A note on sliced inverse regression with regularizations

Summary .   In Li and Yin (2008, Biometrics 64, 124–131), a ridge SIR estimator is introduced as the solution of a minimization problem and computed thanks to an alternating least-squares algorithm. This methodology reveals good performance in practice. In this note, we focus on the theoretical properties of the estimator. It is shown that the minimization problem is degenerated in the sense that only two situations can occur: Either the ridge SIR estimator does not exist or it is zero.     

Mining gene expression profiles: an integrated implementation of kernel principal component analysis and singular value decomposition

The detection of genes that show similar profiles under different experimental conditions is often an initial step in inferring the biological significance of such genes. Visualization tools are used to identify genes with similar profiles in microarray studies. Given the large number of genes recorded in microarray experiments, gene expression data are generally displayed on a low dimensional plot, based on linear methods. However, microarray data show nonlinearity, due to high-order terms of interaction between genes, so alternative approaches, such as kernel methods, may be more appropriate. We introduce a technique that combines kernel principal component analysis (KPCA) and Biplot to visualize gene expression profiles. Our approach relies on the singular value decomposition of the input matrix and incorporates an additional step that involves KPCA. The main properties of our method are the extraction of nonlinear features and the preservation of the input variables (genes) in the output display. We apply this algorithm to colon tumor, leukemia and lymphoma datasets. Our approach reveals the underlying structure of the gene expression profiles and provides a more intuitive understanding of the gene and sample association.     

Thresholding rules for recovering a sparse signal from microarray experiments

We consider array experiments that compare expression levels of a high number of genes in two cell lines with few repetitions and with no subject effect. We develop a statistical model that illustrates under which assumptions thresholding is optimal in the analysis of such microarray data. The results of our model explain the success of the empirical rule of two-fold change. We illustrate a thresholding procedure that is adaptive to the noise level of the experiment, the amount of genes analyzed, and the amount of genes that truly change expression level. This procedure, in a world of perfect knowledge on noise distribution, would allow reconstruction of a sparse signal, minimizing the false discovery rate. Given the amount of information actually available, the thresholding rule described provides a reasonable estimator for the change in expression of any gene in two compared cell lines.     

Hydraulic conductivities of competing root systems of Grevillea robustaand maize in agroforestry

Models of water uptake in mixed stands of vegetation commonly assume that water is partitioned among competing root systems in proportion to relative root length densities. Such an approach assumes implicitly that roots of different species have equivalent hydraulic properties. This was tested for root systems of Grevillea robustaA. Cunn. and maize (Zea maysL.) at a semi-arid site in Kenya. The hydraulic conductances for roots of both species were measured in situat the scale of the whole root or root system using a high pressure flow meter (HPFM). Hydraulic conductivities (_r) were expressed per unit root length. Root lengths were estimated for maize plants by soil coring and for G. robustausing a fractal branching model calibrated against soil coring. Mean _r was 1.88×10^–7 ±0.28×10^–7kg s^–1 MPa^–1 m^–1 for G. robustaand 1.25×10^–7 ±0.13×10^–7kg s^–1 MPa^–1 m^–1 for maize. Values of _r were not significantly different (P<0.05), suggesting that the assumption of hydraulic equivalence for root systems of the two species may be valid, at least when hydrostatic gradients are the major driving force for water uptake. Differences in conductivities between these species could arise, however, because of variation in the hydraulic properties of roots not accounted for here, for example because of root age, phenology or responses to the soil environment.     

兴安落叶松种群格局的分形特征信息维数

种群格局信息维数能够刻划格局强度的尺度变化程度,揭示出种群个体分布的非均匀性状况。采用信息维数对大兴安岭兴安落叶松种群格局的研究显示,各类兴安落叶松林中兴安落叶松种群格局的信息维数大小依次为越桔－兴安落叶松林（１．４８０）＞杜香－兴安落叶松林（１．３７３）＞杜鹃－兴安落叶松林（１．３２１）＞草类－兴安落叶松林（１．０９３）。虽为群落建群种,兴安落叶松种群格局的信息维数较低（＜１．５,远离２）,格局     

Physical and hydrodynamic properties of flocs produced during biological hydrogen production

Dense flocs readily form in continuous culture bioreactors used for hydrogen production, but the fractal and hydrodynamic properties of these flocs have not been previously analyzed. We therefore examined the size distribution, fractal dimension, and hydrodynamic properties of flocs formed in a continuous flow, well-mixed reactor treating synthetic wastewater at a fixed condition of a 4.5 h hydraulic detention time (23 degrees C, pH 5.5). The reactor was operated for a total of 3 months at three different organic loading rates (27, 53, and 80 g-COD/L-d) with influent glucose concentrations of 5, 10, and 15 g-COD/L. At all three loading rates the removal of glucose was nearly complete (98.6-99.4%) and biomass was produced in proportion to the organic loading rate (0.86 +/- 0.11, 2.40 +/- 0.26, and 4.59 +/- 1.55 g/L of MLVSS in the reactor). Overall conversion efficiencies of glucose to hydrogen, evaluated on the basis of a maximum of 4 mol-H2/mol-glucose, increased with organic loading rates in the order 17.7%, 23.1%, and 25.6%. The gas contained 56.1 +/- 4.9% hydrogen, with the balance as carbon dioxide. No methane gas was detected. Under these conditions, flocs were produced with mean sizes that increased with organic loading, in the order 0.12 cm (5 g-COD/L), 0.35 cm (10 g-COD/L), and 0.58 cm (15 g-COD/L). As the average floc size increased, the flocs became on average denser and less fractal, with fractal dimensions increasing from 2.11 +/- 0.17 to 2.48 +/- 0.13. Floc porosities ranged from 0.75-0.96, and resulted in aggregate densities that allowed little intra-aggregate flow through the floc. As a result, average settling velocities were not appreciably larger than those predicted by Stokes' law for spherical, impermeable flocs. Our results demonstrate that dense, relatively impermeable flocs are produced in biohydrogen reactors that have settling properties in reasonable agreement with Stokes' law.     

Space-irrelevant scaling law for fish school sizes

Universal scaling in the power-law size distribution of pelagic fish schools is established. The power-law exponent of size distributions is extracted through the data collapse. The distribution depends on the school size only through the ratio of the size to the expected size of the schools an arbitrary individual engages in. This expected size is linear in the ratio of the spatial population density of fish to the breakup rate of school. By means of extensive numerical simulations, it is verified that the law is completely independent of the dimension of the space in which the fish move. Besides the scaling analysis on school size distributions, the integrity of schools over extended periods of time is discussed.