Reiss, Philip T. Wavelet-domain regression and predictive inference in psychiatric neuroimaging.
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Abstract Article info and citation First page References Supplemental materials Abstract An increasingly important goal of psychiatry is the use of brain imaging data to develop predictive models. Article information Source Ann. Export citation. Export Cancel.
The ADHD consortium: A model to advance the translational potential of neuroimaging in clinical neuroscience. Berlinet, A. If the address matches an existing account you will receive an email with instructions to retrieve your username. Statistics in Medicine Volume 18, Issue 5.
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Citing Literature. We now proceed to enhance our discussion of the MRA structure using wavelets. Hence, an important property of the multiresolution can be written as 1. Since the Wj subspaces are created from wavelets which can be written in terms of scaling functions-see 1. To relate 1. Two spaces R and S are said to be orthogonal, written R.. With this inter-space orthogonality, we further note that 1.
The ideas of the multiresolution analysis with detail spaces between successive levels of approximation are expressed in Figure 1. Though illustrated in this section using only the piecewise constant MRA, these same principles hold for any orthogonal MRA system. The example function from Section 1. Exact values for the coefficients are typically in terms of 1r, 1r 2 , and so the values in the table are just approximate numerical equivalents.
Just as each Fourier coefficient gives information about the frequency content of a function, scaling function coefficients and thus also wavelet coefficients contain information about the function. This concept is covered in more detail in Section 4. One feature in particular of Table 1. This is pointed out briefly here and developed more fully in Chapter 4. In the Haar case, the reason for this is easily seen. In fact, it is easy to generalize this and derive that for any Haar scaling function coefficient, 1.
It may be seen that the expression in 1. This correspondence will also be treated in more detail in Section 4.
The nice thing to notice about 1. Graphs of the detail functions for the example function are provided in Figure 1. The coefficient values can be verified by integration of the product of the wavelets and the function f. Again, there are some interesting things to note from Table 1. The reason for this is that and as a result it is straightforward to see that in general, 1. This is a result of the relationship between the Haar scaling function and the Haar wavelet, expressed in 1.
This simple observation on the Haar example is meant to provide a Table 1. In the Haar case, it is clear that there must be a corresponding reconstruction algorithm relating higher level scaling function coefficients to lower level wavelet and scaling function coefficients. The existence of such an algorithm is merely mentioned here-it is not difficult to derive in the Haar case and it will be treated fully later in the text.
Goals of Multiresolution Analysis The goal of wavelets and multiresolution analysis in fields such as signal processing is to get a representation of a function signal that is written in a parsimonious manner as a sum of its essential components. That is, a parsimonious representation of a function will preserve the interesting features of the original function, but will express the function in terms of a relatively small set of coefficients.
An example of a situation in which such a representation would be desirable is in the field of image analysis, in which researchers have been making widespread use of multiresolution analysis for several years. A black-andwhite image can be expressed in a numerical form as a function f x, y over two dimensions in which the function value f x 0 , Yo represents the "gray scale" value of the image at the point xo, y 0.
This is then discretized to a relatively fine grid. In many images of interest, there are only a few locations in which greater detail would be desired; the greater part of many images consists of large fairly homogeneous areas of a particular shade of gray. One could "sharpen" the image without increasing the amount of information to be stored by using only a very few components to represent the large homo- Wavelets: A Brief Introduction 23 geneous areas, leaving a great many components with which to represent the "action" areas of the image.
The piecewise constant MRA described in Section 1. At each step of increasing decreasing resolution a finer coarser approximation of the original function is created. Moving from a coarse to a fine approximation, or from a fine to a coarse approximation, is known as the "zoom-in, zoom-out" feature of multiresolution analysis.
The "analysis" consists of studying the "detail signals," or the difference in approximations made at adjacent resolution levels. A wavelet which, when appropriately dilated, forms the basis for the detail spaces must be localized in time, in the sense that '!
The oscillating property makes the function a wave, but because it is localized, it becomes a wavelet.
Essential Wavelets for Statistical Applications and Data Analysis
In application, it is typical to start with a low-level approximation and then add in only the higher-level wavelets that correspond to relatively large wavelet coefficients. This is a general description of how a wavelet-based parsimonious representation can be obtained, which will be covered in more detail in Chapter 7. The fundamental principles remain the same. There are a myriad of other wavelet bases. This section will only present a few examples and discuss their general features; a more full treatment and development is reserved for Section 4.
Several families of orthonormal wavelet bases have been developed in recent years. Stromberg developed a family of wavelets that was not noticed much at the time, and, a few years later, unaware of Stromberg's work, Meyer introduced the system now known as the Meyer basis.
Soon thereafter, Battle and Lemarie each independently proposed the same new family of orthogonal wavelets. Thus, the first example in Figure 1. Though not obvious from the picture, these scaling functions and wavelets each have support over the entire real line, although they do have exponential decay. In statistics, we deal with finite data sets, so we might tend to prefer wavelets with compact support. The Haar wavelet does have compact support, but an ideal wavelet would probably be something a bit smoother.
As are the wavelets of the Battle-Lemarie family, the members of the Chui-Wang family of wavelets are based on polynomial B-splines, but have compact support. This family is indexed by m, where m - 1 is the order of the polynomial.
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Elements of the family are shown in Figure 1. Again, a full treatment is reserved for the Chapter 4. These Chui-Wang wavelets are easy to compute and to work with, but a substantial disadvantage of these wavelets at least in standard statistical applications is that the Chui-Wang wavelets do not form an orthonormal basis for L 2 JR. They form instead a semi-orthogonal wavelet basis, but discussion of this is deferred to Chapter 4 as well.