On periodicity in series of related terms

Walker, Gilbert T.

doi:10.1098/rspa.1931.0069

Cited by 261 publications

(40 citation statements)

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“…The earliest discussed estimator is the r 1 estimator (Walker 1931), as implemented in the Yule–Walker model (Yule 1927; Box and Jenkins 1976). However, since several studies have found that the bias of r 1 for small samples is large, especially for data with a positive autocorrelation, various alternatives were proposed (Huitema and McKean 1991, 1994; DeCarlo and Tryon 1993; Arnau and Bono 2001; Solanas et al.…”

Section: Selection Of Estimation Methodsmentioning

confidence: 99%

“…Several estimation methods have been proposed to estimate the AR(1) model. These estimation methods include closed form estimation methods, such as the r 1 estimator (Yule 1927; Walker 1931; Box and Jenkins 1976), C-statistic (Young 1941) and Ordinary Least Squares (OLS) estimator, and iterative estimation methods, such as frequentist Maximum Likelihood Estimation (MLE) and Bayesian Markov Chain Monte Carlo (MCMC) estimation. The performance of the closed form estimation methods in terms of efficiency have been examined and compared in some simulation studies (Huitema and McKean 1991; DeCarlo and Tryon 1993; Arnau and Bono 2001; Solanas et al.…”

Section: Introductionmentioning

confidence: 99%

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A comparative simulation study of AR(1) estimators in short time series

Krone¹,

Albers²,

Timmerman³

2015

Qual Quant

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Various estimators of the autoregressive model exist. We compare their performance in estimating the autocorrelation in short time series. In Study 1, under correct model specification, we compare the frequentist r 1 estimator, C-statistic, ordinary least squares estimator (OLS) and maximum likelihood estimator (MLE), and a Bayesian method, considering flat (Bf) and symmetrized reference (Bsr) priors. In a completely crossed experimental design we vary lengths of time series (i.e., T = 10, 25, 40, 50 and 100) and autocorrelation (from −0.90 to 0.90 with steps of 0.10). The results show a lowest bias for the Bsr, and a lowest variability for r 1. The power in different conditions is highest for Bsr and OLS. For T = 10, the absolute performance of all measurements is poor, as expected. In Study 2, we study robustness of the methods through misspecification by generating the data according to an ARMA(1,1) model, but still analysing the data with an AR(1) model. We use the two methods with the lowest bias for this study, i.e., Bsr and MLE. The bias gets larger when the non-modelled moving average parameter becomes larger. Both the variability and power show dependency on the non-modelled parameter. The differences between the two estimation methods are negligible for all measurements.

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Section: Selection Of Estimation Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A comparative simulation study of AR(1) estimators in short time series

Krone¹,

Albers²,

Timmerman³

2015

Qual Quant

View full text Add to dashboard Cite

show abstract

“…Equation 3.1 can be expanded as a Yule-Walker equation (Yule, 1927;Walker, 1931; Boashash, 1995; Kadtke & Kremliovsky, 1999),

ẋ x_{τ} = a x_{τ}^{2}

. Applying the expectation operator

false〈 F false(t false) false〉 \equiv {lim}_{T \to \infty} false(\frac{1}{T} \int_{0}^{T} F false(t false) normald normalt false)

, we get

a = \frac{false〈 ẋ x_{τ} false〉}{false〈 x_{τ}^{2} false〉} = \frac{false〈 ẋ x_{τ} false〉}{false〈 x^{2} false〉} .

For a harmonic signal with one frequency (this is a special solution of the linear DDE in equation 3.1; see Falbo, 1995, for details), x ( t ) = A cos(ω t + φ).…”

Section: Functional Embedding As a Connection Between Nonlinear Dynmentioning

confidence: 99%

Delay Differential Analysis of Time Series

Lainscsek

Sejnowski

2015

Neural Computation

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Nonlinear dynamical system analysis based on embedding theory has been used for modeling and prediction, but it also has applications to signal detection and classification of time series. An embedding creates a multidimensional geometrical object from a single time series. Traditionally either delay or derivative embeddings have been used. The delay embedding is composed of delayed versions of the signal, and the derivative embedding is composed of successive derivatives of the signal. The delay embedding has been extended to nonuniform embeddings to take multiple timescales into account. Both embeddings provide information on the underlying dynamical system without having direct access to all the system variables. Delay differential analysis is based on functional embeddings, a combination of the derivative embedding with nonuniform delay embeddings. Small delay differential equation (DDE) models that best represent relevant dynamic features of time series data are selected from a pool of candidate models for detection or classification. We show that the properties of DDEs support spectral analysis in the time domain where nonlinear correlation functions are used to detect frequencies, frequency and phase couplings, and bispectra. These can be efficiently computed with short time windows and are robust to noise. For frequency analysis, this framework is a multivariate extension of discrete Fourier transform (DFT), and for higher-order spectra, it is a linear and multivariate alternative to multidimensional fast Fourier transform of multidimensional correlations. This method can be applied to short or sparse time series and can be extended to cross-trial and cross-channel spectra if multiple short data segments of the same experiment are available. Together, this time-domain toolbox provides higher temporal resolution, increased frequency and phase coupling information, and it allows an easy and straightforward implementation of higher-order spectra across time compared with frequency-based methods such as the DFT and cross-spectral analysis.

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“…In this AR model, the curve of the time series was fit by the linear combination of the observed historical values. Walker developed the MA model based on the AR model in 1931 [10]. The MA model used a linear combination of historical random disturbances and prediction errors to obtain the current predictive value.…”

Section: Introductionmentioning

confidence: 99%

Predicting subway passenger flows under different traffic conditions

et al. 2018

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Passenger flow prediction is important for the operation, management, efficiency, and reliability of urban rail transit (subway) system. Here, we employ the large-scale subway smartcard data of Shenzhen, a major city of China, to predict dynamical passenger flows in the subway network. Four classical predictive models: historical average model, multilayer perceptron neural network model, support vector regression model, and gradient boosted regression trees model, were analyzed. Ordinary and anomalous traffic conditions were identified for each subway station by using the density-based spatial clustering of applications with noise (DBSCAN) algorithm. The prediction accuracy of each predictive model was analyzed under ordinary and anomalous traffic conditions to explore the high-performance condition (ordinary traffic condition or anomalous traffic condition) of different predictive models. In addition, we studied how long in advance that passenger flows can be accurately predicted by each predictive model. Our finding highlights the importance of selecting proper models to improve the accuracy of passenger flow prediction, and that inherent patterns of passenger flows are more prominently influencing the accuracy of prediction.

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On periodicity in series of related terms

Cited by 261 publications

References 0 publications

A comparative simulation study of AR(1) estimators in short time series

A comparative simulation study of AR(1) estimators in short time series

Delay Differential Analysis of Time Series

Predicting subway passenger flows under different traffic conditions

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