MATLAB matrix analysis

Equation (27) expresses an error in the dynamic matrix element Lij obtained from full matrix analysis if the error in peak volumes is Aa [50]. It also assumes that volume errors are equal for all peaks and are uncorrelated Aa is volume error normalized to the volume of a single spin at Tm = 0. Modem computer programs (Matlab, Mathematica, Mapple) can calculate the dynamic matrix from eq. (11) directly. 

Cross-relaxation rates and interproton distances in cyclo(Pro-Gly) from the full matrix analysis of NOESY spectrum recorded at Tm = 80 ms and T = 233 K. Cross-relaxation rates are obtained from the volumes shown in table 2 according to eq. (11) by Matlab (Mathworks Inc). Error limits were obtained from eq. (27) with Aa = 0.015 (table 2). 

Example Consider the following example from Noble and Daniel [Applied Linear Algebra, Prentice-Hall (1987)] with the MATLAB commands to do the analysis. Define the following real matrix with m = 3 and n = 2 (whose rank k = 1). 

Note the Scores matrix is referred to as the T matrix in principal components analysis terminology. Let us look at what we have completed so far by showing the SVD calculations in MATLAB as illustrated in Table 22-1. 

Matlab is a matrix oriented language that is just about perfect for most data analysis tasks. Those readers who already know Matlab will agree with that statement. Those who have not used Matlab so far, will be amazed by the ease with which rather sophisticated programs can be developed. This strength of Matlab is a weak point in Excel. While Excel does include matrix operations, they are clumsy and probably for this reason, not well known and used. An additional shortcoming of Excel is the lack of functions for Factor Analysis or the Singular Value Decomposition. Nevertheless, Excel is very powerful and allows the analysis of fairly complex data. 

Prior to analysis, the Raman shift axes of the spectra were calibrated using the Raman spectrum of 4-acetamidophenol. Pretreatment of the raw spectra, such as vector normalization and calculation of derivatives were done using Matlab (The Mathworks, Inc.) or OPUS (Bruker) software. OPUS NT software (Bruker, Ettlingen, Germany) was used to perform the HCA. The first derivatives of the spectra were used over the range from 380 cm-1 to 1700 cm-1. To calculate the distance matrix, Euclidean distances were used and for clustering, Ward s algorithm was applied [59]. 

Seven simulated LC-UV/Vis DAS data matrices were constructed in MATLAB 5.0 (MathWorks Inc., Natick, MA). Each sample forms a 25 x 50 matrix. The simulated LC and spectral profiles are shown in Figure 12.3a and Figure 12.3b, respectively. Spectral and chromatographic profiles are constructed to have a complete overlap of the analyte profile by the interferents. Three of the samples represent pure standards of unit, twice-unit, and thrice-unit concentration. These standards are designated SI, S2, and S3, respectively. Three three-component mixtures of relative concentrations of interferent 1 analyte interferent 2 are 1 1.5 0.5, 2 0.5 2, and 2 2.5 1, and these are employed for all examples. These mixtures are designated Ml, M2, and M3, respectively. An additional two-component mixture, 2 2.5 0, is employed as an example for rank annihilation factor analysis. This sample is designated M4. In most applications, normally distributed random errors are added to each digitized channel of every matrix. These errors are chosen to have a mean of zero and a standard deviation of 0.14, which corresponds to 10% of the mean response of the middle standard. In the rank annihilation factor analysis (RAFA) examples, errors were chosen to simulate noise levels of 2.5 and 5% of the mean response of S2. In some PARAFAC examples, the noise level was chosen to be 30% of the mean response of the second standard. 

For the exactly determined (two-TD or three-TOA) case it is exactly equivalent to the TD solution and therefore we can extend results from TOA or pseudo-range analysis to the TD case. Early LORAN TD analysis was typically done in scalar form before computer programs such as spreadsheets and MatLab made matrix calculations trivial. GPS fix analysis has typically been expressed in matrix form resulting in simpler expressions. 

Figure 9.22 shows the di9 4 step9.m file that contains the code for Multivariate Analysis of Variance, MANOVA (i.e., testing the null hypothesis that the group means are all the same for the n-dimensional multivariate vector, and that any difference observed in the sample stress is due to random chance). The group means must lie in a maximum of dfb-dimensional space, where dfb is the degree of freedom of B matrix. B is the between-groups sum of squares and cross products matrix (see MATLAB online help for manoval). MATLAB manoval will take care of this maximum dimension, dfb. Because d = 1 as calculated by manoval, we cannot reject the hypothesis that the means lie in a 1-D subspace. 

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