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 {1410} hide / / print ref: -0 tags: kernel regression structure discovery fitting gaussian process date: 09-24-2018 22:09 gmt revision:1  [head] Use Gaussian process kernels (squared exponential, periodic, linear, and ratio-quadratic) to model a kernel function, $k(x,x')$ which specifies how similar or correlated outputs $y$ and $y'$ are expected to be at two points \$\$x\$ and $x'$ . By defining the measure of similarity between inputs, the kernel determines the pattern of inductive generalization. This is different than modeling the mapping $y = f(x)$ . It's something more like $y' = N(m(x') + k(x,x'))$ -- check the appendix. See also: http://rsta.royalsocietypublishing.org/content/371/1984/20110550 Gaussian process models use a kernel to define the covariance between any two function values: $Cov(y,y') = k(x,x')$ . This kernel family is closed under addition and multiplication, and provides an interpretable structure. Search for kernel structure greedily & compositionally, then optimize parameters with conjugate gradients with restarts. This seems straightforwardly intuitive... Kernels are scored with the BIC. C.f. {842} -- "Because we learn expressions describing the covariance structure rather than the functions themselves, we are able to capture structure which does not have a simple parametric form." All their figure examples are 1-D time-series, which is kinda boring, but makes sense for creating figures. Tested on multidimensional (d=4) synthetic data too. Not sure how they back out modeling the covariance into actual predictions -- just draw (integrate) from the distribution? {763} hide / / print ref: work-2999 tags: autocorrelation poisson process test neural data ISI synchrony DBS date: 02-16-2012 17:53 gmt revision:5      [head] I recently wrote a matlab script to measure & plot the autocorrelation of a spike train; to test it, I generated a series of timestamps from a homogeneous Poisson process: ```function [x, isi]= homopoisson(length, rate) % function [x, isi]= homopoisson(length, rate) % generate an instance of a poisson point process, unbinned. % length in seconds, rate in spikes/sec. % x is the timestamps, isi is the intervals between them. num = length * rate * 3; isi = -(1/rate).*log(1-rand(num, 1)); x = cumsum(isi); %%find the x that is greater than length. index = find(x > length); x = x(1:index(1,1)-1, 1); isi = isi(1:index(1,1)-1, 1); ``` The autocorrelation of a Poisson process is, as it should be, flat: Above: Red lines are the autocorrelations estimated from shuffled timestamps (e.g. measure the ISIs - interspike intervals - shuffle these, and take the cumsum to generate a new series of timestamps). Hence, red lines are a type of control. Blue lines are the autocorrelations estimated from segments of the full timestamp series. They are used to how stable the autocorrelation is over the recording Black line is the actual autocorrelation estimated from the full timestamp series. The problem with my recordings is that there is generally high long-range correlation, correlation which is destroyed by shuffling. Above is a plot of 1/isi for a noise channel with very high mean 'firing rate' (> 100Hz) in blue. Behind it, in red, is 1/shuffled isi. Noise and changes in the experimental setup (bad!) make the channel very non-stationary. Above is the autocorrelation plotted in the same way as figure 1. Normally, the firing rate is binned at 100Hz and high-pass filtered at 0.005hz so that long-range correlation is removed, but I turned this off for the plot. Note that the suffled data has a number of different offsets, primarily due to differing long-range correlations / nonstationarities. Same plot as figure 3, with highpass filtering turned on. Shuffled data still has far more local correlation - why? The answer seems to be in the relation between individual isis. Shuffling isi order obviuosly does not destroy the distribution of isi, but it does destroy the ordering or pair-wise correlation between isi(n) and isi(n+1). To check this, I plotted these two distributions: -- Original log(isi(n)) vs. log(isi(n+1) -- Shuffled log(isi_shuf(n)) vs. log(isi_shuf(n+1) -- Close-up of log(isi(n)) vs. log(isi(n+1) using alpha-blending for a channel that seems heavily corrupted with electro-cauterizer noise. {735} hide / / print ref: -0 tags: processing javascript vector graphics web date: 05-03-2009 18:20 gmt revision:0 [head] {381} hide / / print ref: notes-0 tags: low-power microprocessor design techniques ieee DSP date: 05-29-2007 03:30 gmt revision:2   [head] http://hardm.ath.cx:88/pdf/lowpowermicrocontrollers.pdf also see IBM's eLite DSP project. {37} hide / / print ref: bookmark-0 tags: Unscented sigma_pint kalman filter speech processing machine_learning SDRE control UKF date: 0-0-2007 0:0 revision:0 [head] http://choosh.ece.ogi.edu/spkf/ excellent resource. Technical Overview ppt presentation, nize. Demos - the Dual parameter estimation is very impressive. SDRE = state-dependent Ricatti Equation Control. http://choosh.ece.ogi.edu/spkf/spkf_files/WAML2003.pdf good stuff = page 40. use sigma-point kalman filter to pass a particles' mean and covariance information through the state transition nonlinearity draw the new particle from this estimated gaussian distribution! thus avoiding particle impovershment!! http://cslu.cse.ogi.edu/nsel/research/ukf.html author: r. van der Merwe