content_image/309420.jpg content_image/309421.jpg
that if the process x ( t ) has at most K peaks and valleys and its derivative x $^{′}$( t ) has at most K zeros, then x ( t ) admits the decomposition 

<img src='content_image/309420.jpg'>

for some p ≤ K + 2 , where the ξ$_{j}$ s are non-random basis functions, the C$_{j}$ s are ran- dom coefficients, and v : I → I is a monotone increasing stochastic process such that E { v ( t ) } = t (or alternatively E { w ( s ) } = s , where w ( t ) is the inverse function of v ( t ) .) We can re-express (3) as 

<img src='content_image/309421.jpg'>

which is just the Karhunen–Lo`eve decomposition of the warp ed process z = x ◦ w . The process w is called the warping process, and it explains the “horizont al” variability in the location of the peaks and valleys of x . 

It is important to point out that the Karhunen–Lo`eve decomp osition (1) is essentially unique (up to the usual indeterminacy of eigenfunctions for multiple eigenvalues), and so is (4) for a given warping process w ; but the warping process w itself is not unique. For a given x ( t ) , different warping processes w ( t ) can be chosen that will give rise to differ- ent decompositions (4). In general, it is not possible to uniquely define what constitutes amplitude variability and what constitutes phase variability for a given process x ( t ) . The approach usually followed in the literature is to specify a warping family W where w ( t ) is constrained to live, and then define as phase variability w hatever is accounted for by the family W and as amplitude variability whatever is accounted for by the residual de- composition (4). This may sound vague, but in fact it is possible to give simple conditions for model (4) to be identifiable given a family W ; see the discussion in Web Appendix E. Some authors choose very rigid warping families W , like linear warping functions (San- 

