Making Sense of Notation Confusion in Statistical Digital Signal Processing

In summary: M$$ are i.i.d. $$\mathcal{N}\left(\mu,\sigma^2\right)$$ makes it clearer. You could also use $$\{\omega\}_{i=1}^M$$ instead of the list format.
  • #1
tworitdash
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I started my research in statistical digital signal processing two years ago, so I need to familiarize myself with all the notations people use in probability and statistics. I come from a deterministic science background. I name my variables based on what they mean. A velocity is a [itex] v [/itex], a position is [itex] p [/itex] and so on.

However, at some point, I had to define a set of variables that are random draws from a distribution. I called the variable [itex] \omega_m [/itex], and there are [itex] M [/itex] of them. So I wrote something like the following.

[itex] \omega_m \sim \mathcal{N}(\mu_{\omega}, \sigma^2_{\omega}) [/itex] and I explained what the variables mean. When I explain this to my colleagues or write it in a forum, some people are upset/ confused with my notation, making it unnecessarily hard (in my opinion) to convey the message (although it is quite clear in my head). For example, for the situation given above, statisticians comment that I should not use [itex] \omega_m [/itex] itself as a variable name because its upper-case letter [itex] \Omega [/itex] represents a parameter for a probability space. Some people even get confused when a random variable of this kind is represented with a lowercase letter. They believe it should always be written in an upper-case letter. So, one possible way to please a statistician is to write it like the following.

$$ X_m \sim \mathcal{N}(\mu_, \sigma^2) $$

So, not only did I get rid of the alphabet [itex] \omega [/itex] (that made sense to me based on the variable I had in mind earlier), but I also made it uppercase.

When I show it to someone familiar with signal processing, they are upset with uppercase letters because they represent matrices, not variables.

For a clean and concise paper, why does it matter that we always have to respect some notation? Does the notation matter if we explain what we are trying to say?

To defend myself: I have read many papers and seen people using very different notations to explain the same thing in the past. For example, a Laplacian was represented as [itex] \Delta [/itex], and sometimes it was represented as [itex] \nabla^2 [/itex].

I ask this here because, as a researcher, most of my time goes into explaining people things based on their understanding, which takes a lot of effort and time. It also sometimes makes me feel dumb, but later I realize that the notations and rules make it difficult (not my understanding of things).
 
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  • #2
tworitdash said:
For a clean and concise paper, why does it matter that we always have to respect some notation? Does the notation matter if we explain what we are trying to sayn?
In the middle of your rant I found a question!!
The answer is easy: communication is a two way street. The purpose of writing a paper is to facilitate same. Why not write your work using only Cyrillic and Arabic? Part of learning a field is to understand conventions. If the work is truly extraordinary, your audience may put up with the inconvenience. Seldom does that work out.
Yes it does matter. For instance I usually like threespace to be denoted x,y,z or ##r,\theta,\phi## and parameter to be ##\alpha,\beta,\gamma## . Otherwise I get cognitive dissonance.
 
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  • #3
It can be a pain to learn but it's a common language to use common symbols.

$$\omega^2_o=(2\pi f_o)^2=\dfrac{1}{LC}$$
You can also have a scratchpad or user profile with handy Greek Letters
- to copy & paste from like Δ Ω σ μ τ ω β δ η ϕθ λ π ζ ∞ ° √ Δ ∂ ∫ Ʃ ± ≈ ≠
 
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  • #4
You use a subscript (m) for the random variable, but not for the distribution. All your R.V.'s have same distribution?
 
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  • #5
mathman said:
You use a subscript (m) for the random variable, but not for the distribution. All your R.V.'s have same distribution?
Yes, they are i.i.d. I didn't mention that in the question.
 
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  • #6
I think one confusing thing about
##
\omega_m = \mathcal{N}\left(\mu_{\omega},\sigma^2_{\omega}\right)
##
is this: the subscript m on the left indicates a sequence of random quantities while the subscript $\omega$ indicates fixed values for the mean and standard deviation [which they apparently are since you said i.i.d].
Saying something like $$\omega_1, \omega_2, \dots, \omega_M$$ are i.i.d. $$\mathcal{N}\left(\mu,\sigma^2\right)$$ makes it clearer. You could also use $$\{\omega\}_{i=1}^M$$ instead of the list format.
In terms of whether to follow notational[sic] conventions from signal processing or from prob/stat: there must be an existing set of standards for this in signal processing? Look for that.
 
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  • #7
statdad said:
I think one confusing thing about
##
\omega_m = \mathcal{N}\left(\mu_{\omega},\sigma^2_{\omega}\right)
##
is this: the subscript m on the left indicates a sequence of random quantities while the subscript $\omega$ indicates fixed values for the mean and standard deviation [which they apparently are since you said i.i.d].
Saying something like $$\omega_1, \omega_2, \dots, \omega_M$$ are i.i.d. $$\mathcal{N}\left(\mu,\sigma^2\right)$$ makes it clearer. You could also use $$\{\omega\}_{i=1}^M$$ instead of the list format.
In terms of whether to follow notational[sic] conventions from signal processing or from prob/stat: there must be an existing set of standards for this in signal processing? Look for that.
Thank you! It makes things clear for me. I will definitely check the standards if there is any.
 

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