Multiple Shrinkage Research Articles

Effect of casting imperfections on the fatigue life of 319-F and A356-T6 Al–Si casting alloys

Casting imperfections, such as porosity, in cast aluminum components greatly influence their fatigue properties. The effect of porosity on the fatigue life of 319-F and A356-T6 aluminum alloys was studied, where the porosity characteristics on the fracture surfaces of fatigue-tested samples were examined using SEM and image analysis. The results show that porosity has the greatest detrimental effect on fatigue life: 92% of all tested samples fractured as a result of porosity which acted as the main crack initiation site. In the absence of casting imperfections, other microstructural aspects such as slip bands may be held responsible (4%). Porosity was investigated in terms of the pore size at the sample fracture surface. It was found that fatigue life decreases as the size of the surface pore increases. A comparison was made between the fatigue behavior of low-pressure-permanent mold-cast 319 alloy and lost foam-cast A356-T6 alloy. The results show that the 319 alloy provides greater fatigue strength compared to the 356 alloy, which may be explained by taking into consideration the nature of the surface porosity (single pore versus multiple shrinkage pores) that initiated the fatigue crack in the two alloys. The microstructural characteristics are of secondary importance in this regard.

Estimating a mean matrix: boosting efficiency by multiple affine shrinkage

The unknown matrix M is the mean of the observed response matrix in a multivariate linear model with independent random errors. This paper constructs regularized estimators of M that dominate, in asymptotic risk, least squares fits to the model and to specified nested submodels. In the first construction, the response matrix is expressed as the sum of orthogonal components determined by the submodels; each component is replaced by an adaptive total least squares fit of possibly lower rank; and these fits are then summed. The second, lower risk, construction differs only in the second step: each orthogonal component is replaced by a modified Efron-Morris fit before summation. Singular value decompositions yield computable formulae for the estimators and their asymptotic and estimated risks. In the asymptotics, the row dimension of M tends to infinity while the column dimension remains fixed. Convergences are uniform when signal-to-noise ratio is bounded.

Bayesian Classification of Tumours by Using Gene Expression Data

SummaryPrecise classification of tumours is critical for the diagnosis and treatment of cancer. Diagnostic pathology has traditionally relied on macroscopic and microscopic histology and tumour morphology as the basis for the classification of tumours. Current classification frameworks, however, cannot discriminate between tumours with similar histopathologic features, which vary in clinical course and in response to treatment. In recent years, there has been a move towards the use of complementary deoxyribonucleic acid microarrays for the classi-fication of tumours. These high throughput assays provide relative messenger ribonucleic acid expression measurements simultaneously for thousands of genes. A key statistical task is to perform classification via different expression patterns. Gene expression profiles may offer more information than classical morphology and may provide an alternative to classical tumour diagnosis schemes. The paper considers several Bayesian classification methods based on reproducing kernel Hilbert spaces for the analysis of microarray data. We consider the logistic likelihood as well as likelihoods related to support vector machine models. It is shown through simulation and examples that support vector machine models with multiple shrinkage parameters produce fewer misclassification errors than several existing classical methods as well as Bayesian methods based on the logistic likelihood or those involving only one shrinkage parameter.

Experimental Study of Self-Similar Crack Distributions

This paper presents laboratory experiments on polyester resin samples subjected to special regimes of curing under which multiple shrinkage cracks with wide distributions of sizes were produced. Loading these samples produced stress-strain curves that followed a power law within certain stress ranges. The results of the tests were analysed using the mechanics of materials with self-similar crack distributions. The analysis showed that the observed power law in the stress-strain relationship indicated that the crack distribution was approximately self-similar. Thus, the described experiments serve as a verification procedure for the proposed formation mechanism of multiscale self-similar crack distribution in brittle materials. This mechanism is based on: (1) the action of self-equilibrating stress fluctuations as initiators and propagators of cracks and (2) the size discriminating action of crack interaction. The concentration factors for the crack distributions have been retrieved from the experimental data.

Multiple shrinkage and subset selection in wavelets

This paper discusses Bayesian methods for multiple shrinkage estimation in wavelets. Wavelets are used in applications for data denoising, via shrinkage of the coefficients towards zero, and for data compression, by shrinkage and setting small coefficients to zero. We approach wavelet shrinkage by using Bayesian hierarchical models, assigning a positive prior probability to the wavelet coefficients being zero. The resulting estimator for the wavelet coefficients is a multiple shrinkage estimator that exhibits a wide variety of nonlinear patterns. We discuss fast computational implementations, with a focus on easy-to-compute analytic approximations as well as importance sampling and Markov chain Monte Carlo methods. Multiple shrinkage estimators prove to have excellent mean squared error performance in reconstructing standard test functions. We demonstrate this in simulated test examples, comparing various implementations of multiple shrinkage to commonly-used shrinkage rules. Finally, we illustrate our approach with an application to the so-called 'glint' data.

Implications from inclusions in topaz for greisenisation and mineralisation in the Hensbarrow topaz granite, Cornwall, England

Textural and geochemical studies of inclusions in topaz from greisens in the Hensbarrow topaz granite stock (St. Austell, Cornwall) are used to constrain the composition of fluids responsible for late stage greisening and mineralisation. The topaz contains an abundant and varied suite of inclusions including aqueous liquid + vapour (L + V), quartz, zinnwaldite, albite, K-feldspar, muscovite, ilmenorutile, apatite, columbite, zircon, varlamoffite [(Sn, Fe)(O, OH)2] and qitianlingite [(Fe+2,Mn+2)2(Nb,Ta)2W+6O10]. Primary L + V inclusions in topaz show relatively high Th (mainly 300 to >500 °C) and a narrow range of salinities (23–30 wt % NaCl equivalent) compared with those in greisen quartz (150–450 °C, 0–50 wt % NaCl equivalent). Textures indicate that topaz formed earlier than quartz and the fluid inclusion data are interpreted as indicating a cooling of the hydrothermal fluids during greisenisation, mixing with meteoric waters and a decrease in pressure causing intermittent boiling. The presence of early-formed albite and K-feldspar as inclusions in the topaz is likely to indicate that the greisen-forming fluid became progressively more acid during greisenisation. The most distinctive inclusions in the topaz are wisp- and bleb-shaped quartz, < 50 μm in size, which show textural characteristics indicating former high degrees of plasticity. They often have multiple shrinkage bubbles at their margins rich in Sn, Fe, Mn, S and Cl and, more rarely, contain euhedral albite, K-feldspar, stannite or pyrrhotite crystals up to 40 μm in size. The quartz inclusions show similar morphologies to inclusions in topaz from quartz-topaz rocks elsewhere which have been interpreted as trapped “silicate melt”. Their compositions are, however, very different to those expected for late stage topaz-normative granitic melts. From their textural and chemical characteristics they are interpreted as representing crystallised silica colloid, probably trapped as a hydro gel during greisenisation. There is also evidence for the colloidal origin of inclusions of varlamoffite in the topaz. These occurrences offer the first reported evidence in natural systems for the formation of colloids in high temperature hydrothermal fluids. Their high ore carrying potential is suggested by the presence of varlamoffite and the occurrence of stannite, pyrrhotite and SnCl within the quartz inclusions.

Prior information in regression: to choose or not to choose?

Stein-rule estimators for the parameters of the linear regression model are simple, nonlinear transformations of the usual least squares/maximum likelihood estimator. Under certain design related conditions Stein-rule estimators are known to be minimax and dominate the MLE under quadratic loss functions. In order to obtain quadratic risk improvements over the MLE using a Stein-rule the user must choose some target value for 3 or more of the parameters. If the chosen target is not reasonably close to the actual value of the parameters, risk gains over the MLE will be small. When several alternative pieces of nonsample information (targets) exist, using the best target results in the greatest overall risk gain. George (1986) has developed a multiple shrinkage estimator which uses all of the possible targets. Each target receives a different weight in the shrinkage estimator. The weights are a function of the target's probable proximity to the true parameter vaue. In this case, no choice need be made. Anothe...

A Bayesian Method for Combining Results from Several Binomial Experiments

Abstract The problem of combining information related to I binomial experiments, each having a distinct probability of success θ i , is considered. Instead of using a standard exchangeable prior for θ; = (θ1, …, θ I ), we propose a more flexible distribution that takes into account various degrees of similarity among the θ i 's. Using ideas developed by Malec and Sedransk, we consider a partition g of the experiments and take the θ i 's belonging to the same partition subset to be exchangeable and the θ i 's belonging to distinct subsets to be independent. Next we perform Bayesian inference on θ; conditional on g. Of course, one is typically uncertain about which partition to use, and so a prior distribution is assigned on a set of plausible partitions g. The final inference on θ; is obtained by combining the conditional inferences according to the posterior distribution of g. The methodology adopted in this article offers a wide flexibility in structuring the dependence among the θ i 's. This allows the estimate of θ i to borrow strength from all other experiments according to an adaptive process governed by the data themselves. The method may be usefully applied to the analysis of binary response variables in the presence of categorical covariates. The latter are used to identify a collection of suitable partitions g, representing factor main effects and interactions, whose relevance will be summarized in the posterior distribution of g. Besides providing novel interpretations on the role played by the various factors, the procedure will also produce parameter estimates that may differ, sometimes in an appreciable manner, from those obtained using more traditional techniques. Finally, three real data sets are used to illustrate the methodology and compare it with other approaches, such as empirical Bayes (both parametric and nonparametric), logistic regression, and multiple shrinkage estimators.

Predicting Risk: Some New Generalizations

Existing adjustment techniques for forecasting systematic risk of individual firms have been based on relatively uniformative prior knowledge about the cross-sectional distribution of risk estimates. This study introduces prior information in the form of size and industry-based cross-sectional distributions of risk estimates. Such information is incorporated into forecasts using familiar and generalized adjustment techniques, the latter being based on recently developed multiple shrinkage methods. Improved forecast performance results.

Multiple shrinkage estimators in multiple linear regression

Consider a standard multiple linear regression model . The usual least squares estimator of β , is dominated by Stein-typed shrinkage estimators; each shrinks to a prior subspace specified by a linear restriction in the form of Cm×pβ = dm×1 where m≤p. However, the reduction in expected sum of prediction squared errors is significant only in a small region in the param-eter space around the prior subspace. The multiple shrinkage estimators, which were first introduced by George (1986), are applied. In multiple linear regression, a multiple shrinkage estimator, which shrinks to more than one prior subspace simultaneously, is shown to dominate . It does not only enlarge the region in the parameter space that gives significant reduction in expected sum of prediction squared errors over that of , but also helps in solving the problem of model selection. A computer simulation is done to show that multiple shrinkage estimators are preferred to estimators given by the stepwise regression procedure in most cases. ...

Minimax estimation of common coefficients of several regression models under quadratic loss

The problem of estimating common coefficients θ=( θ 1… θ p )′ of k regression models with unknown error variances σ 2 1,…, σ 2 k is considered under the quadratic loss ( θ ̂ −θ)′ Q( θ ̂ −θ)/ . max1≤ i≤ k(σ 2 i ) for a positive definite matrix Q . The paper shows the minimaxity of a combined estimator and its admissibility for p= 1,2. Also for p≥3, multiple shrinkage estimators dominating the combined estimator are given.

Combining Minimax Shrinkage Estimators

Abstract When one estimates a multivariate normal mean, the use of Stein estimation entails both the grouping of coordinates and the selection of a set of targets toward which to shrink each group. In this article we propose new minimax multiple shrinkage estimators that allow for multiple specifications of these aspects. We provide examples that are evaluated on real and simulated data, including an estimator that adaptively resolves the issue of combining possibly related estimation problems and an adaptive clustering shrinkage estimator. The construction and properties of these estimators are shown to follow from the application of the multiple shrinkage results of George (1986a) to general partitioned shrinkage estimators.

Minimax Multiple Shrinkage Estimation

For the canonical problem of estimating a multivariate normal mean under squared-error-loss, this article addresses the problem of selecting a minimax shrinkage estimator when vague or conflicting prior information suggests that more than one estimator from a broad class might be effective. For this situation a new class of alternative estimators, called multiple shrinkage estimators, is proposed. These estimators use the data to emulate the behavior and risk properties of the most effective estimator under consideration. Unbiased estimates of risk and sufficient conditions for minimaxity are provided. Bayesian motivations link this construction to posterior means of mixture priors. To illustrate the theory, minimax multiple shrinkage Stein estimators are constructed which can adaptively shrink the data towards any number of points or subspaces.

A formal bayes multiple shrinkage estimator

A new minimax multiple shrinkage estimator is constructed. This estimator which can adaptively shrink towards many subspace targets, is formal Bayes with respect to a mixture of harmonic priors. Unbiased estimates of risk and simulation results suggest that the risk properties of this estimator are very similar to those of the multiple shrinkage Stein estimator proposed by George (1986a). A special case is seen to be admissible.