Distortion Function Research Articles

Transmuted Distortion Functions for Measuring Risks

Transmuted Distortion Functions for Measuring Risks

Optimal reinsurance design under distortion risk measures and reinsurer’s default risk with partial recovery

AbstractReinsurers may default when they have to pay large claims to insurers but are unable to fulfill their obligations due to various reasons such as catastrophic events, underwriting losses, inadequate capitalization, or financial mismanagement. This paper studies the problem of optimal reinsurance design from the perspectives of both the insurer and reinsurer when the insurer faces the potential default risk of the reinsurer. If the insurer aims to minimize the convex distortion risk measure of his retained loss, we prove the optimality of a stop-loss treaty when the promised ceded loss function is charged by the expected value premium principle and the reinsurer offers partial recovery in the event of default. For any fixed premium loading set by the reinsurer, we then derive the explicit expressions of optimal deductible levels for three special distortion functions, including the TVaR, Gini, and PH transform distortion functions. Under these three explicit distortion risk measures adopted by the insurer, we seek the optimal safety loading for the reinsurer by maximizing her net profit where the reserve capital is determined by the TVaR measure and the cost is governed by the expectation. This procedure ultimately leads to the Bowley solution between the insurer and the reinsurer. We provide several numerical examples to illustrate the theoretical findings. Sensitivity analyses demonstrate how different settings of default probability, recovery rate, and safety loading affect the optimal deductible values. Simulation studies are also implemented to analyze the effects induced by the default probability and recovery rate on the Bowley solution.

The Generic Failure of Lower-Semicontinuity for the Linear Distortion Functional

AbstractWe consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, $$n\ge 3$$ n ≥ 3 . The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it ifollows that if $$ \{ f_{n} \}_{n=1}^{\infty } $$ { f n } n = 1 ∞ is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion $$H({f_{n}})$$ H ( f n ) ), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower-semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower-semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence $$ \{f_{n} \}_{n=1}^{\infty } $$ { f n } n = 1 ∞ with $$ {f_{n}}\rightarrow {f}$$ f n → f locally uniformly and with $$\limsup _{n\rightarrow \infty } H( {f_{n}})<H( {f})$$ lim sup n → ∞ H ( f n ) < H ( f ) . Our main result shows this is true for affine mappings. Addressing conjectures of Gehring and Iwaniec we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds: for each $$\alpha <\sqrt{2}$$ α < 2 there is $${f_{n}}\rightarrow {f}$$ f n → f locally uniformly with f affine and $$\begin{aligned} \alpha \; \limsup _{n\rightarrow \infty } H( {f_{n}}) < H( {f}) \end{aligned}$$ α lim sup n → ∞ H ( f n ) < H ( f ) We conjecture $$\sqrt{2}$$ 2 to be best possible.

Stochastic orders and distortion risk contribution ratio measures

Relative spillover effects play a crucial role in the analysis and comparison of systemic risks. This paper introduces a novel approach, referred to as distortion risk contribution ratio measures, for quantifying such effects. Various types of contribution ratio measures are defined based on the newly proposed conditional distortion risk measures by Dhaene et al. (2022), and useful integral-based representations are provided as well. An interesting equivalent characterization result for the convex transform order is also presented, which is not only relevant to proving our main results but also has independent value in other research areas. We then establish comparison results between the distortion risk contribution ratio measures of two different bivariate random vectors with either the same or different copulas. Sufficient conditions are established in terms of stochastic orders, copula functions, distortion functions, and stress levels. Furthermore, we investigate the ordering behaviors of these measures in relation to the interaction between paired risks. Numerical examples are presented to illustrate the conditions and main findings.

Influence of the curing stress effect on the stiffness degradation curve of a silt stabilized with lime and cement

This study investigated the shear modulus degradation curves of reconstituted stabilized soil specimens and their evolution during curing up to 270 days. An experimental protocol is proposed to get as close as possible to the in situ curing conditions of a chemically (lime and hydraulic binder) and mechanically (compaction) improved soil placed in a high-rise geotechnical structure. Innovative solutions now make it possible to build high embankments (> 10 m) in stabilized soil. In the laboratory, a silty soil stabilized with 1% lime and 5% hydraulic binder, which is a classic treatment for linear projects, was cured with a pressure of 300 and 500 kPa, corresponding to a depth in the embankment of 15 and 25 m. These structures are expected to have high mechanical performance and must be able to withstand very small deformations (earthquakes, traffic loads). Experimental campaigns are performed in laboratory using torsion tests with a resonant column apparatus (RCA) to obtain the shear modulus G as a function of distortion. A comparison was made between normalized curing conditions (temperature of 20 °C and atmospheric pressure) and curing with confining pressure (temperature of 20 °C and 300 or 500 kPa pressure). For this aim, 74 torsional RCA tests were conducted on 16 compacted soil specimens. These showed that the curing pressure, applied just after compaction, brings the expected properties more quickly and confers superior properties of around 15% in the long term. A power law was proposed to estimate the Gmax of a stabilized silt specimen as a function of time, based on the Gmax measured during the first two days of curing. This empirical law gives the Gmax over the long term (270 days) for curing with or without confining pressure. Pioneering a combined curing-testing RCA approach, this study elucidates the short and long-term behavior of stabilized silts under realistic curing conditions.

Semi-supervised correction model for turbulence-distorted images

Significant progress has been made in addressing turbulence distortion in recent years, but persistent challenges remain. Firstly, existing methods heavily rely on fully supervised optimization strategies and synthetic datasets, posing difficulties in effectively utilizing unlabeled real data for training. Secondly, most approaches construct networks in a straightforward manner, overlooking the representation model of phase distortion and point spread function (PSF) in spatial and channel dimensions. This oversight restricts the potential for distortion correction. To address these challenges, this paper proposes a semi-supervised atmospheric turbulence correction method based on the mean-teacher framework. Our approach imposes constraints on the unlabeled data of student networks using pseudo-labels generated by teacher networks, thereby enhancing the generalization ability by leveraging information from unlabeled data. Furthermore, we introduce to use no-reference image quality assessment criterion to select the most reliable pseudo-label for each unlabeled sample by predicting physical parameters that indicating the level of degradation. Additionally, we propose to combine sliding window-based self-attention with channel attention to facilitate local-global context interaction. This design is inspired by the representation of phase distortion and PSF, which can be characterized by coefficients and basis functions corresponding to the channel-wise representation of convolutional neural network features. Moreover, the base functions exhibit spatial correlation, akin to Zenike and Airy disks. Experimental results show that the proposed method surpasses state-of-the-art models.

Time consistency of dynamic risk measures and dynamic performance measures generated by distortion functions

. The aim of this work is to study risk measures generated by distortion functions in a dynamic discrete time setup and to investigate the corresponding dynamic coherent acceptability indices (DCAIs) generated by families of such risk measures. First, we show that conditional versions of Choquet integrals are indeed dynamic coherent risk measures (DCRMs) and also introduce the class of dynamic weighted value at risk measures. We prove that these two classes of risk measures coincides. In the spirit of robust representations theorem for DCAIs, we establish some relevant properties of families of DCRMs generated by distortion functions and then define and study the corresponding DCAIs. Second, we study the time consistency of DCRMs and DCAIs generated by distortion functions. In particular, we prove that such DCRMs are sub-martingale time consistent, but they are not super-martingale time consistent. We also show that DCRMs generated by distortion functions are not weakly acceptance-time consistent. We also present several widely used classes of distortion functions and derive some new representations of these distortions.

Composite loss function for 3-D poststack seismic data compression

This work introduces composite functions to compute distortion in volumetric seismic data. Several loss functions, such as those based on Lp-functions, ignore the structure of 3-D seismic data, treating it as a unidimensional vector. Alternatively, applying distinct functions in each axis, properly designed for dimension reduction, can evaluate seismic data error according to its unity and magnitude. We thus propose a novel multidimensional composite loss function to evaluate through dimension reduction, suitable for seismic data compression within a method named 3DSC-GAN. It replaces the usual peak signal-to-noise ratio (PSNR) metric as the distortion function. An extensive study is conducted to analyze potential combinations of functions for the 3-D poststack seismic data compression problem. Results indicate that the new function contributes to improve the neural network learning step. Our method provides superior reconstructions, both quantitatively and qualitatively, when compared to the PSNR metric.

New Bivariate Copulas via Lomax Distribution Generated Distortions

We develop a framework for creating distortion functions that are used to construct new bivariate copulas. It is achieved by transforming non-negative random variables with Lomax-related distributions. In this paper, we apply the distortions to the base copulas of independence, Clayton, Frank, and Gumbel copulas. The properties of the tail dependence coefficient, tail order, and concordance ordering are explored for the new families of distorted copulas. We conducted an empirical study using the daily net returns of Amazon and Google stocks from January 2014 to December 2023. We compared the popular Clayton, Gumbel, Frank, and Gaussian copula models to their corresponding distorted copula models induced by the unit-Lomax and unit-inverse Pareto distortions. The new families of distortion copulas are equipped with additional parameters inherent in the distortion function, providing more flexibility, and are demonstrated to perform better than the base copulas. After analyzing the data, we have found that the joint extremes of Amazon and Google stocks are more likely for high daily net returns than for low daily net returns.

Adaptive HEVC video steganograhpy based on PU partition modes

High EfficiencyVideoCoding (HEVC) −based steganography has gained attention as a prominent research focus. Especially, block structure based HEVC video steganography has received increasing attention due to commendable performance. However, current block structure- based steganography algorithms confront with challenges such as reduced coding efficiency and limited capacity. To avoid these problems, an adaptive video steganography algorithm based on Prediction Unit (PU) partition mode in I-frames is proposed. This is done through the analysis of the block division process and the visual distortion resulting from the modification of the PU partition mode in HEVC. The PU block structure is utilized as steganographic covers, and the Rate Distortion Optimization (RDO) technique is introduced to establish an adaptive distortion function for Syndrome-trellis code (STC). Further comparison is performed between the proposed method and the state-of-the-art steganography algorithms, confirming its advantages in embedding capacity, compression efficiency, visual quality, and resistance to video steganalysis.

Introduction to mean dimension

The purpose of this survey is to explain basic ideas of mean dimension theory. Mean dimension represents a number of parameters per unit time for describing a dynamical system. It was introduced by Gromov in 1999. His motivation is to propose a new approach to geometric analysis over noncompact manifolds. Rather independently of this original motivation, Lindenstrauss and Weiss found applications of mean dimension to classical problems in topological dynamics. In this survey we start from the viewpoint of Lindenstrauss–Weiss and first review the applications to topological dynamics. Next we explain the relation to geometric analysis, in particular holomorphic curve theory. Mean dimension is connected to rate distortion theory, which is a branch of information theory studying the trade-off between distortion and compression rate in lossy data compression. In the last three sections we review a variational principle between mean dimension and rate distortion function and then discuss its possible application to holomorphic curve theory.

Structural Properties of the Wyner-Ziv Rate Distortion Function: Applications for Multivariate Gaussian Sources.

The main focus of this paper is the derivation of the structural properties of the test channels of Wyner's operational information rate distortion function (RDF), R¯(ΔX), for arbitrary abstract sources and, subsequently, the derivation of additional properties for a tuple of multivariate correlated, jointly independent, and identically distributed Gaussian random variables, {Xt,Yt}t=1∞, Xt:Ω→Rnx, Yt:Ω→Rny, with average mean-square error at the decoder and the side information, {Yt}t=1∞, available only at the decoder. For the tuple of multivariate correlated Gaussian sources, we construct optimal test channel realizations which achieve the informational RDF, R¯(ΔX)=▵infM(ΔX)I(X;Z|Y), where M(ΔX) is the set of auxiliary RVs Z such that PZ|X,Y=PZ|X, X^=f(Y,Z), and E{||X-X^||2}≤ΔX. We show the following fundamental structural properties: (1) Optimal test channel realizations that achieve the RDF and satisfy conditional independence, PX|X^,Y,Z=PX|X^,Y=PX|X^,EX|X^,Y,Z=EX|X^=X^. (2) Similarly, for the conditional RDF, RX|Y(ΔX), when the side information is available to both the encoder and the decoder, we show the equality R¯(ΔX)=RX|Y(ΔX). (3) We derive the water-filling solution for RX|Y(ΔX).

Worst-case risk with unspecified risk preferences

In this paper, we study the worst-case distortion risk measure for a given risk when information about distortion functions is partially available. We obtain the explicit forms of the worst-case distortion functions for several different sets of plausible distortion functions. When there is no concavity constraint on distortion functions, the worst-case distortion function is independent of the risk to be measured and the corresponding worst-case distortion risk measure is the weighted average of the VaR's of the risk for all decision makers. When the concavity constraint is imposed on distortion functions and the set of concave distortion functions is defined by the riskiness of one single risk, the explicit form of the worst-case distortion function is obtained, which depends the risk to be measured. When the set of concave distortion functions is defined by the riskiness of multiple risks, we reduce the infinite-dimensional optimization problem to a finite-dimensional optimization problem which can be solved numerically. Finally, we apply the worst-case risk measure to optimal decision making in reinsurance.

On-orbit calibration of space camera lens distortion using a single image

Since space cameras need to withstand the harsh mechanical and thermal conditions in the space environment for a long time, it is necessary to calibrate them in orbit. However, existing calibration methods have various disadvantages, making them impossible to use in orbit. To address this problem, we present an on-orbit calibration of space camera lens distortion with the vanishing points obtained from a single image of the solar panel. First, we propose a parallel-line-extraction method based on collinear constraints to obtain the parallel lines. Then, we train the optimal vanishing point using the common point constraint method. Using the optimal vanishing point, we establish the optimization function of lens distortion based on vanishing point consistency. Finally, we present an improved genetic algorithm to solve the optimization function. Simulations and experiments show that the proposed method is flexible and robust.

Computation of the Fundamental Limits of Data Compression for Certain Nonstationary ARMA Vector Sources

In the present article, the differential entropy rate and the rate distortion function (RDF) are computed for certain nonstationary real Gaussian autoregressive moving average (ARMA) vector sources.

Finite distortion curves: Continuity, differentiability and Lusin’s (N) property

We define finite distortion \omega -curves and we show that for some forms \omega and when the distortion function is sufficiently exponentially integrable, the map is continuous, differentiable almost everywhere and has Lusin’s (N) property. This is achieved through some higher integrability results about finite distortion \omega -curves. It is also shown that this result is sharp both for continuity and for Lusin’s (N) property. We also show that if we assume weak monotonicity for the coordinates of a finite distortion \omega -curve, we obtain continuity.

Random distortion risk measures

This paper presents one type of random risk measures, named as the random distortion risk measure. The random distortion risk measure is a generalization of the traditional deterministic distortion risk measure by randomizing the deterministic distortion function and the risk distribution respectively, where a stochastic distortion is introduced to randomize the distortion function, and a sub-σ-algebra is introduced to illustrate the influence of the known information on the risk distribution. Some theoretical properties of the random distortion risk measure are provided, such as normalization, conditional positive homogeneity, conditional comonotonic additivity, monotonicity in stochastic dominance order, and continuity from below, and a method for specifying the stochastic distortion and the sub-σ-algebra is provided. Based on some stochastic axioms, a representation theorem of the random distortion risk measure is proved. For considering the randomization of a given deterministic distortion risk measure, some families of random distortion risk measures are introduced with the stochastic distortions constructed from a Poisson process, a Brownian motion, and a Dirichlet process, respectively. A numerical analysis is carried out for showing the influence of the stochastic distortion and the sub-σ-algebra by focusing on the sample statistics, empirical distributions, and tail behavior of the random distortion risk measures.

Progressive Histogram Modification for JPEG Reversible Data Hiding

With the development of social network, a large number of private JPEG images are stored in social cloud platform. Correspondingly, the platform embeds user ID or authentication labels to manage these privacy images, preventing them from being arbitrarily accessed or tampered by illegal persons. However, data embedding in JPEG domain inevitably produces irreversible modifications to DCT coefficients, thus resulting in obvious or even serious distortion in the host JPEG images. To address this problem, this paper proposes an efficient JPEG reversible data hiding (RDH) method by constructing progressive two-dimensional histogram mappings. We firstly design distortion function to calculate the cost of each DCT frequency band, and then sort them to build histogram mapping containing a series of coefficient pairs. Subsequently, a progressive mapping mechanism is introduced to maintain most of AC coefficients unchanged. According to the given capacity, this mechanism can adaptively generate an optimum two-dimensional histogram mapping to embed secret messages. Our scheme can achieve an effective balance among embedding capacity, visual quality of the marked image and file size expansion, while keeping high cost-performance complexity. Extensive experiments demonstrate that our method outperforms existing JPEG RDH schemes in terms of visual quality and file size increment of the marked image, and provides an efficient solution for the confidentiality and security access problem of sensitive private image in cloud environment.

A multi-embedding domain video steganography algorithm based on TU partitioning and intra prediction mode

With High Efficiency Video Coding (HEVC) becoming a popular video coding standard in the world, combing the HEVC with data hiding methods is a complex task. In this paper, a multi-embedded video steganography algorithm based on Intra Prediction Mode (IPM) and Transform Unit (TU) partition is proposed. By analyzing the coding characteristics of the TU partition and the IPM, we select embedded regions that are relatively independent as much as possible. Then, suitable distortion functions are designed for different embedding domains. Using the minimum distortion framework of the STC model to embed messages. The experimental results indicate that the proposed method achieves a maximum embedding capacity 1.5 times larger than that of traditional single domain embedding methods, while maintaining good visual quality.

Adaptive HEVC Video Steganography With High Performance Based on Attention-Net and PU Partition Modes

With the increasing popularity of digital video, video steganography has become a hot research topic in the field of covert communication and privacy protection. The existing prediction unit (PU) based video steganography often tends to result in large bit rate increase, which is also easily noticeable to the steganography analyst. To solve this problem, an adaptive steganography for HEVC video based on attention-net and PU partition modes is proposed. First, the distortion of modified PUs is analyzed from the perspective of rate distortion optimization at the group of pictures (GOP) level, and we find that modifying PU will lead to distortion accumulation and abnormal bitrate increase. Therefore, an adaptive distortion function based on the improved rate distortion cost is designed, and the embedding distortion is minimized by using Syndrome-Trellis Code (STC) steganography coding. Meanwhile, a super-resolution convolutional neural network with non-local sparse attention-net filter is proposed to replace the in-loop filter in HEVC to reconstruct the reference frame, thereby reducing the bitrate cost and improving the visual quality of stego-video. Experimental results show that the proposed algorithm can achieve superior perceptual quality and bitrate performance comparing with the sate-of-the-art works.