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This might sound trivial or a simple misunderstanding, but please bear with me as I'm not a Math major.

I want to investigate some aspects of PCA in homogeneous directions and needed simple analytical functions. General solution to the one dimensional wave equation with periodic boundary condition seems reasonable as it's homogeneous and I can use sine and cosine waves in the general form:

$u(x,t)=\sum_i{a_i\sin(f_i(x-ct))+b_i\cos(f_i(x+ct))}$

where $a_i,b_i$ determine amplitudes, $f_i$ represents frequencies and $c$ is the propagation speed.

Now here's my confusion, shouldn't the standard deviation in a homogeneous direction have transitional symmetry? if I use $u(x,t)=sin(3(x-2t))$ this is satisfied, but if I add another term then the std will be a harmonic function of space which is not homogeneous! I cannot figure out what I'm missing here.

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  • $\begingroup$ PCA = Principal Component Analysis en.wikipedia.org/wiki/Principal_component_analysis $\endgroup$
    – YCor
    Commented Apr 20, 2018 at 22:27
  • $\begingroup$ How did you compute the standard deviation in a omogeneous direction? $\endgroup$
    – Luca Ghidelli
    Commented Apr 21, 2018 at 11:50
  • $\begingroup$ @LucaGhidelli: it's per node: $u^\prime(x)=\int{u(x,t)^2dt}$ $\endgroup$
    – anishtain4
    Commented Apr 23, 2018 at 19:37
  • $\begingroup$ I see, I've mistakenly said it's harmonic function of time which is wrong, it's a harmonic function of space. I'll fix it. $\endgroup$
    – anishtain4
    Commented Apr 23, 2018 at 19:38

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So after more readings and talking with my friends in Math department, I was confusing homogeneous equations (zero forcing function) and statistical homogeneity. The general solution is valid for homogeneous wave function, but it is not necessarily statistically homogeneous.

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