Monotonic Transformations Research Articles

Stress-strength reliability for doubly truncated random variables

In the context of reliability and survival analysis, the observed lifetimes are often subject to left, right, or double truncation. Doubly truncated data usually arises when information about the lifetime of a system is available only within a particular time interval. In this paper, we consider the stress-strength reliability measure for the doubly truncated data. Several properties of this measure, including bounds and the effect of monotonic transformations, are discussed. Further, we propose a nonparametric estimator for doubly truncated stress-strength reliability, and its asymptotic properties are studied. A simulation study is also carried out to evaluate the performance of the asymptotic confidence interval and the percentile bootstrap interval for doubly truncated stress-strength reliability. Finally, the proposed estimation procedure is applied to a real data set for illustration.

Efficient parameter estimation for ODE models of cellular processes using semi-quantitative data.

Quantitative dynamical models facilitate the understanding of biological processes and the prediction of their dynamics. The parameters of these models are commonly estimated from experimental data. Yet, experimental data generated from different techniques do not provide direct information about the state of the system but a nonlinear (monotonic) transformation of it. For such semi-quantitative data, when this transformation is unknown, it is not apparent how the model simulations and the experimental data can be compared. We propose a versatile spline-based approach for the integration of a broad spectrum of semi-quantitative data into parameter estimation. We derive analytical formulas for the gradients of the hierarchical objective function and show that this substantially increases the estimation efficiency. Subsequently, we demonstrate that the method allows for the reliable discovery of unknown measurement transformations. Furthermore, we show that this approach can significantly improve the parameter inference based on semi-quantitative data in comparison to available methods. Modelers can easily apply our method by using our implementation in the open-source Python Parameter EStimation TOolbox (pyPESTO) available at https://github.com/ICB-DCM/pyPESTO.

A theory of multivariate stress testing

We present a theoretical framework for stressing multivariate stochastic models. We consider a stress to be a change of measure, placing a higher weight on multivariate scenarios of interest. In particular, a stressing mechanism is a mapping from random vectors to Radon–Nikodym densities. We postulate desirable properties for stressing mechanisms addressing alternative objectives. Consistently with our focus on dependence, we require throughout invariance to monotonic transformations of risk factors. We study in detail the properties of two families of stressing mechanisms, based respectively on mixtures of univariate stresses and on transformations of statistics we call Spearman and Kendall’s cores. Furthermore, we characterize the aggregation properties of those stressing mechanisms, which motivate their use in deriving new capital allocation methods, with properties different to those typically found in the literature. The proposed methods are applied to stress testing and capital allocation, using the simulation model of a UK-based non-life insurer.

Monotonic transformation and recovering the implied stock price process

AbstractThe paper demonstrates the construction of a stock price stochastic process that aligns with observed option prices, drawing inspiration from a manuscript by Professor Erio Castagnoli that employs the concept of a non-linear monotonic transformation of a standard Brownian motion. The paper includes a detailed numerical example that illustrates the implementation of this straightforward and ingenious idea.

Coordinate-wise monotonic transformations enable privacy-preserving age estimation with 3D face point cloud.

The human face is a valuable biomarker of aging, but the collection and use of its image raise significant privacy concerns. Here we present an approach for facial data masking that preserves age-related features using coordinate-wise monotonic transformations. We first develop a deep learning model that estimates age directly from non-registered face point clouds with high accuracy and generalizability. We show that the model learns a highly indistinguishable mapping using faces treated with coordinate-wise monotonic transformations, indicating that the relative positioning of facial information is a low-level biomarker of facial aging. Through visual perception tests and computational 3D face verification experiments, we demonstrate that transformed faces are significantly more difficult to perceive for human but not for machines, except when only the face shape information is accessible. Our study leads to a facial data protection guideline that has the potential to broaden public access to face datasets with minimized privacy risks.

Convergence Rate of the (1+1)-ES on Locally Strongly Convex and Lipschitz Smooth Functions

Evolution strategy (ES) is one of promising classes of algorithms for black-box continuous optimization. Despite its broad successes in applications, theoretical analysis on the speed of its convergence is limited on convex quadratic functions and their monotonic transformation. In this study, an upper bound and a lower bound of the rate of linear convergence of the (1+1)-ES on locally $L$-strongly convex functions with $U$-Lipschitz continuous gradient are derived as $\exp\left(-\Omega_{d\to\infty}\left(\frac{L}{d\cdot U}\right)\right)$ and $\exp\left(-\frac1d\right)$, respectively. Notably, any prior knowledge on the mathematical properties of the objective function such as Lipschitz constant is not given to the algorithm, whereas the existing analyses of derivative-free optimization algorithms require them.

Quantifying and Maximizing the Information Flux in Recurrent Neural Networks.

Free-running recurrent neural networks (RNNs), especially probabilistic models, generate an ongoing information flux that can be quantified with the mutual information I[x→(t),x→(t+1)] between subsequent system states x→. Although previous studies have shown that I depends on the statistics of the network's connection weights, it is unclear how to maximize I systematically and how to quantify the flux in large systems where computing the mutual information becomes intractable. Here, we address these questions using Boltzmann machines as model systems. We find that in networks with moderately strong connections, the mutual information I is approximately a monotonic transformation of the root-mean-square averaged Pearson correlations between neuron pairs, a quantity that can be efficiently computed even in large systems. Furthermore, evolutionary maximization of I[x→(t),x→(t+1)] reveals a general design principle for the weight matrices enabling the systematic construction of systems with a high spontaneous information flux. Finally, we simultaneously maximize information flux and the mean period length of cyclic attractors in the state-space of these dynamical networks. Our results are potentially useful for the construction of RNNs that serve as short-time memories or pattern generators.

A note on interval cumulative past Renyi entropy

In analogy with cumulative residual entropy, Di Crescenzo and Longobardi (2009) proposed cumulative past entropy which served as the basis for many following works. To this end, we define cumulative past Renyi entropy for double truncated random variable which we call as interval cumulative past Renyi entropy (ICPRE). Some properties, such as effect of monotone transformation and bounds of ICPRE, are studied. Furthermore, it is shown that the proposed measure uniquely determines the distribution function and characterizes certain lifetime distributions. Finally, non-parametric estimators of ICPRE are obtained and the best proposed estimator is investigated with the help of simulated data set and real-life data set.

Estimation of correlation coefficient with monotone transformation and multiplicative distortions

This article studies the estimation of the correlation coefficient between unobserved variables of interest. These unobservable variables, mixed with some known monotone link functions, are distorted in multiplicative fashion by an observed confounding variable. We propose four calibration methods for the unobserved variables and the estimation of the correlation coefficient. Theoretical results show that the proposed estimators of correlation coefficient can be asymptotically efficient as if there are no distortions in the variables. Moreover, we suggest an asymptotic normal approximation and an empirical likelihood-based statistic to construct the confidence intervals. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators. These methods are applied to analyze a real dataset for an illustration.

Weighted (residual) varentropy and its applications

In information theory, it is of recent interest to study variability of the uncertainty measures. In this regard, the concept of varentropy has been introduced and studied by several authors in recent past. In this communication, we study the weighted varentropy and weighted residual varentropy. Several theoretical results of these variability measures such as the effect under monotonic transformations and bounds are investigated. Importance of the weighted residual varentropy over the residual varentropy is presented. Further, we study weighted varentropy for coherent systems and weighted residual varentropy for proportional hazard rate models. A kernel-based non-parametric estimator for the weighted residual varentropy is also proposed. The estimation method is illustrated using simulated and two real data sets.

Captive jump processes for bounded random systems with discontinuous dynamics

Stochastic captive jump processes are explicitly constructed in continuous time, whose non-linear dynamics are strictly confined by bounded domains that can be time-dependent. By introducing non-anticipative path-dependency, the framework offers the possibility of generating multiple inner tunnels within a master domain, such that a captive jump process is allowed to proceed either within a single inner tunnel or jump in between tunnels without ever penetrating the outermost shell. If a captive jump process is a continuous martingale or a pure-jump process, the uppermost confining boundary is non-decreasing, and the lowermost confining boundary is non-increasing. Under certain conditions, it can be shown that captive jump processes are invariant under monotonic transformations, enabling one to construct and study systems of increasing complexity using simpler building blocks. Amongst many applications, captive jump processes may be considered to model phenomena such as electrons transitioning from one orbit (valence shell) to another, quantum tunnelling where stochastic wave-functions can “penetrate” inner boundaries (i.e., walls) of potential energy, non-linear dynamical systems involving multiple attractors, and sticky concentration behaviour of pathogens in epidemics. We provide concrete, worked-out examples, and numerical simulations for the dynamics of captive jump processes within different geometries as demonstrations.

Quantifying the direct and indirect interactions for EEG signals by using detrended permutation mutual information

In this paper, we extend detrended cross-correlations analysis (DCCA) and introduce a novel method called detrended permutation mutual information (DPMI) by utilizing information-theoretic tools. The main mechanism of this paper is to use the invariance of the nonlinear monotonic transformation and robustness in permutation mutual information (PMI), which solves the limitations of DCCA only applicable to linear measurements and enhances the robustness of DPMI to noise. We compare the performance of DPMI with DCCA on multifractal binomial measures and Hénon maps for testing the effectiveness of the new method. By analyzing EEG signals obtained from children with attention deficit hyperactivity disorder (ADHD) as well as healthy children, we demonstrate that DPMI surpasses DCCA in processing physiological signals. Furthermore, we conclude that DPMI is more effective in distinguishing between different states within distinct groups. Given that EEG signals in physiological systems are associated with other channel signals, we propose a method called partial DPMI (PDPMI) that combines partial-correlation technique with DPMI. PDPMI can further reveal the direct interactions between two considered signals by eliminating potential influences from other unconsidered signals. In summary, PDPMI serves as a complementary perspective to DPMI, enhancing our understanding in the interactive structure of physiological network.

Multi sensor fusion methods for state of charge estimation of smart lithium-ion batteries

Multi-dimensional sensing is the key characteristic of next generation smart batteries. But the existing researches on multi-sensor fusion methods haven't focused on algorithm mechanism, the global optimal solution has not been built, the superiority has not been proved theoretically. For the first time, this research built the global optimal structure of multi-sensor fusion state estimation algorithm. Specifically, the state of charge (SoC) estimation problem of lithium iron phosphate (LFP) batteries is studied, cooperating with voltage signal, expansion force (EF) signal is introduced. Firstly, a normalization algorithm is introduced to overcome the drift of EF under different cycles and different pre-tightening force. Secondly, the non-monotonic relationship of LFP battery's EF-SoC curve is addressed with a forced monotone transformation method in the transition areas. Then the global optimal multi-sensor fusion method is built, theoretical reductions are carried out to prove the higher precision of multi-sensor fusion methods compared to single-signal methods. Experiments are conducted to verify the effectiveness of the methods, even under the most serious situations, the fusion methods exhibit powerful correction ability, and the root mean square error can be controlled within 3 %. Moreover, the proposed algorithms shows strong tolerance to error sources.

Robust inference for subgroup analysis with general transformation models

A crucial step in developing personalized strategies in precision medicine or precision marketing is to identify the latent subgroups of patients or customers of a heterogeneous population. In this article, we consider a general class of heterogeneous transformation models for subgroup identification, under which an unknown monotonic transformation of the response is linearly related to the covariates via subject-specific regression coefficients with unknown error distribution. This class of models is broad enough to cover many popular models, including the heterogeneous linear model, the heterogeneous Cox’s proportional hazard model and the heterogeneous proportional odds model. Without any priori grouping information, we propose a robust method based on the maximum rank correlation and a concave fusion to automatically determine the number of subgroups, identify the latent subgroup structure, and estimate the subgroup-specific covariate effects simultaneously. We establish the theoretical properties of our proposed estimate under regularity conditions. A random weighting resampling scheme is used for variance estimation. The proposed procedure can be easily extended to handle censored data. Numerical studies including simulations and two real data analysis demonstrate that the proposed method performs reasonably well in practical situations.

Conditional quantiles: An operator-theoretical approach

This paper derives several novel properties of conditional quantiles viewed as nonlinear operators. The results are organized in parallel to the usual properties of the expectation operator. We first define a τ-conditional quantile random set, relative to any sigma-algebra, as a set of solutions of an optimization problem. Then, well-known properties of unconditional quantiles, as translation invariance, comonotonicity, and equivariance to monotone transformations, are generalized to the conditional case. Moreover, a simple proof for Jensen’s inequality for conditional quantiles is provided. We also investigate continuity of conditional quantiles as operators with respect to different topologies and obtain a novel Fatou’s lemma for quantiles. Conditions for continuity in Lp and weak continuity are also derived. Then, the differentiability properties of quantiles are addressed. We demonstrate the validity of Leibniz’s rule for conditional quantiles for the cases of monotone, as well as separable functions. Finally, although the law of iterated quantiles does not hold in general, we characterize the maximum set of random variables for which this law holds, and investigate its consequences for the infinite composition of conditional quantiles.

Endomorphisms of semigroups of monotone transformations

In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain.

Model-free, monotone invariant and computationally efficient feature screening with data-adaptive threshold

Feature screening for ultrahigh-dimensional data, in general, proceeds with two essential steps. The first step is measuring and ranking the marginal dependence between response and covariates, and the second is determining the threshold. We develop a new screening procedure, called SIT-BY procedure, that possesses appealing statistical properties in both steps. By employing sliced independence estimates in the measuring and ranking stage, our proposed procedure requires no model assumptions, remains invariant to monotone transformation, and achieves almost linear computation complexity. Inspired by false discovery rate (FDR) control procedures, we offer a data-adaptive threshold benefit from the asymptotic normality of test statistics. Under moderate conditions, we demonstrate that our procedure can asymptotically control the FDR while maintaining the sure screening property. We investigate the finite sample performance of our proposed procedure via extensive simulations and an application to genome-wide dataset.

Spearman Rank Correlation Screening for Ultrahigh-Dimensional Censored Data

Herein, we propose a Spearman rank correlation-based screening procedure for ultrahigh-dimensional data with censored response cases. The proposed method is model-free without specifying any regression forms of predictors or response variables and is robust under the unknown monotone transformations of these response variable and predictors. The sure-screening and rank-consistency properties are established under some mild regularity conditions. Simulation studies demonstrate that the new screening method performs well in the presence of a heavy-tailed distribution, strongly dependent predictors or outliers, and offers superior performance over the existing nonparametric screening procedures. In particular, the new screening method still works well when a response variable is observed under a high censoring rate. An illustrative example is provided.

Estimating transformations for evaluating diagnostic tests with covariate adjustment.

Receiver operating characteristic analysis is one of the most popular approaches for evaluating and comparing the accuracy of medical diagnostic tests. Although various methodologies have been developed for estimating receiver operating characteristic curves and their associated summary indices, there is no consensus on a single framework that can provide consistent statistical inference while handling the complexities associated with medical data. Such complexities might include non-normal data, covariates that influence the diagnostic potential of a test, ordinal biomarkers or censored data due to instrument detection limits. We propose a regression model for the transformed test results which exploits the invariance of receiver operating characteristic curves to monotonic transformations and accommodates these features. Simulation studies show that the estimates based on transformation models are unbiased and yield coverage at nominal levels. The methodology is applied to a cross-sectional study of metabolic syndrome where we investigate the covariate-specific performance of weight-to-height ratio as a non-invasive diagnostic test. Software implementations for all the methods described in the article are provided in the tram add-on package to the R system for statistical computing and graphics.

The nonparametric Box–Cox model for high-dimensional regression analysis

The mainstream theory for high-dimensional regression assumes that the underlying true model is a low-dimensional linear regression model. On the other hand, a standard technique in regression analysis, even in the traditional low-dimensional setting, is to employ the Box–Cox transformation for reducing anomalies such as non-additivity and heteroscedasticity in linear regression. In this paper, we propose a new high-dimensional regression method based on a nonparametric Box–Cox model with an unspecified monotone transformation function. Model fitting and computation become much more challenging than the usual penalized regression method, and a two-step method is proposed for the estimation of this model in high-dimensional settings. First, we propose a novel technique called composite probit regression (CPR) and use the folded concave penalized CPR for estimating the regression parameters. The strong oracle property of the estimator is established without knowing the nonparametric transformation function. Next, the nonparametric function is estimated by conducting univariate monotone regression. The computation is done efficiently by using a coordinate-majorization-descent algorithm. Extensive simulation studies show that the proposed method performs well in various settings. Our analysis of the supermarket data demonstrates the superior performance of the proposed method over the standard high-dimensional regression method.