Matrix concentration

To obtain this matrix by the multivariate method, we first generate two absorptivity vectors ap and a2j from a known concentration matrix in parts per million... 

State-of-the-art for data evaluation of complex depth profile is the use of factor analysis. The acquired data can be compiled in a two-dimensional data matrix in a manner that the n intensity values N(E) or, in the derivative mode dN( )/d , respectively, of a spectrum recorded in the ith of a total of m sputter cycles are written in the ith column of the data matrix D. For the purpose of factor analysis, it now becomes necessary that the (n X m)-dimensional data matrix D can be expressed as a product of two matrices, i. e. the (n x k)-dimensional spectrum matrix R and the (k x m)-dimensional concentration matrix C, in which R in k columns contains the spectra of k components, and C in k rows contains the concentrations of the respective m sputter cycles, i. e. ... 

As we will soon see, the nature of the work makes it extremely convenient to organize our data into matrices. (If you are not familiar with data matrices, please see the explanation of matrices in Appendix A before continuing.) In particular, it is useful to organize the dependent and independent variables into separate matrices. In the case of spectroscopy, if we measure the absorbance spectra of a number of samples of known composition, we assemble all of these spectra into one matrix which we will call the absorbance matrix. We also assemble all of the concentration values for the sample s components into a separate matrix called the concentration matrix. For those who are keeping score, the absorbance matrix contains the independent variables (also known as the x-data or the x-block), and the concentration matrix contains the dependent variables (also called the y-data or the y-block). 

The first thing we have to decide is whether these matrices should be organized column-wise or row-wise. The spectrum of a single sample consists of the individual absorbance values for each wavelength at which the sample was measured. Should we place this set of absorbance values into the absorbance matrix so that they comprise a column in the matrix, or should we place them into the absorbance matrix so that they comprise a row We have to make the same decision for the concentration matrix. Should the concentration values of the components of each sample be placed into the concentration matrix as a row or as a column in the matrix The decision is totally arbitrary, because we can formulate the various mathematical operations for either row-wise or column-wise data organization. But we do have to choose one or the other. Since Murphy established his laws long before chemometricians came on the scene, it should be no surprise that both conventions are commonly employed throughout the literature ... 

Similarly, a concentration matrix holds the concentration data. The concentrations of the components for each sample are placed into the concentration matrix as a column vector ... 

Where C is the concentration of the c component of sample s. Suppose we were measuring the concentrations of 4 components in each of the 30 samples, above. The concentrations for each sample would be held in a column vector containing 4 concentration values. These 30 column vectors would be assembled into a concentration matrix which would be 4 X 30 in size (4 rows, 30 columns). 

Taken together, the absorbance matrix and the concentration matrix comprise a data set. It is essential that the columns of the absorbance and concentration matrices correspond to the same mixtures. In other words, the sth column of the absorbance matrix must contain the spectrum of the sample... 

We have seen that data matrices are organized into pairs each absorbance matrix is paired with its corresponding concentration matrix. The pair of matrices comprise a data set. Data sets have different names depending on their origin and purpose. 

A data set containing measurements on a set of known samples and used to develop a calibration is called a training set. The known samples are sometimes called the calibration samples. A training set consists of an absorbance matrix containing spectra that are measured as carefully as possible and a concentration matrix containing concentration values determined by a reliable, independent referee method. 

Next, we create a concentration matrix containing mixtures that we will hold in reserve as validation data. We will assemble 10 different validation samples into a concentration matrix called C3. Each of the samples in this validation set will have a random amount of each component determined by choosing numbers randomly from a uniform distribution of random numbers between 0 and 1. 

We will create yet another set of validation data containing samples that have an additional component that was not present in any of the calibration samples. This will allow us to observe what happens when we try to use a calibration to predict the concentrations of an unknown that contains an unexpected interferent. We will assemble 8 of these samples into a concentration matrix called C5. The concentration value for each of the components in each sample will be chosen randomly from a uniform distribution of random numbers between 0 and I. Figure 9 contains multivariate plots of the first three components of the validation sets. 

C is a single column concentration matrix of the form in equation [9]... 

In equation [24], A is generated by multiplying the pure component spectra in the matrix K by the concentration matrix, C, just as was done in equation [20]. But, in this case, C will have a column of concentration values for each sample. Each column of C will generate a corresponding column in A containing the spectrum for that sample. Note that equation [24] can also be written as equation [22]. We can represent equation [24] graphically ... 

To produce a calibration using classical least-squares, we start with a training set consisting of a concentration matrix, C, and an absorbance matrix, A, for known calibration samples. We then solve for the matrix, K. Each column of K will each hold the spectrum of one of the pure components. Since the data in C and A contain noise, there will, in general, be no exact solution for equation [29]. So, we must find the best least-squares solution for equation [29]. In other words, we want to find K such that the sum of the squares of the errors is minimized. The errors are the difference between the measured spectra, A, and the spectra calculated by multiplying K and C ... 

To solve for K, we first post-multiply each side of the equation by CT, the transpose of the concentration matrix. 

Now that we have calculated K we can use it to predict the concentrations in an unknown sample from its measured spectrum. First, we place the spectrum into a new absorbance matrix, Aunk. We can now use equation [29] to give us a new concentration matrix, Cunk, containing the predicted concentration values for the unknown sample. 

Concentration matrix, 7,10 row-wise, 12 training set, 34 validation set, 35 Concentration space, 28 Congruent... 

The metal concentration, matrix, and temperature effects that favor clustering of the cobalt group of metal atoms have been assessed by... 

In Chapter 31 we stated that any data matrix can be decomposed into a product of two other matrices, the score and loading matrix. In some instances another decomposition is possible, e.g. into a product of a concentration matrix and a spectrum matrix. These two matrices have a physical meaning. In this chapter we explain how a loading or a score matrix can be transformed into matrices to which a physical meaning can be attributed. We introduce the subject with an example from environmental chemistry and one from liquid chromatography. 

Spectra at p (=20) wavelengths. Because of the Lambert-Beer law, all measured spectra are linear combinations of the two pure spectra. Together they form a 15x20 data matrix. For example the UV-visible spectra of mixtures of two polycyclic aromatic hydrocarbons (PAH) given in Fig. 34.2 are linear combinations of the pure spectra shown in Fig. 34.3. These mixture spectra define a data matrix X, which can be written as the product of a 15x2 concentration matrix C with the 2x20 matrix of the pure spectra ... 

Note that eq. (36.5) is a collection of many univariate multiple regression models for each wavelength j the multiple regression of the corresponding spectral channel , i.e. Sj, on the concentration matrix C yields a vector of regression coefficients, ky (the yth column of K). For K to be estimable C C must be invertible, i.e. the number of calibration standards should at least be as large as the number of analytes. It is clearly not possible to obtain, directly or indirectly, say 3 pure spectra from recording the spectra of just 1 or 2 standards of known composition. In practice, the condition n>p, or more precisely rank(C)=p, is hardly a restriction. 

By means of the specificity function, concentration-matrix ratios can be estimated for which a reliable determination of analytes may be possible. 

Standard solutions, a solution of nickel in acid with a quoted mass/volume concentration, a solution of sodium hydroxide with a quoted concentration as a molarity and a solution of pesticides with quoted mass/volume concentrations. Matrix RMs - natural materials, river sediment with quoted concentrations of metals, milk powder with a quoted fat content and crab paste with quoted concentrations of trace elements. 

The concentration matrices will be shown as O fringed on each edge by the boundary values written in italic, i.e., as (ny +1) x (nx+ 1) = 6 x 5 matrices, and a symbols with overbar CJ will be used. Given the initial concentration matrix... 

Do you use Certified Reference Materials (CRMs), and if so, how For example, specify the concentrations) matrix type(s) etc. 

Presently, we are able to compute A knowing Y and C. Computing Y knowing C and A is trivial. What about calculating the concentration matrix C, knowing Y and A We could transpose equation (4.50) ... 

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