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I am using R/Python diff() operation. e.g., https://docs.scipy.org/doc/numpy-1.14.0/reference/generated/numpy.diff.html

I would like to know if there is a mathematical function that denotes what diff does?

In other words, assume I have a list of N numbers. How can I denote this using a methematical function?

mylist = [x1, x2, x2, ..., xN]

I am happy to provide more details if needed.

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Assume you want to apply diff to a vector $(x_1, \dots, x_n)$ of length $n$. The result will be the vector $(d_1, \dots, d_{n-1})$ of length $n-1$ with entries

$$ d_i = x_{i+1}-x_i. $$


Some people will use the notation $x_{[i]}$ or similar to indicate the vector $x$ with the $i$th component left out, i.e., $x_{[i]}=(x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n)$. With this, you can write more concisely in vector form

$$ d = x_{[1]} - x_{[n]}.$$

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  • $\begingroup$ Thanks a lot for the great answer. Just wondering how to write this function sd(diff(x))/abs(mean(diff(x))) mathematically. Currently, I am having it as σ(diff(x))/|µ(diff(x))|. Please let me know your thoughts on this. I look forward to hearing from you. Thank you very much :) $\endgroup$ – Emi Apr 21 at 8:32
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    $\begingroup$ That sounds useful enough. You can always write down the formulas for the mean and SD in terms of the differences $d_i$. And if you really want, you can substitute in the $x_i$, but it will get progressively less enlightening. $\endgroup$ – Stephan Kolassa Apr 21 at 8:40
  • $\begingroup$ Interesting notation for leaving out $i$; haven't seen that one before. I have often seen $x_{-i}$ for that $\endgroup$ – duckmayr Apr 21 at 11:43
  • $\begingroup$ @StephanKolassa I did not get what you said. It would be really great if you could show me what you said symbolically. I look forward to hearing from you. Thank you once again :) $\endgroup$ – Emi Apr 21 at 16:17

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