Negative matrix factorization

Lee, D. D., and Seung S. (1999). Learning the parts of objects by non-negative matrix factorization. Nature 401,788-791. 

One method is Non-Negative Matrix Factorization (NMF), which learns to recognize semantic features of the text [34]. A corpus of documents can be summarized by a matrix of words versus documents. This matrix is sparse, with many zero values. The algorithm extracts a set of semantic features, combinations of which can... 

Gao H-T, Li T-H, Chen K, Li W-G, Bi X. Overlapping spectra resolution using non-negative matrix factorization. Talanta 2005 66 65-73. 

A correlation between two variables can be either positive or negative. The interpretation of the factors in terms of patterns in the samples is the most difficult part and not always possible. However, a particular combination of m/z values in a mass spectrum can be indicative of the presence of a chemical compound in the sample. A tool for interpretation, similar to that utilized to interpret canonical variates, is the factor spectrum with intensities Si bij (where Si is the standard deviation of each column in the data matrix X). This spectrum, plotting the values Sj bij at m/z, shows the part of the intensity change described by the factor. In other words, such a spectrum will show those masses that contribute the most to the discrimination of the samples. Because the loadings b j generally can be either positive or negative, the factor spectrum exhibits positive and negative intensities. 

Paatero P, Tapper U, Positive matrix factorization A non-negative factor model with optimal utilization of error estimates of data values, Environmetrics, 1994, 5, 111-126. 

Paatero P, Tapper U (1994) Positive matrix factorization a non-negative factor model with optimal utilization of error estimates of data values. Environmetrics 5 111-126 Park SK, O Neill MS, Stunder BJB et al (2007) Source location of air pollution and cardiac autonomic function trajectory cluster analysis for exposure assessment. J Expos Sci Environ Epidemiol 17 488 97... 

The aim of factor analysis is to calculate a rotation matrix R which rotates the abstract factors (V) (principal components) into interpretable factors. The various algorithms for factor analysis differ in the criterion to calculate the rotation matrix R. Two classes of rotation methods can be distinguished (i) rotation procedures based on general criteria which are not specific for the domain of the data and (ii) rotation procedures which use specific properties of the factors (e.g. non-negativity). 

In previous methods no pre-knowledge of the factors was used to estimate the pure factors. However, in many situations such pre-knowledge is available. For instance, all factors are non-negative and all rows of the data matrix are nonnegative linear combinations of the pure factors. These properties can be exploited to estimate the pure factors. One of the earliest approaches is curve resolution, developed by Lawton and Sylvestre [7], which was applied on two-component systems. Later on, several adaptations have been proposed to solve more complex systems [8-10]. 

The conditions in PHWE are typically harsh and, therefore, the method is not suitable for thermolabile compounds. Analytes may also react with each other or with the water molecules during the extraction. From an analytical point of view the most salient negative factors of SWE in the continuous mode are co-extraction of undesirable components of the matrix (usually polar components) and dilution of the analyte in the extract. This calls for a clean-up and concentration step prior to individual separation and detection of the target compounds. 

The negative reciprocal of phosphorus residence time in each reservoir is found on the diagonal entries of matrix A (Table 7.3). A is factored giving six eigenvalues and six characteristic times of the system as the negative reciprocal of the eigenvalues... 

Because Equation 3-lOla represents a set of homogeneous linear equations, multiplying the solution by a positive or negative factor is still a solution. Therefore, each column vector in Equation 3-lOlc and 3-lOld can be made a unit vector. Then the matrix T is obtained. With this matrix known, diffusion profiles can be calculated by solving Equation 3-99c. 

Up to the scaling, they are co-factors Ag of elements of any column of stoichiometric matrix F (see, for instance, Bykov et ah, 1998 Lazman and Yablonskii, 1991). We can always assign the directions of elementary reactions so that all stoichiometric coefficients are non-negative and this will be assumed later. 

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