Noise as strong as signal

First we will make the problem tougher by boosting the noise level to the point where it is comparable to the signal. This is shown in Figure 3.

 

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Figure 3 Left: data with crosstalk. Right: residuals after attempted crosstalk removal using uniform weights.

Notice that the attempt to remove crosstalk is only partly successful. Interestingly, unlike in Figure 1, the crosstalk retains its original polarity, because of the strong noise. Imagine that the noise $\bold n$ dominated everything: then we would be minimizing something like $(\bold n_v - \alpha \bold n_h) \cdot (\bold n_v - \alpha \bold n_h)$.Assuming the noises were uncorrelated and sample sizes were infinite, then $\bold n_v \cdot \bold n_h=0$, and the best $\alpha$ would be zero. In real life, samples have finite size, so noises are unlikely to be more than roughly orthogonal, and the predicted $\alpha$ in the presence of strong noise is a small number of random polarity. Rerunning the program that produced Figure 3 with different random noise seeds produced results with significantly more and significantly less estimated crosstalk. The results are dominated more by the noise than the difference between $\bold p$ and $\bold s$.More about random fluctuations with finite sample sizes will follow in chapter .