Replica Analysis Research Articles

Precipitates change of ferritic‐martensitic 11 % chromium steel under short‐term creep

AbstractA ferritic‐martensitic (FM) 11 % chromium steel with final heat treatment was subjected to a short‐term creep test at a stress of 150 MPa and 600 °C for 1100 h in order to study the change of precipitates in the steel during the creep test. Except for Nb‐rich metall carbides (MC, M23C6) and Laves phases, Fe‐W‐Cr‐rich M6C (based on Fe3W3C) carbides forming during the creep test were also identified in the crept steel by electron diffraction and x‐ray diffraction in combination with energy dispersive x‐ray analysis of extraction carbon replicas. The identified M6C carbides have a fcc crystal structure, a metallic element composition of approximately 44Fe, 32 W, and 20Cr in atomic %, and large sizes ranging from 100 nm to 300 nm in diameter. The M6C carbides are a dominant phase in the crept steel. M6X precipitates are generally not easy to form during high temperature creep, even if it is a long‐term creep, in ferritic‐martensitic 9–12 % chromium steels with a final heat treatment. The present work provides the evidence for the M6C carbides forming during short‐term creep in ferritic‐martensitic high chromium steels. The formation of the M6C carbides was discussed.

Replica Approach for Minimal Investment Risk with Cost

In the present work, the optimal portfolio minimizing the investment risk with cost is discussed analytically, where this objective function is constructed in terms of two negative aspects of investment, the risk and cost. We note the mathematical similarity between the Hamiltonian in the mean-variance model and the Hamiltonians in the Hopfield model and the Sherrington{Kirkpatrick model and show that we can analyze this portfolio optimization problem by using replica analysis, and derive the minimal investment risk with cost and the investment concentration of the optimal portfolio. Furthermore, we validate our proposed method through numerical simulations.

Validation of the replica trick for simple models

We discuss the replica analytic continuation using several simple models in order to prove mathematically the validity of the replica analysis, which is used in a wide range of fields related to large-scale complex systems. While replica analysis consists of two analytical techniques—the replica trick (or replica analytic continuation) and the thermodynamical limit (and/or order parameter expansion)—we focus our study on replica analytic continuation, which is the mathematical basis of the replica trick. We apply replica analysis to solve a variety of analytical models, and examine the properties of replica analytic continuation. Based on the positive results for these models we propose that replica analytic continuation is a robust procedure in replica analysis.

Approximate message passing for nonconvex sparse regularization with stability and asymptotic analysis

We analyse a linear regression problem with nonconvex regularization called smoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basis for Gaussian random data. We propose an approximate message passing (AMP) algorithm considering nonconvex regularization, namely SCAD-AMP, and analytically show that the stability condition corresponds to the de Almeida–Thouless condition in spin glass literature. Through asymptotic analysis, we show the correspondence between the density evolution of SCAD-AMP and the replica symmetric (RS) solution. Numerical experiments confirm that for a sufficiently large system size, SCAD-AMP achieves the optimal performance predicted by the replica method. Through replica analysis, a phase transition between replica symmetric and replica symmetry breaking (RSB) region is found in the parameter space of SCAD. The appearance of the RS region for a nonconvex penalty is a significant advantage that indicates the region of smooth landscape of the optimization problem. Furthermore, we analytically show that the statistical representation performance of the SCAD penalty is better than that of -based methods, and the minimum representation error under RS assumption is obtained at the edge of the RS/RSB phase. The correspondence between the convergence of the existing coordinate descent algorithm and RS/RSB transition is also indicated.

Macroscopic relationship in primal-dual portfolio optimization problem

In the present paper, using a replica analysis, we examine the portfolio optimization problem handled in previous work and discuss the minimization of investment risk under constraints of budget and expected return for the case that the distribution of the hyperparameters of the mean and variance of the return rate of each asset are not limited to a specific probability family. Findings derived using our proposed method are compared with those in previous work to verify the effectiveness of our proposed method. Further, we derive a Pythagorean theorem of the Sharpe ratio and macroscopic relations of opportunity loss. Using numerical experiments, the effectiveness of our proposed method is demonstrated for a specific situation.

Does $\ell _{p}$ -Minimization Outperform $\ell _{1}$ -Minimization?

In many application areas ranging from bioinformatics to imaging, we are faced with the following question: can we recover a sparse vector $x_{o} \in \mathbb {R}^{N}$ from its undersampled set of noisy observations $y \in \mathbb {R}^{n}$ , $y= A x_{o}+ w$ . The last decade has witnessed a surge of algorithms and theoretical results to address this question. One of the most popular schemes is the $\ell _{p}$ -regularized least squares given by the following formulation: $\hat x(\gamma ,p ) \in {\arg \min }_{x}~({1}/{2})\| {y - Ax} \|_{2}^{2} + \gamma {\| x \|_{p}^{p}}$ , where $p \in [{0, 1}]$ . Among these optimization problems, the case $p = 1$ , also known as LASSO, is the best accepted in practice, for the following two reasons. First, thanks to the extensive studies performed in the fields of high-dimensional statistics and compressed sensing, we have a clear picture of LASSO’s performance. Second, it is convex and efficient algorithms exist for finding its global minima. Unfortunately, neither of the above two properties hold for $0 \leq p . However, they are still appealing because of the following folklores in the high-dimensional statistics. First, $\hat x(\gamma , p )$ is closer to $x_{o}$ than $\hat {x}(\gamma ,1)$ . Second, if we employ iterative methods that aim to converge to a local minima of $ {\arg \min }_{x}~({1}/{2})\| {y - Ax} \|_{2}^{2} + \gamma {\| x \|_{p}^{p}}$ , then under good initialization, these algorithms converge to a solution that is still closer to $x_{o}$ than $\hat {x}(\gamma ,1)$ . In spite of the existence of plenty of empirical results that support these folklore theorems, the theoretical progress to establish them has been very limited. This paper aims to study the above-mentioned folklore theorems and establish their scope of validity. Starting with approximate message passing (AMP) algorithm as a heuristic method for solving $\ell _{p}$ -regularized least squares, we study the following questions. First, what is the impact of initialization on the performance of the algorithm? Second, when does the algorithm recover the sparse signal $x_{o}$ under a “good” initialization? Third, when does the algorithm converge to the sparse signal regardless of the initialization? Studying these questions will not only shed light on the second folklore theorem, but also lead us to the answer the first one, i.e., the performance of the global optima $\hat x(\gamma , p )$ . For that purpose, we employ the replica analysis 1 to show the connection between the solution of AMP and $\hat {x}(\gamma , p)$ in the asymptotic settings. This enables us to compare the accuracy of $\hat x(\gamma ,p )$ and $\hat x(\gamma ,1 )$ . In particular, we will present an accurate characterization of the phase transition and noise sensitivity of $\ell _{p}$ -regularized least squares for every $0 \leq p \leq 1$ . Our results in the noiseless setting confirm that $\ell _{p}$ -regularized least squares (if $\gamma $ is tuned optimally) exhibits the same phase transition for every $0 \leq p and this phase transition is much better than that of LASSO. Furthermore, we show that in the noisy setting, there is a major difference between the performance of $\ell _{p}$ -regularized least squares with different values of $p$ . For instance, we will show that for very small and very large measurement noises, $p = 0$ and $p = 1$ outperform the other values of $p$ , respectively. 1 Replica method is a widely accepted heuristic in statistical physics for analyzing large disordered systems.

Modeling and Quantification of Model-Form Uncertainties in Eigenvalue Computations Using a Stochastic Reduced Model

A feasible, nonparametric, probabilistic approach for modeling and quantifying model-form uncertainties associated with a computational model designed for the solution of a generalized eigenvalue problem is presented. It is based on the construction of a stochastic, projection-based reduced-order model associated with a high-dimensional model using three innovative ideas: 1) the substitution of the deterministic reduced-order basis with a stochastic counterpart featuring a reduced number of hyperparameters, 2) the construction of this stochastic reduced-order basis on a subset of a compact Stiefel manifold to guarantee the linear independence of its column vectors and the satisfaction of any constraints of interest, and 3) the formulation and solution of a reduced-order inverse statistical problem to determine the hyperparameters so that the mean value and statistical fluctuations of the eigenvalues predicted using the stochastic, projection-based reduced-order model match target values obtained from available data. Consequently, the proposed approach for modeling model-form uncertainties can be interpreted as an effective approach for extracting from data fundamental information and/or knowledge that are not captured by a deterministic computational model, and incorporating them in this model. Its potential for quantifying model-form uncertainties in generalized eigencomputations is demonstrated for a natural vibration analysis of a small-scale replica of an X-56-type aircraft made of a composite material for which ground-vibration-test data are available.

Maximizing and minimizing investment concentration with constraints of budget and investment risk

In this paper, as a first step in examining the properties of a feasible portfolio subset that is characterized by budget and risk constraints, we assess the maximum and minimum of the investment concentration using replica analysis. To do this, we apply an analytical approach of statistical mechanics. We note that the optimization problem considered in this paper is the dual problem of the portfolio optimization problem discussed in the literature, and we verify that these optimal solutions are also dual. We also present numerical experiments, in which we use the method of steepest descent that is based on Lagrange’s method of undetermined multipliers, and we compare the numerical results to those obtained by replica analysis in order to assess the effectiveness of our proposed approach.

Replica analysis of overfitting in regression models for time-to-event data

Overfitting, which happens when the number of parameters in a model is too large compared to the number of data points available for determining these parameters, is a serious and growing problem in survival analysis. While modern medicine presents us with data of unprecedented dimensionality, these data cannot yet be used effectively for clinical outcome prediction. Standard error measures in maximum likelihood regression, such as p-values and z-scores, are blind to overfitting, and even for Cox’s proportional hazards model (the main tool of medical statisticians), one finds in literature only rules of thumb on the number of samples required to avoid overfitting. In this paper we present a mathematical theory of overfitting in regression models for time-to-event data, which aims to increase our quantitative understanding of the problem and provide practical tools with which to correct regression outcomes for the impact of overfitting. It is based on the replica method, a statistical mechanical technique for the analysis of heterogeneous many-variable systems that has been used successfully for several decades in physics, biology, and computer science, but not yet in medical statistics. We develop the theory initially for arbitrary regression models for time-to-event data, and verify its predictions in detail for the popular Cox model.

Oral hyaluronan relieves wrinkles: a double-blinded, placebo-controlled study over a 12-week period

BackgroundHyaluronan (HA) has critical moisturizing property and high water retention capacity especially for human skin. This study aimed to evaluate the effect of oral intake of HA.MethodsThe mean molecular weight (MW) of HA is 2 k and 300 k. Sixty Japanese male and female subjects aged 22–59 years who presented with crow’s feet wrinkles were randomly assigned to the HA 2 k or HA 300 k at 120 mg/day or the placebo group. The subjects were administered HA at a rate of 120 mg/day or a placebo for 12 weeks. The skin wrinkles were evaluated by image analysis of skin wrinkle replicas, and their skin condition was evaluated using a questionnaire survey.ResultsDuring the study period, the HA groups showed better level of the whole sulcus volume ratio, wrinkle area ratio, and wrinkle volume ratio than the placebo group. After 8 weeks of ingestion, the HA 300 k group showed significantly diminished wrinkles compared with the placebo group. Skin luster and suppleness significantly improved after 12 weeks in all groups compared with the baseline.ConclusionThe results suggest that oral HA (both HA 2 k and HA 300 k) inhibits skin wrinkles and improves skin condition.

Replica Analysis for Portfolio Optimization with Single-Factor Model

In this paper, we use replica analysis to investigate the influence of correlation among the return rates of assets on the solution of the portfolio optimization problem. We consider the behavior of an optimal solution for the case where the return rate is described with a single-factor model and compare the findings obtained from our proposed methods with correlated return rates with those obtained with independent return rates. We then analytically assess the increase in the investment risk when correlation is included. Furthermore, we also compare our approach with analytical procedures for minimizing the investment risk from operations research.

Emergence of Compositional Representations in Restricted Boltzmann Machines.

Extracting automatically the complex set of features composing real high-dimensional data is crucial for achieving high performance in machine-learning tasks. Restricted Boltzmann machines (RBM) are empirically known to be efficient for this purpose, and to be able to generate distributed and graded representations of the data. We characterize the structural conditions (sparsity of the weights, low effective temperature, nonlinearities in the activation functions of hidden units, and adaptation of fields maintaining the activity in the visible layer) allowing RBM to operate in such a compositional phase. Evidence is provided by the replica analysis of an adequate statistical ensemble of random RBMs and by RBM trained on the handwritten digits data set MNIST.

Minimal investment risk of a portfolio optimization problem with budget and investment concentration constraints

In the present paper, the minimal investment risk for a portfolio optimization problem with imposed budget and investment concentration constraints is considered using replica analysis. Since the minimal investment risk is influenced by the investment concentration constraint (as well as the budget constraint), it is intuitive that the minimal investment risk for the problem with an investment concentration constraint can be larger than that without the constraint (that is, with only the budget constraint). Moreover, a numerical experiment shows the effectiveness of our proposed analysis. In contrast, the standard operations research approach failed to identify accurately the minimal investment risk of the portfolio optimization problem.

Portfolio optimization problem with nonidentical variances of asset returns using statistical mechanical informatics

The portfolio optimization problem in which the variances of the return rates of assets are not identical is analyzed in this paper using the methodology of statistical mechanical informatics, specifically, replica analysis. We defined two characteristic quantities of an optimal portfolio, namely, minimal investment risk and investment concentration, in order to solve the portfolio optimization problem and analytically determined their asymptotical behaviors using replica analysis. Numerical experiments were also performed, and a comparison between the results of our simulation and those obtained via replica analysis validated our proposed method.

Quantitative cross-linking/mass spectrometry reveals subtle protein conformational changes.

Quantitative cross-linking/mass spectrometry (QCLMS) probes protein structural dynamics in solution by quantitatively comparing the yields of cross-links between different conformational statuses. We have used QCLMS to understand the final maturation step of the proteasome lid and also to elucidate the structure of complement C3(H2O). Here we benchmark our workflow using a structurally well-described reference system, the human complement protein C3 and its activated cleavage product C3b. We found that small local conformational changes affect the yields of cross-linking residues that are near in space while larger conformational changes affect the detectability of cross-links. Distinguishing between minor and major changes required robust analysis based on replica analysis and a label-swapping procedure. By providing workflow, code of practice and a framework for semi-automated data processing, we lay the foundation for QCLMS as a tool to monitor the domain choreography that drives binary switching in many protein-protein interaction networks.

Replica analysis for the duality of the portfolio optimization problem.

In the present paper, the primal-dual problem consisting of the investment risk minimization problem and the expected return maximization problem in the mean-variance model is discussed using replica analysis. As a natural extension of the investment risk minimization problem under only a budget constraint that we analyzed in a previous study, we herein consider a primal-dual problem in which the investment risk minimization problem with budget and expected return constraints is regarded as the primal problem, and the expected return maximization problem with budget and investment risk constraints is regarded as the dual problem. With respect to these optimal problems, we analyze a quenched disordered system involving both of these optimization problems using the approach developed in statistical mechanical informatics and confirm that both optimal portfolios can possess the primal-dual structure. Finally, the results of numerical simulations are shown to validate the effectiveness of the proposed method.

Anti-wrinkle effects of Sargassum muticum ethyl acetate fraction on ultraviolet B-irradiated hairless mouse skin and mechanistic evaluation in the human HaCaT keratinocyte cell line.

The present study investigated the photoprotective properties of the ethyl acetate fraction of Sargassum muticum (SME) against ultraviolet B (UVB)-induced skin damage and photoaging in a mouse model. HR-1 strain hairless male mice were divided into three groups: An untreated control group, a UVB-irradiated vehicle group and a UVB-irradiated SME group. The UVB-irradiated mice in the SME group were orally administered with SME (100 mg/kg body weight in 0.1 ml water per day) and then exposed to radiation at a dose of 60–120 mJ/cm2. Wrinkle formation and skin damage were evaluated by analysis of skin replicas, epidermal thickness and collagen fiber integrity in the dermal connective tissue. The mechanism underlying the action of SME was also investigated in the human HaCaT keratinocyte cell line following exposure of the cells to UVB at a dose of 30 mJ/cm2. The protein expression levels and activity of matrix metalloproteinase-1 (MMP-1), and the binding of activator protein-1 (AP-1) to the MMP-1 promoter were assessed in the HaCaT cells using western blot analysis, an MMP-1 fluorescent assay and a chromatin immune-precipitation assay, respectively. The results showed that the mean length and depth of the wrinkles in the UVB-exposed hairless mice were significantly improved by oral administration of SME, which also prevented the increase in epidermal thickness triggered by UVB irradiation. Furthermore, a marked increase in collagen bundle formation was observed in the UVB-treated mice with SME administration. SME pretreatment also significantly inhibited the UVB-induced upregulation in the expression and activity of MMP-1 in the cultured HaCaT keratinocytes, and the UVB-enhanced association of AP-1 with the MMP-1 promoter. These results suggested that SME may be useful as an anti-photoaging resource for the skin.

Performance Limits for Noisy Multimeasurement Vector Problems

Compressed sensing (CS) demonstrates that sparse signals can be estimated from under-determined linear systems. Distributed CS (DCS) further reduces the number of measurements by considering joint sparsity within signal ensembles. DCS with jointly sparse signals has applications in multi-sensor acoustic sensing, magnetic resonance imaging with multiple coils, remote sensing, and array signal processing. Multi-measurement vector (MMV) problems consider the estimation of jointly sparse signals under the DCS framework. Two related MMV settings are studied. In the first setting, each signal vector is measured by a different independent and identically distributed (i.i.d.) measurement matrix, while in the second setting, all signal vectors are measured by the same i.i.d. matrix. Replica analysis is performed for these two MMV settings, and the minimum mean squared error (MMSE), which turns out to be identical for both settings, is obtained as a function of the noise variance and number of measurements. To showcase the application of MMV models, the MMSE's of complex CS problems with both real and complex measurement matrices are also analyzed. Multiple performance regions for MMV are identified where the MMSE behaves differently as a function of the noise variance and the number of measurements. Belief propagation (BP) is a CS signal estimation framework that often achieves the MMSE asymptotically. A phase transition for BP is identified. This phase transition, verified by numerical results, separates the regions where BP achieves the MMSE and where it is suboptimal. Numerical results also illustrate that more signal vectors in the jointly sparse signal ensemble lead to a better phase transition.

Alloy for resistance to polythionic acid stress corrosion cracking for hydroprocessing applications

In certain process units, such as hydrocracking, soda ash washing (neutralization) of austenitic stainless steel is required during turnarounds to mitigate the potential for polythionic acid stress corrosion cracking (PTA SCC). Soda ash washing can be a costly and time consuming endeavor for the refiner. This paper introduces a grade of austenitic stainless steel, a proprietary version of Type 347LN, that does not sensitize with long-term exposure to elevated temperature, thus rendering it immune to PTA SCC. ASTM A262 Practice A corrosion test results will be presented for samples isothermally aged at 565°C for a duration of up to 10,000h. These data will be compared to samples of conventional Type 347, Type 321, and Type 304H similarly aged. Photomicrographs will be shown that demonstrate the lack of grain boundary sensitization, and also the lack of grain boundary ditching in the oxalic acid test. Replica analysis of a heater tube from commercial service at 460°C average tube wall temperature for 12years showing no evidence of sensitization or grain boundary precipitation will also be presented.

The Time Machine? Using Replica Analysis to Understand Merchant Ships and the Development of the British Atlantic,1600-1800

Pour actualiser et élargir la perspective de l’histoire théorique sur la technologie des navires marchands européens, il faut analyser les idées associées à l’histoire des pays côtiers de l’Atlantique, l’histoire économique maritime, l’archéologie nautique, l’étude des cultures matérielles, l’histoire de la technologie et l’histoire technique des navires. La présente étude vise à compléter et à évaluer la recherche archivistique dans les documents officiels des marchands et des constructeurs de navires, ainsi que l’analyse technique des plans de bâtiment de navires existants à l’aide d’applications en architecture navale et en génie maritime. Proposant de recourir à des répliques de navires et aux expériences de ceux qui les exploitent, selon les méthodes de l’archéologie expérimentale, l’étude présente quelques résultats préliminaires. À la suite des connaissances déjà acquises, il est difficile de demeurer satisfait des suppositions héritées du passé au sujet de cette technologie sans procéder à de nouvelles enquêtes.