Math

>> "Although we have powerful computing and mathematical tools, perhaps there is some value in taking Diotellevi warning to heart: your machine may bring you delirium instead of ecstasy. Inexpensive high speed computing, modern statistics and mathematics sometimes seem like the modern version of rune marked bones, thrown by a shaman in cave lit by a smoky tallow lamp. It remains to be seen whether computing and mathematics are more reliable than the bones at unmasking truth and predicting the future. In theory statistics give us tools that tell us how successful our techniques are. But statistical tests rarely give us a final answer." -- From <Wavelet and Signal Processing>

Walking Randomly is more fun than getting someplace in a straight line: the power of randomness. Now getting interested in the complex system like percolation and fractals which I learned basics in my master and phd training. In this arena, many interesting topics about large sparse matrix arise.
Fourier, Wavelet and Resting Brain (WRB) : in plan to looking at. The 1/f noise stochastic process would be the first class here.
Computations for Resting Brain Dynamics (RBD) : this should be my another new start to exploring neuroscience from a theoretical or computational perspectives. As a complex network or system, human brain collects attractive mathematics so much.
Independent Component Analysis (ICA) : this is the start where I am attempting to look what is the neuroscience like. Basically, the representative mathematics for ICA is matrix theory. Although ICA is stated as one of multiple variates statistical methods, in mathematics, it is just a linear system extending to the specific set of variables, i.e. random variables, or in some cases, it is called as functional variable. It is a statistical tool which represents a set of random variables as a linear combination of some statistically independent non-Gaussian random variables. Please refer to the below equation for the initial impression. The real challenge is how to dig out the (a_ij) matrix and signals (s_i) vector without any information about that when only output signals (x_i) were observed. Most solutions to ICA have to add a prior about the relationships among hidden signals and further used it as an object function to get an approximate solution by using some random approaches. These approaches make the solution is not unique determined for every calculation on the same data. It is bad to evaluate the clinical evaluations of the differences in (a_ij) or (s_i) among several groups. Therefore, the best understanding of ICA is as an approach like exploring a unkown mine.

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