claudeMar 18, 2020

The #VectorAutoRegression stuff I've been doing can be summarized as
$$ y_t = \sum_{i=1}^p A_i y_{t - i} $$
where each $y_t$ is a $D$-vector and each $A_i$ is a $D \times D$ matrix.

Given an input series of $y$ values, the $A_i$ can be calculated by #LeastSquares minimization:

$$
Y = [ y_p ; y_1; \ldots ; y_{T - 1} ] \\
X = [ y_{p-1}, y_{p-2}, \ldots, y_0 ; y_p, y_{p-1}, \ldots, y_1 ; \ldots ; y_{T-2}, y_{T-3}, \ldots, y_{T-1 - p} ] \\
A = \left(X^T X\right)^{-1} X^T Y
$$
where the matrices have these initial dimensions:
Y : (T - p) × D
X : (T - p) × (p × D)
A : (p × D) × D
then reshape $A$ to $p × (D × D)$

In my case all values are complex (and ^T is conjugate transpose) because each $y$ is an FFT of a block of input data - I'm using FFT size $256$ (making $D = 129 = 256/2+1$ unique bins for real input) overlapped 4x.

Show threadclaudeMar 18, 2020

#VectorAutoRegression is a pure feedback model, similar to poles in the pole-zero representation of z-transform filters. To add zeros to the model is quite simple:

$$ y_t = \sum_{i=1}^p A_i y_{t - i} + \sum_{i=0}^q B_i x_{t-i}$$

and then add the $x$ vectors and $B$ matrices to the least squares stuff. I think the jargon for this is #VARMA, for Vector Auto-Regressive Moving Average.

I worked it through and coded up most of it before realizing a #FatalFlaw : I need the $x$ input data corresponding to the $y$ output data to do the regression to estimate the $A$ and $B$ parameter matrices, while so far I've just been using WAV files as $y$ series, which means I don't have any $x$.

Show threadclaudeMar 18, 2020

Now thinking of using #Timidity software MIDI synthesizer to generate $(x,y)$ pairs: X could be a series of notes played with a piano and Y the same played with a guitar, hard-panned left and right so they are synchronized.

The #GeneralMIDI instrument list is at
https://www.midi.org/specifications-old/item/gm-level-1-sound-set

GM 1 Sound Set

THE MIDI ASSOCIATION, a global community of people who work, play and create with MIDI and the central repository of information about anything related to MIDI.

Show threadclaudeMar 18, 2020

First successful #VARMA test with small P, Q. (Follows about 25 unsuccessful tests.)

Audio attached has 3 parts, the first is the target sound (Y), the second is the input sound (X), the third is the VARMA model applied to X (normalized in volume) which should hopefully sound like Y.

P = 0 (one matrix for input mapping, no real moving average here...)
Q = 1 (one matrix for auto-regressive feedback)
D = 129 (number of dimensions for FFT vectors)
T = 18103 (number of input vectors)

Computed #SpectralRadius of the feedback matrix was 0.9144670749524257, less than 1 so it's stable (good).

The peak level of the output audio was 0.148676, significantly quieter than I expected (most of my previous tests using Python statsmodels VAR had gains around 1.0e+7 or so...).

Show threadclaudeMar 18, 2020
Just realized the input X and output Y need not be equally dimensioned. So maybe I could make something that's more like a synthesizer than an audio filter...
Show threadclaude

But I think this whole system I've been working on today is #LinearTimeInvariant, so, not too much scope for interesting behaviour... e.g. with two control inputs, the output is simply a mix of two impulse responses, which isn't so amazing.

I think the power of other algorithms like neural networks comes mainly from the non-linearities.

Mar 18, 2020 at 11:03pmWeb
Show threadclaudeMar 19, 2020
Maybe it's not Linear after all, because linear systems can't add frequencies that are not already there, while matrix transformations operating across FFT bins can certainly do that.