Ratio of expected values of squares of errors in power quantities, in dB

An amplitude quantity $$a$$ is estimated by two estimators $$\hat a_1$$ and $$\hat a_2.$$ The error of estimator $$\hat a_2$$ is compared to that of $$\hat a_1$$. One possible comparison is the ratio of expected values of absolute error:

$$\frac{\operatorname{E}[|a - \hat a_2|]}{\operatorname{E}[|a - \hat a_1|]}.$$

This is a ratio of amplitude quantitities and can be expressed in decibels as:

$$20 \log_{10}\left(\frac{\operatorname{E}[|a - \hat a_2|]}{\operatorname{E}[|a - \hat a_1|]}\right) \text{ dB.}$$

Another possible comparison is the ratio of expected values of square error:

$$\frac{\operatorname{E}[(a - \hat a_2)^2]}{\operatorname{E}[(a - \hat a_1)^2]}.$$

This is a ratio of power quantities and can be expressed in decibels as:

$$10 \log_{10}\left(\frac{\operatorname{E}[(a - \hat a_2)^2]}{\operatorname{E}[(a - \hat a_1)^2]}\right) \text{ dB.}$$

The squares of the estimators can be used to estimate $$a^2.$$ The estimators $$\hat a_1^2$$ and $$\hat a_2^2$$ can be compared by the ratio of expected values of square error:

$$\frac{\operatorname{E}[(a^2 - \hat a_2^2)^2]}{\operatorname{E}[(a^2 - \hat a_1^2)^2]}.$$

This is a ratio of squares of power quantities. How should it be expressed in decibels?

$$10 \log_{10}\left(\frac{\operatorname{E}[(a^2 - \hat a_2^2)^2]}{\operatorname{E}[(a^2 - \hat a_1^2)^2]}\right) \text{ dB,}$$

or perhaps:

$$5 \log_{10}\left(\frac{\operatorname{E}[(a^2 - \hat a_2^2)^2]}{\operatorname{E}[(a^2 - \hat a_1^2)^2]}\right) \text{ dB} = 10 \log_{10}\left(\frac{\sqrt{\operatorname{E}[(a^2 - \hat a_2^2)^2]}}{\sqrt{\operatorname{E}[(a^2 - \hat a_1^2)^2]}}\right) \text{ dB},$$

where the square roots are power quantities, or is there no commonly agreed convention?

Your question raises many very present concerns about measuring differences, and optimizing, in signal/image processing. Measuring can help:

• to compare different outcomes from two different processes,
• to optimize processing per se (for denoising, restoration).

Most use cost functions related to $$p$$-norms (when $$p\ge 1$$) or quasi-norms (when $$0): $$\mathcal{C}_p(a,\hat{a})=\left(\sum_k|a_k-\hat{a}_k|^p\right)^{1/p}\,.$$ Those functions are symmetric, and are consistent with the physical units of $$a$$. The behavior of $$\mathcal{C}_p$$ depends on power $$p$$: classical norm inequalities tells you that $$\mathcal{C}_p(a,\hat{a})\ge \mathcal{C}_q(a,\hat{a})$$ when $$p\le q$$. Those (quasi-)norms are $$1$$-homogeneous. Using different $$p$$ or $$q$$ can be useful, especially in a sparsity context, see for instance references in: Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed \ell_1/\ell_2 Regularization. Here is a rendition of classical penalties: Thus, quasi-norm or norm ratios are used to compare estimates, in a unitless fashion. The $$\ell_p$$ signal-to-noise ratios are typically of the shape:

$$\frac{\mathcal{C}_p(a,0)}{\mathcal{C}_p(a,\hat{a})}\,.$$

They are unit-less (ratios of same units). To better express orders of magnitude, one usually take logarithms. The logarithms can also been interpreted as unit-less measure. Interestingly, either $$\log$$ or $$\ell_p/\ell_q$$ ratios (like $$\ell_1/\ell_2$$) have been used a non-convex penalties in signal restoration.

Your functions with power $$\alpha$$ can be written as:

$$B\log_b \frac{\mathcal{C}_p(a^\alpha,\hat{a}_2^\alpha)}{\mathcal{C}_p(a^\alpha,\hat{a}_1^\alpha)}\,.$$

Now, those logarithms of ratios are unit-less, and are usually employed in a relative way, for comparison only. With $$B>0$$, $$b>1$$, $$t \to B \log_b (t)$$ is a monotonic increasing function. So, for value comparison only, we don't really care about the values for $$b$$ and $$B$$. However, for consistency in dB scale (as Olli Niemitalo already answered properly), signal-to-noise ratios tend to be computed in $$10\log_{10}$$ of energy measures, so homogeneously:

• $$20\log_{10}(|a|)$$
• $$10\log_{10}(|a|^2)$$
• $$5\log_{10}(|a|^4)$$

make sense. However, it is not easy to compare them objectively, but they can be plotted on similar graph.