Data smoothing function

As a simple illustration of this philosophy, we analyze one aspect of the following data-smoothing device. The input to this device is a time function X(t) and the output Y(t) is given by the formula... 

A smooth function f(x) is used quite often to describe a set of data, (x y,). (x2,y2),. . .,(x j,yN). Fitting of a smooth curve, f(x), through these points is usually done for interpolation or visual purposes (Sutton and MacGregor, 1977). This is called curve fitting. The parameters defining the curve are calculated by minimizing a measure of the closeness of fit such as the function S(k)= [y j - f(x j ) 2. ... 

Additional software has been developed to merge data from various data collection steps and to model the data using suitable statistical distribution functions. We are working on software to perform corrections for absorption, specimen shape, and misalignment. Library routines for 2-diraensional data smoothing and integration are being adapted to the calculation of orientation functions and other moments of the probability distributions. 

If you encounter these functions, you can reformulate them as equivalent smooth functions by introducing additional constraints and variables. For example, consider the problem of fitting a model to n data points by minimizing the sum of weighted absolute errors between the measured and model outputs. This can be formulated as follows ... 

Given a finite number of measurements at a given latitude (90° — 6) and longitude on the surface of the Earth, we look for a smooth function that could be fitted to the data and represent their variations to within any desired precision. Spherical harmonics are suitable because they make an orthogonal set of functions which can... 

In the case of finite star chains with very high functionality, the units are concentrated near and in the star core. Therefore, their theoretical behavior can approximately be described by a rigid sphere [2]. The form factor of a sphere presents a series of oscillations. The experimental data of stars with 128 arms [67] show a smooth function covering the first two oscillations of the sphere, followed by a peak coincident with the third oscillation and the asymptotic behavior for high q previously described for stars of lower functionalities. It seems that the chain resembles a soft spherical core with a peripheral region of considerably smaller density. 

Although numerical differentiation is considered as a routine step in signal processing, our discussion tries to emphasize that its results heavily depend on the choice of the interpolating or smoothing function. Different methods may lead to much deviating estimates. Nevertheless, from frequently sampled data we may be able to locate extrema or inflection points by numerical differentiation, since zero-crossing of the first or second derivatives is somewhat more reliable than their values. 

Weiss (1970a) reviewed and evaluated the literature data then available for N2, 02, and Ar and fitted them to a smooth functional dependence. His results are reported in terms of the Bunsen coefficient j5, defined as the quantity of gas (in cm3 STP) dissolved in unit volume (1 cm3) under unit partial pressure (1 atm). Pure water data were fitted to an integrated van t Hoff equation ... 

Each type of smoothing function removes different features in the data and often a combination of several approaches is recommended especially for real world problems. Dealing with outliers is an important issue sometimes these points are due to measurement errors. Many processes take time to deviate from the expected value, and a sudden glitch in the system unlikely to be a real effect. Often a combination of filters is recommend, for example a five point median smoothing followed by a three point Hanning window. These methods are very easy to implement computationally and it is possible to view the results of different filters simultaneously. 

Calculate three and five point median smoothing functions (denoted by 3 and 5 ) on the data (to do this, replace each point by the median of a span of N points), and plot die resultant graphs. 

Re-smoodi die three point median smoothed data by a furdier diree point median smoothing function (denoted by 33 ) and dien further by a Hanning window of die form Xj = 0.25a, + 0.5a-, + 0.25.v,+ (denoted by 33H ), plotting bodi graphs as appropriate. 

Numerical methods can be apphed to discrete (finite) data sets in order to carry ont such procedures as differentiation, integration, solution of algebraic and differential eqna-tions, and data smoothing. Analytical methods, which deal with continnons functions, are exact or at least capable of being carried ont to any arbitrary precision. Nirmerical methods applied to experimental data are necessarily approximate, being limited by the finite nnm-ber of data points employed and their precision. 

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