X-RAY SPECTROMETRY X-Ray Spectrom. 2003; 32: 423–427 Published online 30 September 2003 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/xrs.662 Determination of some rare earth elements by EDXRF and artificial neural networks Fernando Schimidt, 1 Lorena Cornejo-Ponce, 2 Maria Izabel M. S. Bueno 1* and Ronei J. Poppi 1 1 Instituto de Qu´ ımica, Universidade Estadual de Campinas, Campinas, Brazil 2 Facultad de Ciencias, Universidad de Tarapac ´ a, Tarapac ´ a, Chile Received 7 November 2001; Accepted 10 March 2003 This paper describes the simultaneous determination of Pr, Nd and Sm by EDXRF spectrometry using mixtures of oxides of these metals in a silica matrix. The data were treated by distinct neural network algorithms: back-propagation (BP), Levenberg–Marquardt (LM) and two variations of back-propagation (called BP-SC, single component, and BP-MC, multiple component), using results from the PLS model (partial least square regression) for comparison. The best applied model was the BP-SC neural network, which produced relative standard errors of prediction of 17.5% for Pr, 12.5% for Nd and 12.6% for Sm. Copyright  2003 John Wiley & Sons, Ltd. INTRODUCTION Energy-dispersive x-ray fluorescence (EDXRF) spectrometry is a very versatile analytical technique, which allows simulta- neous and non-destructive determinations in solid and liquid samples without the need for complex preparations. Data treatments are usually made through analytical curves where the spectral lines corresponding to the elements of interest are integrated. When it is required to correlate spectral lines that show matrix effects or have spectral interferences, more complex mathematical methods, such as artificial neural networks, 1 must be used. Neural networks describe groups of mathematical algo- rithms that imitate the acquisition and processing of infor- mation by the human brain. 1,2 The neural network can be considered as a processing box, accepting a series of input data and producing one or more outputs. 1 These algo- rithms use some predefined mathematical parameters and are trained using a group of standards. During the process, weights are attributed for each neuron as a function of values used in the training step. This process is repeated in several cycles (Fig. 1). An input layer, an intermediate layer and an output layer generally form neural networks. They can be used in several situations, such as treatment of very com- plex data, data contaminated with noise or incomplete and non-linear data. Two algorithms to train the neural network were studied in this work, back-propagation (BP) and Leven- berg–Marquardt (LM), which differ in the methodology for the correction of the weights. Two variations of BP, the BP-SC and BP-MC, were tested. SC means ‘single compo- nent’ because the model was built for just one output for each application, in other words, the model calculates the concentration of only one metal each time it is processed, L Correspondence to: Maria Izabel M. S. Bueno, Instituto de Qu´ ımica, Universidade Estadual de Campinas, Campinas, Brazil. E-mail: bell@iqm.unicamp.br output = w 1 s 1 + w 2 s 2 + ... + w i s i + ... + w n s n S n output S 1 S 2 ... Figure 1. Representation of an artificial neuron. All input data S i are multiplied by mathematical weights w i generating output data. for example. Luo et al. 3 studied this procedure and obtained better results than with the standard algorithm, since BP-SC needs fewer weight attributions to be calculated and the optimum parameters can be found more easily. MC means ‘multiple component,’ when the neural net has more than one output. The main difference between SC and MC is basi- cally the choice of a smaller root mean square (r.m.s.) error for each element, as will be seen later. The BP algorithm makes the correction of weights through the method of descending gradients. 1,2 The main weight correction equation can be written as w k D D C w k1 ⊲1⊳ where: w k represents the difference between the initial and final weights for the kth iteration,  is called the ‘learning rate’ and determines how fast the changes of weight corrections should be implemented in the iteration cycles, D is the derivative of error calculated through the chosen transfer functions, known as the gradient descent method, and gives the minimum error direction at the error surface, and  is called the ‘momentum’ and represents a constant that is multiplied by the error variation before iteration (w k1 ). It Copyright  2003 John Wiley & Sons, Ltd.