Functional data analysis (FDA) is a branch of statistics that analyses data providing information about curves, surfaces or anything else varying over a continuum. In its most general form, under an FDA framework, each sample element of functional data is considered to be a random function. The physical continuum over which these functions are defined is often time, but may also be spatial location, wavelength, probability, etc. Intrinsically, functional data are infinite dimensional. The high intrinsic dimensionality of these data brings challenges for theory as well as computation, where these challenges vary with how the functional data were sampled. However, the high or infinite dimensional structure of the data is a rich source of information and there are many interesting challenges for research and data analysis.
History Functional data analysis has roots going back to work by Grenander and Karhunen in the 1940s and 1950s. They considered the decomposition of square-integrable continuous time stochastic process into eigencomponents, now known as the Karhunen-Loève decomposition. A rigorous analysis of functional principal components analysis was done in the 1970s by Kleffe, Dauxois and Pousse including results about the asymptotic distribution of the eigenvalues. More recently in the 1990s and 2000s the field has focused more on applications and understanding the effects of dense and sparse observations schemes. The term "Functional Data Analysis" was coined by James O. Ramsay.
Mathematical formalism Random functions can be viewed as random elements taking values in a Hilbert space, or as a stochastic process. The former is mathematically convenient, whereas the latter is somewhat more suitable from an applied perspective. These two approaches coincide if the random functions are continuous and a condition called mean-squared continuity is satisfied.
Hilbertian random variables In the Hilbert space viewpoint, one considers an H {\displaystyle H} -valued random element X {\displaystyle X} , where H {\displaystyle H} is a separable Hilbert space such as the space of square-integrable functions L 2 [ 0 , 1 ] {\displaystyle L^{2}[0,1]} . Under the integrability condition that E ‖ X ‖ L 2 2 = E ( ∫ 0 1 | X ( t ) | 2 d t ) < ∞ {\displaystyle \mathbb {E} \|X\|_{L^{2}}^{2}=\mathbb {E} (\int _{0}^{1}|X(t)|^{2}dt)<\infty } , one can define the mean of X {\displaystyle X} as the unique element μ ∈ H {\displaystyle \mu \in H} satisfying
E ⟨ X , h ⟩ = ⟨ μ , h ⟩ , h ∈ H . {\displaystyle \mathbb {E} \langle X,h\rangle =\langle \mu ,h\rangle ,\qquad h\in H.}
This formulation is the Pettis integral but the mean can also be defined as Bochner integral μ = E X {\displaystyle \mu =\mathbb {E} X} . Under the integrability condition that E ‖ X ‖ L 2 2 {\displaystyle \mathbb {E} \|X\|_{L^{2}}^{2}} is finite, the covariance operator of X {\displaystyle X} is a linear operator C : H → H {\displaystyle {\mathcal {C}}:H\to H} that is uniquely defined by the relation
C h = E [ ⟨ h , X − μ ⟩ ( X − μ ) ] , h ∈ H , {\displaystyle {\mathcal {C}}h=\mathbb {E} [\langle h,X-\mu \rangle (X-\mu )],\qquad h\in H,}
or, in tensor form, C = E [ ( X − μ ) ⊗ ( X − μ ) ] {\displaystyle {\mathcal {C}}=\mathbb {E} [(X-\mu )\otimes (X-\mu )]} . The spectral theorem allows to decompose X {\displaystyle X} as the Karhunen-Loève decomposition
X = μ + ∑ i = 1 ∞ ⟨ X , φ i ⟩ φ i , {\displaystyle X=\mu +\sum _{i=1}^{\infty }\langle X,\varphi _{i}\rangle \varphi _{i},}
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