Fundamental concepts of functional data analysis

Transcription

1 Fundamental concepts of functional data analysis Department of Statistics, Colorado State University

2 Examples of functional data Time in minutes The horizontal component of the magnetic field measured at Honolulu magnetic observatory.

3 St. Johns log10(mm) Jan Mar May Jul Sep Nov Day Average log precipitation at St. Johns, Canada. The thick line shows a smooth.

4 height (cm) Smoothed growth curves of 54 girls age age

5 Expansions Smoothing and dimension reduction have been major themes of FDA research. Basis expansion: X n (t) M c nm B m (t), 1 n N. m=1 Derivatives can be computed.

6 Acceleration Age Acceleration curves of 54 girls with the mean function in bold.

7 Acceleration Age Continuously registered acceleration curves of the 54 girls.

8 Statistical models Functional response model: Y i = ψ(s)x i (s)ds +ε i, 1 i N. Function on function regression: Y i (t) = ψ(t,s)x i (s)ds +ε i (t). Autoregression: X i (t) = ψ(t,s)x i 1 (s)ds +ε i (t). Dimension of functional parameters, e.g. ψ, is larger than the sample size N. Regularization has been a major theme of FDA research.

9 Octane rating Gasoline sample Octane ratings of 60 gasoline samples.

10 Spectrum Difference Wavelength Wavelength Left: Near infrared spectrum of a gasoline sample with index 1; Right: differences between the spectrum of the samples with indexes 2 and 1 (continuous) and 5 and 1 (dashed).

11 Time series of functions USA: male death rates (2003) Log death rate Age

12 USA: male death rates ( ) Log death rate Age Predicted US male log mortality rate curves in rainbow code; years close to 2011 are in red, those close to 2040 in violet.

13 Forecasted fertility rates ( ) Births per 1,000 females Age Recursive Forecasts Births per 1,000 females Age

14 Spatial functional data 35 Canadian weather stations, Calgary marked with a square.

15 True temperature function Kriged function Day True and predicted (kriged) temperature functions at Calgary.

16 Some new directions Clustering and classification of huge sets of functions (spectral profiles of stars, expression profiles of proteins) High dimensional functional regressions (Y i (t) protein expressions, X g (s) gene expressions, 1 g G, G can be several thousand for one i.) Manifold domains (second generation FD) (physical domains (brain), domains generated by restrictions) Multilevel dependence (patient, visit, records from brain regions, scalar covariates) Connectivity and Network identification (possibly multilevel) (Several types of measurements of brain activity, which regions interact at which levels?) Extreme events in set of functions indexed by time and space (X n (s,t), temperature in year n at location s on day t, probabilities of heat waves or droughts).

17 Examples of asymptotic techniques Functional observations are assumed to be functions in an L 2 space (so that variance-like objects can be defined). ( { ν p (X) = E X 2 (t)dt} p/2 ) 1/p <. If ν 2 2 (X) = E X 2 (t)dt <, we say that X is square integrable. We must often assume that ν 4 (X) <.

18 Dependence in time series Bernoulli shifts: X n = f(ε n,ε n 1,...), the ε i are iid elements taking values in a measurable space S, and f is a measurable function f : S L 2. Suppose {ε i } are independent copies of {ε i} and set X (m) n Approximability condition: = f(ε n,ε n 1,...,ε n m+1,ε n m.ε n m 1,...) m=1 ( ν p Xn X n (m) ) <.

19 Estimation of second order structure Covariance function (EX i = 0): Sample covariance function: c(t,s) = cov(x 1 (t),x 1 (s)) ĉ(t,s) = 1 N N (X n (t) X N (t))(x n (s) X N (s)). n=1 Under approximability with p = 4, E Ĉ C 2 S = O(N 1 ). (Hilbert Schmidt norm) Consistency, with the same rate, of the eigenfunctions follows. These eigenfunctions, called functional principal components often form an optimal basis for expansions.

20 Invariance principle Functional partial sum process: V N (x,t) = 1 Nx X n (t), 0 t,x 1. N n=1 Under approximability with p = 2+δ, there are Gaussian processes Γ N (x,t) such that for every N sup V N (x, ) Γ N (x, ) 2 = o p (1). 0 x 1 {Γ N (x,t),0 x,t 1} = D {Γ(x,t),0 x,t 1}, Γ(x,t) = λ 1/2 i W i (x)φ i (t). i=1 The W i are independent standard Wiener processes, λ i and φ i are eigenvalues and the eigenfunction of the long run covariance function (different from c above).