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I am learning Data Mining/Statistics currently but having a bit of trouble understanding normalized vectors and how to interpret them.

I have a vector $x = (1, 2, 3, 4)$, I took 1-norm and 2-norm of this vector with 1-norm = 10, 2-norm = 30.

I am now trying to interpret and understand what each of these normalizations tell me, or what they mean, and I don't understand how to interpret these values.

Any help is much appreciated.

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    $\begingroup$ If your vector is non-negative numbers, then you can interpret the entries as weights, and the 1-norm normalized vector as frequencies (e.g. in your case if the integers are counts, then dividing by the sum gives probabilities). For an arbitrary (+/-) vector, the 2-norm normalized vector is a point on the unit hypersphere, and can be interpreted as the tip of a "direction vector" which starts at the origin. $\endgroup$
    – GeoMatt22
    Sep 19 '16 at 2:56
  • $\begingroup$ Why 30 and not its root? $\endgroup$
    – Michael M
    Sep 19 '16 at 6:41
  • $\begingroup$ @MichaelM $||x_p|| = (|x_1|^p + |x_2|^p + ... + |x_n|^p)$ is p-norm definition, so 2-norm is $||x_2|| = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$. Right? I'm still learning so I may be incorrect. $\endgroup$ Sep 19 '16 at 14:01
  • $\begingroup$ Usually, it is used with $p$-th root, but it is not too important of course. Check en.wikipedia.org/wiki/Norm_(mathematics) $\endgroup$
    – Michael M
    Sep 19 '16 at 14:29
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To understand $||x_p||$ (the p-norm) in the explanation below, the formula for $||x_p||$ is $$||x_p|| = (|x_1|^p + |x_2|^p + ... + |x_n|^p)^{1/p}$$


To interpret $||x_1||$, take $\frac{x_i}{||x_1||}$, then you will get the vector (for the given example) $(\frac{1}{10}, \frac{2}{10}, \frac{3}{10}, \frac{4}{10})$.

Adding each of the entries in the now-normalized vector will equal $1$. This means you can interpret 1-norm as probability.

To interpret $||x_2||$, take $\frac{x_i}{||x_2||}$, then you will get the vector (for the given example) $(\frac{1}{30}, \frac{4}{30}, \frac{9}{30}, \frac{16}{30})$.

The 2-norm normalized vector can be interpreted as the tip of a "direction vector" of the unit hypersphere, which starts at the origin.


Thanks to GeoMatt22 for the help on understanding this.

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