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I think generally normalized quantities maintain their original units whereas in your ratio case the unit is dropped. Units in the abstract sense I guess, such

I think generally normalized quantities maintain their original units whereas in your ratio case the unit is dropped. Units in the abstract sense I guess, such as this is a measurement along this vector or this is a ratio of any vector.

quietbritishjim 6y ago
I think it's the opposite way round: normalisation results in units being dropped. The typical, most common, definition of normalisation is value ÷ max possible value, giving a result in [0,1]. (More general definitions of normalisation exist e.g. if you rescale so the standard deviation is a fixed value, or even use non linear rescaling, that could count, but never mind all that.) The parent comment's example of "position along width ÷ total width" certainly fits that bill. Whenever you divide something by the max of that something, the max is going to have the same units as the original value and you're bound to end up cancelling them. Or put another way, if you rescale 10cm into 0.5, it's certainly not 0.5cm so the units are either dropped or, at least, changed e.g. you could argue you've got 0.5x where x is the unit equal to 20cm.
- f154hfds 6y ago
  Hmm when I think of 'normalizing' I don't think of dividing by a max at all - in my experience it is more taking a quantity (perhaps in English measurement) and transferring to a more 'standard' unit (say metric). In general I don't think normalization always includes a sense of being in a bounded interval. From a mathematical perspective you could perhaps say normalization is achieved by multiplying your quantity by a 1D operator. You can't change the dimensionality this way, but are certainly changing 'units' a la mm in the x direction -> m in the x direction for example. I guess what I'm saying is that 'normalized' and the like are not the best fit for the SO question. If I could take my own shot at the SO challenge from a mathematical perspective it would perhaps be sigmoid. Where the result of a normalization function takes a 1D value and maps it to a similar 1D value, the sigmoid takes a 1D value and maps it to a similar 1D value between (0, 1). So if I want to drop the previous information and only keep the resulting map, I can say 'this is my sigmoided value' - IE it is impossible for it to be outside of that range. Unfortunately sigmoid also connotates a differentiable curve which is extraneous information... https://en.wikipedia.org/wiki/Normalization_(statistics) https://en.wikipedia.org/wiki/Normalization_(statistics) https://en.wikipedia.org/wiki/Sigmoid_function https://en.wikipedia.org/wiki/Sigmoid_function
  - quietbritishjim 6y ago
    Of course the literal meaning of "normalisation" is to make more "normal", and that can mean almost anything at all. Even the Wikipedia article you linked to starts with "normalization can have a range of meanings". If you have heard that word most used with one meaning and I have heard it most used with another meaning then that doesn't invalidate either of those definitions. The Wikipedia article you linked to gives two very broad definitions in the lead. The definition covered in the first paragraph is "adjusting values measured on different scales to a notionally common scale", which seems to be what you're talking about. The definition covered in the second paragraph is "the creation of shifted and scaled versions of statistics", in particular "some types of normalization involve only a rescaling, to arrive at values relative to some size variable". I'm not comfortable with the use of "of statistics" in that second definition: the very first example is in the article standard score [1], which is about a rescaled element of the population, not a rescaled statistic. In any case, outside of statistics, a rescaling is a common meaning for this word, and a rescaled statistic is clearly just a special case of this. I think it was clear from the context that we were talking about this meaning originally. By far the most common case of this is a linear rescaling (including translation) to [0,1] but I was already up front that this is just a special case. As for your sigmoid comment, I may have misunderstood but it sounds like you're saying that if you have a variable in the range in [0,1] then it can be described as the result of a sigmoid function. My objection to this is the same as my original objection to calling such as variable "normalised": it is a confusing variable name to use unless you actually did get it by applying a sigmoid function to something, not just because it holds a value that could hypothetically be obtained from a sigmoid function (but you didn't). [1] https://en.wikipedia.org/wiki/Standard_score https://en.wikipedia.org/wiki/Standard_score

quietbritishjim 6y ago

I think it's the opposite way round: normalisation results in units being dropped. The typical, most common, definition of normalisation is value ÷ max possible value, giving a result in [0,1]. (More general definitions of normalisation exist e.g. if you rescale so the standard deviation is a fixed value, or even use non linear rescaling, that could count, but never mind all that.) The parent comment's example of "position along width ÷ total width" certainly fits that bill. Whenever you divide something by the max of that something, the max is going to have the same units as the original value and you're bound to end up cancelling them. Or put another way, if you rescale 10cm into 0.5, it's certainly not 0.5cm so the units are either dropped or, at least, changed e.g. you could argue you've got 0.5x where x is the unit equal to 20cm.

f154hfds 6y ago
Hmm when I think of 'normalizing' I don't think of dividing by a max at all - in my experience it is more taking a quantity (perhaps in English measurement) and transferring to a more 'standard' unit (say metric). In general I don't think normalization always includes a sense of being in a bounded interval. From a mathematical perspective you could perhaps say normalization is achieved by multiplying your quantity by a 1D operator. You can't change the dimensionality this way, but are certainly changing 'units' a la mm in the x direction -> m in the x direction for example. I guess what I'm saying is that 'normalized' and the like are not the best fit for the SO question. If I could take my own shot at the SO challenge from a mathematical perspective it would perhaps be sigmoid. Where the result of a normalization function takes a 1D value and maps it to a similar 1D value, the sigmoid takes a 1D value and maps it to a similar 1D value between (0, 1). So if I want to drop the previous information and only keep the resulting map, I can say 'this is my sigmoided value' - IE it is impossible for it to be outside of that range. Unfortunately sigmoid also connotates a differentiable curve which is extraneous information... https://en.wikipedia.org/wiki/Normalization_(statistics) https://en.wikipedia.org/wiki/Normalization_(statistics) https://en.wikipedia.org/wiki/Sigmoid_function https://en.wikipedia.org/wiki/Sigmoid_function
- quietbritishjim 6y ago
  Of course the literal meaning of "normalisation" is to make more "normal", and that can mean almost anything at all. Even the Wikipedia article you linked to starts with "normalization can have a range of meanings". If you have heard that word most used with one meaning and I have heard it most used with another meaning then that doesn't invalidate either of those definitions. The Wikipedia article you linked to gives two very broad definitions in the lead. The definition covered in the first paragraph is "adjusting values measured on different scales to a notionally common scale", which seems to be what you're talking about. The definition covered in the second paragraph is "the creation of shifted and scaled versions of statistics", in particular "some types of normalization involve only a rescaling, to arrive at values relative to some size variable". I'm not comfortable with the use of "of statistics" in that second definition: the very first example is in the article standard score [1], which is about a rescaled element of the population, not a rescaled statistic. In any case, outside of statistics, a rescaling is a common meaning for this word, and a rescaled statistic is clearly just a special case of this. I think it was clear from the context that we were talking about this meaning originally. By far the most common case of this is a linear rescaling (including translation) to [0,1] but I was already up front that this is just a special case. As for your sigmoid comment, I may have misunderstood but it sounds like you're saying that if you have a variable in the range in [0,1] then it can be described as the result of a sigmoid function. My objection to this is the same as my original objection to calling such as variable "normalised": it is a confusing variable name to use unless you actually did get it by applying a sigmoid function to something, not just because it holds a value that could hypothetically be obtained from a sigmoid function (but you didn't). [1] https://en.wikipedia.org/wiki/Standard_score https://en.wikipedia.org/wiki/Standard_score