Machine Learning: fundamental algorithms
Propagation of uncertainty
For the covariance, refer to page:
Given a function of
nn
non-correlated variables
f(x1,x2,,xn)f(x_1, x_2, \ldots, x_n)
, such that each variable
xix_i
has its own uncertainty
Δxi\Delta x_i
, we can compute the uncertainty on
ff
as
Δf(x1,x2,,xn,Δx1,,Δxn)=[i=1i=nfxiΔxi]12 .\Delta f(x_1, x_2, \ldots, x_n, \Delta x_1, \ldots, \Delta x_n) = \left[\sum_{i=1}^{i=n} \frac{\partial f}{\partial x_i} \Delta x_i\right]^{\frac{1}{2}} \ .
If the variables are correlated, then there is the covariance term:
Δf(x1,x2,,xn,Δx1,,Δxn)=[i=1nk=1nfxifxkCik]1/2 ,\Delta f(x_1, x_2, \ldots, x_n, \Delta x_1, \ldots, \Delta x_n) = \left[\sum_{i=1}^{n} \sum_{k=1}^n \frac{\partial f}{\partial x_i} \frac{\partial f}{\partial x_k} C_{ik}\right]^{1/2} \ ,
where
CikC_{ik}
is the covariance matrix.
In the most common cases:
  • f=x±y(Δf)2=(Δx)2+(Δy)2±2Cxyf = x \pm y \Rightarrow (\Delta f)^2 = (\Delta x)^2 + (\Delta y)^2 \pm 2 C_{xy}
  • f=cxΔf=cΔxf = cx \Rightarrow \Delta f = c \Delta x
  • f=xy(Δf)2=y2(Δx)2+x2(Δy)2+2xyCxyf = xy \Rightarrow (\Delta f)^2 = y^2 (\Delta x)^2 + x^2 (\Delta y)^2 + 2 xy C_{xy}
  • f=xy(Δf)2=(Δx)2y2+x2y4(Δy)22xy3Cxyf = \frac{x}{y} \Rightarrow (\Delta f)^2 = \frac{(\Delta x)^2}{y^2} + \frac{x^2}{y^4} (\Delta y)^2 - 2 \frac{x}{y^3} C_{xy}
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