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  "body": "\n## Definition\n\nLet $\\mathcal T = [a, b]$ with $a \u003c b$, and let $H = L^2(\\mathcal T)$ be the [[Hilbert space|HM2HHE3J51]] of [[square-integrable|SASP14MA1Z]] real functions on $\\mathcal T$, with inner product $\\langle f, g \\rangle = \\int_{\\mathcal T} f(t) g(t) \\, dt$.\n*Functional data analysis* is the statistical analysis of data $X_1, \\dots, X_n$ that are realisations of a random element $X$ of $H$.\nSuch an $X$ is a [[stochastic process|0QH5ENKZM1]] $(X(t))_{t \\in \\mathcal T}$ with paths in $H$.\nIts basic parameters, when $E\\norm{X}^2 \u003c \\infty$, are:\n\n1. the mean function $\\mu(t) = E[X(t)]$, an [[expected value|0WQNRCNY3M]] for each $t$;\n2. the [[covariance|0KBFAWYRTE]] function $C(s, t) = \\operatorname{Cov}(X(s), X(t))$;\n3. the covariance operator $\\mathcal C \\colon H \\to H$, $(\\mathcal C f)(t) = \\int_{\\mathcal T} C(s, t) f(s) \\, ds$.\n\n## Remarks\n\nThe operator $\\mathcal C$ is compact, self-adjoint and positive semidefinite, so it has an orthonormal basis of eigenfunctions $\\phi_k$ with eigenvalues $\\lambda_1 \\ge \\lambda_2 \\ge \\dots \\ge 0$.\nFunctional principal component analysis estimates these and writes $X = \\mu + \\sum_k \\xi_k \\phi_k$ with uncorrelated scores $\\xi_k = \\langle X - \\mu, \\phi_k \\rangle$ of variance $\\lambda_k$.\nThe sample versions are $\\hat\\mu = \\frac1n \\sum_i X_i$ and $\\hat C(s, t) = \\frac1n \\sum_i (X_i(s) - \\hat\\mu(s))(X_i(t) - \\hat\\mu(t))$.\n\nExample: let $X(t) = \\xi \\sqrt{2} \\sin(\\pi t)$ on $[0, 1]$, with $\\xi$ a real random variable of mean $0$ and variance $1$.\nThen $\\mu = 0$ and $C(s, t) = 2 \\sin(\\pi s) \\sin(\\pi t)$, so $\\mathcal C$ has the single nonzero eigenvalue $1$, with eigenfunction $\\sqrt{2} \\sin(\\pi t)$ of norm $1$.\n\nNon-example: a table of the heights of children measured once each, at a single age, is ordinary multivariate data, not functional data, since no observation is a function of $t$.\n",
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