Popular variance

Another parameter is called the standard deviation, which is designated as O. The square of the standard deviation is used frequently and is called the popular variance, O". Basically, the standard deviation is a quantity which measures the spread or dispersion of the distribution from its mean [L. If the spread is broad, then the standard deviation will be larger than if it were more constrained. 

W. Mendenhall, Introduction to EinearMode/s and the Design andAna/ysis of Experiments, Duxbury Press, Belmont, Calif., 1968. This book provides an introduction to basic concepts and the most popular experimental designs without going into extensive detail. In contrast to most other books, the emphasis in the development of many of the underlying models and analysis methods is on a regression, rather than an analysis-of-variance, viewpoint. 

A general comment that affects all statistical multivariate data analysis techniques, namely that each of the variables should be given equal chance to influence the outcome of the analysis. This can be achieved by scaling the variables in appropriative way. One popular method for scaling variables is autoscaling, whereby the variance of each variable is adjusted to 1. 

In the last decade several other multivariable controllers have been proposed. We will briefly discuss two of the most popular in the sections below. Other multivariable controllers that will not be discussed but are worthy of mention are minimum variance controllers (see Bergh and MacGregor, lEC Research, Vol. 26, 1987, p. 1558) and extended horizon controllers (see Ydstie, Kershenbaum, and Sargent, AIChE J., Vol. 31, 1985, p. 1771). 

The relative % standard deviation (Equation 4.5) is also called the coefficient of variance, c.v. Relative standard deviation relates the standard deviation to the value of the mean and represents a practical and popular expression of data quality. Again, for an entire population of samples, s is replaced by c. 

It can be shown that the variance of the coefficient estimates, b, of P is o (X X). Furthermore, the variance of the predicted response at any setting of the variables is also a function of (X X)". Thus one way to choose good values for the elements of X is to choose them so that (X X) is, in some sense, small . A number of criteria have been developed, the most popular of which are ... 

The parameters A,k and b must be estimated from sr The general problem of parameter estimation is to estimate a parameter, 0, given a number of samples, x,-, drawn from a population that has a probability distribution P(x, 0). It can be shown that there is a minimum variance bound (MVB), known as the Cramer-Rao inequality, that limits the accuracy of any method of estimating 0 [55]. There are a number of methods that approach the MVB and give unbiased estimates of 0 for large sample sizes [55]. Among the more popular of these methods are maximum likelihood estimators (MLE) and least-squares estimation (LS). The MLE... 

The most commonly employed univariate statistical methods are analysis of variance (ANOVA) and Student s r-test [8]. These methods are parametric, that is, they require that the populations studied be approximately normally distributed. Some non-parametric methods are also popular, as, f r example, Kruskal-Wallis ANOVA and Mann-Whitney s U-test [9]. A key feature of univariate statistical methods is that data are analysed one variable at a rime (OVAT). This means that any information contained in the relation between the variables is not included in the OVAT analysis. Univariate methods are the most commonly used methods, irrespective of the nature of the data. Thus, in a recent issue of the European Journal of Pharmacology (Vol. 137), 20 out of 23 research reports used multivariate measurement. However, all of them were analysed by univariate methods. 

The most popular method of reporting variability is the sample variance, defined as ... 

Statistical methods are the most popular techniques for EN analysis. The potential difference and coupling current signals are monitored with time. The signals are then treated as statistical fluctuations about a mean level. Amplitudes are calculated as the standard deviations root-mean-square (rms) of the variance according to (for the potential noise)... 

Complementary to graphical data summaries are numerical summarizations. For the simple case of data collected under a single set of conditions, the most commonly used measures deal with the location/center of the data set and the variability/spread of the data. The (arithmetic) mean and the median are the most popular measures of location, and the variance and its square root, the standard deviation, are the most widely used measures of internal variability in a data set. 

The probability density function, written as pif), describes the fraction of time that the fluctuating variable/ takes on a value between/ and/ + A/. The concept is illustrated in Fig. 5.7. The fluctuating values off are shown on the right side while p(f) is shown on the left side. The shape of the PDF depends on the nature of the turbulent fluctuations of/. Several different mathematical functions have been proposed to express the PDF. In presumed PDF methods, these different mathematical functions, such as clipped normal distribution, spiked distribution, double delta function and beta distribution, are assumed to represent the fluctuations in reactive mixing. The latter two are among the more popular distributions and are shown in Fig. 5.8. The double delta function is most readily computed, while the beta function is considered to be a better representation of experimentally observed PDF. The shape of these functions depends solely on the mean mixture fraction and its variance. The beta function is given as... 

The calibration model referred to a partial least squares regression (PLSR) is a relatively modem technique, developed and popularized in analytical science by Wold. The method differs from PCR by including the dependent variable in the data compression and decomposition operations, i.e. both y and x data are actively used in the data analysis. This action serves to minimize the potential effects of jc variables having large variances but which are irrelevant to the calibration model. The simultaneous use of X and y information makes the method more complex than PCR as two loading vectors are required to provide orthogonality of the factors. 

Normalization is a very important step, as it aims to reduce experimental variance. Normalization is most often performed by dividing each spectrum by a normalization factor (Figure 2G). The most popular normalization factor is calculated as the total ion count (TIC), which is the sum of all ion intensities in a spectrum. Several studies discovered that in MSI the assumptions for TIC applicability hold true only for very homogeneous tissues. In heterogeneous samples, more robust normalization factors based on the median or the TIC with exclusion of very localized mass signals have been proposed (35-37). 

M. H. Quenouille introduced the jackknife (JKK) in 1949 (12) and it was later popularized by Tukey in 1958, who first used the term (13). Quenouille s motivation was to construct an estimator of bias that would have broad applicability. The JKK has been applied to bias correction, the estimation of variance, and standard error of variables (4,12-16). Thus, for pharmacometrics it has the potential for improving models and has been applied in the assessment of PMM reliability (17). The JKK may not be employed as a method for model validation. 

Autocorrelation in data affects the accuracy of the charts developed based on the iid assumption. One way to reduce the impact of autocorrelation is to estimate the value of the observation from a model and compute the error between the measured and estimated values. The errors, also called residuals, are assumed to have a Normal distribution with zero mean. Consequently regular SPM charts such as Shewhart or CUSUM charts could be used on the residuals to monitor process behavior. This method relies on the existence of a process model that can predict the observations at each sampling time. Various techniques for empirical model development are presented in Chapter 4. The most popular modeling technique for SPM has been time series models [1, 202] outlined in Section 4.4, because they have been used extensively in the statistics community, but in reality any dynamic model could be used to estimate the observations. If a good process model is available, the prediction errors (residual) e k) = y k)—y k) can be used to monitor the process status. If the model provides accurate predictions, the residuals have a Normal distribution and are independently distributed with mean zero and constant variance (equal to the prediction error variance). 

The most popular tool for monitoring single-loop feedback and feedforward/feedback controllers is based on relative performance with respect to minimum variance control (MVC) [53, 102[. The idea is not to implement MVC but to use the variance of the controlled output variable that would be obtained if MVC were used as the reference point. The variation of the inflation of the controlled output variance indicates if the process is operating as expected or not. Furthermore, if the variance with a MVC is larger than what could be tolerated, this indicates the need for modification of operating conditions or process. 

Since yMst is a random variable, SPM tools can be used to detect statistically significant changes. histXk) is highly autocorrelated. Use of traditional SPM charts for autocorrelated variables may yield erroneous results. An alternative SPM method for autocorrelated data is based on the development of a time series model, generation of the residuals between the values predicted by the model and the measured values, and monitoring of the residuals [1]. The residuals should be approximately normally and independently distributed with zero-mean and constant-variance if the time series model provides an accurate description of process behavior. Therefore, popular univariate SPM charts (such as x-chart, CUSUM, and EWMA charts) are applicable to the residuals. Residuals-based SPM is used to monitor lhist k). An AR model is used for representing st k) ... 

---