Signal Processing and Bootstrap
Astronomers have developed their ways of processing signals almost independent to but sometimes collaboratively with engineers, although the fundamental of signal processing is same: extracting information. Doubtlessly, these two parallel roads of astronomers’ and engineers’ have been pointing opposite directions: one toward the sky and the other to the earth. Nevertheless, without an intensive argument, we could say that somewhat statistics has played the medium of signal processing for both scientists and engineers. This particular issue of IEEE signal processing magazine may shed lights for astronomers interested in signal processing and statistics outside the astronomical society.
IEEE Signal Processing Magazine Jul. 2007 Vol 24 Issue 4: Bootstrap methods in signal processing
This link will show the table of contents and provide links to articles; however, the access to papers requires IEEE Xplore subscription via libraries or individual IEEE memberships). Here, I’d like to attempt to introduce some articles and tutorials.
Special topic on bootstrap:
The guest editors (A.M. Zoubir & D.R. Iskander)^{[1]} open the issue by providing the rationale, the occasional invalid Gaussian noise assumption, and the consequential complex modeling in their editorial opening, Bootstrap Methods in Signal Processing. A practical approach has been Monte Carlo simulations but the cost of repeating experiments is problematic. The suggested alternative is the bootstrap, which provides tools for designing detectors for various signals subject to noise or interference from unknown distributions. It is said that the bootstrap is a computer-intensive tool for answering inferential questions and this issue serves as tutorials that introduce this computationally intensive statistical method to the signal processing community.
The first tutorial is written by those two guest editors: Bootstrap Methods and Applications, which begins with the list of bootstrap methods and emphasizes its resilience. It discusses the number of bootstrap samples to compensate a simulation (Monte Carlo) error to a statistical error and the sampling methods for dependent data with real examples. The flowchart from Fig. 9 provides the guideline for how to use the bootstrap methods as a summary.
The title of the second tutorial is Jackknifing Multitaper Spectrum Estimates (D.J. Thomson), which introduces the jackknife, multitaper estimates of spectra, and applying the former to the latter with real data sets. The author added the reason for his preference of jackknife to bootstrap and discussed the underline assumptions on resampling methods.
Instead of listing all articles from the special issue, a few astrostatistically notable articles are chosen:
- Bootstrap-Inspired Techniques in Computational Intelligence (R. Polikar) explains the bootstrap for estimating errors, algorithms of bagging, boosting, and AdaBoost, and other bootstrap inspired techniques in ensemble systems with a discussion of missing.
- Bootstrap for Empirical Multifractal Analysis (H. Wendt, P. Abry & S. Jaffard) explains block bootstrap methods for dependent data, bootstrap confidence limits, bootstrap hypothesis testing in addition to multifractal analysis. Due to the personal lack of familiarity in wavelet leaders, instead of paraphrasing, the article’s conclusion is intentionally replaced with quoting sentences:
First, besides being mathematically well-grounded with respect to multifractal analysis, wavelet leaders exhibit significantly enhanced statistical performance compared to wavelet coefficients. … Second, bootstrap procedures provide practitioners with satisfactory confidence limits and hypothesis test p-values for multifractal parameters. Third, the computationally cheap percentile method achieves already excellent performance for both confidence limits and tests.
- Wild Bootstrap Test (J. Franke & S. Halim) discusses the residual-based nonparametric tests and the wild bootstrap for regression models, applicable to signal/image analysis. Their test checks the differences between two irregular signals/images.
- Nonparametric Estimates of Biological Transducer Functions (D.H.Foster & K.Zychaluk) I like the part where they discuss generalized linear model (GLM) that is useful to expend the techniques of model fitting/model estimation in astronomy beyond gaussian and least square. They also mentioned that the bootstrap is simpler for getting confidence intervals.
- Bootstrap Particle Filtering (J.V.Candy) It is a very pleasant reading for Bayesian signal processing and particle filter. It overviews MCMC and state space model, and explains resampling as a remedy to overcome the shortcomings of importance sampling in signal processing.
- Compressive sensing. (R.G.Baranuik)
A lecture note presents a new method to capture and represent compressible signals at a rate significantly below the Nyquist rate. This method employs nonadaptive linear projections that preserve the structure of the signal;
I do wish this brief summary assists you selecting a few interesting articles.
- They wrote a book, the bootstrap and its application in signal processing.[↩]
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