Update FormalDescriptions.md

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. 
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+* $`w_{i} = (t_i - t_{i-1})`$ / `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`
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