Makin' Pha-ses

Makin' Pha-ses, indeed [1]. Here is some "interesting" trace from two forms of noise that demonstrates a phenomenon called 1-D stochastic resonance [2]. This will generate a noisy signal with three distinct phases from two sources of noise (white and black):

1) generate white noise (a randomized sine wave over a 1,000 point interval). White noise is the kind of static you may have encountered while tuning an analog radio or television [3].

2) generate black noise (1/f³ deterministic signal over a 1,000 point interval). Black noise is the kind of noise you might encounter when modeling the frequency of natural disasters or blackbody radiation, and consists of mostly silence.

3) convolve both sources. Results in a signal with three distinct phases (over a 2,000 point interval). Demo conducted and plots made in Matlab.

The resulting plots are shown below. The associated Matlab code is located in my Github repository [4].

Also, even though this is tangentially related (e.g. making independent components), I will nevertheless post it here. I have a new tutorial up on Figshare [5] concerning how to conduct independent component analysis (ICA) and how to extract ICs from real-time thermocycling curves (e.g. qRT-PCR).

In keeping with the theme of the post, however, the Fast ICA algorithm does use white noise (a whitening matrix) to derive the ICs.

NOTES:

[1] an obscure reference from Saturday Night Live ("Makin' Copies" with Rob Schneider).

[2] Wellens, T., Shatokhin, V., and Buchleitner, A.   Stochastic Resonance. Reports on Progress in Physics, 67(1), 45 (2004) AND Balenzuela, P. Braun, H., and Chialvo, D.R.   The Ghost of Stochastic Resonance: An Introductory Review. arXiv, 1110.0136 (2011).

[3] For an artistic take on television static, please see TV Static Photos by Tom Moody and Ray Rapp. Examples of white noise in 2-D.

[4] this is part of work I am doing on an idea I am currently calling "noise strategies", a hybrid approach that merges game theory with the physics of noise. More on this in future posts.

[5] Alicea, B.   Independent features of quantified thermocycling reactions (qRT-PCR). Figshare, doi:10.6084/m9.figshare.649432 (2013).

The analysis was done using FastICA 2.5 for MATLAB. For details, see the following paper: Hyvarinen, A. and Oja, E. Independent Component Analysis: Algorithms  and Application. Neural Networks, 13(4-5), 411-430. (2000).