Performance variance

Flexible budgeting is more widely used than static budgeting despite certain logic difficulties. This is so because production in many cases is seasonal and the use of a static production norm might distort evaluation of performance. Variances are the difference between the actual costs expended and the budgeted costs expec ted. Variances are unfavorable if positive and favorable if negative. Any variance should be explained and, if necessaiy, controlled the largest variance should be considered first. 

Linear regression is undoubtedly the most widely used statistical method in quantitative analysis (Fig. 21.3). This approach is used when the signal y as a function of the concentration x is linear. It stems from the principle that if many samples are used (generally dilutions of a stock solution), it becomes possible to perform variance analysis and estimate calibration error or systematic errors. 

The chapters in Characterization and Catalyst Development An Interactive Approach, assembled from both academic and industrial contributors, give a unique perspective on catalyst development Some chapters thoroughly characterize the catalyst prior to plant evaluation, whereas others utilize characterization to explain performance variances. Some new types of catalysts incorporated into this volume include the preparation of novel catalyst supports based on alumina and hydrous titanates. Attrition-resistant catalysts and ultrafine ceramics were prepared by modified spray-drying methods. New catalyst compositions based on vanadium-containing anionic clays were proposed for oxidation. A recently commercialized catalyst based on magnesium spinel was proposed for use in the abatement of sulfur oxide pollutants in fluid... 

Flynn ReaTworld performance variance perhaps, but not reflected in IQ scores. Maynard Smith Is it irreligious to assert that perhaps IQ tests aren t a very good measure of performance ... 

Traditional approaches in process performance evaluation rely on characteristics and time trends of critical process variables such as controlled variables and manipulated variables. Ranges of variation of these variables, their frequency of reaching hard constraints, or any abnormal trends in their behavior have been used by many experienced plant personnel to track process performance. Variances of these variables and their histograms have also been used. More formal techniques for process performance evaluation rely on the extension of statistical process control (SPC) to continuous processes. 

In theory, independent input parameters are required for performing variance-based SA. It is not clear what happens with the SA algorithms in the presence of dependencies between the inpnt parameters or to what extent the results can be interpreted. 

Monte Carlo simulation (MC) is the most widely used method for calculation of performance variance. It can be applied when many classes of parameters are simultaneously uncertain, and it is easy to understand (22). 

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... 

Selection alone cannot achieve an optimization towards the solution With mere scicction performed over a number of generations, one would get a population which comprises only the best chromosome of the original population. Therefore, an operator has to be applied which causes variance within the population, This is achieved by the application of genetic operators such as the crossover and the mutation operators. 

The t test can be applied to differences between pairs of observations. Perhaps only a single pair can be performed at one time, or possibly one wishes to compare two methods using samples of differing analytical content. It is still necessary that the two methods possess the same inherent standard deviation. An average difference d calculated, and individual deviations from d are used to evaluate the variance of the differences. 

Some of these devices have a respectable quantum efficiency of charge generation and collection, approaching 0.4 (20). The nature of the polymeric binder has a large effect on the device performance (21), and so does the quaUty and source of the dye (22). Sensitivity to the environment and fabrication methods results in some irreproducibiUties and batch-to-batch variances. However, the main advantage of the ZnO-based photoreceptor paper is its very low cost. 

One measure of the performance of a control system is the variance of the controlled variable from the target. Both improving the control svstem and reducing the disturbances will lead to a lower variance in the controlled variable. 

From Table 9-39 we find that the flexible budgeted overhead cost for a produc tion rate of 12 million kg per month is 190,000. The corresponding variance is 186,000 minus 190,000, or — 4,000, which is favorable Because 4,000 less was spent than was anticipated. Thus, the use of flexible budgeting makes this particular performance look better without changing either the production rate or a single cost of the planned budget. 

Work done by Wiesner [6] is a much more accurate approach. The subject has also been reported on more recently by Simon and Bulskamper [71. They generally agree with Wiesner that the variance of performance with Reynolds number was more true at low value that at high values. The additional influence above a Reynolds number of 10 is not much. It would appear that if a very close guarantee depended on the Reynolds number to get the compressor within the acceptance range (if the Reynolds number was high to begin with), the vendor would be rather desperate. 

Having established that a finite volume of sample causes peak dispersion and that it is highly desirable to limit that dispersion to a level that does not impair the performance of the column, the maximum sample volume that can be tolerated can be evaluated by employing the principle of the summation of variances. Let a volume (Vi) be injected onto a column. This sample volume (Vi) will be dispersed on the front of the column in the form of a rectangular distribution. The eluted peak will have an overall variance that consists of that produced by the column and other parts of the mobile phase conduit system plus that due to the dispersion from the finite sample volume. For convenience, the dispersion contributed by parts of the mobile phase system, other than the column (except for that from the finite sample volume), will be considered negligible. In most well-designed chromatographic systems, this will be true, particularly for well-packed GC and LC columns. However, for open tubular columns in GC, and possibly microbore columns in LC, where peak volumes can be extremely small, this may not necessarily be true, and other extra-column dispersion sources may need to be taken into account. It is now possible to apply the principle of the summation of variances to the effect of sample volume. 

Variances in resin performance and capacities can be expected from normal annual attrition rates of ion-exchange resins. Typical attrition losses that can be expected include (1) Strong cation resin 3 percent per year for three years or 1,000,000 gals/ cu.ft (2) Strong anion resin 25 percent per year for two years or 1,000,000 gals/ cu.ft (3) Weak cation/anion 10 percent per year for two years or 750,000 gals/ cu. ft. A steady falloff of resin-exchange capacity is a matter of concern to the operator and is due to several conditions ... 

Create stability and minimize variance Eliminate complexity and reduce processing time Benchmark current performance Focus attention on quality... 

Traditionally, column efficiency or plate counts in column chromatography were used to quantify how well a column was performing. This does not tell the entire story for GPC, however, because the ability of a column set to separate peaks is dependent on the molecular weight of the molecules one is trying to separate. We, therefore, chose both column efficiency and a parameter that we simply refer to as D a, where Di is the slope of the relationship between the log of the molecular weight of the narrow molecular weight polystyrene standards and the elution volume, and tris simply the band-broadening parameter (4), i.e., the square root of the peak variance. 

Where f(x) is tlie probability of x successes in n performances. One can show that the expected value of the random variable X is np and its variance is npq. As a simple example of tlie binomial distribution, consider tlie probability distribution of tlie number of defectives in a sample of 5 items drawn with replacement from a lot of 1000 items, 50 of which are defective. Associate success with drawing a defective item from tlie lot. Tlien the result of each drawing can be classified success (defective item) or failure (non-defective item). The sample of items is drawn witli replacement (i.e., each item in tlie sample is relumed before tlie next is drawn from tlie lot tlierefore the probability of success remains constant at 0.05. Substituting in Eq. (20.5.2) tlie values n = 5, p = 0.05, and q = 0.95 yields... 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

The Standard Error of Prediction (SEP) is supposed to refer uniquely to those situations when a calibration is generated with one data set and evaluated for its predictive performance with an independent data set. Unfortunately, there are times when the term SEP is wrongly applied to the errors in predicting y variables of the same data set which was used to generate the calibration. Thus, when we encounter the term SEP, it is important to examine the context in order to verify that the term is being used correctly. SEP is simply the square root of the Variance of Prediction, s2. The RMSEP (see below) is sometimes wrongly called the SEP. Fortunately, the difference between the two is usually negligible. 

Variance scaling is performed on a variable by variable basis. In other words, we would variance scale a set the concentration values of a data set on a component by component basis. Starting with the first component, we compute the total variance of the concentrations of that component. There are several variations on variance scaling. First, we will consider the most the method which adjusts all the variables to exactly unit variance. To do this we compute the variance of the variable, and then use the variance to scale all the concentrations of all the samples so that the new variance for the component is equal to unity. 

Normalization is performed on a sample by sample basis. For example, to normalize a spectrum in a data set, we first sum the squares of all of the absorbance values for all of the wavelengths in that spectrum. Then, we divide the absorbance value at each wavelength in the spectrum by the square root of this sum of squares. Figure C7 shows the same data from Figure Cl after variance scaling Figure C8 shows the mean centered data from Figure C2 after variance... 

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