Transformations of data

Step 1. Generate the finite, discrete dyadic wavelet transform of data using Mallat and Zhong s (1992) cubic spline wavelet (Fig. 8c). 

In the introduction to Part A we discussed the arch of knowledge [1] (see Fig. 28.1), which represents the cycle of acquiring new knowledge by experimentation and the processing of the data obtained from the experiments. Part A focused mainly on the first step of the arch a proper design of the experiment based on the hypothesis to be tested, evaluation and optimization of the experiments, with the accent on univariate techniques. In Part B we concentrate on the second and third steps of the arch, the transformation of data and results into information and the combination of information into knowledge, with the emphasis on multivariate techniques. 

A special type of data pre-treatment is the transformation of data into a smaller number of new variables. Principal components analysis is a natural example and we have treated it in Section 36.2.3 as PCR. Another way to summarize a spectrum in a few terms is through Fourier analysis. McClure [29] has shown how a NIR... 

Almost any transformation of data is changing the weight of inherent features that we want to know about. A striking example is the simple logarithmic representation of scattering data and the related distortion of the error-bar spread (p. 124, Fig. 8.11). As we are interested in structure, we should fit data that present an undistorted view of structure. Our X-ray instrument has already transformed structure information into a scattering pattern, and we have to ask what we should do with the pattern before fitting - leave it as it is or transform it back ... 

Ordinary least squares regression requires constant variance across the range of data. This has typically not been satisfied with chromatographic data ( 4,9,10 ). Some have adjusted data to constant variance by a weighted least squares method ( ) The other general adjustment method has been by transformation of data. The log-log transformation is commonly used ( 9,10 ). One author compares the robustness of nonweighted, weighted linear, and maximum likelihood estimation methods ( ). Another has... 

The direct linear plot (O Figure 4-6) requires no transformation of data, and this approach is arguably the most appropriate for determining and values. Individual values of [S] are plotted as their negatives on the x-axis and measured values of v are plotted on the axis. Corresponding pairs of ( — [S], 0)... 

Fitting velocity data directly to a hyperbohc curve has several advantages over linear methods, transformed or otherwise. The major advantages are that no transformation of data is necessary, curves are fitted easily with currently available graphing software, and variations in behavior from a simple Michaelis-Menten one-substrate equation usually result in an equation which still describes a hyperbola, thus requiring no change in the analytical approach. 

In Eq. 13.15, the squared standard deviations (variances) act as weights of the squared residuals. The standard deviations of the measurements are usually not known, and therefore an arbitrary choice is necessary. It should be stressed that this choice may have a large influence of the final best set of parameters. The scheme for appropriate weighting and, if appropriate, transformation of data (for example logarithmic transformation to fulfil the requirement of homoscedastic variance) should be based on reasonable assumptions with respect to the error distribution in the data, for example as obtained during validation of the plasma concentration assay. The choice should be checked afterwards, according to the procedures for the evaluation of goodness-of-fit (Section 13.2.8.5). 

Transformations of data are also used to linearize relationships involving the variable in question. For example, the relationship between solids and viscosity of a resin solution may be sharply curved the relationship between solids and log viscosity is a straight line. 

Figure 19-9 (a) Spectrophotometric titration of 30.0 mL of EDTA in acetate buffer with CuS04 in the same buffer. Upper curve [EDTA] = [Cu2 ] = 5.00 mM. Lower curve [EDTA] = [Cu2 ] = 2.50 mM. The absorbance has not been corrected in any way. (b) Transformation of data into mole fraction format. The absorbance of free CuS04 at the same formal concentration has been subtracted from each point in panel a. EDTA is transparent at this wavelength. [From L D. Hill and P MacCarthy, Novel Approach to Job s Method Chem. Ed. 1986,63, 162.]... 

There are many commercial programs available for fitting data to theoretical curves. They are extremely powerful but very dangerous. Used properly, they fit data directly to nonlinear curves without the need for transformation of data to linear equations that can distort the statistics, and they allow data to be fitted to complex equations that could not be solved by hand. But a computer just gives the best fit to the particular equation, y = f(x), that you choose, and your data might not follow that equation. 

Figure 4.10 Representation of the transformation of data from a single-column data string to a matrix form, based on the sampling frequency and modulation time. The data points acquired for each modulation period are placed in a separate row of the matrix. The matrix data are then in a suitable format to read into an appropriate plotting package such as the Transform program.

In order to cast results of other studies in terms of our model, some transformation of data has been necessary. In some cases we have had to estimate activity coefficients in solutions, and this has been done via the Davies equation ( ),... 

Besides the scientific questions related to the coupling of models, the interaction of the numerical models is a big technical challenge. The transformation of data at different temporal and spatial resolutions as well as computational efficiency, memory consumption, data storage capacity, meta-data communication and code management are issues which have to be addressed. 

The problem in both methods is the error propagation. If an error exists in the measurement, this error will be submitted to the transformation as well. A second problem arises in the variances. Usually the variances of measurement in TLC are constant within the calibration range. The transformation of data will lead to inhomogeneous variances and this is the reason for unreliable regression analysis. 

The quality of the end result fundamentally depends on the sample, the data, the measurement process, the transformation of data to information and, finally, the display and transformation of the information. It is obvious that the role of the instrument in providing the integrity of data is also fundamental to the end result. Written requirements for instrumental performance are not sufficient for assuring the reliability of the result unless they are tested and validated. 

The parametric method is much more complicated than the simple nonparametric method and requires computer software. The method is presented here under separate headings for testing of type of distribution, transformation of data, and the estimation of percentiles and their confidence intervals. 

Principle components analysis (PCA), a form of factor analysis (FA), is one of the most common unsupervised methods used in the analysis of NMR data. Also known as Eigenanalysis or principal factor analysis (PEA), this method involves the transformation of data matrix D into an orthogonal basis set which describes the variance within the data set. The data matrix D can be described as the product of a scores matrix T, and a loading matrix P,... 

Table C.2 Transformation of Data Centered at 50 s to a New Coordinate System...

XML is becoming increasingly more important as a mechanism to exchange data between applications15. Because many types of bioinformatics analyses require iterative application of tools in a pipeline, a mechanism to transfer information between stages of a pipeline is required. In the past these were often proprietary formats, and it was necessary to write specialized tools to transform the output of one tool into the input of the next. XML and the use of style sheets to drive transformations promises to at least make the parsing and transformation of data metadata driven as opposed to customized for each new application pipeline. 

Figure 4.8 Scatter plot of simulated data from a Michaelis-Menten model with Vmax — 100 and Km — 20 (top) and Lineweaver-Burke transformation of data (bottom). Stochastic variability was added by assuming normally distributed constant variability with a standard deviation of 3.

Transformation of data for distributions that are not parametric, data can be converted to log,o or log values to obtain a log-normal distribution prior to applying a parametric method of comparison such as t-test or ANOVA. 

Scheme 2 Schematic representation of (a) discrete wavelet transform of data set, X, from time domain to wavelet domain, W, and (hj matrix Wcontaining wavelet coefficients sorted according to their contribution to the data variance.

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