home/regression-approach.md

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 title: regression approach - powerlaw
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 ## Dependency
 
 It is assumed the data can be described by a power law, expressed either as Y or X dependent (referring to the second and first axis on a fatigue curve).
@@ -51,12 +42,41 @@ A tollerance band states that a certain percentage of the data lies within or ab
 
 There exist a multitude of methods as to identify these limits such as assuming the data follows a normal- or Weibull distribution. However, in this scenario the upper- and lower quantiles in the distribution are used to identify said limits. The following steps explains the algorithm step by step
 <br>
+<br>
 **Step 1**
+<br>
 Set quantile of interest 
-
 ```math
 \alpha_{0.75} = 0.75, \quad \alpha_{0.95} = 0.95
 ```
 
+**Step 2**
+<br>
+Compute error limit
+
+```math
+\Gamma_{0.75} = 1 - \alpha_{0.75}, \quad \Gamma_{0.95} = 1 - \alpha_{0.95}
+``` 
+
+**Step 3**
+<br>
+Employ the interval on the active data-range
+
+```math
+I_{0.75} = \textrm{int}\left(\textrm{round}\left(k\cdot\Gamma_{0.75}\right)\right), \quad 
+I_{0.95} = \textrm{int}\left(\textrm{round}\left(k\cdot\Gamma_{0.95}\right)\right)
+```
+
+where $k$ is the number of points in the data-series.
+
+
+**Step 4**
+<br>
+Perform _minmax_ normalization. By doing so ensures that the features used by the model have similar scales and aid in faster convergence and accuracy.
+
+```math
+X_{\textrm{scaled}} = \frac{X_i - X_{\textrm{min}}}{X_{\textrm{max}} - X_{\textrm{min}}}, \quad
+Y_{\textrm{scaled}} = \frac{Y_i - Y_{\textrm{min}}}{Y_{\textrm{max}} - Y_{\textrm{min}}}
+```