Univariate data analysis

A typical multivariate set of data is generated when performing Py-MS. Each Py-MS trace is formed from a considerable number of peaks and each peak can be considered a measurable variable. The results from a set of samples to be evaluated by considering all these variables will generate a typical data matrix. The data matrix can be studied using multivariate data analysis. However, simple univariate data analysis can be applicable by selecting for analysis one single m/z value. 

The usefulness of univariate data analysis seems to be limited when analyzing Py-MS results because in each spectrum there is a significant number of peaks, each representing a measurement. However, if one mass peak is properly selected from the Py-MS spectrum, this can be used as a unique measurement for the given specimen, and univariate data analysis can be quite informative. Also, any peak can be selected separately, one at a time, for evaluation. This type of approach is less informative than multivariate data analysis, but has the advantage of being simpler. 

Examination of individual parameters provided only a limited and somewhat distorted view of the SAM microcosm response to Jet-A. The univariate data analysis did indeed show that there were some significant responses to the toxicant by individual taxa and chemistry however, the responses were scattered over time and did not present a logical, coherent pattern. Furthermore, the individual responses detected were typified by wild swings in a taxon s population density over time. 

Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

The overall objective of the system is to map from three types of numeric input process data into, generally, one to three root causes out of the possible 300. The data available include numeric information from sensors, product-specific numeric information such as molecular weight and area under peak from gel permeation chromatography (GPC) analysis of the product, and additional information from the GPC in the form of variances in expected shapes of traces. The plant also uses univariate statistical methods for data analysis of numeric product information. 

This is for univariate data what happens in the case of multivariate (multiwavelength) spectroscopic analysis. The same thing, only worse. To calculate the effects rigorously and quantitatively is an extremely difficult exercise for the multivariate case, because not only are the errors themselves are involved, but in addition the correlation stmcture of the data exacerbates the effects. Qualitatively we can note that, just as in the univariate case, the presence of error in the absorbance data will bias the coefficient(s) toward zero , to use the formal statistical description. In the multivariate case, however, each coefficient will be biased by different amounts, reflecting the different amounts of noise (or error, more generally) affecting the data at different wavelengths. As mentioned above, these... 

Depending on the data structure, different types of models are possible to be applied for data analysis. Thus, when data are ordered in one direction, linear univariant models can be applied (see (1)), and nonlinear models as well (see (2)). For data ordered in two directions, bilinear models can be applied (see (3)) or nonbilinear models. Finally, for data ordered in three directions, trilinear models can be applied (see (4)) or, failing that, nontrilinear models. 

Graphical and statistical data analysis will be carried out at various scales (regional, States/Northern Territory, and National). Non-parametric univariate and multivariate analysis along with the production of geochemical maps will be carried out. 

We will skip (1) and (2) above as methods not to be preferred as global analyses. Graphical displays have tremendous values as exploratory data analysis (EDA) techniques with the type of data one encounters in these studies. For formal analyses, one could weigh univariate repeated and other factorial designs against their true multivariate counterparts. 

Fourier analysis (Bloomfield, 1976) is most frequently a univariate method used for either simplifying data (which is the basis for its inclusion in this chapter) or for modeling. It can, however, also be a multivariate technique for data analysis. 

Some of the above plots can be combined in one graphical display, like onedimensional scatter plot, histogram, probability density plot, and boxplot. Figure 1.7 shows this so-called edaplot (exploratory data analysis plot) (Reimann et al. 2008). It provides deeper insight into the univariate data distribution The single groups are... 

More complex than vectors or matrices (X, X andy, X and Y) are three-way data or multiway data (Smilde et al. 2004). Univariate data can be considered as one-way data (one measurement per sample, a vector of numbers) two-way data are obtained for instance by measuring a spectrum for each sample (matrix, two-dimensional array, classical multivariate data analysis) three-way data are obtained by measuring a spectrum under several conditions for each sample (a matrix for each sample, three-dimensional array). This concept can be generalized to multiway data. 

The initial multivariate analysis consisted of a principal component analysis on the raw data to determine if any obvious relationships were overlooked by univariate statistical analysis. The data base was reviewed and records containing missing data elements were deleted. The data was run through the Statistical Analysis System (SAS) procedure PRINCOMP and the results were evaluated. 

Thus, the use of univariate criteria is advisable only when the number of variables is very large, or in a preliminary data analysis, because sometimes it is possible to find one or two variables that give enough information to solve the classification problem. In this way, Van der Greef et al. showed that Rhone and Bordeaux wines are almost completely separated in the plane of the masses 300 and 240. 

This method cannot solve distributions such as those of Fig. 36 however, because of its simplicity, we think that it can be recommended in preliminary data analysis, at least as an improvement in comparison with the univariate method. 

This chapter constitutes an attempt to demonstrate the utility of multivariate statistics in several stages of the scientific process. As a provocation, it is suggested that the multivariate approach (in experimental design, in data description and in data analysis) will always be more informative and make generalizations more valid than the univariate approach. Finally, the multivariate strategy can be really enjoyable, not the least for its capacity to reveal hidden treasures in data that in a univariate analysis look like a set of random numbers. 

Sinclair, A. J. (1983). Univariate Analysis. 57-81 in Statistics and Data Analysis in Geochemical Prospecting. In Handbook ofExploration Geochemistry (R. J. Howarth, ed.), Vol. 2. Elsevier, Amsterdam. 

Statistical methods based on histograms, cumulative frequency probability curves (see above), univariate and multivariate data analysis (Miesh, 1981 Sinclair, 1974, 1976, 1991 Stanley, 1987) are widely used to separate geochemical baseline (natural and/or anthropogenic) values from anomalies. 

In addition to univariate statistical analysis, the data were also examined by means of multivariate statistical techniques. In particular, R-mode factor analysis was used, which is a very effective tool to interpret anomalies and to help identify their sources. Factor analysis allows grouping of anomalies by compatible geochemical associations from a geologic-mineralogical point of view, the presence of mineralizing processes, or processes connected to the surface environment. Based on this analysis, six meaningful chemical associations were identified (Fig. 15.8). 

As stated in the introduction (Section 7.1), this chapter is about the analysis of multivariate data in kinetics, i.e., measurements at many wavelengths. Compared with univariate data this has two important consequences (a) there is much more data to be analyzed and (b) there are many more parameters to be fitted. 

Exploratory data analysis (EDA). This analysis, also called pretreatment of data , is essential to avoid wrong or obvious conclusions. The EDA objective is to obtain the maximum useful information from each piece of chemico-physical data because the perception and experience of a researcher cannot be sufficient to single out all the significant information. This step comprises descriptive univariate statistical algorithms (e.g. mean, normality assumption, skewness, kurtosis, variance, coefficient of variation), detection of outliers, cleansing of data matrix, measures of the analytical method quality (e.g. precision, sensibility, robustness, uncertainty, traceability) (Eurachem, 1998) and the use of basic algorithms such as box-and-whisker, stem-and-leaf, etc. 

Univariate calibration involves relating two single variables to each other, and is often called linear regression. It is easy to perform using most data analysis packages. 

In addition to the discrepancies generated as a result of study definition (univariate discrepancies), discrepancies may also arise when a batch validation detects data inconsistencies (univariate and multivariate discrepancies). Discrepancies are also identified by a visual review of the data, e.g., monitoring lists, SDV review. Discrepancies may also be created by people responsible for data analysis (e.g., statisticians, pharmacoeconomists, clinical pharmacologists). All discrepancies and data fields requiring verification or clarification are tracked using the clinical database. 

Py-MS data analysis with univariate statistical techniques. 

Before discussing some applications, a few basic aspects on univariate statistics will be presented. A large amount of information exists regarding this field, and more details can be found in the original literature (e.g. [70,71]). Also a variety of computer packages performing statistical data analysis is available (e.g. [71a]). 

Several applications of univariate statistical analysis for data evaluation in Py-MS are known [73]. One such application is the evaluation of reproducibility of a replicate of an analysis for the peak intensity at a given m/z value. If a series of measurements are made on identical specimens, this will provide a sample xi, X2...Xn. This sample will allow the calculation of parameters such as the mean m and the standard deviation s. By comparing the value s for different m/z values it is possible to select those m/z that are more reproducible (smaller s). 

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