Regularized least squares

Almost all earth sciences inverse problems are nonlinear and involve a large number of unknown parameters, making the application of analytical inversion methods quite restrictive. In practice, most analytical methods are local in nature and rely on a linearized form of the problem equations, adopting an iterative procedure which typically employs partial derivatives in order to optimize the starting (initial) model by minimizing a misfit (penalty) function. Unfortunately, especially for highly non-linear cases, the final model strongly depends on the initial model, hence it is prone to solution-entrapment in local minima of the misfit function, while the derivative calculation is often computationally inefficient and creates instabilities when numerical approximations are used. An alternative is to employ global techniques which do not rely on partial derivatives, are independent of the misfit form and are computationally robust. Such methods employ pseudo-randomly generated models (sampling an appropriately selected section of the model space) which are assessed in terms of their data-fit. A typical example is the class of methods known as genetic algorithms (GA), which achieves the aforementioned approximation through model representation and manipulations, and has attracted the attention of the earth sciences community during the last decade, with several applications already presented for several geophysical problems. In this paper, we examine the efficiency of the combination of the typical regularized least-squares and genetic methods for a typical seismic tomography problem. The proposed approach combines a local (LOM) and a global (GOM) optimization method, in an attempt to overcome the limitations of each individual approach, such as local minima and slow convergence, respectively. The potential of both optimization methods is tested and compared, both independently and jointly, using the several test models and synthetic refraction travel-time date sets that employ the same experimental geometry, wavelength and geometrical characteristics of the model anomalies. Moreover, real data from a crosswell tomographic project for the subsurface mapping of an ancient wall foundation are used for testing the efficiency of the proposed algorithm. The results show that the combined use of both methods can exploit the benefits of each approach, leading to improved final models and producing realistic velocity models, without significantly increasing the required computation time.

In this paper we focus on the prediction of Crohn's disease (CD) susceptibility by analyzing SNP profiles for a number of defined or suggested gene polymorphisms. We assess the correlation between genetic markers and the phenotype by using well-founded methods and procedures developed in the field of statistical learning theory. To this end, we use a sample generated by a case-control study composed of 178 CD patients and 127 healthy controls. The genetic profile of each subject is composed of 16 genetic variants distributed over 11 genes. We find that regularized least squares (RLS) classifiers predict Crohn's disease with a statistically significant accuracy of 62% ( p = 0.018), significantly increasing the diagnostic accuracy by at least 10% compared to that obtained with the more largely confirmed gene involved in CD predisposition, namely CARD15. This also demonstrates that our sample size is adequate for accurate and significant prediction estimates. The strength of this methodology, in contrast to classical statistical methods, is that it accounts simultaneously for the effect of several genetics markers and their possible interactions. The findings of this study show that RLS methodology is able to increase the diagnostic accuracy of CD prediction by contemporary evaluation of a large number of gene polymorphisms. This approach may be particularly useful in large-scale population screening programs, and when evaluating large datasets of gene polymorphisms (i.e. chips, microarrays). Moreover, it could shed more light on possible candidate genes with a weak genetic contribution, and for evaluating gene-gene and gene-phenotype interactions by analyzing populations with a reasonably small sample size.

In this paper we present a method for the statistical assessment of cancer predictors which make use of gene expression profiles. The methodology is applied to a new data set of microarray gene expression data collected in Casa Sollievo della Sofferenza Hospital, Foggia -Italy. The data set is made up of normal (22) and tumor (25) specimens extracted from 25 patients affected by colon cancer. We propose to give answers to some questions which are relevant for the automatic diagnosis of cancer such as: Is the size of the available data set sufficient to build accurate classifiers? What is the statistical significance of the associated error rates? In what ways can accuracy be considered dependant on the adopted classification scheme? How many genes are correlated with the pathology and how many are sufficient for an accurate colon cancer classification? The method we propose answers these questions whilst avoiding the potential pitfalls hidden in the analysis and interpretation of microarray data.

This paper focuses on why the regular least–squares fitting technique is unstable when used to fit exponential functions to signal waveforms, since such functions are highly correlated. It talks about alternative approaches, such as the search method, which has a slow convergence rate of 1/N 1/M , for M parameters, where N is the number of computations performed. We have used the Monte Carlo method, utilizing both search and random walk, to devise a stable least–squares fitting algorithm that converges rapidly at a rate 1/N 1/2 , regardless of the number of parameters used in fitting the waveforms. The Monte Carlo approach has been tested for computed data—with and without noise, and by fitting actual experimental signal waveforms associated with optogalvanic transitions recorded with a hollow cathode discharge tube containing a mixture of neon (Ne) and carbon monoxide (CO) gases, and has yielded excellent results, making the developed algorithm both stable and fast for today's ...

1] We explore the effects of sampling density, signal/noise ratios, and position-dependent measurement errors on deconvolution calculations for u channel magnetometer data, using a combination of experimental and numerical approaches. Experiments involve a synthetic sample set made by setting hydraulic cement in a 30-cm u channel and slicing the hardened material into ∼2-cm lengths, and a natural lake sediment u channel sample. The cement segments can be magnetized and measured individually, and reassembled for continuous u channel measurement and deconvolution; the lake sediment channel was first measured continuously and then sliced into discrete samples for individual measurement. Each continuous data set was deconvolved using the ABIC minimization code of and two new approaches that we have developed, using singular-value decomposition and regularized least squares. These involve somewhat different methods to stabilize the inverse calculations and different criteria for identifying the optimum solution, but we find in all of our experiments that the three methods converge to essentially identical solutions. Repeat scans in several experiments show that measurement errors are not distributed with position-independent variance; errors in setting/determining the u channel position (standard deviation ∼0.2 mm) translate in regions of strong gradients into measurement uncertainties much larger than those due to instrument noise and drift. When we incorporate these depth-dependent measurement uncertainties into the deconvolution calculations, the resulting models show decreased stability and accuracy compared to inversions assuming depth-independent measurement errors. The cement experiments involved varying directions and uniform intensities downcore, and very good accuracy was obtained using all of the methods when the signal/noise ratio was greater than a few hundred and the sampling interval no larger than half the length scale of magnetization changes. Addition of synthetic noise or reduction of sampling density decreased the resolution and accuracy of all the methods equally. The sediment-core experiment involved uniform (axial) magnetization direction and strongly varying intensities downcore. Intensity variations are well resolved and directions are accurate to within about 5 degrees, with errors attributable to omission and/or inaccurate calibration of cross terms in the instrument response function.