Number of measurements required

CALCULATING THE NUMBER OF MEASUREMENTS REQUIRED TO ESTABLISH A MEAN VALUE (OR ANALYTICAL RESULT) WITH A PRESCRIBED UNCERTAINTY (ACCURACY)... 

If error is random and follows probabilistic (normally distributed) variance phenomena, we must be able to make additional measurements to reduce the measurement noise or variability. This is certainly true in the real world to some extent. Most of us having some basic statistical training will recall the concept of calculating the number of measurements required to establish a mean value (or analytical result) with a prescribed accuracy. For this calculation one would designate the allowable error (e), and a probability (or risk) that a measured value (m) would be different by an amount (d). 

Different types of equipment, depending on the resources available and the number of measurements required, can be used for determination of an OUR versus time curve. A rather simple and manually operated type was used by Bjerre et al. (1995). A still relatively inexpensive apparatus, simple to operate automatically, was designed by Tanaka and Hvitved-Jacobsen (1998). However, this type may introduce a minor error at especially low OUR values because of a potential release of oxygen into a headspace of nitrogen gas in the reactor. 

However, it is still necessary to consider whether it is possible to perform easily all transport-property measurements that industry requires. Let us assume that we need to measure only three properties at just 10 temperatures and 10 pressures, for 15 pure fluids and all their possible multicomponent mixtures, at five compositions in the liquid and gas phases. Then, the total number of measurements required is of the order of 10 (3x10x10x32766x5x2). If one further assumes that one can perform three measurements per day, then it is obvious that even for the above program of measurements, 90.000 man-years will be required and it considered only 15 materials from among the set of several thousand involved routinely in industry. It is clear that measurement alone cannot serve the industrial appetite for transport property data. Nevertheless, accurate measurements will continue to be vital for the verification of theory and the validation of predication. 

As we shall see later, the more measurements that are made, the more reliable will be the measure of precision. The number of measurements required will depend on the accuracy required and on the known reproducibility of the method. 

Density can be measured in the laboratory in a number of different ways depending on the need for accuracy and the number of measurements required. Solution density can be easily estimated with reasonable accuracy by weighing a known volume of solution. Very precise instruments for the measurement of density that work employing a vibrating quartz element in a tube are sold by the Mettler Company (Hightstown, New Jersey). The period of vibration of the element is proportional to the density of the material placed in the tube. With careful calibration and temperature control the accuracy of these instruments ranges from 1 x 10 to 1 X 10 g/cm. It is possible to use these instruments for on-line solution density measurement of fluid in a crystallizer (Rush 1991). 

Many research programs driven by systems biology face common problems. Complex systems with many variables lead to a noisy environment, making it difficult to achieve accuracy and sensitivity of measurements. The large number of measurements required makes it imperative to keep the cost per measurement as low as possible. Also, as information becomes increasingly viewed as a commodity of value, rights and restrictions to data access come with associated costs and complications. 

A typical state space model for stand-alone GPS would have 8 states, the spatial coordinates and their velocities, and the clock offset and frequency. The individual pseudo-range measurements can be processed sequentially, which means that the Kalman gains can be calculated as scalars without the need for matrix inversions. There is no minimum number of measurements required to obtain an updated position estimate. The measurements are processed in an optimum fashion and if not enough for good geometry, the estimate of state error variance [P (fc)] will grow. If two sateUites are available, the clock bias terms are just propagated forward via the state transition matrix. 

The program preparation software can analyze the conductor pattern on the board and, noting that short circuits should occur only between physically adjacent conductors, reduce the number of measurements required. This is termed adjacency analysis.To achieve further time savings, the testing department might use an indirect measurement method. Please refer to Sec. 38.4.4. 

Sample Containers More saiTiple containers will be required for a complex test than are typically used for normal operation. The number and type of sample containers must be gathered in advance, recognizing the number of measurements that 1 be required. The sample containers should be tagged for the sample location, type, and conditions. 

Motivation Unit tests require a substantial investment in time and resources to complete successfully. This is the case whether the test is a straightforward analysis of pump performance or a complex analysis of an integrated reactor and separation train. The uncertainties in the measurements, the likelihood that different underlying problems lead to the same symptoms, and the multiple interpretations of unit performance are barriers against accurate understanding of the unit operation. The goal of any unit test should be to maximize the success (i.e., to describe accurately unit performance) while minimizing the resources necessary to arrive at the description and the subsequent recommendations. The number of measurements and the number of trials should be selected so that they are minimized. 

Identifying the minimum number of specific measurements containing the most information such that the model parameters are uniquely estimated requires that the model and parameter estimates be known in advance. Repeated unit tests and model building exercises will ultimately lead to the appropriate measurements. However, for the first unit test in absence of a model, the identification of the minimum number of measurements is not possible. 

The conditions of total liquid reflux in a column also represent the minimum number of plates required for a given separation. Under such conditions the column has zero production of product, and infinite heat requirements, and Lj/Vs = 1.0 as shown in Figure 8-15. This is the limiting condition for the number of trays and is a convenient measure of the complexity or difficulty of separation. 

For routine monitoring, 800 lines of resolution are recommended. Higher resolution may be needed for root-cause analysis, but this requires substantially more memory in both the analyzer and host computer. While the latter is not a major problem, higher resolution reduces the number of measurement points that can be acquired with an analyzer without transferring acquired data to the host computer. This can greatly increase the time required to complete a measurement route and should be avoided when possible. 

The most popular, and also a very accurate, experimental method for measuring nonselective spin-lattice relaxation-rates is the inversion recovery (180°-r-90°-AT-PD)NT pulse sequence. Here, t is the variable parameter, the little t between pulses, AT is the acquisition time, PD is the pulse delay, set such that AT-I- PD s 5 x T, and NT is the total number of transients required for an acceptable signal-to-noise ratio. Sequential application of a series of two-pulse sequences, each using a different pulsespacing, t, gives a series of partially relaxed spectra. Values of Rj can... 

One of the main determinants of the number of subjects required to reach the desired statistical power is the precision of the measurement tool utilized. More precise measurements will reduce the number of subjects required. As an example, if a study is being conducted to assess the influence of a dietary supplement on body fat, several measurement tools could be used to assess this outcome. These tools range from low levels of cost and precision (e.g. skinfold measurements) to moderate levels (e.g. bioelectrical impedance) to high levels of cost and precision (dual x-ray absorptiometry - DXA). A study that uses skinfold measurements to measure the outcome will require many more subjects than one which employs DXA. Therefore, it is often less expensive in total to utilize a more expensive measurement tool, because the more precise tool will allow the study to have sufficient power with a smaller number of subjects. 

Verification implies that the laboratory investigates trueness and precision in particular. Elements which should be included in a full validation of an analytical method are specificity, calibration curve, precision between laboratories and/or precision within laboratories, trueness, measuring range, LOD, LOQ, robustness and sensitivity. The numbers of analyses required by the NMKL standard and the criteria for the adoption of quantitative methods are summarized in Table 10. 

In coulometry, one measures the number of coulombs required to convert the analyte specifically and completely by means of direct or indirect electrolysis. 

Context-dependent situations often lead to a large-scale input dimension. Because the required number of training examples increases with the number of measured variables or features, reducing the input dimensionality may improve system performance. In addition, decision discriminants will be less complex (because of fewer dimensions in the data) and more easily determined. The reduction in dimensionality can be most readily achieved by eliminating redundancy in the data so that only the most relevant features are used for mapping to a given set of labels. 

EELS spectrometers have to satisfy a number of stringent requirements. First, the primary electrons should be monochromatic, with as little spreading in energy as possible, preferably around 1 meV or better (1 meV = 8 cm ). Second, the energy of the scattered electrons should be measured with an accuracy of 1 meV or better. Third, the low energy electrons must be effectively shielded from magnetic fields. The resolution of EELS has steadily been improved from typically 50-100 cm-1 around 1975 to better than 10 cnf1 for the spectrometers that are available in 2000. The latter value comes close to the line width of a molecular vibration. 

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