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Building prediction models from mass-spectrometry data is challenging due to the abundance of correlated features with varying degrees of zero-inflation, leading to a common interest in reducing the features to a concise predictor set with good predictive performance. In this study, we formally established and examined regularized regression approaches, designed to address zero-inflated and correlated predictors. In particular, we describe a novel two-stage regularized regression approach (ridge-garrote) explicitly modelling zero-inflated predictors using two component variables, comprising a ridge estimator in the first stage and subsequently applying a nonnegative garrote estimator in the second stage. We contrasted ridge-garrote with one-stage methods (ridge, lasso) and other two-stage regularized regression approaches (lasso-ridge, ridge-lasso) for zero-inflated predictors. We assessed the predictive performance and predictor selection properties of these methods in a comparative simulation study and a real-data case study to predict kidney function using peptidomic features derived from mass-spectrometry. In the simulation study, the predictive performance of all assessed approaches was comparable, yet the ridge-garrote approach consistently selected more parsimonious models compared to its competitors in most scenarios. While lasso-ridge achieved higher predictive accuracy than its competitors, it exhibited high variability in the number of selected predictors. Ridge-lasso exhibited slightly superior predictive accuracy than ridge-garrote but at the expense of selecting more noise predictors. Overall, ridge emerged as a favourable option when variable selection is not a primary concern, while ridge-garrote demonstrated notable practical utility in selecting a parsimonious set of predictors, with only minimal compromise in predictive accuracy.

Statistical techniques are needed to analyse data structures with complex dependencies such that clinically useful information can be extracted. Individual-specific networks, which capture dependencies in complex biological systems, are often summarized by graph-theoretical features. These features, which lend themselves to outcome modelling, can be subject to high variability due to arbitrary decisions in network inference and noise. Correlation-based adjacency matrices often need to be sparsified before meaningful graph-theoretical features can be extracted, requiring the data analysts to determine an optimal threshold.. To address this issue, we propose to incorporate a flexible weighting function over the full range of possible thresholds to capture the variability of graph-theoretical features over the threshold domain. The potential of this approach, which extends concepts from functional data analysis to a graph-theoretical setting, is explored in a plasmode simulation study using real functional magnetic resonance imaging (fMRI) data from the Autism Brain Imaging Data Exchange (ABIDE) Preprocessed initiative. The simulations show that our modelling approach yields accurate estimates of the functional form of the weight function, improves inference efficiency, and achieves a comparable or reduced root mean square prediction error compared to competitor modelling approaches. This assertion holds true in settings where both complex functional forms underlie the outcome-generating process and a universal threshold value is employed. We demonstrate the practical utility of our approach by using resting-state fMRI data to predict biological age in children. Our study establishes the flexible modelling approach as a statistically principled, serious competitor to ad-hoc methods with superior performance.

In this note we continue the study of nonlocal interaction dynamics on a sequence of infinite graphs, extending the results of [Esposito et. al 2023+] to an arbitrary number of species. Our analysis relies on the observation that the graph dynamics form a gradient flow with respect to a non-symmetric Finslerian gradient structure. Keeping the nonlocal interaction energy fixed, while localising the graph structure, we are able to prove evolutionary {\Gamma}-convergence to an Otto-Wassertein-type gradient flow with a tensor-weighted, yet symmetric, inner product. As a byproduct this implies the existence of solutions to the multi-species non-local (cross-)interacation system on the tensor-weighted Euclidean space, Comment: arXiv admin note: text overlap with arXiv:2306.03475

We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretisation for the equation under study., Comment: 50 pages. Added result on diagonal limits. Comments welcome

We propose a framework for fitting fractional polynomials models as special cases of Bayesian Generalized Nonlinear Models, applying an adapted version of the Genetically Modified Mode Jumping Markov Chain Monte Carlo algorithm. The universality of the Bayesian Generalized Nonlinear Models allows us to employ a Bayesian version of the fractional polynomials models in any supervised learning task, including regression, classification, and time-to-event data analysis. We show through a simulation study that our novel approach performs similarly to the classical frequentist fractional polynomials approach in terms of variable selection, identification of the true functional forms, and prediction ability, while providing, in contrast to its frequentist version, a coherent inference framework. Real data examples provide further evidence in favor of our approach and show its flexibility., Comment: 18 pages, 9 tables, 3 figures

Although the biostatistical scientific literature publishes new methods at a very high rate, many of these developments are not trustworthy enough to be adopted by the scientific community. We propose a framework to think about how a piece of methodological work contributes to the evidence base for a method. Similarly to the well-known phases of clinical research in drug development, we define four phases of methodological research. These four phases cover (I) providing logical reasoning and proofs, (II) providing empirical evidence, first in a narrow target setting, then (III) in an extended range of settings and for various outcomes, accompanied by appropriate application examples, and (IV) investigations that establish a method as sufficiently well-understood to know when it is preferred over others and when it is not. We provide basic definitions of the four phases but acknowledge that more work is needed to facilitate unambiguous classification of studies into phases. Methodological developments that have undergone all four proposed phases are still rare, but we give two examples with references. Our concept rebalances the emphasis to studies in phase III and IV, i.e., carefully planned methods comparison studies and studies that explore the empirical properties of existing methods in a wider range of problems., Comment: 14 pages

We explore the dynamical behavior and energetic properties of a model of two species that interact nonlocally on finite graphs. The authors recently introduced the model in the context of nonquadratic Finslerian gradient flows on generalized graphs featuring nonlinear mobilities. In a continuous and local setting, this class of systems exhibits a wide variety of patterns, including mixing of the two species, partial engulfment, or phase separation. This work showcases how this rich behavior carries over to the graph structure. We present analytical and numerical evidence thereof., Comment: arXiv admin note: substantial text overlap with arXiv:2107.11289

We study the evolution of a system of two species with nonlinear mobility and nonlocal interactions on a graph whose vertices are given by an arbitrary, positive measure. To this end, we extend a recently introduced $2$-Wasserstein-type quasi-metric on generalized graphs, which is based on an upwind-interpolation, to the case of two-species systems, concave, nonlinear mobilities, and $p\ne 2$. We provide a rigorous interpretation of the interaction system as a gradient flow in the Finslerian setting, arising from the new quasi-metric.

For finite samples with binary outcomes penalized logistic regression such as ridge logistic regression (RR) has the potential of achieving smaller mean squared errors (MSE) of coefficients and predictions than maximum likelihood estimation. There is evidence, however, that RR is sensitive to small or sparse data situations, yielding poor performance in individual datasets. In this paper, we elaborate this issue further by performing a comprehensive simulation study, investigating the performance of RR in comparison to Firth's correction that has been shown to perform well in low-dimensional settings. Performance of RR strongly depends on the choice of complexity parameter that is usually tuned by minimizing some measure of the out-of-sample prediction error or information criterion. Alternatively, it may be determined according to prior assumptions about true effects. As shown in our simulation and illustrated by a data example, values optimized in small or sparse datasets are negatively correlated with optimal values and suffer from substantial variability which translates into large MSE of coefficients and large variability of calibration slopes. In contrast, if the degree of shrinkage is pre-specified, accurate coefficients and predictions can be obtained even in non-ideal settings such as encountered in the context of rare outcomes or sparse predictors.

Penalized logistic regression methods are frequently used to investigate the relationship between a binary outcome and a set of explanatory variables. The model performance can be assessed by measures such as the concordance statistic (c-statistic), the discrimination slope and the Brier score. Often, data resampling techniques, e.g. crossvalidation, are employed to correct for optimism in these model performance criteria. Especially with small samples or a rare binary outcome variable, leave-one-out crossvalidation is a popular choice. Using simulations and a real data example, we compared the effect of different resampling techniques on the estimation of c-statistics, discrimination slopes and Brier scores for three estimators of logistic regression models, including the maximum likelihood and two maximum penalized-likelihood estimators. Our simulation study confirms earlier studies reporting that leave-one-out crossvalidated c-statistics can be strongly biased towards zero. In addition, our study reveals that this bias is more pronounced for estimators shrinking predicted probabilities towards the observed event rate, such as ridge regression. Leave-one-out crossvalidation also provided pessimistic estimates of the discrimination slope but nearly unbiased estimates of the Brier score. We recommend to use leave-pair-out crossvalidation, five-fold crossvalidation with repetition, the enhanced or the .632+ bootstrap to estimate c-statistics and leave-pair-out or five-fold crossvalidation to estimate discrimination slopes.

Firth-type logistic regression has become a standard approach for the analysis of binary outcomes with small samples. Whereas it reduces the bias in maximum likelihood estimates of coefficients, bias towards 1/2 is introduced in the predicted probabilities. The stronger the imbalance of the outcome, the more severe is the bias in the predicted probabilities. We propose two simple modifications of Firth-type logistic regression resulting in unbiased predicted probabilities. The first corrects the predicted probabilities by a post-hoc adjustment of the intercept. The other is based on an alternative formulation of Firth-types estimation as an iterative data augmentation procedure. Our suggested modification consists in introducing an indicator variable which distinguishes between original and pseudo observations in the augmented data. In a comprehensive simulation study these approaches are compared to other attempts to improve predictions based on Firth-type penalization and to other published penalization strategies intended for routine use. For instance, we consider a recently suggested compromise between maximum likelihood and Firth-type logistic regression. Simulation results are scrutinized both with regard to prediction and regression coefficients. Finally, the methods considered are illustrated and compared for a study on arterial closure devices in minimally invasive cardiac surgery.

Separation in logistic regression is a common problem causing failure of the iterative estimation process when finding maximum likelihood estimates. Firth's correction (FC) was proposed as a solution, providing estimates also in presence of separation. In this paper we evaluate whether ridge regression (RR) could be considered instead, specifically, if it could reduce the mean squared error (MSE) of coefficient estimates in comparison to FC. In RR the tuning parameter determining the penalty strength is usually obtained by minimizing some measure of the out-of-sample prediction error or information criterion. However, in presence of separation tuning these measures can yield an optimized value of zero (no shrinkage), and hence cannot provide a universal solution. We derive a new bootstrap based tuning criterion $B$ that always leads to shrinkage. Moreover, we demonstrate how valid inference can be obtained by combining resampled profile penalized likelihood functions. Our approach is illustrated in an example from oncology and its performance is compared to FC in a simulation study. Our simulations showed that in analyses of small and sparse datasets and with many correlated covariates $B$-tuned RR can yield coefficient estimates with MSE smaller than FC and confidence intervals that approximately achieve nominal coverage probabilities., Comment: 45 pages, 6 figures

Variable selection for regression models plays a key role in the analysis of biomedical data. However, inference after selection is not covered by classical statistical frequentist theory which assumes a fixed set of covariates in the model. We review two interpretations of inference after selection: the full model view, in which the parameters of interest are those of the full model on all predictors, and then focus on the submodel view, in which the parameters of interest are those of the selected model only. In the context of L1-penalized regression we compare proposals for submodel inference (selective inference) via confidence intervals available to applied researchers via software packages using a simulation study inspired by real data commonly seen in biomedical studies. Furthermore, we present an exemplary application of these methods to a publicly available dataset to discuss their practical usability. Our findings indicate that the frequentist properties of selective confidence intervals are generally acceptable, but desired coverage levels are not guaranteed in all scenarios except for the most conservative methods. The choice of inference method potentially has a large impact on the resulting interval estimates, thereby necessitating that the user is acutely aware of the goal of inference in order to interpret and communicate the results. Currently available software packages are not yet very user friendly or robust which might affect their use in practice. In summary, we find submodel inference after selection useful for experienced statisticians to assess the importance of individual selected predictors in future applications., Comment: 42 pages, 19 figures, 7 tables

Future quantum repeater architectures, capable of efficiently distributing information encoded in quantum states of light over large distances, will benefit from multiplexed photonic quantum memories. In this work we demonstrate a temporally multiplexed quantum repeater node in a laser-cooled cloud of $^{87}$Rb atoms. We employ the DLCZ protocol where pairs of photons and single collective spin excitations (so called spin waves) are created in several temporal modes using a train of write pulses. To make the spin waves created in different temporal modes distinguishable and enable selective readout, we control the dephasing and rephasing of the spin waves by a magnetic field gradient, which induces a controlled reversible inhomogeneous broadening of the involved atomic hyperfine levels. We demonstrate that by embedding the atomic ensemble inside a low finesse optical cavity, the additional noise generated in multi-mode operation is strongly suppressed. By employing feed forward readout, we demonstrate distinguishable retrieval of up to 10 temporal modes. For each mode, we prove non-classical correlations between the first and second photon. Furthermore, an enhancement in rates of correlated photon pairs is observed as we increase the number of temporal modes stored in the memory. The reported capability is a key element of a quantum repeater architecture based on multiplexed quantum memories.

How to select variables and identify functional forms for continuous variables is a key concern when creating a multivariable model. Ad hoc 'traditional' approaches to variable selection have been in use for at least 50 years. Similarly, methods for determining functional forms for continuous variables were first suggested many years ago. More recently, many alternative approaches to address these two challenges have been proposed, but knowledge of their properties and meaningful comparisons between them are scarce. To define a state-of-the-art and to provide evidence-supported guidance to researchers who have only a basic level of statistical knowledge many outstanding issues in multivariable modelling remain. Our main aims are to identify and illustrate such gaps in the literature and present them at a moderate technical level to the wide community of practitioners, researchers and students of statistics. We briefly discuss general issues in building descriptive regression models, strategies for variable selection, different ways of choosing functional forms for continuous variables, and methods for combining the selection of variables and functions. We discuss two examples, taken from the medical literature, to illustrate problems in the practice of modelling. Our overview revealed that there is not yet enough evidence on which to base recommendations for the selection of variables and functional forms in multivariable analysis. Such evidence may come from comparisons between alternative methods. In particular, we highlight seven important topics that require further investigation and make suggestions for the direction of further research.

Interfacing fundamentally different quantum systems is key to build future hybrid quantum networks. Such heterogeneous networks offer superior capabilities compared to their homogeneous counterparts as they merge individual advantages of disparate quantum nodes in a single network architecture. However, only very few investigations on optical hybrid-interconnections have been carried out due to the high fundamental and technological challenges, which involve e.g. wavelength and bandwidth matching of the interfacing photons. Here we report the first optical quantum interconnection between two disparate matter quantum systems with photon storage capabilities. We show that a quantum state can be faithfully transferred between a cold atomic ensemble and a rare-earth doped crystal via a single photon at telecommunication wavelength, using cascaded quantum frequency conversion. We first demonstrate that quantum correlations between a photon and a single collective spin excitation in the cold atomic ensemble can be transferred onto the solid-state system. We also show that single-photon time-bin qubits generated in the cold atomic ensemble can be converted, stored and retrieved from the crystal with a conditional qubit fidelity of more than $85\%$. Our results open prospects to optically connect quantum nodes with different capabilities and represent an important step towards the realization of large-scale hybrid quantum networks.

Strong interaction between two single photons is a long standing and important goal in quantum photonics. This would enable a new regime of nonlinear optics and unlock several applications in quantum information science, including photonic quantum gates and deterministic Bell-state measurements. In the context of quantum networks, it would be important to achieve interactions between single photons from independent photon pairs storable in quantum memories. So far, most experiments showing nonlinearities at the single-photon level have used weak classical input light. Here, we demonstrate the storage and retrieval of a paired single photon emitted by an ensemble quantum memory in a strongly nonlinear medium based on highly excited Rydberg atoms. We show that nonclassical correlations between the two photons persist after retrieval from the Rydberg ensemble. Our result is an important step towards deterministic photon-photon interactions, and may enable deterministic Bell-state measurements with multimode quantum memories.

Entanglement between light and matter combines the advantage of long distance transmission of photonic qubits with the storage and processing capabilities of atomic qubits. To distribute photonic states efficiently over long distances several schemes to encode qubits have been investigated -- time-bin encoding being particularly promising due to its robustness against decoherence in optical fibers. Here, we demonstrate the generation of entanglement between a photonic time-bin qubit and a single collective atomic spin excitation (spin-wave) in a cold atomic ensemble, followed by the mapping of the atomic qubit onto another photonic qubit. A magnetic field that induces a periodic dephasing and rephasing of the atomic excitation ensures the temporal distinguishability of the two time-bins and plays a central role in the entanglement generation. To analyse the generated quantum state, we use largely imbalanced Mach-Zehnder interferometers to perform projective measurements in different qubit bases and verify the entanglement by violating a CHSH Bell inequality.

We report on quantum frequency conversion of memory-compatible narrow-bandwidth photons at 606 nm to the telecom C-band at 1552$\,$nm. The 200$\,$ns long photons, compatible with Praseodymium-based solid-state quantum memories are frequency converted using a single-step difference frequency generation process in a periodically poled Lithium Niobate waveguide. We characterize the noise processes involved in the conversion and by applying strong spectral filtering of the noise, we demonstrate high signal-to-noise ratio conversion at the single photon level (SNR$\,>\,$100 for a mean input photon number per pulse of 1). We finally observe that a memory compatible heralded single photon with a bandwidth of 1.8$\,$MHz, obtained from a spontaneous parametric down conversion pair source, still shows a strong non-classical behavior after conversion. We first demonstrate that correlations between heralding and converted heralded photons stay in the non-classical regime. Moreover, we measured the heralded autocorrelation function of the heralded photon using the converter device as a frequency-domain beam splitter, yielding a value of $0.19\pm0.07$. The presented work represents a step towards the connection of several quantum memory systems emitting narrow-band visible photons to the telecommunication wavelengths.