Statistical selection

When we consider a process with only one input and one output variable, the experimental analysis of the process must contain enough data to describe the relationship between the dependent variable y and the independent variable V. This relation can be obtained only if the data collected result from the evolution of one stationary process, and then supplementary experimental data can be necessary to demonstrate that the process is really in a stationary state. 

As an actual process, we can consider the case of an isothermal and isobaric reactor working at steady state, where the input variable is the reactant s concentration and the output process variable (dependent variable) is the transformation degree. In this case, the values of the data collected are reported in Table 5.2. We can observe that we have the proposed input values (a prefixed set-point of the measurements) and the measured input values. 

Current number for input Proposed input value of x (set point) Measured x value Measured y value 

In experimental research, each studied case is generally characterized by the measurement of x (x values) and y (y values). Each chain of x and each chain of y represents a statistical selection because these chains must be extracted from a very large number of possibilities (tvhich can be defined as populations). However, for simplification purposes in the example above (Table 5.2), we have limited the input and output variables to only 5 selections. To begin the analysis, the researcher has to answer to this first question what values must be used for x (and corresponding y) when we start analysing of the identification of the coefficients by a regression function Because the normal equation system (5.9) requires the same number of x and y values, we can observe that the data from Table 5.2 cannot be used as presented for this purpose. To prepare these data for the mentioned scope, we observe that, for each proposed x value (x = 13.5 g/1, x=20 g/1, x = 27 g/1, X = 34 g/1, X = 41 g/1), several measurements are available these values can be summed into one by means of the corresponding mean values. So, for each type of X data, we use a mean value, where, for example, i = 5 for the first case (proposed X = 13.5 g/1), i = 3 for the third case, etc. The same procedure will be applied for y where, for example, i = 4 for the first case, i = 6 for the second case, etc. 

---

The selection of the 1980-82 measurements (Swedjemark and MjOnes, 1984) was made on dwellings built before 1976 and with the aim of determining dose distributions and the collective dose to the Swedish population from the exposure of the short-lived radon decay products. This was done by using the statistical selection made by the National Institute for Building Research intended for an energy study of the Swedish stock of houses. From a selection of 3 100 houses in 103 municipalities, 2 900 were inspected. The data was found to be in substantial conformity with data from the land register and the population census of 1975. For the study of the radon concentration 752 dwellings were selected at random. 

Unlike many dosage form specifications, the sterility specification is an absolute value. A product is either sterile or nonsterile. Historically, judgment of sterility has relied on an official compendial sterility test however, end-product sterility testing suffers from a myriad of limitations [1-4], The most obvious limitation is the nature of the sterility test. It is a destructive test thus, it depends on the statistical selection of a random sample of the whole lot. Uncertainty will always exist as to whether or not the sample unequivocally represents the whole. If it were known that one unit out of 1000 units was contaminated (i.e., contamination rate = 0.1%) and 20 units were randomly sampled out of those 1000 units, the probability of that one contaminated unit being included in those 20 samples is 0.02 [5], In other words, the chances are only 2% that the contaminated unit would be selected as part of the 20 representative samples of the whole 1000-unit lot. 

Fish and Wildlife. A survey of 2,3,7,8-TCDD contamination in benthic (bottom feeding) and predator fish from major U.S. watersheds was conducted for the EPA National Dioxin Study (Kuehl et al. 1989). It was observed that 17 of 90 (19%) samples collected at sites statistically selected by the EPA had detectable levels of 2,3,7,8-TCDD, whereas 95 of 305 (31%) samples from sites chosen by EPA regional laboratories had detectable levels (detection limits 0.5-2 ppt on a wet weight basis). Of the 112 sites where 2,3,7,8-TCDD was detected, 74 samples (67%) were below 5 ppt, 34 samples (32%) were between 5 and 25 ppt, and 4 samples (1%) were above 25 ppt. A subset of samples collected at sites near the discharges from pulp/paper manufacturing facilities (n=28) had a higher frequency of... 

In a process, when the value domain of each of the independent variables is the same in the passive and in the active experiments simultaneously, two identical statistical models are expected. The model is thus obtained from a statistical selection and its different states are represented by the response curves, which combine the input parameters for each of the output parameters. 

The system above contains N equations and consequently it will produce a single real solution for Pq, Pi,P (n unknowns). It is necessary to specify that the size of the statistical selection, here represented by Ne, must be appreciable. Moreover, whenever the regression coefficients have to be identified, Ne must be greater than n. This system (5.9) is frequently called system of normal equations [5.4, 5.12-5.14]. 

The purpose of this section is to show the calculation of the confidence interval for the variance in an actual example. The statistical data used for this example are given in Table 5.3. In this table, the statistically measured real input concentrations and the associated output reactant transformation degrees are given for five proposed concentrations of the limiting reactant in the reactor feed. Table 5.3 also contains the values of the computed variances for each statistical selection. The confidence interval for each mean value from Table 5.3 has to be calculated according to the procedure established in steps 6-10 from the algorithm shown in Section 5.2.2.1. In this example, the number of measurements for each experiment is small, thus the estimation of the mean value is difficult. Therefore, we... 

When the preliminary steps of the statistical model have been accomplished, the researchers must focus their attention on the problem of correlation between dependent and independent variables (see Fig. 5.1). At this stage, they must use the description and the statistical selections of the process, so as to propose a model state with a mathematical expression showing the relation between each of the dependent variables and all independent variables (relation (5.3)). During this selection, the researchers might erroneously use two restrictions Firstly, they may tend to introduce a limitation concerning the degree of the polynomial that describes the relation between the dependent variable y( and the independent variables Xj, j = l,n Secondly, they may tend to extract some independent variables or terms which show the effect of the interactions between two or more independent variables on the dependent variable from the above mentioned relationship. 

We observe that the covariance indicator (cov(x,y)) has an expression which is similar to the dispersion of a statistical selection datum near the mean value (Eq. (5.11)). It is important to specify that the notion of variance (or dispersion) differs completely from the notion of covariance. 

Experimental design. If we have selected u subset of structures from our collection, we can use any of several. statistical measures to quantify the diversity of our subset. Further, there are statistical selection procedures that, if followed, will make it more likely (but not guarantee) that wc pick more and more diverse subsets. Tliese are optintiMion procedures, and they are widely used in the design of experiments, statistical surveys. etc. An example of a common procedure is one called O-optinial selection, which is designed In pick a subset of points that are as widely scparalc d from each other as possible. 

The deconvolution methods are multi-wavelength procedures which can be classified with regard to the selection procedure of reference spectra. These spectra can be chosen from specific compounds (Maier, 1981), from independent spectra of real samples (Thomas et al., 1993), statistically selected (Gallot and Thomas, 1993) or from a mixed choice of spectra of specific compounds and of real samples. Reference spectra are not universal recently, according to the complexity of the composition of wastewater, SECOMAM has developed UVPro software based on advanced UV spectral deconvolution (Patent 00402038-4, 17 July 2000) which allows creating dedicated models and determination of reference spectra from a set of studied water and wastewater UV spectra an automatic calibration step is carried using parameters values obtained by standard or reference method. Deconvolution is used in order to find a linear relation between measured and UV estimated values for any parameter. 

Findings. Of the statistically selected soil sites, 142 of 200 rural and 221 of 300 urban sites were sampled. The remaining 58 rural sites and 79 urban sites could not be sampled because of difficulty in locating the site (130 sites) or because permission to collect a sample was denied (7 sites). Of the 100 statistically selected fish sites, 90 were sampled. The remaining ten sites could not be sampled because of lack of water, fish, or success in catching fish at the time sampling was attempted. 

After discussing the dynamics taking place within the potential wells, we will come back in Section VI to the question how to understand the statistical-selective nature of reaction processes from the viewpoint of Hamilton chaos. 

A crude criterion for identifying the genes that discriminate between all the classes is the f-test. The f-statistic selects features under the following ranking criterion ... 

Bechhofer R. E., Santner, T. J., and Goldsman, D. M., Design and Analysis of Experiments for Statistical Selection, Screening, and Multiple Comparisons, Wiley Series in Probability and Statistics, John Wiley Sons. New York, 1995. 

Azobenzene chromophores can be oriented using polarized light (Yu and Ikeda, 2004 Ichimura, 2000) via a statistical selection process, described schematically in Fig. 1.9. Azobenzenes preferentially absorb light polarized along their transition dipole axis (long axis of the azomolecule). The probability of absorption varies as cos cj), where 4> is the angle between the light polarization and the azo dipole axis. 

---