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The story behind the curve: How functional data reveals patterns we can't see?
Functional Data Analysis (FDA) is a branch of statistics that analyzes curves, surfaces, or other continuously changing information. Under the FDA framework, each functional data sample is treated as a random function. These random functions are usually defined on the physical continuum of time, but can also be on other continuums such as spatial location, wavelength, or probability. Since functional data are infinite-dimensional in nature, their high-dimensional structure provides us with a rich source of information, which also poses many research and data analysis challenges.
Functional data has infinite dimensions, which not only outlines rich data structures, but also adds complexity to data understanding.
History of functional data analysis
The roots of functional data analysis can be traced to the work of Grenander and Karhunen in the 1940s and 1950s. They considered the eigendecomposition of square-integrable continuous-time stochastic processes, a method now known as Karhunen-Loève decomposition. In the 1970s, Kleffe, Dauxois and Pousse conducted a rigorous theoretical analysis of functional principal component analysis and obtained some results on the asymptotic distribution of eigenvalues. In the 1990s and 2000s, the focus of the field gradually turned to applications, especially the effects of dense and sparse observation schemes. James O. Ramsay first proposed the term "functional data analysis".
Mathematical Formalism
Random functions can be viewed as random elements taking values in a Hilbert space, or as a stochastic process. The two methods complement each other, but the Hilbert space perspective is mathematically more convenient, while stochastic processes are more suitable for practical applications. The mean square continuity condition also requires the random function to be stable to a certain extent.
Random functions provide a variety of perspectives in different applications, making data analysis methods more abundant.
Functional data design
Functional data are viewed as realizations of stochastic processes. For each subject, sample observations can be measured on a dense grid, which is mathematically convenient but difficult to achieve in reality. Taking the Berkeley Growth Study as an example, these data demonstrate functional data analysis under a dense design. In practical applications, samples may also be sparse and may be affected by measurement noise.
Functional principal component analysis
Functional principal component analysis (FPCA) is the most commonly used tool in the FDA, in part because FPCA can reduce the dimensionality of essentially infinite-dimensional functional data to fractions of finite-dimensional random vectors. This method introduces the concept of principal component analysis into functional data by unfolding random trajectories on a functional basis and is able to capture the main sources of variation among random variables.
With the growth of data, functional data analysis is increasingly used in various fields, such as biomedical data, environmental monitoring data, etc. In these areas, the underlying patterns in functional data not only help us understand the data themselves, but also help us make better predictions.
However, as technology develops and analysis models increase in complexity, the challenges of functional data analysis continue to evolve. Does this mean that functional data will open up new horizons and possibilities in future research?
