On the transformation of raw time series data: A review

Effective time-series analysis is based on the assumption that the series under investigation is a realisation of a "stationary" stochastic process. In practice, such a stable series can generally only be obtained after some appropriate transformation of the raw data. Two types of non-stationarity can be removed by, respectively, linear and non-linear transformation. These are "homogeneous" non-stationarity and variance instability. The first can be dealt with by backshift operator methods, whilst the second is usually carried out by the approach of Box and Cox, though an easier way is given. The loss of optimal properties, on transforming back to the original situation, can be offset by suitably biasing the results.     

Non-stationary partition modeling of geostatistical data for malaria risk mapping

The most common assumption in geostatistical modeling of malaria is stationarity, that is spatial correlation is a function of the separation vector between locations. However, local factors (environmental or human-related activities) may influence geographical dependence in malaria transmission differently at different locations, introducing non-stationarity. Ignoring this characteristic in malaria spatial modeling may lead to inaccurate estimates of the standard errors for both the covariate effects and the predictions. In this paper, a model based on random Voronoi tessellation that takes into account non-stationarity was developed. In particular, the spatial domain was partitioned into sub-regions (tiles), a stationary spatial process was assumed within each tile and between-tile correlation was taken into account. The number and configuration of the sub-regions are treated as random parameters in the model and inference is made using reversible jump Markov chain Monte Carlo simulation. This methodology was applied to analyze malaria survey data from Mali and to produce a country-level smooth map of malaria risk.     

The Effect of Non-Stationarity on Extreme Sea-Level Estimation

The sea-level is the composition of astronomical tidal and meteorological surge processes. It exhibits temporal non-stationarity due to a combination of long-term trend in the mean level, the deterministic tidal component, surge seasonality and interactions between the tide and surge. We assess the effect of these non-stationarities on the estimation of the distribution of extreme sea-levels. This is important for coastal flood assessment as the traditional method of analysis assumes that, once the trend has been removed, extreme sea-levels are from a stationary sequence. We compare the traditional approach with a recently proposed alternative that incorporates the knowledge of the tidal component and its associated interactions, by applying them to 22 UK data sites and through a simulation study. Our main finding is that if the tidal non-stationarity is ignored then a substantial underestimation of extreme sea-levels results for most sites. In contrast, if surge seasonality and the tide–surge interaction are not modelled the traditional approach produces little additional bias. The alternative method is found to perform well but requires substantially more statistical modelling and better data quality.     

Violent crime and incentives in the long-run: evidence from England and Wales

This study uses recent advances in time-series econometrics to investigate the non-stationarity and co-integration properties of violent crime series in England and Wales. In particular, we estimate the long-run impact of economic conditions, beer consumption and various deterrents on different categories of recorded violent crime. The results suggest that a long-run causal model exists for only minor crimes of violence, with beer consumption being a predominant factor.     

Detection and correction of artificial shifts in climate series

Summary.  Many long instrumental climate records are available and might provide useful information in climate research. These series are usually affected by artificial shifts, due to changes in the conditions of measurement and various kinds of spurious data. A comparison with surrounding weather-stations by means of a suitable two-factor model allows us to check the reliability of the series. An adapted penalized log-likelihood procedure is used to detect an unknown number of breaks and outliers. An example concerning temperature series from France confirms that a systematic comparison of the series together is valuable and allows us to correct the data even when no reliable series can be taken as a reference.     

Subsampling Variance Estimation for Non-stationary Spatial Lattice Data

ABSTRACT.  Most proposed subsampling and resampling methods in the literature assume stationary data. In many empirical applications, however, the hypothesis of stationarity can easily be rejected. In this paper, we demonstrate that moment and variance estimators based on the subsampling methodology can also be employed for different types of non-stationarity data. Consistency of estimators are demonstrated under mild moment and mixing conditions. Rates of convergence are provided, giving guidance for the appropriate choice of subshape size. Results from a small simulation study on finite-sample properties are also reported.     

Divisive clustering of high dimensional data streams

Clustering streaming data is gaining importance as automatic data acquisition technologies are deployed in diverse applications. We propose a fully incremental projected divisive clustering method for high-dimensional data streams that is motivated by high density clustering. The method is capable of identifying clusters in arbitrary subspaces, estimating the number of clusters, and detecting changes in the data distribution which necessitate a revision of the model. The empirical evaluation of the proposed method on numerous real and simulated datasets shows that it is scalable in dimension and number of clusters, is robust to noisy and irrelevant features, and is capable of handling a variety of types of non-stationarity.     

Non-parametric Curve Estimation by Wavelet Thresholding with Locally Stationary Errors

An important aspect in the modelling of biological phenomena in living organisms, whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc., is time variation in the data. Thus, the recovery of a 'smooth' regression or trend function from noisy time-varying sampled data becomes a problem of particular interest. Here we use non-linear wavelet thresholding to estimate a regression or a trend function in the presence of additive noise which, in contrast to most existing models, does not need to be stationary. (Here, non-stationarity means that the spectral behaviour of the noise is allowed to change slowly over time). We develop a procedure to adapt existing threshold rules to such situations, e.g. that of a time-varying variance in the errors. Moreover, in the model of curve estimation for functions belonging to a Besov class with locally stationary errors, we derive a near-optimal rate for the -risk between the unknown function and our soft or hard threshold estimator, which holds in the general case of an error distribution with bounded cumulants. In the case of Gaussian errors, a lower bound on the asymptotic minimax rate in the wavelet coefficient domain is also obtained. Also it is argued that a stronger adaptivity result is possible by the use of a particular location and level dependent threshold obtained by minimizing Stein's unbiased estimate of the risk. In this respect, our work generalizes previous results, which cover the situation of correlated, but stationary errors. A natural application of our approach is the estimation of the trend function of non-stationary time series under the model of local stationarity. The method is illustrated on both an interesting simulated example and a biostatistical data-set, measurements of sheep luteinizing hormone, which exhibits a clear non-stationarity in its variance.     

Long-term demographic interactions in precensus England

"Modern time series methods are applied to the analysis of annual demographic data for England, 1541-1800. Evidence is found of non-stationarity in the series and of co-integration among the series. Building on economic models of historical demography, optimal inferential procedures are implemented to estimate the structural parameters of long-term equilibria among the variables. Evidence is found for a small, but significant, Malthusian 'preventive check' as well as interactions between fertility, mortality and nuptiality that are consistent with the predictions often made in demographic studies. Tentative experiments to detect the influence of environmental factors fail to reveal any significant impact on the estimates obtained."     

Estimation of Time-Varying Long Memory Parameter Using Wavelet Method

Stationary long memory processes have been extensively studied over the past decades. When we deal with financial, economic, or environmental data, seasonality and time-varying long-range dependence can often be observed and thus some kind of non-stationarity exists. To take into account this phenomenon, we propose a new class of stochastic processes: locally stationary k-factor Gegenbauer process. We present a procedure to estimate consistently the time-varying parameters by applying discrete wavelet packet transform. The robustness of the algorithm is investigated through a simulation study. And we apply our methods on Nikkei Stock Average 225 (NSA 225) index series.     

Global and local tests to assess stationarity of Markov transition models

We present global and local likelihood-based tests to evaluate stationarity in transition models. Three motivational studies are considered. A simulation study was carried out to assess the performance of the proposed tests. The results showed that they present good performance with the control of the type-I error, especially for ordinal responses, and control of the type-II error, especially for the nominal case. Asymptotically they are close to the classical test performance. They can be executed in a single framework without the need to estimate the transition probabilities, incorporating both categorical and continuous covariates, and used to identify sources of non-stationarity.     

State‐space model for proxy‐based millennial reconstruction

It is important to study historical temperature time series prior to the industrial revolution so that one can view the current global warming trend from a long‐term historical perspective. Because there are no instrumental records of such historical temperature data, climatologists have been interested in reconstructing historical temperatures using various proxy time series. In this paper, the authors examine a state‐space model approach for historical temperature reconstruction which not only makes use of the proxy data but also information on external forcings. A challenge in the implementation of this approach is the estimation of the parameters in the state‐space model. The authors developed two maximum likelihood methods for parameter estimation and studied the efficiency and asymptotic properties of the associated estimators through a combination of theoretical and numerical investigations. The Canadian Journal of Statistics 38: 488–505; 2010 © 2010 Crown in the right of Canada     

中国区域经济增长路径的差异性研究——基于ACF检验的新证据

运用考虑结构变化的自相关函数（ACF）方法,利用子样本抽样和稳态自助法抽样,依据相对最优的原理构建了ACF的最优置信区间,使用1953—2008年中国省市自治区数据,对中国区域经济增长路径的实际特征进行了经验分析。研究的结论表明,中国区域经济增长的转移动态路径具有显著的差异性,区域经济增长均呈现出非线性振荡的转移特征,对数线性化的新古典索罗增长模型并不能刻画出中国区域经济转移动态特征。     

Estimation and prediction of time-varying GARCH models through a state-space representation: a computational approach

We propose a state-space approach for GARCH models with time-varying parameters able to deal with non-stationarity that is usually observed in a wide variety of time series. The parameters of the non-stationary model are allowed to vary smoothly over time through non-negative deterministic functions. We implement the estimation of the time-varying parameters in the time domain through Kalman filter recursive equations, finding a state-space representation of a class of time-varying GARCH models. We provide prediction intervals for time-varying GARCH models and, additionally, we propose a simple methodology for handling missing values. Finally, the proposed methodology is applied to the Chilean Stock Market (IPSA) and to the American Standard&Poor's 500 index (S&P500).     

Structure identification and variable selection in geographically weighted regression models

Geographically weighted regression (GWR) is an important tool for exploring spatial non-stationarity of a regression relationship, in which whether a regression coefficient really varies over space is especially important in drawing valid conclusions on the spatial variation characteristics of the regression relationship. This paper proposes a so-called GWGlasso method for structure identification and variable selection in GWR models. This method penalizes the loss function of the local-linear estimation of the GWR model by the coefficients and their partial derivatives in the way of the adaptive group lasso and can simultaneously identify spatially varying coefficients, nonzero constant coefficients and zero coefficients. Simulation experiments are further conducted to assess the performance of the proposed method and the Dublin voter turnout data set is analysed to demonstrate its application.     

A Non-stationary Cox Model

The purpose of this paper is to consider the problem of statistical inference about a hazard rate function that is specified as the product of a parametric regression part and a non-parametric baseline hazard. Unlike Cox's proportional hazard model, the baseline hazard not only depends on the duration variable, but also on the starting date of the phenomenon of interest. We propose a new estimator of the regression parameter which allows for non-stationarity in the hazard rate. We show that it is asymptotically normal at root- n and that its asymptotic variance attains the information bound for estimation of the regression coefficient. We also consider an estimator of the integrated baseline hazard, and determine its asymptotic properties. The finite sample performance of our estimators are studied.     

Nonparametric panel stationarity testing with an application to crude oil production

A nonparametric panel stationarity test is proposed which offers the advantage of not requiring prior specification of the trend function for each of the series in the panel. A bootstrap implementation of the test is outlined and its finite sample performance is analyzed via Monte Carlo simulations. An application is also included where the proposed test is used to analyze the stochastic properties of monthly crude oil production for a panel of 20 -both OPEC and non-OPEC- countries from 1973 to 2015. Our analysis detects strong evidence of non-stationarity, both globally and group-wise. Results have implications for the effectiveness of government intervention and stabilization policies.KEYWORDS: Stationarity testing, panel, nonparametric, bootstrap, oil production     

Statistics for Extreme Sea Currents

Estimates of various characteristics of extreme sea currents, such as speeds and their directions, are required when designing offshore structures. This paper extends standard statistical methods for extreme values to handle the directionality, temporal dependence and tidal non-stationarity that are present in sea current extremes. The methods are applied to a short period of data from the Inner Dowsing Light Tower in the North Sea. Substantial benefits, over existing methods, are obtained from our analysis of the sea current by decomposing it into tide and surge currents. In particular, we find that at the Inner Dowsing the strong directionality in extreme sea current speeds is completely explained by the tidal current and directionality in the non-extreme surge currents. This finding aids model fitting and extrapolation.     

Structural changes and unit roots in non-stationary time series

The problem of testing hypotheses of a unit root and a structural change in one-dimensional time series is considered. A non-parametric two-step method for solution of the problem is proposed. The method is based upon the modified Kolmogorov-Smirnov statistic. At the first step of this method the hypothesis of stationarity of an obtained sample is tested against a unified alternative of a statistical non-stationarity of a time series (a unit root or a structural change). At the second step of the proposed method, in case of rejecting the stationarity hypothesis at the first step, the hypothesis of an unknown structural change is tested against the alternative of a unit root. We prove that probabilities of errors (false classification of hypotheses) of the proposed method converge to zero as the sample size tends to infinity.     

The analysis of extreme pollution levels: A case study

Our case study focuses on Milan. Italian law specifies strict guidelines for the permissibility of high levels of a variety of air pollutants in cities. In Milan, a highly sophisticated network of recording stations has been constructed to monitor pollutant levels. The aim of this paper is to obtain a summary of the temporal behaviour of the pollutant series, with particular reference to extreme levels. Simple exploratory analysis reveals a number of sources of stochastic variation and possible dependence on covariate effects, which are subsequently modelled, exploiting recent developments in the modelling and inference for temporal extremes. Using this methodology, we examine the issues of data trends, non-stationarity, meteorological effects and temporal dependence, all of which have substantive implications in the design of pollution control regulations. Moreover, the asymptotic basis of these extreme value models justifies the interpretation of our results, even at levels that are exceptionally high.