Lectures 4,5 (hand in attempts at lecture Thurs 4 Mar)
(a) The following link will give you access to the time series of X-ray emission by a particular source in the sky. Your task is to denoise this time series by wavelet methods. [Note: in contrast to the example discussed by Dubok de Wit, the noise here is at shorter scales (higher frequncies) and the useful signal at longer scales (lower frequencies).]
Link: http://xte.mit.edu/asmlc/srcs/x0512-401.html In its Data window please leave the days blank and select: "One-day Average Light Curve" and for data columns "Sum Band Intensity" and "Sum Band Uncertainties" then click "Retrieve...". You should end up with a text file in which the SECOND number on each line is the X-ray intensity for each successive day (averaged over that day), for each of 4717 successive days, and the THIRD number is an uncertainty in that measurement.
(i) Produce a simple graph of the series with time average removed.
(ii) Satisfy yourself that the rms (Root Mean Square) value of the uncertainty is quite close to 0.5. Generate a model error series comprising independent gaussian dsitributed random numbers with mean zero and standard deviation 0.5, and graph this for comparison with (i).
(iii) Compare scalograms [intensity maps of the power of the wavelet transform of] your series from (i) and (ii) and comment as to which parts of the scalogram of (i) appear significant on this basis. Please show results for the Haar wavelet. Open-endedly: do you find any other wavelet more informative?
(iv) Produce a graph of the significant denoised time dependence. Please show result for the Haar wavelet, plus any you find does better.
[Note: I have deliberately made the question answerable using the Haar wavelet alone: it is so simple you could programme its calculation directly (I just did so). However it would be much more productive to master the Wavelet Toolkit in MATLAB and thereby access other wavelet choices.]
(b) on scaling to follow.....