Error Estimation and Adaptivity for Differential Equations with Multiple Scales in Time

We consider systems of ordinary differential equations with multiple scales in time. In general, we are interested in the long time horizon of a slow variable that is coupled to solution components that act on a fast scale. Although the fast scale variables are essential for the dynamics of the coupled problem, they are often of no interest in themselves. Recently, we have proposed a temporal multiscale approach that fits into the framework of the heterogeneous multiscale method and that allows for efficient simulations with significant speedups. Fast and slow scales are decoupled by introducing local averages and by replacing fast scale contributions by localized periodic-in-time problems. Here, we generalize this multiscale approach to a larger class of problems, but in particular, we derive an a posteriori error estimator based on the dual weighted residual method that allows for a splitting of the error into averaging error, error on the slow scale and error on the fast scale. We demonstrate the accuracy of the error estimator and also its use for adaptive control of a numerical multiscale scheme.

Transitional SAX Representation for Knowledge Discovery for Time Series

Numerous dimensionality-reducing representations of time series have been proposed in data mining and have proved to be useful, especially in handling a high volume of time series data. Among them, widely used symbolic representations such as symbolic aggregate approximation and piecewise aggregate approximation focus on information of local averages of time series. To compensate for such methods, several attempts were made to include trend information. However, the included trend information is quite simple, leading to great information loss. Such information is hardly extendable, so adjusting the level of simplicity to a higher complexity is difficult. In this paper, we propose a new symbolic representation method called transitional symbolic aggregate approximation that incorporates transitional information into symbolic aggregate approximations. We show that the proposed method, satisfying a lower bound of the Euclidean distance, is able to preserve meaningful information, including dynamic trend transitions in segmented time series, while still reducing dimensionality. We also show that this method is advantageous from theoretical aspects of interpretability, and practical and superior in terms of time-series classification tasks when compared with existing symbolic representation methods.

Anomaly analysis on an open DNS dataset

The increasing availability of open data and the demand to understand better the nature of anomalies and the causes underlying them in modern systems is encouraging researchers to analyse open datasets in various ways. These include both quantitative and qualitative methods. We show here how quantitative methods, such as timeline, local averages and exponentially weighted moving average analyses, led in this work to the discovery of three anomalies in a large open DNS dataset published by the Los Alamos National Laboratory.

Anomaly analysis on an open DNS dataset

The increasing availability of open data and the demand to understand better the nature of anomalies and the causes underlying them in modern systems is encouraging researchers to analyse open datasets in various ways. These include both quantitative and qualitative methods. We show here how quantitative methods, such as timeline, local averages and exponentially weighted moving average analyses, led in this work to the discovery of three anomalies in a large open DNS dataset published by the Los Alamos National Laboratory.