X.6 Non-Negative Matrix Factorization a.k.a. Multivariate Curve Resolution (MCR) when applied to spectroscopy. Goal: Decompose a set of spectra into their pure-component spectra. Method: Assuming the pure component spectra are known (or guessed), determine the concentrations of each in each spectrum, removing negative entries. From the known (or guessed) set of concentrations, determine the spectra of each component, removing negative entries. Iterate to convergence. 4.1 : 1/14

When to consider MCR/NNMF Multiple spectra are available, each with varying (but not necessarily known) contributions from all the components. The obtained spectra can be described as a linear combination of pure component spectra. 4.1 : 1/14

Overview 1. Decide the number of species present in your sample. Select a few of the spectra at random (same number as your species) and assign them as pure component spectra to get the ball rolling. 2. Fit each spectrum in the set as a linear combination of the pure component spectra, and replace negative concentrations with zeros. 3. Using these concentrations, solve for the set of pure component spectra, setting all negative amplitudes in the spectra to zero. 4. Iterate between steps 2 and 3 until no negative values are present in either the concentration or the pure component spectra. 4.1 : 1/14

The Maths Remember this? If we have guesses for the pure component spectra, we can invert the problem and solve for the concentrations. 4.1 : 1/14

The Maths In MCR and NNMF, D = the set of n measured multi-component spectra of length N are given by an (Nn) data matrix D  = the (Nm) matrix of m pure component spectra C = the (mn) set of corresponding concentrations m = assumed number of pure components Each column of C describes the combination s of pure components required to recover the corresponding column in D. Each column of D is a measured spectrum Each column of  is a pure component spectrum 4.1 : 1/14

The Maths Part 1. If the pure spectra are known (or guessed), the concentrations can be isolated by matrix inversion. Since negative concentrations correspond to nonphysical results, set all negative entries to zero. 4.1 : 1/14

The Maths Part 2. Use the recovered nonnegative concentrations to solve for the set of pure component spectra . Since negative intensities/absorbances correspond to nonphysical results, set all negative entries to zero. 4.1 : 1/14

Uncertainties For a given measurement i, the standard deviations are calculated in the usual way: Note: N is the number of elements in the spectrum, and n is the number of unique pure components. The variance in the concentration for the species in row r for the ith measurement is given by the following: 4.1 : 1/14