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I have $N$ time series each of which can be modeled as $$y_{kt}=Ax_{kt}+b+\varepsilon_{kt}\quad(1\le k\le N,1\le t\le T),$$ where $x_{kt}\sim\text{Pois}(\lambda\Delta t)$ and $\varepsilon_{kt}\sim N(0,\sigma^2)$. Parameters $A$, $b$ and $\sigma^2$ are unknown and I want to extract the uncontamined version of all $y_{kt}$. Several techniques come into my mind, like ICA (two sources, uncontamined data and noise) or wavelet (perform DWT on data and keep the largest coefficients, for example, or use wavelet shrinkage, etc.). But I am not sure about that:

  1. Can ICA handle poisson noise inherent in poisson processes (which is described by $\lambda$ but not $\varepsilon_{ki}$)?
  2. Will wavelet denoising ruin the properties of the model?

Or are there better choices? Any hints are appreciated.

ziyuang
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  • What do you observe? Only $y_{kt}$? What is the relationship between $x_{kt}$ and $\varepsilon_{kt}$? Are they independent? – mpiktas Jul 18 '11 at 12:02
  • $x_{kt}$ itself is the count of events. $A$ and $b$ stand for the measurement. And $\varepsilon_{kt}$ is the Gaussian noise, which is independent of $x_{kt}$, presumptively. – ziyuang Jul 18 '11 at 12:23
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    If you have both y and x, I see no reason why simple linear regression should not work. – mpiktas Jul 18 '11 at 13:56
  • But I don't have x's, all I have are y's (the measured data). I get a $y$ at every $t$. – ziyuang Jul 18 '11 at 14:23
  • Interesting problem. Is it a single $\lambda$ for all $k$? – Memming Sep 03 '13 at 02:46

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