# Applied Functional Data Analysis (Springer Series in by James O. Ramsay, Bernard W. Silverman

By James O. Ramsay, Bernard W. Silverman

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If a triangular basis is used, we could use a roughness penalty based on ﬁrst derivatives, but the principle is the same. One speciﬁc feature of the general approach we have described is that it does not matter if the various functional data in the sample are not observed at the same evaluation points—the procedure will refer all the diﬀerent functional data to the same basis, regardless of the evaluation points at which each has been observed. 3 Smoothing the sample mean function Now we move on to the calculation of the smoothed overall mean and to smoothed principal components analysis.

6) As usual, the parameter α ≥ 0 controls the amount of smoothing inherent in the procedure. A roughness penalty is also incorporated into the additional constraints on the second-, third-, and higher-order smoothed principal components. 6) and the additional constraint ξ2 (t)ξ1 (t)dt + α ξ2 (t)ξ1 (t)dt = 0. 5), but with corresponding extra terms taking the roughness penalty into account. This will ensure that the estimated components satisfy the condition ξi (t)ξj (t)dt + α ξi (t)ξj (t)dt = 0 for all i and j with i = j.

M (t), for example, such that any function of interest can be expanded in terms of the functions βj (t). 8) j=1 then the vector of m coeﬃcients ξ = (ξ1 , . . , ξm ) speciﬁes the function. Storing functional data in terms of an appropriate basis is a key step in most functional data analyses. Very often, the basis is deﬁned implicitly within the procedure and there is no need for the user to be aware of it. 14. In mathematical terms, the basis functions δi (t) 2. 14. Three triangular basis functions.