I am trying to learn how to calculate DFA from wikipedia
https://en.wikipedia.org/wiki/Detrended_fluctuation_analysis
First:
$X_t=\sum_{i=1}^t (x_i-\langle x\rangle)$
Where $X_t$ = cumulative sum, and $\langle x\rangle$ = mean value of time series, and $t\in {\mathbb {N}}$
My interpretation using arbitrary numbers $({1,2,3,4,5,6,7,8,9,10,11,12})$
$\langle x\rangle$ = 6.5
Deviations = $-5.5,-4.5,-3.5,-2.5,-1.5,-.5,.5,1.5,2.5,3.5,4.5,6.5$
Therefore, cumulative sum $X_t$ = $0$
Obviously the sum of deviations from mean will awlays be 0 so I must be missing something here.
Second:
This part confused me even more because it says Next, $X_{t}$ is divided into time windows of length $n$ samples each, and a local least squares straight-line fit (the local trend) is calculated by minimising the squared errors within each time window. Let $Y_{t}$ indicate the resulting piecewise sequence of straight-line fits. Then, the root-mean-square deviation from the trend, the fluctuation, is calculated:
${\displaystyle F(n)={\sqrt {{\frac {1}{N}}\sum _{t=1}^{N}\left(X_{t}-Y_{t}\right)^{2}}}.}$
What defines the length of the time windows and if we are getting multiple least squares regression lines as from my interpretation "$X_t$ is divided into time windows of lengh $n$" then how would one calculate a log-log graph of $F_n$ agains $n$?