home/regression-approach.md

@@ -2,6 +2,7 @@
 title: regression approach - powerlaw
 ---
 
+
 ## 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).
@@ -76,15 +77,14 @@ where $k$ is the number of points in the data-series.
 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}}}
+X_{s} = \frac{X_i - X_{\textrm{min}}}{X_{\textrm{max}} - X_{\textrm{min}}}, \quad
+X_{s} = \frac{Y_i - Y_{\textrm{min}}}{Y_{\textrm{max}} - Y_{\textrm{min}}}
 ```
 
+where the lower index s is an abbreviation for scaled.
+
 **Step 5**
-<br>
-<div align="center">
-![quantile_example](uploads/c949d51e14598b30283f3adf7eef84d8/quantile_example.png){width=45%}
-</div>
+
 Compute the error/difference between the model and the data 
 
 ```math
@@ -93,10 +93,16 @@ E = Y_{\textrm{p}} - Y_{\textrm{d}}
 Extract the data with a positive error. We are only interested in the positive difference, as these are an indication of the points below the mean line.
 
 ```math
-idx = 
+E = E\left[E > 0\right], \quad X_{s} = X_{s}[E>0], \quad Y_{s} = Y_{s}[E>0]
 ```
 
 
+<br>
+<div align="center">
+![quantile_example](uploads/c949d51e14598b30283f3adf7eef84d8/quantile_example.png){width=25%}
+</div>
+
+
 as the data-points w below the mean
 
  as these are the ones below the mean line.