A financial time series, such as a stock market index, foreign exchange rate, or a commodity price, fluctuates heavily and shows scaling behaviors. Scaling and multi-scaling behaviors are measured for a nonstationary time series, such as stock market indices, high-frequency stock prices of individual stocks, or the volatility time series of a stock index. We review the fractality, multi-scaling, and multifractality of the financial time series of a stock market. We introduce a detrended fluctuation analysis of the financial time series to extract fluctuation patterns. Multifractality is measured using various methods, such as generalized Hurst exponents, the generalized partition function method, a detrended fluctuation analysis, the detrended moving average method, and a wavelet transformation.

中文翻译：

---

股票市场非平稳金融时间序列中的分形和多重分形

金融时间序列，例如股票市场指数、外汇汇率或商品价格，波动很大并表现出缩放行为。标度和多标度行为用于衡量非平稳时间序列，例如股票市场指数、个股的高频股价或股票指数的波动时间序列。我们回顾了股票市场金融时间序列的分形、多重尺度和多重分形。我们引入了金融时间序列的去趋势波动分析来提取波动模式。多重分形是使用各种方法测量的，例如广义 Hurst 指数、广义分区函数方法、去趋势波动分析、去趋势移动平均法和小波变换。