For ICA this measure is based on nonGaussianity, such as kurtosis, and the axes are not necessarily orthogonal. For PCA the measure we use to discover the axes is variance and leads to a set of orthogonal axes (because the data are decorrelated in a second order sense and the dot product of any pair of the newly discovered axes is zero). That is, we do not look for specific pre-defined components, such as the energy at a specific frequency, but rather, we allow the data to determine the components.
Since we discover, rather than define the the new axes, this process is known as blind source separation.
The sources are the data projected onto each of the new axes. (Therefore, they may overlap in the frequency domain in contrast to Fourier techniques.) We then define some measure of independence and attempt to decorrelate the data by maximising this measure for (or between) projections onto each axis of the new space which we have transformed the data into. 1 sources in the data, but do not assume their exact properties. We assume there are a set of independent 1 (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. In PCA and ICA we attempt to find a set of axes which are independent of one another in some sense. 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. This is true for both ICA and PCA as well as Fourier-based techniques. 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. The power (amplitude squared) along certain frequency vectors is therefore high, meaning we have a strong component in the signal at that frequency. For example, by calculating the power spectrum of a segment of data, we hope to see peaks at certain frequencies. That is, the direction of projection increases the signal-to-noise ratio (SNR) for a particular signal source. 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. 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. The axes onto which the data are projected are therefore discovered. Another important difference between these statistical techniques and Fourier-based techniques 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 analyzed.
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. Both of these techniques utilize a representation of the data in a statistical domain rather than a time or frequency domain. In particular, we will examine the techniques of Principal Component Analysis (PCA) using Singular Value Decomposition (SVD), and Independent Component Analysis (ICA).
Measurement artifact meaning series#
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 facilitate systematic methods of either removing (filtering) or adding (interpolating) data.