Signal-To-Noise ratio is defined as

$$SNR = \frac{P_{signal}}{P_{noise}}$$

For my application, I find the following metric to be more useful

$$\frac{P_{signal}}{P_{signal} + P_{noise}}$$

Does it have a name?

Context: I have a function $F$ that analyses a dataset and outputs a number. I want to test how the output of that function depends on the added noise. For this purpose, I create a surrogate dataset $X$ with zero noise, and white noise $N$, and zscore each of them. Then, I produce the noisy dataset

$$Y(\alpha) = \alpha X + (1 - \alpha)N$$

for multiple values of $\alpha \in [0, 1]$. I finally proceed to plot $F(Y)$ as a function of $\alpha$. I want to know what is a good name for the variable $\alpha$. I am tempted to call it SNR, but that is not true according to definition of SNR. I could change the definition of $Y$ such that the free parameter would be exactly SNR, but then its limits would be $[0, \infty)$, which is less convenient than $\alpha$ to sample and plot on x-axis.

  • $\begingroup$ not from the top of my head. Also, albeit easier to measure, it's usually not a useful measure - with SNR you can make statements on the ability of a medium to transport data, with S/(S+N) not so much. So, the subtraction of signal power from the overall receive power is usually a good "investment". $\endgroup$ Jun 7 '20 at 22:10
  • $\begingroup$ You shouldn't call it SNR, but it is the fraction of the total power that belongs to the signal so you could work with that and come up with something nice $\endgroup$
    – Engineer
    Jun 7 '20 at 23:54
  • $\begingroup$ that's referred to as the kalman gain in a local level model but, since you're not dealing with a kalman filter, I'm not sure if that interpretation is okay because your coefficients are not assumed to have noise in them ( I don't think) . See this for details. Around page 32 is where they describe $k_t$. creates.au.dk/fileadmin/user_upload/2016Aarhus_Day1.pdf $\endgroup$
    – mark leeds
    Jun 8 '20 at 2:03
  • $\begingroup$ Note that, if you were in a bayesian regression model setting with a normal prior and a normal likelihood, then a/a+b and b/a+b ( where a and b are referred to in the link ) have bayesian update interpretations. see page 6 of this. www-math.mit.edu/~dav/05.dir/class15-prep.pdf $\endgroup$
    – mark leeds
    Jun 8 '20 at 2:16

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