Update BreakDetection.md

docs/funcs/BreakDetection.md

@@ -16,7 +16,7 @@ breaks_spektrumBased(thresh_rel=0.1, thresh_abs=0.01,
 |-----------------------|---------------------------------------------------------------|---------------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------|
 | thresh_rel            | float                                                         | `0.1`         | Minimum relative difference between two values to consider the latter as a break candidate. See condition (1)                                                         |
 | thresh_abs            | float                                                         | `0.01`        | Minimum absolute difference between two values to consider the latter as a break candidate. See condition (2)                                                         |
-| first_der_factor      | float                                                         | `10`          | Multiplication factor for arithmetic mean of the first derivatives surrounding a break candidate. See condition (3).                          |
+| first_der_factor      | float                                                         | `10`          | Multiplication factor for the arithmetic mean of the first derivatives surrounding a break candidate. See condition (3).                          |
 | first_der_window      | [offset string](docs/ParameterDescriptions.md#offset-strings) | `"12h"`       | Window around a break candidate for which the arithmetic mean is calculated. See condition (3)                                                          |
 | scnd_der_ratio_range  | float                                                         | `0.05`        | Range of the area, covering all the values of the second derivatives quotient, that are regarded "sufficiently close to 1" for signifying a break. See condition (5). |
 | scnd_der_ratio_thresh | float                                                         | `10.0`        | Threshold for the ratio of the second derivatives succeeding a break. See condition (5).                                                                              |
@@ -30,19 +30,27 @@ The function flags breaks (jumps/drops) by evaluating the derivatives of a time
 A value $`x_k`$ of a time series $`x_t`$ with timestamps $`t_i`$, is considered to be a break, if:
 
 1. $`x_k`$ represents a sufficiently large relative jump:
-    * $`|\frac{x_k - x_{k-1}}{x_k}| >`$ `thresh_rel`
+
+   $`|\frac{x_k - x_{k-1}}{x_k}| >`$ `thresh_rel`
+
 2. $`x_k`$ represents a sufficient absolute jump:
-    * $`|x_k - x_{k-1}| >`$ `thresh_abs`
-3. The dataset $`X = x_i, ..., x_{k-1}, x_{k+1}, ..., x_j`$, with  
-   $`|t_{k-1} - t_i| = |t_j - t_{k+1}| =`$ `first_der_window` fulfills the following condition:
+
+   $`|x_k - x_{k-1}| >`$ `thresh_abs`
+
+3. The dataset $`X = x_i, ..., x_{k-1}, x_{k+1}, ..., x_j`$, with $`|t_{k-1} - t_i| = |t_j - t_{k+1}| =`$ `first_der_window`
+   fulfills the following condition:
    
-   $`|x'_k| >`$ `first_der_factor` $` \times \bar{X} `$
+   $`|x'_k| >`$ `first_der_factor` $` \cdot \bar{X} `$
    
    where $`\bar{X}`$ denotes the arithmetic mean of $`X`$.
+
 4. The ratio (last/this) of the second derivatives is close to 1:
-    * $` 1 -`$ `scnd_der_ratio_range` $`< |\frac{x''_{k-1}}{x_{k''}}| < 1 + `$`scnd_der_ratio_range`
+
+   $` 1 -`$ `scnd_der_ratio_range` $`< |\frac{x''_{k-1}}{x_{k''}}| < 1 + `$`scnd_der_ratio_range`
+
 5. The ratio (this/next) of the second derivatives is sufficiently height:
-    * $`|\frac{x''_{k}}{x''_{k+1}}| > `$`scnd_der_ratio_thresh`
+
+   $`|\frac{x''_{k}}{x''_{k+1}}| > `$`scnd_der_ratio_thresh`
 
 NOTE:
 - Only works for time series