RNA baseline prediction covariance

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 $x_i=T_i-T_0$. ThenThis 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.^[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 $20\times 10^{-6}$, while their covariance is $-12\times 10^{-6}$. 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:Its standard uncertainty is 0.008. Ignoring covariance would give $\sqrt{40\times 10^{-6}}\approx 0.00632$, 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.