How to extract trend from highly nonlinear and nonstationary data is an important problem that has many practical applications ranging from bio-medical signal analysis to econometrics, finance, and geophysical fluid dynamics. We review some exisiting methodologies in defining trend and instantaneous frequency in data analysis. Many of these methods use pre-determined basis and is not completely adaptive. They tend to introduce artificial harmonics in the decomposion of the data. Various attempts to preserve the temportal locality property of the data introduce problems of their own. Here we discuss how adaptive data analysis can be formulated as a nonlinear optimization problem in which we look for a sparse representation of data in some unknown basis which is derived from the physical data. We will show that this formulation has some beautiful mathematical structure and can be considered as a nonlinear version of compressed sensing.