On the asymptotic behavior of the eigenvalue distribution of block correlation matrices of high-dimensional time series

We consider linear spectral statistics built from the block-normalized correlation matrix of a set of [Formula: see text] mutually independent scalar time series. This matrix is composed of [Formula: see text] blocks. Each block has size [Formula: see text] and contains the sample cross-correlation measured at [Formula: see text] consecutive time lags between each pair of time series. Let [Formula: see text] denote the total number of consecutively observed windows that are used to estimate these correlation matrices. We analyze the asymptotic regime where [Formula: see text] while [Formula: see text], [Formula: see text]. We study the behavior of linear statistics of the eigenvalues of this block correlation matrix under these asymptotic conditions and show that the empirical eigenvalue distribution converges to a Marcenko–Pastur distribution. Our results are potentially useful in order to address the problem of testing whether a large number of time series are uncorrelated or not.

A Recursive Characteristics Analysis-Based Stationarity Evaluation Model for Friction-Induced Attractors in the Sliding Friction Process

In this study, the coefficient of friction (COF) signals throughout the running-in process were examined by sliding a ring against a static disc. By reconstructing the scalar time-series into multi-dimensional phase spaces, friction-induced attractors were obtained and quantified by recursive characteristics analysis, which can effectively realize the running-in status identification. Moreover, a recursive characteristics analysis-based evaluation model was established to investigate the stationarity of the friction-induced attractors based on the recurrence quantification analysis (RQA) measures. The analyses of the numerically simulated signals and experimental results indicate that the extracted model is an intuitive and effective method. Furthermore, to improve the stationary of the friction-induced attractors, the normal pressure should be low, whereas the relative sliding velocities should be increased appropriately. These results would contribute to the revelation of the recursive characteristics of the tribosystem and the improvement of the stationarity of friction-induced attractors.

Model forecast error correction based on the Local Dynamical Analog method: an example application to the ENSO forecast

<p>Numerical forecasts always have associated errors. Analogue correction methods combine numerical simulations with statistical analyses to reduce model forecast errors. However, identifying appropriate analogues remains a challenging task. Here, we use the Local Dynamical Analog (LDA) method to locate analogues and correct model forecast errors. As an example, an ENSO model forecast error correction experiment confirms that the LDA method locates more dynamical analogues of states of interest and better corrects forecast errors than do other methods. This is because the LDA method ensures similarity of the initial states and the evolution of both states. In addition, the LDA method can be applied using a scalar time series, which reduces the complexity of the dynamical system. Model forecast error correction using the LDA method provides a new approach to correcting state-dependent model errors and can be readily integrated with other advanced models.</p>

Stability of exoplanetary systems retrieved from scalar time series

We propose a novel method applied to extrasolar planetary dynamics to describe the system stability. The observations in this field serve the measurements mainly of radial velocity, transit time, and/or celestial position. These scalar time series are used to build up the high-dimensional phase space trajectory representing the dynamical evolution of planetary motion. The framework of nonlinear time series analysis and Poincaré recurrences allows us to transform the obtained univariate signals into complex networks whose topology carries the dynamical properties of the underlying system. The network-based analysis is able to distinguish the regular and chaotic behaviour not only for synthetic inputs but also for noisy and irregularly sampled real world observations. The proposed scheme does not require neither n-body integration nor best fitting planetary model to perform the stability investigation, therefore, the computation time can be reduced drastically compared to those of the standard numerical methods.

Machine Learning of Time Series Using Time-Delay Embedding and Precision Annealing

Tasking machine learning to predict segments of a time series requires estimating the parameters of a ML model with input/output pairs from the time series. We borrow two techniques used in statistical data assimilation in order to accomplish this task: time-delay embedding to prepare our input data and precision annealing as a training method. The precision annealing approach identifies the global minimum of the action ([Formula: see text]). In this way, we are able to identify the number of training pairs required to produce good generalizations (predictions) for the time series. We proceed from a scalar time series [Formula: see text] and, using methods of nonlinear time series analysis, show how to produce a [Formula: see text]-dimensional time-delay embedding space in which the time series has no false neighbors as does the observed [Formula: see text] time series. In that [Formula: see text]-dimensional space, we explore the use of feedforward multilayer perceptrons as network models operating on [Formula: see text]-dimensional input and producing [Formula: see text]-dimensional outputs.

Multivariate Graph-Based Analysis on Coefficient of Friction Signal During the Friction Process

To describe the dynamic evolutionary law and tribological behavior of the tribopair AISI 52100-AISI 1045, rotational experiments were conducted by sliding a disk against a static pin. The multidimensional phase spaces were reconstructed based on the scalar time-series by the time-delay embedding technique, and the multivariate graph-based method was used to visualize the overall picture of the phase space. The evolution of radar plots and the corresponding multivariate graph centrobaric trajectory (MGCT) is consistent with the description of “running-in, steady-state and increasing friction stages,” and can serve as effective indicators for the friction state transitions. Results show that the radar plot can inform quantitative interpretations of friction process identification. Therefore, the multivariate graph-based method is a useful approach to characterize the nonlinear dynamics of tribological behaviors.

Modeling recovery curves with application to prostatectomy

Summary In many clinical settings, a patient outcome takes the form of a scalar time series with a recovery curve shape, which is characterized by a sharp drop due to a disruptive event (e.g., surgery) and subsequent monotonic smooth rise towards an asymptotic level not exceeding the pre-event value. We propose a Bayesian model that predicts recovery curves based on information available before the disruptive event. A recovery curve of interest is the quantified sexual function of prostate cancer patients after prostatectomy surgery. We illustrate the utility of our model as a pre-treatment medical decision aid, producing personalized predictions that are both interpretable and accurate. We uncover covariate relationships that agree with and supplement that in existing medical literature.