标题： 在噪声存在下分析标量图脊 显示英文标题

标题： Analyzing Scalogram Ridges in the Presence of Noise

摘要： 虽然由解析小波变换（AWT）的平方模确定的小波标度图中的脊线在时间序列分析和时频分析中得到了广泛应用，但它们在噪声环境中的行为仍未被充分探索。 本文填补了这一空白。 我们将脊线定义为沿尺度轴由标度图局部极大值点形成的路径，并分析信号受到平稳高斯噪声污染时其性质。 除了为随机过程的AWT建立若干关键性质外，我们还研究了标度图中由此产生的随机脊点的概率特性。 具体而言，我们证明了单个时间点处脊点的独特性，并证明了脊点集合的上半连续性。 此外，我们推导了噪声信号与干净信号的脊点偏差超过指定阈值的概率界。 这些界以与信噪比相关的常数形式表达，并在信号满足自适应谐波模型且受到平稳高斯过程污染的情况下适用。 为了实现这些结果，我们推导了非平稳高斯过程复模的最大不等式，利用了Borell-TIS不等式和Dudley定理等经典工具，这可能具有独立的兴趣。 显示英文摘要

摘要： While ridges in the scalogram, determined by the squared modulus of analytic wavelet transform (AWT), have been widely applied in time series analysis and time-frequency analysis, their behavior in noisy environments remains underexplored. We fill in this gap in this paper. We define ridges as paths formed by local maximizers of the scalogram along the scale axis, and analyze their properties when the signal is contaminated by stationary Gaussian noise. In addition to establishing several key properties of the AWT for random processes, we investigate the probabilistic characteristics of the resulting random ridge points in the scalogram. Specifically, we establish the uniqueness property of the ridge point at individual time instances and prove the upper hemicontinuity of the set of ridge points. Furthermore, we derive bounds on the probability that the deviation between the ridge points of noisy and clean signals exceeds a specified threshold. These bounds are expressed in terms of constants related to the signal-to-noise ratio and apply when the signal satisfies the adaptive harmonic model and is corrupted by a stationary Gaussian process. To achieve these results, we derive maximal inequalities for the complex modulus of nonstationary Gaussian processes, leveraging classical tools such as the Borell-TIS inequality and Dudley's theorem, which might be of independent interest.