sufficient conditions for the Mittag–Leffler stability of their states are derived. The main advantages of the proposed approach are using of fractional-order derivatives, whose nonlocal property makes the fractional calculus a suitable tool for modeling actual financial systems as well as using of impulsive perturbations which give an opportunity to control the dynamic behavior of the model. The modeling approach proposed in this article can be applied to investigate macroeconomic systems.

scholarly journals The KM-Algorithm Identifies Regulated Genes in Time Series Expression Data

2009 ◽  
Vol 2009 ◽  
pp. 1-10
Author(s):  
Martina Bremer ◽  
R. W. Doerge
Keyword(s):  
Gene Expression ◽  
Time Series ◽  
Statistical Power ◽  
Regulatory Networks ◽  
Real Data ◽  
Model Parameters ◽  
Expression Data ◽  
Data Set ◽  
Gene Expression Time Series ◽  
Expression Time

We present a statistical method to rank observed genes in gene expression time series experiments according to their degree of regulation in a biological process. The ranking may be used to focus on specific genes or to select meaningful subsets of genes from which gene regulatory networks can be built. Our approach is based on a state space model that incorporates hidden regulators of gene expression. Kalman (K) smoothing and maximum (M) likelihood estimation techniques are used to derive optimal estimates of the model parameters upon which a proposed regulation criterion is based. The statistical power of the proposed algorithm is investigated, and a real data set is analyzed for the purpose of identifying regulated genes in time dependent gene expression data. This statistical approach supports the concept that meaningful biological conclusions can be drawn from gene expression time series experiments by focusing on strong regulation rather than large expression values.

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