Number Of Principal Components Research Articles

Sensitivity of Local Daily Temperature Change Estimates to the Selection of Downscaling Models and Predictors

Abstract A number of statistical downscaling models are applied to the Canadian Climate Centre general circulation model (CCCM) outputs to provide climate change estimates for local daily surface temperature at a network of 39 stations in central and western Europe. Several different linear downscaling methods (multiple linear regression of gridded values, multiple linear regression of principal components, canonical correlation analysis) are applied to a number of sets of predictors (500- and 1000-hPa heights, 850-hPa temperature, 1000–500-hPa thickness, and their combinations) defined on a grid over the Euro–Atlantic domain. The temperature change estimates are shown to vary widely among the methods as well as among the predictors, both in their areal mean, spatial pattern, and elevation dependence in the Alpine region. The most counterintuitive result is the dependence of the estimated warming on the number of principal components (PCs) of predictors: the larger the number of PCs, the higher the warmin...

Statistical properties of dense interacting boson systems with one- plus two-body random matrix ensembles

Statistical properties like nearest neighbor spacing distribution (NNSD), number of principal components (NPC), information entropy ( S info) and number entropy ( S occu) have been studied for interacting boson systems using one-body ( H 1) and an embedded GOE of two-body ( H 2) interactions. The NNSD for the bosonic ensembles moves steadily from Poisson to Wigner form as the 2-body interaction strength λ in the H= H 1+ λH 2 is varied. Critical strength λ C for the transition, agrees with Jacquad and Shepelyansky criterion. For NPC, S info and S occu, there is another transition point λ F , as in fermionic ensembles. Bosonic ensembles in the dense limit tend to be ergodic with increasing the number of single-particle states.

Simple improvement of consumer fit in external preference mapping

In the common implementation of external preference mapping consumer preferences are fitted as polynomial functions of the first two principal components (PCs) of the sensory data. A major weakness of the method is the relatively small number of consumers that can be significantly fitted. Several researchers have proposed to improve the consumer fit by including higher-numbered PCs in the analysis. We have explored the possibility of including higher-numbered PCs while restricting the model choice to the simplest polynomial function, i.e. the (linear) vector model. In addition, we have developed a heuristic decision rule for determining the number of PCs to keep in the fit. A practical example is discussed where the consumer fit improved from 51% (two PCs, polynomial) to 80% (five PCs, vector).

Model Updating for the Identification of NIR Spectra from a Pharmaceutical Excipient

The effect of model updating on the identification of a pharmaceutical excipient based on its near-infrared (NIR) spectra has been investigated. A pragmatic updating approach, consisting of adding stepwise newly available samples to the training set and rebuilding the classification model, was applied. Its performance is compared for three pattern recognition methods: the wavelength distance method, the Mahalanobis distance method, and the SIMCA (soft independent modeling of class analogy) residual variance method. For the wavelength distance method, the updating approach is straightforward. In the case of the multivariate classification methods, which are based on a certain number of significant principal components (PCs), the selection of the number of PCs included in the model must be performed with care, as this number has a major impact on the classification results.

Selection of the Number of Principal Components: The Variance of the Reconstruction Error Criterion with a Comparison to Other Methods

One of the main difficulties in using principal component analysis (PCA) is the selection of the number of principal components (PCs). There exist a plethora of methods to calculate the number of PCs, but most of them use monotonically increasing or decreasing indices. Therefore, the decision to choose the number of principal components is very subjective. In this paper, we present a method based on the variance of the reconstruction error to select the number of PCs. This method demonstrates a minimum over the number of PCs. Conditions are given under which this minimum corresponds to the true number of PCs. Ten other methods available in the signal processing and chemometrics literature are overviewed and compared with the proposed method. Three data sets are used to test the different methods for selecting the number of PCs: two of them are real process data and the other one is a batch reactor simulation.

Decision criteria for soft independent modelling of class analogy applied to near infrared data

SIMCA (soft independent modelling of class analogy) is a well known pattern recognition method which describes each class separately in a principal components (PC) space. New objects are considered to belong to the class if their Euclidean distance towards the constructed PC space is not significantly larger than the Euclidean distance of the class objects towards their PC space. The large number of wrongly rejected objects ( α-error), which is a known problem of SIMCA, was examined. Using scores, predicted by leave-one-out cross-validation, instead of the original scores, obtained after PCA on the class objects, to compute the distance towards the class model clearly improves the decision criterion of the original SIMCA for a data set consisting of near infrared (NIR) spectra of tablets. The original SIMCA and modifications using different distance measures as defined by Hawkins (Mahalanobis distance) and Gnanadesikan were compared with respect to classification and their robustness towards the number of PCs selected to describe the different classes. SIMCA modified with the Mahalanobis distance was found to be a good alternative of the original SIMCA which, for the presented NIR data set, seems to be more robust for finding outliers when the exact number of PCs to build the model is not known.

Spurious principal components in the set of spectra subjected to disturbances: I. Presentation of the problem

It has been shown that a kind of imperfections which can be met in spectrometers, or small shifts of the recorded lines, can influence properties of spectra sets in such a way that the number of principal components (PCs) necessary to describe the non-noisy part of the total variance of the set increases compared to the number when the disturbances are absent. The number of additional (spurious) PCs depends on the magnitude of disturbances and on the signal-to-noise ratio (SNR). The mechanism which generates spurious PCs is discussed, and a criterion enabling an evaluation when the effect becomes important, is given. Numerical simulations of the effect enabled the inference of an empirical relationship between the number of PCs, the magnitude of imperfections, and SNR. Finally, a procedure is proposed to check if a spectrometer is free of the kind of disturbances discussed.

Local prediction models by principal component regression

Principal component regression (PCR) is widely used for analytical calibration. In most applications of PCR, principal components (PCs) are included in regression models in sequence according to respective variances. A suitable number of PCs is determined by the predictive ability obtained through cross-validation within the calibration samples or by relying on an external validation sample set. It has recently been reported that such a strategy of selecting PCs in sequence according to variance, i.e., top-down selection, may not necessarily result in the best prediction model and some alternative strategies have been proposed. Top-down selection and other suggested selection methods aim to build a global calibration model to predict all future samples. These approaches do not take into consideration measurement information of prediction samples in forming the global model. In actuality, the best model can be different for individual prediction samples. In this paper, a strategy is proposed to build local (sample-dependent) models based on PCR. Several data sets have been investigated with up to 40% improvement in prediction errors compared to the conventional top-down selection strategy for global models.

Calibration in near Infrared Diffuse Reflectance Spectroscopy. A Comparative Study of Various Methods

The results obtained by implementing Principal Component Regression (PCR) according to three different criteria for choosing principal components (PCs), and those provided by Partial Least-Squares Regression (PSLR), in the determination of the active compound in a pharmaceutical preparation by near infrared diffuse reflectance spectroscopy are compared. The PCR-top down criterion used is commonly implemented in commercially available software: it selects consecutive PCs beginning with that possessing the largest eigenvalue. The other two criteria used do not assume the PCs with the largest eigenvalues to be the best predictors for the response variable; rather, the PCR-correlation criterion chooses only those PCs exhibiting the highest correlation with the response variable, and the PCR-best subset criterion selects those that provide the lowest predicted residual sum of squares ( PRESS) for an external prediction set. All the calibration methods tested exhibited a similar predictive ability (prediction errors ranged from 1.34% to 1.49%); however, the number of PCs used in the regression varied among them. The PLSR technique did not excel the methods based on selecting the best PCs for regression. Also, the PCR-correlation and PCR-best subset methods provided the same results and used fewer PCs than the PCR-top down method.

APPLICATION OF PRINCIPAL COMPONENT ANALYSIS FOR PARSIMONIOUS SUMMARIZATION OF DEA INPUTS AND/OR OUTPUTS

In Data Envelopment Analysis (DEA), when there are more inputs and outputs, there are more efficient Decision Making Units (DMus). For example, if the specific inputs or outputs advantageous for a particular DMU are used, the DMU will become efficient. Usually the variables used as inputs or outputs are correlated. Therefore, the inputs and outputs should be selected appropriately by experts who know their characteristics very well. People who are less familiar with those characteristics require tools to assist in the selection. We propose using principal component analysis as a means of weighting inputs and/or outputs and summarizing parsimoniously them rather than selecting them. A basic model and its modification are proposed. In principal component analysis, many weights for the variables that define principal components (PCs) have negative values. This may cause a negative integrated input that is a denominator of the objective function in fractional programming. The denominator should be positive. In the basic model, a condition that the denominator must be positive is added. When the number of PCs is less than the number of original variables, a part of original information is neglected. In the modified model, a part of the neglected information is also used.

Random error bias in principal component analysis. Part I. derivation of theoretical predictions

Principal component analysis (PCA) or singular value decomposition (SVD) are multivariate techniques that are often used to compress large data matrices to a relevant size. Subsequent data analysis then proceeds with the model representation of the data. In this first paper expressions are derived for the prediction of the bias in the eigenvalues of PCA and singular values of SVD that results from random measurement errors in the data. Theoretical expressions for the prediction of this “random error bias” have been given in the statistics literature. These results are, however, restricted to the case that only one principal component (PC) is significant. The first objective of this paper is to extend these results to an arbitrary number of significant PCs. For the generalization Malinowski's error functions are used. A signal-to-noise ratio is defined that describes the error situation for each individual PC. This definition enhances the interpretability of the derived expressions. The adequacy of the derived expressions is tested by a limited Monte Carlo study. This finally leads to the second objective of this paper. Simulation results are always restricted to the class of data that is well represented in the study. Thus rather than giving extensive simulation results it is outlined how the validation and evaluation of theoretical predictions can proceed for a specific application in practice.