A nonparametric procedure to detect periods in time series

Suppose that for a given time series the experimenter knows that it has a certain periodic property and that he wishes to find out the length of the period. For this problem a nonparametric procedure is proposed. It consists of a new smoothing technique based on Kendall's Tau and a specific counting method. The procedure is studied under a simple model of periodic time series which are composed of periodic (deterministic) functions, a linear trend and exchangeable (stochastic) sequences. The performance of the procedure is illustrated by a simple example.     

Detecting abrupt changes in a piecewise locally stationary time series

Non-stationary time series arise in many settings, such as seismology, speech-processing, and finance. In many of these settings we are interested in points where a model of local stationarity is violated. We consider the problem of how to detect these change-points, which we identify by finding sharp changes in the time-varying power spectrum. Several different methods are considered, and we find that the symmetrized Kullback-Leibler information discrimination performs best in simulation studies. We derive asymptotic normality of our test statistic, and consistency of estimated change-point locations. We then demonstrate the technique on the problem of detecting arrival phases in earthquakes.     

Runs tests for assessing volatility forecastability in financial time series

In this work we refine a nonparametric methodology firstly applied in Christoffersen and Diebold [Review of Economics and Statistics 82 (2000) 12] for assessing volatility forecastability in financial time series based on discretization and on the use of runs tests. Empirical results are provided for SP500 and MIB30 indexes that lead naturally to a discretized one-period Markov chain. The results are confirmed with other persistence measures and their robustness is studied via numerical simulation.     

A note on the time series which is the product of two stationary time series

The time series […,x_-1y_-1,x₀y₀,x₁y₁,…]> which is the product of two stationary time series x_t and y_t is studied. Such sequences arise in the study of nonlinear time series, censored time series, amplitude modulated time series, time series with random parameters, and time series with missing observations. The mean and autocovariance function of the product sequence are derived.     

Detecting high-dimensional determinism in time series with application to human movement data

We numerically investigate the ability of a statistic to detect determinism in time series generated by high-dimensional continuous chaotic systems. This recently introduced statistic (denoted V_E2) is derived from the averaged false nearest neighbors method for analyzing data. Using surrogate data tests, we show that the proposed statistic is able to discriminate high-dimensional chaotic data from their stochastic counterparts. By analyzing the effect of the length of the available data, we show that the proposed criterion is efficient for relatively short time series. Finally, we apply the method to real-world data from biomechanics, namely postural sway time series. In this case, the results led us to exclude the hypothesis of nonlinear deterministic underlying dynamics for the observed phenomena.     

Estimation of a time series model from unequally spaced data

A process generated by a stochastic differential equation driven by pure noise is sampled at irregular intervals. A model for the sampled sequence is deduced. We describe a maximum likelihood procedure for estimating the parameters and establish the strong consistency and asymptotic normality of the estimates. The use of the model in prediction is considered. Simplifications in the case of periodic sampling are explored.     

Forecasting seasonal time series data: a Bayesian model averaging approach

A flexible Bayesian periodic autoregressive model is used for the prediction of quarterly and monthly time series data. As the unknown autoregressive lag order, the occurrence of structural breaks and their respective break dates are common sources of uncertainty these are treated as random quantities within the Bayesian framework. Since no analytical expressions for the corresponding marginal posterior predictive distributions exist a Markov Chain Monte Carlo approach based on data augmentation is proposed. Its performance is demonstrated in Monte Carlo experiments. Instead of resorting to a model selection approach by choosing a particular candidate model for prediction, a forecasting approach based on Bayesian model averaging is used in order to account for model uncertainty and to improve forecasting accuracy. For model diagnosis a Bayesian sign test is introduced to compare the predictive accuracy of different forecasting models in terms of statistical significance. In an empirical application, using monthly unemployment rates of Germany, the performance of the model averaging prediction approach is compared to those of model selected Bayesian and classical (non)periodic time series models.     

Robust interpolation of random fields homogeneous in time and isotropic on a sphere,which are observed with noise

We study the problem of optimal linear estimation of the functional $$A_N \xi = \sum\limits_{k = 0}^{\rm N} {\int\limits_{S_n } {a(k,x)\xi (k,x)m_n (dx),} }$$ , which depends on unknown values of a random field ξ(k, x),k?Z,x?S _n homogeneous in time and isotropic on a sphereS _n, by observations of the field ξ(k,x)+η(k,x) with k? Z{0, 1, ...,N},x?S_n (here, η (k, x) is a random field uncorrelated with ξ(k, x), homogeneous in time, and isotropic on a sphere S_n). We obtain formulas for calculation of the mean square error and spectral characteristic of the optimal estimate of the functionalA _Nξ. The least favorable spectral densities and minimax (robust) spectral characteristics are found for optimal estimates of the functionalA _Nξ.     

The validity of least squares estimation in a time series model using the bootstrap methodology

The boostrap methodology may be used for estimating standard errors of the estimated parameters in a time series model. The idea is to approximate the theoretical error distribution by the residual distribution. The main objective of this article is to demonstrate the use of the bootstrap to attach standard errors to coefficient estimates in a second-order auto-regressive model fitted by least squares estimation. A comparison of the conventional and bootstrap methodology is made. A numerical result shows that the traditional least squares asymptotic formula for estimating standard errors appear to overestimate the true standard errors. But there are two problems in the simulation world of bootstrap for the autoregressive model of order two: (1) the first two observations y₁ and y₂ have been fixed, and (2) the residuals have not been inflated. After these two factors are considered in the trial and bootstrap experiment, both the conventional least squares and bootstrap estimates of the standard errors appear to be performing quite well.     

Estimation of a monotonous function which is observed in a white noise

Monotone functions on a segment are observed in the Gaussian white noise. The Maximum Likelihood Estimator of such a function is piecewise continious. A limit theorem on the convergence rate for such an estimator is proved. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 124–133.     

在线时间序列搜索的风险补偿模型

对于在线时间序列搜索问题,在假设对未来信息有一定的预期下,提出了在线时间序列搜索的风险补偿模型,进一步研究了模型的求解,给出了模型的一个最优策略,并通过数值计算讨论了最优策略的补偿函数随参数变化规律.数值实验结果表明,随着风险容忍度的增大与预期区间下限的增大,补偿函数均增大且趋于收敛;随着预期概率的增大与预期区间上限的减少,补偿函数分别增大.研究结果丰富了在线时间序列搜索的理论且具有实际应用价值.     

Estimation of Lyapunov spectrum and model selection for a chaotic time series

The estimation of the Lyapunov spectrum for a chaotic time series is discussed in this study. Three models: the local linear (LL) model; the local polynomial (LP) model and the global radial basis function (RBF) model, are compared for estimating the Lyapunov spectrum in this study. The number of neighbors for training the LL model and the LP model; the number of centers for building the RBF model, have been determined by the generalized degree of freedom for a chaotic time series. The above models have been applied to three artificial chaotic time series and two real-world time series, the numerical results show that the model-chosen LL model provides more accurate estimation than other models for clean data set while the RBF model behaves more robust to noise than other models for noisy data set.     

Choosing the optimal model parameters for Granger causality in application to time series with main timescale

The problem of determining the presence and direction of coupling between experimentally observed time series is of immediate interest in many relevant areas of knowledge. One of the approaches to its solution is the method of nonlinear Granger causality. The algorithm is based on the construction of predictive models and its effectiveness depends on the proper selection of model parameters.The most important of them for signals with a characteristic time scale fluctuations are the time lag used in the reconstruction of the state vector, and the range forecast. In this paper, we propose two criteria for evaluating performance of the method of nonlinear Granger causality, which allows one to select the lag and range forecast and achieves the best sensitivity and specificity. The sensitivity is determined by range of weakness the method can detect and specificity means the ability to avoid false positive results. Because of the proposed criteria on the example of several unidirectionally coupled reference systems were received practical advice on the selection of the following model parameters: lag and range forecast.     

A single-index model procedure for interpolation intervals in time series

In this paper we propose a procedure that uses a single-index model to construct interpolation intervals for a general class of linear processes. We present an extensive Monte Carlo experiment which studies the finite sample properties of this procedure. Finally, we illustrate the performance of the proposed method with a real data example.     

A time series model of CDS sequences in complete genome

A time series model of CDS sequences in complete genome is proposed. A map of DNA sequence to integer sequence is given. The correlation dimensions and Hurst exponents of CDS sequences in complete genome of bacteria are calculated. Using the average of correlation dimensions, some interesting results are obtained.     

Analytical approximation of the transition density in a local volatility model

We present a simplified approach to the analytical approximation of the transition density related to a general local volatility model. The methodology is sufficiently flexible to be extended to time-dependent coefficients, multi-dimensional stochastic volatility models, degenerate parabolic PDEs related to Asian options and also to include jumps.     

A central limit theorem for estimation in Gaussian stationary time series observed at unequally spaced times

The central limit theorem is proved for estimates of parameters which specify the covariance structure of a zero mean, stationary, Gaussian, discrete time series observed at unequally spaced times. The estimates considered are obtained by a single iteration from consistent estimates. The result also applies to the maximum likelihood estimate if it is consistent although consistency is not proved here. The essential condition on the sampling times is that the finite sample information matrix, when divided by the sample size, has a limit which is nonsingular and has finite norm. Some examples are presented to illustrate this condition.     

Pricing derivatives in a regime switching market with time inhomogenous volatility

This paper studies pricing derivatives in a componentwise semi-Markov (CSM) modulated market. We consider a financial market where the asset price dynamics follows a regime switching geometric Brownian motion model in which the coefficients depend on finitely many age-dependent semi-Markov processes. We further allow the volatility coefficient to depend on time explicitly. Under these market assumptions, we study locally risk minimizing pricing of a class of European options. It is shown that the price function can be obtained by solving a non-local Black–Scholes–Merton-type PDE. We establish existence and uniqueness of a classical solution to the Cauchy problem. We also find another characterization of price function via a system of Volterra integral equation of second kind. This alternative representation leads to computationally efficient methods for finding price and hedging. An explicit expression of quadratic residual risk is also obtained.     

Using the pointwise dimension to filter time series

This paper provides an insight into dimension analysis from time series. In particular, we propose a procedure based on the pointwise dimension in order to extract, for each embedding dimension, the subset of points in the phase space (and the corresponding ones in the time series) which give rise to the scaling behaviour. We may consider the output time series as the result of a filtering process, based on correlations of points in the phase-space domain. Furthermore, the procedure gives the statistics of points which determine the scaling behaviour.