Point-Wise Dimension Estimate of Nonstationary, Multi-Episode, Time-Series and Application to Gearbox Signal

Volume 7A: 17th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc99/vib-8328 ◽

1999 ◽

Author(s):

D. C. Lin ◽

B. J. Augustine ◽

M. F. Golnaraghi

Keyword(s):

Time Series ◽

First Passage Time ◽

Passage Time ◽

Nonstationary Time Series ◽

Stationary Time Series ◽

Dimension Estimate ◽

Estimation Algorithms ◽

Data Segment ◽

Multiple Episode ◽

Autocorrelation Time

Abstract Dimensions of nonstationary time series is studied. The nonstationarity is considered to be due to multiple episode where an episode is a piece of stationary time series. The dimension estimation algorithms in the literature can be naturally extended to study multi-episode time series by restricting the calculation on data segment of pre-determined length. Inevitably, more than one episode will be included in the segment. This work focuses on finding when such dimension estimate has a local interpretation as the dimension of the episode. It was found that the local interpretation is valid if there is a large enough difference in the autocorrelation time of the episodes. This is termed EES. In practice, the average first passage time of the reconstructed “orbit” can be used to determine EES. Numerical evidence of these results are given and the application to the mechanical gearbox signal are shown.

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Level-crossing probabilities and first-passage times for linear processes

Advances in Applied Probability ◽

10.1239/aap/1086957589 ◽

2004 ◽

Vol 36 (2) ◽

pp. 643-666 ◽

Cited By ~ 5

Author(s):

Gopal K. Basak ◽

Kwok-Wah Remus Ho

Keyword(s):

Time Series ◽

First Passage Time ◽

Passage Time ◽

Time Series Models ◽

System Control ◽

Level Crossing ◽

First Passage ◽

Linear Processes ◽

Integral Equation Approach ◽

Crossing Probabilities

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.

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Comparison of Jump-Diffusion Parameters Using Passage Times Estimation

Journal of Applied Mathematics ◽

10.1155/2014/975418 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5 ◽

Cited By ~ 1

Author(s):

K. Khaldi ◽

K. Djeddour ◽

S. Meddahi

Keyword(s):

Time Series ◽

First Passage Time ◽

Jump Diffusion ◽

Passage Time ◽

Share Price ◽

First Passage ◽

Diffusion Parameters ◽

Stochastic Jump ◽

Moments Method ◽

Passage Times

The main purposes of this paper are two contributions: (1) it presents a new method, which is the first passage time generalized for all passage times (PT method), in order to estimate the parameters of stochastic jump-diffusion process. (2) It compares in a time series model, share price of gold, the empirical results of the estimation and forecasts obtained with the PT method and those obtained by the moments method applied to the MJD model.

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Level-crossing probabilities and first-passage times for linear processes

Advances in Applied Probability ◽

10.1017/s0001867800013641 ◽

2004 ◽

Vol 36 (02) ◽

pp. 643-666 ◽

Cited By ~ 1

Author(s):

Gopal K. Basak ◽

Kwok-Wah Remus Ho

Keyword(s):

Time Series ◽

First Passage Time ◽

Passage Time ◽

Time Series Models ◽

System Control ◽

Level Crossing ◽

First Passage ◽

Linear Processes ◽

Integral Equation Approach ◽

Crossing Probabilities

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Estimation of AR (2) Model with Dependent Errors for Unbounded Stationary and Nonstationary Time Series

Journal of University of Shanghai for Science and Technology ◽

10.51201/jusst/21/121027 ◽

2021 ◽

Vol 23 (12) ◽

pp. 417-422

Author(s):

Prof. Ahmed Amin EL- Sheikh ◽

◽

Mohammed Ahmed Farouk Ahmed ◽

Keyword(s):

Time Series ◽

Sample Size ◽

Covariance Matrix ◽

Nonstationary Time Series ◽

Stationary Time Series ◽

Times Series ◽

The Third ◽

Stationary Conditions ◽

Simulation Results ◽

Better Than

In this paper the GLS and the ML estimators, the variance-covariance matrix, the unbiased for the GLS and the ML estimators of parameters of AR (2) model with constant in case of dependent errors have been derived, the simulation results shown that the values of MSE and Thiel’s U in case of unbounded stationary time series for all sample size T are less than the values of MSE and Thiel’s U in case of unbounded nonstationary time series which approved that the results for unbounded stationary times series are better than the results for unbounded nonstationary times series, and the simulation results for unbounded nonstationary time series shown that by using the measurement of MSE the best case among of all cases of nonstationary which gives the smallest values of MSE is case four when the first and the second conditions of stationary conditions for AR (2) model are exists, while by using the measurement of Thiel’s U the best case among of all cases of nonstationary which gives the smallest values of Thiel’s U is case six when the second and the third conditions of stationary conditions for AR (2) model are exists.

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A SARIMA and Adjusted SARIMA Models in a Seasonal Nonstationary Time Series; Evidence of Enugu Monthly Rainfall

European Journal of Mathematics and Statistics ◽

10.24018/ejmath.2021.2.1.15 ◽

2021 ◽

Vol 2 (1) ◽

pp. 13-18

Author(s):

Chibuzo Gabriel Amaefula

Keyword(s):

Time Series ◽

Least Squares ◽

Diagnostic Test ◽

Nonstationary Time Series ◽

Stationary Time Series ◽

Seasonal Characteristics ◽

Parameter Redundancy ◽

Least Squares Estimates ◽

Underlying Variable ◽

Sarima Models

The paper compares SARIMA and adjusted SARIMA(ASARIMA) in a regular stationary series where the underlying variable is seasonally nonstationary. Adopting empirical rainfall data and Box-Jenkins iterative algorithm that calculates least squares estimates, Out of 11 sub-classes of SARIMA and 7 sub-classes of ASARIMA models, AIC chose ASARIMA(2,1,1)12 over all sub-classes of SARIMA(p,0,q)x(P,1,Q)12 identified. Diagnostic test indicates absence of autocorrelation up to the 48th lag. The forecast values generated by the fitted model are closely related to the actual values. Hence, ASARIMA can be recommended for regular stationary time series with seasonal characteristics and where parameter redundancy and large sum of square errors are penalized.

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A wavelet-based approach for imputation in nonstationary multivariate time series

Statistics and Computing ◽

10.1007/s11222-021-09998-2 ◽

2021 ◽

Vol 31 (2) ◽

Author(s):

Rebecca E. Wilson ◽

Idris A. Eckley ◽

Matthew A. Nunes ◽

Timothy Park

Keyword(s):

Time Series ◽

Health Monitoring ◽

Multivariate Time Series ◽

Nonstationary Time Series ◽

Stationary Time Series ◽

Data Imputation ◽

Missing Observations ◽

Novel Method ◽

Monitoring Application ◽

Successful Analysis

AbstractMany multivariate time series observed in practice are second order nonstationary, i.e. their covariance properties vary over time. In addition, missing observations in such data are encountered in many applications of interest, due to recording failures or sensor dropout, hindering successful analysis. This article introduces a novel method for data imputation in multivariate nonstationary time series, based on the so-called locally stationary wavelet modelling paradigm. Our methodology is shown to perform well across a range of simulation scenarios, with a variety of missingness structures, as well as being competitive in the stationary time series setting. We also demonstrate our technique on data arising in a health monitoring application.

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