Sampling (statistics)

Summary

In statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population, and thus, it can provide insights in cases where it is infeasible to measure an entire population. Each observation measures one or more properties (such as weight, location, colour or mass) of independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified sampling. Results from probability theory and statistical theory are employed to guide the practice. In business and medical research, sampling is widely used for gathering information about a population. Acc

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Sampling theory has experienced a strong research revival over the past decade, which led to a generalization of Shannon's original theory and development of more advanced formulations with immediate relevance to signal processing and communications. For example, it was recently shown that it is possible to develop exact sampling schemes for a large class of non-bandlimited signals, namely, certain signals with finite rate of innovation. A common feature of such signals is that they have a parametric representation with a finite number of degrees of freedom and can be perfectly reconstructed from a finite number of samples. The goal of this thesis is to advance the sampling theory for signals of finite rate of innovation and consider its possible extensions and applications. In the first part of the thesis, we revisit the sampling problem for certain classes of such signals, including non-uniform splines and piecewise polynomials, and develop improved schemes that allow for stable and precise reconstruction in the presence of noise. Specifically, we develop a subspace approach to signal reconstruction, which converts a nonlinear estimation problem into the simpler problem of estimating the parameters of a linear model. This provides an elegant and robust framework for solving a large class of sampling problems, while offering more flexibility than the traditional scheme for bandlimited signals. In the second part of the thesis, we focus on applications of our results to certain classes of nonlinear estimation problems encountered in wideband communication systems, most notably ultra-wideband (UWB) systems, where the bandwidth used for transmission is much larger than the bandwidth or rate of information being sent. We develop several frequency domain methods for channel estimation and synchronization in UWB systems, which yield high-resolution estimates of all relevant channel parameters by sampling a received signal below the traditional Nyquist rate. We also propose algorithms that are suitable for identification of more realistic UWB channel models, where a received signal is made up of pulses with different pulse shapes. Finally, we extend our results to multidimensional signals, and develop exact sampling schemes for certain classes of parametric non-bandlimited 2-D signals, such as sets of 2-D Diracs, polygons or signals with polynomial boundaries.

EPFL2004

Bayesian risk analysis of financial time series

Christophe Osinski

This thesis is a contribution to financial statistics. One of the principal concerns of investors is the evaluation of portfolio risk. The notion of risk is vague, but in finance it is always linked to possible losses. In this thesis, we present some measures allowing the valuation of risk with the help of Bayesian methods. An exploratory analysis of data is presented to describe the sampling properties of financial time series. This analysis allows us to understand the origins of the daily returns studied in this thesis. Moreover, a discussion of different models is presented. These models make strong assumptions on investor behaviour, which are not always satisfied. This exploratory analysis shows some differences between the behaviour anticipated under equilibrium models, and that of real data. The Bayesian approach has been chosen because it allows one to incorporate all the variability, in particular that associated with model choice. The models studied in this thesis allow one to take heteroskedasticity into account, as well as particular shapes of the tails of returns. ARCH type models and models based on extreme value theory are studied. One original aspect of this thesis is its use of Bayesian analysis to detect change points in financial time series. We suppose that a market has two phases, and that it switches from a state to the other at random. Another new contribution is a model integrating heteroskedasticity and time dependence of extreme values, by superposition of the model proposed by Bortot and Coles (2003) and a GARCH process. This thesis uses simulation intensively for the estimation of risk measures. The drawback of simulation is the amount of time needed to obtain accurate estimates. However, simulation allows one to produce results when direct calculation is not feasible. For example, simulation allows one to compute risk estimates for time horizons greater than one day. The methods presented in this thesis are illustrated on simulated data, and on real data from European and American markets. This thesis involved the construction of a library containing C and S code to perform risk analysis using GARCH and extreme value theory models. The results show that model uncertainty can be incorporated, and that risk measures for time horizons greater than one can be obtained by simulation. The methods presented in this thesis have a natural representation involving conditioning. Thus, they permit the computation of both conditional and unconditional risk estimates. Three methods are described: the GARCH method; the two-state GARCH method; and the HBC method. Unconditional risk estimation using the GARCH method is satisfactory on data which seem stationary, but not reliable on data which are non-stationary, such as data with change points. The two-state GARCH model does a little better, but gives very satisfactory results when the risk is estimated conditionally on time. The HBC method does not give satisfactory results.

EPFL2006

Shift-Invariant and Sampling Spaces Associated With the Fractional Fourier Transform Domain

Ayush Bhandari

Shift-invariant spaces play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. A special class of the shift-invariant spaces is the class of sampling spaces in which functions are determined by their values on a discrete set of points. One of the vital tools used in the study of sampling spaces is the Zak transform. The Zak transform is also related to the Poisson summation formula and a common thread between all these notions is the Fourier transform.

2012

Bayesian risk analysis of financial time series

Christophe Osinski

EPFL2006

EPFL2004