1. Background

The background of this course is calculus, basic complex analysis, andprobability.

After completed course, the students are expected to be ableto:

• understand how to quantify the pattern ofsignals,
• evaluate the impact of deformation on the extracted features,and
• characterize random processes by the scatteringtransform.

2. Outline

This course will introduce the wavelet-based scattering transform. In 2012, Stephane Mallat proposed this transformation, which is computed through a cascade of wavelet transforms and modulus nonlinearities. Its structure enables us to retrieve the lost information caused by the poolingoperatorandprovidesuswithaninterpretablefeatureextractionmethod.Weplantostudy itsinvarianceproperties,includingthetranslationinvarianceandthedeformationstability.Next, we will introduce some stochastic processes, including the Poisson processes, the fractional Brownian motions, and the stochastic self-similar processes. The scattering transform of these stochastic processes have different statistical properties, which provide us a measurement of the appearance of multiscale activities among random processes. Finally, we will introduce its application in the sleep stage classification. The progress of this course is listed asfollows.

The 1st week: Effects of time warping on the Fourier coefficients

The 2nd week: Relationship between the Fourier transform and the wavelet transform The 3rd week: Structure of the scattering transform (ST)

The 4th week: Translation invariance of ST The 5-6th week: Deformation stability of ST The 7th week: Poisson processes

The8thweek:Statisticalpropertyofthesecond-orderSTofthePoisson processes The 9th week: Fractional Brownian motions(FBM)

The 10th week: Statistical property of the second-order ST of the FBMs The 11th week: Stochastic self-similar processes (SSP)

The 12th week: Statistical property of the second-order ST of the SSPs

The 13th week: Application of the ST on the feature extraction of physiological signals The 14th week: Dimension reduction methods (Diffusion maps)

The15thweek:Automaticsleepscoringsystem basedontheSTandthediffusion maps The 16th week:Discussion