Review of Chemometrics Applied to Spectroscopy: 1985-95, Part 3 — Multi-way Analysis

INTRODUCTION

In this third part of a review on chemometrics in spectroscopy we will describe a recent methodology that has attracted increasing interest in spectroscopy. namely multi-way analysis. The application of multi-way analysis in spectroscopy is still relatively new. hence many methodological improvements are being investigated currently. Part of thls review will also be used to describe the algorithmic improvements gained the last decade.     

Generalized Cornish-Fisher expansions

Cornish-Fisher expansions about the normal distribution provide accurate approximations for distributions of estimates and also for the level in the nominal error of confidence intervals. However, there is an advantage is expanding about a skew distribution like the chi-square, since the first order approximations become second order if the skewness is matched. Higher order approximations are also simplified. We demonstrate the method by approximating the distribution of standardized and Studentized linear combinations of means.     

Eigenvalues of the difference and product of projections

There is an isomorphism between the matrices over the Boolean algebra of subsets of a k-element set and the k-tuples of Boolean binary (i.e. zero-one) matrices. This isomorphism allows many problems concerning nonbinary Boolean matrices to the referred to the binary ease. However, there are some features of the general (i.e. nonbinary) case that have not been mentioned, although they differ somewhat from the binary case. We exhibit characterizations of the linear operators that preserve several invariants of matrices over finite Boolean algebras to illustrate the differences and similarities of the general vs. the binary cases. We employ a canonical form that is useful in applying the isomorphism.     

An anisotropic enhanced thermal conductivity approach for modelling laser melt pools for Ni-base super alloys

Ni-base super alloys are extensively used in high temperature gas turbine engines and energy industries. Due to the high replacement costs of these components, there are huge economic benefits of repairing these components. Laser direct metal deposition processes (LDMD) based on laser cladding, laser fusion welding, and laser surface melting are some of the processes which are used to repair these high value components. Precise control of these processes is important to achieve the desired microstructure, stress distribution, distortions due to thermal stresses and other important output variables. Modelling of these processes is therefore an extremely important activity for achieving any degree of control/optimisation. However, modelling of these processes is not straight-forward due to melt pool flows dominated by Marangoni and buoyancy driven convection. Detailed CFD models are required for accurate prediction of melt pool geometry. But these models are computationally expensive and require greater expertise. To simplify and speed up the modelling process, many researchers have used the isotropic enhanced thermal conductivity approach to account for melt pool convection. A recent study on mild steel has highlighted that isotropic enhanced thermal conductivity approach is not able to accurately predict the melt pool geometry. Based on these findings a new approach namely anisotropic enhanced thermal conductivity approach has been developed. This paper presents an analysis on the effectiveness of the isotropic and anisotropic enhanced thermal conductivity approaches for laser melting of Inconel 718 using numerical technique. Experimental melt pool geometry has been compared with modelling results. It has been found that the isotropic enhanced thermal conductivity approach is not able to accurately predict the melt pool geometry, whereas anisotropic enhanced thermal conductivity approach gives good agreement with the experimental results.     

Expansions about the Gamma for the Distribution and Quantiles of a Standard Estimate

We give expansions for the distribution, density, and quantiles of an estimate, building on results of Cornish, Fisher, Hill, Davis and the authors. The estimate is assumed to be non-lattice with the standard expansions for its cumulants. By expanding about a skew variable with matched skewness, one can drastically reduce the number of terms needed for a given level of accuracy. The building blocks generalize the Hermite polynomials. We demonstrate with expansions about the gamma.