229 research outputs found

    On the convergence of the chi square and noncentral chi square distributions to the normal distribution

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    A simple and novel asymptotic bound for the maximum error resulting from the use of the central limit theorem to approximate the distribution of chi square and noncentral chi square random variables is derived. The bound enables the quick calculation of the number of degrees of freedom required to ensure a given approximation error, and is significantly tighter than bounds derived using the Berry-Esseen theorem. An application to widely-used approximations for the decision probabilities of energy detectors is also provided

    Co-channel interference in heterogeneous networks: Rician/Rayleigh scenario

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    The purpose of this study is to analyze interference for a macro cellular network h which embeds femtocells and/or relay nodes. Femtocells and relay nodes are being introduced to improve coverage and capacity of conventional cellular networks. However, their introduction comes with additional interference between the uacrocell and femtocell/relay node, and between individual femtocells/relay nodes. Hence, there is a need for analytical expressions for the performance measures such that outage probability and average capacity so that the effect of co-channe1 interference is understood thoroughly. In this thesis, analytical and numerical results are presented for outage probability and average capacity in a Rician/Rayleigh Scenario. In this scenario, the desired signal experiences Rician fading channel and the interfering signals experience Rayleigh fading channels. Closed-form and infinite series expressions are found for the outage probability and an infinite sum expression is found for the average capacity. The analysis approximates SINR by its tight upper bound SIR; therefore, the outage probability result is a tight lower bound and the average capacity result is a tight upper bound of their corresponding results without the approximation. Moreover, the thesis, as a background work, includes detailed studies of the relevant probability distributions used to model fading channels and radio link performance metrics in the absence of interference

    Leveraging Privacy In Data Analysis

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    Data analysis is inherently adaptive, where previous results may influence which tests are carried out on a single dataset as part of a series of exploratory analyses. Unfortunately, classical statistical tools break down once the choice of analysis may depend on the dataset, which leads to overfitting and spurious conclusions. In this dissertation we put constraints on what type of analyses can be used adaptively on the same dataset in order to ensure valid conclusions are made. Following a line of work initiated from Dwork et al. [2015], we focus on extending the connection between differential privacy and adaptive data analysis. Our first contribution follows work presented in Rogers et al. [2016]. We generalize and unify previous works in the area by showing that the generalization properties of (approximately) differentially private algorithms can be used to give valid p-value corrections in adaptive hypothesis testing while recovering results for statistical and low-sensitivity queries. One of the main benefits of differential privacy is that it composes, i.e. the combination of several differentially private algorithms is itself differentially private and the privacy parameters degrade sublinearly. However, we can only apply the composition theorems when the privacy parameters are all fixed up front. Our second contribution then presents a framework for obtaining composition theorems when the privacy parameters, along with the number of procedures that are to be used, need not be fixed up front and can be adjusted adaptively Rogers et al. [2016]. These contributions are only useful if there actually exists some differentially private procedures that a data analyst would want to use. Hence, we present differentially private hypothesis tests for categorical data based on the classical chi-square hypothesis tests (Gaboardi et al. [2016], Kifer Rogers [2017])

    Bayesian model calibration for diblock copolymer thin film self-assembly using power spectrum of microscopy data

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    Identifying parameters of computational models from experimental data, or model calibration, is fundamental for assessing and improving the predictability and reliability of computer simulations. In this work, we propose a method for Bayesian calibration of models that predict morphological patterns of diblock copolymer (Di-BCP) thin film self-assembly while accounting for various sources of uncertainties in pattern formation and data acquisition. This method extracts the azimuthally-averaged power spectrum (AAPS) of the top-down microscopy characterization of Di-BCP thin film patterns as summary statistics for Bayesian inference of model parameters via the pseudo-marginal method. We derive the analytical and approximate form of a conditional likelihood for the AAPS of image data. We demonstrate that AAPS-based image data reduction retains the mutual information, particularly on important length scales, between image data and model parameters while being relatively agnostic to the aleatoric uncertainties associated with the random long-range disorder of Di-BCP patterns. Additionally, we propose a phase-informed prior distribution for Bayesian model calibration. Furthermore, reducing image data to AAPS enables us to efficiently build surrogate models to accelerate the proposed Bayesian model calibration procedure. We present the formulation and training of two multi-layer perceptrons for approximating the parameter-to-spectrum map, which enables fast integrated likelihood evaluations. We validate the proposed Bayesian model calibration method through numerical examples, for which the neural network surrogate delivers a fivefold reduction of the number of model simulations performed for a single calibration task

    Analysing energy detector diversity receivers for spectrum sensing

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    The analysis of energy detector systems is a well studied topic in the literature: numerous models have been derived describing the behaviour of single and multiple antenna architectures operating in a variety of radio environments. However, in many cases of interest, these models are not in a closed form and so their evaluation requires the use of numerical methods. In general, these are computationally expensive, which can cause difficulties in certain scenarios, such as in the optimisation of device parameters on low cost hardware. The problem becomes acute in situations where the signal to noise ratio is small and reliable detection is to be ensured or where the number of samples of the received signal is large. Furthermore, due to the analytic complexity of the models, further insight into the behaviour of various system parameters of interest is not readily apparent. In this thesis, an approximation based approach is taken towards the analysis of such systems. By focusing on the situations where exact analyses become complicated, and making a small number of astute simplifications to the underlying mathematical models, it is possible to derive novel, accurate and compact descriptions of system behaviour. Approximations are derived for the analysis of energy detectors with single and multiple antennae operating on additive white Gaussian noise (AWGN) and independent and identically distributed Rayleigh, Nakagami-m and Rice channels; in the multiple antenna case, approximations are derived for systems with maximal ratio combiner (MRC), equal gain combiner (EGC) and square law combiner (SLC) diversity. In each case, error bounds are derived describing the maximum error resulting from the use of the approximations. In addition, it is demonstrated that the derived approximations require fewer computations of simple functions than any of the exact models available in the literature. Consequently, the regions of applicability of the approximations directly complement the regions of applicability of the available exact models. Further novel approximations for other system parameters of interest, such as sample complexity, minimum detectable signal to noise ratio and diversity gain, are also derived. In the course of the analysis, a novel theorem describing the convergence of the chi square, noncentral chi square and gamma distributions towards the normal distribution is derived. The theorem describes a tight upper bound on the error resulting from the application of the central limit theorem to random variables of the aforementioned distributions and gives a much better description of the resulting error than existing Berry-Esseen type bounds. A second novel theorem, providing an upper bound on the maximum error resulting from the use of the central limit theorem to approximate the noncentral chi square distribution where the noncentrality parameter is a multiple of the number of degrees of freedom, is also derived
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