arXiv subjects

Alexander Aue

Publications and source records attributed to Alexander Aue.

At least 19 recordsLinked to original sources

arXiv2026

Geometric Regime--Switching Diffusions on Stratified Riemannian Spaces with an Application to Covariance Matrices

We construct geometric regime-switching diffusions, a class of Markov processes on locally compact stratified Riemannian state spaces. In contrast with classical regime-switching and stochastic hybrid diffusions, the regimes are not external labels, but intrinsic strata of a singular geometric state space. Changes of regime may therefore change dimension, rank, geometry or combinatorial type while the state space maintains its ambient topology. On each stratum the motion is a conservative Feller diffusion, while inter-stratum transitions are specified by state-dependent jump rates and landing kernels along a directed graph. We characterize the process through a martingale problem on a natural stratified core. Under a uniform bound on the total jump rate, we construct a conservative c\`adl\`ag strong Markov process by combining the stratumwise diffusions with a Poisson thinning mechanism. Uniqueness is proved using an auxiliary disjoint-union topology and a bounded perturbation argument. Standard Foster--Lyapunov conditions for the extended generator give positive Harris recurrence, uniqueness of the invariant probability measure and, under aperiodicity, $V$-uniform geometric ergodicity. The framework is applied to the cone of positive semidefinite covariance matrices, stratified by rank. The resulting process combines fixed-rank covariance diffusions with stochastic rank changes and is $V$-uniformly geometrically ergodic.

math.PR↗

arXiv2026

An operator-level ARCH Model

AutoRegressive Conditional Heteroscedasticity (ARCH) models are standard for modeling time series exhibiting volatility, with a rich literature in univariate and multivariate settings. In recent years, these models have been extended to function spaces. However, functional ARCH and generalized ARCH (GARCH) processes established in the literature have thus far been restricted to model ``pointwise'' variances. In this paper, we propose a new ARCH framework for data residing in general separable Hilbert spaces that accounts for the full evolution of the conditional covariance operator. We define a general operator-level ARCH model. For a simplified Constant Conditional Correlation version of the model, we establish conditions under which such models admit strictly and weakly stationary solutions, finite moments, and weak serial dependence. Additionally, we derive consistent Yule--Walker-type estimators of the infinite-dimensional model parameters. The practical relevance of the model is illustrated through simulations and a data application to high-frequency cumulative intraday returns.

stat.ME↗

arXiv2026

SIGMA: Scalable Spectral Insights for LLM Model Collapse

The rapid adoption of synthetic data for training Large Language Models (LLMs) has introduced the technical challenge of "model collapse"-a degenerative process where recursive training on model-generated content leads to a contraction of distributional variance and representational quality. While the phenomenology of collapse is increasingly evident, rigorous methods to quantify and predict its onset in high-dimensional spaces remain elusive. In this paper, we introduce SIGMA (Spectral Inequalities for Gram Matrix Analysis), a unified framework that benchmarks model collapse through the spectral lens of the embedding Gram matrix. By deriving and utilizing deterministic and stochastic bounds on the matrix's spectrum, SIGMA provides a mathematically grounded metric to track the contraction of the representation space. Crucially, our stochastic formulation enables scalable estimation of these bounds, making the framework applicable to large-scale foundation models where full eigendecomposition is intractable. We demonstrate that SIGMA effectively captures the transition towards degenerate states, offering both theoretical insights into the mechanics of collapse and a practical, scalable tool for monitoring the health of recursive training pipelines.

cs.LG↗

arXiv2025

Bitcoin Forecasting with Classical Time Series Models on Prices and Volatility

This paper evaluates the performance of classical time series models in forecasting Bitcoin prices, focusing on ARIMA, SARIMA, GARCH, and EGARCH. Daily price data from 2010 to 2020 were analyzed, with models trained on the first 90 percent and tested on the final 10 percent. Forecast accuracy was assessed using MAE, RMSE, AIC, and BIC. The results show that ARIMA provided the strongest forecasts for short-run log-price dynamics, while EGARCH offered the best fit for volatility by capturing asymmetry in responses to shocks. These findings suggest that despite Bitcoin's extreme volatility, classical time series models remain valuable for short-run forecasting. The study contributes to understanding cryptocurrency predictability and sets the stage for future work integrating machine learning and macroeconomic variables.

q-fin.ST↗

arXiv2025

Taming Variability: Randomized and Bootstrapped Conformal Risk Control for LLMs

We transform the randomness of LLMs into precise assurances using an actuator at the API interface that applies a user-defined risk constraint in finite samples via Conformal Risk Control (CRC). This label-free and model-agnostic actuator manages ship/abstain/escalate actions based solely on a scalar score from opaque outputs. We enhance CRC's computational efficiency and robustness through Batched Bootstrap CRC (BB-CRC) and Randomized Batched Weighted-Average CRC (RBWA-CRC), reducing calibration calls and stabilizing thresholds while maintaining statistical validity. Additionally, we present a semantic quantification method grounded in gram matrix geometry, resulting in interpretable signal and metric design. Together these pieces deliver principled randomness control for LLM hallucination mitigation and LLM-as-judge reliability. Our framework is assessed using four datasets, demonstrating its efficacy in enhancing factual accuracy and measuring LLM-as-judge performance, yielding a simplified and computationally efficient control layer that converts variability into statistical validity.

stat.ME↗

arXiv2025

Unsupervised Conformal Inference: Bootstrapping and Alignment to Control LLM Uncertainty

Deploying black-box LLMs requires managing uncertainty in the absence of token-level probability or true labels. We propose introducing an unsupervised conformal inference framework for generation, which integrates: generative models, incorporating: (i) an LLM-compatible atypical score derived from response-embedding Gram matrix, (ii) UCP combined with a bootstrapping variant (BB-UCP) that aggregates residuals to refine quantile precision while maintaining distribution-free, finite-sample coverage, and (iii) conformal alignment, which calibrates a single strictness parameter $\tau$ so a user predicate (e.g., factuality lift) holds on unseen batches with probability $\ge 1-\alpha$. Across different benchmark datasets, our gates achieve close-to-nominal coverage and provide tighter, more stable thresholds than split UCP, while consistently reducing the severity of hallucination, outperforming lightweight per-response detectors with similar computational demands. The result is a label-free, API-compatible gate for test-time filtering that turns geometric signals into calibrated, goal-aligned decisions.

stat.ML↗

arXiv2025

Spectral estimation for high-dimensional linear processes

We propose a novel estimation procedure for certain spectral distributions associated with a class of high dimensional linear time series. The processes under consideration are of the form $X_t = \sum_{\ell=0}^\infty \mathbf{A}_\ell Z_{t-\ell}$ with iid innovations $(Z_t)$. The key structural assumption is that the coefficient matrices and the variance of the innovations are simultaneously diagonalizable in a common orthonormal basis. We develop a strategy for estimating the joint spectral distribution of the coefficient matrices and the innovation variance by making use of the asymptotic behavior of the eigenvalues of appropriately weighted integrals of the sample periodogram. Throughout we work under the asymptotic regime $p,n \to \infty$, such that $p/n\to c \in (0,\infty)$, where $p$ is the dimension and $n$ is the sample size. Under this setting, we first establish a weak limit for the empirical distribution of eigenvalues of the aforementioned integrated sample periodograms. This result is proved using techniques from random matrix theory, in particular the characterization of weak convergence by means of the Stieltjes transform of relevant distributions. We utilize this result to develop an estimator of the joint spectral distribution of the coefficient matrices, by minimizing an $L^\kappa$ discrepancy measure, for $\kappa \geq 1$, between the empirical and limiting Stieltjes transforms of the integrated sample periodograms. This is accomplished by assuming that the joint spectral distribution is a discrete mixture of point masses. We also prove consistency of the estimator corresponding to the $L^2$ discrepancy measure. We illustrate the methodology through simulations and an application to stock price data from the S\&P 500 series.

math.ST↗

arXiv2024

Estimating invertible processes in Hilbert spaces, with applications to functional ARMA processes

Invertible processes are central to functional time series analysis, making the estimation of their defining operators a key problem. While asymptotic error bounds have been established for specific ARMA models on $L^2[0,1]$, a general theoretical framework has not yet been considered. This paper fills in this gap by deriving consistent estimators for the operators characterizing the invertible representation of a functional time series with white noise innovations in a general separable Hilbert space. Under mild conditions covering a broad class of functional time series, we establish explicit asymptotic error bounds, with rates determined by operator smoothness and eigenvalue decay. These results further provide consistency-rate estimates for operators in Hilbert space-valued causal linear processes, including functional MA, AR, and ARMA models of arbitrary order.

math.ST↗

arXiv2024

Benchmarking M6 Competitors: An Analysis of Financial Metrics and Discussion of Incentives

The M6 Competition assessed the performance of competitors using a ranked probability score and an information ratio (IR). While these metrics do well at picking the winners in the competition, crucial questions remain for investors with longer-term incentives. To address these questions, we compare the competitors' performance to a number of conventional (long-only) and alternative indices using standard industry metrics. We apply factor models to measure the competitors' value-adds above industry-standard benchmarks and find that competitors with more extreme performance are less dependent on the benchmarks. We also uncover that most competitors could not generate significant out-performance compared to randomly selected long-only and long-short portfolios but did generate out-performance compared to short-only portfolios. We further introduce two new strategies by picking the competitors with the best (Superstars) and worst (Superlosers) recent performance and show that it is challenging to identify skill amongst investment managers. We also discuss the incentives of winning the competition compared to professional investors, where investors wish to maximize fees over an extended period of time.

q-fin.PM↗

arXiv2022

Functional time series prediction under partial observation of the future curve

This paper tackles one of the most fundamental goals in functional time series analysis which is to provide reliable predictions for future functions. Existing methods for predicting a complete future functional observation use only completely observed trajectories. We develop a new method, called partial functional prediction (PFP), which uses both completely observed trajectories and partial information (available partial data) on the trajectory to be predicted. The PFP method includes an automatic selection criterion for tuning parameters based on minimizing the prediction error, and the convergence rate of the PFP prediction is established. Simulation studies demonstrate that incorporating partially observed trajectory in the prediction outperforms existing methods with respect to mean squared prediction error. The PFP method is illustrated to be superior in the analysis of environmental data and traffic flow data.

stat.ME↗

arXiv2020

Simultaneous Detection of Multiple Change Points and Community Structures in Time Series of Networks

In many complex systems, networks and graphs arise in a natural manner. Often, time evolving behavior can be easily found and modeled using time-series methodology. Amongst others, two common research problems in network analysis are community detection and change-point detection. Community detection aims at finding specific sub-structures within the networks, and change-point detection tries to find the time points at which sub-structures change. We propose a novel methodology to detect both community structures and change points simultaneously based on a model selection framework in which the Minimum Description Length Principle (MDL) is utilized as minimizing objective criterion. The promising practical performance of the proposed method is illustrated via a series of numerical experiments and real data analysis.

cs.SI↗

arXiv2019

Two-sample tests for relevant differences in the eigenfunctions of covariance operators

This paper deals with two-sample tests for functional time series data, which have become widely available in conjunction with the advent of modern complex observation systems. Here, particular interest is in evaluating whether two sets of functional time series observations share the shape of their primary modes of variation as encoded by the eigenfunctions of the respective covariance operators. To this end, a novel testing approach is introduced that connects with, and extends, existing literature in two main ways. First, tests are set up in the relevant testing framework, where interest is not in testing an exact null hypothesis but rather in detecting deviations deemed sufficiently relevant, with relevance determined by the practitioner and perhaps guided by domain experts. Second, the proposed test statistics rely on a self-normalization principle that helps to avoid the notoriously difficult task of estimating the long-run covariance structure of the underlying functional time series. The main theoretical result of this paper is the derivation of the large-sample behavior of the proposed test statistics. Empirical evidence, indicating that the proposed procedures work well in finite samples and compare favorably with competing methods, is provided through a simulation study, and an application to annual temperature data.

math.ST↗

arXiv2019

Testing for stationarity of functional time series in the frequency domain

Interest in functional time series has spiked in the recent past with papers covering both methodology and applications being published at a much increased pace. This article contributes to the research in this area by proposing a new stationarity test for functional time series based on frequency domain methods. The proposed test statistics is based on joint dimension reduction via functional principal components analysis across the spectral density operators at all Fourier frequencies, explicitly allowing for frequency-dependent levels of truncation to adapt to the dynamics of the underlying functional time series. The properties of the test are derived both under the null hypothesis of stationary functional time series and under the smooth alternative of locally stationary functional time series. The methodology is theoretically justified through asymptotic results. Evidence from simulation studies and an application to annual temperature curves suggests that the test works well in finite samples.

stat.ME↗

arXiv2019

Bootstrapping spectral statistics in high dimensions

Statistics derived from the eigenvalues of sample covariance matrices are called spectral statistics, and they play a central role in multivariate testing. Although bootstrap methods are an established approach to approximating the laws of spectral statistics in low-dimensional problems, these methods are relatively unexplored in the high-dimensional setting. The aim of this paper is to focus on linear spectral statistics as a class of prototypes for developing a new bootstrap in high-dimensions --- and we refer to this method as the Spectral Bootstrap. In essence, the method originates from the parametric bootstrap, and is motivated by the notion that, in high dimensions, it is difficult to obtain a non-parametric approximation to the full data-generating distribution. From a practical standpoint, the method is easy to use, and allows the user to circumvent the difficulties of complex asymptotic formulas for linear spectral statistics. In addition to proving the consistency of the proposed method, we provide encouraging empirical results in a variety of settings. Lastly, and perhaps most interestingly, we show through simulations that the method can be applied successfully to statistics outside the class of linear spectral statistics, such as the largest sample eigenvalue and others.

stat.ME↗

arXiv2018

High-dimensional general linear hypothesis tests via non-linear spectral shrinkage

We are interested in testing general linear hypotheses in a high-dimensional multivariate linear regression model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA and others as special cases. A family of rotation-invariant tests is proposed that involves a flexible spectral shrinkage scheme applied to the sample error covariance matrix. The asymptotic normality of the test statistic under the null hypothesis is derived in the setting where dimensionality is comparable to sample sizes, assuming the existence of certain moments for the observations. The asymptotic power of the proposed test is studied under various local alternatives. The power characteristics are then utilized to propose a data-driven selection of the spectral shrinkage function. As an illustration of the general theory, we construct a family of tests involving ridge-type regularization and suggest possible extensions to more complex regularizers. A simulation study is carried out to examine the numerical performance of the proposed tests.

stat.ME↗

arXiv2018

Structural break analysis for spectrum and trace of covariance operators?

This paper deals with analyzing structural breaks in the covariance operator of sequentially observed functional data. For this purpose, procedures are developed to segment an observed stretch of curves into periods for which second-order stationarity may be reasonably assumed. The proposed methods are based on measuring the fluctuations of sample eigenvalues, either individually or jointly, and traces of the sample covariance operator computed from segments of the data. To implement the tests, new limit results are introduced that deal with the large-sample behavior of vector-valued processes built from partial sample eigenvalue estimates. These results in turn enable the calibration of the tests to a prescribed asymptotic level. A simulation study and an application to Australian annual minimum temperature curves confirm that the proposed methods work well in finite samples. The application suggests that the variation in annual minimum temperature underwent a structural break in the 1950s, after which typical fluctuations from the generally increasing trendstarted to be significantly smaller.

stat.ME↗

arXiv2017

Functional data analysis in the Banach space of continuous functions

Functional data analysis is typically conducted within the $L^2$-Hilbert space framework. There is by now a fully developed statistical toolbox allowing for the principled application of the functional data machinery to real-world problems, often based on dimension reduction techniques such as functional principal component analysis. At the same time, there have recently been a number of publications that sidestep dimension reduction steps and focus on a fully functional $L^2$-methodology. This paper goes one step further and develops data analysis methodology for functional time series in the space of all continuous functions. The work is motivated by the fact that objects with rather different shapes may still have a small $L^2$-distance and are therefore identified as similar when using an $L^2$-metric. However, in applications it is often desirable to use metrics reflecting the visualization of the curves in the statistical analysis. The methodological contributions are focused on developing two-sample and change-point tests as well as confidence bands, as these procedures appear do be conducive to the proposed setting. Particular interest is put on relevant differences; that is, on not trying to test for exact equality, but rather for pre-specified deviations under the null hypothesis. The procedures are justified through large-sample theory. To ensure practicability, non-standard bootstrap procedures are developed and investigated addressing particular features that arise in the problem of testing relevant hypotheses. The finite sample properties are explored through a simulation study and an application to annual temperature profiles.

math.ST↗

arXiv2017

Detecting and dating structural breaks in functional data without dimension reduction

Methodology is proposed to uncover structural breaks in functional data that is "fully functional" in the sense that it does not rely on dimension reduction techniques. A thorough asymptotic theory is developed for a fully functional break detection procedure as well as for a break date estimator, assuming a fixed break size and a shrinking break size. The latter result is utilized to derive confidence intervals for the unknown break date. The main results highlight that the fully functional procedures perform best under conditions when analogous fPCA based estimators are at their worst, namely when the feature of interest is orthogonal to the leading principal components of the data. The theoretical findings are confirmed by means of a Monte Carlo simulation study in finite samples. An application to annual temperature curves illustrates the practical relevance of the proposed procedures.

stat.ME↗