diff --git a/docs/funcs/FormalDescriptions.md b/docs/funcs/FormalDescriptions.md index d037145f59d102700b73d64a5acae95df84192b0..d9be8ab097866f9d658d998514084a6e16ed1ac0 100644 --- a/docs/funcs/FormalDescriptions.md +++ b/docs/funcs/FormalDescriptions.md @@ -5,6 +5,7 @@ A collection of detailed mathematical descriptions. ## Index - [spikes_flagRaise](#spikes_flagraise) +- [spikes_flagSpektrumBased](#spikes_flagspektrumbased) ## spikes_flagRaise @@ -21,4 +22,25 @@ timestamps $`t_i`$, is flagged a rise, if: * $`t_k - t_{k-1} > `$ `min_slope_weight`*`intended_freq` The weighted average $`\mu^*`$ was calculated with weights $`w_{i}`$, defined by: -* $`w_{i} = (t_i - t_{i-1})`$ / `intended_freq`, if $`(t_i - t_{i-1})`$ < `intended_freq` and $`w_i =1`$ otherwise. \ No newline at end of file +* $`w_{i} = (t_i - t_{i-1})`$ / `intended_freq`, if $`(t_i - t_{i-1})`$ < `intended_freq` and $`w_i =1`$ otherwise. + + + +The value $`x_{k}`$ of a time series $`x_t`$ with +timestamps $`t_i`$ is considered a spikes, if: + + +## spikes_flagSpektrumBased + + +1. The quotient to its preceding data point exceeds a certain bound: + * $` |\frac{x_k}{x_{k-1}}| > 1 + `$ `raise_factor`, or + * $` |\frac{x_k}{x_{k-1}}| < 1 - `$ `raise_factor` +2. The quotient of the second derivative $`x''`$, at the preceding + and subsequent timestamps is close enough to 1: + * $` |\frac{x''_{k-1}}{x''_{k+1}} | > 1 - `$ `deriv_factor`, and + * $` |\frac{x''_{k-1}}{x''_{k+1}} | < 1 + `$ `deriv_factor` +3. The dataset $`X = x_i, ..., x_{k-1}, x_{k+1}, ..., x_j`$, with + $`|t_{k-1} - t_i| = |t_j - t_{k+1}| =`$ `noise_window` fulfills the + following condition: + `noise_func`$`(X) <`$ `noise_thresh` \ No newline at end of file