Some quick thoughts on projections

Suppose S(t) is a musical signal- piece of an instrument -say violin. Suppose we know that  this signal is composed of 5 different notes of violin-call them n1,n2….n5. So how do you find which notes were being played in the signal at any given time ? [Assume n1 to n5 span S(t) completely]

This, roughly, was the first question in my MLSP class. The way this is solved is by finding the spectrum for all of the notes using STFT. Then STFT is calculated at each time point for the aggregate signal (aka a spectrogram). Then the STFT at each time stamp is projected on to the STFT of the notes. If the magnitude of the projections are above a certain (empirical) threshold, then the note is played at that time stamp, otherwise it is not. The reason for empirical threshold arises from the fact that the notes aren’t orthogonal to one another. If the notes were orthogonal then the notes that don’t contribute to the signal at that time would have a zero magnitude. I am not completely sure, but I think if they are orthonormal, the projections on to notes that contribute would have a value of 1. [Mario ?/ Emre?]

Point is, Energy disaggregation is the same problem. In your ideal case the aggregate signal would be projected on to all of  the appliance signatures, and the ones that are OFF would have a magnitude of zero. Problem is, there is no reason the signatures would be orthonormal to each other. And once you try to create an orthonormal basis (say through PCA or LDA) you lose the simple contributes/doesn’t contribute classification.

Maybe- once we calculate the orthonormal bases for appliance signatures, we could project the appliance signatures into these bases as well. This way you know which of the signatures contribute how much to the basis components [say the first Principal Component]. After that we can project the aggregate signal into the orthonormal basis and know exactly what bases contribute to the signal.

Then you go and look at the projection of the signatures into these bases and see which signatures contribute to these bases  [There has to be some statistical way to do this]. Besides, it sounds reasonable that the new set of orthonormal basis (orthonormal signatures) would still span the same space as the one spanned by the signatures.

Either I am getting a better understanding of projections, or I am ruining my understanding by over thinking things in terms of NILM.

One thought on “Some quick thoughts on projections

  1. A few things:
    - These are pretty much the same thoughts that were going through my head when I did that homework three years ago.
    - I’ll leave it up to you to run through the math required to answer your question about orthonormal projections.
    - Regarding your last statement, I encourage you to think about this a little more to determine if its really feasible to apply this in the NILM domain. While you’re thinking about it, consider this: how does the spectrum of tones generated by real musical instruments compare to the spectrum of a AC voltage and/or current signal measured on a circuit with an electrical load? If you take one STFT vector from each one of these cases, which one would be more sparse? How would this sparseness affect your ability to generate orthogonal bases that can allow you to separate the tones/loads?

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