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An error in the Gaussian random variate generator by Kinderman and Ramage is described that results in the generation of random variates with an incorrect distribution. An additional statement that corrects the original algorithm is given. / Series: Preprint Series / Department of Applied Statistics and Data Processing

In the analysis of spatio-temporal data, statistical inference based on the Gaussian assumption is ubiquitous due to its many attractive properties. However, data collected from different fields of science rarely meet the assumption of Gaussianity. One option is to apply a monotonic transformation to the data such that the transformed data have a distribution that is close to Gaussian. In this thesis, we focus on a flexible two-parameter family of transformations, the Tukey g-and-h (TGH) transformation. This family has the desirable properties that the two parameters g ∈ R and h ≥ 0 involved control skewness and tail-heaviness of the distribution, respectively. Applying the TGH transformation to a standard normal distribution results in the univariate TGH distribution. Extensions to the multivariate case and to a spatial process were developed recently. In this thesis, motivated by the need to exploit wind as renewable energy, we tackle the challenges of modeling big spatio-temporal data that are non-Gaussian by applying the TGH transformation to different types of Gaussian processes: spatial (random field), temporal (time series), spatio-temporal, and their multivariate extensions. We explore various aspects of spatio-temporal data modeling techniques using transformed Gaussian processes with the TGH transformation. First, we use the TGH transformation to generate non-Gaussian spatial data with the Matérn covariance function, and study the effect of non-Gaussianity on Gaussian likelihood inference for the parameters in the Matérn covariance via a sophisticatedly designed simulation study. Second, we build two autoregressive time series models using the TGH transformation. One model is applied to a dataset of observational wind speeds and shows advantaged in accurate forecasting; the other model is used to fit wind speed data from a climate model on gridded locations covering Saudi Arabia and to Gaussianize the data for each location. Third, we develop a parsimonious spatio-temporal model for time series data on a spatial grid and utilize the aforementioned Gaussianized climate model wind speed data to fit the latent Gaussian spatio-temporal process. Finally, we discuss issues under a unified framework of modeling multivariate trans-Gaussian processes and adopt one of the TGH autoregressive models to build a stochastic generator for global wind speed.

D.Phil. (Mathematical Statistics) / Fractional Brownian motion and its increment process, fractional Gaussian noise, are syn- onymous with the concept of long range dependence. A strictly stationary time series is said to exhibit long range dependence or long memory if its autocorrelations decrease to zero as a power of the lag, but their sum over all lags is not absolutely convergent. This phenomenon has been observed in numerous scientific areas such as hydrology, ethernet traffic data, stock returns and exchange rates, to name just a few. The extent of long memory dependence is characterized by the value of the so called Hurst exponent or Hurst coefficient H. Approximate normality and unbiasedness of the maximum likelihood estimate of H hold reasonably well for sample sizes as small as 20 if the mean and scale parameters are known. We show in a Monte Carlo study that if the latter two parameters are unknown, the bias and variance of the maximum likelihood estimate of H both increase substantially. We also show that the bias can be reduced by using a jackknife or parametric bootstrap proce- dure. However, in very large samples, maximum likelihood estimation becomes problematic because of the large dimension of the covariance matrix that must be inverted. We consider an approach for estimating the Hurst exponent by taking first order differ- ences of fractional Gaussian noise. We find that this differenced process has short memory and that, consequently, we may assume approximate independence between the estimates of the Hurst exponents in disjoint blocks of data. We split the data into a number of con- tiguous blocks, each containing a relatively small number of observations. Computation of the likelihood function in a block then presents no computational problem. We form a pseudo likelihood function consisting of the product of the likelihood functions in each of the blocks and provide a formula for the standard error of the resulting estimator of H. This formula is shown in a Monte Carlo study to provide a good approximation to the true standard error. Application of the methodology is illustrated in two data sets. The long memory property of a time series is primarily characterized by H. In general, such series are exceptionally long, therefore it is natural to enquire whether or not H remains constant over the full extent of the time series. We propose a number of tests for the hypothesis that H remains constant, against an alternative of a change in one or more values of H. Formulas are given to enable calculation of asymptotic p-values. We also propose a permutational procedure for evaluating exact p-values. The proposed tests are applied to three sets of data.

Despite the impressive performance of deep neural networks (DNNs) on numerous vision tasks, they still exhibit yet-to-understand uncouth behaviours. One puzzling behaviour is the reaction of DNNs to various noise attacks, where it has been shown that there exist small adversarial noise that can result in a severe degradation in the performance of DNNs. To rigorously treat this, we derive exact analytic expressions for the first and second moments (mean and variance) of a small piecewise linear (PL) network with a single rectified linear unit (ReLU) layer subject to general Gaussian input. We experimentally show that these expressions are tight under simple linearizations of deeper PL-DNNs, especially popular architectures in the literature (e.g. LeNet and AlexNet). Extensive experiments on image classification show that these expressions can be used to study the behaviour of the output mean of the logits for each class, the inter-class confusion and the pixel-level spatial noise sensitivity of the network. Moreover, we show how these expressions can be used to systematically construct targeted and non-targeted adversarial attacks. Then, we proposed a special estimator DNN, named mixture of linearizations (MoL), and derived the analytic expressions for its output mean and variance, as well. We employed these expressions to train the model to be particularly robust against Gaussian attacks without the need for data augmentation. Upon training this network on a loss that is consolidated with the derived output probabilistic moments, the network is not only robust under very high variance Gaussian attacks but is also as robust as networks that are trained with 20 fold data augmentation.

Phase Retrieval is an inference problem where one seeks to recover an unknown complex-valued 𝓃-dimensional signal vector from the magnitudes of 𝓶 linear measurements. The linear measurements are specified using a 𝓶 × 𝓃 sensing matrix. This problem is a mathematical model for imaging systems arising in X-ray crystallography and other applications where it is infeasible to acquire the phase of the measurements. This dissertation presents some results regarding the analysis of this problem in the high-dimensional asymptotic regime where the number of measurements and the signal dimension diverge proportionally so that their ratio remains fixed. A limitation of existing high-dimensional analyses of this problem is that they model the sensing matrix as a random matrix with independent and identically (i.i.d.) distributed Gaussian entries. In practice, this matrix is highly structured with limited randomness. This work studies a correction to the i.i.d. Gaussian sensing model, known as the sub-sampled Haar sensing model which faithfully captures a crucial orthogonality property of realistic sensing matrices. The first result of this thesis provides a precise asymptotic characterization of the performance of commonly used spectral estimators for phase retrieval in the sub-sampled Haar sensing model. This result can be leveraged to tune certain parameters involved in the spectral estimator optimally. The second part of this dissertation studies the information-theoretic limits for better-than-random (or weak) recovery in the sub-sampled Haar sensing model. The main result in this part shows that appropriately tuned spectral methods achieve weak recovery with the information-theoretically optimal number of measurements. Simulations indicate that the performance curves derived for the sub-sampled Haar sensing model accurately describe the empirical performance curves for realistic sensing matrices such as randomly sub-sampled Fourier sensing matrices and Coded Diffraction Pattern (CDP) sensing matrices. The final part of this dissertation tries to provide a mathematical understanding of this empirical universality phenomenon: For the real-valued version of the phase retrieval problem, the main result of the final part proves that the dynamics of a class of iterative algorithms, called Linearized Approximate Message Passing schemes, are asymptotically identical in the sub-sampled Haar sensing model and a real-valued analog of the sub-sampled Fourier sensing model.

Philosophiae Doctor - PhD / The faint radio population consisting of star forming galaxies (SFG) and radio-quiet active galactic nuclei (AGN) is important in the study of galaxy evolution. However, the bulk of the faint population is below the detection threshold of the current radio surveys. I study this population through a Bayesian-stacking technique that I have adapted to probe the radio luminosity function (RLF) below the typical 5σ detection threshold. The technique works by fitting RLF models to radio flux densities extracted at the position of galaxies selected from an auxiliary catalogue. I test the technique by adding Gaussian noise (σ) to simulated data and the RLF models are in agreement with the simulated data for up to three orders of magnitude (3 dex) below the detection threshold (5σ). The source of radio emission from radio quiet quasars (subset of AGN) is widely debated. I apply the technique to 1.4-GHz flux densities from the Faint Images of the Radio Sky at Twenty-cm survey (FIRST) at the positions of the optical quasars from the Sloan Digital Sky Survey (SDSS). The RLF models are constrained to 2 dex below the FIRST detection threshold. I found that the radio luminosity where radio-quiet quasars emerge coincides with the luminosity where SFGs are expected to start to dominate the RLF. This Implies that the radio emission of radio-quiet quasars and radio-quiet AGN, in general, could have a significant contribution from star formation in the host galaxies.

Based on the geometric analysis of Gaussian elimination (GE) found in Neal and Poole (Linear Algebra Appl. 173 (1992) 239-264) and Poole and Neal (Linear Algebra Appl. 149 (1991) 249-272; 162-164 (1992) 309-324), a new pivoting strategy, Rook's pivoting (RP), was introduced in Neal and Poole (Linear Algebra Appl. 173 (1992) 239-264) which encourages stability in the back-substitution phase of GE while controlling the growth of round-off error during the sweep-out. In fact, Foster (J. Comput. Appl. Math. 86 (1997) 177-194) has previously shown that RP, as with complete pivoting, cannot have exponential growth error. Empirical evidence presented in Neal and Poole (Linear Algebra Appl. 173 (1992) 239-264) showed that RP produces computed solutions with consistently greater accuracy than partial pivoting. That is, Rook's pivoting is, on average, more accurate than partial pivoting, with comparable costs. Moreover, the overhead to implement Rook's pivoting in a scalar or serial environment is only about three times the overhead to implement partial pivoting. The theoretical proof establishing this fact is presented here, and is empirically confirmed in this paper and supported in Foster (J. Comput. Appl. Math. 86 (1997) 177-194).

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We observe a stochastic process 𝑌 on [0,1]^𝑑 (𝑑 ≥ 1) satisfying 𝑑𝑌(𝑡)=𝑛¹/²𝑓(𝑡)𝑑𝑡 + 𝑑𝑊(𝑡), 𝑡 ∈ [0,1]^𝑑, where 𝑛 ≥ 1 is a given scale parameter (`sample size'), 𝑊 is the standard Brownian sheet on [0,1]^𝑑 and 𝑓 ∈ L₁([0,1]^𝑑) is the unknown function of interest. We propose a multivariate multiscale statistic in this setting and prove that the statistic attains a subexponential tail bound; this extends the work of 'Dumbgen and Spokoiny (2001)' who proposed the analogous statistic for 𝑑=1. In the process, we generalize Theorem 6.1 of 'Dumbgen and Spokoiny (2001)' about stochastic processes with sub-Gaussian increments on a pseudometric space, which is of independent interest. We use the proposed multiscale statistic to construct optimal tests (in an asymptotic minimax sense) for testing 𝑓 = 0 versus (i) appropriate Hölder classes of functions, and (ii) alternatives of the form 𝑓 = 𝜇_𝑛𝕀_{𝐵_𝑛}$, where 𝐵_𝑛 is an axis-aligned hyperrectangle in [0,1]^𝑑 and 𝜇_𝑛 ∈ ℝ; 𝜇_𝑛 and 𝐵_𝑛 unknown. In Chapter 3 we use this proposed multiscale statistics to construct honest confidence bands for multivariate shape-restricted regression including monotone and convex functions.