检测数据集中的非线性。傅里叶相位图的加权标度指数表征
Detecting non-linearities in data sets. Characterization of Fourier phase maps using the Weighted Scaling Indices
摘要：
我们提出了一种方法，用于检测数据集的非线性特征的傅立叶相位图的结构特征的表征的基础上。傅立叶相位图是点$M= \{(ϕ_{\vec{k}}, ϕ_{\vec{k} + \vecΔ})\}$的2D集合，其中$ ϕ_{\vec{k}}$是数据集的傅立叶变换的$k$模式的相位，并且$\vecΔ$是相移。使用加权缩放指数的频谱来分析由此在该空间上呈现的信息，以检测任何尺度$\vecΔ$下的相位耦合。我们提出了一个统计检验的显着性的基础上，从原始数据和代理实现创建的相位图的属性的比较。我们已经将我们的方法应用到洛伦兹系统和道琼斯指数的对数股票收益率。高维数据的应用是直接的。结果表明，无论是洛伦兹系统和道琼斯时间序列表现出显着的非线性行为的签名。
英文摘要：
We present a methodology for detecting non-linearities in data sets based on the characterization of the structural features of the Fourier phase maps. A Fourier phase map is a 2D set of points $M= \{(ϕ_{\vec{k}}, ϕ_{\vec{k} + \vecΔ})\}$, where $ ϕ_{\vec{k}}$ is the phase of the $k$-mode of the Fourier transform of the data set and $\vecΔ$ a phase shift. The information thus rendered on this space is analyzed using the spectrum of weighted scaling indices to detect phase coupling at any scale $\vecΔ$. We propose a statistical test of significance based on the comparison of the properties of phase maps created from both the original data and surrogate realizations. We have applied our method to the Lorenz system and the logarithmic stock returns of the Dow Jones index. Applications to higher dimensional data are straightforward. The results indicate that both the Lorenz system and the Dow Jones time series exhibit significant signatures of non-linear behavior.
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