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Sample text
Shorter term perturbations are also visible, such as World War II and the end of the Vietnam War in 1974. • On the shortest scale there is seasonal variation over an annual cycle that tends to repeat itself. 6, suggests that the index varies fairly smoothly and regularly within each year. The solid line is a smooth of these data using the roughness penalty method described in Chapter 5. 5 can possibly reveal. This curve oscillates three times during the year, with the size of the oscillation being smallest in spring, larger in the summer, and largest in the autumn.
2 Some properties of functional data The basic philosophy of functional data analysis is to think of observed data functions as single entities, rather than merely as a sequence of individual observations. The term functional in reference to observed data refers to the intrinsic structure of the data rather than to their explicit form. In practice, functional data are usually observed and recorded discretely as n pairs (tj , yj ), and yj is a snapshot of the function at time tj , possibly blurred by measurement error.
3 The interplay between smooth and noisy variation Smoothness, in the sense of possessing a certain number of derivatives, is a property of the latent function x, and may not be at all obvious in the raw 40 3. From functional data to smooth functions data vector y = (y1 , . . , yn ) owing to the presence of observational error or noise that is superimposed on the underlying signal by aspects of the measurement process. 1) where the noise, disturbance, error, perturbation or otherwise exogenous term j contributes a roughness to the raw data.