= RNA baseline prediction covariance
{c}
<RNA baseline prediction covariance> quantifies how errors in fitted <RNA melting baselines> link their predicted values at different temperatures. Reusing the same fit parameters creates shared uncertainty; predictions are not independent merely because they are evaluated at different temperatures.
For a linear <RNA baseline extrapolation> $A_F(T)=a+b(T-T_0)$, put $x_i=T_i-T_0$. Then
$$
C_{ij}=\operatorname{Cov}(A_F(T_i),A_F(T_j))
=u_a^2+(x_i+x_j)\operatorname{Cov}(a,b)+x_ix_ju_b^2.
$$
This relation follows from the covariance of two linear combinations. The diagonal $C_{ii}$ gives a prediction variance; the off-diagonal entries determine how predictions move together.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[§5.2, correlated input quantities.]{ref}
An original example assumes independent parameters with $u_a=0.002$ and $u_b=0.0002\,\mathrm{K^{-1}}$. At temperatures $20\,\mathrm K$ below and above $T_0$, both prediction variances are $20\times10^{-6}$, while their covariance is $-12\times10^{-6}$. The <correlation coefficient> is therefore $-0.6$. A slope error moves the predictions in opposite directions around the reference point, outweighing the common intercept error.
The difference between those two predictions cancels the intercept exactly:
$$
A_F(T_0+20\,\mathrm K)-A_F(T_0-20\,\mathrm K)=40\,\mathrm K\,b.
$$
Its <standard uncertainty> is 0.008. Ignoring covariance would give $\sqrt{40\times10^{-6}}\approx0.00632$, which is too small for this model. Shared fitted inputs can produce negative as well as positive prediction correlations.
In vector form, a prediction <covariance matrix> is $C=X\Sigma_\theta X^{\mathsf T}$, with rows of $X$ equal to $(1,x_i)$ and parameter covariance $\Sigma_\theta$. A two-parameter fit supplies at most two independent shared-error directions, even if evaluated at a hundred temperatures. Additional independent readout errors contribute their own covariance.
When these predictions are used to infer populations in <RNA optical melting>, propagate their joint uncertainty rather than only separate error bars. This accounting does not test whether a linear baseline is chemically adequate; curvature or an incorrect pure-state model needs separate evidence.
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