Given the following:
a. x0 = known scalar
b. x1 = known scalar
c. x2 = unknown scalar
d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where
$g[x0] = x1+ zg$
$g[x1] = x2 + zg$ where $\mathbf{zg}$ is possibly non-gaussian noise
and
e. a known nonlinear function $\mathbf{f}$ where
$f[x0] = x2 + zf$
in a mse sense where $\mathbf{zf}$ is possibly gaussian noise
which given a,b,c,d above implies (with variations d/t introduction of noise)
- $f[x0] \approx g \circ g [x0] \approx g[x1] $
- $f[x1] \approx g \circ g \circ g[x0] \approx g \circ g[x1] $
- $f \circ f[x0] \approx g \circ g \circ f[x0] \approx g \circ g \circ g[x1] $
- $f \circ f[x1] \approx g \circ g \circ f[x1] \approx g \circ g \circ g \circ g[x1] $
What mathematical tools [Fourier/other transforms, ODE/SDE Theorems, Geometric Algebra, etc]
can I use to leverage the compositional relationships above and gain insight into
i. the value $x2 \approx g[x1] $
ii. the relationship between $\mathbf{g}$ and $\mathbf{f}$
iii. transforms I can do on $\mathbf{f}$ to get closer to the true $\mathbf{g}$?
The motivation being that f[x0] is a suboptimal estimate, and there may be a transform T that takes advantage of the relationships and reduces noise such that
$\mathbb{E}( |g[x1]- \mathbf{T}[x0,x1,f[x0],f[x1],f[f[x0]]]|) < \mathbb{E}( |g[x1] - f[x0]|)$
or even
$\mathbb{E}( |g[x1]- \mathbf{f} \circ \mathbf{T}[x0,x1,f[x0],f[x1],f[f[x0]]]|) < \mathbb{E}( |g[x1] - f[x0]|)$