This article develops a statistical test for the null hypothesis of strict stationarity of a discrete time stochastic process. When the null hypothesis is true, the second order cumulant spectrum is zero at all the discrete Fourier frequency pairs present in the principal domain of the cumulant spectrum. The test uses a frame (window) averaged sample estimate of the second order cumulant spectrum to build a test statistic that has an asymptotic complex standard normal distribution. We derive the test statistic, study the size and power properties of the test, and demonstrate its implementation with intraday stock market return data. The test has conservative size properties and good power to detect varying variance and unit root in the presence of varying variance.
The PHR second order cumulant spectrum based test of strict stationarity is different from widely used tests for structural change of a linear time series model. For example, @cite_15 proposed a control chart to detect small shifts in the mean of a stochastic time series modeled as a white noise process plus a mean parameter that can be constant or shifting. Their proposed chart draws no conclusions on the stationarity of the process.
@cite_15
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We introduce a test for strict stationarity based on the fluctuations of the quantiles of the data, and we show that this test has power against the alternative hypothesis of unconditional heteroskedasticity while other tests for first order (level) stationarity as the KPSS test (, 1992) and, its robust version, the IKPSS test (de , 2007) have low power against this alternative of time-varying variance. Moreover, our test has power against the alternative hypothesis of time-varying kurtosis, while the test for second order (covariance) stationarity introduced by Xiao and Lima (2007) has power close to size against this alternative. Stationarity is a cornerstone property that facilitates the analysis and processing of random signals in the time domain. Although time-varying signals are abundant in nature, in many practical scenarios, the information of interest resides in more irregular graph domains. This lack of regularity hampers the generalization of the classical notion of stationarity to graph signals. This paper proposes a definition of weak stationarity for random graph signals that takes into account the structure of the graph where the random process takes place, while inheriting many of the meaningful properties of the classical time domain definition. Provided that the topology of the graph can be described by a normal matrix, stationary graph processes can be modeled as the output of a linear graph filter applied to a white input. This is shown equivalent to requiring the correlation matrix to be diagonalized by the graph Fourier transform; a fact that is leveraged to define a notion of power spectral density (PSD). Properties of the graph PSD are analyzed and a number of methods for its estimation are proposed. This includes generalizations of nonparametric approaches such as periodograms, window-based average periodograms, and filter banks, as well as parametric approaches, using moving-average, autoregressive, and ARMA processes. Graph stationarity and graph PSD estimation are investigated numerically for synthetic and real-world graph signals. A recently introduced latent feature learning technique for time-varying dynamic phenomena analysis is the so-called slow feature analysis (SFA). SFA is a deterministic component analysis technique for multidimensional sequences that, by minimizing the variance of the first-order time derivative approximation of the latent variables, finds uncorrelated projections that extract slowly varying features ordered by their temporal consistency and constancy. In this paper, we propose a number of extensions in both the deterministic and the probabilistic SFA optimization frameworks. In particular, we derive a novel deterministic SFA algorithm that is able to identify linear projections that extract the common slowest varying features of two or more sequences. In addition, we propose an expectation maximization (EM) algorithm to perform inference in a probabilistic formulation of SFA and similarly extend it in order to handle two and more time-varying data sequences. Moreover, we demonstrate that the probabilistic SFA (EM-SFA) algorithm that discovers the common slowest varying latent space of multiple sequences can be combined with dynamic time warping techniques for robust sequence time-alignment. The proposed SFA algorithms were applied for facial behavior analysis, demonstrating their usefulness and appropriateness for this task. Abstract This paper proposes a stochastic and dynamic mixed-integer linear program (SD-MILP) for optimal coordinated bidding of a risk-averse profit-maximizing hydropower producer. The day-ahead, intra-day, and real-time markets are considered. To model and predict day-ahead, intra-day, and real-time prices, the Holt–Winter (HW) and the Generalized Autoregressive Conditional Heteroscedastic (GARCH) predictors are combined using a proposed Markov switch. The discrete behavior of intra-day and real-time prices is modeled as different Markov states. The proposed Markov-based HW–GARCH model with a standard scenario generation-reduction technique is used to capture the uncertainty in day-ahead, intra-day, and real-time prices. The time-dependent conditional value at risk (T-CVaR) is proposed to model the risk of trading in different considered markets. The convex combination of the expected profit and T-CVaR is used as the objective of SD-MILP. The Markov-based HW–GARCH is modeled in Matlab and the SD-MILP is coded in GAMS. The Markov-based HW–GARCH predictor and the SD-MILP are used to develop the bidding curve of a three-reservoir hydropower producer using the electricity prices from the Nordic power market. To further examine the developed models a seven-reservoir hydropower producer is also studied. For these two cases, the coordinated bidding curves are derived and discussed.