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ch15_bss

ch15_bss - MIT OpenCourseWare http/ocw.mit.edu HST.582J...

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MIT OpenCourseWare http://ocw.mit.edu HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing Spring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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HST-582J/6.555J/16.456J Biomedical Signal and Image Processing Spring 2008 Chapter 15 - BLIND SOURCE SEPARATION: Principal & Independent Component Analysis c G.D. Clifford 2005-2008 Introduction In this chapter we will examine how we can generalize the idea of transforming a time series into an alternative representation, such as the Fourier (frequency) domain, to facil - itate systematic methods of either removing (ﬁltering) or adding (interpolating) data. In particular, we will examine the techniques of Principal Component Analysis (PCA) using Singular Value Decomposition (SVD), and Independent Component Analysis (ICA). Both of these techniques utilize a representation of the data in a statistical domain rather than a time or frequency domain. That is, the data are projected onto a new set of axes that fulﬁll some statistical criterion, which implies independence, rather than a set of axes that represent discrete frequencies such as with the Fourier transform, where the independence is assumed . Another important difference between these statistical techniques and Fourier-based tech - niques is that the Fourier components onto which a data segment is projected are ﬁxed, whereas PCA- or ICA-based transformations depend on the structure of the data being ana - lyzed. The axes onto which the data are projected are therefore discovered . If the structure of the data (or rather the statistics of the underlying sources) changes over time, then the axes onto which the data are projected will change too 1 . Any projection onto another set of axes (or into another space) is essentially a method for separating the data out into separate components or sources which will hopefully allow us to see important structure more clearly in a particular projection. That is, the direction of projection increases the signal-to-noise ratio (SNR) for a particular signal source. For example, by calculating the power spectrum of a segment of data, we hope to see peaks at certain frequencies. The power (amplitude squared) along certain frequency vectors is therefore high, meaning we have a strong component in the signal at that frequency. By discarding the projections that correspond to the unwanted sources (such as the noise or artifact sources) and inverting the transformation, we effectively perform a ﬁltering of the recorded observation. This is true for both ICA and PCA as well as Fourier-based techniques. However, one important difference between these techniques is that Fourier techniques assume that the projections onto each frequency component are independent of the other frequency components. In PCA and ICA we attempt to ﬁnd a set of axes which are independent of one another in some sense. We assume there are a set of independent (The structure of the data can change because existing sources are non-stationary, new signal sources manifest, or the manner in which the sources interact at the sensor changes.
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ch15_bss - MIT OpenCourseWare http/ocw.mit.edu HST.582J...

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