Order-preserving pattern (OPP) mining is a type of sequential pattern mining method in which a group of ranks of time series is used to represent an OPP. This approach can discover frequent trends in time series. Existing OPP mining algorithms consider data points at different time to be equally important; however, newer data usually have a more significant impact, while older data have a weaker impact. We therefore introduce the forgetting mechanism into OPP mining to reduce the importance of older data. This paper explores the mining of OPPs with forgetting mechanism (OPF) and proposes an algorithm called OPF-Miner that can discover frequent OPFs. OPF-Miner performs two tasks, candidate pattern generation and support calculation. In candidate pattern generation, OPF-Miner employs a maximal support priority strategy and a group pattern fusion strategy to avoid redundant pattern fusions. For support calculation, we propose an algorithm called support calculation with forgetting mechanism, which uses prefix and suffix pattern pruning strategies to avoid redundant support calculations. The experiments are conducted on nine datasets and 12 alternative algorithms. The results verify that OPF-Miner is superior to other competitive algorithms. More importantly, OPF-Miner yields good clustering performance for time series, since the forgetting mechanism is employed.

Wave equation is a commonly used tool for describing various kinds of wave phenomena in nature such as sound wave, water wave, electromagnetic wave and string vibration. It is a second-order partial differential equation and describe the wave propagation without noises. However, real world is filled with noises everywhere. So deterministic wave equation is not enough to model some problems with additive noises. As a remedy method, stochastic wave equation driven by Wiener process was presented where the noise is considered random and modeled by using Wiener process. Except for randomness, uncertainty associated belief degrees is another different type of indeterministic phenomenon. For modeling the wave phenomena with uncertain noises, this paper aims at deriving an uncertain wave equation driven by Liu process, which is a type of partial differential equation. Here, Liu process is a Lipschitz continuous uncertain process with stationary and independent increments. Then, we prove the existence and uniqueness of the solution of an uncertain wave equation. Additionally, we give the inverse uncertainty distribution of a solution of an uncertain wave equation.

Entropy is commonly used as a way to measure the uncertainty of random variables. In uncertain set theory, a concept of entropy for uncertain sets has been defined by using logarithm. However, such an entropy fails to measure the uncertain degree of some uncertain sets. This paper aims at proposing a concept of elliptic entropy for uncertain sets and investigating its properties such as translation invariance and positive linearity. It also provides some formulas for calculating the elliptic entropy via inverse membership functions. Additionally, elliptic relative entropy for uncertain sets is presented as a measure of the difference between two membership functions, and some applications are considered in portfolio selection and clustering.

A random variable is a measurable function from an uncertainty space to the set of real numbers, which is used to model randomness. An uncertain variable is a measurable function from uncertainty space to the set of real numbers, which is used to describe uncertainty. However, randomness and uncertainty often simultaneously appear in a complex system. Uncertain random variable provides a useful tool to handle such a hybrid case. This concept integrates random variable and uncertain variable into a broader view. For uncertain random variables, a basic and important topic is to discuss the convergence of its sequence. Specifically, this paper focuses on studying the convergence in distribution for a sequence of uncertain random variables without a common chance distribution.