A Tale of Two Matrix Factorizations

In statistical practice, rectangular tables of numeric data are commonplace, and are often analyzed using dimension-reduction methods like the singular value decomposition and its close cousin, principal component analysis (PCA). This analysis produces score and loading matrices representing the rows and the columns of the original table and these matrices may be used for both prediction purposes and to gain structural understanding of the data. In some tables, the data entries are necessarily nonnegative (apart, perhaps, from some small random noise), and so the matrix factors meant to represent them should arguably also contain only nonnegative elements. This thinking, and the desire for parsimony, underlies such techniques as rotating factors in a search for “simple structure.” These attempts to transform score or loading matrices of mixed sign into nonnegative, parsimonious forms are, however, indirect and at best imperfect. The recent development of nonnegative matrix factorization, or NMF, is an attractive alternative. Rather than attempt to transform a loading or score matrix of mixed signs into one with only nonnegative elements, it directly seeks matrix factors containing only nonnegative elements. The resulting factorization often leads to substantial improvements in interpretability of the factors. We illustrate this potential by synthetic examples and a real dataset. The question of exactly when NMF is effective is not fully resolved, but some indicators of its domain of success are given. It is pointed out that the NMF factors can be used in much the same way as those coming from PCA for such tasks as ordination, clustering, and prediction. Supplementary materials for this article are available online.     

Adaptive Sequential Preventive Maintenance Policy and Bayesian Consideration

This article proposes an adaptive sequential preventive maintenance (PM) policy for which an improvement factor is newly introduced to measure the PM effect at each PM. For this model, the PM actions are conducted at different time intervals so that an adaptive method needs to be utilized to determine the optimal PM times minimizing the expected cost rate per unit time. At each PM, the hazard rate is reduced by an amount affected by the improvement factor which depends on the number of PM's preceding the current one. We derive mathematical formulas to evaluate the expected cost rate per unit time by incorporating the PM cost, repair cost, and replacement cost. Assuming that the failure times follow a Weibull distribution, we propose an optimal sequential PM policy by minimizing the expected cost rate. Furthermore, we consider Bayesian aspects for the sequential PM policy to discuss its optimality. The effect of some parameters and the functional forms of improvement factor on the optimal PM policy is measured numerically by sensibility analysis and some numerical examples are presented for illustrative purposes.     

A Lasso-type Robust Variable Selection for Time-Course Microarray Data

Lasso has been widely used for variable selection because of its sparsity, and a number of its extensions have been developed. In this article, we propose a robust variant of Lasso for the time-course multivariate response, and develop an algorithm which transforms the optimization into a sequence of ridge regressions. The proposed method enables us to effectively handle multivariate responses and employs a basis representation of the regression parameters to reduce the dimensionality. We assess the proposed method through simulation and apply it to the microarray data.     

A note on the effect of simple equicorrelation in detecting a spurious multivariate observation

This paper examines the robustness of the multivariate version of Grubs' (1950) procedure for detecting an outlier in a sample of n independent observations against equicorrelation of the observations. It is shown that the robustness of the univariate test to equicorrelation extends to the multivariate test in that the distribution of the maximum squared radii-test for a multivariate oulier in identical for both the independent and siaply equicorrelated data models.     

Aboriginal socio-economic characteristics: Issues affecting the interpretation of statistics

Journal of Population Research - As this brief discussion suggests, census statistics are useful as an indicator of some socio-economic characteristics of the Aboriginal population, particularly...     

Advances towards a critical criminology

Theory and Society -     

Investigation and improvement of a process for vacuum-formed ceramic fibre composites, using a fractional 2n experimental design

Refractory ceramic fibre products are used extensively as high temperature furnace linings, because of their properties of a lower thermal conductivity, improved resistance to thermal shock, cheaper installation and maintenance costs, and reduced wall thickness when compared with traditional firebrick linings. Vacuum forming of the ceramic fibre enables modular linings for small laboratory furnaces to be produced efficiently and cost effectively. This process involves drawing a slurry of ceramic fibre, binder and water through a porous mould, and subsequently oven drying the finished product. The product mass, which is strongly related to its relevant thermal and structural properties, was identified as a suitable response variable. A resolution IV fractional 2n factorial design was employed to investigate the effects of 15 key process factors on the product mass in just 32 experimental runs. Analysis of the results revealed that only two of the 15 process factors investigated in the experiment significantly affected the mean product mass, while none of these factors significantly affected the variability of the product mass. The experimental design is described, and its results and their implications are presented and discussed.