Observations and variables

It is useful at this stage to distinguish three kinds of observation that can be made in scientific investigations which result in different forms of variable. 

Nominal variables These are variables that can only be classified into groups, but not ranked. An assessment, e.g., of eye colour in a population group would yield information about the frequency of occurrence of different colours, but the variable itself (in this case colour) cannot be quantified on any meaningful scale, and certainly not averaged. 

Ordinal variables these are variables that can be classified and ranked, but the individual results lack numerical precision. An example might be a survey of patients responses to a specific treatment which could be classified as much better, better, unchanged, worse, or much worse. These responses form a meaningful scale, and the number of responses in each group can be subjected to different forms of statistical analysis. 

Metrical variables here the observed quantity is a variable which can be given a number, and ranked and analysed quantitatively. An example would be the fluorescence emission intensity of a sample. If this measurement was repeated many times the results would be similar, but would deviate from 

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From equations 3a, 5 and 6 and from conditions (i) and (ii) it can be derived that the total variation of all PC variables is given by the sum "=1 K> while the relative contribution of PCr in the total variation is K Y = i K- The first p principal components, where p is usually 2 or 3, should explain more than 80% of the total variation of the data set, thus t kr/Y = i K 0-8. However, the nature of the observations and variables does not always present such a convenient situation. 

Figure 5 shows the relationship between observations and variables for a chance correlation probability of 1 or less for various r2 levels. The linear relationships shown are each statistically significant at the p < 0.0001 level. This graph permits the determination of the number of observations required to screen, for example, ten variables while keeping the probability of encountering a chance correlation with r2 0.8 at the 1 level or less. From the graph it can be estimated that this number of observations is about 20. For r2 0.9> the number required is less,... 

In data sets with a large number of variables, collinear data and missing values, projection models based on latent structures, such as Principal Component Analysis (PC A) (6) (7) (1) and Partial Least Squares (PLS) (8) (9) (10), are valuable tools within EDA. Projection models and the set of tools used in combination simplify the analysis of complex data sets, pointing out to special observations (outliers), clusters of similar observations, groups of related variables, and crossed relationships between specific observations and variables. All this information is of paramount importance to improve data knowledge. 

II), is used to investigate the relationships between variables in projection subspaces. The second one is an extension of MEDA, named observation-based MEDA or oMEDA (33), to discover the relationships between observations and variables. The EDA approach based on PCA/PLS with scores and loading plots, MEDA and oMEDA is illustrated with several real examples from the chemometrics field. 

This chapter is organized as follows. Section 2 briefly discusses the importance of subspace models and score plots to explore the data distribution. Section 3 is devoted to the investigation of the relationship among variables in a data set. Section 4 studies the relationship between observations and variables in latent subspaces. Section 5 presents a EDA case study of Quantitative Structure-Activity Relationship (QSAR) modelling and... 

Camacho J. Observation-based missing data methods for exploratory data analysis to unveil the connection between observations and variables in latent subspace models Journal of Chemometrics 25 (2011) 592 - 600. 

If the differenee between the aetual and observed state variables is... 

Comparing the system shown in Figure 8.12 with the original PD eontroller given in Example 5.10, the state feedbaek system may be eonsidered to be a PD eontroller where the proportional term uses measured output variables and the derivative term uses observed state variables. 

The production of some products can be controlled simply by inspection after the product has been produced. In other cases, as with the continuous production of food and drugs, you may need to monitor certain process parameters to be sure of producing conforming product. By observing the variability of certain parameters using control charts, you can determine whether the process is under control within the specified limits. 

Xjj is the ith observation of variable Xj. yi is the ith observation of variable y. y, is the ith value of the dependent variable calculated with the model function and the final least-squares parameter estimates. 

The calculated values y of the dependent variable are then found, for jc, corresponding to the experimental observations, from the model equation (2-71). The quantity ct, the variance of the observations y is calculated with Eq. (2-90), where the denominator is the degrees of freedom of a system with n observations and four parameters. 

With the best observing conditions, it is possible for the trained observer to compete with photoelectric colorimeters for detection of small color differences in samples which can be observed simultaneously. However, the human observer cannot ordinarily make accurate color comparisons over a period of time if memory of sample color is involved. This factor and others, such as variability among observers and color blindness, make it important to control or eliminate the subjective factor in color grading. In this respect, objective methods, which make use of instruments such as spectrophotometers or carefully calibrated colorimeters with conditions of observation carefully standardized, provide the most reliable means of obtaining precise color measurements. 

In a further study, Brubaker et have reported on the effects of the addition of chloride ion to the sulphate exchange system at virtually constant ionic strength (3.68 M sulphate and hydrogen-ion concentrations. For the concentration ratio [C1 ]/[T1(III)] of 9.2x10" to 9.5 at 24.9 °C results analogous to the effect observed in perchlorate media were obtained. The minimum in the rate corresponded to a ratio of 2.5. Results were also presented for the conditions, constant [CI ] and variable [804 ] and [If"] ( = 3.68 M). Brubaker et al have suggested that the exchange paths most likely to occur in sulphate media are... 

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