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by N basis function expansions (B-spline, Fourier or some other appropriate basis) and then base their analysis on the coefficients of the basis function expansions. The selection of the number and position of the basis functions is clearly an important question in these methods. Furby et al. (1990) use a small number of adaptively selected B-spline basis functions while Ramsay and Silverman (1997, 1999) use a generous number of basis functions to model the data, and then apply a roughness penalty to the coefficients of the basis function expansions. As the number of basis functions is large, the basis function coefficients show serial correlation. Ramsay and Silverman thus substitute the continuous optimization problem of finding the penalty coefficient X for the more difficult discrete optimization problem of determining the number of knots. a function for modeling the We consider the implications of treating the data variance. For the vector valued case we have z ( t ) = p ( t ) E where E C) and
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(11.1) (11.2)
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where A is a diagonal matrix containing the eigenvalues of C and the columns of Z contain the corresponding eigenvectors. We can then write x ( t ) = p ( t ) where IE(E,) = 0, IE(e;) = X, and IE(ctcJ) = 0 for # j . Similarly, for the function valued case we have z(t) = p ( t ) where the same conditions on E % hold above, but now the c ( t ) are eigenfunctions of the covariance function
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(11.3) and the Ct(t) are orthogonal in the L2 sense (Rice and Silverman, 1991). We see an example of this in Figure 11.1 which shows data from the HyMap example described in Section 11.3.3 (p. 193). A single observation is plotted and it appears reasonable to model the observations, in this case, a mean curve plus a sum of curves rather than a mean curve plus uncorrelated Gaussian noise. The figure also shows the mean curve p ( t ) plus and minus the first two eigenfunctions, calculated using the methodology of Ramsay and Silverman (1997). Note that, as is common in applications, the first eigenfunction appears t o be largely due to the magnitude of the observations. The HyMap data is a hyper-spectral image, with multiple bands for every pixel. Figure 11.2 (p. 182) displays the first three eigenfunctions (or smoothed eigenvectors). We briefly consider how estimates of these eigenfunctions were arrived at. We consider the functional case and write the inner-product
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The case where is a vector is then notationally similar in terms of inner-products. The first principal component is found by arg max
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where C is the covariance of the mean centered data. This can be written as (11.4)
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Figure 11.1 The data is the HyMap example (described in Section 11.3.3, p. 193). The plot shows the mean curve p ( t ) plus and minus the first two eigen-functions of the covariance function. For comparison, a single observation (shifted upwards by 0.3) is also plotted.
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as any scaling of is then found by
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< does not affect the value of (11.4). T h e j t h principal component
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which becomes a matrix equation when the integral is solved by a numerical procedure (Ramsay and Silverman, 1997, 56) as will generally be the case. We note that the model in which ~ ( t is a smooth function plus additive noise ) is incorporated into this model by observing that the effect of white noise is t o add a constant t o each eigenvalue. T h e covariance function white noise is the Dirac - s) so (11.5) becomes delta function
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which, by the properties of the delta function, gives
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A functional principal component image of the HyMap data for Toolibin
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which is (11.5) with a constant value of added to each eigenvalue. Returning to the functional case we can penalize
(11.6)
where is the second derivative operator. We require the boundary conditions that = D3c = 0 a t the ends of the interval3. With these conditions it can be shown, by a repeated integration by parts, that is self-adjoint* so that