Applications of maximum likelihood principal component analysis: incomplete data sets and calibration transfer

The application of a new method to the multivariate analysis of incomplete data sets is described. The new method, called maximum likelihood principal component analysis (MLPCA), is analogous to conventional principal component analysis (PCA), but incorporates measurement error variance information in the decomposition of multivariate data. Missing measurements can be handled in a reliable and simple manner by assigning large measurement uncertainties to them. The problem of missing data is pervasive in chemistry, and MLPCA is applied to three sets of experimental data to illustrate its utility. For exploratory data analysis, a data set from the analysis of archeological artifacts is used to show that the principal components extracted by MLPCA retain much of the original information even when a significant number of measurements are missing. Maximum likelihood projections of censored data can often preserve original clusters among the samples and can, through the propagation of error, indicate which samples are likely to be projected erroneously. To demonstrate its utility in modeling applications, MLPCA is also applied in the development of a model for chromatographic retention based on a data set which is only 80% complete. MLPCA can predict missing values and assign error estimates to these points. Finally, the problem of calibration transfer between instruments can be regarded as a missing data problem in which entire spectra are missing on the ‘slave’ instrument. Using NIR spectra obtained from two instruments, it is shown that spectra on the slave instrument can be predicted from a small subset of calibration transfer samples even if a different wavelength range is employed. Concentration prediction errors obtained by this approach were comparable to cross-validation errors obtained for the slave instrument when all spectra were available.     

Palladium salicylaldimine complexes containing boronate esters

Reactions of salicylaldehydes with boronate ester derivatives of aniline have been examined. Addition of these Schiff base ligands to palladium acetate or Na₂PdCl₄ afforded novel boron-containing trans-bis(N-arylsalicylaldiminato) palladium complexes.Condensation of salicylaldehyde (2-HOC₆H₄C(O)H) with H₂NC₆H₄Bpin (pin=1,2-O₂C₂Me₄) afforded the boron-containing Schiff bases, 2-HOC₆H₄C(H)=NC₆H₄Bpin (1–3a). Similar reactivity with 2-hydroxy-5-nitrobenzaldehyde and 2-hydroxy-1-naphthaldehyde gave the corresponding Schiff bases (1-3b) and (1-3c), respectively. Reaction of Schiff bases (2) and (3) with palladium acetate or Na₂PdCl₄ afforded complexes of the type PdL₂ (4,5), where L=deprotonated Schiff base. The molecular structure of the nitro-salicylaldehyde 4-Bpin palladium complex (5b) was characterized by an X-ray diffraction study. All new palladium compounds have been characterized fully and tested for their antifungal activity against Aspergillus niger and Aspergillus flavus.     

Asymptotic expansions in limit theorems for stochastic processes. – III

For families of processes with independent increments (t), 0tT, with frequent small jumps, limit theorems for expectations of the functionals F([0, T]) are proved of the form where _diD, are positive numbers, A _di are linear integro-differential or differential operators acting on functionals, and some differentiability conditions are imposed on the functional F. The case of power tails of the jump distribution is considered.Mathematics Subject Classification (2000):60F17     

Fast and simple methods for the optimization of kurtosis used as a projection pursuit index

As a powerful method for exploratory data analysis, projection pursuit (PP) often outperforms principal component analysis (PCA) to discover important data structure. PP was proposed in 1970s but has not been widely used in chemistry largely because of the difficulty in the optimization of projection indices. In this work, new algorithms, referred as "quasi-power methods", are proposed to optimize kurtosis as a projection index. The new algorithms are simple, fast, and stable, which makes the search for the global optimum more efficient in the presence of multiple local optima. Maximization of kurtosis is helpful in the detection of outliers, while minimization tends to reveal clusters in the data, so the ability to search separately for the maximum and minimum of kurtosis is desirable. The proposed algorithms can search for either with only minor changes. Unlike other methods, no optimization of step size is required and sphering or whitening of the data is not necessary. Both univariate and multivariate kurtosis can be optimized by the proposed algorithms. The performance of the algorithms is evaluated using three simulated data sets and its utility is demonstrated with three experimental data sets relevant to analytical chemistry.     

Bootstrap method for the estimation of measurement uncertainty in spotted dual-color DNA microarrays

DNA microarrays permit the measurement of gene expression across the entire genome of an organism, but the quality of the thousands of measurements is highly variable. For spotted dual-color microarrays the situation is complicated by the use of ratio measurements. Studies have shown that measurement errors can be described by multiplicative and additive terms, with the latter dominating for low-intensity measurements. In this work, a measurement-error model is presented that partitions the variance into general experimental sources and sources associated with the calculation of the ratio from noisy pixel data. The former is described by a proportional (multiplicative) structure, while the latter is estimated using a statistical bootstrap method. The model is validated using simulations and three experimental data sets. Monte-Carlo fits of the model to data from duplicate experiments are excellent, but suggest that the bootstrap estimates, while proportionately correct, may be underestimated. The bootstrap standard error estimates are particularly useful in determining the reliability of individual microarray spots without the need for replicate spotting. This information can be used in screening or weighting the measurements.     

Preparation and Crystal Structure of PbRe2O6. An Example of the Re2O10 Unit with No Metal-Metal Bonding

Single crystals of PbRe₂O₆ were obtained from a stoichiometric mixture of PbO and ReO₃ at 500°C. The structure was determined by Patterson methods from X-ray four-circle diffractometer data using 2 270 (392 unique) reflections; R_W(F) = 0.036. PbRe₂O₆ crystallizes in the trigonal space group R3m with a = 10.359(6) Å, c = 11.092(6) Å and Z = 9. The compound is isotypic with the trigonal modification of PbNb₂O₆ and contains an Re₂O₁₀ unit. Unlike other compounds containing this species, there is no metal-metal bonding.     

Procrustes rotation as a diagnostic tool for projection pursuit analysis

Projection pursuit (PP) is an effective exploratory data analysis tool because it optimizes the projection of high dimensional data using distributional characteristics rather than variance or distance metrics. The recent development of fast and simple PP algorithms based on minimization of kurtosis for clustering data has made this powerful tool more accessible, but under conditions where the sample-to-variable ratio is small, PP fails due to opportunistic overfitting of random correlations to limiting distributional targets. Therefore, some kind of variable compression or data regularization is required in these cases. However, this introduces an additional parameter whose optimization is manually time consuming and subject to bias. The present work describes the use of Procrustes analysis as diagnostic tool that can be used to evaluate the results of PP analysis in an efficient manner. Through Procrustes rotation, the similarity of different PP projections can be examined in an automated fashion with “Procrustes maps” to establish regions of stable projections as a function of the parameter to be optimized. The application of this diagnostic is demonstrated using principal components analysis to compress FTIR spectra from ink samples of ten different brands of pen, and also in conjunction with regularized PP for soybean disease classification.     

Asymptotic expansions in limit theorems for stochastic processes. II

. For a certain class of families of stochastic processes η^ε(t), 0≤t≤T, constructed starting from sums of independent random variables, limit theorems for expectations of functionals F(η^ε[0,T]) are proved of the form

Generalized error-dependent prediction uncertainty in multivariate calibration

Most of the current expressions used to calculate figures of merit in multivariate calibration have been derived assuming independent and identically distributed (iid) measurement errors. However, it is well known that this condition is not always valid for real data sets, where the existence of many external factors can lead to correlated and/or heteroscedastic noise structures. In this report, the influence of the deviations from the classical iid paradigm is analyzed in the context of error propagation theory. New expressions have been derived to calculate sample dependent prediction standard errors under different scenarios. These expressions allow for a quantitative study of the influence of the different sources of instrumental error affecting the system under analysis. Significant differences are observed when the prediction error is estimated in each of the studied scenarios using the most popular first-order multivariate algorithms, under both simulated and experimental conditions.     

Small mass asymptotic for the motion with vanishing friction

We consider the small mass asymptotic (Smoluchowski–Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and the boundary condition of this limiting Markov process and prove the convergence.