tailieunhanh - Independent component analysis P15

EXTENSIONS AND RELATED METHODS Independent Component Analysis. Aapo Hyv¨ rinen, Juha Karhunen, Erkki Oja a Copyright 2001 John Wiley & Sons, Inc. ISBNs: 0-471-40540-X (Hardback); 0-471-22131-7 (Electronic) 15 Noisy ICA In real life, there is always some kind of noise present in the observations. Noise can correspond to actual physical noise in the measuring devices, or to inaccuracies of the model used. Therefore, it has been proposed that the independent component analysis (ICA) model should include a noise term as well. In this chapter, we consider different methods for estimating the ICA model when noise is present. However, estimation of the mixing matrix. | Independent Component Analysis. Aapo Hyvarinen Juha Karhunen Erkki Oja Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-40540-X Hardback 0-471-22131-7 Electronic Part III EXTENSIONS AND RELATED METHODS Independent Component Analysis. Aapo Hyvarinen Juha Karhunen Erkki Oja Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-40540-X Hardback 0-471-22131-7 Electronic 15 Noisy ICA In real life there is always some kind of noise present in the observations. Noise can correspond to actual physical noise in the measuring devices or to inaccuracies of the model used. Therefore it has been proposed that the independent component analysis ICA model should include a noise term as well. In this chapter we consider different methods for estimating the ICA model when noise is present. However estimation of the mixing matrix seems to be quite difficult when noise is present. It could be argued that in practice a better approach could often be to reduce noise in the data before performing ICA. For example simple filtering of time-signals is often very useful in this respect and so is dimension reduction by principal component analysis PCA see Sections and . In noisy ICA we also encounter a new problem estimation of the noise-free realizations of the independent components ICs . The noisy model is not invertible and therefore estimation of the noise-free components requires new methods. This problem leads to some interesting forms of denoising. DEFINITION Here we extend the basic ICA model to the situation where noise is present. The noise is assumed to be additive. This is a rather realistic assumption standard in factor analysis and signal processing and allows for a simple formulation of the noisy model. Thus the noisy ICA model can be expressed as x As n 293 294 NOISYICA where n ni nn Tis the noise vector. Some further assumptions on the noise are usually made. In particular it is assumed that 1. The noise is independent from the independent components. 2. The