Statistics numerical operations

Finally, Sect. 20.5 completes this chapter by giving examples of statistical calculations and trivial numerical operations encotmtered in daily practice. It includes how to treat an apparent outlier. 

Applied Mathematics Branch of mathematics devoted to developing and applying mathematical methods, including math-modeling techniques, to solve scientific, engineering, industrial, and social problems. Areas of focus include ordinary and partial differential equations, statistics, probability, operational analysis, optimization theory, solid mechanics, fluid mechanics, numerical analysis, and scientific computing. 

Unknown Statistical Distributions Sixth, despite these problems, it is necessaiy that these data be used to control the plant and develop models to improve the operation. Sophisticated numerical and statistical methods have been developed to account for random... 

The objective is to estimate, numerically, the probability that a system composed of many components will fail. The obvious question is, "Why don t you just estimate the failure rate of the system from operating experience " There are three reasons IJ the system may not exist, so new data are not available, 2) the injuries and fatalities from the developmental learning experience are unacceptable - the risk must be known ahead of time, and 3) by designing redundancy, the probability of the system failing can be made acceptably remote in which case system failure data caimot be collected directly. The only practical way uses part failure statistics in a system model to estimate the system s reliability. 

Failure rates are computed by dividing the total number of failures for the equipment population under study by the equipment s total exposure hours (for time-related rates) or by the total demands upon the equipment (for demand-related rates). In plant operations, there are a large number of unmeasured and varying influences on both numerator and denominator throughout the study period or during data processing. Accordingly, a statistical approach is necessary to develop failure rates that represent the true values. 

Preprocessing is the operation which precedes the extraction of latent vectors from the data. It is an operation which is carried out on all the elements of an original data table X and which produces a transformed data table Z. We will discuss six common methods of preprocessing, including the trivial case in which the original data are left unchanged. The effects of each of these six types of preprocessing will be illustrated numerically by means of the small 4x3 data table from the study of trace elements in atmospheric samples which has been used in previous sections (Table 31.1). The various effects of the transformations can be observed from the two summary statistics (mean and norm). These statistics include the vector of column-means m and the vector of column-norms of the transformed data table Z ... 

In the present time with almost unlimited computer facilities in the analytical laboratory, analytical chemists should be able to obtain substantial benefits from the application of time series, information theory, multivariate statistics, a.o. factor analysis and pattern recognition, operations research, numerical analysis, linear algebra, computer science, artificial intelligence, etc. This is in fact what chemo-metricians have been doing for the past decades. 

In semi-automatic moulding the operator may inspect each unit for features including colour, dispersion and uniformity of colour, standard and uniformity of gloss or mattness, and the presence or otherwise of defects like silver streaks or stringing. However, it is preferable if assessments can be made in numerical terms, expressed in units that may be compared quantitatively and subjected easily to statistical treatment and analysis. It is often convenient in the course of... 

The BLAS are a collection of 38 FORTRAN callable subroutines that peform many of the basic operations of numerical linear algebra. Contact International Mathematical and Statistical Libraries, (IMSL) for more information. 

Chemistry, and in particular physical and analytical chemistry, often requires a numerical or statistical approach. Not only is mathematical modelling an important aid to understanding, but computations are often needed to turn raw data into meaningful information or to compare them with other data sets. Moreover, calculations are part of laboratory routine, perhaps required for making up solutions of known concentration (see p. 170 and below) or for the calibration of an analytical instrument (see p. 171). In research, trial calculations can reveal what input data are required and where errors in their measurement might be amplified in the final result, e.g. flame atomic absorption spectrometer (see Chapter 27). Table 39.7 Sets of numbers and operations ... 

The probability function is used in numerous statistical analyses of the experiment, only one of which we describe here. We calcnlate X, the mean, or average, valne of the random variable, by multiplying each possible outcome by its probability and summing over all the possible ontcomes. In onr compact form this operation for the probability function in Fignre C.5 is... 

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