The power law is useful in describing count phenomena such as network degrees and word frequencies. With a single parameter, it captures the main feature that the frequencies are linear on the log-log scale. Nevertheless, there have been criticisms of the power law, for example that a threshold needs to be pre-selected without its uncertainty quantified, that the power law is simply inadequate, and that subsequent hypothesis tests are required to determine whether the data could have come from the power law. We propose a modelling framework that combines two different generalisations of the power law, namely the generalised Pareto distribution and the Zipf-polylog distribution, to resolve these issues. The proposed mixture distributions are shown to fit the data well and quantify the threshold uncertainty in a natural way. A model selection step embedded in the Bayesian inference algorithm further answers the question whether the power law is adequate.
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Measurement and comparison of distributional shift and its relation to rarity, poverty, and scarcity
The comparison of frequency distributions is a common statistical task with broad applications. However, existing measures do not explicitly quantify the magnitude and direction by which one distribution is shifted relative to another. In the present study, we define distributional shift (DS) as the concentration of frequencies towards the lowest discrete class, e.g., the left-most bin of a histogram. We measure DS via the sum of cumulative frequencies and define relative distributional shift (RDS) as the difference in DS between distributions. Using simulated random sampling, we show that RDS is highly related to measures that are widely used to compare frequency distributions. Focusing on specific applications, we show that DS and RDS provide insights into healthcare billing distributions, ecological species-abundance distributions, and economic distributions of wealth. RDS has the unique advantage of being a signed (i.e., directional) measure based on a simple difference in an intuitive property that, in turn, serves as a measure of rarity, poverty, and scarcity.
We consider arbitrary bounded discrete time series. From its statistical feature, without any use of the Fourier transform, we find an almost periodic function which suitably characterizes the corresponding time series.
This paper investigates extremal quantiles under two-way cluster dependence. We demonstrate that the limiting distribution of the unconditional intermediate order quantiles in the tails converges to a Gaussian distribution. This is remarkable as two-way cluster dependence entails potential non-Gaussianity in general, but extremal quantiles do not suffer from this issue. Building upon this result, we extend our analysis to extremal quantile regressions of intermediate order.
The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of L\'evy walks - a stochastic process that originates from the famous continuous-time random walks. It is usually defined as the Fourier-Laplace multiplier, therefore, it can be thought of as a pseudo-differential operator. In this paper, we show that there exists a local representation in time and space, pointwise, of the fractional material derivative. This allows us to define it on a space of locally integrable functions which is larger than the original one in which Fourier and Laplace transform exist as functions. We consider several typical differential equations involving the fractional material derivative and provide conditions for their solutions to exist. In some cases, the analytical solution can be found. For the general initial value problem, we devise a finite volume method and prove its stability, convergence, and conservation of probability. Numerical illustrations verify our analytical findings. Moreover, our numerical experiments show superiority in the computation time of the proposed numerical scheme over a Monte Carlo method applied to the problem of probability density function's derivation.
Several subjective proposals have been made for interpreting the strength of evidence in likelihood ratios and Bayes factors. I identify a more objective scaling by modelling the effect of evidence on belief. The resulting scale with base 3.73 aligns with previous proposals and may partly explain intuitions.
A simple, recently observed generalization of the classical Singleton bound to list-decoding asserts that rate $R$ codes are not list-decodable using list-size $L$ beyond an error fraction $\frac{L}{L+1} (1-R)$ (the Singleton bound being the case of $L=1$, i.e., unique decoding). We prove that in order to approach this bound for any fixed $L >1$, one needs exponential alphabets. Specifically, for every $L>1$ and $R\in(0,1)$, if a rate $R$ code can be list-of-$L$ decoded up to error fraction $\frac{L}{L+1} (1-R -\varepsilon)$, then its alphabet must have size at least $\exp(\Omega_{L,R}(1/\varepsilon))$. This is in sharp contrast to the situation for unique decoding where certain families of rate $R$ algebraic-geometry (AG) codes over an alphabet of size $O(1/\varepsilon^2)$ are unique-decodable up to error fraction $(1-R-\varepsilon)/2$. Our bounds hold even for subconstant $\varepsilon\ge 1/n$, implying that any code exactly achieving the $L$-th generalized Singleton bound requires alphabet size $2^{\Omega_{L,R}(n)}$. Previously this was only known only for $L=2$ under the additional assumptions that the code is both linear and MDS. Our lower bound is tight up to constant factors in the exponent -- with high probability random codes (or, as shown recently, even random linear codes) over $\exp(O_L(1/\varepsilon))$-sized alphabets, can be list-of-$L$ decoded up to error fraction $\frac{L}{L+1} (1-R -\varepsilon)$.
In this short note we formulate a stabilizer formalism in the language of noncommutative graphs. The classes of noncommutative graphs we consider are obtained via unitary representations of compact groups, and suitably chosen operators on finite-dimensional Hilbert spaces. Furthermore, in this framework, we generalize previous results in this area for determining when such noncommutative graphs have anticliques.
In large-scale, data-driven applications, parameters are often only known approximately due to noise and limited data samples. In this paper, we focus on high-dimensional optimization problems with linear constraints under uncertain conditions. To find high quality solutions for which the violation of the true constraints is limited, we develop a linear shrinkage method that blends random matrix theory and robust optimization principles. It aims to minimize the Frobenius distance between the estimated and the true parameter matrix, especially when dealing with a large and comparable number of constraints and variables. This data-driven method excels in simulations, showing superior noise resilience and more stable performance in both obtaining high quality solutions and adhering to the true constraints compared to traditional robust optimization. Our findings highlight the effectiveness of our method in improving the robustness and reliability of optimization in high-dimensional, data-driven scenarios.
With the growing demand of mineral consumption, the management of the mining waste is crucial. Cemented paste backfill (CPB) is one of the techniques developed by the mining industry to fill the voids generated by the excavation of underground spaces. The CPB process is the subject of various studies aimed at optimizing its implementation in the field. In this article, we focus on the modelling of the backfill phase where it has been shown in [Vigneaux et al., Cem. Concr. Res. 164 (2023) 107038] that a viscoplastic lubrication model can be used to describe CPB experiments. The aim here is to propose an accelerated method for performing the parameters' estimation of the properties of the paste (typically its rheological properties), with an inverse problem procedure based on observed height profiles of the paste. The inversion procedure is based on a metamodel built from an initial partial differential equation model, thanks to a Polynomial Chaos Expansion coupled with a Principal Component Analysis.
Data augmentation (DA) is a powerful workhorse for bolstering performance in modern machine learning. Specific augmentations like translations and scaling in computer vision are traditionally believed to improve generalization by generating new (artificial) data from the same distribution. However, this traditional viewpoint does not explain the success of prevalent augmentations in modern machine learning (e.g. randomized masking, cutout, mixup), that greatly alter the training data distribution. In this work, we develop a new theoretical framework to characterize the impact of a general class of DA on underparameterized and overparameterized linear model generalization. Our framework reveals that DA induces implicit spectral regularization through a combination of two distinct effects: a) manipulating the relative proportion of eigenvalues of the data covariance matrix in a training-data-dependent manner, and b) uniformly boosting the entire spectrum of the data covariance matrix through ridge regression. These effects, when applied to popular augmentations, give rise to a wide variety of phenomena, including discrepancies in generalization between over-parameterized and under-parameterized regimes and differences between regression and classification tasks. Our framework highlights the nuanced and sometimes surprising impacts of DA on generalization, and serves as a testbed for novel augmentation design.
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